Extragalactic
Astronomy
and
Cosmology
Peter Schneider
Extragalactic
Astronomy
and Cosmology
An Introduction
Peter Schneider
Extragalactic Astronomy
and Cosmology
An Introduction
With 446 figures, including 266 color figures
4y Springer
Prof. Dr. Peter Schneider
Argelander Institut i'iir Astronomic
Universitat Bonn
AufdemHiigel71
D-53121 Bonn, Germany
e mail: iiclcr@astro.uni-bonn.de
Library of Congress Control Number: 2006931134
ISBN-10 3-540-33174-3
Springer Berlin Heidelberg New York
ISBN-13 978-3-540-33174-2
Springer Berlin Heidelberg New York
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Co\ er:T he cox cr shows ;m 1 1ST image of the cluster KXJ 13 17-1 I 15.
the most X, ray luminous cluster ol galaxies known, 'flic large number
ol gra\ nationally lensed arcs, ol which only two ol them have been
detected from ground based imaging previously, clearly shows that
this redshift z = 0.45 cluster is a very massive one, its mass being
dominated b\ dark matter. The data ha\e been processed by Tim
Schrabback and Thomas Lrhen. Argelander Institut I'iir Astronomic
of Bonn University.
se of general de si hi i\ e nam di unes, trademarks,
this publication docs not imply. c\en in the absence of a specific
that such names are exempt from the rele\ant protecti\e
regulations and therefore free for general use.
Typesetting and production:
LE-TgX Jelonek. Schmidt & \ tickler GbK. Leipzig. Germain
Cover design: Erich Kirchner, Heidelberg
Printed on acid free paper :^/:> I41A L -543210
Preface
This book began as a series of lecture notes for an in-
troductory astronomy course I have been teaching at
the University of Bonn since 2001. This annual lec-
ture course is aimed at students in the first phase of their
studies. Most are enrolled in physics degrees and choose
astronomy as one of their subjects. This series of lec-
tures forms the second part of the introductory course,
and since the majority of students have previously at-
tended the first part, I therefore assume that they have
acquired a basic knowledge of astronomical nomencla-
ture and conventions, as well as of the basic properties
of stars. Thus, in this part of the course, I concentrate
mainly on extragalactic astronomy and cosmology, be-
ginning with a discussion of our Milky Way as a typical
(spiral) galaxy. To extend the potential readership of
this book to a larger audience, the basics of astronomy
and relevant facts about radiation fields and stars are
summarized in the appendix.
The goal of the lecture course, and thus also of this
book, is to confront physics students with astronomy
early in their studies. Since their knowledge of physics
is limited in their first year, many aspects of the material
covered here need to be explained with simplified argu-
ments. However, it is surprising to what extent modern
extragalactic astronomy can be treated with such ar-
guments. All the material in this book is covered in
the lecture course, though not all details are written up
here. I believe that only by covering this wide range
of topics can the students be guided to the forefront of
our present astrophysical knowledge. Hence, they learn
a lot about issues which are currently not settled and
under intense discussion. It is also this aspect which
I consider of great importance for the role of astronomy
in the framework of a physics program, since in most
other sub-disciplines of physics the limits of our cur-
rent knowledge are approached only at a later stage in
the student's education.
In particular, the topic of cosmology is usually met
with interest by students. Despite the large amount of
material, most of them are able to digest and under-
stand what they are taught, as evidenced from the oral
examinations following this course - and this is not
small-number statistics: my colleague Klaas de Boer
and I together grade about 100 oral examinations per
year, covering both parts of the introductory course.
Some critical comments coming from students concern
the extent of the material as well as its level. However,
I do not see a rational reason why the level of an astron-
omy lecture should be lower than that of one in physics
or mathematics.
Why did I turn this into a book? When preparing the
concept for my lecture course, I soon noticed that there
is no book which I can (or want to) follow. In particular,
there are only a few astronomy textbooks in German,
and they do not treat extragalactic astronomy and cos-
mology nearly to the extent and depth as I wanted for
this course. Also, the choice of books on these top-
ics in English is fairly limited - whereas a number of
excellent introductory textbooks exist, most shy away
from technical treatments of issues. However, many as-
pects can be explained better if a technical argument is
also given. Thus I hope that this text presents a field
of modern astrophysics at a level suitable for the afore-
mentioned group of people. A further goal is to cover
extragalactic astronomy to a level such that the reader
should feel comfortable turning to more professional
literature.
When being introduced to astronomy, students face
two different problems simultaneously. On the one
hand, they should learn to understand astrophysical
arguments - such as those leading to the conclusion
that the central engine in AGNs is a black hole. On
the other hand, they are confronted with a multitude
of new terms, concepts, and classifications, many of
which can only be considered as historical burdens. Ex-
amples here are the classification of supernovae which,
although based on observational criteria, do not agree
with our current understanding of the supernova phe-
nomenon, and the classification of the various types of
AGNs. In the lectures, I have tried to separate these
two issues, clearly indicating when facts are presented
where the students should "just take note", or when as-
trophysical connections are uncovered which help to
understand the properties of cosmic objects. The lat-
ter aspects are discussed in considerably more detail.
I hope this distinction can still be clearly seen in this
written version.
The order of the material in the course and in this
book accounts for the fact that students in their first year
of physics studies have a steeply rising learning curve;
hence, I have tried to order the material partly accord-
ing to its difficulty. For example, homogeneous world
models are described first, whereas only later are the
processes of structure formation discussed, motivated
in the meantime by the treatment of galaxy clusters.
The topic and size of this book imply the necessity
of a selection of topics. I want to apologize here to all
of those colleagues whose favorite subject is not cov-
ered at the depth that they feel it deserves. I also took
the freedom to elaborate on my own research topic -
gravitational lensing - somewhat disproportionately. If
it requires a justification: the basic equations of grav-
itational lensing are sufficiently simple that they and
their consequences can be explained at an early stage in
astronomy education.
With a field developing as quickly as the subject of
this book, it is unavoidable that parts of the text will be-
come somewhat out-of-date quickly. I have attempted to
include some of the most recent results of the respective
topics, but there are obvious limits. For example, just
three weeks before the first half of the manuscript was
sent to the publisher the three-year results from WMAP
were published. Since these results are compatible with
the earlier one-year data, I decided not to include them
in this text.
Many students are not only interested in the physical
aspects of astronomy, they are also passionate obser-
vational astronomers. Many of them have been active
in astronomy for years and are fascinated by phenom-
ena occurring beyond the Earth. I have tried to provide
a glimpse of this fascination at some points in the lecture
course, for instance through some historical details, by
discussing specific observations or instruments, or by
highlighting some of the great achievements of modern
cosmology. At such points, the text may deviate from
the more traditional "scholarly" style.
Producing the lecture notes, and their extension to
a textbook, would have been impossible without the
active help of several students and colleagues, whom
I want to thank here. Jan Hartlap, Elisabeth Krause and
Anja von der Linden made numerous suggestions for
improving the text, produced graphics or searched for
figures, and TgXed tables - deep thanks go to them.
Oliver Czoske, Thomas Erben and Patrick Simon read
the whole German version of the text in detail and
made numerous constructive comments which led to
a clear improvement of the text. Klaas de Boer and
Thomas Reiprich read and commented on parts of this
text. Searching for the sources of the figures, Leonardo
Castaneda, Martin Kilbinger, Jasmin Pierloz and Pe-
ter Watts provided valuable help. A first version of the
English translation of the book was produced by Ole
Markgraf, and I thank him for this heroic task. Further-
more, Kathleen Schriifer, Catherine Vlahakis and Peter
Watts read the English version and made zillions of
suggestions and corrections - 1 am very grateful to their
invaluable help. Thomas Erben, Mischa Schirmer and
Tim Schrabback produced the cover image very quickly
after our HST data of the cluster RXJ 1347- 1 145 were
taken. Finally, I thank all my colleagues and students
who provided encouragement and support for finishing
this book.
The collaboration with Springer- Verlag was very
fruitful. Thanks to Wolf Beiglbock and Ramon Khanna
for their encouragement and constructive collaboration.
Bea Laier offered to contact authors and publishers to
get the copyrights for reproducing figures - without
her invaluable help, the publication of the book would
have been delayed substantially. The interaction with
LE-Tr-X, where the book was produced, and in particular
with Uwe Matrisch, was constructive as well.
Furthermore, I thank all those colleagues who
granted permission to reproduce their figures here, as
well as the public relations departments of astronomical
organizations and institutes who, through their excellent
work in communicating astronomical knowledge to the
general public, play an invaluable role in our profes-
sion. In addition, they provide a rich source of pictorial
material of which I made ample use for this book. Repre-
sentative of those, I would like to mention the European
Southern Observatory (ESO), the Space Telescope Sci-
ence Institute (STScI), the NASA/SAO/CXC archive
for Chandra data and the Legacy Archive for Microwave
Background Data Analysis (LAMBDA).
List of Contents
Introduction
and Overview
2. The Milky Way
as a Galaxy
1.2.1
1.2.2
1.2.3
1.2.4
1.2.5
1.2.6
1.2.7
1.2.8
1.3
1.3.1
1.3.2
1.3.3
1.3.4
1.3.5
1.3.6
2.2
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
2.2.6
2.2.7
2.3
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
2.3.6
2.3.7
2.4
2.4.1
2.4.2
Introduction 1
Overview 4
Our Milky Way as a Galaxy 4
The World of Galaxies 7
The Hubble Expansion of the Universe 8
Active Galaxies and Starburst Galaxies 10
Voids, Clusters of Galaxies, and Dark Matter 12
World Models and the Thermal History of the Universe 14
Structure Formation and Galaxy Evolution 17
Cosmology as a Triumph of the Human Mind 17
The Tools of Extragalactic Astronomy 18
Radio Telescopes 19
Infrared Telescopes 22
Optical Telescopes 25
UV Telescopes 30
X-Ray Telescopes 31
Gamma-Ray Telescopes 32
Galactic Coordinates 35
Determination of Distances Within Our Galaxy 36
Trigonometric Parallax 37
Proper Motions 38
Moving Cluster Parallax 38
Photometric Distance; Extinction and Reddening 39
Spectroscopic Distance 43
Distances of Visual Binary Stars 43
Distances of Pulsating Stars 43
The Structure of the Galaxy 44
The Galactic Disk: Distribution of Stars 46
The Galactic Disk: Chemical Composition and Age 47
The Galactic Disk: Dust and Gas 50
Cosmic Rays 51
The Galactic Bulge 54
The Visible Halo 55
The Distance to the Galactic Center 56
Kinematics of the Galaxy 57
Determination of the Velocity of the Sun 57
The Rotation Curve of the Galaxy 59
The Galactic Microlensing Effect:
The Quest for Compact Dark Matter 64
List of Contents
2.5.1 The Gravitational Lensing Effect I 64
2.5.2 Galactic Microlensing Effect 69
2.5.3 Surveys and Results 72
2.5.4 Variations and Extensions 75
2.6 The Galactic Center 77
2.6.1 Where is the Galactic Center? 78
2.6.2 The Central Star Cluster 78
2.6.3 A Black Hole in the Center of the Milky Way 80
2.6.4 Flares from the Galactic Center 82
2.6.5 The Proper Motion of Sgr A* 83
2.6.6 Hypervelocity Stars in the Galaxy 84
3. The World of Galaxies 3.1 Classification 88
3.1.1 Morphological Classification: The Hubble Sequence 88
3.1.2 Other Types of Galaxies 89
3.2 Elliptical Galaxies 90
3.2.1 Classification 90
3.2.2 Brightness Profile 90
3.2.3 Composition of Elliptical Galaxies 92
3.2.4 Dynamics of Elliptical Galaxies 93
3.2.5 Indicators of a Complex Evolution 95
3.3 Spiral Galaxies 98
3.3.1 Trends in the Sequence of Spirals 98
3.3.2 Brightness Profile 98
3.3.3 Rotation Curves and Dark Matter 100
3.3.4 Stellar Populations and Gas Fraction 102
3.3.5 Spiral Structure 103
3.3.6 Corona in Spirals? 103
3.4 Scaling Relations 104
3.4.1 The Tully-Fisher Relation 104
3.4.2 The Faber-Jackson Relation 107
3.4.3 The Fundamental Plane 107
3.4.4 The D n -a Relation 108
3.5 Black Holes in the Centers of Galaxies 109
3.5.1 The Search for Supermassive Black Holes 109
3.5.2 Examples for SMBHs in Galaxies 110
3.5.3 Correlation Between SMBH Mass and Galaxy Properties 111
3.6 Extragalactic Distance Determination 114
3.6.1 Distance of the LMC 115
3.6.2 The Cepheid Distance 115
3.6.3 Secondary Distance Indicators 116
3.7 Luminosity Function of Galaxies 117
3.7.1 The Schechter Luminosity Function 118
3.7.2 The Bimodal Color Distribution of Galaxies 119
List of Contents
4. Cosmology I:
Homogeneous Isotropic
World Models
3.9
3.9.1
3.9.2
3.9.3
3.9.4
3.9.5
3.9.6
3.9.7
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
4.3
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.4
4.4.1
4.4.2
4.4.3
4.4.4
4.4.5
4.4.6
4.5
4.5.1
4.5.2
Galaxies as Gravitational Lenses 121
The Gravitational Lensing Effect - Part II 121
Simple Models 123
Examples for Gravitational Lenses 125
Applications of the Lens Effect 130
Population Synthesis 132
Model Assumptions 132
Evolutionary Tracks in the HRD; Integrated Spectrum 133
Color Evolution 135
Star Formation History and Galaxy Colors 136
Metallicity, Dust, and Hll Regions 136
Summary 136
The Spectra of Galaxies 137
Chemical Evolution of Galaxies 138
Introduction and Fundamental Observations 141
Fundamental Cosmological Observations 142
Simple Conclusions 142
An Expanding Universe 145
Newtonian Cosmology 146
Kinematics of the Universe 146
Dynamics of the Expansion 147
Modifications due to General Relativity 148
The Components of Matter in the Universe 149
"Derivation" of the Expansion Equation 150
Discussion of the Expansion Equations 150
Consequences of the Friedmann Expansion 152
The Necessity of a Big Bang 152
Redshift 155
Distances in Cosmology 157
Special Case: The Einstein-de Sitter Model 159
Summary 160
Thermal History of the Universe 160
Expansion in the Radiation-Dominated Phase 161
Decoupling of Neutrinos 161
Pair Annihilation 162
Primordial Nucleosynthesis 163
Recombination 166
Summary 169
Achievements and Problems of the Standard Model 169
Achievements 169
Problems of the Standard Model 170
Extension of the Standard Model: Inflation 173
List of Contents
5. Active Galactic Nuclei
6. Clusters and Groups
of Galaxies
5.1.1
5.1.2
5.1.3
5.1.4
5.2
5.2.1
5.2.2
5.2.3
5.2.4
5.2.5
5.3
5.3.1
5.3.2
5.3.3
5.3.4
5.3.5
5.4.1
5.4.2
5.4.3
5.4.4
5.4.5
5.4.6
5.5
5.5.1
5.5.2
5.5.3
5.5.4
5.6
5.6.1
5.6.2
5.6.3
Introduction
Brief History of AGNs
Fundamental Properties of Quasars
Quasars as Radio Sources: Synchrotron Radia
Broad Emission Lines
AGN Zoology
Quasi-Stellar Objects
Seyfert Galaxies
Radio Galaxies
Optically Violently Variables
BL Lac Objects
The Central Engine: A Black Hole .
Why a Black Hole?
Accretion
Superluminal Motion
Further Arguments for SMBHs
A First Mass Estimate for the SMBH:
The Eddington Luminosity
93
Components of an AGN 195
The IR, Optical, and UV Continuum
The Broad Emission Lines
Narrow Emission Lines 201
X-Ray Emission 201
The Host Galaxy 202
The Black Hole Mass in AGNs 204
Family Relations of AGNs 207
Unified Models 207
Beaming 210
Beaming on Large Scales 211
Jets at Higher Frequencies 212
AGNs and Cosmology 215
The K-Correction 215
The Luminosity Function of Quasars 216
Quasar Absorption Lines 219
6.1 The Local Group 224
6.1.1 Phenomenology 224
6.1.2 Mass Estimate 225
6.1.3 Other Components of the Local Group 227
6.2 Galaxies in Clusters and Groups 228
6.2.1 The Abell Catalog 228
6.2.2 Luminosity Function of Cluster Galaxies 230
6.2.3 Morphological Classification of Clusters 231
List of Contents
Cosmology II:
Inhomogeneities in the
6.2.4 Spatial Distribution of Galaxies 231
6.2.5 Dynamical Mass of Clusters 233
6.2.6 Additional Remarks on Cluster Dynamics 234
6.2.7 Intergalactic Stars in Clusters of Galaxies 236
6.2.8 Galaxy Groups 237
6.2.9 The Morphology-Density Relation 239
6.3 X-Ray Radiation from Clusters of Galaxies 242
6.3.1 General Properties of the X-Ray Radiation 242
6.3.2 Models of the X-Ray Emission 246
6.3.3 Cooling Flows 248
6.3.4 The Sunyaev-Zeldovich Effect 252
6.3.5 X-Ray Catalogs of Clusters 255
6.4 Scaling Relations for Clusters of Galaxies 256
6.4. 1 Mass-Temperature Relation 256
6.4.2 Mass- Velocity Dispersion Relation 257
6.4.3 Mass-Luminosity Relation 258
6.4.4 Near-Infrared Luminosity as Mass Indicator 259
6.5 Clusters of Galaxies as Gravitational Lenses 260
6.5.1 Luminous Arcs 260
6.5.2 The Weak Gravitational Lens Effect 264
6.6 Evolutionary Effects 270
7.1 Introduction 277
7.2 Gravitational Instability 278
7.2.1 Overview 278
7.2.2 Linear Perturbation Theory 279
7.3 Description of Density Fluctuations 282
7.3.1 Correlation Functions 283
7.3.2 The Power Spectrum 284
7.4 Evolution of Density Fluctuations 285
7.4. 1 The Initial Power Spectrum 285
7.4.2 Growth of Density Perturbations 286
7.5 Non-Linear Structure Evolution 289
7.5.1 Model of Spherical Collapse 289
7.5.2 Number Density of Dark Matter Halos 291
7.5.3 Numerical Simulations of Structure Formation 293
7.5.4 Profile of Dark Matter Halos 298
7.5.5 The Substructure Problem 302
7.6 Peculiar Velocities 306
7.7 Origin of the Density Fluctuations 307
List of Contents
8. Cosmology III:
The Cosmological
Parameters
The Universe
at High Redshift
Redshift Surveys of Galaxies 309
Introduction 309
Redshift Surveys 310
Determination of the Power Spectrum 313
Effect of Peculiar Velocities 316
Angular Correlations of Galaxies 318
Cosmic Peculiar Velocities 319
8.2 Cosmological Parameters from Clusters of Galaxies 321
8.2.1 Number Density 322
8.2.2 Mass-to-Light Ratio 322
8.2.3 Baryon Content 323
8.2.4 The LSS of Clusters of Galaxies 323
8.3 High-Redshift Supernovae
and the Cosmological Constant 324
8.3.1 Are SNIa Standard Candles? 324
8.3.2 Observing SNe la at High Redshifts 325
8.3.3 Results 326
8.3.4 Discussion 328
Cosmic Shear .
. 329
8.5 Origin of the Lyman-a Forest 331
8.5.1 The Homogeneous Intergalactic Medium 331
8.5.2 Phenomenology of the Lyman-a Forest 332
8.5.3 Models of the Lyman-a Forest 333
8.5.4 The Lya Forest as Cosmological Tool 335
8.6 Angular Fluctuations
of the Cosmic Microwave Background 336
8.6.1 Origin of the Anisotropy: Overview 336
8.6.2 Description of the Cosmic Microwave
Background Anisotropy 338
8.6.3 The Fluctuation Spectrum 339
8.6.4 Observations of the Cosmic Microwave
Background Anisotropy 341
8.6.5 WMAP: Precision Measurements
of the Cosmic Microwave Background Anisotropy 345
8.7 Cosmological Parameters 349
8.7. 1 Cosmological Parameters with WMAP 349
8.7.2 Cosmic Harmony 352
9.1 Galaxies at High Redshift 356
9.1.1 Lyman-Break Galaxies (LBGs) 356
9.1.2 Photometric Redshift 362
9.1.3 Hubble Deep Field(s) 364
9.1.4 Natural Telescopes 367
List of Contents
9.2 New Types of Galaxies 369
9.2.1 Starburst Galaxies 369
9.2.2 Extremely Red Objects (EROs) 371
9.2.3 Submillimeter Sources: A View Through Thick Dust 374
9.2.4 Damped Lyman- Alpha Systems 377
9.2.5 Lyman-Alpha Blobs 378
9.3 Background Radiation at Smaller Wavelengths 379
9.3.1 The IR Background 380
9.3.2 The X-Ray Background 380
9.4 Reionization of the Universe 382
9.4.1 The First Stars 383
9.4.2 The Reionization Process 385
9.5 The Cosmic Star-Formation History 387
9.5.1 Indicators of Star Formation 387
9.5.2 Redshift Dependence of the Star Formation:
The Madau Diagram 389
9.6 Galaxy Formation and Evolution 390
9.6.1 Expectations from Structure Formation 391
9.6.2 Formation of Elliptical Galaxies 392
9.6.3 Semi- Analytic Models 395
9.6.4 Cosmic Downsizing 400
9.7 Gamma-Ray Bursts 402
io. Outlook 407
Appendix
A. The Electromagnetic A.l Parameters of the Radiation Field 417
Radiation Field »„„,..„,.
A.2 Radiative Transfer 417
A.3 Blackbody Radiation 418
A.4 The Magnitude Scale 420
A.4. 1 Apparent Magnitude 420
A.4.2 Filters and Colors 420
A.4.3 Absolute Magnitude 422
A.4.4 Bolometric Parameters 422
B. Properties of Stars B.l The Parameters of Stars 425
B.2 Spectral Class, Luminosity Class,
and the Hertzsprung-Russell Diagram 425
B.3 Structure and Evolution of Stars 427
C. Units and Constants 431
List of Contents
D. Recommended D.l General Textbooks 433
D.2 More Specific Literature 433
D.3 Review Articles, Current Literature, and Journals 434
E. Acronyms Used 437
F. Figure Credits 441
Subject Index 453
i. Introduction and Overview
1.1 Introduction
The Milky Way, the galaxy in which we live, is but one
of many galaxies. As a matter of fact, the Milky Way,
also called the Galaxy, is a fairly average representative
of the class of spiral galaxies. Two other examples of
spiral galaxies are shown in Fig. 1 . 1 and Fig. 1 .2, one of
which we are viewing from above (face-on), the other
from the side (edge-on). These are all stellar systems in
which the majority of stars are confined to a relatively
thin disk. In our own Galaxy, this disk can be seen as
the band of stars stretched across the night sky, which
led to it being named the Milky Way. Besides such
disk galaxies, there is a second major class of luminous
stellar systems, the elliptical galaxies. Their properties
differ in many respects from those of the spirals.
It was less than a hundred years ago that a;
first realized that objects exist outside our Milky Way
and that our world is significantly larger than the size of
the Milky Way. In fact, galaxies are mere islands in the
Universe: the diameter of our Galaxy 1 (and other galax-
ies) is much smaller than the average separation between
luminous galaxies. The discovery of the existence of
other stellar systems and their variety of morphologies
raised the question of the origin and evolution of these
galaxies. Is there anything between the galaxies, or is
it just empty space? Are there any other cosmic bodies
besides galaxies? Questions like these motivated us to
explore the Universe as a whole and its evolution. Is our
"Milky Way" and "Galaxy synonymously
Fig. 1.1. The spiral galaxy NGC
1232 may resemble our Milky Way
if it were to be observed from
"above" (face-on). This image, ob-
served with the VLT, has a size of
6.'8 x 6.'8, corresponding to a lin-
ear size of 60 kpc at its distance
of 30Mpc. If this was our Gal-
axy, our Sun would be located at
a distance of 8.0 kpc from the cen-
ter, orbiting around it at a speed
of ~ 220km/s. A full revolution
would take us about 230 x 10 6
years. The bright knots seen along
the spiral arms of this galaxy are
clusters of newly-formed stars, sim-
ilar to bright young star clusters in
our Milky Way. The different, more
reddish, color of the inner part of
this galaxy indicates that the aver-
age age of the stars there is higher
than in the outer parts. The small
galaxy at the lower left edge of the
image is a companion galaxy that is
distorted by the gravitational tidal
forces caused by the spiral galaxy
Peter Schneider,
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Fig. 1.2. We see the spiral galaxy NGC 4013 from the side
(edge-on); an observer looking at the Milky Way from a di-
rection which lies in the plane of the stellar disk ("from the
side") may have a view like this. The disk is clearly visible,
with its central region obscured by a layer of dust. One also
sees the central bulge of the galaxy. As will be discussed at
length later on, spiral galaxies like this one are surrounded by
a halo of matter which is observed only through its gravita-
lional action, e.g., by affecting the velocity of stars and ga\
rotating around the center of the galaxy
Universe finite or infinite? Does it change over time?
Does it have a beginning and an end? Mankind has long
been fascinated by these questions about the origin and
the history of our world. But for only a few decades have
we been able to approach these questions in an empir-
ical manner. As we shall discuss in this book, many of
the questions have now been answered. However, each
answer raises yet more questions, as we aim towards
an ever increasing understanding of the physics of the
Universe.
The stars in our Galaxy have very different ages.
The oldest stars are about 12 billion years old, whereas
in some regions stars are still being born today: for
instance in the well-known Orion nebula. Obviously,
the stellar content of our Galaxy has changed over time.
To understand the formation and evolution of the Galaxy
a view of its (and thus our own) past would be useful.
Unfortunately, this is physically impossible. However,
due to the finite speed of light, we see objects at large
distances in an earlier state, as they were in the past. One
can now try to identify and analyze such distant galaxies,
which may have been the progenitors of galaxies like
our own Galaxy, in this way reconstructing the main
aspects of the history of the Milky Way. We will never
know the exact initial conditions that led to the evolution
of the Milky Way, but we may be able to find some
characteristic conditions. Emerging from such initial
states, cosmic evolution should produce galaxies similar
to our own, which we would then be able to observe from
the outside. On the other hand, only within our own
Galaxy can we study the physics of galaxy evolution
We are currently witnessing an epoch of tremendous
discoveries in astronomy. The technical capabilities in
observation and data reduction are currently evolving at
an enormous pace. Two examples taken from ground-
based optical astronomy should serve to illustrate this.
In 1993 the first 10-m class telescope, the Keck
telescope, was commissioned, the first increase in
light-collecting power of optical telescopes since the
completion of the 5-m mirror on Mt. Palomar in 1948.
Now, just a decade later, about ten telescopes of the
10-m class are in use, and even more are soon to come.
In recent years, our capabilities to find very distant, and
thus very dim, objects and to examine them in detail
have improved immensely thanks to the capability of
these large optical telescopes.
A second example is the technical evolution and size
of optical detectors. Since the introduction of CCDs
in astronomical observations at the end of the 1970s,
which replaced photographic plates as optical detec-
tors, the sensitivity, accuracy, and data rate of optical
observations have increased enormously. At the end of
the 1980s, a camera with 1000 x 1000 pixels (picture
elements) was considered a wide-field instrument. In
2003 a camera called Megacam began operating; it has
(18 000) 2 pixels and images a square degree of the sky
at a sampling rate of 0"2 in a single exposure. Such
a camera produces roughly 100 GB of data every night,
the reduction of which requires fast computers and vast
storage capacities. But it is not only optical astronomy
that is in a phase of major development; there has also
been huge progress in instrumentation in other wave-
bands. Space-based observing platforms are playing
1.1 Introduction
a crucial role in this. We will consider this topic in
Sect. 1.3.
These technical advances have led to a vast increase
in knowledge and insight in astronomy, especially in ex-
tragalactic astronomy and cosmology. Large telescopes
and sensitive instruments have opened up a window to
the distant Universe. Since any observation of distant
objects is inevitably also a view into the past, due to the
finite speed of light, studying objects in the early Uni-
verse has become possible. Today, we can study galaxies
which emitted the light we observe at a time when the
Universe was less than 10% of its current age; these gal-
axies are therefore in a very early evolutionary stage.
We are thus able to observe the evolution of galaxies
throughout the past history of the Universe. We have
the opportunity to study the history of galaxies and thus
that of our own Milky Way. We can examine at which
epoch most of the stars that we observe today in the
local Universe have formed because the history of star
formation can be traced back to early epochs. In fact,
it has been found that star formation is largely hidden
from our eyes and only observable with space-based
telescopes operating in the far-infrared waveband.
One of the most fascinating discoveries of recent
years is that most galaxies harbor a black hole in their
center, with a characteristic mass of millions or even
billions of solar masses - so-called supermassive black
holes. Although as soon as the first quasars were found
in 1963 it was proposed that only processes around
a supermassive black hole would be able to produce the
huge amount of energy emitted by these ultra-luminous
objects, the idea that such black holes exist in normal
galaxies is fairly recent. Even more surprising was the
finding that the black hole mass is closely related to
the other properties of its parent galaxy, thus providing
a clear indication that the evolution of supermassive
black holes is closely linked to that of their host galaxies.
Detailed studies of individual galaxies and of asso-
ciations of galaxies, which are called galaxy groups or
clusters of galaxies, led to the surprising result that these
objects contain considerably more mass than is visible
in the form of stars and gas. Analyses of the dynamics
of galaxies and clusters show that only 10-20% of their
mass consists of stars, gas and dust that we are able to
observe in emission or absorption. The largest fraction
of their mass, however, is invisible. Hence, this hidden
mass is called dark matter. We know of its presence
only through its gravitational effects. The dominance
of dark matter in galaxies and galaxy clusters was es-
tablished in recent years from observations with radio,
optical and X-ray telescopes, and it was also confirmed
and quantified by other methods. However, we do not
know what this dark matter consists of; the unambigu-
ous evidence for its existence is called the "dark matter
problem".
The nature of dark matter is one of the central ques-
tions not only in astrophysics but also poses a challenge
to fundamental physics, unless the "dark matter prob-
lem" has an astronomical solution. Does dark matter
consist of non-luminous celestial bodies, for instance
burned-out stars? Or is it a new kind of matter? Have
astronomers indirectly proven the existence of a new el-
ementary particle which has thus far escaped detection
in terrestrial laboratories? If dark matter indeed con-
sists of a new kind of elementary particle, which is the
common presumption today, it should exist in the Milky
Way as well, in our immediate vicinity. Therefore, ex-
periments which try to directly detect the constituents of
dark matter with highly sensitive and sophisticated de-
tectors have been set up in underground laboratories.
Physicists and astronomers are eagerly awaiting the
commissioning of the Large Hadron Collider (LHC),
a particle accelerator at the European CERN research
center which, from 2007 on, will produce particles at
significantly higher energies than accessible today. The
hope is to find an elementary particle that could serve
as a candidate constituent of dark matter.
Without doubt, the most important development in
recent years is the establishment of a standard model of
cosmology, i.e., the science of the Universe as a whole.
The Universe is known to expand and it has a finite age;
we now believe that we know its age with a precision of
as little as a few percent — it is fo = 13.7 Gyr. The Uni-
verse has evolved from a very dense and very hot state,
the Big Bang, expanding and cooling over time. Even
today, echoes of the Big Bang can be observed, for ex-
ample in the form of the cosmic microwave background
radiation. Accurate observations of this background ra-
diation, emitted some 380 000 years after the Big Bang,
have made an important contribution to what we know
today about the composition of the Universe. However,
these results raise more questions than they answer:
only ~ 4% of the energy content of the Universe can
be accounted for by matter which is well-known from
1. Introduction and Overviev
other fields of physics, the baryonic mailer lhat consists
mainly of atomic nuclei and electrons. About 25% of
the Universe consists of dark matter, as we already dis-
cussed in the context of galaxies and galaxy clusters.
Recent observational results have shown that the mean
density of dark matter dominates over that of baryonic
matter also on cosmic scales.
Even more surprising than the existence of dark mat-
ter is the discovery that about 70% of the Universe
consists of something that today is called vacuum en-
ergy, or dark energy, and that is closely related to the
cosmological constant introduced by Albert Einstein.
The fact that various names do exist for it by no means
implies that we have any idea what this dark energy
is. It reveals its existence exclusively in its effect on
cosmic expansion, and it even dominates the expansion
dynamics at the current epoch. Any efforts to estimate
the density of dark energy from fundamental physics
have failed hopelessly. An estimate of the vacuum en-
ergy density using quantum mechanics results in a value
that is roughly 120 orders of magnitude larger than the
value derived from cosmology. For the foreseeable fu-
ture observational cosmology will be the only empirical
probe for dark energy, and an understanding of its phys-
ical nature will probably take a substantial amount of
time. The existence of dark energy may well pose the
greatest challenge to fundamental physics today.
In this book we will present a discussion of the ex-
tragalactic objects found in astronomy, starting with
the Milky Way which, being a typical spiral galaxy, is
considered a prototype of this class of stellar systems.
The other central topic in this book is a presentation
of modern astrophysical cosmology, which has expe-
rienced tremendous advances in recent years. Methods
and results will be discussed in parallel. Besides pro-
viding an impression of the fascination that arises from
astronomical observations and cosmological insights,
astronomical methods and physical considerations will
be our prime focus. We will start in the next section
with a concise overview of the fields of extragalactic
astronomy and cosmology. This is, on the one hand, in-
tended to whet the reader's appetite and curiosity, and
on the other hand to introduce some facts and technical
terms that will be needed in what follows but which are
discussed in detail only later in the book. In Sect. 1.3
we will describe some of the most important telescopes
used in extragalactic astronomy today.
1.2 Overview
1.2.1 Our Milky Way as a Galaxy
The Milky Way is the only galaxy which we are able
to examine in detail. We can resolve individual stars
and analyze them spectroscopically. We can perform
detailed studies of the interstellar medium (ISM), such
as the properties of molecular clouds and star-forming
regions. We can quantitatively examine extinction and
reddening by dust. Furthermore, we can observe the
local dynamics of stars and gas clouds as well as the
properties of satellite galaxies (such the Magellanic
Clouds). Finally, the Galactic center at a distance of
only 8 kpc 2 gives us the unique opportunity to examine
the central region of a galaxy at very high resolution.
Only through a detailed understanding of our own Gal-
axy can we hope to understand the properties of other
galaxies. Of course, we implicitly assume that the phys-
ical processes taking place in other galaxies obey the
same laws of physics that apply to us. If this were not
the case, we would barely have a chance to understand
the physics of other objects in the Universe, let alone the
Universe as a whole. We will return to this point shortly.
We will first discuss the properties of our own Galaxy.
One of the main problems here, and in astronomy in
general, is the determination of the distance to an object.
Thus we will start by considering this topic. From the
analysis of the distribution of stars and gas in the Milky
Way we will then derive its structure. It is found that the
Galaxy consists of several distinct components:
• a thin disk of stars and gas with a radius of about
20 kpc and a scale-height of about 300 pc, which
also hosts the Sun;
• a ~ 1 kpc thick disk, which contains a different stellar
population compared to the thin disk;
• a central bulge, as is also found in other spiral
galaxies;
• and a nearly spherical halo which contains most of
the globular clusters and some old stars.
Figure 1.3 shows a schematic view of our Milky Way
and its various components. For a better visual impres-
sion, Figs. 1.1 and 1.2 show two spiral galaxies, the
-One parsec (! pc) is the common unit ol distance in astronomy.
with 1 pc = 3.086 x 10 18 cm. Also used are 1 kpc = 10 3 pc, 1 Mpc =
10 6 pc, 1 Gpc = 10 9 pc. Other commonly used unit
listed in Appendix C.
%
Halo
Center
Bulge
Sun
1
^jisk
8.5 kpc
•
30kpc
•
Globular __
cluster
r _^__^^Observed
| 200
\ v sun is -220 km/s
-
>.
Difference:
Dark Matter halo
8 150
v sun should be ^^^^^^
"
>
-160 km/s
■8 100
Visible matter only"
50
n
Fig. 1.3. Schematic structure of the Milky Way consisting
of the disk, the central bulge with the Galactic center, and
the spherical halo in which most of ihc globular clusters arc
located. The Sun orbits around the Galactic center at a distance
of about 8 kpc
former viewed from "above" (face-on) and the latter
from the "side" (edge-on). In the former case, the spi-
ral structure, from which this kind of galaxy derives
its name, is clearly visible. The bright knots in the spi-
ral arms are regions where young, luminous stars have
recently formed. The image shows an obvious color
gradient: the galaxy is redder in the center and bluest in
the spiral arms - while star formation is currently tak-
ing place in the spiral arms, we find mainly old stars
towards the center, especially in the bulge.
The Galactic disk rotates, with rotational velocity
V(R) depending on the distance R from the center. We
can estimate the mass of the Galaxy from the distri-
bution of the stellar light and the mean mass-to-light
ratio of the stellar population, since gas and dust repre-
sent less than ~ 10% of the mass of the stars. From this
mass estimate we can predict the rotational velocity as
a function of radius simply from Newtonian mechanics.
However, the observed rotational velocity of the Sun
around the Galactic center is significantly higher than
would be expected from the observed mass distribution.
If M(Ro) is the mass inside a sphere around the Gal-
actic center with radius Rq « 8 kpc, then the rotational
15
20
25
Distance to Center (kpc)
Fig. 1.4. The upper curve is the observed rotation curve V(R)
of our Galaxy, i.e., the rotational velocity of stars and gas
around the Galactic centei as a function of their galacto-centric
distance. The lower curve is the rotation curve that we would
predict based solely on the observed stellar mass of the Gal-
axy. The difference between these two curves is ascribed to
the presence of dark matter, in which the Milky Way disk is
embedded
velocity from Newtonian mechanics 3 is
GM(Rq)
From the visible matter in stars we would expect
a rotational velocity of ~ 160 km/s, but we observe
Vo ~ 220 km/s (see Fig. 1.4). This, and the shape of
the rotation curve V(R) for larger distances R from the
Galactic center, indicates that our Galaxy contains sig-
nificantly more mass than is visible in the form of stars. 4
This additional mass is called dark matter. Its physical
nature is still unknown. The main candidates are weakly
interacting elementary particles like those postulated by
some elementary particle theories, but they have yet not
been detected in the laboratory. Macroscopic objects
(i.e., celestial bodies) are also in principle possible can-
didates if they emit very little light. We will discuss ex-
periments which allow us to identify such macroscopic
'We use standard nutation: (', is the New Ionian gravitational constant,
c the speed of light.
Strictl) speaking. ( 1. 1 ) is \alidonly I or a spherically swnmetnc mass
distribution. However, the rotational velocity for an oblate density
distribution does not diner much, so we can use this relation as an
approximation.
1. Introduction and Overviev
objects and come to the conclusion that the solution of
the dark matter problem probably can not be found in
astronomy, but rather most likely in particle physics.
The stars in the various components of our Gal-
axy have different properties regarding their age and
their chemical composition. By interpreting this fact
one can infer some aspects of the evolution of the
Galaxy. The relatively young age of the stars in the
thin disk, compared to that of the older population in
the bulge, suggests different phases in the formation
and evolution of the Milky Way. Indeed, our Galaxy is
a highly dynamic object that is still changing today. We
see cold gas falling into the Galactic disk and hot gas
outflowing. Currently the small neighboring Sagittarius
dwarf galaxy is being torn apart in the tidal gravita-
tional field of the Milky Way and will merge with it in
the (cosmologically speaking) near future.
One cannot see far through the disk of the Galaxy
at optical wavelengths due to extinction by dust. There-
fore, the immediate vicinity of the Galactic center can
be examined only in other wavebands, especially the
infrared (IR) and the radio parts of the electromag-
netic spectrum (see also Fig. 1.5). The Galactic center
is a highly complex region but we have been able to
study it in recent years thanks to various substantial im-
provements in IR observations regarding sensitivity and
angular resolution. Proper motions, i.e., changes of the
positions on the sky with time, of bright stars close to
the center have been observed. They enable us to deter-
mine the mass M in a volume of radius ~ 0. 1 pc to be
M(0.1 pc) - 3 x 10 6 M Q . Although the data do not al-
low us to make a totally unambiguous interpretation of
this mass concentration there is no plausible alternative
to the conclusion that the center of the Milky Way har-
bors a supermassive black hole (SMBH) of roughly this
mass. And yet this SMBH is far less massive than the
ones that have been found in many other galaxies.
Unfortunately, we are unable to look at our Galaxy
from the outside. This view from the inside renders it
difficult to observe the global properties of the Milky
Way. The structure and geometry of the Galaxy, e.g., its
spiral arms, are hard to identify from our location. In
addition, the extinction by dust hides large parts of the
Galaxy from our view (see Fig. 1.6), so that the global
Fig. 1.5. The Galactic disk ob-
served in nine different wavebands.
Its appearance differs strongly in
the various images; for example,
the distribution of atomic hydrogen
and of molecular gas is much more
concentrated towards the Galactic
plane than the distribution of stars
observed in the near-infrared, the
latter clearly showing the presence
of a central bulge. The absorp-
tion by dust at optical wavelengths
is also clearly visible and can be
compared to that in Fig. 1.2
Fig. 1.6. The galaxy Dwingeloo 1 is only five times more
distant than our closest large neighboring galaxy, Andromeda.
yet it was not discovered until the 1990s because it hides
behind the Galactic center. The absorption in this direction and
numerous bright stars prevented it being discovered earlier.
The figure shows an image observed with the Isaac Newton
Telescope in the V-, R-, and I-bands
parameters of the Milky Way (like its total luminosity)
are difficult to measure. These parameters are estimated
much better from outside, i.e., in other similar spiral gal-
axies. In order to understand the large-scale properties
of our Galaxy, a comparison with similar galaxies which
we can examine in their entirety is extremely helpful.
Only by combining the study of the Milky Way with
that of other galaxies can we hope to fully understand
the physical nature of galaxies and their evolution.
1.2.2 The World of Galaxies
Next we will discuss the properties of other galaxies.
The two main types of galaxies are spirals (like the
Milky Way, see also Fig. 1.7) and elliptical galaxies
(Fig. 1.8). Besides these, there are additional classes
such as irregular and dwarf galaxies, active galaxies, and
starburst galaxies, where the latter have a very high star-
formation rate in comparison to normal galaxies. These
classes differ not only in their morphology, which forms
the basis for their classification, but also in their physical
properties such as color (indicating a different stellar
content), internal reddening (depending on their dust
Fig. 1.7. NGC 2997 is a typical spiral gal-
axy, with its disk inclined by about 45° with
respect to the line-of-sight. Like most spi-
ral galaxies it has two spiral arms; they are
significantly bluer than other parts of the
galaxy. This is caused by ongoing star for-
mation in these regions so that young, hot
and thus blue stars are present in the arms,
whereas the center of the galaxy, especially
the bulge, consists mainly of old stars
Fig. 1.8. M87 is a very luminous elliptical galaxy in the
center of the Virgo Cluster, at a distance of about 18Mpc.
The diameter of the visible part of this galaxy is about
40kpc; it is significantly more massive than the Milky Way
(M > 3 x 10 12 M Q ). We will frequently refer to this galaxy:
it is not only an excellent example of a central cluster galaxy
but also a representative of the family of "active galaxies". It
is a strong radio emitter (radio astronomers also know it as
Virgo A), and it has an optical jet in its center
content), amount of interstellar gas, star-formation rate,
etc. Galaxies of different morphologies have evolved in
different ways.
Spiral galaxies are stellar systems in which active star
formation is still taking place today, whereas elliptical
galaxies consist mainly of old stars - their star forma-
tion was terminated a long time ago. The SO galaxies,
an intermediate type, show a disk similar to that of spi-
ral galaxies but like ellipticals they consist mainly of old
stars, i.e., stars of low mass and low temperature. Ellipti-
cals and SO galaxies together are often called early-type
galaxies, whereas spirals are termed late-type galaxies.
These names do not imply any interpretation but exist
only for historical reasons.
The disks of spiral galaxies rotate differentially. As
for the Milky Way, one can determine the mass from
the rotational velocity using the Kepler law (1.1). One
finds that, contrary to the expectation from the distribu-
tion of light, the rotation curve does not decline at larger
distances from the center. Like our own Galaxy, spiral
galaxies contain a tunic amount of dark matter; the vis-
ible mailer is embedded in a halo oj dark mailer. We can
only get rough estimates of the extent of this halo, but
there are strong indications that it is substantially largei
than the extent of the visual matter. For instance, the ro-
tation curve is flat up to the largest radii where one still
finds gas to measure the velocity. Studying dark n
in elliptical galaxies is more complicated, but the exis-
tence of dark halos has also been proven for ellipticals
The Hertzsprung-Russell diagram of stars, or thei:
color-magnitude diagram (see Appendix B), has turned
out to be the most important diagram in stellar
trophysics. The fact that most stars are aligned along
a one-dimensional sequence, the main sequence, led to
the conclusion that, for main-sequence stars, the lumi-
nosity and the surface temperature are not independent
parameters. Instead, the properties of such stars are in
principle characterized by only a single parameter: the
stellar mass. We will also see that the various proper-
ties of galaxies are not independent parameters. Rather,
dynamical properties (such as the rotational velocity
of spirals) are closely related to the luminosity. These
scaling relations are of similar importance to the study
of galaxies as the Hertzsprung-Russell diagram is for
stars. In addition, they turn out to be very convenient
tools for the determination of galaxy distances.
Like our Milky Way, other galaxies also seem to har-
bor a SMBH in their center. We obtained the astonishing
result that the mass of such a SMBH is closely related
to the velocity distribution of stars in elliptical galax-
ies or in the bulge of spirals. The physical reason for
this close correlation is as yet unknown, but it strongly
suggests a joint evolution of galaxies and their SMBHs.
1.2.3 The Hubble Expansion of the Universe
The radial velocity of galaxies, measured by means of
the Doppler shift of spectral lines (Fig. 1.9), is positive
for nearly all galaxies, i.e., they appear to be moving
away from us. In 1928, Edwin Hubble discovered that
Galaxy,
Part of Estimated Distance Redshift
Cluster in: (mega pars ecs)
□ H +K
■inHHflnn
Fig. 1.9. The spectra of galaxies show char-
acteristic spectral lines, e.g., the H + Klines
of calcium. These lines, however, do not
appear at the wavelengths measured in the
laboratory hut are in general lifted towards
longer wavelengths. This is shown here for
a set of sample galaxies, with distance in
creasing from top to bottom. The shift in
the lines, interpreted as being due to the
Doppler effect, allows us to determine the
relative radial velocity - the larger it is,
the more distant the galaxy is. The discrete
lines above and below the spectra are for
calibration purposes only
this escape velocity v increases with the distance of
the galaxy. He identified a linear relation between the
radial velocity v and the distance D of galaxies, called
a Hubble law,
(1.2)
--H D
where Hq is a constant. If we plot the radial velocity of
galaxies against their distance, as is done in the Hubble
diagram of Fig. 1.10, the resulting points are approxi-
mated by a straight line, with the slope being determined
by the constant of proportionality, Hq, which is called
the Hubble constant. The fact that all galaxies seem
to move away from us with a velocity which increases
linearly with their distance is interpreted such that the
Universe is expanding. We will see later that this Hub-
ble expansion of the Universe is a natural property of
cosmological world models.
The value of Ho has been determined with ap-
preciable precision only in recent years, yielding the
conservative estimate
eOkms-'Mpc" 1 <H < 80 km s" 1 Mpc" 1 , (1.3)
obtained from several different methods which will be
discussed later. The error margins vary for the differ-
1. Introduction and Overviev
g 500 KM
8
o. 4^T
V**
*
10 6 Parsec
Entfernung
Fig. 1.10. The original 1929 version of the
Hubble diagram shows the radial velocity of
galaxies as a function of their distance. The
reader may notice that the velocity axis is
labeled with errornous units - of course they
should read km/s. While the radial (escape)
velocity is easily measured by means of the
Dopplci shift in spectral lines, an accurate
determination of distances is much more
difficult; we will discuss methods of dis-
tance determination for galaxies in Sect. 3.6.
Hubble has underestimated the distances
considerably, resulting in too high a value
for the Hubble constant. Only very few
and very close galaxies show a blueshii't,
i.e., they move towards us; one of these is
Andromeda (= M31)
ent methods and also for different authors. The main
problem in determining Hq is in measuring the absolute
distance of galaxies, whereas Doppler shifts are easily
measurable. If one assumes (1.2) to be valid, the radial
velocity of a galaxy is a measure of its distance. One
defines the redshift, z, of an object from the wavelength
shift in spectral lines,
. ,_X ohs -X
A, l
= (l + z)A. ,
(1.4)
with An denoting the wavelength of a spectral t
in the rest-frame of the emitter and A obs the observed
wavelength. For instance, the Lyman-a transition, i.e.,
the transition from the first excited level to the ground
state in the hydrogen atom is at A. = 1216 A. For small
redshifts,
v^zc, (1.5)
whereas this relation has to be modified for large red-
shifts, together with the interpretation of the redshift
itself. 5 Combining (1.2) and (1.5), we obtain
£>% — ss3000z/j _1 Mpc, (1.6)
Ho
"What is ohser\ed is the wavelength shift of spectral lines. Depend
ing on the context, it is interpreted either as a radial velocity of
a source moving away from us - for instance, if we measure the
radial velocity oi stars in the Milky Way - or as a cosmologieal
escape velocity, as is the case for the Hubble law. It is in prin-
eipl i n blc lo distingui i Ih < ntci'i itioi
because a galaxy not only take-, pari in the cosmic expansion but it
where the uncertainty in determining Ho is parametrized
by the scaled Hubble constant h, defined as
H = hl00kms
Mpc
(1.7)
Distance determinations based on redshift therefore al-
ways contain a factor of h~ l , as seen in (1.6). It needs
to be emphasized once more that (1.5) and (1.6) are
valid only for z <?C 1 ; the generalization for larger red-
shifts will be discussed in Sect. 4.3. Nevertheless, z is
also a measure of distance for large redshifts.
1.2.4 Active Galaxies and Starburst Galaxies
A special class of galaxies are the so-called active gal-
axies which have a very strong energy source in their
center (active galactic nucleus, AGN). The best-known
representatives of these AGNs are the quasars, ob-
jects typically at high redshift and with quite exotic
properties. Their spectrum shows strong emission lines
which can be extremely broad, with a relative width of
AX/X ~ 0.03. The line width is caused by very high
can. in addition, tunc a so called peculiar velocity. We will there
lore use the word- "Doppler shift" and "redshift". respectively, and
"radial velocity" depending on the context, but always keeping in
mind that both are measured b\ the shift of spectral lines. Only
when observing the distant Universe where the Doppler shift is
fully dominated by the cosmic expansion will we exclusively call
it "redshift".
random velocities of the gas which emits these line: if
we interpret the line width as due to Doppler broadening
resulting from the superposition of lines of emitting gas
with a very broad velocity distribution, we obtain veloc-
ities of typically An ~ 10 000 km/s. The central source
of these objects is much brighter than the other parts of
the galaxy, making these sources appear nearly point-
like on optical images. Only with the Hubble Space
Telescope (HST) did astronomers succeed in detecting
structure in the optical emission for a large sample of
quasars (Fig. 1.11).
Many properties of quasars resemble those of Seyfert
type I galaxies, which are galaxies with a very luminous
nucleus and very broad emission lines. For this reason,
quasars are often interpreted as extreme members of
(his class. The total luminosity of quasars is extremely
large, with some of them emitting more than a thou-
sand times the luminosity of our Galaxy. In addition,
this radiation must originate from a very small spatial
region whose size can be estimated, e.g., from the vari-
ability time-scale of the source. Due to these and other
properties which will be discussed in Chap. 5, it is con-
cluded that the nuclei of active galaxies must contain
a supermassive black hole as the central powerhouse.
The radiation is produced by matter falling towards this
black hole, a process called accretion, thereby convert-
ing its gravitational potential energy into kinetic energy.
If this kinetic energy is then transformed into internal
energy (i.e., heat) as happens in the so-called accre-
tion disk due to friction, it can get radiated away. This
is in fact an extremely efficient process of energy pro-
duction. For a given mass, the accretion onto a black
hole is about 10 times more efficient than the nuclear
fusion of hydrogen into helium. AGNs often emit radi-
ation across a very large portion of the electromagnetic
spectrum, from radio up to X-ray and gamma radiation.
Spiral galaxies still form stars today; indeed star
formation is a common phenomenon in galaxies. In ad-
dition, there are galaxies with a considerably higher star-
formation rate than "normal" spirals. These galaxies are
undergoing a burst of star formation and are thus known
as siarlnirst i>nhixies. Their star-formation rates are typ-
ically between 10 and 300M Q /yr, whereas our Milky
Way gives birth to about 2M /yr of new stars. This
vigorous star formation often takes place in localized
regions, e.g., in the vicinity of the center of the respec-
tive galaxy. Starbursts are substantially affected, if not
triggered, by disturbances in the gravitational field of
the galaxy, such as those caused by galaxy interactions.
Such starburst galaxies (see Fig. 1.12) are extremely lu-
minous in the far-infrared (FIR); they emit up to 98% of
their total luminosity in this part of the spectrum. This
happens by dust emission: dust in these galaxies ab-
sorbs a large proportion of the energetic UV radiation
Fig. 1.11. The quasar PKS 2349 is located at the center of
a galaxy, its host galaxy. The diffraction spikes (diffiaciion
patterns caused bj the suspension of the telescope's secondary
mirroi i in the middle of the object show that the center of the
galaxy contains a point source, the actual quasar, which is
significant!} brighter than its host galaxy. The galaxy shows
clear signs of distortion, visible as large and thin tidal tails.
1 he tails are caused by a neighboring galaxy that is visible in
the right-hand image, just above the quasar; it is about the size
of die Lai la llanic CI >ud hi isarhosi ilaxi ire often
distorted or in the process of merging with other galaxies. The
two images shown here differ in their brightness contrast
Fig. 1.12. Arp 220 is the most luminous object in the local
l'iii\erse. Originalh cataloged as a peculiar galaxy, the in-
frared satellite IRAS later discovered its enormous luminosity
in the infrared (IR). Arp 220 is the prototype of ultra-luminous
infrared galaxies (ULIRGs). This near-IR image taken with
the Hubble Space Telescope (HST) unveils the structure of
this object. With two colliding spiral galaxies in the center of
Arp 220, the disturbances in the interstellar medium caused
by this collision trigger a starburst. Dust in the galaxy ab-
sorbs most of the ultraviolet (UV) radiation from the young
hot stars and re-emits it in the IR
produced in the star-formation region and then re-emits
this energy in the form of thermal radiation in the FIR.
1.2.5 Voids, Clusters of Galaxies, and Dark Matter
The likelihood of galaxies interacting (Fig. 1.13) is
enhanced by the fact that galaxies are not randomly
distributed in space. The projection of galaxies on the
celestrial sphere, for instance, shows a distinct structure.
In addition, measuring the distances of galaxies allows
a determination of their three-dimensional distribution.
One finds a strong correlation of the galaxy positions.
There are regions in space that have a very high galaxy
density, but also regions where nearly no galaxies are
seen at all. The latter are called voids. Such voids can
have diameters of up to 30 h~ l Mpc.
Clusters of galaxies are gravitationally bound sys-
tems of a hundred or more galaxies in a volume of
diameter ~ 2 h' 1 Mpc. Clusters predominantly contain
early-type galaxies, so there is not much star formation
taking place any more. Some clusters of galaxies seem
to be circular in projection, others have a highly ellipti-
cal or irregular distribution of galaxies; some even have
more than one center. The cluster of galaxies closest to
us is the Virgo Cluster, at a distance of ~ 18 Mpc; it is
a cluster with an irregular galaxy distribution. The clos-
est regular cluster is Coma, at a distance of ~ 90 Mpc. 6
Coma (Fig. 1.14) contains about 1000 luminous
galaxies, of which 85% are early-type galaxies.
i of these two clusters are not determined from red-
by direct methods that will be discussed in
Scci 3«v such direct measurements are one of the most successful
methods of determining the Hubble constant.
Fig. 1.13. Two spiral galaxies interacting
with each other. NGC 2207 (on the left)
and IC 2163 are not only close neighbors
in projection: the strong gravitational tidal
interaction they are exerting on each other
is clearly visible in the pronounced tidal
arms, particularly visible to the right of the
right hand galaxy. Furthermore, a bridge of
stars is seen to connect these two galaxies,
also due to tidal gra\ national i'oiees. This
image was taken with the Hubble Space
Telescope
Fig. 1.14. The Coma cluster of galaxies, at a distance of
roughly 90 Mpc from us, is the closes) massive regular cluster
of galaxies. Almost all objects visible in this image are gal-
axies associated w ith Ihe cluster - Coma contains more than
a thousand luminous galaxies
In 1933, Fritz Zwicky measured the radial velocities
of the galaxies in Coma and found that they have a dis-
persion of about 1000 km/s. From the total luminosity
of all its galaxies the mass of the cluster can be esti-
mated. If the stars in the cluster galaxies have an average
mass-to-light ratio (M/L) similar to that of our Sun,
we would conclude M = (M /L )L. However, stars
in early-type galaxies are on average slightly less mas-
sive than the Sun and thus have a slightly higher M/L. 7
Thus, the above mass estimate needs to be increased by
a factor of ~ 10.
Zwicky then estimated the mass of the cluster by
multiplying the luminosity of its member galaxies with
the mass-to-light ratio. From this mass and the size
of the cluster, he could then estimate the velocity that
a galaxy needs to have in order to escape from the
gravitational field of the cluster -the escape velocity. He
found that the characteristic peculiar velocity of cluster
galaxies (i.e., the velocity relative to the mean velocity)
is substantially larger than this escape velocity. In this
case, the galaxies of the cluster would fly apart on a time-
scale of about 10 9 years - the time it takes a galaxy to
7 In Chap. 3 we will see that for stars in spiral galaxies M/L
3Af IL. mi axcrage. while for those in elliptical galaxies a larger
wiluc of A// Z. 10.1/ IL.. applies. Here and throughout this book.
mass-to-light ratios are quoted in Solar units.
cross through the cluster once - and, consequently, the
cluster would dissolve. However, since Coma seems to
be a relaxed cluster, i.e., it is in equilibrium and thus
its age is definitely larger than the dynamical time-scale
of 10 9 years, Zwicky concluded that the Coma cluster
contains significantly more mass than the sum of the
masses of its galaxies. Using the virial theorem 8 he
was able to estimate the mass of the cluster from the
velocity distribution of the galaxies. This was the first
clear indicator of the existence of dark matter.
X-ray satellites later revealed that clusters of galaxies
are strong sources of X-ray radiation. They contain hot
gas. with temperatures ranging from 10 7 K up to 10 8 K
(Fig. 1.15). This gas temperature is another measure for
the depth of the cluster's potential well, since the hot-
ter the gas is, the deeper the potential well has to be to
prevent the gas from escaping via evaporation. Mass es-
timates based on the X-ray temperature result in values
that are comparable to those from the velocity dispersion
of the cluster galaxies, clearly confirming the hypothe-
sis of the existence of dark matter in clusters. A third
method for determining cluster masses, the so-called
gravitational lensing effect, utilizes the fact thai light
is deflected in a gravitational field. The angle through
which light rays are bent due to the presence of a massive
object depends on the mass of that object. From obser-
vation and analysis of the gravitational lensing effect
in clusters of galaxies, cluster masses are derived that
are in agreement with those from the two other meth-
ods. Therefore, clusters of galaxies are a second class of
cosmic objects whose mass is dominated by dark matter.
Clusters of galaxies are cosmologically young struc-
tures. Their dynamical time-scale, i.e., the time in which
the mass distribution in a cluster settles into an equilib-
rium state, is estimated as the time it takes a member
galaxy to fully cross the cluster once. With a charac-
teristic velocity of v ~ 1000 km/s and a diameter of
2R ~ 2 Mpc one thus finds
'dyn
- 2 x 10 9 yr .
(1.9)
8 The virial theorem in its simplest form says that, for an isolated
dynamical system in a stationary state of equilibrium, the kinetic
energy is just half the potential energy.
Ekin=i|Epot|. (1.8)
In particular, the system's total energy is E tot = E^ n + E pot =
». 1.15. The Hydra A cluster of galaxies. The left-hand fi
tire shows an optical image, the one on the right an image
taken with the X-ray satellite Chandra. The cluster has a red-
shift of z « 0.054 and is thus located at a distance of about
250 Mpc. The X-ray emission originates from gas at a temper-
ature of 40 x 10 6 K which fills the space between the cluster
galaxies. In the center of the cluster, the gas is cooler by about
15%
As we will later see, the Universe is about 14 x 10 9 years
old. During this time galaxies have not had a chance to
cross the cluster many times. Therefore, clusters still
contain, at least in principle, information about their
initial state. Most clusters have not had the time to fully
relax and evolve into a state of equilibrium that would be
largely independent of their initial conditions. Compar-
ing this with the time taken for the Sun to rotate around
the center of the Milky Way - about 2 x 10 8 years - gal-
axies thus have had plenty of time to reach their state of
equilibrium.
Besides massive clusters of galaxies there are also
galaxy groups, which sometimes contain only a few
luminous galaxies. Our Milky Way is part of such
a group, the Local Group, which also contains M3 1 (An-
dromeda) which is another dominant galaxy, as well as
some far less luminous galaxies such as the Magellanic
Clouds. Some groups of galaxies are very compact, i.e.,
their galaxies are confined within a very small volume
(Fig. 1.16). Interactions between these galaxies cause
the lifetimes of many such groups to be much smaller
than the age of the Universe, and the galaxies in such
groups will merge.
1.2.6 World Models and the Thermal History
of the Universe
Quasars, clusters of galaxies, and nowadays even sin-
gle galaxies are also found at very high redshifts where
the simple Hubble law (1.2) is no longer valid. It is
therefore necessary to generalize the distance-redshift
relation. This requires considering world models as
a whole, which are also called cosmological models.
The dominant force in the Universe is gravitation. On
the one hand, weak and strong interactions both have
an extremely small (subatomic) range, and on the other
hand, electromagnetic interactions do not play a role on
large scales since the matter in the Universe is on av-
erage electrically neutral. Indeed, if it was not, currents
would immediately flow to balance net charge densi-
ties. The accepted theory of gravitation is the theory of
General Relativity (GR), formulated by Albert Einstein
in 1915.
Based on the two postulates that (1) our place in
the Universe is not distinguished from other locations
and that (2) the distribution of matter around us is
isotropic, at least on large scales, one <
Fig. 1.16. The galaxy group HCG87 belongs to the class of
so-called compact groups. In this HST image we can see three
massive galaxies belonging to this group: an edge-on spi-
ral in the lower part of the image, an elliptical galaxy to
the lower right, and another spiral in the upper part. The
small spiral in the center is a background object and there-
fore does not belong to the group. The two lower galaxies
have an active galactic nucleus, whereas the upper spiral
seems to be undergoing a phase of star formation. The gal-
axies in this group are so close together that in projeciion
they appear to touch. Between the galaxies, gas streams can
be detected. The galaxies are disturbing each other, which
could be the cause of the nuclear activity and star formation.
The galaxies are bound in a common gravitational potential
and will heavily interfere and presumably merge on a cos-
mologically small time-scale, which means in only a few
orbits, with an orbit taking about 10 8 years. Such merging
processes are of utmost importance for the evolution of the
galaxy population
homogeneous and isotropic world models (so-called
Friedmann-Lemaitre models) that obey the laws of
General Relativity. Expanding world models that con-
tain the Hubble expansion result from this theory
naturally. Essentially, these models are characterized
by three parameters:
• the current expansion rate of the Universe, i.e., the
Hubble constant Ho;
» the current mean matter density of the Universe
p m , often parametrized by the dimensionless density
parameter
8ttG
£2 m
J /-/ (T
(1.10)
• and the density of the so-called vacuum energy, de-
scribed by the cosmological constant A or by the
corresponding density parameter of the vacuum
A
Q A = r. (1.11)
3H*
The cosmological constant was originally introduced
by Einstein to allow stationary world models within
GR. After the discovery of the Hubble expansion he
called the introduction of A into his equations his
greatest blunder. In quantum mechanics A attains a dif-
ferent interpretation, that of an energy density of the
vacuum.
The values of the cosmological parameters are known
quite accurately today (see Chap. 8), with values of
Q m «» 0.3 and Q A ss 0.7. The discovery of a non-
vanishing Q A came completely unexpectedly. To date,
all attempts have failed to compute a reasonable value
for Q A from quantum mechanics. By that we mean
a value which has the same order-of-magnitude as the
one we derive from cosmological observations. In fact,
simple and plausible estimates lead to a value of A that
is ~ 10 120 times larger than that obtained from obser-
vation, a tremendously bad estimate indeed. This huge
discrepancy is probably one of the biggest challenges
in fundamental physics today.
According to the Friedmann-Lemaitre models, the
Universe used to be smaller and hotter in the past, and
it has continuously cooled down in the course of expan-
sion. We are able to trace back the cosmic expansion
under the assumption that the known laws of physics
were also valid in the past. From that we get the Big
Bang model of the Universe, according to which our
Universe has evolved out of a very dense and very
hot state, the so-called Big Bang. This world model
makes a number of predictions that have been verified
convincingly:
1. About 1/4 of the baryonic matter in the Universe
should consist of helium which formed about 3 min
after the Big Bang, while most of the rest consists
1. Introduction and Overviev
of hydrogen. This is indeed the case: the mass frac-
tion of helium in metal-poor objects, whose chemical
composition has not been significantly modified by
processes of stellar evolution, is about 24%.
2. From the exact fraction of helium one can derive
the number of neutrino families - the more neutrino
species that exist, the larger the fraction of helium
will be. From this, it was derived in 1981 that there
are 3 kinds of neutrinos. This result was later con-
firmed by particle accelerator experiments.
3. Thermal radiation from the hot early phase of the
Universe should still be measurable today. Predicted
in 1946 by George Gamow, it was discovered by Arno
Penzias and Robert Wilson in 1965. The correspond-
ing photons have propagated freely after the Universe
cooled down to about 3000 K and the plasma con-
stituents combined to neutral atoms, an epoch called
recombination. As a result of cosmic expansion, this
radiation has cooled down to about 7q «s 2.73 K. This
microwave radiation is nearly perfectly isotropic,
once we subtract the radiation which is emitted
locally by the Milky Way (see Fig. 1.17). Indeed,
measurements from the COBE satellite showed that
the cosmic microwave background (CMB) is the
most accurate blackbody spectrum ever measured.
4. Today's structures in the Universe have evolved out
of very small density fluctuations in the early cos-
mos. The seeds of structure formation must have
already been present in the early phases of cosmic
evolution. These density fluctuations should also be
visible as small temperature fluctuations in the mi-
crowave background emitted about 380 000 years
after the Big Bang at the epoch of recombination. In
fact, COBE was the first to observe these predicted
anisotropies (see Fig. 1.17). Later experiments, es-
pecially the WMAP satellite, observed the structure
of the microwave background at much improved an-
gular resolution and verified the theory of structure
formation in the Universe in detail (see Sect. 8.6).
With these predictions so impressively confirmed, in
this book we will exclusively consider this cosmolog-
ical model; currently there is no competing model of
the Universe that could explain these very basic cosmo-
logical observations in such a natural way. In addition,
this model does not seem to contradict any fundamental
observation in cosmology. However, as the e
Fig. 1.17. Temperature distribution of the cos
background on the sky as measured by the COBE satellite.
The uppermost image shows a dipole distribution; it origi-
nates from the Earth's motion relative to the rest-frame of the
CMB. We move at a speed of ~ 600 km/s relative to that sys-
tem, which leads to a dipole anisotropy with an amplitude
of AT/T ~ v/c ~ 2 x 10" 3 due to the Doppler effect. If this
dipole contribution is subtracted, we get the map in the mid-
dle which clearly shows the emission from the Galactic disk.
hi i In mi i i] In i an i ni p ii I i i di i' ibui ■ ii
(it is not a blackbody of T ~ 3 K), it can also be subtracted
to get the temperature map at the bottom. These are the pri-
mordial fluctuations of the CMB, with an amplitude of about
A77r~2x 10" 5
a non-vanishing vacuum energy density shows, together
with a matter density p m that is about six times the mean
baryon density in the Universe (which can be derived
from the abundance of the chemical elements formed in
the Big Bang), the physical nature of about 95% of the
content of our Universe is not yet understood.
The CMB photons we receive today had their last
physical interaction with matter when the Universe was
about 3.8 x 10 5 years old. Also, the most distant galax-
ies and quasars known today (at z ~ 6.5) are strikingly
young - we see them at a time when the Universe was
less than a tenth of its current age. The exact relation
between the age of the Universe at the time of the light
emission and the redshift depends on the cosmologi-
cal parameters Hq, Q m , and Q A . In the special case
that Q m — 1 and Q A — 0, called the Einstein-de Sitter
model, one obtains
Kz) = -
1
3// (l + z) 3/2 '
In particular, the age of the Universe today (i.i
is, according to this model,
(1.12)
itz = 0)
(1.13)
The Einstein-de Sitter (EdS) model is the simplest
world model and we will sometimes use it as a refer-
ence, but recent observations suggest that Q m < 1 and
Qa > 0. The mean density of the Universe in the EdS
model is
3#n
hence it is really, really small.
1.2.7 Structure Formation and Galaxy Evolution
The low amplitude of the CMB anisotropies implies
that the inhomogeneities must have been very small
at the epoch of recombination, whereas today's Uni-
verse features very large density fluctuations, at least on
scales of clusters of galaxies. Hence, the density field
of the cosmic matter must have evolved. This structure
evolution occurs because of gravitational instability, in
that an overdense region will expand more slowly than
the mean Universe due to its self-gravity. Therefore,
any relative overdensity becomes amplified in time. The
growth of density fluctuations in time will then cause
the formation of large-scale structures, and the gravita-
tional instability is also responsible for the formation of
galaxies and clusters. Our world model sketched above
predicts the abundance of galaxy clusters as a function
of redshift, which can be compared with the observed
cluster counts. This comparison can then be used to
determine cosmological parameters.
Another essential conclusion from the smallness of
the CMB anisotropies is the existence of dark matter
on cosmic scales. The major fraction of cosmic matter
is dark matter. The baryonic contribution to the matter
density is < 20% and to the total energy density < 5%.
The energy density of the Universe is dominated by the
vacuum energy.
Unfortunately, the spatial distribution of dark mat-
ter on large scales is not directly observable. We only
observe galaxies or, more precisely, their stars and
gas. One might expect that galaxies would be located
preferentially where the dark matter density is high.
However, it is by no means clear that local fluctuations
of the galaxy number density are strictly proportional
to the density of dark matter. The relation between the
dark and luminous matter distributions is currently only
approximately understood.
Eventually, this relation has to result from a detailed
understanding of galaxy formation and evolution. Lo-
cations with a high density of dark matter can support
the formation of galaxies. Thus we will have to examine
how galaxies form and why there are different kinds of
galaxies. In other words, what decides whether a form-
ing galaxy will become an elliptical or a spiral? This
question has not been definitively answered yet, but it is
supposed that ellipticals can form only by the merging
of galaxies. Indeed, the standard model of the Universe
predicts that small galaxies will form first; larger galax-
ies will be formed later through the ongoing merger of
smaller ones.
The evolution of galaxies can actually be observed
directly. Galaxies at high redshift (i.e., cosmological ly
young galaxies) are in general smaller and bluer, and the
star-formation rate was significantly higher in the earlier
Universe than it is today. The change in the mean color
of galaxies as a function of redshift can be understood as
a combination of changes in the star formation processes
and an aging of the stellar population.
1.2.8 Cosmology as a Triumph
of the Human Mind
Cosmology, extragalactic astronomy, and astrophysics
as a whole are a heroic undertaking of the human mind
and a triumph of physics. To understand the Universe we
1. Introduction and Overviev
apply physical laws that were found empirically under
completely different circumstances. All the known laws
of physics were derived "today" and, except for Gen-
eral Relativity, are based on experiments on a laboratory
scale or, at most, on observations in the Solar System,
such as Kepler's laws which formed the foundation for
the Newtonian theory of gravitation. Is there any a pri-
ori reason to assume that these laws are also valid in
other regions of the Universe or at completely different
times? However, this is apparently indeed the case: nu-
clear reactions in the early Universe seem to obey the
same laws of strong interaction that are measured to-
day in our laboratories, since otherwise the prediction
of a 25% mass fraction of helium would not be possi-
ble. Quantum mechanics, describing the wavelengths of
atomic transitions, also seems to be valid at very large
distances - since even the most distant objects show
emission lines in their spectra with frequency ratios
(which are described by the laws of quantum mechanics )
identical to those in nearby objects.
By far the greatest achievement is General Relativ-
ity. It was originally formulated by Albert Einstein since
his Special Theory of Relativity did not allow him to
incorporate Newtonian gravitation. No empirical find-
ings were known at that time (1915) which would not
have been explained by the Newtonian theory of gravity.
Nevertheless, Einstein developed a totally new theory
of gravitation for purely theoretical reasons. The first
success of this theory was the correct description of the
gravitational deflection of light by the Sun, measured
in 1919, and of the perihelion rotation of Mercury. 9
His theory permits a description of the expanding Uni-
verse, which became necessary after Hubble's discovery
in 1928. Only with the help of this theory can we re-
construct the history of the Universe back into the past.
Today this history seems to be well understood up to the
time when the Universe was about 10~ 6 s old and had
a temperature of about 10 13 K. Particle physics models
allow an extrapolation to even earlier epochs.
The cosmological predictions discussed above are
based on General Relativity describing an expanding
Universe, therefore providing a test of Einstein's the-
ory. On the other hand, General Relativity also describes
much smaller systems and with much stronger gravita-
fjas already known in 1915, but it was not clear whether it
lot have any other explanation. c.t\, a quadnipole moment of
is distribution of the Sun.
tional fields, such as neutron stars and black holes. With
the discovery of a binary system consisting of two neu-
tron stars, the binary pulsar PSR 1913+16, in the last
~ 25 years very accurate tests of General Relativity
have become possible. For example, the observed peri-
helion rotation in this binary system and the shrinking
of the binary orbit over time due to the radiation of en-
ergy by gravitational waves is very accurately described
by General Relativity. Together, General Relativity has
been successfully tested on length-scales from 10 11 cm
(the characteristic scale of the binary pulsar) to 10 28 cm
(the size of the visible Universe), that is over more than
10 17 orders of magnitude - an impressive result indeed!
1.3 The Tools of Extragalactic
Astronomy
Extragalactic sources - galaxies, quasars, clusters of
galaxies - are at large distances. This means that in
general they appear to be faint even if they are intrinsi-
cally luminous. They are also seen to have a very small
angular size despite their possibly large linear extent.
In fact, just three extragalactic sources are visible to the
naked eye: the Andromeda galaxy (M3 1) and the Large
and Small Magellanic Clouds. Thus for extragalactic as-
tronomy, telescopes are needed that have large apertures
(photon collecting area) and a high angular resolution.
This applies to all wavebands, from radio astronomy to
gamma ray astronomy.
The properties of astronomical telescopes and their
instruments can be judged by different criteria, and we
will briefly describe the most important ones. The sen-
sitivity specifies how dim a source can be and still be
observable in a given integration time. The sensitivity
depends on the aperture of the telescope as well as on the
efficiency of the instrument and the sensitivity of the de-
tector. The sensitivity of optical telescopes, for instance,
was increased by a large factor when CCDs replaced
photographic plates as detectors in the early 1980s. The
sensitivity also depends on the sky background, i.e.,
the brightness of the sky caused by non-astronomical
sources. Artificial light in inhabited regions has forced
optical telescopes to retreat into more and more remote
areas of the world where light pollution is minimized.
Radio astronomers have similar problems caused by ra-
dio emission from the telecommunication infrastructure
1.3 The Tools of Extragalactic Astronomy
of modern civilization. The angular resolution of a tele-
scope specifies down to which angular separation two
sources in the sky can still be separated by the detec-
tor. For diffraction-limited observations like those made
with radio telescopes or space-born telescopes, the an-
gular resolution A6 is limited by the diameter D of the
telescope. For a wavelength k one has A6 — k/D. For
optical and near-infrared observations from the ground,
the angular resolution is in general limited by turbu-
lence in the atmosphere, which explains the choice of
high mountain tops as sites for optical telescopes. These
atmospheric turbulences cause, due to scintillation, the
smearing of the images of astronomical sources, an ef-
fect that is called seeing. In interferometry, where one
combines radiation detected by several telescopes, the
angular resolution is limited by the spatial separation of
the telescopes. The spectral resolution of an instrument
specifies its capability to separate different wavelengths.
The throughput of a telescope/instrument system is of
particular importance in large sky surveys. For instance,
the efficiency of photometric surveys depends on the
number of spectra that can be observed simultaneously.
Special multiplex spectrographs have been constructed
for such tasks. Likewise, the efficiency of photomet-
ric surveys depends on the region of sky that can be
observed simultaneously, i.e., the field-of-view of the
camera. Finally, the efficiency of observations also de-
pends on factors like the number of clear nights at an
astronomical site, the fraction of an observing night in
which actual science data is taken, the fraction of time
an instrument cannot be used due to technical problems,
the stability of the instrumental set-up (which deter-
mines the time required for calibration measurements),
and many other such aspects.
In the rest of this section some telescopes will be
presented that are of special relevance to extragalac-
tic astronomy and to which we will frequently refer
throughout the course of this book.
1.3.1 Radio Telescopes
With the exception of optical wavelengths, the Earth's
atmosphere is transparent only for very large wave-
lengths - radio waves. The radio window of the
atmosphere is cut off towards lower frequencies, at
about v ~ 10 MHz, because radiation of a wavelength
larger than k ~ 30 m is reflected by the Earth's iono-
sphere and therefore cannot reach the ground. Below
k ~ 5 mm radiation is increasingly absorbed by oxygen
and water vapor in the atmosphere. Therefore, below
about A. ~ 0.3 mm ground-based observations are no
longer possible.
Mankind became aware of cosmic radio radiation -
in the early 1930s - only when noise in radio antennae
was found that would not vanish, no matter how quiet
the device was made. In order to identify the source
of this noise the AT&T Bell Labs hired Karl Jansky,
who constructed a movable antenna called "Jansky's
Merry-Go-Round" (Fig. 1.18).
After some months Jansky had identified, besides
thunderstorms, one source of interference that rose and
set every day. However, it did not follow the course of
the Sun which was originally suspected to be the source.
Rather, it followed the stars. Jansky finally discovered
that the signal originated from the direction of the center
of the Milky Way. He published his result in 1933, but
this publication also marked the end of his career as the
world's first radio astronomer.
Inspired by Jansky's discovery, Grote Reber was the
first to carry out real astronomy with radio waves. When
AT&T refused to employ him, he built his own radio
"dish" in his garden, with a diameter of nearly 10 m.
Between 1938 and 1943, Reber compiled the first sky
maps in the radio domain. Besides strong radiation from
the center of the Milky Way he also identified sources
in Cygnus and in Cassiopeia. Through Reber's research
and publications radio astronomy became an accepted
field of science after World War II.
The largest single-dish radio telescope is the Arecibo
telescope, shown in Fig. 1.19. Due to its enormous area,
and thus high sensitivity, this telescope, among other
achievements, detected the first pulsar in a binary sys-
tem, which is used as an important test laboratory for
General Relativity (see Sect. 7.7). Also, the first extra-
solar planet, in orbit around a pulsar, was discovered
with the Arecibo telescope. For extragalactic astron-
omy Arecibo plays an important role in measuring the
redshifts and line widths of spiral galaxies, both deter-
mined from the 2 1 -cm emission line of neutral hydrogen
(see Sect. 3.4).
The Effelsberg 100-m radio telescope of the Max-
Planck-Institut fiir Radioastronomie was, for many
years, the world's largest fully steerable radio telescope,
but since 2000 this title has been claimed by the new
Fig. 1.18. "Jansky's Merry-Go-Round". By
turning the structure in an azimuthal direc-
tion, a rough estimate of the position of
radio sources could be obtained
^^^m
§31? :
m
f^Sti
?: Vivf
V,\ -
?z s&i^
jjJSB
Fig. 1.19. With a diameter of 305 m, the
Arecibo telescope in Puerto Rico is the
largest single-dish telescope in the world; it
may also be known from the James Bond
movie "Goldeneye". The disadvantage of
its construction is its lack of steerability.
Tracking of sources is only possible within
narrow limits by moving the secondary
Green Bank Telescope (see Fig. 1.20) after the old one
collapsed in 1988. With Effelsberg, for example, star-
formation regions can be investigated. Using molecular
line spectroscopy, one can measure their densities and
temperatures. Magnetic fields also play a role in star for-
mation, though many details still need to be clarified.
By measuring the polarized radio flux, Effelsberg has
mapped the magnetic fields of numerous spiral galaxies.
In addition, due to its huge collecting area Effelsberg
plays an important role in interferometry at very long
baselines (see below).
Because of the long wavelength, the angular res-
olution of even large radio telescopes is fairly low,
compared to optical telescopes. For this reason, radio
Fig. 1.20. The world's two largest fully steerable radio tele-
scopes. Left: The 100-m telescope in Effelsberg. It was
commissioned in 1972 and is used in the wavelength range
from 3.5 mm to 35 cm. Eighteen different detector systems
are necessary for this. Right: The Green Bank Telescope. It
does not have a rotationally symmetric mirror; one axis has a
diameter of 100 m and the other 110 m
astronomers soon began utilizing interferometric meth-
ods, where the signals obtained by several telescopes are
correlated to get an interference pattern. One can then
reconstruct the structure of the source from this pattern
using Fourier transformation. With this method one gets
the same resolution as one would achieve with a single
telescope of a diameter corresponding to the maximum
pair separation of the individual telescopes used.
Following the first interferometric measurements in
England (around 1960) and the construction of the large
Westerbork Synthesis Radio Telescope in the Nether-
lands (around 1970), at the end of the 1970s the Very
Large Array (VLA) in New Mexico (see Fig. 1.21) be-
gan operating. With the VLA one achieved an angular
resolution in the radio domain comparable to that of
optical telescopes at that time. For the first time, this al-
lowed the combination of radio and optical images with
the same resolution and thus the study of cosmic sources
over a range of several clearly separated wavelength
regimes. With the advent of the VLA radio astronomy
experienced an enormous breakthrough, particularly in
the study of AGNs. It became possible to examine the
large extended jets of quasars and radio galaxies in de-
tail (see Sect. 5.1.2). Other radio interferometers must
also be mentioned here, such as the British MERLIN,
where seven telescopes with a maximum separation of
230 km are combined.
In the radio domain it is also possible to interconnect
completely independent and diverse antennae to form
an interferometer. For example, in Very Long Base-
line Interferometry (VLBI) radio telescopes on different
continents are used simultaneously. These frequently
also include Effelsberg and the VLA. In 1995 a sys-
tem of ten identical 25 -m antennae was set up in the
USA, exclusively to be used in VLBI, the Very Long
Baseline Array (VLBA). Angular resolutions of bet-
ter than a milliarcsecond (mas) can be achieved with
VLBI. Therefore, in extragalactic astronomy VLBI is
Fig. 1.21. The Very Large Array (VLA)
in New Mexico consists of 27 antennae
with a diameter of 25 m each that can be
moved on rails. It is used in four different
configurations that vary in the separation
of the telescopes; switching configurations
takes about two weeks
particularly used in the study of AGNs. With VLBI we
have learned a great deal about the central regions of
AGNs, such as the occurrence of apparent superluminal
velocities in these sources.
Some of the radio telescopes described above are
also capable of observing in the millimeter regime.
For shorter wavelengths the surfaces of the anten-
nae are typically too coarse, so that special telescopes
are needed for wavelengths of 1 mm and below. The
30-m telescope on Pico Veleta (Fig. 1.22), with its
exact surface shape, allows observations in the mil-
limeter range. It is particularly used for molecular
spectroscopy at these frequencies. Furthermore, impor-
tant observations of high-redshift galaxies at 1 .2 mm
have been made with this telescope using the bolometer
camera MAMBO. Similar observations are also con-
ducted with the SCUBA (Submillimeter Common-User
Bolometer Array) camera at the James Clerk Maxwell
Telescope (JCMT; Fig. 1.23) on Mauna Kea, Hawaii.
Due to its size and excellent location, the JCMT is
arguably the most productive telescope in the submil-
limeter range; it is operated at wavelengths between
3 mm and 0.3 mm. With the SCUBA-camera, operating
at 850 |xm (0.85 mm), we can observe star-formation re-
gions in distant galaxies for which the optical emission
is nearly completely absorbed by dust in these sources.
These dusty star-forming galaxies can be observed in the
(sub (millimeter regime of the electromagnetic spec-
trum even out to large redshifts, as will be discussed in
Sect. 9.2.3.
To measure the tiny temperature fluctuations of the
cosmic microwave background radiation one needs
extremely stable observing conditions and low-noise
detectors. In order to avoid the thermal radiation of
the atmosphere as much as possible, balloons and
satellites were constructed to operate instruments at
very high altitude or in space. The American COBE
(Cosmic Background Explorer) satellite measured the
anisotropies of the CMB for the first time, at wave-
lengths of a few millimeters. In addition, the frequency
spectrum of the CMB was precisely measured with
instruments on COBE. The WMAP (Wilkinson Mi-
crowave Anisotropy Probe) satellite obtained, like
COBE, a map of the full sky in the microwave regime,
but at a significantly improved angular resolution and
sensitivity. The first results from WMAP, published in
February 2003, were an enormously important mile-
stone for cosmology, as will be discussed in Sect. 8.6.5.
Besides observing the CMB these missions are also of
great importance for millimeter astronomy; these satel-
lites not only measure the cosmic background radiation
but of course also the microwave radiation of the Milky
Way and of other galaxies.
1.3.2 Infrared Telescopes
In the wavelength range 1 |im < X < 300 |im, observa-
tions from the Earth's surface are always subject to very
difficult conditions, if they are possible at all. The at-
mosphere has some windows in the near-infrared (NIR,
1.3 The Tools of Extragalactic Astronomy
Fig. 1.22. The 30-m telescope on Pico
Veleta was designed for observations in
the millimeter range of the spectrum. This
telescope, like all millimeter telescopes,
is located on a mountain to minimize the
column density of water in the atmosphere
Fig. 1.23. The JCMT has a 15-m dish. It
is protected by the largest single piece of
Gore-Tex, which has a transmissivity of
97% at submillimeter wavelengths
1 |xm < X < 2.4 |xm) which render ground-based ob-
servations possible. In the mid-infrared (MIR, 2.4 |xm
<A<20|xm) and far-infrared (FIR, 20 |xm < k <
300 [im) regimes, observations need to be carried out
from outside the atmosphere, i.e., using balloons, high-
flying airplanes, or satellites. The instruments have to
be cooled to very low temperatures, otherwise their own
thermal radiation would outshine any signal.
The first noteworthy observations in the far-infrared
were made by the Kuiper Airborne Observatory (KAO),
an airplane equipped with a 91 -cm mirror which oper-
ated at altitudes up to 15 km. However, the breakthrough
for IR astronomy had to wait until the launch of IRAS,
the InfraRed Astronomical Satellite (Fig. 1 .24). In 1983,
with its 60-cm telescope, IRAS compiled the first IR
map of the sky at 12, 25, 60, and 100 |xm, at an angular
Fig. 1.24. The left-hand picture shows an artist's impression
of IRAS in orbit. The project was a cooperation of the Nether-
lands, the USA, and Great Britain. IRAS was launched in 1 983
and operated for 10 months; after that the supply of liquid he-
lium, needed to cool the detectors, was exhausted. During this
time IRAS scanned 96% of the sky at four wavelengths. The
ISO satellite, shown on the right, was an ESA project and ob-
served between 1995 and 1998. Compared to IRAS it covered
a larger wavelength range, had a better angular resolution and
a thousand times higher sensitivity
resolution of 30" (2') at 12 |xm (100 |xm). It discovered
about a quarter of a million point sources as well as
about 20 000 extended sources. The positional accu-
racy for point sources of better than ~ 20" allowed an
identification of these sources at optical wavelengths.
Arguably the most important discovery by IRAS was
the identification of galaxies which emit the major frac-
tion of their energy in the FIR part of the spectrum.
These sources, often called IRAS galaxies, have a very
high star-formation rate where the UV light of the
young stars is absorbed by dust and then re-emitted
as thermal radiation in the FIR. IRAS discovered about
75 000 of these so-called ultra-luminous IR galaxies
(ULIRGs).
In contrast to the IRAS mission with its prime task
of mapping the full sky, the Infrared Space Observa-
tory ISO (Fig. 1.24) was dedicated to observations of
selected objects and sky regions in a wavelength range
2.5-240 |xm. Although the telescope had the same di-
ameter as IRAS its angular resolution at 12 (xm was
about a hundred times better than that of IRAS, since
the latter was limited by the size of the detector ele-
ments. The sensitivity of ISO topped that of IRAS by
a factor ~ 1000. ISO carried four instruments: two cam-
eras and two spectrographs. Among the most important
results from ISO in the extragalactic domain are the
spatially-resolved observations of the dust-enshrouded
star- formation regions of ULIRGs. Although the mis-
sion itself came to an end, the scientific analysis of the
data continues on a large scale, since to date the ISO
data are still unique in the infrared.
In 2003 a new infrared satellite was launched (the
Spitzer Space Telescope) with capabilities that by
far outperform those of ISO. With its 85-cm tele-
scope, Spitzer observes at wavelengths between 3.6
and 160 \im. Its IRAC (Infrared Array Camera) camera,
operating at wavelengths below ~ 9 (im, has a field-of-
view of 5.'2 x 5.'2 and 256 x 256 pixels, significantly
more than the 32 x 32 pixels of ISOCAM on ISO
that had a comparable wavelength coverage. The spec-
tral resolution of the IRS (Infrared Spectrograph)
instrument in the MIR is about R — X/AX ~ 100.
1.3 The Tools of Extragalactic Astronomy
1.3.3 Optical Telescopes
The atmosphere is largely transparent in the optical part
of the electromagnetic spectrum (0.3 |im < X < 1 |im),
and thus we are able to conduct observations from the
ground. Since for the atmospheric windows in the NIR
one normally uses the same telescopes as for optical
astronomy, we will thus not distinguish between these
two ranges here.
Although optical astronomy has been pursued for
many decades, it has evolved very rapidly in recent
years. This is linked to a large number of technical
achievements. A good illustration of this is the 10-m
Keck telescope which was put into operation in 1993;
this was the first optical telescope with a mirror diam-
eter of more than 6 m. Constructing telescopes of this
size became possible by the development of adaptive
optics, a method to control the surface of the mirror.
A mirror of this size no longer has a stable shape but is
affected, e.g., by gravitational deformation as the tele-
scope is steered. It was also realized that part of the air
turbulence that generates the seeing is caused by the
telescope and its dome itself. By improving the thermal
condition of telescopes and dome structures a reduction
of the seeing could be achieved. The aforementioned
replacement of photographic plates by CCDs, together
with improvements to the latter, resulted in a vastly
enhanced quantum efficiency of ~ 70% (at maximum
even more than 90%), barely leaving room for further
improvements.
The throughput of optical telescopes has been
immensely increased by designing wide-field CCD
cameras, the largest of which nowadays have a field-of-
view of a square degree and ~ 16 000 x 16 000 pixels,
with a pixel scale of ~ 0"2. Furthermore, multi-object
spectrographs have been built which allow us to observe
the spectra of a large number of objects simultane-
ously. The largest of them are able to get spectra
for several hundred sources in one exposure. Finally,
with the Hubble Space Telescope the angular resolu-
tion of optical observations was increased by a factor
of ~ 10. Further developments that will revolutionize
the field even more, such as interferometry in the near
IR/optical and adaptive optics, will soon be added to
these achievements.
Currently, about 13 optical telescopes of the 4-m
class exist worldwide. They differ mainly in their
location and their instrumentation. For example, the
Canada-France-Hawaii Telescope (CFHT) on Mauna
Kea (Fig. 1.25) has been a leader in wide-field photom-
etry for many years, due to its extraordinarily good
seeing. This is again emphasized by the installation
of Megacam, a camera with 18 000x 18 000 pixels.
The Anglo-Australian Telescope (AAT) in Australia,
in contrast, has distinctly worse seeing and has there-
fore specialized, among other things, in multi-object
Fig. 1.25. Telescopes at the summit of
Mauna Kea, Hawaii, at an altitude of
4200 m. The cylindrical dome to the left
and below the center of the image contains
the Subaru 8-m telescope; just behind it
are the two 10-m Keck telescopes. The two
large domes at the back house the Canada-
France-Hawaii telescope (CFHT, 3.6 m)
and the 8-m Gemini North. The telescope
at the lower right is the 15-m James Clerk
Maxwell submillimeter telescope (JCMT)
1. Introduction and Overview
spectroscopy, for which the 2dF (two-degree field) in-
strument was constructed. Most of these telescopes are
also equipped with NIR instruments. The New Tech-
nology Telescope (NTT, see Fig. 1.26) is especially
noteworthy due to its SOFI camera, a near-IR instru-
ment that has a large field-of-view of ~ 5' x 5' and an
excellent image quality.
Hubble Space Telescope. To avoid the greatest problem
in ground-based optical astronomy, the rocket scientist
Hermann Oberth had already speculated in the 1920s
about telescopes in space which would not be affected
by the influence of the Earth's atmosphere. In 1946 the
astronomer Lyman Spitzer took up this issue again and
discussed the possibilities for the realization of such
a project.
Shortly after NASA was founded in 1958, the con-
struction of a large telescope in space was declared
a long-term goal. After several feasibility studies and
ESAs agreement to join the project, the HST was finally
built. However, the launch was delayed by the explosion
of the space shuttle Challenger in 1986, so that it did not
take place until April 24, 1990. An unpleasant surprise
came as soon as the first images were taken: it was found
that the 2.4-m main mirror was ground into the wrong
shape. This problem was remedied in December 1993
during the first "servicing mission" (a series of Space
Shuttle missions to the HST; see Fig. 1.27), when a cor-
rection lens was installed. After this, the HST became
one of the most successful and best-known scientific
instruments.
The refurbished HST has two optical cameras, the
WFPC2 (Wide-Field and Planetary Camera) and, since
2002, the ACS (Advanced Camera for Surveys). The lat-
ter has a field-of-view of 3. '4 x 3. '4, about twice as large
as WFPC2, and 4000 x 4000 pixels. Another instrument
was STIS (Space Telescope Imaging Spectrograph),
operating mainly in the UV and at short optical wave-
lengths. Due to a defect it was shut down in 2004. The
HST also carries a NIR instrument, NICMOS (Near
Infrared Camera and Multi-Object Spectrograph). The
greatly reduced thermal radiation, compared to that
on the surface of the Earth, led to progress in NIR
astronomy, albeit with a very small field-of-view.
HST has provided important insights into our Solar
System and the formation of stars, but it has achieved
milestones in extragalactic astronomy. With HST ob-
servations of the nucleus of M87 (Fig. 1.8), one has
derived from the Doppler shift of the gas emission that
the center of this galaxy contains a black hole of two
billion solar masses. HST has also proven that black
holes exist in other galaxies and AGNs. The enormously
improved angular resolution has allowed us to study gal-
axies to a hitherto unknown level of detail. In this book
we will frequently report on results that were achieved
with HST.
Arguably the most important contribution of the HST
to extragalactic astronomy are the Hubble Deep Fields.
Fig. 1.26. The La Silla Observatory of ESO
in Chile. On the peak in the middle, one can
see the New Technology Telescope (NTT),
a 3.5-m prototype of the VLT The silvery
shining dome to its left is the MPG/ESO
2.2-m telescope that is currently equipped
with the Wide-Field Imager, a 8096 2 pixel
camera with a 0.5° field-of-view. The pic-
ture was taken from the location of the
3.6-m telescope, the largest on La Silla
Fig. 1.27. Left: The HST mounted on the manipulator arm of the data released one month later. To compile this multicolor
the Space Shuttle during oneof the repair missions. Right: The image, which at that time was the deepest image of the sky,
Hubble Deep Field (North) was taken in December 1995 and images from four different filters were combined
Scientists managed to convince Robert Williams, then
director of the Space Telescope Science Institute, to
use the HST to take a very deep image in an empty
region of the sky, a field with (nearly) no foreground
stars and without any known clusters of galaxies. At
that time it was not clear whether anything interesting
at all would come from these observations. Using the
observing time that is allocated to the Director, the "di-
rector's discretionary time", in December 1995 HST
was pointed at such a field in the Big Dipper, taking
data for 10 days. The outcome was the Hubble Deep
Field North (HDFN), one of the most important as-
tronomical data sets, displayed in Fig. 1.27. From the
HDFN and its southern counterpart, the HDFS, one
obtains information about the early states of galaxies
and their evolution. One of the first conclusions was
that most of the early galaxies are classified as irreg-
ulars. In 2002, the Hubble Ultra-Deep Field (HUDF)
was observed with the then newly installed ACS cam-
era. Not only did it cover about twice the area of the
HDFN but it was even deeper, by about one magni-
tude, owing to the higher sensitivity of ACS compared
to WFPC2.
Large Telescopes. For more than 40 years the 5-m tele-
scope on Mt. Palomar was the largest telescope in the
western world - the Russian 6-m telescope suffered
from major problems from the outset. The year 1993
saw the birth of a new class of telescope, of which the
two Keck telescopes (see Fig. 1 .28) were the first, each
with a mirror diameter of 10 m.
The site of the two Kecks at the summit of Mauna
Kea (at an altitude of 4200 m) provides ideal ob-
serving conditions for many nights per year. This
summit is now home to several large telescopes. The
new Japanese telescope Subaru, and Gemini North
are also located here, as well as the aforementioned
CFHT and JCMT. The significant increase in sensi-
tivity obtained by Keck, especially in spectroscopy,
permitted completely new insights, for instance through
absorption line spectroscopy of quasars. Keck was also
essential for the spectroscopic verification of innumer-
Fig. 1.28. The two Keck telescopes c
Mauna Kea. With Keck I the era of large
telescopes was heralded in 1993
J V
ft
BjjAntofagasta
I 5 "
; vT
."?
Mfe-T/ ;■■ Arna-or,
)
° )■■ [
l , P — J
^^-^ A
| ~ — — a-
Fig. 1.29. The left panel shows a map of the location of the
VLT on Cerro Paranal. It can be reached via Antofagasta,
about a two-hour flight north of Santiago de Chile. Then an-
other three-hour trip by car through a desert (see Fig. 1.30)
brings one to the site. Paranal is shown on the right during
i phase; in the foreground we can see the con-
i camp. The top of the mountain was flattened to get
a leveled space (of diameter ~ 300 m) large enough to ac-
commodate the telescopes and the facilities used for optical
interferometry (VLTI)
able galaxies of redshift z > 3, which are normally
so dim that they cannot be examined with smaller
telescopes.
The largest ground-based telescope project to date
was the construction of the Very Large Telescope (VLT)
of the European Southern Observatory (ESO), consist-
1.3 The Tools of Extragalactic Astronomy
ing of four telescopes each with a diameter of 8.2 m.
ESO already operates the La Silla Observatory in Chile
(see Fig. 1.26), but a better location was found for the
VLT, the Cerro Paranal (at an altitude of 2600 m). This
mountain is located in the Atacama desert, one of the
driest regions on Earth. To build the telescopes on the
mountain a substantial part of the mountain top first had
to be cut off (Fig. 1.29).
In contrast to the Keck telescopes, which have
a primary mirror that is segmented into 36 hexago-
nal elements, the mirrors of the VLT are monolithic,
i.e., they consist of a single piece. However, they are
very thin compared to the 5-m mirror on Mt. Palomar,
far too thin to be intrinsically stable against gravity
and other effects such as thermal deformations. There-
fore, as for the Kecks, the shape of the mirrors has to
be controlled electronically (see Fig. 1.30, right). The
monolithic structure of the VLT mirrors results in better
image quality than that of the Keck telescopes, resulting
in an appreciably simpler point-spread function.
Each of the four telescopes has three accessible foci;
this way, 12 different instruments can be installed at
the VLT at any time. Switching between the three
instruments is done with a deflection mirror. The perma-
nent installation of the instruments allows their stable
operation.
The VLT (Fig. 1.31) also marks the beginning of
a new form of ground-based observation with large op-
tical telescopes. Whereas until recently an astronomer
proposing an observation was assigned a certain number
and dates of nights in which she could observe with the
telescope, the VLT is mainly operated in the so-called
service mode. The observations are performed by lo-
cal astronomers according to detailed specifications that
must be provided by the principal investigator of the ob-
serving program and the data are then transmitted to the
astronomer at her home institution. A significant advan-
tage of this procedure is that one can better account for
special requirements for observing conditions. For ex-
ample, observations that require very good seeing can
be carried out during the appropriate atmospheric con-
ditions. With service observing the chances of getting
a useful data set are increased. At present about half
of the observations with the VLT are performed in ser-
vice mode. Another aspect of service observing is that
the astronomer does not have to make the long journey
(see Fig. 1 .30), at the expense of also missing out on the
adventure and experience of observing.
Fig. 1.30. Left: Transport of one of the VLT mirrors from
Antofagasta to Paranal. The route passes through an extremelj
dry desert, and large parts of the road are not paved. The
VLT thus clearly demonstrates that astronomers search for
ever more remote locations to get the best possible observing
conditions. Right: The active optics system at the VLT. Each
mirror is supported at 150 points; at these points, the mirror
is adjusted to correct for deformations. The primary mirror is
always shaped such that the light is focused in an optimal way,
with its form being corrected for the changing gravitational
forces when the telescope changes the pointing direction. In
adaptive optics, in contrast to active optics, the wavefront is
controlled: the mirrors are deformed with high frequencies
in such a way that the wavefront is as planar as possible
after passing through the optical system. In this way one can
correct for the permanently changing atmospheric conditions
and achieve images at diffraction-limited resolution, though
only across a fairly small region of the focal plane
Fig. 1.31. The Paranal Observatory after
completion of the domes for the four VLT
unit telescopes. The tracks seen in the fore-
ground were installed for additional smaller
telescopes that are now jointly used with
the VLT unit telescopes for interferometric
observations in the NIR
1.3.4 uv Telescopes
Radiation with a wavelength shorter than k < 0.3 |tm =
3000 A cannot penetrate the Earth's atmosphere but is
instead absorbed by the ozone layer, whereas radiation
at wavelengths below 912 A is absorbed by neutral hy-
drogen in the interstellar medium. The range between
these two wavelengths is the UV part of the spectrum,
in which observation is only possible from space.
The Copernicus satellite (also known as the Or-
biting Astronomical Observatory 3, OAO-3) was the
first long-term orbital mission designed to observe
high-resolution spectra at ultraviolet wavelengths. In
addition, the satellite contained an X-ray detector.
Launched on 21 August, 1972, it obtained UV spec-
tra of 551 sources until its decommissioning in 1981.
Among the achievements of the Copernicus mission are
the first detection of interstellar molecular hydrogen H 2
and of CO, and measurements of the composition of the
interstellar medium as well as of the distribution of Ovi,
i.e., five-time ionized oxygen.
The IUE (International Ultraviolet Explorer) op-
erated between 1978 and 1996 and proved to be
a remarkably productive observatory. During its more
than 18 years of observations more than 10 5 spectra
of galactic and extragalactic sources were obtained.
In particular, the IUE contributed substantially to our
knowledge of AGN.
The HST, with its much larger aperture, marks
the next substantial step in UV astronomy, although
no UV instrument is operational onboard HST af-
ter the failure of STIS in 2004. Many new insights
were gained with the HST, especially through spec-
troscopy of quasars in the UV, insights into both the
the quasars themselves and, through the absorption
lines in their spectra, into the intergalactic medium
along the line-of-sight towards the sources. In 1999
the FUSE (Far Ultraviolet Spectroscopic Explorer)
satellite was launched. From UV spectroscopy of
absorption lines in luminous quasars this satellite
provided us with a plethora of information on the
state and chemical composition of the intergalactic
medium.
1.3 The Tools of Extragalactic Astronomy
While the majority of observations with UV satellites
were dedicated to high-resolution spectroscopy of stars
and AGNs, the prime purpose of the GALEX satellite
mission, launched in 2003, is to compile an extended
photometric survey. GALEX observes at wavelengths
1350 A < X < 2830 A and will perform a complete sky
survey as well as observe selected regions of the sky
with a longer exposure time. In addition, it will perform
several spectroscopic surveys. The results from GALEX
will be of great importance, especially for the study of
the star-formation rate in nearby and distant galaxies.
1.3.5 X-Ray Telescopes
As mentioned before, interstellar gas absorbs radiation
at wavelengths shortward of 912 A, the so-called Ly-
man edge. This corresponds to the ionization energy of
hydrogen in its ground state, which is 13.6 eV. Only at
energies about ten times this value does the ISM be-
come transparent again and this denotes the low-energy
limit of the domain of X-ray astronomy. Typically, X-
ray astronomers do not measure the frequency of light
in Hertz (or the wavelength in u_m), but instead photons
are characterized by their energy, measured in electron
volts (eV).
The birth of X-ray astronomy was in the 1960s.
Rocket and balloon-mounted telescopes which were
originally only supposed to observe the Sun in X-rays
also received signals from outside the Solar System.
UHURU, the first satellite to observe exclusively the
cosmic X-ray radiation, compiled the first X-ray map
of the sky, discovering about 340 sources. This cata-
log of point sources was expanded in several follow-up
missions, especially by NASA's High-Energy Astro-
physical Observatory (HEAO-1) which also detected
a diffuse X-ray background radiation. On HEAO-2, also
known as the Einstein satellite, the first Wolter telescope
was used for imaging, increasing the sensitivity by a fac-
tor of nearly a thousand compared to earlier missions.
The Einstein observatory also marked a revolution in
X-ray astronomy because of its high angular resolution,
about 2" in the range of 0.1 to 4 keV. Among the great
discoveries of the Einstein satellite is the X-ray emis-
sion of many clusters of galaxies that traces the presence
of hot gas in the space between the cluster galaxies. The
total mass of this gas significantly exceeds the mass
of the stars in the cluster galaxies and therefore repre-
sents the main contribution to the baryonic content of
clusters.
The next major step in X-ray astronomy was ROSAT
(ROentgen SATellite; Fig. 1 .32), launched in 1990. Dur-
ing the first six months of its nine-year mission ROSAT
produced an all-sky map at far higher resolution than
UHURU; this is called the ROSAT All Sky Survey.
More than 10 5 individual sources were detected in
this survey, the majority of them being AGNs. In
the subsequent period of pointed observations ROSAT
examined, among other types of sources, clusters of
galaxies and AGNs. One of its instruments (PSPC)
provided spectral information in the range between
0.1 and 2.4 keV at an angular resolution of ~ 20",
while the other (HRI) instrument had a much better
angular resolution (~ 3") but did not provide any spec-
tral information. The Japanese X-ray satellite ASCA
(Advanced Satellite for Cosmology and Astrophysics),
launched in 1993, was able to observe in a significantly
higher energy range 0.5-12 keV and provided spectra
of higher energy resolution, though at reduced angular
resolution.
Since 1999 two new powerful satellites are in
operation: NASA's Chandra observatory and ESA's
XMM-Newton (X-ray Multi-Mirror Mission; see
Fig. 1.32). Both have a large photon-collecting area and
a high angular resolution, and they also set new stan-
dards in X-ray spectroscopy. Compared to ROSAT, the
energy range accessible with these two satellites has
been extended to 0.1-10 keV. The angular resolution of
Chandra is about 0'.'5 and thus, for the first time, com-
parable to that of optical telescopes. This high angular
resolution already led to major discoveries in the early
years of operation. For instance, well-defined sharp
structures in the X-ray emission from gas in clusters of
galaxies were discovered, and X-ray radiation from the
jets of AGNs which had been previously observed in the
radio was detected. Furthermore, Chandra discovered
a class of X-ray sources, termed ultra-luminous compact
X-ray sources (ULXs), in which we may be observing
the formation of black holes (Sect. 9.6). XMM-Newton
has a larger sensitivity compared to Chandra, however
at a somewhat smaller angular resolution. Among the
most important observations of XMM-Newton at the be-
ginning of its operation was the spectroscopy of AGNs
and of clusters of galaxies.
Fig. 1.32. Left: ROSAT, a German-US-British cooperation,
was in orbit from 1990 to 1999 and observed in the energy
range between 0.1 and 2.5 keV (soft X-ray). Upper right:
Chandra was launched in July 1999. The energy range of
its instruments lies between 0.1 and lOkeV. Its highly el-
liptical orbit permits long uninterrupted exposures. Lower
right: XMM-Newton was launched in December 1999 and
is planned to be used for 10 years. Observations are car-
ried out with three telescopes at energies between 0.1 and
15keV
1.3.6 Gamma-Ray Telescopes
The existence of gamma radiation was first postulated
in the 1950s. This radiation is absorbed by the atmo-
sphere, which is fortunate for the lifeforms on Earth.
The first observations, carried out from balloons, rock-
ets, and satellites, have yielded flux levels of less than
100 photons. Those gamma photons had energies in the
GeV range and above.
Detailed observations became possible with the satel-
lites SAS-2 and COS-B. They compiled a map of
the galaxy, confirmed the existence of a gamma back-
ground radiation, and for the first time observed pulsars
in the gamma range. The first Gamma Ray Bursts
(GRB), extremely bright and short-duration flashes
on the gamma-ray sky, were detected in the 1970s
by military satellites. Only the Italian-Dutch satellite
Beppo-SAX (1996 to 2002) managed to localize a GRB
with sufficient accuracy to allow an identification of the
source in other wavebands, and thus to reveal its phys-
ical nature; we will come back to this subject later, in
Sect. 9.7.
An enormous advance in high-energy astronomy was
made with the launch of the Compton Gamma Ray Ob-
servatory (CGRO; Fig. 1.33) in 1991; the observatory
was operational for nine years. It carried four differ-
ent instruments, among them the Burst And Transient
Source Experiment (BATSE) and the Energetic Gamma
Ray Experiment Telescope (EGRET). During its life-
time BATSE discovered more than 2000 GRBs and
contributed substantially to the understanding of the
nature of these mysterious gamma-ray flashes. EGRET
Fig. 1.33. The left image shows the Compton Gamma Ray Ob-
servatory (CGRO) mounted on the Space Shuttle manipulator
arm. This NASA satellite carried out observations between
1991 and 2000. It was finally shut down after a gyroscope
failed, and it burned up in the Earth's atmosphere in a c
trolled re-entry. ESA's Integral observatory, in operation si
2002, is shown on the right
discovered many AGNs at very high energies above
20 MeV, which hints at extreme processes taking place
in these objects.
The successor of the CGRO, the Integral satellite,
was put into orbit as an ES A mission by a Russian Proton
rocket at the end of 2002. At a weight of two tons, it is the
heaviest ES A satellite that has been launched thus far. It
is primarily observing at energies of 15 keV to 10 MeV
in the gamma range, but has additional ii
observation in the optical and X-ray regimes.
2. The Milky Way as a Galaxy
The Earth is orbiting around the Sun, which itself is
orbiting around the center of the Milky Way. Our Milky
Way, the Galaxy, is the only galaxy in which we are able
to study astrophysical processes in detail. Therefore, our
journey through extragalactic astronomy will begin in
our home Galaxy, with which we first need to become
familiar before we are ready to take off into the depths
of the Universe. Knowing the properties of the Milky
Way is indispensable for understanding other galaxies.
2.1 Galactic Coordinates
On a clear night, and sufficiently far away from cities,
one can see the magnificent band of the Milky Way
on the sky (Fig. 2.1). This observation suggests that the
distribution of light, i.e., that of the stars in the Galaxy,
is predominantly that of a thin disk. A detailed analy-
sis of the geometry of the distribution of stars and gas
confirms this impression. This geometry of the Galaxy
suggests the introduction of two specially adapted co-
ordinate systems which are particularly convenient for
quantitative descriptions.
Spherical Galactic Coordinates (£, fa). We consider
a spherical coordinate system, with its center being
"here", at the location of the Sun (see Fig. 2.2). The
Galactic plane is the plane of the Galactic disk, i.e., it
is parallel to the band of the Milky Way. The two Gal-
actic coordinates £ and b are angular coordinates on
the sphere. Here, b denotes the Galactic latitude, the
angular distance of a source from the Galactic plane,
with b € [-90°, +90°]. The great circle b = 0° is then
located in the plane of the Galactic disk. The direc-
tion b = 90° is perpendicular to the disk and denotes
the North Galactic Pole (NGP), while b = -90° marks
the direction to the South Galactic Pole (SGP). The
second angular coordinate is the Galactic longitude £,
with £ e [0°, 360°]. It measures the angular separation
between the position of a source, projected perpendic-
ularly onto the Galactic disk (see Fig. 2.2), and the
Galactic center, which itself has angular coordinates
b = 0° and £ = 0°. Given I and b for a source, its loca-
tion on the sky is fully specified. In order to specify its
three-dimensional location, the distance of that source
from us is also needed.
The conversion of the positions of sources given in
Galactic coordinates (b, £) to that in equatorial coordi-
nates (a, S) and vice versa is obtained from the rotation
between these two coordinate systems, and is described
by spherical trigonometry. ' The necessary formulae can
be found in numerous standard texts. We will not re-
produce them here, since nowadays this transformation
is done nearly exclusively using computer programs.
Instead, we will give some examples. The following
figures refer to the Epoch 2000: due to the precession
1 The equatorial coordinates are defined by the direction of the Earth's
rotation axis and h\ the rotation of the Earth. The intersections of the
Earth's axis and lite sphere define the northern and southern poles. The
treat circles on the sphere through these two poles, die meridian'., are
cutxes of constant rr..:hi n.sccii.iion <_<. Ctines perpendicular to them
and parallel to the projection of the Earth's equator onto the sk\ are
curves of constant declination 8, with the poles located at 8 = ±90°.
Fig. 2.1. An unusual op-
tical image of the Milky
Way. This total view of
the Galaxy is composed
of a large number of
individual images
Peter Schneider, The Milky Way as a Galaxy.
DOI: 10.1007/1 1614371_2© Springer- VerlagBi
2. The Milky Way as a Galaxy
Fig. 2.2. The Sun is at the origin of the Galactic coordinate
system. The directions to the Galactic center and to the North
Galactic Pole (NGP) are indicated and are located at I — 0°
and b — 0°, and at b = 90°, respectively
of the rotation axis of the Earth, the equatorial coor-
dinate system changes with time, and is updated from
time to time. The position of the Galactic center (at
t = 0° = b) is a = 17 h 45.6 m , S = -28°56.'2 in equato-
rial coordinates. This immediately implies that at the La
Silla Observatory, located at geographic latitude —29°,
the Galactic center is found near the zenith at local
midnight in May/June. Because of the high stellar den-
sity in the Galactic disk and the large extinction due
to dust this is therefore not a good season for extra-
galactic observations from La Silla. The North Galactic
Pole has coordinates a NGP = 192.85948° % 12 h 51 m ,
<5 NGP = 27.128 25°%27°7.'7.
Zone of Avoidance. As already mentioned, the absorp-
tion by dust and the presence of numerous bright stars
render optical observations of extragalactic sources in
the direction of the disk difficult. The best observing
conditions are found at large \b\, while it is very hard
to do extragalactic astronomy in the optical regime at
\b\ < 10°; this region is therefore often called the "Zone
of Avoidance". An illustrative example is the galaxy
Dwingeloo 1 , which was already mentioned in Sect. 1 . 1
(see Fig. 1.6). This galaxy was only discovered in the
1990s despite being in our immediate vicinity: it is
located at low \b\, right in the Zone of Avoidance.
Cylindrical Galactic Coordinates (R, 0, z). The an-
gular coordinates introduced above are well suited to
describing the angular position of a source relative to the
Galactic disk. However, we will now introduce another
three-dimensional coordinate system for the descrip-
tion of the Milky Way geometry that will prove very
convenient in the study of the kinematic and dynamic
properties of the Milky Way. It is a cylindrical coor-
dinate system, with the Galactic center at the origin
(see also Fig. 2.13). The radial coordinate R measures
the distance of an object from the Galactic center in
the disk, and z specifies the height above the disk (ob-
jects with negative z are thus located below the Galactic
disk, i.e., south of it). For instance, the Sun has a dis-
tance from the Galactic center of R «s 8 kpc. The angle
9 specifies the angular separation of an object in the disk
relative to the position of the Sun, seen from the Gal-
actic center. The distance of an object with coordinates
R,9,z from the Galactic center is then ^/R 2 + z 2 , inde-
pendent of 6. If the matter distribution in the Milky Way
were axially symmetric, the density would then depend
only on R and z, but not on 6. Since this assumption
is a good approximation, this coordinate system is very
well suited for the physical description of the Galaxy.
2.2 Determination of Distances
Within Our Galaxy
A central problem in astronomy is the estimation of dis-
tances. The position of sources on the sphere gives us
a two-dimensional picture. To obtain three-dimensional
information, measurements of distances are required.
Furthermore, we need to know the distance to a source
if we want to draw conclusions about its physical param-
eters. For example, we can directly observe the angular
diameter of an object, but to derive the physical size we
need to know its distance. Another example is the de-
termination of the luminosity L of a source, which can
be derived from the observed flux S only by means of
its distance D, using
L=4ttSD 2 . (2.1)
It is useful to consider the dimensions of the physical
parameters in this equation. The unit of the luminosity
is[L] — ergs" 1 , and that of the flux [S] = ergs _1 cm" 2 .
The flux is the energy passing through a unit area per
unit time (see Appendix A). Of course, the physical
properties of a source are characterized by the lumi-
2.2 Determination of Distances Within Our Galaxy
nosity L and not by the flux S, which depends on its
distance from the Sun.
In the following section we will review various meth-
ods for the estimation of distances in our Milky Way,
postponing the discussion of methods for estimating
extragalactic distances to Sect. 3.6.
2.2.1 Trigonometric Parallax
The most important method of distance determination
is the trigonometric parallax, and not only from a his-
torical point-of-view. This method is based on a purely
geometric effect and is therefore independent of any
physical assumptions. Due to the motion of the Earth
around the Sun the positions of nearby stars on the
sphere change relative to those of very distant sources
(e.g., extragalactic objects such as quasars). The latter
therefore define a fixed reference frame on the sphere
(see Fig. 2.3). In the course of a year the apparent po-
sition of a nearby star follows an ellipse on the sphere,
the semimajor axis of which is called the parallax p.
The axis ratio of this ellipse depends on the direc-
tion of the star relative to the ecliptic (the plane that
is defined by the orbits of the planets) and is of no
further interest. The parallax depends on the radius r
of the Earth's orbit, hence on the Earth-Sun distance
which is, by definition, one astronomical unit. 2 Further-
more, the parallax depends on the distance D of the
D
where we used p <£ 1 in the last step, and p is measured
in radians as usual. The trigonometric parallax is also
used to define the common unit of distance in astron-
omy: one parsec (pc) is the distance of a hypothetical
source for which the parallax is exactly p — 1". With
the conversion of arcseconds to radians (1" « 4.848 x
10 -6 radians) one gets p/l" — 206 265 p, which for
a parsec yields
1 pc = 206 265 AU = 3.086 x 10 18 cm . (2.3)
Fig. 2.3. Illustration of the parallax effect: in the course of the
Earth's orbit around the Sun the apparent positions of nearby
stars on the sky seem to change relative to those of very distant
background sources
The distance corresponding to a measured parallax is
then calculated as
D= {{-)
pc
(2.4)
To determine the parallax p, precise measurements of
the position of an object at different times are needed,
spread over a year, allowing us to measure the ellipse
drawn on the sphere by the object's apparent posi-
tion. For ground-based observation the accuracy of this
method is limited by the atmosphere. The seeing causes
a blurring of the images of astronomical sources and
thus limits the accuracy of position measurements. From
the ground this method is therefore limited to parallaxes
larger than «^0"01, implying that the trigonometric
parallax yields distances to stars only within ~ 30 pc.
An extension of this method towards smaller p, and
thus larger distances, became possible with the as-
trometric satellite HIPPARCOS. It operated between
November 1989 and March 1993 and measured the po-
sitions and trigonometric parallaxes of about 120 000
bright stars, with a precision of ~ 0"001 for the brighter
targets. With HIPPARCOS the method of trigonomet-
2. The Milky Way as a Galaxy
ric parallax could be extended to stars up to distances
of ~ 300 pc. The satellite GAIA, the successor mis-
sion to HIPPARCOS, is scheduled to be launched in
2012. GAIA will compile a catalog of ~ 10 9 stars up
to V *« 20 in four broad-band and eleven narrow-band
filters. It will measure parallaxes for these stars with
an accuracy of ~ 2 x 10~ 4 arcsec, with the accuracy
for brighter stars even being considerably better. GAIA
will thus determine the distances for ~ 2 x 10 8 stars
with a precision of 10%, and tangential velocities (see
next section) with a precision of better than 3 km/s.
The trigonometric parallax method forms the basis
of nearly all distance determinations owing to its purely
geometrical nature. For example, using this method the
distances to nearby stars have been determined, allow-
ing the production of the Hertzsprung-Russell diagram
(see Appendix B.2). Hence, all distance measures that
are based on the properties of stars, such as will be
described below, are calibrated by the trigonometric
parallax.
2.2.2 Proper Motions
Stars are moving relative to us or, more precisely, rel-
ative to the Sun. To study the kinematics of the Milky
Way we need to be able to measure the velocities of
stars. The radial component v t of the velocity is easily
obtained from the Doppler shift of spectral lines,
AX
(2.5)
where X is the rest-frame wavelength of an atomic
transition and AX = A b s — Xq the Doppler shift of the
wavelength due to the radial velocity of the source. The
sign of the radial velocity is defined such that v r >
corresponds to a motion away from us, i.e., to a redshift
of spectral lines.
In contrast, the determination of the other two veloc-
ity components is much more difficult. The tangential
component, v t , of the velocity can be obtained from the
proper motion of an object. In addition to the motion
caused by the parallax, stars also change their posi-
tions on the sphere as a function of time because of
the transverse component of their velocity relative to
the Sun. The proper motion fi is thus an angular veloc-
ity, e.g., measured in milliarcseconds per year (mas/yr).
This angular velocity is linked to the tangential velocity
component via
4.74 f^fji
(2.6)
Therefore, one can calculate the tangential velocity from
the proper motion and the distance. If the latter is derived
from the trigonometric parallax, (2.6) and (2.4) can be
combined to yield
- = 4.74
(2.7)
km/s " ' \l"/yrj
HIPPARCOS measured proper motions for ~ 10 5 stars
with an accuracy of up to a few mas/yr; however, they
can be translated into physical velocities only if their
distance is known.
Of course, the proper motion has two components,
corresponding to the absolute value of the angular ve-
locity and its direction on the sphere. Together with v T
this determines the three-dimensional velocity vector.
Correcting for the known velocity of the Earth around
the Sun, one can then compute the velocity vector v
of the star relative to the Sun, called the heliocentric
velocity.
2.2.3 Moving Cluster Parallax
The stars in an (open) star cluster all have a very similar
spatial velocity. This implies that their proper motion
vectors should be similar. To what extent the proper
motions are aligned depends on the angular extent of the
star cluster on the sphere. Like two railway tracks that
run parallel but do not appear parallel to us, the vectors
of proper motions in a star cluster also do not appear
parallel. They are directed towards a convergence point,
as depicted in Fig. 2.4. We shall demonstrate next how
to use this effect to determine the distance to a star
cluster.
We consider a star cluster and assume that all stars
have the same spatial velocity v. The position of the i-th
star as a function of time is then described by
r,(f) = ii + vt, (2.8)
where r, is the current position if we identify the origin
of time, t — 0, with "today". The direction of a star
2.2 Determination of Distances Within Our Galaxy
Fig. 2.4. The moving cluster parallax is a projection effect,
similar to that known from viewing railway tracks. The di-
rections of velocity vectors pointing away from us seem to
u i i i mi i it th i) i nee point. The connect-
ing line from the observer to the convergence point is parallel
to the velocity vector of the star cluster
relative to us is described by the unit vector
From this, one infers that for large times, t -> oo, the
direction vectors are identical for all stars in the cluster,
«;(/)-
\v\
(2.10)
Hence for large times all stars will appear at the same
point n conv : the convergence point. This only depends
on the direction of the velocity vector of the star cluster.
In other words, the direction vector of the stars is such
that they are all moving towards the convergence point.
Thus, « conv (and hence v/\v\) can be measured from
the direction of the proper motions of the stars in the
cluster, and so can v/ 1 v \ . On the other hand, one compo-
nent of v can be determined from the (easily measured)
radial velocity v r . With these two observables the three-
dimensional velocity vector v is completely determined,
as is easily demonstrated: let x[r be the angle between the
line-of-sight n towards a star in the cluster and v. The
angle V is directly read off from the direction vector n
and the convergence point, cos i\t — n ■ v/\ v\ — w conv • n.
With ii = |v| one then obtains
v r — v cos \j/ , D t = v sin i/r ,
This means that the tangential velocity v t can be mea-
sured without determining the distance to the stars in
the cluster. On the other hand, (2.6) defines a relation
between the proper motion, the distance, and v t . Hence,
a distance determination for the star is now possible with
v t u r tani/f v r tani[r
M= D = D ~* D= Jl ' ( ' )
This method yields accurate distance estimates of star
clusters within ~ 200 pc. The accuracy depends on the
measurability of the proper motions. Furthermore, the
cluster should cover a sufficiently large area on the sky
for the convergence point to be well defined. For the
distance estimate, one can then take the average over
a large number of stars in the cluster if one assumes that
the spatial extent of the cluster is much smaller than its
distance to us. Targets for applying this method are the
Hyades, a cluster of about 200 stars at a mean distance
of D « 45 pc, the Ursa-Major group of about 60 stars
at D« 24 pc, and the Pleiades with about 600 stars at
D^ 130 pc.
Historically the distance determination to the
Hyades, using the moving cluster parallax, was ex-
tremely important because it defined the scale to all
other, larger distances. Its constituent stars of known
distance are used to construct a calibrated Hertzsprung-
Russell diagram which forms the basis for determining
the distance to other star clusters, as will be discussed in
Sect. 2.2.4. In other words, it is the lowest rung of the so-
called distance ladder that we will discuss in Sect. 3.6.
With HIPPARCOS, however, the distance to the Hyades
stars could also be measured using the trigonometric
parallax, yielding more accurate values. HIPPARCOS
was even able to differentiate the "near" from the "far"
side of the cluster - this star cluster is too close for the
assumption of an approximately equal distance of all
its stars to be still valid. A recent value for the mean
distance of the Hyades is
D H ,
les = 46.3 ± 0.3 pc.
(2.13)
2.2.4 Photometric Distance;
Extinction and Reddening
Most stars in the color-magnitude diagram are located
along the main sequence. This enables us to com-
pile a calibrated main sequence of those stars whose
2. The Milky Way as a Galaxy
trigonometric parallaxes are measured, thus with known
distances. Utilizing photometric methods, it is then pos-
sible to derive the distance to a star cluster, as we will
demonstrate in the following.
The stars of a star cluster define their own main
sequence (color-magnitude diagrams for some star
clusters are displayed in Fig. 2.5); since they are all
located at the same distance, their main sequence is al-
ready defined in a color-magnitude diagram in which
only apparent magnitudes are plotted. This cluster
main sequence can then be fitted to a calibrated main
sequence 3 by a suitable choice of the distance, i.e., by
adjusting the distance modulus m — M,
m - M = 5 log (D/pc) - 5 ,
where m and M denote the apparent and absolute
magnitude, respectively.
In reality this method cannot be applied so easily
since the position of a star on the main sequence does
not only depend on its mass but also on its age and
metallicity. Furthermore, only stars of luminosity class
V (i.e., dwarf stars) define the main sequence, but with-
out spectroscopic data it is not possible to determine the
luminosity class.
Extinction and Reddening. Another major problem is
extinction. Absorption and scattering of light by dust af-
fect the relation of absolute to apparent magnitude: for
a given M, the apparent magnitude m becomes larger
(fainter) in the case of absorption, making the source
appear dimmer. Also, since extinction depends on wave-
length, the spectrum of the source is modified and the
observed color of the star changes. Because extinction
by dust is always associated with such a change in color,
one can estimate the absorption - provided one has suf-
ficient information on the intrinsic color of a source or
of an ensemble of sources. We will now demonstrate
how this method can be used to estimate the distance to
a star cluster.
We consider the equation of radiative transfer for
pure absorption or scattering (see Appendix A),
dl v
',,./,■
(2.14)
absolute magnitudes are plotted
a color-magnitude diagrai
Color Index (B -V)
Fig. 2.5. Color-magnitude diagram (CWID) for different sun
clusters. Such a diagram can be used for the distance deter-
mination of star clusters because the absolute magnitudes of
main-sequence stars are known (by calibration with nearby
clusters, especially the Hyades). One can thus determine the
distance modulus by vertically "shifting" the main sequence.
Also, the age of a star cluster can be estimated from a CMD:
luminous main-sequence stars have a shorter lifetime on the
main sequence than less luminous ones. The turn-off point in
the stellar sequence away from the main sequence therefore
corresponds to that stellar mass for which the lifetime on the
main < n in < n I th of th tan hi lei . i ordin 1
the age is specified on the right axis as a function of the posi-
tion of the turn-off point; the Sun will leave the main sequence
ai'tci about 10 x 10 9 years
where /„ denotes the specific intensity at frequency v, k v
the absorption coefficient, and s the distance coordinate
along the light beam. The absorption coefficient has the
dimension of an inverse length. Equation (2.14) says
that the amount by which the intensity of a light beam
is diminished on a path of length ds is proportional to
the original intensity and to the path length ds. The
absorption coefficient is thus defined as the constant of
proportionality. In other words, on the distance interval
ds, a fraction k v ds of all photons at frequency v is
absorbed or scattered out of the beam. The solution of
the transport equation (2.14) is obtained by writing it
in the form d In I v — d I v /I v = —k v ds and integrating
from to s,
2.2 Determination of Distances Within Our Galaxy
In I v (s) - In / v (0) = - / ds' «„(/) = -r v (j) ,
where in the last step we defined the optical depth,
which depends on frequency. This yields
The specific intensity is thus reduced by a factor e~ T
compared to the case of no absorption taking place.
Accordingly, for the flux we obtain
(2.16)
S v = S v (0)e- r "°
where S v is the flux measured by the observer at a dis-
tance s from the source, and 5 V (0) is the flux of the
source without absorption. Because of the relation be-
tween flux and magnitude m — —2.5 log S + const, or
S oc 10~°' 4m , one has
_!L_ = 10 -0.4(m-m ) = e -T„ = 10 -log(e)T w
S v ,0
where in the last step we introduced the factor of pro-
portionality R x between the extinction coefficient and
the color excess, which depends only on the properties
of the dust and the choice of the filters. Usually, one
uses a blue and a visual filter (see Appendix A.4.2 for
a description of the filters commonly used) and writes
A V = R V E(B- V) .
For example, for dust in o
characteristic relation
ar Milky Way w
(2.20)
e have the
A v = (3.l±0.l)E(B-
-V) .
(2.21)
This relation is not a universal law, but the factor of pro-
portionality depends on the properties of the dust. They
are determined, e.g., by the chemical composition and
the size distribution of the dust grains. Fig. 2.6 shows the
wavelength dependence of the extinction coefficient for
different kinds of dust, corresponding to different val-
ues of R V - In the optical part of the spectrum we have
A v :=m-m Q = -2.5 log(5 w /S v , )
= 2.5 log(e) r y = 1 .086r y . (2. 17)
Here, A v is the extinction coefficient describing the
change of apparent magnitude m compared to that with-
out absorption, niQ. Since the absorption coefficient k v
depends on frequency, absorption is always linked to
a change in color. This is described by the color excess
which is defined as follows:
E(X-Y) ■- A x - A Y = (X- X )-(Y -Y )
= (X-Y)-(X-Y) . (2.18)
The color excess describes the change of the color index
(X — Y), measured in two filters X and Y that define the
corresponding spectral windows by their transmission
curves. The ratio A X /A Y — t V (X)/x V (y) depends only on
the optical properties of the dust or, more specifically,
on the ratio of the absorption coefficients in the two
frequency bands X and Y considered here. Thus, the
color excess is proportional to the extinction coefficient,
E(X-Y) = A X -A Y
1-
X-'(urrr 1 )
Fig. 2.6. Wavelength dependence of the extinction coefficient
A v , normalized to the extinction coefficient A/ at X = 9000 A.
Different kinds of clouds, characterized by the value of R v ,
i.e., by the reddening law, are shown. On the x-axis we
have plotted the inverse wavelength, so that the frequency
increases to the right. 'The solid fine specifics the mean Galac-
tic extinction curve. The extinction coefficient, as determined
from the observation of an individual star, is also shown;
clearly the observed law deviates from the model in some
details. The figure insert shows a detailed plot at relatively
I >' i length in ili I! i in of th \ ti in it tl
wavelengths the extinction depends only weakly on the value
of fly
Fig. 2.7. These images of the molecular cloud Barnard 68
show the effects of extinction and reddening: the left im-
age is a composite of exposures in the filters B, V, and I.
At the center of the cloud essentially all the light from the
background stars is absorbed. Near the edge it is dimmed
ibly shifted to the red. In the right-hand image ob-
in the filters B, I, and K have been combined
(red is assigned here to the near-infrared K-band filter); we
can clearly see that the cloud is more transparent at longer
wavelengths
approximately r„ oc v, i.e., blue light is absorbed (or
scattered) more strongly than red light. The extinction
therefore always causes a reddening. 4
In the Solar neighborhood the extinction coefficient
for sources in the disk is about
Ay ss lmag -
kpc
(2.221
but this relation is at best a rough approximation, since
the absorption coefficient can show strong local devi-
ations from this law, for instance in the direction of
molecular clouds (see, e.g., Fig. 2.7).
Color-color diagram. We now return to the distance
determination for a star cluster. As a first step in this
measurement, it is necessary to determine the degree
of extinction, which can only be done by analyzing
the reddening. The stars of the cluster are plotted in
a color-color diagram, for example by plotting the col-
ors (U - B) and (B - V) on the two axes (see Fig. 2.8).
A color-color diagram also shows a main sequence
along which the majority of the stars are aligned. The
wavelength-dependent extinction causes a reddening in
hoik colors. This shifts the positions of the stars in the
4 With what we have just learned we can readily a:
ol why i he sky is blue and the selling San red.
er the question
diagram. The direction of the reddening vector depends
only on the properties of the dust and is here assumed
to be known, whereas the amplitude of the shift de-
pends on the extinction coefficient. In a similar way to
the CMD, this amplitude can now be determined if one
has access to a calibrated, unreddened main sequence
for the color-color diagram which can be obtained from
the examination of nearby stars. From the relative shift
of the main sequence in the two diagrams one can then
derive the reddening and thus the extinction. The essen-
tial point here is the fact that the color-color diagram is
independent of the distance.
This then defines the procedure for the distance deter-
mination of a star cluster using photometry: in the first
step we determine the reddening E(B — V), and thus
with (2.21) also Ay, by shifting the main sequence in
a color-color diagram along the reddening vector until it
matches a calibrated main sequence. In the second step
the distance modulus is then determined by vertically
(i.e., in the direction of M) shifting the main sequence
in the color-magnitude diagram until it matches a cal-
ibrated main sequence. From this, the distance is then
obtained according to
m - M = 5 log(Z)/ 1 pc) - 5 + A
(2.23)
2.2 Determination of Distances Within Our Galaxy
V - A v - M v = 5 log (D/pc) - 5
2.2.6 Distances of Visual Binary Stars
Kepler's third law for a two-body problem,
G(m
U)
specifies the relation between the orbital period P of
a binary star, the masses m, of the two components,
and the semimajor axis a of the ellipse. The latter is
defined by the distance vector between the two stars
in the course of one period. This law can be used to
determine the distance to a visual binary star. For such
a system, the period P and the angular diameter 26
of the orbit are direct observables. If one additionally
knows the mass of the two stars, for instance from their
spectral classification, a can be determined according to
(2.25), and from this the distance follows with D — a/6.
Fig. 2.8. Color-color diagram for main-sequence stars. Spec-
iral types and absolute magnitudes are specified. Black bodies
I 77 10 3 K) would be located along the upper line. Interstellar
reddening shifts the measured stellar locations parallel to the
reddening vector indicated by the arrow
2.2.5 Spectroscopic Distance
From the spectrum of a star, the spectral type as well
as the luminosity class can be determined. The former
is determined from the strength of various absorption
lines in the spectrum, while the latter is obtained from
the width of the lines. From the line width the surface
gravity of the star can be derived, and from that its ra-
dius (more precisely, M/R 2 ). From the spectral type and
the luminosity class the position of the star in the HRD
follows unambiguously. By means of stellar evolution
models, the absolute magnitude M v can then be de-
termined. Furthermore, the comparison of the observed
color with that expected from theory yields the color ex-
cess E(B — V), and from that we obtain A v . With this
information we are then able to determine the distance
usiii"
2.2.7 Distances of Pulsating Stars
Several types of pulsating stars show periodic changes in
their brightnesses, where the period of a star is related
to its mass, and thus to its luminosity. This period-
luminosity (PL) relation is ideally suited for distance
measurements: since the determination of the period is
independent of distance, one can obtain the luminosity
directly from the period. The distance is thus directly de-
rived from the measured magnitude using (2.24), if the
extinction can be determined from color measurements.
The existence of a relation between the luminosity
and the pulsation period can be expected from simple
physical considerations. Pulsations are essentially ra-
dial density waves inside a star that propagate with the
speed of sound, c s . Thus, one can expect that the pe-
riod is comparable to the sound crossing time through
the star, P ~ R/c s . The speed of sound c s in a gas is of
the same order of magnitude as the thermal velocity of
the gas particles, so that k B T ~ m p c 2 , where m p is the
proton mass (and thus a characteristic mass of particles
in the stellar plasma) and k B is Boltzmann's constant.
According to the virial theorem, one expects that the
2. The Milky Way as a Galaxy
gravitational binding energy of the star is about twice
the kinetic (i.e., thermal) energy, so that for a proton
GMm p
Combin
obtain
-k B T.
lg these relations, for the pulsation period v
I RJW- p tf3/2
= <xp~'
(2.26)
where p is the mean density of the star. This is a remark-
able result - the pulsation period depends only on the
mean density. Furthermore, the stellar luminosity is re-
lated to its mass by approximately L oc M 3 . If we now
consider stars of equal effective temperature T e g (where
L on R 2 Tt f ), we find that
sJM
(2.21)
which is the relation between period and luminosity that
we were aiming for.
One finds that a well-defined period-luminosity
relation exists for three types of pulsating stars:
• 8 Cepheid stars (classical Cepheids). These are young
stars found in the disk population (close to the Gal-
actic plane) and in young star clusters. Owing to
their position in or near the disk, extinction always
plays a role in the determination of their lumi-
nosity. To minimize the effect of extinction it is
particularly useful to look at the period-luminosity
relation in the near-IR (e.g., in the K-band at
X ~ 2.4 |xm). Furthermore, the scatter around the
period-luminosity relation is smaller for longer
wavelengths of the applied filter, as is also shown
in Fig. 2.9. The period-luminosity relation is also
steeper for longer wavelengths, resulting in a more
accurate determination of the absolute magnitude.
• W Virginis stars, also called Population II Cepheids
(we will explain the term of stellar populations in
Sect. 2.3.2). These are low-mass, metal-poor stars
located in the halo of the Galaxy, in globular clusters,
and near the Galactic center.
• RR Lyrae stars. These are likewise Population II stars
and thus metal-poor. They are found in the halo, in
globular clusters, and in the Galactic bulge. Their ab-
solute magnitudes are confined to a narrow interval,
M v e [0.5, 1.0], withamean value of about0.6. This
obviously makes them very good distance indicators.
More precise predictions of their magnitudes are pos-
sible with the following dependence on metallicity
and period:
(M*) = -(2.0±0.3)log(/yW)
+ (0.06 ± 0.04) [Fe/H]- 0.7 ±0.1 .
Metallicity. In the last equation, the metallicity of a star
was introduced, which needs to be defined. In astro-
physics, all chemical elements heavier than helium are
called metals. These elements, with the exception of
some traces of lithium, were not produced in the early
Universe but rather later in the interior of stars. The
metallicity is thus also a measure of the chemical evolu-
tion and enrichment of matter in a star or gas cloud. For
an element X, the metallicity index of a star is defined as
™-*G£M£).
thus it is the logarithm of the ratio of the fraction of X
relative to hydrogen in the star and in the Sun, where
n is the number density of the species considered. For
example, [Fe/H] = — 1 means that iron has only a tenth
of its Solar abundance. The metallicity Z is the total
mass fraction of all elements heavier than helium; the
Sun has Z « 0.02, meaning that about 98% of the Solar
mass are contributed by hydrogen and helium.
The period-luminosity relations are not only of sig-
nificant importance for distance determination within
our Galaxy. They also play an important role in ex-
tragalactic astronomy, since by far the most luminous
of the three types of pulsating stars listed above, the
Cepheids, are also found and observed in other gal-
axies; they therefore enable us to directly determine
the distances of other galaxies, which is essential for
measuring the Hubble constant. These aspects will be
discussed in detail in Sect. 3.6.
2.3 The Structure of the Galaxy
Roughly speaking, the Galaxy consists of the disk, the
central bulge, and the Galactic halo - a roughly spherical
2.3 The Structure of the Galaxy
M B °
:
:
-,«^S^*^
-
^^^r^-"
-
,-
t.'s
* .
^--*
'
M B °
= (-2.757±0.112) log P + (-0.472i0.133); a = 0.27, N =
53
M v °
A,
^&0^'
£&
* .
^*^r^ J
^<T^
'*
M,,°
= (-3.14H0.100) log P + (-0.826±0.119);a= 0.24, N
53 ~
A
M°
•^^
. •••
^
*r*
<■
(-3.408i0.095) log P + (-1 .325+0. 1 1 4); a = 0.23, N =
53
Fig. 2.9. Period-I uminosily relation for Gal-
actic Cepheids. measured in three different
filters bands (B, V, and I, from top to
bottom). The absolute magnitudes were
corrected for extinction by using colors.
The period is given in days. Open and
solid circles denote data for those Cepheids
for which distances were estimated us-
ing different methods; the three objects
marked by triangles have a variable pe-
riod and are discarded in the derivation
of the period-luminosity relation. The lat-
ter is indicated by the solid line, with its
parametrisation specified in the plots. The
broken lines indicate the uncertainty range
of the period-luminosity relation. The slope
of the period-luminosity relation it
if one moves to redder filters
distribution of stars and globular clusters that surrounds
the disk. The disk, whose stars form the visible band
of the Milky Way, contains spiral arms similar to those
observed in other galaxies. The Sun, together with its
planets, orbits around the Galactic center on an approx-
imately circular orbit. The distance ^?o to the Galactic
center is not very well known, as we will discuss later.
To have a reference value, the International Astronom-
ical Union (IAU) officially defined the value of Ro in
1985,
= 8.5 kpc official value, IAU 1985 . (2.30)
More recent examinations have, however, found that
the real value is slightly smaller, Rq « 8.0 kpc. The di-
ameter of the disk of stars, gas, and dust is ~ 40 kpc.
2. The Milky Way as a Galaxy
A schematic depiction of our Galaxy is shown in
Fig. 1.3. Its most important structural parameters are
listed in Table 2.1.
2.3.1 The Galactic Disk: Distribution of Stars
By measuring the distances of stars in the Solar neigh-
borhood one can determine the three-dimensional stellar
distribution. From these investigations, one finds that
there are different stellar components, as we will discuss
below. For each of them, the number density in the direc-
tion perpendicular to the Galactic disk is approximately
described by an exponential law,
«(z)cxexp
(2.31s
where the scale-height h specifies the thickness of the
respective component. One finds that h varies between
different populations of stars, motivating the definition
of different components of the Galactic disk. In princi-
ple, three components need to be distinguished: (1) The
young thin disk contains the largest fraction of gas and
dust in the Galaxy, and in this region star formation is
still taking place today. The youngest stars are found in
the young thin disk, which has a scale-height of about
hyid ~ 100 pc. (2) The old thin disk is thicker and has
a scale-height of about h old ~ 325 pc. (3) The thick disk
has a scale-height of h thick ~ 1.5 kpc. The thick disk
contributes only about 2% to the total mass density in the
Galactic plane at z = 0. This separation into three disk
components is rather coarse and can be further refined
if one uses a finer classification of stellar populations.
Molecular gas, out of which new stars are born, has
the smallest scale-height, h mo \ ~ 65 pc, followed by the
atomic gas. This can be clearly seen by comparing
the distributions of atomic and molecular hydrogen in
Fig. 1.5. The younger a stellar population is, the smaller
its scale-height. Another characterization of the differ-
ent stellar populations can be made with respect to the
velocity dispersion of the stars, i.e., the amplitude of
the components of their random motions. As a first
approximation, the stars in the disk move around the
Galactic center on circular orbits. However, these orbits
are not perfectly circular: besides the orbital velocity
(which is about 220km/s in the Solar vicinity), they
have additional random velocity components.
The formal definition of the components of the veloc-
ity dispersion is as follows: let /(t>)d 3 i> be the number
density of stars (of a given population) at a fixed loca-
tion, with velocities in a volume element d 3 u around v
in the vector space of velocities. If we use Cartesian co-
ordinates, for example v = (v\, t>2, 1)3), then f(v)d 3 v is
the number of stars with the «-th velocity component
in the interval [u,-, v t + dt>,], and d 3 i> = duidi^di^. The
mean velocity (v) of the population then follows from
this distribution via
(v}=n~ l jd 3 vf(v) v, where n = jd\f(v)
(2.32)
denotes the total number density of stars in the pop-
ulation. The velocity dispersion a then describes the
mean squared deviations of the velocities from (v). For
Table 2.1. Parameters and characteristic values for the compo- to 1/e of its central value. a z is the velocity dispersion ir
nents of the Milky Way. The scale-height denotes the distance direction perpendicular to the disk
from the Galactic plane at which the density has decreased
2.3 The Structure of the Galaxy
a component i of the velocity vector, the dispersion or,
is defined as
cr, 2 = {(«!- <»«)) 2 ) = W-(««) 2 )
= n~ l fd 3 vf(v)(v*-(Vi) 2 ) . (2.33)
:_■'
The larger 07 is, the broader the distribution of the
stochastic motions. We note that the same concept ap-
plies to the velocity distribution of molecules in a gas.
The mean velocity (v) at each point defines the bulk
velocity of the gas, e.g., the wind speed in the atmo-
sphere, whereas the velocity dispersion is caused by
thermal motion of the molecules and is determined by
the temperature of the gas.
The random motion of the stars in the direction
perpendicular to the disk is the reason for the finite
thickness of the population; it is similar to a thermal
distribution. Accordingly, it has the effect of a pressure,
the so-called dynamical pressure of the distribution.
This pressure determines the scale-height of the dis-
tribution, which corresponds to the law of atmospheres.
The larger the dynamical pressure, i.e., the larger the
velocity dispersion a, perpendicular to the disk, the
larger the scale-height h will be. The analysis of stars
in the Solar neighborhood yields a z ~ 16 km/s for stars
younger than ~ 3 Gyr, corresponding to a scale-height
of ft ~ 250 pc, whereas stars older than ~ 6 Gyr have
a scale-height of ~ 350 pc and a velocity dispersion of
a z ~ 25 km/s.
The density distribution of the total star population,
obtained from counts and distance determir
stars, is to a good approximation described by
not smooth at z — 0; it has a kink at this point and it is
therefore unphysical. To get a smooth distribution which
follows the exponential law for large z and is smooth in
the plane of the disk, the distribution is slightly modi-
fied. As an example, for the luminosity density of the
old thin disk (that is proportional to the number density
of the stars), we can write:
L„c-'
n(R, z) = «o (e- |z|/,! *m +0 .02e- |z|// " hi *)
-Izl/Vick\ e -R/h R
(2.34)
here, R and z are the cylinder coordinates introduced
above (see Sect. 2.1), with the origin at the Galactic
center, and ft t hin ^ ^otd ^ 325 pc is the scale-height of
the thin disk. The distribution in the radial direction can
also be well described by an exponential law, where
h R ^3.5kpc denotes the scale-length of the Galactic
disk. The normalization of the distribution is determined
by the density n «s 0.2 stars/pc 3 in the Solar neighbor-
hood, for stars in the range of absolute magnitudes of
4.5 <M V < 9.5. The distribution described by (2.34) is
with h, = 2h thin and L % O.O5L /pc 3 . The Sun is
a member of the young thin disk and is located above
the plane of the disk, at z = 30 pc.
2.3.2 The Galactic Disk:
Chemical Composition and Age
Stellar Populations. The chemical composition of stars
in the thin and the thick disks differs: we observe the
clear tendency that stars in the thin disk have a higher
metallicity than those in the thick disk. In contrast, the
metallicity of stars in the Galactic halo and in the bulge
is smaller. To paraphrase these trends, one distinguishes
between stars of Population I (Pop I) which have a Solar-
like metallicity (Z ~ 0.02) and are mainly located in
the thin disk, and stars of Population II (Pop II) that
are metal-poor (Z ~ 0.001) and predominantly found
in the thick disk, in the halo, and in the bulge. In reality,
stars cover a wide range in Z, and the figures above
are only characteristic values. For stellar populations
a somewhat finer separation was also introduced, such
as "extreme Population I", "intermediate Population II",
and so on. The populations also differ in age (stars of
Pop I are younger than those of Pop II), in scale-height
(as mentioned above), and in the velocity dispersion
perpendicular to the disk (a z is larger for Pop II stars
than for Pop I stars).
We shall now attempt to understand the origin of
these different metallicities and their relation to the
scale-height and to age. We start with a brief discus-
sion of the phenomenon that is the main reason for the
metal enrichment of the interstellar medium.
Metallicity and Supernovae. Supernovae (SNe) are
explosive events. Within a few days, a SN can reach
2. The Milky Way as a Galaxy
a luminosity of 10 9 L o , which is a considerable fraction
of the total luminosity of a galaxy; after that the lumi-
nosity decreases again with a time-scale of weeks. In the
explosion, a star is disrupted and (most of) the matter of
the star is driven into the interstellar medium, enriching
it with metals that were produced in the course of stellar
evolution or in the process of the supernova explosion.
Classification of Supernovae. Depending on their
spectral properties, SNe are divided into several classes.
SNe of Type I do not show any Balmer lines of hydro-
gen in their spectrum, in contrast to those of Type II.
A further subdivision of Type I SNe is based on spec-
tral properties: SNe la show strong emission of Sill X
6150 A whereas no Sill at all is visible in spectra of
Type Ib,c. Our current understanding of the supernova
phenomenon differs from this spectral classification. 5
Following various observational results and also the-
oretical analyses, we are confident today that SNe la
are a phenomenon which is intrinsically different from
the other supernova types. For this interpretation, it is
of particular importance that SNe la are found in all
types of galaxies, whereas we observe SNe II and SNe
Ib,c only in spiral and irregular galaxies, and here only
in those regions in which blue stars predominate. As
we will see in Chap. 3, the stellar population in ellip-
tical galaxies consists almost exclusively of old stars,
while spirals also contain young stars. From this ob-
servational fact it is concluded that the phenomenon of
SNe II and SNe Ib,c is linked to a young stellar popula-
tion, whereas SNe la occur in older stellar populations.
We shall discuss the two classes of supernovae next.
Core-Collapse Supernovae. SNe II and SNe Ib,c are
the final stages in the evolution of massive (> 8M Q )
stars. Inside these stars, ever heavier elements are gener-
ated by fusion: once all the hydrogen is used up, helium
will be burned, then carbon, oxygen, etc. This chain
comes to an end when the iron nucleus is reached, the
atomic nucleus with the highest binding energy per nu-
5 This notation scheme (Type la, Type II, and so i
i i i i l ii i ii ii i i i i ii i i i li
no physical interpretation is available at that tunc. Other examples arc
the spectral classes ol stars, which aic nol named in alphabetical order
i rd i 111 ii ii i n ii | ii rthedixisi I
galaxies into T\ pe I and Type 2. Once such a notation is established
ii often becomes permanent even if a later physical understanding of
the phenomenon suggest-, a more meaningful classification.
cleon. After this no more energy can be gained from
fusion to heavier elements so that the pressure, which
is normally balancing the gravitational force in the star,
can no longer be maintained. The star will thus collapse
under its own gravity. This gravitational collapse will
proceed until the innermost region reaches a density
about three times the density of an atomic nucleus. At
this point the so-called rebounce occurs: a shock wave
runs towards the surface, thereby heating the infalling
material, and the star explodes. In the center, a compact
object probably remains - a neutron star or, possibly,
depending on the mass of the iron core, a black hole.
Such neutron stars are visible as pulsars 6 at the location
of some historically observed SNe, the most famous of
which is the Crab pulsar which has been identified with
a supernovae explosion seen by Chinese astronomers in
1054. Presumably all neutron stars have been formed in
such core-collapse supernovae.
The major fraction of the binding energy released
in the formation of the compact object is emitted in
the form of neutrinos: about 3 x 10 53 erg. Underground
neutrino detectors were able to trace about 10 neutrinos
originating from SN 1987 A in the Large Magellanic
Cloud. Due to the high density inside the star after
the collapse, even neutrinos, despite their very small
cross-section, are absorbed and scattered, so that part
of their outward-directed momentum contributes to the
explosion of the stellar envelope. This shell expands at
v ~ lOOOOkm/s, corresponding to a kinetic energy of
£kin ~ 10 51 erg. Of this, only about 10 49 erg is converted
into photons in the hot envelope and then emitted - the
energy of a SN that is visible in photons is thus only
a small fraction of the total energy produced.
Owing to the various stages of nuclear fusion in the
progenitor star, the chemical elements are arranged in
shells: the light elements (H, He) in the outer shells, and
the heavier elements (C, O, Ne, Mg, Si, Ar, Ca, Fe, Ni) in
the inner ones - see Fig. 2.10. The explosion ejects them
into the interstellar medium which is thus chemically
enriched. It is important to note that mainly nuclei with
an even number of protons and neutrons are formed.
This is a consequence of the nuclear reaction chains
''Pulsars are sources which show a verx regular periodic radiation.
most often seen at radio frequencies. Their periods lie in the range
from ~ 10 -3 s (millisecond pulsars) to ~ 5 s. Their pulse period is
identilied as the rotational period of the neutron star- an object with
about one Solar mass and a radius of ~ 10 km. The matter density in
neutron stars is about the same as that in atomic nuclei.
2.3 The Structure of the Galaxy
Fig. 2.10. Chemical shell st
sive star at the end of its life. The elements
that have been formed in the various stages
oi'lhc nuclear burninu arc ordered in a siruc
ture resembling that of an onion. This is the
initial condition for a supernova explosion
25 M/M,.
involved, where successive nuclei in this chain are ob-
tained by adding an ce-particle (or 4 He-nucleus), i.e.,
two protons and two neutrons. Such elements are there-
fore called a-elements. The dominance of a-elements
in the chemical abundance of the interstellar medium
is thus a clear indication of nuclear fusion occurring in
the He-rich zones of stars where the hydrogen has been
burnt.
Supernovae Type la. SNe la are most likely the ex-
plosions of white dwarfs (WDs). These compact stars
which form the final evolutionary stages of less mas-
sive stars no longer maintain their internal pressure by
nuclear fusion. Rather, they are stabilized by the degen-
eracy pressure of the electrons - a quantum mechanical
phenomenon. Such a white dwarf can only be stable
if its mass does not exceed a limiting mass, the Chan-
el rasekhar mass; it has a value of Mqu ^ 1.44M©. For
M > Mch, the degeneracy pressure can no longer bal-
ance the gravitational force. If matter falls onto a WD
with mass below Mch, as may happen by accretion
in close binary systems, its mass will slowly increase
and approach the limiting mass. At about M « 1.3M©,
carbon burning will ignite in its interior, transforming
about half of the star into iron-group elements, i.e.,
iron, cobalt, and nickel. The resulting explosion of the
star will enrich the ISM with ~ 0.6M Q of Fe, while
the WD itself will be torn apart completely, leaving no
remnant star.
Since the initial conditions are probably very homo-
geneous for the class of SNe la (defined by the limiting
mass prior to the trigger of the explosion), they are good
candidates for standard candles: all SNe la have approx-
imately the same luminosity. As we will discuss later
(see Sect. 8.3.1), this is not really the case, but neverthe-
less SNe la play an important role in the cosmological
distance determination, and thus in the determination of
cosmological parameters.
This interpretation of the different types of SNe ex-
plains why one finds core-collapse SNe only in galaxies
in which star formation occurs. They are the final stages
of massive, i.e., young, stars which have a lifetime of
not more than 2 x 10 7 yr. By contrast, SNe la can occur
in all types of galaxies.
In addition to SNe, metal enrichment of the interstel-
lar medium (ISM) also takes place in other stages of
stellar evolution, by stellar winds or during phases in
which stars eject part of their envelope which is then
visible, e.g., as a planetary nebula. If the matter in the
star has been mixed by convection prior to such a phase,
so that the metals newly formed by nuclear fusion in the
interior have been transported towards the surface of the
star, these metals will then be released into the ISM.
Age-Metallicity Relation. Assuming that at the begin-
ning of its evolution the Milky Way had a chemical
composition with only low metal content, the metal-
licity should be strongly related to the age of a stellar
population. With each new generation of stars, more
metals are produced and ejected into the ISM, partially
by stellar winds, but mainly by SN explosions. Stars that
are formed later should therefore have a higher metal
content than those that were formed in the early phase
of the Galaxy. One would therefore expect that a re-
lation should exists between the age of a star and its
metallicity.
For instance, under this assumption [Fe/H] can be
used as an age indicator for a stellar population, with the
iron predominantly being produced and ejected in SNe
of type la. Therefore, newly formed stars have a higher
fraction of iron when they are born than their prede-
cessors, and the youngest stars should have the highest
2. The Milky Way as a Galaxy
iron abundance. Indeed one finds [Fe/H] = —4.5 for
extremely old stars (i.e., 3 x 10~ 5 of the Solar iron abun-
dance), whereas very young stars have [Fe/H] = 1, so
their metallicity can significantly exceed that of the Sun.
However, this age-metallicity relation is not very
tight. On the one hand, SNe la occur only > 10 9 years
after the formation of a stellar population. The exact
time-span is not known because even if one accepts the
scenario for SN la described above, it is unclear in what
form and in what systems the accretion of material onto
the white dwarf takes place and how long it typically
takes until the limiting mass is reached. On the other
hand, the mixing of the SN ejecta in the ISM occurs only
locally, so that large inhomogeneities of the [Fe/H] ra-
tio may be present in the ISM, and thus even for stars
of the same age. An alternative measure for metallic-
ity is [O/H], because oxygen, which is an a-element,
is produced and ejected mainly in supernova explo-
sions of massive stars. These begin only ~ 10 7 yr after
the formation of a stellar population, which is virtually
instantaneous.
Characteristic values for the metallicity are —0.5
< [Fe/H] < 0.3 in the thin disk, while for the thick disk
-1.0 < [Fe/H] < -0.4 is typical. From this, one can
deduce that stars in the thin disk must be significantly
younger on average than those in the thick disk. This
result can now be interpreted using the age-metallicity
relation. Either star formation has started earlier, or
ceased earlier, in the thick disk than in the thin disk,
or stars that originally belonged to the thin disk have
migrated into the thick disk. The second alternative is
favored for various reasons. It would be hard to under-
stand why molecular gas, out of which stars are formed,
was much more broadly distributed in earlier times than
it is today, where we find it well concentrated near the
Galactic plane. In addition, the widening of an initially
narrow stellar distribution in time is also expected. The
matter distribution in the disk is not homogeneous and,
along their orbits around the Galactic center, stars expe-
rience this inhomogeneous gravitational field caused by
other stars, spiral arms, and massive molecular clouds.
Stellar orbits are perturbed by such fluctuations, i.e.,
they gain a random velocity component perpendicular
to the disk from local inhomogeneities of the gravita-
tional field. In other words, the velocity dispersion a z of
a stellar population grows in time, and the scale-height
of a population increases. In contrast to stars, the gas
keeps its narrow distribution around the Galactic plane
due to internal friction.
This interpretation is, however, not unambiguous.
Another scenario for the formation of the thick disk
is also possible, where the stars of the thick disk were
formed outside the Milky Way and only became con-
stituents of the disk later, through accretion of satellite
galaxies. This model is supported, among other reasons,
by the fact that the rotational velocity of the thick disk
around the Galactic center is smaller by ~ 50 km/s than
that of the thin disk. In other spirals, in which a thick
disk component was found and kinematically analyzed,
the discrepancy between the rotation curves of the thick
and thin disks is sometimes even stronger. In one case,
the thick disk has been observed to rotate around the
center of the galaxy in the opposite direction to the gas
disk. In such a case, the aforementioned model of the
evolution of the thick disk by kinematic heating of stars
would definitely not apply.
Mass-to-Light Ratio. The total stellar mass of the thin
disk is ~ 6 x 10 10 M o , to which ~ 0.5 x 1O 1O M in the
form of dust and gas has to be added. The luminosity of
the stars in the thin disk is L B « 1.8 x 1O 1O L -T o g emer '
this yields a mass-to-light ratio of
The M/L ratio in the thick disk is higher. For this com-
ponent, one has M ~ 3 x 10 9 M o and L s «2x 10 8 L o ,
so that M/L B ~ 15 in Solar units. The thick disk thus
does not play any significant role for the total mass
budget of the Galactic disk, and even less for its total
luminosity. On the other hand, the thick disk is invalu-
able for the diagnosis of the dynamical evolution of the
disk. If the Milky Way were to be observed from the
outside, one would find a M/L value for the disk of
about 4 in Solar units; this is a characteristic value for
spiral galaxies.
2.3.3 The Galactic Disk: Dust and Gas
The spiral structure of the Milky Way and other spiral
galaxies is delineated by very young objects like O- and
2.3 The Structure of the Galaxy
B-stars and Hll regions. 7 This is the reason why spi-
ral arms appear blue. Obviously, star formation in our
Milky Way takes place mainly in the spiral arms. Here,
the molecular clouds - gas clouds which are sufficiently
dense and cool for molecules to form in large abun-
dance - contract under their own gravity and form new
stars. The spiral arms are much less prominent in red
light. Emission in the red is dominated by an older stel-
lar population, and these old stars have had time to move
away from the spiral arms. The Sun is located close to,
but not in, a spiral arm - the so-called Orion arm.
Observing the gas in the Galaxy is made possible
mainly by the 21 -cm line emission of Hi (neutral atomic
hydrogen) and by the emission of CO, the second-most
abundant molecule after H2 (molecular hydrogen). H2
is a symmetric molecule and thus has no electric dipole
moment, which is the reason why it does not radiate
strongly. In most cases it is assumed that the ratio of
CO to H2 is a universal constant (called the "X-factor").
Under this assumption, the distribution of CO can be
converted into that of the molecular gas. The Milky Way
is optically thin at 21 cm, i.e., 21 -cm radiation is not
absorbed along its path from the source to the observer.
With radio-astronomical methods it is thus possible to
observe atomic gas throughout the entire Galaxy.
To examine the distribution of dust, two options are
available. First, dust is detected by the extinction it
causes. This effect can be analyzed quantitatively, for in-
stance by star counts or by investigating the reddening of
stars (an example of this can be seen in Fig. 2.7). Second,
dust emits thermal radiation observable in the FIR part
of the spectrum, which was mapped by several satellites
such as IRAS and COBE. By combining the sky maps of
these two satellites at different frequencies the Galactic
distribution of dust was determined. The dust tempera-
ture varies in a relatively narrow range between ~ 17 K
and ~ 21 K, but even across this small range, the dust
emission varies, for fixed column density, by a factor
~ 5 at a wavelength of 100 ujti. Therefore, one needs
to combine maps at different frequencies in order to de-
termine column densities and temperatures. In addition,
the zodiacal light caused by the reflection of solar radia-
tion by dust inside our Solar system has to be subtracted
iin regions arc nearly spherical regions of full) ionized hydrogen
(thus the name lln region) surrounding a young hot star which pho
toi 1 li ' iic lii nil 1 in mi 1 1 1 11. 1 1 hii 1 ,n L. Imcr
lines of hydrogen are strongest.
before the Galactic FIR emission can be analyzed. This
is possible with multifrequency data because of the dif-
ferent spectral shapes. The resulting distribution of dust
is displayed in Fig. 2.11. It shows the concentration of
dust around the Galactic plane, as well as large-scale
anisotropies at high Galactic latitudes. The dust map
shown here is routinely used for extinction correction
when observing extragalactic sources.
Besides a strong concentration towards the Galac-
tic plane, gas and dust are preferentially found in spiral
arms where they serve as raw material for star formation.
Molecular hydrogen (H2) and dust are generally found
at 3 kpc < R < 8 kpc, within |z| < 90 pc of both sides of
the Galactic plane. In contrast, the distribution of atomic
hydrogen (Hi) is observed out to much larger distances
from the Galactic center (R < 25 kpc), with a scale-
height of ~ 160 pc inside the Solar orbit, R < R Q . At
larger distances from the Galactic center, R > 12 kpc,
the scale-height increases substantially to ~ 1 kpc. The
gaseous disk is warped at these large radii though the
origin of this warp is unclear. For example, it may
be caused by the gravitational field of the Magellanic
Clouds. The total mass in the two components of hydro-
gen is about M(Hl) « 4 x 10 9 M o andM(H 2 )« 1O 9 M ,
respectively, i.e., the gas mass in our Galaxy is less than
~ 10% of the stellar mass. The density of the gas in the
Solar neighborhood is about p(gas) ~ 0.04M G /pc 3 .
2.3.4 Cosmic Rays
The Magnetic Field of the Galaxy. Like many other
cosmic objects, the Milky Way has a magnetic field. The
properties of this field can be analyzed using a variety
of methods and we list some of them in the following.
• Polarization of stellar light. The light of distant stars
is partially polarized, with the degree of polarization
being strongly related to the extinction, or reddening,
of the star. This hints at the polarization being linked
to the dust causing the extinction. The light scat-
tered by dust particles is partially linearly polarized,
with the direction of polarization depending on the
alignment of the dust grains. If their orientation were
random, the superposition of the scattered radiation
from different dust particles would add up to a van-
ishing net polarization. However, a net polar
2. The Milky Way as a Galaxy
Fig. 2.11. Distribution of dust in the Galaxy, derived from
a combination of IRAS and COBE sky maps. The northern
Galactic sky in Galactic coordinates is displayed on ihc left.
the southern on the right. We can clearly see
tion of dust towards the Galactic plane, as well as regions
with a very low column density of dust; these regions in the
sky are particularly well suited for very deep extragalactic
observations
is measured, so the orientation of dust particles can-
not be random, rather it must be coherent on large
scales. Such a coherent alignment is provided by
a large-scale magnetic field, whereby the o
of dust particles, measurable from the polar
direction, indicates the (projected) direction of the
magnetic field.
> The Zeeman effect. The energy levels in an atom
change if the atom is placed in a magnetic field. Of
particular importance in the present context is the fact
that the 21 -cm transition line of neutral hydrogen is
split in a magnetic field. Because the amplitude of
the line split is proportional to the strength of the
magnetic field, the field strength can be determined
from observations of this Zeeman effect.
> Synchrotron radiation. When relativistic electrons
move in a magnetic field they are subject to the
Lorentz force. The corresponding acceleration is per-
pendicular both to the velocity vector of the particles
and to the magnetic field vector. As a result, the elec-
trons follow a helical (i.e., corkscrew) track, which is
a superposition of circular orbits perpendicular to the
field lines and a linear motion along the field. Since
accelerated charges emit electromagnetic radiation,
this helical movement is the source of the so-called
synchrotron radiation (which will be discussed in
more detail in Sect. 5.1.2). This radiation, which is
observable at radio frequencies, is linearly polarized,
with the direction of the polarization depending on
the direction of the magnetic field.
< Faraday rotation. If polarized radiation passes
through a magnetized plasma, the direction of the
polarization rotates. The rotation angle depends
quadratically on the wavelength of the radiation,
A6» = RMA 2
(2.37)
The rotation measure RM is the integral along the
line-of-sight towards the source over the electron
density and the component By of the magnetic field
n of the line-of-sight,
rad [ d£ n e B»
4 = 81 — j / ^7^'
cm z J pc cm -3 G
The dependence of the rotation angle (2.37) on X al-
lows us to determine the rotation measure RM, and
thus to estimate the product of electron density and
magnetic field. If the former is known, one imme-
diately gets information about B. By measuring the
RM for sources in different directions and at differ-
ent distances the magnetic field of the Galaxy can be
mapped.
2.3 The Structure of the Galaxy
From applying the methods discussed above, we know
that a magnetic field exists in the disk of our Milky Way.
This field has a strength of about 4 x 10~ 6 G and mainly
follows the spiral arms.
Cosmic Rays. We obtain most of the information about
our Universe from the electromagnetic radiation that
we observe. However, we receive an additional radia-
tion component, the energetic cosmic rays. They consist
primarily of electrically charged particles, mainly elec-
trons and nuclei. In addition to the particle radiation that
is produced in energetic processes at the Solar surface,
a much more energetic cosmic ray component exists that
can only originate in sources outside the Solar system.
The energy spectrum of the cosmic rays is, to
a good approximation, a power law: the flux of par-
ticles with energy larger than E can be written as
S(> E) oc E~ q , with q «s 1.7. However, the slope of
the spectrum changes slightly, but significantly, at some
energy scales: at E ~ 10 15 eV the spectrum becomes
steeper, and at E > 10 18 eV it flattens again. 8 Measure-
ments of the spectrum at these high energies are rather
uncertain, however, because of the strongly decreasing
flux with increasing energy. This implies that only very
few particles are detected.
Cosmic Ray Acceleration and Confinement. To ac-
celerate particles to such high energies, highly energetic
processes are necessary. For energies below 10 15 eV,
very convincing arguments suggest SN remnants as
the sites of the acceleration. The SN explosion drives
a shock front 9 into the ISM with an initial velocity of
~ lOOOOkm/s. Plasma processes in a shock front can
accelerate some particles to very high energies. The
theory of this diffuse shock acceleration predicts (hat
8 These energies should be compared with those reached in par-
ticle accelerators: LEP at CERN reached ~ lOOGeV = 10" eV.
Ilcncc. cosmic accelerators arc much more efficient than man made
machines.
'Shock fronts are surfaces in a gas flow where the parameters oi
state lor the gas. such as pressure, density, and temperature, change
diseontinuously. The standard example for a shock front is the bang in
an explosion, where a spherical shock ua\c propagates outwards from
the point of explosion. Another example is the some boom caused, for
npl i irplai u it in peed ing the velocity ol
sound. Such shock fronts are solutions oi the h\ drodynamic equations.
They occur frequently in astrophysics, e.g., in explosion phenomena
such as supernoxae or in rapid (i.e.. supersonic) Hows such as those
we will discuss in the context of AGNs.
the resulting energy spectrum of the particles follows
a power law, the slope of which depends only on the
strength of the shock (i.e., the ratio of the densities on
both sides of the shock front). This power law agrees
very well with the slope of the observed cosmic ray spec-
trum, if additional propagation processes in the Milky
Way are taken into account. The presence of very en-
ergetic electrons in SN remnants is observed directly
by their synchrotron emission, so that the slope of the
produced spectrum is also directly observable.
Accelerated particles then propagate through the
Galaxy where, due to the magnetic field, they move
along complicated helical tracks. Therefore, the direc-
tion from which a particle arrives at Earth cannot be
identified with the direction to its source of origin.
The magnetic field is also the reason why particles do
not leave the Milky Way along a straight path, but in-
stead are stored for a long time (~ 10 7 yr) before they
eventually diffuse out, an effect also called confinement.
The sources of the particles with energy between
~ 10 15 eV and ~ 10 18 eV are likewise presumed to be
located inside our Milky Way, because the magnetic
field is sufficiently strong to confine them in the Gal-
axy. However, SN remnants are not likely sources for
particles at these energies; in fact, the origin of these
rays is largely unknown. Particles with energies larger
than ~ 10 18 eV are probably of extragalactic origin. The
radius of the helical tracks in the magnetic field of the
Galaxy, i.e., their Larmor radius, is larger than the ra-
dius of the Milky Way itself, so they cannot be confined.
Their origin is also unknown, but AGNs are the most
probable source of these particles. Finally, one of the
largest puzzles of high-energy astrophysics is the ori-
gin of cosmic rays with E > 10 19 eV. The energy of
these particles is so large that they are able to inter-
act with the cosmic microwave background to produce
pions and other particles, losing much of their energy
in this process. These particles cannot propagate much
further than ~ 10 Mpc through the Universe before they
lose most of their energy. This implies that their accel-
eration sites should be located in the close vicinity of
the Milky Way. Since the curvature of the orbits of such
highly energetic particles is very small, it should, in
principle, be possible to identify their origin: there are
not many AGNs within 10 Mpc that are promising can-
didates for the origin of these ultra-high-energy cosmic
rays. However, the observed number of these particles
2. The Milky Way as a Galaxy
is so small that no reliable information on these sources
has thus far been obtained.
Energy Density. It is interesting to realize that the en-
ergy densities of cosmic rays, the magnetic field, the
turbulent energy of the ISM, and the electromagnetic
radiation of the stars are about the same - as if an equi-
librium between these different components has been
established. Since these components interact with each
other - e.g., the turbulent motions of the ISM can am-
plify the magnetic field, and vice versa, the magnetic
field affects the velocity of the ISM and of cosmic rays -
it is not improbable that these interaction processes can
establish an equipartition of the energy densities.
Gamma Radiation from the Milky Way. The Milky
Way emits y -radiation, as can be seen in Fig. 1.5. There
is diffuse y-ray emission which can be traced back to
the cosmic rays in the Galaxy. When these energetic
particles collide with nuclei in the interstellar medium,
radiation is released. This gives rise to a continuum ra-
diation which closely follows a power-law spectrum,
such that the observed flux S v is oc v~ a , with a ~ 2. The
quantitative analysis of the distribution of this emis-
sion provides the most important information about the
spatial distribution of cosmic rays in the Milky Way.
with an energy corresponding to the rest-mass energy of
an electron, i.e., 511 keV. 10 This annihilation radiation
was identified first in the 1970s. With the Integral satel-
lite, its emission morphology has been mapped with an
angular resolution of ~ 3°. The 511 keV line emission
is detected both from the Galactic disk and the bulge.
The angular resolution is not sufficient to tell whether
the annihilation line traces the young stellar popula-
tion (i.e., the thin disk) or the older population in the
thick disk. However, one can compare the distribution
of the annihilation radiation with that of Al 26 and other
radioactive species. In about 85% of all decays Al 26
emits a positron. If this positron annihilates close to its
production site one can predict the expected annihila-
tion radiation from the distribution of the 1.809 MeV
line. In fact, the intensity and angular distribution of
the 511 keV line from the disk is compatible with this
scenario for the generation of positrons.
The origin of the annihilation radiation from the
bulge, which has a luminosity larger than that from
the disk by a factor ~ 5, is unknown. One needs to find
a plausible source for the production of positrons in
the bulge. There is no unique answer to this problem at
present, but Type la supernovae and energetic processes
near low-mass X-ray binaries are prime candidates for
this source.
Gamma-Ray Lines. In addition to the continuum ra-
diation, one also observes line radiation in y-rays, at
energies below ~ 10 MeV. The first detected and most
prominent line has an energy of 1.809 MeV and corre-
sponds to a radioactive decay of the Al 26 nucleus. The
spatial distribution of this emission is strongly concen-
trated towards the Galactic disk and thus follows the
young stellar population in the Milky Way. Since the
lifetime of the Al 26 nucleus is short (~ 10 6 yr), it must
be produced near the emission site, which then implies
that it is produced by the young stellar population. It
is formed in hot stars and released to the interstellar
medium either through stellar winds or core-collapse
supernovae. Gamma lines from other radioactive nuclei
have been detected as well.
Annihilation Radiation from the Galaxy. Further-
more, line radiation with an energy of 5 1 1 keV has been
detected in the Galaxy. This line is produced when an
electron and a positron annihilate into two photons, each
2.3.5 The Galactic Bulge
The Galactic bulge is the central thickening of our Gal-
axy. Figure 1.2 shows another spiral galaxy from its
side, with its bulge clearly visible. The characteristic
scale-length of the bulge is ~ 1 kpc. Owing to the strong
extinction in the disk, the bulge is best observed in the
IR, for instance with the IRAS and COBE satellites.
The extinction to the Galactic Center in the visual is
A v ~ 28 mag. However, some lines-of-sight close to the
Galactic center exist where Ay is significantly smaller,
so that observations in optical and near IR light are pos-
sible, e.g., in Baade's window, located about 4° below
the Galactic center at I ~ 1°, for which A v ~ 2 mag
(also see Sect. 2.6).
From the observations by COBE, and also from Gal-
actic microlensing experiments (see Sect. 2.5), we know
10 In addition to the two photon annihilation, there is also an annihila-
tion channel in which three photons are produced: the corresponding
radiation forms a continuum spectrum, i.e.. no spectral lines.
2.3 The Structure of the Galaxy
that our bulge has the shape of a bar, with the major axis
pointing away from us by about 30°. The scale-height
of the bulge is ~ 400 pc, with an axis ratio of ~ 0.6.
As is the case for the exponential profiles that de-
scribe the light distribution in the disk, the functional
form of the brightness distribution in the bulge is also
suggested from observations of other spiral galaxies.
The profiles of their bulges, observed from the outside,
are much better determined than in our Galaxy where
we are located amid its stars.
The de Vaucouleurs Profile. The brightness profile
of our bulge can be approximated by the de Vau-
couleurs law which describes the surface brightness I
as a function of the distance R from the center,
log (M) = -3.3307
Vr\" 4 1
with I(R) being the measured surface brightness, e.g.,
in [I] = L Q /pc 2 . R e is the effective radius, defined such
that half of the luminosity is emitted from within R e ,
«e OO
dRR I(R) = - / dR R I(R) . (2.40)
This definition of R e also leads to the numerical fac-
tor on the right-hand side of (2.39). As one can easily
see from (2.39), / e = I(R e ) is the surface brightness
at the effective radius. An alternative form of the de
Vaucouleurs law is
I(R) = 7 e exp (-7.669 [(R/R s ) 1/4 - l])
(2.41)
Because of its mathematical form, it is also called an r l,A
law. The r 1/4 law falls off significantly more slowly than
an exponential law for large R. For the Galactic bulge,
one finds an effective radius of R e «s 0.7 kpc. With the
de Vaucouleurs profile, a relation between luminosity,
effective radius, and surface brightness is obtained by
integrating over the surface brightness,
f dR2jrRI(R) = 7.2157tI e Rl
Stellar Age Distribution in the Bulge. The stars in
the bulge cover a large range in metallicity, —1 <
[Fe/H] < +1, with a mean of about 0.3, i.e., the mean
metallicity is about twice that of the Sun. This high
metallicity hints at a contribution by a rather young
population, whereas the color of the bulge stars points
towards a predominantly old stellar population. The
bulge also contains about 10° M in molecular gas. On
the other hand, one finds very metal-poor RR Lyrae
stars, i.e., old stars. However, the distinction in mem-
bership between bulge and disk stars is not easy, so it
is possible that the young component may actually be
part of the inner disk.
The mass of the bulge is about Mb u i ge ~ 10 10 M G and
its luminosity is L B ,buige ~ 3 x 10 9 L Q > which results in
a mass-to-light ratio of
M
M,
in the bulge
very similar to that of the thin disk.
2.3.6 The Visible Halo
The visible halo of our Galaxy consists of about 150
globular clusters and field stars with a high velocity
component perpendicular to the Galactic plane. A glob-
ular cluster is a collection of typically several hundred
thousand stars, contained within a spherical region of ra-
dius ~ 20 pc. The stars in the cluster are gravitational]}
bound and orbit in the common gravitational field. The
old globular clusters with [Fe/H] < — 0.8 have an ap-
proximately spherical distribution around the Galactic
center. A second population of globular clusters exists
that contains younger stars with a higher metallicity,
[Fe/H] > —0.8. They have a more oblate geometri-
cal distribution and are possibly part of the thick disk
because they show roughly the same scale-height.
Most globular clusters are at a distance of r < 35 kpc
(with r = V ' R 2 + z 2 ) from the Galactic center, but some
are also found at r > 60 kpc. At these distances it is hard
to judge whether these objects are part of the Galaxy
or whether they have been captured from a neighboring
galaxy, such as the Magellanic Clouds. Also, field stars
have been found at distances out to r ~ 50 kpc, which
is the reason why one assumes a characteristic value of
'"halo ~ 50 kpc for the extent of the visible halo.
2. The Milky Way as a Galaxy
The density distribution of metal-poor globular
clusters and field stars in the halo is described by
n(r) <xr
(2.44)
Alternatively, one can fit a de Vaucouleurs profile to the
density distribution, which results in an effective radius
ofr e ~2.7kpc.
At large distances from the disk, neutral hydrogen
is also found, in the form of clouds. Most of these
clouds, visible in 21 -cm line emission, have a nega-
tive radial velocity, i.e., they are moving towards us,
with velocities of up to u r ~ —400 km/s. These high-
velocity cloui Is I H '• '•' i 1 1 -annot be following the general
Galactic rotation. We have virtually no means of de-
termining the distances of these clouds, and thus their
origin and nature are still subject to discussion. There is
one exception, however: the Magellanic Stream is a nar-
row band of Hi emission which follows the Magellanic
Clouds along their orbit around the Galaxy (also see
Fig. 6.6). This gas stream may be the result of a close
encounter of the Magellanic Clouds with the Milky Way
in the past. The (tidal) gravitational force that the Milky
Way had imposed on our neighboring galaxies in such
an encounter could strip away part of the interstellar gas
from them.
at relatively large Galactic latitudes where they are not
too strongly affected by extinction. As was discussed
in Sect. 2.2, the distance determination of globular clus-
ters is possible using photometric methods. On the other
hand, one also finds RR Lyrae stars in globular clusters
to which the period-luminosity relation can be applied.
Therefore, the spatial distribution of the globular clus-
ters can be determined. However, at about 150, the
number of known globular clusters is relatively small,
resulting in a fairly large statistical error for the deter-
mination of the common center. Much more numerous
are the RR Lyrae field stars in the halo, for which dis-
tances can be measured using the period-luminosity
relation. The statistical error in determining the cen-
ter of their distribution is therefore much smaller. On
the other hand, this distance to the Galactic center is
based only on the calibration of the period-luminosity
relation, and any uncertainty in this will propagate into
a systematic error on Rq. Effects of the extinction add
to this. However, such effects can be minimized by ob-
serving the RR Lyrae stars in the NIR, which in addition
benefits from the narrower luminosity distribution of RR
Lyrae stars in this wavelength regime. These analyses
yield a value of R « 8.0 kpc (see Fig. 2.12).
2.3.7 The Distance to the Galactic Center
As already mentioned, our distance from the Galactic
center is rather difficult to measure and thus not very
precisely known. The general problem with such a mea-
surement is the high extinction in the disk, prohibiting
measurements of the distance of individual stars close
to the Galactic center. Thus, one has to rely on more
indirect methods, and the most important ones will be
outlined here.
The visible halo of our Milky Way is populated by
globular clusters and also by field stars. They have
a spherical, or, more generally, a spheroidal distribution.
The center of this distribution is obviously the center of
gravity of the Milky Way, around which the halo objects
are moving. If one measures the three-dimensional dis-
tribution of the halo population, the geometrical center
of this distribution should correspond to the Galactic
center.
This method can indeed be applied because, due to
their extended distribution, halo objects can be observed
log 10 d(kpc)
Fig. 2.12. The number of RR Lyrae stars as a function of dis-
tance, measured in a direction dial 1 loscly passes die Galactic
center, at I = 0° and b = -8°. If we assume a spherically
s) nimctric distribution of die RR L\ rae stars, concentrated to
wards the center, the distance to the Galactic center can be
identified with the maximum of this distribution
2.4 Kinematics of the Galaxy
2.4 Kinematics of the Galaxy
2.4.1 Determination of the Velocity of the Sun
Unlike a solid body, the Galaxy rotates differentially.
This means that the angular velocity is a function of the
distance R from the Galactic center. Seen from above,
i.e., from the NGP, the rotation is clockwise. To de-
scribe the velocity field quantitatively we will in the
following introduce velocity components in the coordi-
nate system (R, 9, z), as shown in Fig. 2.13. An object
following a track \R(t), 0(f). z(t)] then has the velocity
components
U I:
d K
dt '
For example, the Sun is not moving on a simple circular
orbit around the Galactic center, but currently inwards,
U < 0, and with W > 0, so that it is moving away from
the Galactic plane.
In this section we will examine the rotation of the
Milky Way. We start with the determination of the ve-
locity components of the Sun. Then we will consider
the rotation curve of the Galaxy, which describes the
rotational velocity V(R) as a function of the distance
R from the Galactic center. We will find the intrigu-
ing result that the velocity V does not decline towards
large distances, but that it virtually remains constant.
Because this result is of extraordinary importance, we
will discuss the methods needed to derive it in some
detail.
Fig.2.13. Cylindrical coordinate system (R,0,z) with the
Galactic center at its origin. Note that 9 increases in the
clockwise direction if the disk is viewed from above. The
corresponding velocity components (U, V, W) of a star are
indicated
Local Standard of Rest. To link local n
to the Galactic coordinate system (R, 9, z), the local
standard of rest is defined. It is a fictitious rest-frame
in which velocities are measured. For this purpose, we
consider a point that is located today at the position of
the Sun and that moves along a perfectly circular orbit in
the plane of the Galactic disk. The velocity components
in the LSR are then by definition,
(2.46)
t/ LS R = 0,
Vlsr = Vh .
Wlsr^O
CA5)
with Vq = V(Ro) being the orbital velocity at the loca-
tion of the Sun. Although the LSR changes over time,
the time-scale of this change is so large (the orbital
period is ~ 230 x 10 6 yr) that this effect is negligible.
Peculiar Velocity. The velocity of an object relative to
the LSR is called its peculiar velocity. It is denoted by
v, and its components are given as
= (U-U LSR , V-Vlsr, W-W L sr)
= (U, V-V , W)
(2.47)
The velocity of the Sun relative to the LSR is denoted by
v Q . If v Q is known, any velocity measured relative to the
Sun can be converted into a velocity relative to the LSR:
let Ad be the velocity of a star relative to the Sun, which
is directly measurable using the methods discussed in
Sect. 2.2, then the peculiar velocity of this star is
= Vq + Av.
(2.48
Peculiar Velocity of the Sun. We consider now an
semble of stars in the immediate vicinity of the Sun, and
assume the Galaxy to be axially symmetric and stE
ary. Under these assumptions, the number of stars that
move outwards to larger radii R equals the number ol
stars moving inwards. Likewise, as many stars move up-
wards through the Galactic plane as downwards. If these
conditions are not satisfied, the assumption of a station-
ary distribution would be violated. The mean values ol
the corresponding peculiar velocity components must
therefore vanish,
(«)=0, (w)=0,
(2.49)
2. The Milky Way as a Galaxy
where the brackets denote an average over the ensemble
considered. The analog argument is not valid for the i;
component because the mean value of v depends on the
distribution of the orbits: if only circular orbits in the
disk existed, we would also have (v) — (this is trivial,
since then all stars would have v — 0), but this is not
the case. From a statistical consideration of the orbits in
the framework of stellar dynamics, one deduces that (v)
is closely linked to the radial velocity dispersion of the
stars: the larger it is, the more (v) deviates from zero.
One finds that
(v) =
(2.50)
where C is a positive constant that depends on the den-
sity distribution and on the local velocity distribution
of the stars. The sign in (2.50) follows from noting that
a circular orbit has a higher tangential velocity than el-
liptical orbits, which in addition have a non-zero radial
component. Equation (2.50) expresses the fact that the
mean rotational velocity of a stellar population around
the Galactic center deviates from the corresponding cir-
cular orbit velocity, and that the deviation is stronger for
a larger radial velocity dispersion. This phenomenon is
also known as asymmetric drift. From the mean of (2.48)
over the ensemble considered and by using (2.49) and
(2.50), one obtains
= {-(Au),-C{ U 2 )-(Av),-(Aw))
(2.51)
One still needs to determine the constant C in order
to make use of this relation. This is done by consider-
ing different stellar populations and measuring (w 2 ) and
(Av) separately for each of them. If these two quanti-
ties are then plotted in a diagram (see Fig. 2. 14), a linear
relation is obtained, as expected from (2.50). The slope
C can be determined directly from this diagram. Fur-
thermore, from the intersection with the (At;) -axis, v Q
is readily read off. The other velocity components in
(2.51) follow by simply averaging, yielding the result:
i; G = (-10, 5,7)km/s
LSR. However, we have not yet discussed how V , the
rotational velocity of the LSR itself, is determined.
Velocity Dispersion of Stars. The dispersion of the stel-
lar velocities relative to the LSR can now be determined,
i.e., the mean square deviation of their velocities from
the velocity of the LSR. For young stars (A stars, for
example), this dispersion happens to be small. For older
K giants it is larger, and is larger still for old, metal-
poor red dwarf stars. We observe a very well-defined
velocity-metallicity relation. When this is combined
with the age-metallicity relation it appears that the
oldest stars have the highest peculiar velocities. This
effect is observed in all three coordinates. This re-
sult is in agreement with the relation between the age
of a stellar population and its scale-height (discussed
in Sect. 2.3.1), the latter being linked to the velocity
dispersion via tr,.
Asymmetric Drift. If one considers high-velocity stars,
only a few are found that have v > 65 km/s and which
are thus moving much faster around the Galactic center
than the LSR. However, quite a few stars are found that
have v < —250 km/s, so their orbital velocity is oppo-
site to the direction of rotation of the LSR. Plotted in
a (w — v) -diagram, a distribution is found which is nar-
rowly concentrated around u — km/s = v for young
stars, as already mentioned above, and which gets in-
creasingly wider for older stars. For the oldest stars,
(2.52)
Hence, the Sun is currently moving inwards, upwards,
and faster than it would on a circular orbit at its location.
We have therefore determined v Q , so we are now able
to analyze any measured stellar velocities relative to the
u 2 )(km 2 s
Fig. 2.14. The velocity components ( Ad) = (t;) — vq are plot-
ted against (w 2 ) for stars in the Solar neighborhood. Because
of the linear relation, vq can be read off from the ii
with the x-axis, and C from the slope
2.4 Kinematics of the Galaxy
which belong to the halo population, one obtains a cir-
cular envelope with its center located at u — km/s and
v s» -220 km/s (see Fig. 2.15). If we assume that the
Galactic halo, to which these high- velocity stars belong,
does not rotate (or only very slowly), this asymmetry in
the ^-distribution can only be caused by the rotation of
the LSR. The center of the envelope then has to be at
— Vq. This yields the orbital velocity of the LSR
V Q = V(R ) = 220 km/s
(2.53)
Knowing this velocity, we can then compute the mass
of the Galaxy inside the Solar orbit. A circular orbit
is characterized by an equilibrium between centrifugal
and gravitational acceleration, V 2 /R — GM(< R)/R 2 ,
so that
Furthermore, for the orbital period of the LSR, which is
similar to that of the Sun, one obtains
2ttRq _
2.4.2 The Rotation Curve of the Galaxy
From observations of the velocity of stars or gas around
the Galactic center, the rotational velocity V can be
determined as a function of the distance R from the
Galactic center. In this section, we will describe methods
to determine this rotation curve and discuss the result.
We consider an object at distance R from the Galactic
center which moves along a circular orbit in the Galactic
plane, has a distance D from the Sun, and is located at
a Galactic longitude £ (see Fig. 2.16). In a Cartesian
coordinate system with the Galactic center at the origin,
the positional and velocity vectors (we only consider
the two components in the Galactic plane because we
assume a motion in the plane) are given by
-R( &in0 ),
\cos9 /
5 9 denotes the
> seen from the
11 in Fig. 2.16 it
/ D sinl \
\Ro-Dcosl) '
= V(R)
I cos 9 \
\-sin9) '
where 9 denotes the angle between the Sun and the ob-
ject as seen from the Galactic center. From the geometry
shown in Fig. 2.16 it follows that
Hence, during the lifetime of the Solar System, esti-
mated to be ~ 4.6 x 10 9 yr, it has completed about 20
orbits around the Galactic center.
If we now identify the t
ponents of r, we obtain
:> expressions for the com-
sm6 = (D/R)sm£,
cos 9 = (R /R) - (D/R) cos I .
50 100 150
Fig. 2.15. The motion of the Sun around the
Galactic center is reflected in the asymmetric
drift: while young stars in the Solar vicinity
have velocities very similar to the Solar ve-
locity. i.e.. small relative velocities, members
of other populations (and of other Milky Way
components) have different velocities - e.g.,
for halo objects v = —220 km/s on average.
Thus, different velocity ellipses show up in a
(// — c)-diagram
2. The Milky Way as a Galaxy
If we disregard the difference between the velocities of
the Sun and the LSR we get V Q «a y LSR = ( Vq, 0) in this
coordinate system. Thus the relative velocity between
the object and the Sun is, in Cartesian coordinates,
the formalism of differen-
= (n-Qo)R smi,
AV= V-V
/ V (J?o/*) - V (D/R) cos € - V \
\ -V(D/R) sin^ / '
With the angular velocity defined as
V(R)
Q(R) = "V ' (2 " 56)
we obtain for the relative velocity
l (a-G )-G D cosl\
-DQ sin£ /
where Qq = Vq/Rq is the angular velocity of the Sun.
The radial and tangential velocities of this relative mo-
tion then follow by projection of A V along the direction
parallel or perpendicular, respectively, to the separation
_ / *o(*
v t = A V • ( C ° S l ) = (Q - S2 )Ro cos £ - Q D
\ sin€ /
A purely geometric derivation of these relations is given
in Fig. 2.16.
Rotation Curve near R ; Oort Constants. Using
(2.57) one can derive the angular velocity by means
of measuring v r , but not the radius R to which it cor-
responds. Therefore, by measuring the radial velocity
alone Q(R) cannot be determined. If one measures
v r and, in addition, the proper motion \i = v t /D of
stars, then Q and D can be determined from the
equations above, and from D and £ one obtains
R = JrI + D 2 -2R q Dcos£. The effects of extinction
prohibits the use of this method for large distances
D, since we have considered objects in the Galac-
tic disk. For small distances D <$C Ro, which implies
\R — Rq\ <£ Rq, we can make a local approximation by
evaluating the expressions above only up to first order
in (R — Ro)/Rq. In this linear approximation we get
Q-Q Q
\~dR~)
(R-Ro),
(259)
where the derivative has to be evaluated z
Hence
v r = (R-R )( R Q sm£,
and furthermore, with (2.56),
R o(-
Ro \ (dV
' ~R \\d~R
Ro '
in zeroth order in (R — R )/R . Combining the last two
equations yields
■[(
dV\
in analogy to this, we obtain for the tangential velocity
*=KS) -^\(R-Ro)co & i-Q D.
\dRj
R
(2.61)
For \R - R Q \ «: R it follows that R - R^ D cos I; if
we insert this into (2.60) and (2.61) we get
I v, % A D sin 21 , v t % A D cos U + B D I ,
where A and B are the Oort
2l\dR), Ro R \
2l\dRj lf
The radial and tangential velocity fields relative to the
Sun show a sine curve with period it, where u t and u r
are phase-shifted by it /A. This behavior of the velocity
field in the Solar neighborhood is indeed observed (see
Fig. 2.17). By fitting the data for v T (l) and v t (l) for stars
of equal distance D one can determine A and B, and
iliii-
(-)
\dRj }R
45 90 135 180 225 270 315 360
Gal. Longitude (deg)
Fig. 2.17. The radial velocity v r of stars at a fixed distance
D is proportional to sin 21':. the tangential velocity D t is a lin-
ear function of cos2£. From the amplitude of the oscillating
curves and from the mean value of v t the Oort constants A
and B can be derived, respectively (see (2.62))
The Oort constants thus yield the angular velocity of
the Solar orbit and its derivative, and therefore the
local kinematical information. If our Galaxy was ro-
tating rigidly so that Q was independent of the radius,
A = would follow. But the Milky Way rotates differ-
entially, i.e., the angular velocity depends on the radius.
Measurements yield the following values for A and B,
= (14.8 ±0.8) kms-'kpc" 1 ,
= (-12.4±0.6) kms -1 kpc~'
Galactic Rotation Curve for R < R ; Tangent Point
Method. To measure the rotation curve for radii that are
significantly smaller than Rq, one has to turn to large
wavelengths due to extinction in the disk. Usually the
21 -cm emission line of neutral hydrogen is used, which
can be observed over large distances, or the emission of
CO in molecular gas. These gas components are found
throughout the disk and are strongly concentrated to-
wards the plane. Furthermore, the radial velocity can
easily be measured from the Doppler effect. However,
since the distance to a hydrogen cloud cannot be deter-
mined directly, a method is needed to link the measured
radial velocities to the distance of the gas from the Gal-
actic center. For this purpose the tangent point method
is used.
2. The Milky Way as a Galaxy
Consider a line-of-sight at fixed Galactic longitude I,
with cos£ > (thus "inwards"). The radial velocity v r
along this line-of-sight for objects moving on circu-
lar orbits is a function of the distance D, according to
(2.57). If £2(R) is a monotonically decreasing function,
v t attains a maximum where the line-of-sight is tangent
to the local orbit, and thus its distance R from the Gal-
actic center attains the minimum value R m i„. This is the
case at
= Rq cos I,
(see Fig. 2.18). The maximum radial velocity there,
according to (2.57), is
Ur.max = [«(/?min) " «o] #0 sin I
= V(R mia )-V sini, (2.67)
so that from the measured value of v r<m . dx as a func-
tion of direction £, the rotation curve inside Rq can be
determined,
V(R)=(^)vo + v r , max (sinl = R/R ) . (2.68)
In the optical regime of the spectrum this method
can only be applied locally, i.e., for small D, due to
extinction. This is the case if one observes in a direc-
tion nearly tangential to the orbit of the Sun, i.e., if
0<n/2-l«. 1 or 0<£-3tt/2« 1, or |sin£| % 1,
so that Rq — R m i n <JC Ro- In this case we get, to first
order in (R - R^n), using (2.66),
/dV\
V(R miB )^V + — (Rnin-Ro)
/dV\
= Vb-( — I Ro(l-sinl), (2.69)
\dRj
A'n
so that with (2.6/)
(.2.66)
— h(si+—
= 2AR Q (l-
nO
(2.70)
where (2.63) was used in the last step. This relation can
also be used for determining the Oort constant A.
To determine V(R) for smaller R by employing the
tangent point method, we have to observe in wavelength
regimes in which the Galactic plane is transparent, us-
ing radio emission lines of gas. In Fig. 2.18, a typical
intensity profile of the 21 -cm line along a line-of-
sight is sketched; according to the Doppler effect this
can be converted directly into a velocity profile us-
ing v T = (k — Ao)/Ao. It consists of several maxima that
originate in individual gas clouds. The radial velocity of
each cloud is defined by its distance R from the Galactic
Fig. 2.18. The ISM is optically thin for
21-cm radiation, and thus we receive the
21-cm emission of HI regions from every-
where in the Galaxy. Due to the motion of
an HI cloud relative to us, the wavelength
is shifted. This can be used to measure
the radial velocity of the cloud. With the
assumption that the gas is moving on a cir-
cular orbit around the Galactic center, one
expects that for the cloud in the tangent
point (cloud 4), the full velocity is pro-
jected along the line-of-sight so that this
cloud will therefore have the largest radial
velocity. If the distance of the Sun to the
Galactic center is known, the velocity of
a cloud and its distance from the Galactic
center can then be determined
2.4 Kinematics of the Galaxy
center (if the gas follows the Galactic rotation), so that
the largest radial velocity will occur for gas closest to
the tangent point, which will be identified with u r ,max CO-
Figure 2.19 shows the observed intensity profile of the
12 CO line as a function of the Galactic longitude, from
which the rotation curve for R < R can be read off.
With the tangent point method, applied to the 2 1 -cm
line of neutral hydrogen or to radio emission lines
of molecular gas, the rotation curve of the Galaxy
inside the Solar orbit can be measured.
Rotation Curve for R > R . The tangent point
method cannot be applied for R > Rq because for lines-
of-sight at jt/2 < £ < 3jt/2, the radial velocity u r attains
no maximum. In this case, the line-of-sight is nowhere
parallel to a circular orbit.
Measuring V(R) for R > R requires measuring v r
for objects whose distance can be determined directly,
e.g., Cepheids, for which the period-luminosity relation
(Sect. 2.2.7) is used, or O- and B-stars in Hll regions.
With £ and D known, R can then be calculated, and
with (2.57) we obtain Q(R) or V(R), respectively. Any
object with known D and v T thus contributes one data
point to the Galactic rotation curve. Since the distance
estimates of individual objects are always affected by
uncertainties, the rotation curve for large values of R is
less accurately known than that inside the Solar circle.
It turns out that the rotation curve for R > Ro does
not decline outwards (see Fig. 2.20) as we would ex-
pect from the distribution of visible matter in the Milky
Way. Both the stellar density and the gas density of
the Galaxy decline exponentially for large R - e.g., see
(2.34). This steep radial decline of the visible matter
density should imply that M(R), the mass inside R, is
nearly constant for R > Ro, from which a velocity pro-
file like V oc R~ ]/2 would follow, according to Kepler's
law. However, this is not the case: V(R) is virtually con-
stant for R > R , indicating that M(R) <x R. Thus, to
get a constant rotational velocity of the Galaxy much
more matter has to be present than we observe in gas
and stars.
The Milky Way contains, besides stars and gas,
an additional component of matter that dominates
the mass at R> Rq but which has not yet been
observed directly. Its presence is known only by its
gravitational effect - hence, it is called dark matter.
In Sect. 3.3.3 we will see that this is a common phe-
nomenon. The rotation curves of spiral galaxies are flat
at large radii up to the maximum radius at which it can
be measured; spiral galaxies contain dark mailer.
-
i
-
'
s?%
j
v.v$S
'0itfi
Fig. 2.19. 12 CO emission of molecular gas
in the Galactic disk. For each I, the intensity
of the emission in the I - v r plane is plot-
ted, integrated over the range -2° < b < 2°
(i.e., very close to the middle of the plane).
Since v r depends on the distance along each
line-of-sight, characterized by (., this dia
gram contains information on the rotation
curve of the Galaxy as well as on the spa-
tial distribution of the gas. The maximum
velocity at each I is rather well defined and
forms the basis for the tangent point method
2. The Milky Way as a Galaxy
I
Tif
i
' i
Fig. 2.20. Rotation curve of the Milky
Way. Inside the "Solar circle", that is at
R < Ro, the radial velocity is determined
quite accurately using the tangent point
method; the measurements outside have
8 10 12
Radius (kpc)
The nature of dark matter is thus far unknown; in
principle, we can distinguish two totally different kinds
of dark matter candidates:
• Astrophysical dark matter, consisting of compact
objects - e.g., faint stars like white dwarfs, brown
dwarfs, black holes, etc. Such objects were assigned
the name MACHOs, which stands for "MAssive
Compact Halo Objects".
• Particle physics dark matter, consisting of elemen-
tary particles which have thus far escaped detection
in accelerator laboratories.
Although the origin of astrophysical dark matter would
be difficult to understand (not least because of the
baryon abundance in the Universe - see Sect. 4.4.4 - and
because of the metal abundance in the ISM), a direct
distinction between the two alternatives through ob-
servation would be of great interest. In the following
section we will describe a method which is able to
probe whether the dark matter in our Galaxy consists of
MACHOs.
2.5 The Galactic Microlensing Effect:
The Quest for Compact
Dark Matter
In 1986, Bohdan Paczyriski proposed to test the possi-
ble presence of MACHOs by performing microlensing
experiments. As we will soon see, this was a daring idea
at that time, but since then such experiments have been
carried out. In this section we will mainly summarize
and discuss the results of these searches for MACHOs.
We will start with a description of the microlensing ef-
fect and then proceed with its specific application to the
search for MACHOs.
2.5.1 The Gravitational Lensing Effect I
Einstein's Deflection Angle. Light, just like 11
/'(/nicies, is dej'leclc d in a gravitational field. 'This is one
of the specific predictions by Einstein's theory of grav-
ity, General Relativity. Quantitatively it predicts that
a light beam which passes a point mass M at a distance
£ is deflected by an angle a, which amounts to
AGM
The deflection law (2.71) is valid as long as a«l,
which is the case for weak gravitational fields. If we
now set M = M Q , R = R Q in the foregoing equation,
we obtain
I & ^i: ; 74j
for the light deflection at the limb of the Sun. This de-
flection of light was measured during a Solar eclipse in
1919 from the shift of the apparent positions of stars
2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
close to the shaded Solar disk. Its agreement with the
value predicted by Einstein made him world-famous
over night, because this was the first real and challeng-
ing test of General Relativity. Although the precision
of the measured value back then was only ~ 30%, it
was sufficient to confirm Einstein's theory. By now the
law (2.71) has been measured in the Solar System with
a precision of about 0.1%, and Einstein's prediction has
been confirmed.
Not long after the discovery of gravitational light
deflection at the Sun, the following scenario was con-
sidered. If the deflection of the light were sufficiently
strong, light from a very distant source could be vis-
ible at two positions in the sky: one light ray could
pass a mass concentration, located between us and the
source, "to the right", and the second one "to the left", as
sketched in Fig. 2.21. The astrophysical consequence of
this gravitational light deflection is also called the grav-
itational lens effect. We will discuss various aspects of
Source -^
Observer
Fig. 2.21. Sketch of a gravitational lens system. If a sufficiently
massive mass concentration is located between us and a distant
source, it may happen that we observe this source at two
different positions on the sphere
the lens effect in the course of this book, and we will
review its astrophysical applications.
The Sun is not able to cause multiple images of dis-
tant sources. The maximum deflection angle a Q is much
smaller than the angular radius of the Sun, so that two
beams of light that pass the Sun to the left and to the
righl cannot converge by light deflection at the position
of the Earth. Given its radius, the Sun is too close to pro-
duce multiple images, since its angular radius is (far)
larger than the deflection angle a©. However, the light
deflection by more distant stars (or other massive ce-
lestial bodies) can produce multiple images of sources
located behind them.
Lens Geometry. The geometry of a gravitational lens
system is depicted in Fig. 2.22. We consider light rays
from a source at distance D s from us that pass a mass
concentration (called a lens or deflector) at a separa-
tion §. The deflector is at a distance Da from us. In Fig.
2.22 y) denotes the true, two-dimensional position of the
source in the source plane, and /J is the true angular po-
sition of the source, that is the angular position at which
it would be observed in the absence of light deflection,
p = J- . (2.72)
The position of the light ray in the lens plane is denoted
by |, and is the corresponding angular position,
Hence, is the observed position of the source on the
sphere relative to the position of the "center of the lens"
which we have chosen as the origin of the coordinate
system, £ — 0. Dd s is the distance of the source plane
from the lens plane. As long as the relevant distances are
much smaller than the "radius of the Universe" c/Hq,
which is certainly the case within our Galaxy and in
the Local Group, we have Dd s — D s — Da. However,
this relation is no longer valid for cosmologically dis-
tant sources and lenses; we will come back to this in
Sect. 4.3.3.
Lens Equation. From Fig. 2.22 we can deduce the con-
dition that a light ray from the source will reach us from
the direction (or §),
D s .
(2 /3)
A
■$-Ai.&($),
(2.74)
The deflection angle a(6) depends on the mass distri-
bution of the deflector. We will discuss the deflection
angle for an arbitrary density distribution of a lens in
Sect. 3.8. Here we will first concentrate on point masses,
which is - in most cases - a good approximation for the
lensing effect on stars.
For a point mass, we get - see (2.71)
Observer
Fig. 2.22. Geometry of a gravitational lens system. Consider
a source to be located at a distance D s from us and a mass
it distance D d . An optical axis is defined that
ts the observer and the center of the mass concentration;
n will intersect the so-called source plane, a plane
perpendicular to the optical axis at the distance of the source.
Accordingly, the lens plane is the plane perpendicular lo ihc
line of sight to the mass concentration at distance Dd from
us. The intersections of the optical axis and the planes are
chosen as (he origins of the respective coordinate systems.
Let the source be at the point ij in the source plane; a light
beam ih n n lo . in i is h o 111 opli il i i ini i i I (he-
lens plane at the point % and is deflected by an angle &(%). All
these quantities are two-dimensional vectors. The condition
that the source is observed in the direction 9 is given by
the lens equation (2.74) which follows from the theorem of
intersecting lines
r, after dividing by D s and using (2.72) and (2.73):
(2.75)
-£«*»>.
Due to the factor multiplying the deflection angle in
(2.75), it is convenient to define the reduced deflection
angle
\u(9)\
D ds 4GM
or, if we account for the direction of the deflection (the
deflection angle always points towards the point mass),
4 G M D ds
c 2 D s D d W
Multiple Images of a source occur if the lens equa-
tion (2.77) has multiple solutions 0, for a (true) source
position fi - in this case, the source is observed at the
positions 0, on the sphere.
Explicit Solution of the Lens Equation for a Point
Mass. The lens equation for a point mass is simple
enough to be solved analytically which means that for
each source position ft the respective image positions 0,
can be determined. If we define the so-called Einstein
angle of the lens,
E :=
J4GM D ds
V c2 As D d
then the lens equation (2.77) for the point-mass lens
with a deflection angle (2.78) can be written as
p = 0-6l
Obviously, E is a characteristic angle in this equation,
so that for practical reasons we will use the scaling
so that the lens equation (2.75) attains the simple form Hence the lens equation simplifies t(
I p = 9-a(9) I. (2.77) y=x--^.
2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
After multiplication with a
equation, whose solut
this becomes a quadratic
From this solution of the lens equation one can
immediately draw a number of conclusions:
• For each source position y, the lens equation for
a point-mass lens has two solutions - any source is
(formally, at least) imaged twice. The reason for this
is the divergence of the deflection angle for 6^0.
This divergence does not occur in reality because of
the finite geometric extent of the lens (e.g., the radius
of the star), as the solutions are of course physically
relevant only if £ = D d 9 E \x\ is larger than the radius
of the star. We need to point out again that we ex-
plicitly exclude the case of strong gravitational fields
such as the light deflection near a black hole or a neu-
tron star, for which the equation for the deflection
angle has to be modified.
• The two images jc, are collinear with the lens and the
source. In other words, the observer, lens, and source
define a plane, and light rays from the source that
reach the observer are located in this plane as well.
One of the two images is located on the same side of
the lens as the source (jc • y > 0), the second image is
located on the other side - as is already indicated in
Fig. 2.21.
• If y = 0, so that the source is positioned exactly be-
hind the lens, the full circle |jc| = 1, or \0\ = 9 E , is
a solution of the lens equation (2.80) - the source
is seen as a circular image. In this case, the source,
lens, and observer no longer define a plane, and the
problem becomes axially symmetric. Such a circular
image is called an Einstein ring. Ring-shaped im-
ages have indeed been observed, as we will discuss
in Sect. 3.8.3.
• The angular diameter of this ring is then 20 E .
From the solution (2.81), one can easily see that
the distance between the two images is about
Ax = |*i — X2I > 2 (as long as \y\ < 1), hence
A0 > 26» E ;
the Einstein angle thus specifies the characteristic
image separation. Situations with \y\ » 1, and hence
angular separations significantly larger than 29 E , are
astrophysically of only minor relevance, as will
shown below.
Magnification: The Principle. Light beams are not
only deflected as a whole, but they are also subject
to differential deflection. For instance, those rays of
a light beam (also called light bundle) that are closer to
the lens are deflected more than rays at the other side of
the beam. The differential deflection is an effect of the
tidal component of the deflection angle; this is sketched
in Fig. 2.23. By differential deflection, the solid an-
gle which the image of the source subtends on the sky
changes. Let co s be the solid angle the source would
subtend if no lens were present, and a> the observed
solid angle of the image of the source in the presence
of a deflector. Since gravitational light deflection is not
linked to emission or absorption of radiation, the sur-
Fig. 2.23. Light beams are deflected differentially, leading to
changes of the shape and the cross-sectional area of the beam.
As a consequence, the observed solid angle subtended by the
source, as seen by the observer, is modified by gravitational
light deflection. In the example shown, the observed solid an-
gle Ai/D^ is larger than the one subtended by the undeflected
source, As/D^ - the image of the source is thus magnified
2. The Milky Way as a Galaxy
face brightness (or specific intensity) is preserved. The
flux of a source is given as the product of surface bright-
ness and solid angle. Since the former of the two factors
is unchanged by light deflection, but the solid angle
changes, the observed flux of the source is modified. If
Sq is the flux of the unlensed source and S the flux of
an image of the source, then
^ : =^=-
describes the change in flux that is caused by a magnifi-
cation (or a diminution) of the image of a source. Obvi-
ously, the magnification is a purely geometrical effect.
Magnification for "Small" Sources. For sources and
images that are much smaller than the characteristic
scale of the lens, the magnification [i is given by the
differential area distortion of the lens mapping (2.77),
--S) =Kt)
7 ^ + JS±±^\.
have magnification either larger or less than unity, de-
pending on y. The magnification of the two images is
illustrated in Fig. 2.24, while Fig. 2.25 shows the mag-
nification for several different source positions y. For
y Js> 1, one has /z+ > 1 and ^_ ~ 0, from which we
draw the following conclusion: if the source and lens
are not sufficiently well aligned, the secondary image is
strongly demagnified and the primary image has mag-
nification very close to unity. For this reason, situations
with y ;§> 1 are of little relevance since then essentially
only one image is observed which has about the same
flux as the unlensed source.
For y -> 0, the two magnifications diverge,
fx± -»■ oo. The reason for this is purely geometric: in
this case, out of a zero-dimensional point source a one-
dimensional image, the Einstein ring, is formed. This
divergence is not physical, of course, since infinite mag-
nifications do not occur in reality. The magnifications
remain finite even for y = 0, for two reasons. First, real
sources have a finite extent, and for these the magnifi-
Hence for small sources, the ratio of solid angles of the
lensed image and the unlensed source is described by
the determinant of the local Jacobi matrix. 11
The magnification can therefore be calculated for
each individual image of the source, and the total mag-
nification of a source, given by the ratio of the sum of
the fluxes of the individual images and the flux of the
unlensed source, is the sum of the magnifications for
the individual images.
Magnification for the Point-Mass Lens. For a point-
mass lens, the magnifications for the two images (2.81)
From this it follows that for the "+"-image /x + > 1 for
all source positions y = \y\, whereas the "—"-image can
1 I I i mi i ii i i i , h i 1 1 i in ii ii in one
spatial dimension to higher dimensional mappings. Consider a scalar
mapping v - vt.v): through litis mapping, a "small" interval A.v is
mapped onto a small interval Ay, where Ay ~ (dy/dx) Ax. The
Jacobian determinant occurring in (2. S3) generalizes this result for
a two-dimensional mapping from the lens plane to the source plane.
Fig. 2.24. Illustration of the lens mapping by a point mass M.
The unlensed source 5 and the two images I\ and h of the
lensed source are shown. We see that the two images have
a solid angle different from the unlensed source and they also
have a different shape. The dashed circle shows the Einstein
radius of the lens
2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
Fig. 2.25. Image of a circular source with
a radial brightness profile - indicated by
colors - for different relative positions of
the lens and source, y decreases from left to
right; in the rightmost figure y = and an
Einstein ring is formed
cation is finite. Second, even if one had a point source,
wave effects of the light (interference) would lead to
a finite value of [i. The total magnification of a point
source by a point-mass lens follows from the sum of the
magnifications (2.84),
2.5.2 Galactic Microlensing Effect
After these theoretical considerations we will now re-
turn to the starting point of our discussion, employing
the lensing effect as a potential diagnostic for dark mat-
ter in our Milky Way, if this dark matter were to consist
of compact mass concentrations, e.g., very faint stars.
Image Splitting. Considering a star in our Galaxy as
the lens, (2.79) yields the Einstein angle
Magnification. Bohdan Paczyriski pointed out in 1986
that, although image splitting was unobservable, the
magnification by the lens should nevertheless be mea-
surable. To do this, we have to realize that the absolute
magnification is observable only if the unlensed flux of
the source is known - which is not the case, of course
(for nearly all sources). However, the magnification,
and therefore also the observed flux, changes with time
by the relative motion of source, lens, and ourselves.
Therefore, the flux is a function of time, caused by the
time-dependent magnification.
Characteristic Time-Scale of the Variation. Let v be
a typical transverse velocity of the lens, then its angular
velocity is
6>= — = 4.22 mas yr" 1 ( )( — ) ,
D d J V200km/s/V10kpc/
(2.87)
if we consider the source and the observer to be at rest.
The characteristic time-scale of the variability is then
given by
9 E = 0.902 mas
' VIOkpcJ \~~dJ
Since the angular separation A0 of the two images is
about 20 E , the typical image splittings are about a mil-
liarcsecond (mas) for lens systems including Galactic
stars; such angular separations are as yet not observable
with optical telescopes. This insight made Einstein be-
lieve in 1936, after he conducted a detailed quantitative
analysis of gravitational lensing by point masses, that
the lens effect will not be observable. 12
, E :=*=0.214„(iL)" 2 (^" 2
r.-^i
\ i > 2 /
\200km/s/
~~ (2.88)
This time-scale is of the order of a month for lenses
with M ~ M Q and typical Galactic velocities. Hence,
12 The
microlens" has its origin in the angular scale (2.86)
in the context of the lens effect on quasars by stars
cosmological distance:., for which one obtain*, image splittings of
2. The Milky Way as a Galaxy
the effect is measurable in principle. In the general case
that source, lens, and observer are all moving, v has
to be considered as an effective velocity. Alternatively,
the motion of the source in the source plane can be
considered.
Light Curves. In most cases, the relative motion can be
considered linear, so that the position of the source in
the source plane can be written as
Using the scaled position y — fi/9 E , for y = | v| \
obtain
where p — y m ; n specifies the minimum distance from
the optical axis, and f max is the time at which y — p
attains this minimum value, thus when the magnification
fi — fi(p) — fi max is maximized. From this, and using
(2.85), one obtains the light curve
S(t) = Sofi(y(t)) = So
y 2 (t) + 2
y(t)Vy 2 (t) + 4
Examples for such light curves are shown in Fig. 2.26.
They depend on only four parameters: the flux of the
unlensed source So, the time of maximum magnifica-
tion f max , the smallest distance of the source from the
optical axis p, and the characteristic time-scale t E . All
these values are directly observable in a light curve. One
obtains t m . dx from the time of the maximum of the light
curve, So is the flux that is measured for very large
and small times, So — S(t -> ±oo), or So » S(t) for
\t — t m - dx \ J5> t E . Furthermore, p follows from the max-
imum magnification fi m . dx = S max /So by inversion of
(2.85), and t E from the width of the light curve.
Only t E contains information of astrophysical rele-
vance, because the time of the maximum, the unlensed
flux of the source, and the minimum separation p
provide no information about the lens. Since t E oc
*JM D A /v, this time-scale contains the combined in-
formation on the lens mass, the distances to the lens and
the source, and the transverse velocity: Only the com-
bination ti_ ex y'M {),;/>: can be derived from the light
curve, but not muss, distance, or velocity individually.
Paczyriski's idea can be expressed as follows: if the
halo of our Milky Way consists (partially) of compact
objects, a distant compact source should, from time to
time, be lensed by one of these MACHOs and thus show
characteristic changes in flux, corresponding to a light
curve similar to those in Fig. 2.26. The number density
of MACHOs is proportional to the probability or abun-
dance of lens events, and the characteristic mass of the
MACHOs is proportional to the square of the typical
variation time-scale t E . All one has to do is measure
the light curves of a sufficiently large number of back-
ground sources and extract all lens events from those
light curves to obtain information on the population of
potential MACHOs in the halo. A given halo model
predicts the spatial density distribution and the distribu-
tion of velocities of the MACHOs and can therefore be
compared to the observations in a statistical way. How-
ever, one faces the problem that the abundance of such
lensing events is very small.
Probability of a Lens Event. In practice, a system of
a foreground object and a background source is con-
sidered a lens system if p < 1 and hence /x max > 3/V5
s» 1.34, i.e., if the relative trajectory of the source passes
within the Einstein circle of the lens.
If the dark halo of the Milky Way consisted solely of
MACHOs, the probability that a very distant source is
lensed (in the sense of |/J| < 6 E ) would be ~ 10~ 7 , where
the exact value depends on the direction to the source.
At any one time, one of ~ 10 7 distant sources would be
located within the Einstein radius of a MACHO in our
halo. The immediate consequence of this is that the light
curves of millions of sources have to be monitored to
detect this effect. Furthermore, these sources have to be
located within a relatively small region on the sphere to
keep the total solid angle that has to be photometrically
monitored relatively small. This condition is needed to
limit the required observing time, so that many such
sources should be present within the field-of-view of
the camera used. The stars of the Magellanic Clouds
are well suited for such an experiment: they are close
together on the sphere, but can still be resolved into
individual stars.
Problems, and their Solution. From this observational
strategy, a large number of problems arise immediately;
they were discussed in Paczyhski's original paper. First,
2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
...ooooCpooooaoooooooooooo..,.
Fig. 2.26. Illustration of a Galactic microlensing event: In the
upper left panel a source (depicted by the open circles) moves
behind a point mass lens: for each source position two images
of the source are formed, which are indicated by the black
ellipses. The identification of the corresponding image pah
with the source position follows from the fact that, in pro-
jection, the source, the lens, and the two images are located
on a straight line, which is indicated for one source position.
The dashed circle represents the Einstein ring. In the lower left
panel, different trajectories of the source are shown, each chat
acterized by the smallest projected separation p to the lens.
The light curves resulting from these relative motions, which
can be calculated using equation (2.90), are then shown in the
right hand panel for different values of p. The smaller p is,
the larger the maximum magnification will be, here measured
in magnitudes
the photometry of so many sources over many epochs
produces a huge amount of data that need to be handled;
they have to be stored and reduced. Second, one has the
problem of "crowding": the stars in the Magellanic
Clouds are densely packed on the sky, which renders
the photometry of individual stars difficult. Third, stars
also show intrinsic variability - about 1% of all stars
are variable. This intrinsic variability has to be distin-
guished from that due to the lens effect. Due to the
small abundance of the latter, selecting the lens events
is comparable to searching for a needle in a haystack.
Finally, it should be mentioned that one has to ensure
that the experiment is indeed sensitive enough to detect
lens events. A "calibration experiment" would therefore
be desirable.
Faced with these problems, it seemed daring to seri-
ously think about the realization of such an observing
program. However, a fortunate event helped, in the mag-
2. The Milky Way as a Galaxy
nificent time of the easing of tension between the US
and the Soviet Union, and their respective allies, at the
end of the 1980s. Physicists and astrophysicists, partly
occupied with issues concerning national security, then
saw an opportunity to meet new challenges. In addition,
scientists in national laboratories had much better ac-
cess to sufficient computing power and storage capacity
than those in other research institutes, attenuating some
of the aforementioned problems. While the expected
data volume was still a major problem in 1986, it could
be handled a few years later. Also, wide-field cameras
were constructed, with which large areas of the sky
could be observed simultaneously. Software was devel-
oped which specializes in the photometry of objects in
crowded fields, so that light curves could be measured
even if individual stars in the image were no longer
cleanly separated.
To distinguish between lensing events and intrin-
sic variablity of stars, we note that the microlensing
lighl curves have a characteristic shape that is described
by only four parameters. The light curves should be
symmetric and achromatic because gravitational light
deflection is independent of the frequency of the radia-
tion. Furthermore, due to the small lensing probability,
any source should experience at most one microlens-
ing event and show a constant flux before and after,
whereas intrinsic variations of stars are often periodic
and in nearly all cases chromatic.
And finally a control experiment could be performed:
the lensing probability in the direction of the Galactic
bulge is known, or at least, we can obtain a lower limit
for it from the observed density of stars in the disk. If
a microlens experiment is carried out in the dir
of the Galactic bulge, we have to find lens e
experiment is sufficiently s
2.5.3 Surveys and Results
In the early 1990s, two collaborations (MACHO and
EROS) began the search for microlensing events to-
wards the Magellanic clouds. Another group (OGLE)
started searching in an area of the Galactic bulge. Fields
in the respective survey regions were observed regu-
larly, typically once every night if weather conditions
permitted. From the photometry of the stars in the fields,
lighl curves for many millions of stars were generated
and then checked for microlensing events.
First Detections. In 1993, all three groups reported their
first results. The MACHO collaboration found one event
in the Large Magellanic Cloud (LMC), the EROS group
two events, and the OGLE group observed one event in
the bulge. The light curve of the first MACHO event
is plotted in Fig. 2.27. It was observed in two different
filters, and the fit to the data, which corresponds to
a standard light curve (2.90), is the same for both filters,
proving that the event is achromatic. Together with the
quality of the fit to the data, this is very strong evidence
for the microlensing nature of the event.
Statistical Results. In the years since 1993, all three
aforementioned teams have proceeded with their ob-
daysfrom 2 Jan 1992
Fig. 2.27. Light curve of the first observed microlensing evenl
in the LMC, in two broad-band filters. The solid curve is
the best-fit microlens lighl curve as described by (2.90), with
Umax = 6.86. The ratio of the magnifications in both filters is
displayed at the bottom, and it is compatible with 1. Some of
the data points deviate significantly from the curve; this means
that either the errors in the measurements were underesti-
mated, or this event is more complicated than one described
by a point-mass lens - see Sect. 2.5.4
2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
servations and analysis (Fig. 2.28), and more groups
have begun the search for microlensing events, choos-
ing various lines-of-sight. The most important results
from these experiments can be summarized as follows:
About 20 events have been found in the direction
of the Magellanic Clouds, and of the order a thousand
in the direction of the bulge. The statistical analysis of
the data revealed the lensing probability towards the
bulge to be higher than originally expected. This can
be explained by the fact that our Galaxy features a bar
(see Chap. 3). This bar was also observed in IR maps
such as those made by the COBE satellite. The events
in the direction of the bulge are dominated by lenses
that are part of the bulge themselves, and their column
density is increased by the bar-like shape of the bulge.
On the other hand, the lens probability in the direction of
the Magellanic Clouds is smaller than expected for the
case where the dark halo consists solely of MACHOs.
Based on the analysis of the MACHO collaboration,
the observed statistics of lensing events towards the
Magellanic Clouds is best explained if about 20% of the
halo mass consists of MACHOs, with a characteristic
mass of about M ~ 0.5M o (see Fig. 2.29).
Interpretation and Discussion. This latter result is not
easy to interpret and came as a real surprise. If a result
compatible with ~ 100% had been found, it would have
been obvious to conclude that the dark matter in our
Milky Way consists of compact objects. Otherwise, if
very few lensing events had been found, it would have
been clear that MACHOs do not contribute significantly
to the dark matter. But a value of 20% does not allow
any unambiguous interpretation. Taken at face value,
the result from the MACHO group would imply that the
total mass of MACHOs in the Milky Way halo is about
the same as that in the stellar disk.
Furthermore, the estimated mass scale is hard to un-
derstand: what could be the nature of MACHOs with
M = 0.5M o ? Normal stars can be excluded, because
they would be far too luminous not to be observed.
White dwarfs are also unsuitable candidates, because to
Fig. 2.28. In this 8= x 8'"' image of the LMC, 30 fields are
marked in red which the MACHO group has searched for mi-
crolensing events during the ~ 5.5 years of their experiment;
images were taken in two filters to test for achromalicin . The
positions of 17 microlens c\ ents are marked by yellow circles;
these have been subject to statistical analysis
Fig. 2.29. Probability contours for a specific halo model as
a function of the characteristic MACHO mass M (here denoted
by m) and the mass fraction / of MACHOs in the halo. The
halo of the LMC was either taken into account as an additional
source for microlenses (lmc halo) or not (no lmc halo), and two
different selection criteria (A,B) for the statistically complete
microlensing sample have been used. In all cases, M ~ 0.5Mq
and / ~ 0.2 are the best-fit values
2. The Milky Way as a Galaxy
produce such a large number of white dwarfs as a fi-
nal stage of stellar evolution, the total star formation
in our Milky Way, integrated over its lifetime, needs
to be significantly larger than normally assumed. In
this case, many more massive stars would also have
formed, which would then have released the metals they
produced into the ISM, both by stellar winds and in su-
pernova explosions. In such a scenario, the metal content
of the ISM would therefore be distinctly higher than is
actually observed. The only possibility of escaping this
argument is with the hypothesis that the mass function
of newly formed stars (the initial mass function, IMF)
was different in the early phase of the Milky Way com-
pared to that observed today. The IMF that needs to be
assumed in this case is such that for each star of interme-
diate mass which evolves into a white dwarf, far fewer
high-mass stars, responsible for the metal enrichment
of the ISM, must have formed in the past compared
to today. However, we lack a plausible physical model
for such a scenario, and it is in conflict with the star-
formation history that we observe in the high-redshift
Universe (see Chap. 9).
Neutron stars can be excluded as well, because
they are too massive (typically > 1M Q ); in addition,
they are formed in supernova explosions, implying that
the aforementioned metallicity problem would be even
greater for neutron stars. Would stellar-mass black holes
be an alternative? The answer to this question depends
on how they are formed. They could not originate in SN
explosions, again because of the metallicity problem.
If they had formed in a very early phase of the Uni-
verse (they are then called primordial black holes), this
would be an imaginable, though perhaps quite exotic,
alternative.
However, we have strong indications that the
interpretation of the MACHO results is not as straight-
forward as described above. Some doubts have been
raised as to whether all events reported as being due to
microlensing are in fact caused by this effect. In fact,
one of the microlensing source stars identified by the
MACHO group showed another bump seven years after
the first event. Given the extremely small likelihood of
two microlensing events happening to a single source
this is almost certainly a star with unusual variability.
As argued previously, by means of t E we only mea-
sure a combination of lens mass, transverse velocity,
and distance. The result given in Fig. 2.29 is therefore
based on the statistical analysis of the lensing events
in the framework of a halo model that describes the
shape and the radial density profile of the halo. How-
ever, microlensing events have been observed for which
more than just t E can be determined - e.g., events in
which the lens is a binary star, or those for which t E
is larger than a few months. In this case, the orbit of
the Earth around the Sun, which is not a linear motion,
has a noticeable effect, causing deviations from the
standard curve. Such parallax events have indeed been
observed. 13 Three events are known in the direction of
the Magellanic Clouds in which more than just t E could
be measured. In all three cases the lenses are most likely
located in the Magellanic Clouds themselves (an effect
called self-lensing) and not in the halo of the Milky
Way. If for those three cases, where the degeneracy be-
tween lens mass, distance, and transverse velocity can
be broken, the respective lenses are not MACHOs in the
Galactic halo, we might then suspect that in most of the
other microlensing events the lens is not a MACHO ei-
ther. Therefore, it is currently unclear how to interpret
the results of the microlensing surveys. In particular, it
is unclear to what extent self-lensing contributes to the
results. Furthermore, the quantitative results depend on
the halo model.
The EROS collaboration used an observation strategy
which was sightly different from that of the MACHO
group, by observing a number of fields in very short time
intervals. Since the duration of a lensing event depends
on the mass of the lens as At oc M x/1 - see (2.88) - they
were also able to probe very small MACHO masses.
The absence of lensing events of very short duration
then allowed them to derive limits for the mass fraction
of such low-mass MACHOs, as is shown in Fig. 2.30.
Despite this unsettled situation concerning the inter-
pretation of the MACHO results, we have to emphasize
that the microlensing surveys have been enormously
successful experiments because they accomplished
exactly what was expected at the beginning of the ob-
servations. They measured the lensing probability in
the direction of the Magellanic Clouds and the Galactic
bulge. The fact that the distribution of the lenses differs
from that expected by no means diminishes the success
of these surveys.
2.5 The Galactic Microlensing Effect: The Quest for Compact Dark Matter
Fig. 2.30. From observations by the EROS collaboration,
a large mass range for MACHO candidates can be excluded.
The maximum allowed fraction of the halo mass contained
in MACHOs is plotted as a function of the MACHO mass
M, as an upper limit with 95% confidence. A standard model
for the mass distribution in the Galactic halo was assumed
which describes the rotation curve of the Milky Way quite
well. The various curves show different phases of the EROS
experiment. They are plotted separately for observations in
10 2
the deflector M
the directions of the LMC and the SMC. The experiment
EROS 1 searched for microlensing events on short time-
scales but did not find any; this yields the upper limits at
small masses. Upper limits al larger masses were obtained
by the EROS 2 experiment. The thick solid curve represents
the upper limit derived from combining the individual ex-
periments, [f nol a single MACHO event had been found
the upper limit would have been described by the dotted
2.5.4 Variations and Extensions
Besides the search for MACHOs, microlensing surveys
have yielded other important results and will continue to
do so in the future. For instance, the distribution of stars
in the Galaxy can be measured by analyzing the lens-
ing probability as a function of direction. Thousands
of variable stars have been newly discovered and accu-
rately monitored; the extensive and publicly accessible
databases of the surveys form an invaluable resource
for stellar astrophysics. Furthermore, globular clusters
in the LMC have been identified from these photometric
observations.
For some lensing events, the radius and the surface
structure of distant stars can be measured with very high
precision. This is possible because the magnification fi
depends on the position of the source. Situations can oc-
cur, for example where a binary star acts as a lens (see
Fig. 2.31), in which the dependence of the magnifica-
tion on the position in the source plane is very sensitive.
Since the source - the star - is in motion relative to the
line-of-sight between Earth and the lens, its different
regions are subject to different magnification, depend-
ing on the time-dependent source position. A detailed
analysis of the light curve of such events then enables
us to reconstruct the light distribution on the surface of
the star. The light curve of one such event is shown in
Fig. 2.32.
For these lensing events the source can no longer
be assumed to be a point source. Rather, the details of
the light curve are determined by its light distribution.
Therefore, another length-scale appears in the system,
the radius of the star. This length-scale shows up in the
corresponding microlensing light curve, as can be seen
in Fig. 2.32, by the time-scale which characterizes the
width of the peaks in the light curve - it is directly re-
lated to the ratio of the stellar radius and the transverse
velocity of the lens. With this new scale, the degener-
acy between M, v, and Da is partially broken, so that
these special events provide more information than the
"classical" ones.
In fact, the light curve in Fig. 2.27 is probably not
caused by a single lens star, but instead by additional
slight disturbances from a companion star. This would
2. The Milky Way as a Galaxy
Fig. 2.31. If a binary star acts as a lens, significantly more com-
plicaied light curves can be generated. In the left-hand panel
tracks are plotted for five different relative motions of a back-
ground source; the dashed curve is the so-called critical curve.
formally defined by det(3/?/30) = 0, and the solid line is die
corresponding image of the critical curve in the source plat
called a
. Light c
tracks arc plotted in the right-hand panel. If the source crosses
die caustic, die magnification \jl becomes very large - formally
infinite if the source was point-like. Since it has a finite extent,
/x has to be finite as well; from the maximum fj. during caus-
tic crossing, the radius of the source can be determined, and
of the surface brightness across
s corresponding to these five the stellar disk, an effect known as limb darkening
explain the deviation of the observed light curve from
a simple model light curve. However, the sampling in
time of this particular light curve is not sufficient to
determine the parameters of the binary system.
By now, detailed light curves with very good time
coverage have been measured, which was made possi-
ble with an alarm system. The data from those groups
searching for microlensing events are analyzed imme-
diately after observations, and potential candidates for
interesting events are published on the Internet. Other
groups (such as the PLANET collaboration, for ex-
ample) then follow-up these systems with very good
time coverage by using several telescopes spread over
a large range in geographical longitude. This makes
around-the-clock observations of the event possible.
Using this method light curves of extremely high qual-
ity have been measured. These groups hope to detect
extra-solar planets by characteristic deviations in these
light curves. Indeed, these microlens observations may
be the most realistic (and cheapest) option for finding
low-mass planets. Other methods for finding extra-solar
planets, such as the search for small periodic changes
of the radial velocity of stars which is caused by the
gravitational pull of their orbiting planet, are mostly
sensitive to high-mass planets. Whereas such surveys
Days since UT31.0 May
20 3D
970 980 990 1000
Julian Date -2-150000.
Fig. 2.32. Light curve of an event in which the lens was a bi-
nary star. The MACHO group discovered this "binary event".
Members of the PLANET collaboration obtained this data us-
ing four different telescopes (in Chile, Tasmania, and South
Africa). The second caustic crossing is highly resolved (dis-
played in the small diagram I and allows us to draw conclusions
about the size and the brightness distribution of the source star.
The two curves show the fits of a binary lens to the data
2.6 The Galactic Center
have been extremely successful in the past decade, hav-
ing detected far more than one hundred planets around
other stars, the characteristic mass of these planets is
that of Jupiter, i.e., ~ 1000 times more massive than
the Earth. At least two planet-mass companions to lens
stars have already been discovered through microlens-
ing, one of them having a mass of only six times that of
the Earth.
Pixel Lensing. An extension to the microlensing search
was suggested in the form of the so-called pixel lensing
method. Instead of measuring the light curve of a sin-
gle star one records the brightness of groups of stars
that are positioned closely together on the sky. This
method is applicable in situations where the density of
source stars is very high, such as for the stars in the
Andromeda galaxy (M31), which cannot be resolved
individually. If one star is magnified by a microlens-
ing event, the brightness in the corresponding region
changes in a characteristic way, similar to that in the
lensing events discussed above. To identify such events,
the magnification needs to be relatively large, because
only then can the light of the lensed star dominate over
the local brightness in the region, so that the event can be
recognized. On the other hand, the number of photomet-
rically monitored stars (per solid angle) is larger than in
surveys where single stars are observed, so that events of
larger magnification are also more abundant. By now,
several groups have successfully started to search for
microlensing events in M31. The quantitative analy-
sis of these surveys is more complicated than for the
surveys targeting the Magellanic Clouds. However, the
M31 experiments are equally sensitive to both MA-
CHOs in the halo of our Milky Way and in that of M3 1 .
Therefore, these surveys promise to finally resolve the
question of whether part of the dark matter consists of
MACHOs.
Annihilation Radiation due to Dark Matter? The
5 1 1 keV annihilation radiation from the Galactic bulge,
discussed in Sect. 2.3.4 above, has been suggested to
be related to dark matter particles. Depending on the
density of dark matter in the center of the Galaxy, as
well as on the cross-section of the constituent particles
of the dark matter (if it is indeed due to elementary
particles), these particles can annihilate. In this process,
positrons might be released which can then annihilate
with the electrons of the interstellar medium. However,
in order for this to be the source of the 511 keV line
radiation, the dark matter particles must have rather
"exotic" properties.
2.6 The Galactic Center
The Galactic center (GC, see Fig. 2.33) is not observable
at optical wavelengths, because the extinction in the V-
band is ~ 28 mag. Our information about the GC has
Fig. 2.33. Optical image in the direction of the Galactic center.
■Marked are some Messier objects: gas nebulae such as M8,
M16, M17, M20; open star clusters such as M6, M7, M18,
M2 1 . M23, M24, and M25; globular clusters such as M9, M22,
M28, M54, M69, and M70. Also marked is the Galactic center,
as well as the Galactic plane, which is indicated by a line.
Baade":. Window can be easily recognized, a direction in which
the extinction is significantly lower than in nearby directions,
so that a clear increase in stellar density is visible there. This
is the reason why the microlensing observations towards the
Galactic center were preferably done in Baade's Window
2. The Milky Way as a Galaxy
been obtained from radio-, IR-, and X-ray radiation.
Since the GC is nearby, and thus serves as a prototype
of the central regions of galaxies, its observation is of
great interest for our understanding of the processes
taking place in the centers of galaxies.
2.6.1 Where is the Galactic Center?
The question of where the center of our Milky Way is
located is by no means trivial, because the term "center"
is in fact not well-defined. Is it the center of mass of the
Galaxy, or the point around which the stars and the gas
are orbiting? And how could we pinpoint this "center"
accurately? Fortunately, the center can nevertheless be
localized because, as we will see below, a distinct source
exists that is readily identified as the Galactic center.
Radio observations in the direction of the GC show
a relatively complex structure, as is displayed in Fig.
2.34.
A central disk of Hi gas exists at radii from several
100 pc up to about 1 kpc. Its rotational velocity yields
a mass estimate M(R) for R > 100 pc. Furthermore,
radio filaments are observed which extend perpendicu-
larly to the Galactic plane, and also a large number of
supernova remnants are seen. Within about 2 kpc from
the center, roughly 3 x 1O 7 M of atomic hydrogen is
found. Optical images show regions close to the GC
towards which the extinction is significantly lower. The
best known of these is Baade's window - most of the
microlensing surveys towards the bulge are conducted
in this region. In addition, a fairly large number of glob-
ular dusters and gas nebulae are observed towards the
central region. X-ra\ imagi (Fig 2.35) show numerous
X-ray binaries, as well as diffuse emission by hot gas.
The innermost 8 pc contain the radio source Sgr A
(Sagittarius A), which itself consists of different
components:
• A circumnuclear molecular ring, shaped like a torus,
which extends between radii of 2 pc < R < 8 pc and
is inclined by about 20° relative to the Galactic
disk. The rotational velocity of this ring is about
~ 1 10 km/s, nearly independent of R. This ring has
a sharp inner boundary; this cannot be the result of an
equilibrium flow, because internal turbulent motions
would quickly (on a time-scale of ~ 10 5 yr) erase
this boundary. Probably, it is evidence of an ener-
getic event that occurred in the Galactic center within
the past ~ 10 5 years. This interpretation is also sup-
ported by other observations, e.g., by a dumpiness
in density and temperature.
• Sgr A East, a non-thermal (synchrotron) source of
shell-like structure. Presumably this is a supernova
remnant (SNR), with an age between 100 and 5000
years.
• Sgr A West is located about 1.'5 away from Sgr
A East. It is a thermal source, an unusual Hll region
with a spiral-like structure.
• Sgr A* is a strong compact radio source close to the
center of Sgr A West. Recent observations with mm-
VLBI show that its extent is smaller than 3 AU. The
radio luminosity is L rac | ~ 2 x 10 34 erg/s. Except for
the emission in the mm and cm domain, Sgr A* is
a weak source. Since other galaxies often have a com-
pact radio source in their center, Sgr A* is an excellent
candidate for being the center of our Milky Way.
Through observations of stars which contain a radio
maser 14 source, the astrometry of the GC in the radio
domain was matched to that in the IR, i.e., the position
of Sgr A* is also known in the IR. 15 The uncertainty in
the relative positions between radio and IR observations
is only ~ 30 mas - at a presumed distance of the GC of
8 kpc, one arcsecond corresponds to 0.0388 pc, or about
8000 AU.
2.6.2 The Central Star Cluster
Density Distribution. Observations in the K-band
(X ~ 2 |xm) show a compact star cluster that is cen-
tered on Sgr A*. Its density behaves like «r u in the
distance range 0. 1 pc < r < 1 pc. The number density
14 Masers are regions of stimulated non thermal emission which show
;i very high surface brightness. The maser phenomenon is similar to
that of lasers, except that the former radiate in the microwave regime
of the spectrum. Maser. are sometimes found in the atmospheres of
15 One problem in the combined analysis of data taken in different
wavelength bands is that astrometry in each individual wavelength
band can be performed with a very high precision - e.g.. individu
ally in the radio and the IK band - however, the relative astrometry
between these bands is less well known. To stack maps of different
I i I i i I i i i i I
alive astrometry is essential. Tins can be gained if a population of
compact source', exists that is observable in both wavelength domains
and for which accurate positions can be measured.
Fig. 2.34. Left: A VLA wide-field image of the region around where the red dot marks Sgr A*. Center right: Sgr A West,
the Galactic center, with a large number of sources identified. as seen in a 6-cm continuum VLA image. Lower right: the
Upper right: a 20 cm continuum VLA image of Sgr A East, circumnuclear ring in HCN line ei
of stars in its inner region is so large that close stellar
encounters are common. It can be estimated that a star
has a close encounter about every ~ 10 6 years. Thus, it
is expected that the distribution of the stars is "thermal-
ized", which means that the local velocity distribution
of the stars is the same everywhere, i.e., it is close to
a Maxwellian distribution with a constant velocity dis-
persion. For such an isothermal distribution we expect
a density profile n oc r~ 2 , which is in good agreement
with the observation.
However, another observational result yields a strik-
ing and interesting discrepancy with respect to the idea
of an isothermal distribution. Instead of the expected
constant dispersion a of the radial velocities of the stars,
a strong radial dependence is observed: a increases to-
wards smaller r. For example, one finds a ~ 55 km/s at
Fig. 2.35. Mosaic of X-ray images of the Galactic center, taken
by the Chandra satellite. The image covers an area of about
130 pc x 300 pc (48' x 120'). The actual GC, in which a su-
permassive black hole is suspected to reside, is located in
the white region near the center of the image. Furthermore,
on this image hundreds of white dwarfs, neutron stars, and
black holes are visible that radiate in the X-ray regime due
to accretion phenomena (accreting X-ray binaries). Colors
code the photon energy, from low energy (red) to high energy
(blue). The diffuse emission, predominantly red in this im-
age, originates in diffuse hot gas with a temperature of about
T ~ 10 7 K
r = 5pc, buter ~ 180 km/s at r — 0.15 pc. This discrep-
ancy indicates that the gravitational potential in which
the stars are moving is generated not only by themselves.
According to the virial theorem, the strong increase of
a for small r implies the presence of a central mass
n the star cluster.
Proper Motions. Since the middle of the 1990s, proper
motions of stars in this star cluster have also been mea-
sured, using the methods of speckle interferometry and
adaptive optics. These produce images at diffraction-
limited angular resolution, about ~ 0'.'15 in the K-band
at the ESO/NTT (3.5 m) and about ~ 0'.'05 at the Keck
(10 m). Proper motions are currently known for about
1000 stars within ~ 10" of Sgr A*. This breakthrough
was achieved independently by two groups, whose re-
sults are in excellent agreement. For more than 20 stars
within ~ 5" of Sgr A* both proper motions and radial ve-
locities, and therefore their three-dimensional velocities
are known. The radial and tangential velocity disper-
sions resulting from these measurements are in good
mutual agreement. Thus, it can be concluded that a ba-
sically isotropic distribution of the stellar orbits exists,
simplifying the study of the dynamics of this stellar
cluster.
The Origin of Very Massive Stars near the Galactic
Center. One of the unsolved problems is the presence
of these massive stars close to the Galactic center. One
finds that most of the innermost stars are main-sequence
B-stars. Their small lifetime of ~ 10 8 yr probably im-
plies that these stars were born close to the Galactic
center. This, however, is very difficult to understand.
Both the strong tidal gravitational field of the central
black hole (see below) and the presumably strong mag-
netic field in this region will prevent the "standard"
star-formation picture of a collapsing molecular cloud:
the former effect tends to disrupt such a cloud while
the latter stabilizes it against gravitational contraction.
Several solutions to this problem have been suggested,
such as a scenario in which the stars are born at larger
distances from the Galactic center and then brought
there by dynamical processes, involving strong gravita-
tional scattering events. However, none of these models
appears satisfactory at present.
2.6.3 A Black Hole in the Center of the Milky Way
Some stars within 0"6 of Sgr A* have a proper motion
of more than 1000 km/s, as shown in Fig. 2.36. For
instance, the star S 1 has a separation of only 0'.' 1 from
Sgr A* and shows proper motion of 1470 km/s at the
epoch displayed in Fig. 2.36. Combining the velocity
dispersions in radial and tangential directions reveals
it to be increasing according to the Kepler law for the
presence of a point mass, a oc r~ 1/2 down tor ~ 0.01 pc.
By now, the acceleration of some stars in the star
cluster has also been measured, i.e., the change of proper
2.6 The Galactic Center
Fig. 2.36. Proper motions of stars in the
central region of the GC. The differently
colored arrows denote different types of
stars. The small image shows the proper
motions in the Sgr A* star cluster within half
an arcsecond from Sgr A*; the fastest star
(SI) has a proper motion of ~ 1500 km/s
(from Genzel, 2000, astro-ph/0008 1 19
motion with time. From these measurements Sgr A* in-
deed emerges as the focus of the orbits and thus as the
center of mass. Figure 2.37 shows the orbits of some
stars around Sgr A*. The star S2 could be observed
during a major fraction of its orbit, where a maximum
velocity of more than 5000 km/s was found. The eccen-
tricity of the orbit of S2 is 0.87, and its orbital period
is ~ 15.7 yr. The minimum separation of this star from
Sgr A* is only 6 x 10" 4 pc, or about 100 AU.
From the observed kinematics, the enclosed mass
M(r) can be calculated, see Fig. 2.38. The correspond-
ing analysis yields that M(r) is basically constant over
the range 0.01 pc < r < 0.5 pc. This exciting result
clearly indicates the presence of a point mass, for which
a mass of
M=(3.6±0.4)x 10 b M o (2.91)
is determined. For larger radii, the mass of the star clus-
ter dominates; it nearly follows an isothermal density
distribution with a core radius of ~ 0.34 pc and a central
density of 3.6 x 10 6 M G /pc 3 . This result is also compat-
ible with the kinematics of the gas in the center of the
Galaxy. However, stars are much better kinematic indi-
cators because gas can be affected by magnetic fields,
viscosity, and various other processes besides gravity.
The kinematics of stars in the central star cluster
of the Galaxy shows that our Milky Way contains
a mass concentration in which ~ 3 x 10 6 M o are
concentrated within a region smaller than 0.01 pc.
This is most probably a black hole in the center
of our Galaxy at the position of the compact radio
source Sgr A*.
Why a Black Hole? We have interpreted the central
mass concentration as a black hole; this requires some
further explanation:
• The energy for the central activity in quasars, radio
galaxies, and other AGNs is produced by accretion of
gas onto a supermassive black hole (SMBH); we will
discuss this in more detail in Sect. 5.3. Thus we know
that at least a subclass of galaxies contains a central
SMBH. Furthermore, we will see in Sect. 3.5 that
many "normal" galaxies, especially ellipticals, har-
bor a black hole in their center. The presence of
a black hole in the center of our own Galaxy would
therefore not be something unusual.
• To bring the radial mass profile M(r) into accordance
with an extended mass distribution, its density dis-
tribution must be very strongly concentrated, with
a density profile steeper than ex r~ 4 ; otherwise the
2. The Milky Way as a Galaxy
Fig. 2.37. At left, the or-
bit of the star S2 around
Sgr A* is shown as de-
termined by two different
observing campaigns. The
position of Sgr A* is
indicated by the black cir-
cled cross. The individual
points along the orbit are
identified by the epoch
of the observation. The
right-hand image shows
the orbits of some other
stars for which accelera-
tions have already been
measured
0.1 0.05 -0.05 -0.1
Right Ascension ["]
Right Ascension ["]
mass profile M(r) would not be as flat as observed in
Fig. 2.38. Hence, this hypothetical mass distribution
must be vastly different from the expected isother-
mal distribution which has a mass profile oc r~ 2 , as
discussed in Sect. 2.6.2. However, observations of
the stellar distribution provide no indication of an
inwardly increasing density of the star cluster with
such a steep profile.
• Even if such an ultra-dense star cluster (with a central
density of > 4 x 10 12 M G /pc 3 ) were present it could
not be stable, but instead would dissolve within ~ 10 7
years through frequent stellar collisions.
• Sgr A* itself has a proper motion of less than
20km/s. It is therefore the dynamic center of
the Milky Way. Due to the large velocities of its
surrounding stars, one would derive a mass of
M ^> 10 3 M o for the radio source, assuming equipar-
tition of energy (see also Sect. 2.6.5). Together with
the tight upper limits for its extent, a lower limit for
the density of 10 18 M o /pc 3 can then be obtained.
Following the stellar orbits in forthcoming years will
further complete our picture of the mass distribution in
the GC.
We have to emphasize at this point that the gravi-
tational effect of the black hole on the motion of stars
and gas is constrained to the innermost region of the
Milky Way. As one can see from Fig. 2.38, the gravita-
tional field of the SMBH dominates the rotation curve
of the Galaxy only for R < 2 pc - this is the very reason
why the detection of the SMBH is so difficult. At larger
radii, the presence of the SMBH is of no relevance for
the rotation curve of the Milky Way.
2.6.4 Flares from the Galactic Center
In 2000, the X-ray satellite Chandra discovered a power-
ful X-ray flare from Sgr A*. This event lasted for about
three hours, and the X-ray flux increased by a factor
of 50 during this period. XMM-Newton confirmed the
existence of X-ray flares, recording one where the lu-
minosity increased by a factor of ~ 200. Combining the
flare duration of a few hours with the short time-scale
of variability of a few minutes indicates that the emis-
sion must originate from a very small source, not larger
than ~ 10 13 cm in size.
Monitoring Sgr A* in the NIR, flare emission was
also found in this spectral regime. These NIR flares
are more frequent than in X-rays, occurring several
times per day. Furthermore, the NIR emission seems
to show some sort of periodicity of ~ 17 min, which is
most likely to be identified with an orbital motion of
the emitting material around the SMBH. Indeed, a re-
analysis of the X-ray light curve shows some hint of
2.6 The Galactic Center
radius (light hours)
10 2 10 3 10 4
0.001 0.01
radius (parsec)
Fig. 2.38. Determination of the mass M(f) within a radius r
from Sgr A*, as measured by the radial \ elocities and proper
motions of stars in the central cluster. Mass estimates obtained
from individual stars (S14, S2, S12) are given by the points
with error bars for small r. The other data points were derived
from the kinematic analysis of the observed proper motions
of the stars, where different methods have been applied. As
can be seen, these methods produce results that are mutu-
ally compatible, so that the mass profile plotted here can be
regarded to be robust. The solid curve is the best-fit model,
representing a point mass of 2.9 x 10 6 M Q plus a star cluster
with a central density of 3.6 x 10 6 M©/pc 3 (the mass profile
of this star cluster is indicated by the dash-dotted curve). The
dashed curve shows the mass profile of a hypothetical clus-
ter with a very steep profile, n oc r~ 5 , and a central density of
2.2 x 10 17 M o pc" 3
the same modulation time-scale. Observing the Galac-
tic center simultaneously in the NIR and the X-rays
revealed a clear correlation of the corresponding light
curves; for example, simultaneous flares were found in
these two wavelength regimes. These flares have sim-
ilar light profiles, indicating a similar origin of their
radiation. The consequences of these observations for
the nature of the central black hole will be discussed
in Sect. 5.4.6, after we have introduced the concept of
black holes in a bit more detail. Flares were also ob-
served at mm-wavelengths; their time-scale appears to
be longer than that at higher frequencies, as expected
if the emission comes from a more extended source
component.
2.6.5 The Proper Motion of Sgr A*
From a series of VLBI observations of the position of
Sgr A*, covering eight years, the proper motion of this
compact radio source was measured with very high pre-
cision. To do this, the position of Sgr A* was determined
relative to two compact extragalactic radio sources. Due
to their large distances these are not expected to show
2. The Milky Way as a Galaxy
any proper motion, and the VLBI measurements show
that their separation vector is indeed constant over time.
The position of Sgr A* over the observing period is
plotted in Fig. 2.39.
From the plot, we can conclude that the observed
proper motion of Sgr A* is essentially parallel to the
Galactic plane. The proper motion perpendicular to the
Galactic plane is about 0.2mas/yr, compared to the
proper motion in the Galactic plane of 6.4 mas/yr. If
R = (8.0 ±0.5) kpc is assumed for the distance to the
GC, this proper motion translates into an orbital velocity
of (241 ± 15) km/s, where the uncertainty is dominated
by the exact value of Ro (the error in the measurement
alone would yield an uncertainty of only 1 km/s). This
proper motion is easily explained by the Solar orbital
East Offset (mas)
Fig. 2.39. The position of Sgr A* at different epochs, relath e
to the position in 1996. To a very good approximation the
motion is linear, as indicated by the dashed best-fit straight
line. In comparison, the solid line shows the orientation of the
Galactic plane
motion around the GC, i.e., i
no hint of a non-zero velocity of the radio source Sgr A*
itself. In fact, the small deviation of the proper mo-
tion from the orientation of the Galactic plane can be
explained by the peculiar velocity of the Sun relative
to the LSR (see Sect. 2.4.1). If this is taken into ac-
count, a velocity perpendicular to the Galactic disk of
v± = (-0.4 ±0.9) km/s is obtained for Sgr A*. The
component of the orbital velocity within the disk has
a much larger uncertainty because we know neither Rq
nor the rotational velocity V °f the LSR very precisely.
The small upper limit for v±_ suggests, however, that the
motion in the disk should also be very small. Under the
(therefore plausible) assumption that Sgr A* has no pe-
culiar velocity, the ratio Rq/Vq can be determined from
these measurements with an as yet unmatched precision.
What also makes this observation so impressive is
that from it we can directly derive a lower limit for the
mass of Sgr A* . Since this radio source is surrounded
by ~ 10 6 stars within a sphere of radius ~ 1 pc, the net
acceleration towards the center is not vanishing, even in
the case of a statistically isotropic distribution of stars.
Rather, due to the discrete nature of the mass distribu-
tion, a stochastic force exists that changes with time
because of the orbital motion of the stars. The radio
source is accelerated by this force, causing a motion
of Sgr A* which becomes larger the smaller the mass
of the source. The very strong limits to the velocity of
Sgr A* enable us to derive a lower limit for its mass of
0.4 x 10 6 M o . This mass limit is significantly lower than
the mass of the SMBH that was derived from the stellar
orbits, but it is the mass of the radio source itself. Al-
though we have excellent reasons to assume that Sgr A*
coincides with the SMBH, this new observation is the
first proof for a large mass of the radio source itself.
2.6.6 Hypervelocity Stars in the Galaxy
Discovery. In 2005, a Galactic star was discovered
which travels with a velocity of at least 700 km/s rela-
tive to the Galactic rest-frame. This B-star has a distance
of ~ 110 kpc from the Galactic center, and its actual
space velocity depends on its transverse motion which
has not be yet been measured, due to the large distance of
the object from us. The velocity of this star is so large
that it exceeds the escape velocity from the Galaxy;
2.6 The Galactic Center
hence, this star is gravitationally unbound to the Milky
Way. Within one year after this first discovery, four
more such hypervelocity stars have been discovered, all
of them early-type stars (O- or B-stars) with Galactic
rest-frame velocities in excess of 500 km/s. They will
all escape the gravitational potential of the Galaxy.
Acceleration of Hypervelocity Stars. The fact that the
hypervelocity stars are gravitationally unbound to the
Milky Way implies that they must have been accelerated
very recently, i.e., less than a crossing time through the
Galaxy ago. In addition, since they are early-type stars,
they must have been accelerated within the lifetime of
such stars. The acceleration mechanism must be of grav-
itational origin and is related to the dynamical instability
of N-body systems, with N > 2. A pair of objects will
orbit in their joint gravitational field, either on bound or-
bits (ellipses) or unbound ones (gravitational scattering
on hyperbolic orbits); in the former case, the system is
stable and the two masses will orbit around each other
literally forever. If more than two masses are involved
this is no longer the case - such a system is inherently
unstable. Consider three masses, initially bound to each
other, orbiting around their center-of-mass. In general,
their orbits will not be ellipses but are more compli-
cated; in particular, they are not periodic. Such a system
is, mathematically speaking, chaotic. A chaotic system
is characterized by the property that the state of a sys-
tem at time t depends very sensitively on the initial
conditions set at time t\ < t. Whereas for a dynamically
stable system the positions and velocities of the masses
at time t are changed only a little if their initial con-
ditions are slightly varied (e.g., by giving one of the
masses a slightly larger velocity), in a chaotic, dynam-
ically unstable system even tiny changes in the initial
conditions can lead to completely different states at later
times. Any JV-body system with N > 2 is dynamically
unstable.
Back to our three-body system. The three masses
may orbit around each other for an extended period of
time, but their gravitational interaction may then change
the state of the system suddenly, in that one of the three
masses attains a sufficiently high velocity relative to the
other two and may escape to infinity, whereas the other
two masses form a binary system. What was a bound
system initially may become an unbound system later
on. This behavior may appear unphysical at first sight -
where does the energy come from to eject one of the
stars? Is this process violating energy conservation?
Of course not! The trick lies in the properties of
gravity: a binary has negative binding energy, and the
more negative, the tighter the binary orbit. By three-
body interactions, the orbit of two masses can become
tighter (one says that the binary "hardens"), and the
corresponding excess energy is transferred to the third
mass which may then become gravitationally unbound.
In fact, a single binary of compact stars can in princi-
ple take up all the binding energy of a star cluster and
"evaporate" all other stars.
This discussion then leads to the explanation of hy-
pervelocity stars. The characteristic escape velocity of
the "third mass" will be the orbital velocity of the three-
body system before the escape. The only place in our
Milky Way where orbital velocities are as high as that
observed for the hypervelocity stars is the Galactic cen-
ter. In fact, the travel time of a star with current velocity
of ~ 600 km/s from the Galactic center to Galactro-
centric distances of ~ 80 kpc is of order 10 8 yr, slightly
shorter than the main-sequence lifetime of a B-star. Fur-
thermore, most of the bright stars in the central 1" of
the Galactic center region are B-stars. Therefore, the
immediate environment of the central black hole is the
natural origin for these hypervelocity stars. Indeed, long
before their discovery the existence of such stars was
predicted. When a binary star gets close to the black
hole, this three-body interaction can lead to the ejec-
tion of one of the two stars into an unbound orbit. Thus,
the existence of hypervelocity stars can be considered
as an additional piece of evidence for the presence of
a central black hole in our Galaxy.
3. The World of Galaxies
The insight that our Milky Way is just one of many gal-
axies in the Universe is less than 100 years old, despite
the fact that many had already been known for a long
time. The catalog by Charles Messier (1730-1817), for
instance, lists 103 diffuse objects. Among them M31,
the Andromeda galaxy, is listed as the 31st entry in
the Messier catalog. Later, this catalog was extended
to 110 objects. John Dreyer (1852-1926) published
the New General Catalog (NGC) that contains nearly
8000 objects, most of them galaxies. In 1912, Vesto
Slipher found that the spiral nebulae are rotating, using
spectroscopic analysis. But the nature of these extended
sources, then called nebulae, was still unknown at that
time; it was unclear whether they are part of our Milky
Way or outside it.
The year 1920 saw a public debate (the Great Debate)
between Harlow Shapley and Heber Curtis. Shapley
believed that the nebulae are part of our Milky Way,
whereas Curtis was convinced that the nebulae must
be objects located outside the Galaxy. The arguments
which the two opponents brought forward were partly
based on assumptions which later turned out to be in-
valid, as well as on incorrect data. We will not go into the
details of their arguments which were partially linked
to the assumed size of the Milky Way since, only a few
years later, the question of the nature of the nebulae was
resolved.
In 1925, Edwin Hubble discovered Cepheids in An-
dromeda (M31). Using the period-luminosity relation
for these pulsating stars (see Sect. 2.2.7) he derived
a distance of 285 kpc. This value is a factor of ~ 3
smaller than the distance of M3 1 known today, but it
provided clear evidence that M31, and thus also other
spiral nebulae, must be extragalactic. This then imme-
diately implied that they consist of innumerable stars,
like our Milky Way. Hubble's results were considered
conclusive by his contemporaries and marked the begin-
ning of extragalactic astronomy. It is not coincidental
that at this time George Hale began to arrange the fund-
ing for an ambitious project. In 1928 he obtained six
Fig. 3.1. Galaxies occur in differ-
ent shapes and sizes, and often they
are grouped together in groups or
clusters. This cluster, ACO 3341, at
a redshift of z = 0.037, contains nu-
merous galaxies of different types
and luminosities
Peter Schneider, The World of Galaxies.
In: Peter Schneider, Extragalactic Astronomy and Cosmology, pp. 87-140 (2006)
DOI: 10.1007/1 1614371_3 © Springer- Verlag Berlin Heidelberg 2006
3. The World of Galaxies
million dollars for the construction of the 5-m telescope
on Mt. Palomar which was completed in 1949.
This chapter is about galaxies. We will confine the
consideration here to "normal" galaxies in the local
Universe; galaxies at large distances, some of which are
in a very early evolutionary state, will be discussed in
Chap. 9, and active galaxies, like quasars for example,
will be discussed later in Chap. 5.
3.1 Classification
The classification of objects depends on the type of ob-
servation according to which this classification is made.
This is also the case for galaxies. Historically, optical
photometry was the method used to observe galaxies.
Thus, the morphological classification defined by Hub-
ble is still the best-known today. Besides morphological
criteria, color indices, spectroscopic parameters (based
on emission or absorption lines), the broad-band spec-
tral distribution (galaxies with/without radio- and/or
X-ray emission), as well as other features may also be
used.
3.1.1 Morphological Classification:
The Hubble Sequence
Figure 3.2 shows the classification scheme defined by
Hubble. According to this, three main types of galaxies
exist:
• Elliptical galaxies (E's) are galaxies that have nearly
elliptical isophotes 1 without any clearly defined
'isophotes are
then its isophou
ntours along which the surface brightness of
int. If the light profile of a galaxy is elliptical,
re ellipses.
structure. They are subdivided according to their
ellipticity e=\— b/a, where a and b denote the
semimajor and the semiminor axes, respectively. El-
lipticals are found over a relatively broad range in
ellipticity, < e < 0.7. The notation Em is commonly
used to classify the ellipticals with respect to e,
with n = 10e; i.e., an E4 galaxy has an axis ratio
of b/a — 0.6, and EO's have circular isophotes.
1 Spiral galaxies consist of a disk with spiral arm struc-
ture and a central bulge. They are divided into two
subclasses: normal spirals (S"s) and barred spirals
(SB's). In each of these subclasses, a sequence is de-
fined that is ordered according to the brightness ratio
of bulge and disk, and that is denoted by a, ab, b,
be, c, cd, d. Objects along this sequence are often re-
ferred to as being either an early-type or a late-type;
hence, an Sa galaxy is an early-type spiral, and an
SBc galaxy is a late-type barred spiral. We stress ex-
plicitly that this nomenclature is not a statement of
the evolutionary stage of the objects but is merely
a nomenclature of purely historical origin.
< Irregular galaxies (Irr's) are galaxies with only weak
(Irr I) or no (Irr II) regular structure. The classi-
fication of Irr's is often refined. In particular, the
sequence of spirals is extended to the classes Sdm,
Sm, Im, and Ir (m stands for Magellanic; the Large
Magellanic Cloud is of type SBm).
i SO galaxies are a transition between ellipticals and
spirals. They are also called lenticulars as they are
lentil-shaped galaxies which are likewise subdivided
into SO and SBO, depending on whether or not they
show a bar. They contain a bulge and a large en-
veloping region of relatively unstructured brightness
which often appears like a disk without spiral arms.
Ellipticals and SO galaxies are referred to as early-
type galaxies, spirals as late-type galaxies. As before,
; "tuning fork" for galaxy
3.1 Classification
these names are only historical and are not meant to
describe an evolutionary track!
Obviously, the morphological classification is at least
partially affected by projection effects. If, for instance,
the spatial shape of an elliptical galaxy is a triaxial
ellipsoid, then the observed ellipticity e will depend on
its orientation with respect to the line-of-sight. Also,
it will be difficult to identify a bar in a spiral that is
observed from its side ("edge-on").
Besides the aforementioned main types of galaxy
morphologies, others exist which do not fit into the
Hubble scheme. Many of these are presumably caused
by interaction between galaxies (see below). Further-
more, we observe galaxies with radiation characteristics
that differ significantly from the spectral behavior of
"normal" galaxies. These galaxies will be discussed
3.1.2 Other Types of Galaxies
The light from "normal" galaxies is emitted mainly by
stars. Therefore, the spectral distribution of the radiation
from such galaxies is in principle a superposition of
the spectra of their stellar population. The spectrum of
stars is, to a first approximation, described by a Planck
function (see Appendix A) that depends only on the
star's surface temperature. A typical stellar population
covers a temperature range from a few thousand Kelvin
up to a few tens of thousand Kelvin. Since the Planck
function has a well-localized maximum and from there
steeply declines to both sides, most of the energy of
such "normal" galaxies is emitted in a relatively narrow
frequency interval that is located in the optical and NIR
sections of the spectrum.
In addition to these, other galaxies exist whose spec-
tral distribution cannot be described by a superposition
of stellar spectra. One example is the class of active
galaxies which generate a significant fraction of their
luminosity from gravitational energy that is released in
the infall of matter onto a supermassive black hole, as
was mentioned in Sect. 1.2.4. The activity of such ob-
jects can be recognized in various ways. For example,
some of them are very luminous in the radio and/or
in the X-ray portion of the spectrum (see Fig. 3.3), or
they show strong emission lines with a width of several
thousand km/s if the line width is interpreted as due
to Doppler broadening, i.e., AA./A. = Av/c. In many
cases, by far the largest fraction of luminosity is pro-
duced in a very small central region: the active galactic
nucleus (AGN) that gave this class of galaxies its name.
In quasars, the central luminosity can be of the order of
~ 10 13 L Q , about a thousand times as luminous as the to-
tal luminosity of our Milky Way. We will discuss active
galaxies, their phenomena, and their physical properties
in detail in Chap. 5.
Another type of galaxy also has spectral properties
that differ significantly from those of "normal" galax ies,
namely the starburst galaxies. Normal spiral galaxies
like our Milky Way form new stars at a star-formation
rate of ~ 3M Q /yr which can be derived, for instance,
from the B aimer lines of hydrogen generated in the
Hll regions around young, hot stars. By contrast, el-
liptical galaxies show only marginal star formation or
none at all. However, there are galaxies which have
a much higher star-formation rate, reaching values of
10 is 10 2o
Frequency (Hz)
Fig. 3.3. The spectrum of a quasar ( 3C273 1
in comparison to that of an elliptical gal
axv. While the radiation from the elliptical
is concentrated in a narrow range span
ning less than two decades in frequency,
the emission from the quasar is observed
over the full range of the electromagnetic
spectrum, and the energy per logarithmic
frequency interval is roughly constant. This
demonstrates that the light from the quasar
cannot be interpreted as a superposition of
stellar spectra, but instead has to be gener-
ated by completely different sources and by
different radiation mechanisms
3. The World of Galaxies
lOOMg/yr and more. If many young stars are formed we
would expect these starburst galaxies to radiate strongly
in the blue or in the UV part of the spectrum, corre-
sponding to the maximum of the Planck function for
the most massive and most luminous stars. This ex-
pectation is not fully met though: star formation takes
place in the interior of dense molecular clouds which
often also contain large amounts of dust. If the major
part of star formation is hidden from our direct view
by layers of absorbing dust, these galaxies will not be
very prominent in blue light. However, the strong ra-
diation from the young, luminous stars heats the dust;
the absorbed stellar light is then emitted in the form of
thermal dust emission in the infrared and submillimeter
regions of the electromagnetic spectrum - these gal-
axies can thus be extremely luminous in the IR. They
are called ultra-luminous infrared galaxies (ULIRGs).
We will describe the phenomena of starburst galaxies
in more detail in Sect. 9.2.1. Of special interest is the
discovery that the star-formation rate of galaxies seems
to be closely related to interactions between galaxies
- many ULIRGs are strongly interacting galaxies (see
Fig. 3.4).
3.2 Elliptical Galaxies
3.2.1 Classification
The general term "elliptical galaxies" (or ellipticals, for
short) covers a broad class of galaxies which differ in
their luminosities and sizes - some of them are displayed
in Fig. 3.5. A rough subdivision is as follows:
• Normal ellipticals. This class includes giant ellipti-
cals (gE's), those of intermediate luminosity (E's),
and compact ellipticals (cE's), covering a range in
absolute magnitudes from M B ~ —23 to M B ~ —15.
In addition, SO galaxies are often assigned to this
class of early-type galaxies.
• Dwarf ellipticals (dE's). These differ from the cE's
in that they have a significantly smaller surface
brightness and a lower metallicity.
• cD galaxies. These are extremely luminous (up to
M B ~ —25) and large (up to R < 1 Mpc) galaxies
that are only found near the centers of dense clusters
of galaxies. Their surface brightness is very high
close to the center, they have an extended diffuse
envelope, and they have a very high M/L ratio.
• Blue compact dwarf galaxies. These "blue compact
dwarfs" (BCD's) are clearly bluer (with (B — V) be-
tween 0.0 and 0.3) than the other ellipticals, and
contain an appreciable amount of gas in comparison.
• Dwarf spheroidals (dSph's) exhibit a very low lu-
minosity and surface brightness. They have been
observed down to M B ~ — 8. Due to these proper-
ties, they have thus far only been observed in the
Local Group.
Thus elliptical galaxies span an enormous range (more
than 10 6 ) in luminosity and mass, as is shown by the
compilation in Table 3.1.
3.2.2 Brightness Profile
The brightness profiles of normal E's and cD's follow
a de Vaucouleurs profile (see (2.39) or (2.41), respec-
tively) over a wide range in radius, as is demonstrated
in Fig. 3.6. The effective radius R e is strongly corre-
lated with the absolute magnitude M B , as can be seen
in Fig. 3.7, with rather little scatter. In comparison, the
dE's and the dSph's clearly follow a different distribu-
tion. Owing to the relation (2.42) between luminosity,
Fig. 3.5. Different types of elliptical galaxies. Upper left: the the galaxy Leo I belongs to the nine known dwarf spheroidals
cD galaxy M87 in the center of the Virgo galaxy cluster; in the Local Group; lower right: NGC 1705, a dwarf irregular,
upper right: Centaurus A, a giant elliptical galaxy with a very shows indications of massive star formation - a super star
distinct dust disk and an active galactic nucleus; lower left: cluster and strong galactic winds
3. The World of Galaxies
Table 3.1. Characteristic values for elliptical galaxies. D25
denotes the diameter al which the surface brightness has de-
creased to 25 B-mag/arcsec 2 , 5n is the "specific frequency".
•e for the number of globular clusters in relation to the
visual luminosity (see (3.13)), and M/L is the mass-to- light
ratio in Solar units (the values of this table are taken from the
book by Carroll & Ostlie, 1996)
SO
cD
E
dE
dSph
BCD
M B
-17 to -22
-22 to -25
-15 to -23
-13 to -19
-8 to -15
-1410-17
M(M Q )
10 10 to 10 12
10" to 10 14
10 8 to 10 13
10 7 to 10"
10 7 to 10 8
-10"
D 25 (kpc)
10-100
300-1000
1-200
1-10
0.1-0.5
<3
(M/L B )
-10
> 100
10-100
1-10
5-100
0.1-10
(Sn)
~5
-15
-5
4.8=1=1.0
Fig. 3.6. Surface brightness profile of the galaxy NGC 4472,
fitted by a de Vaucouleurs profile. The de Vaucouleurs pro-
file describes a linear relation between the logarithm of the
intensity (i.e., linear on a magnitude scale) and r 1 / 4 ; for this
reason, it is also called an r 1/,4 -law
effective radius and central surface brightness, an anal-
ogous relation exists for the average surface brightness
/u. a ve (unit: B — mag/arcsec 2 ) within R e as a function of
M B . In particular, the surface brightness in normal E's
decreases with increasing luminosity, while it increases
for dE's and dSph's.
Yet another way of expressing this correlation is
by eliminating the absolute luminosity, thus obtain-
ing a relation between effective radius R e and surface
brightness /Lt ave . This form is then called the Kormendy
relation.
The de Vaucouleurs profile provides the best fits for
normal E's, whereas for E's with exceptionally high (or
low) luminosity the profile decreases more slowly (or
rapidly) for larger radii. The profile of cD's extends
much farther out and is not properly described by a de
Vaucouleurs profile (Fig. 3.8), except in its innermost
part. It appears that cD's are similar to E's but embed-
ded in a very extended, luminous halo. Since cD's are
only found in the centers of massive clusters of galax-
ies, a connection must exist between this morphology
and the environment of these galaxies. In contrast to
these classes of ellipticals, diffuse dE's are often better
described by an exponential profile.
3.2.3 Composition of Elliptical Galaxies
Except for the BCD's, elliptical galaxies appear red
when observed in the optical, which suggests an old
stellar population. It was once believed that ellipticals
contain neither gas nor dust, but these components have
now been found, though at a much lower mass-fraction
than in spirals. For example, in some ellipticals hot
gas (~ 10 7 K) has been detected by its X-ray emission.
Furthermore, Ha emission lines of warm gas (~ 10 4 K)
have been observed, as well as cold gas (~ 100 K) in the
Hi (21 -cm) and CO molecular lines. Many of the nor-
mal ellipticals contain visible amounts of dust, partially
manifested as a dust disk. The metallicity of ellipticals
and SO galaxies increases towards the galaxy center, as
derived from color gradients. Also in SO galaxies the
bulge appears redder than the disk. The Spitzer Space
Telescope, launched in 2003, has detected a spatially
extended distribution of warm dust in SO galaxies, or-
ganized in some sort of spiral structure. Cold dust has
also been found in ellipticals and SO galaxies.
This composition of ellipticals clearly differs from
that of spiral galaxies and needs to be explained by mod-
3.2 Elliptical Galaxies
Fig. 3.7. Left panel: effective radius R e versus absolute mag-
nitude M B ; the correlation for normal ellipticals is different
from that of dwarfs. Right panel: average surface brightness
, e versus M B ; for normal ellipticals, the surface bright-
is decreases with increasing luminosity while for dwarfs it
1 i i i 1 i
\_
A2670 CD
-
— \
, M e = 23 - 11
—
j-
\ log r e = 1.47
-i
r
v«.
-!
:
\ ""■■■.
:
r
\ '■.
-
■ i i
I . . i \ .
\ t :
30
r 1/4 (kpc)
Fig. 3.8. Comparison of the brightness profile of a cD galaxy,
the central galaxy of the cluster of galaxies Abell 2670. u ith
a de Vaucouleurs profile. The light excess for large radii is
clearly visible
els of the formation and evolution of galaxies. We will
see later that the cosmic evolution of elliptical galaxies
is also observed to be different from that of spirals.
3.2.4 Dynamics of Elliptical Galaxies
Analyzing the morphology of elliptical galaxies raises
a simple question: Why are ellipticals not round? A sim-
ple explanation would be rotational flattening, i.e., as in
a rotating self-gravitating gas ball, the stellar distribu-
tion bulges outwards at the equator due to centrifugal
forces, as is also the case for the Earth. If this explana-
tion were correct, the rotational velocity i> rot , which is
measurable in the relative Doppler shift of absorption
lines, would have to be of about the same magnitude
as the velocity dispersion of the stars er„ that is mea-
surable through the Doppler broadening of lines. More
precisely, by means of stellar dynamics one can show
that for the rotational flattening of an axially symmetric,
oblate 2 galaxy, the relation
(3.1)
has to be satisfied, where "iso" indicates the assumption
of an isotropic velocity distribution of the stars. How-
ever, for luminous ellipticals one finds that, in general,
frot <C a v , so that rotation cannot be the major cause of
their ellipticity (see Fig. 3.9). In addition, many ellip-
ticals arc presumably triaxial, so that no unambiguous
rotation axis is defined. Thus, luminous ellipticals are
in general not rotationally flattened. For less luminous
ellipticals and for the bulges of disk galaxies, however,
rotational flattening can play an important role. The
question remains of how to explain a stable elliptical
distribution of stars without rotation.
2 If a >b> c denote the lengths oft the major axes of an ellipsoid, then
ii is called an oblate spheroid (— rotational ellipsoid) if a — h > c.
whereas ,1 prolate spheroid is specilied h\ a ■ h- <■. If all three axes
are different, it is called triaxial ellipsoid.
3. The World of Galaxies
Fig. 3.9. The rotation parameter (^f)/(^f). of elliptical
galaxies, here denoted by (V/ct)*, plotted as a function of
absolute magnitude. Dots denote elliptical galaxies, crosses
the bulges of disk galaxies
The brightness profile of an elliptical galaxy is de-
fined by the distribution of its stellar orbits. Let us
assume that the gravitational potential is given. The stars
are then placed into this potential, with the initial posi-
tions and velocities following a specified distribution. If
this distribution is not isotropic in velocity space, the re-
sulting light distribution will in general not be spherical.
For instance, one could imagine that the orbital planes
of the stars have a preferred direction, but that an equal
number of stars exists with positive and negative angu-
lar momentum L z , so that the total stellar distribution
has no angular momentum and therefore does not rotate.
Each star moves along its orbit in the gravitational po-
tential, where the orbits are in general not closed. If an
initial distribution of stellar orbits is chosen such that the
statistical properties of the distribution of the orbits are
invariant in time, then one will obtain a stationary sys-
tem. If, in addition, the distribution is chosen such that
the respective mass distribution of the stars will generate
exactly the originally chosen gravitational potential, one
arrives at a self-gravitating equilibrium system. In gen-
eral, it is a difficult mathematical problem to construct
such self-gravitating equilibrium systems.
Relaxation Time-Scale. The question now arises
whether such an equilibrium system can also be sta-
ble in time. One might expect that close encounters of
pairs of stars would cause a noticeable disturbance in the
distribution of orbits. These pair-wise collisions could
then lead to a "thermalization" of the stellar orbits. 3
To examine this question we need to estimate the time-
scale for such collisions and the changes in direction
they cause.
For this purpose, we consider the relaxation time-
scale by pair collisions in a system of N stars of mass m ,
lolal muss M = Nm, extent R, and a mean stellar density
of n = 3N/(4jtR 3 ). We define the relaxation time f relax
as the characteristic time in which a star changes its
velocity direction by ~ 90° due to pair collisions with
other stars. By simple calculation (see below), we find
that
ta-»^. (3-2)
vlnN
cross |nA'
(3.3)
where f cross = R/v is the crossing time-scale, i.e., the
time it takes a star to cross the stellar system. If we now
consider a typical galaxy, with f cross ~ 10 8 yr, N ~ 10 12
(thus In N ~ 30), then we find that the relaxation time is
much longer than the age of the Universe. This means
that pair collisions do not plax any role in the evolution
of stellar orbits. The dynamics of the orbits are deter-
mined solely by the large-scale gravitational field of the
galaxy. In Sect. 7.5.1, we will describe a process called
violent relaxation which most likely plays a central
role in the formation of galaxies and which is proba-
bly also responsible for the stellar orbits establishing an
equilibrium configuration.
The stars behave like a collisionless gas: elliptical
galaxies are stabilized by (dynamical) pressure, and
they are elliptical because the stellar distribution is
■'Noli- that in a gas like air, scattering between molecules occurs
frequently, which drives the velocity distribution of the molecules
towards an isotropic Maxwellian. i.e.. the thermal, distribution.
3.2 Elliptical Galaxies
anisotropic in velocity space. This corresponds to an
anisotropic pressure - where we recall that the pressure
of a gas is nothing but the momentum transport of gas
particles due to their thermal motions.
Derivation of the Collisional Relaxation Time-Scale.
We consider a star passing by another one, with the im-
pact parameter b being the minimum distance between
the two. From gravitational deflection, the star attains
a velocity component perpendicular to the incoming
direction of
)(?)■
IGm
~b^~
Fig. 3.10. Sketch related to the derivation of the dynamical
parameters within db of b are located in a cylindrical
shell of volume {2itb db) (vt), so that
{\v ± \ 2 (f)) = J2
- 2n
f db
(3.6*
where a is the acceleration at closest separation and Af
the "duration of the collision", estimated as Af = 2b /v
(see Fig. 3.10). Equation (3.4) can be derived more
rigorously by integrating the perpendicular accelera-
tion along the orbit. A star undergoes many collisions,
through which the perpendicular velocity components
will accumulate; these form two-dimensional vectors
perpendicular to the original direction. After a time t we
have v±(t) — J2i v ± ■ The expectation value of this vec-
tor is (dj_ (?)} = J^ ( . ( vf ) = since the directions of the
individual v^ are random. But the mean square velocity
perpendicular to the incoming direction does not vanish,
where we set lvf-vlj = for i^j because the
directions of different collisions are assumed to be un-
correlated. The velocity v± performs a so-called random
walk. To compute the sum, we convert it into an integral
where we have to integrate over all collision parame-
ters b. During time t, all collision partners with impact
At
The integral cannot be performed from to oo. Thus,
it has to be cut off at b m { n and /3 max and then yields
m (^max/^min)- Due to the finite size of the stellar distri-
bution, b m . dx — R is a natural choice. Furthermore, our
approximation which led to (3.4) will certainly break
down if i/j_ is of the same order of magnitude as v;
hence we choose b m { n — 2Gm/v 2 . With this, we obtain
b m zx/b min = Rv 2 /(2Gm). The exact choice of the in-
tegration limits is not important, since b m \ a and b max
appear only logarithmically. Next, using the virial theo-
rem, | Spot I = 2£"kin, and thus GM/R — v 2 for a typical
star, we get b max /b m i n « N. Thus,
{\v ± \ 2 (t)) = 2jt
We define the relaxation time f re i ax by ( | i^x 1 2 (f re iax)) = v 2 -,
i.e., the time after which the perpendicular velocity
roughly equals the infall velocity:
f relax
- — ( — ) —
2itnv \2GmJ InN
~ 2imv \2Rm) lnJV ~ ~v \n~N '
from which we finally obtain (3.3).
3.2.5 Indicators of a Complex Evolution
The isophotes (that is, the curves of constant surface
brightness) of many of the normal elliptical galax-
ies are well approximated by ellipses. These elliptical
isophotes with different surface brightnesses are con-
centric to high accuracy, with the deviation of the
isophote's center from the center of the galaxy being
typically < 1 % of its extent. However, in many cases
the ellipticity varies with radius, so that the value for
€ is not a constant. In addition, many ellipticals show
a so-called isophote twist: the orientation of the semi-
major axis of the isophotes changes with the radius.
3. The World of Galaxies
This indicates that elliptical galaxies are not spheroidal,
but triaxial systems (or that there is some intrinsic twist
of their axes).
Although the light distribution of ellipticals appears
rather simple at first glance, a more thorough analysis
reveals that the kinematics can be quite complicated. For
example, dust disks are not necessarily perpendicular to
any of the principal axes, and the dust disk may rotate in
a direction opposite to the galactic rotation. In addition,
ellipticals may also contain (weak) stellar disks.
Boxiness and Diskiness. The so-called boxiness par-
ameter describes the deviation of the isophotes' shape
from that of an ellipse. Consider the shape of an
isophote. If it is described by an ellipse, then after
a suitable choice of the coordinate system, x —a cos t,
6 2 = b sin t , where a and b are the two semi-axes of
the ellipse and t e [0, 2jt] parametrizes the curve. The
distance r{t) of a point from the center is
~^b~ 2
- cos(2f) .
Deviations of the isophote shape from this ellipse
are now expanded in a Taylor series, where the term
ex cos (4?) describes the lowest-order correction that
preserves the symmetry of the ellipse with respect to re-
flection in the two coordinate axes. The modified curve
is then described by
9(f) = 1
CI4 cos(4f)
K0~
)(
/? sili /
(3.9)
with r(f) as defined above. The parameter an thus
describes a deviation from an ellipse: if CI4 > 0, the
isophote appears more disk-like, and if CI4 < 0, it be-
comes rather boxy (see Fig. 3.1 1). In elliptical galaxies
we typically find |a 4 /a| ~ 0.01, thus only a small
deviation from the elliptical form.
Correlations of a 4 with Other Properties of Ellip-
ticals. Surprisingly, we find that the parameter a^/a is
strongly correlated with other properties of ellipticals
(see Fig. 3.12). The ratio (^\ I (^f). (upper left
in Fig. 3.12) is of order unity for disky ellipses (a 4 > 0)
and, in general, significantly smaller than 1 for boxy
ellipticals. From this we conclude that "diskies" are
in part rotationally supported, whereas the flattening
Fig. 3.11. Sketch to illustrate boxiness and diskiness. The solid
red curve shows an ellipse (04 = 0), the green dashed curve
a disky ellipse (04 > 0), and the blue dotted curve a boxy
ellipse U/4 < 0). In elliptical galaxies, the deviations in the
shape of the isophotes from an ellipse are considerably smaller
than in this sketch
of "boxies" is mainly caused by the anisotropic dis-
tribution of their stellar orbits in velocity space. The
mass-to-light ratio is also correlated with 04: boxies
(diskies) have a value of M/L in their core which
is larger (smaller) than the mean elliptical of com-
parable luminosity. A very strong correlation exists
between a 4 /a and the radio luminosity of ellipticals:
while diskies are weak radio emitters, boxies show
a broad distribution in L ra dio- These correlations are
also seen in the X-ray luminosity, since diskies are
weak X-ray emitters and boxies have a broad distribu-
tion in L x . This bimodality becomes even more obvious
if the radiation contributed by compact sources (e.g.,
X-ray binary stars) is subtracted from the total X-ray
luminosity, thus considering only the diffuse X-ray
emission. Ellipticals with a different sign of a 4 also dif-
fer in the kinematics of their stars: boxies often have
cores spinning against the general direction of rota-
tion (counter-rotating cores), which is rarely observed
in diskies.
About 70% of the ellipticals are diskies. The transi-
tion between diskies and SO galaxies may be continuous
along a sequence of varying disk-to-bulge ratio.
Shells and Ripples. In about 40% of the early-type
galaxies that are not member galaxies of a cluster,
sharp discontinuities in the surface brightness are found,
3.2 Elliptical Galaxies
5H++*
i Rotation Parameter
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i
6 (M/L v ) c - ■_
J* 0.2
Elhpticity
I ■ | ■ ■ l ~ l f I ■ H | ■ 1 1 1 1 1 ■ 1 1 1 ■ I
Radio Luminosity
Fig. 3.12. Correlations of a/^/a with some
other properties of elliptical galaxies.
IO(V;i \)/a (corresponding to cn/a) de-
scribes the deviation of the isophote shape
from an ellipse in percent. Negative val-
ues denote boxy ellipticals, positive values
disky ellipticals. The upper left panel
shows the rotation parameter discussed
in Sect. 3.2.4; at the lower left, the devi-
ation from the average mass-to-light ratio is
shown. The upper right panel shows the el-
lipti ii mi thi lo i ,i 'In panel displays
the radio luminosity at 1.4 GHz. Obviously,
there is a correlation of all these parameters
with the boxiness parameter
a kind of shell structure ("shells" or "ripples"). They
are visible as elliptical arcs curving around the center
of the galaxy (see Fig. 3.13). Such sharp edges can only
be formed if the corresponding distribution of stars is
"cold", i.e., they must have a very small velocity dis-
persion, since otherwise such coherent structures would
smear out on a very short time-scale. As a compari-
son, we can consider disk galaxies that likewise contain
sharp structures, namely the thin stellar disk. Indeed,
the stars in the disk have a very small velocity disper-
sion, ~ 20 km/s, compared to the rotational velocity of
typically 200 km/s.
These peculiarities in ellipticals are not uncommon.
Indicators for shells can be found in about half of the
early-type galaxies, and about a third of them show boxy
isophotes.
Fig. 3.13. In the galaxy NGC 474, here
shown in two images of different contrast,
a number of sharp edged elliptical arcs are
\ isible around the center of the galaxy, the
so-called ripples or shells. The displayed
image corresponds to a linear scale of about
90kpc
3. The World of Galaxies
Boxiness, counter-rotating cores, and shells and
ripples are all indicators of a complex evolution
that is probably caused by past mergers with other
galaxies.
We will proceed with a discussion of this interpre-
tation in Chap. 9.
3.3 Spiral Galaxies
3.3.1 Trends in the Sequence of Spirals
Looking at the sequence of early-type spirals (i.e., Sa's
or SBa's) to late-type spirals, we find a number of dif-
ferences that can be used for classification (see also
Fig. 3.14):
• a decreasing luminosity ratio of bulge and disk, with
^buige/idisk ~ 0.3 for Sa's and ~ 0.05 for Sc's;
• an increasing opening angle of the spiral arms, from
~ 6° for Sa's to ~ 18° for Sc's;
• and an increasing brightnes s structure along the spiral
arms: Sa's have a "smooth" distribution of stars along
the spiral arms, whereas the light distribution in the
spiral arms of Sc's is resolved into bright knots of
stars and Hll regions.
Compared to ellipticals, the spirals cover a distinctly
smaller range in absolute magnitude (and mass). They
are limited to -16 > M B > -23 and 1O 9 M < M <
1O 12 M , respectively. Characteristic parameters of the
various types of spirals are compiled in Table 3.2.
Bars are common in spiral galaxies, with ~ 70%
of all disk galaxies containing a large-scale stellar bar.
Such a bar perturbs the axial symmetry of the gravita-
tional potential in a galaxy, which may have a number
of consequences. One of them is that this perturbation
can lead to a redistribution of angular momentum of
the stars, gas, and dark matter. In addition, by perturb-
ing the orbits, gas can be driven towards the center of
the galaxy which may have important consequences for
triggering nuclear activity (see Chap. 5).
3.3.2 Brightness Profile
The light profile of the bulge of spirals is described by
a de Vaucouleurs profile to a good approximation - see
(2.39) and (2.41) - while the disk follows an exponential
brightness profile, as is the case for our Milky Way. Ex-
pressing these distributions of the surface brightness in
IX oc —2.5 log(7), measured in mag/arcsec 2 , we obtain
/*bulge(*)= Me + 8.3268
' R \
A*diak(*) = /*0 + 1.091
Table 3.2. Characteristic values l'oi spiral galaxies. V' 1!UA is the
maximum rotation velocity, inns characterizing the flat part
of the rotation curve. The opening angle is the angle under
which the spiral arms branch off, i.e., the angle between the
tangent to the spiral arms and the circle around the center of the
galaxy running through this tangential point. .V N is the specific
abundance of globular clusters as defined in (3.13). The values
in this table are taken from the book by Carroll & Ostlie ( 1 996 1
Sa
Sb
Sc
Sd/Sm
Im/Ir
M B
-17 to -23
-17 to -23
-16 to -22
-15 to -20
-13 to -18
M(M e )
10 9 -10 12
10 9 -10 12
10 9 -10 12
10 8 -10 10
10 8 -10 10
(italge/itotJs
0.3
0.13
0.05
-
-
Diam. (Z> 25 ,kpc)
5-100
5-100
5-100
0.5-50
0.5-50
(M/L B )(M /L Q )
6.2 ±0.6
4.5 ±0.4
2.6±0.2
~1
~1
(VmaxXkms- 1 )
299
222
175
-
-
163-367
144-330
99-304
50-70
Opening angle
-6°
-12°
-18°
-
-
mm (magarcsec- 2 )
21.52±0.39
21.52±0.39
21.52±0.39
22.61 ±0.47
22.61 ±0.47
(B-V)
0.75
0.64
0.52
0.47
0.37
(M gas /M tot )
0.04
0.08
0.16
0.25 (Scd)
-
(Mh 2 /M H i>
2.2 ±0.6 (Sab)
l.8±0.3
0.73 ±0.1 3
0.I9±0.10
-
(Sn)
1.2±0.2
1.2 ±0.2
0.5 ±0.2
0.5 ±0.2
-
Fig. 3.14. Types of spiral galaxies. Top left: M94, an Sj
galaxy. Top middle: M51, an Sbc galaxy. Top right: M101, i
Sc galaxy. Lower left: M83, an SBa galaxy. Lower middle:
NGC 1365, an SBb galaxy. Lower right: M58, an SBc galaxy
Here, fi e is the surface brightness at the effective ra-
dius R e which is defined such that half of the luminosity
is emitted within R e (see (2.40)). The central surface
brightness and the scale-length of the disk are denoted
by fMo and h r , respectively. It has to be noted that fio
is not directly measurable since /u-o is not the central
surface brightness of the galaxy, only that of its disk
component. To determine /i. , the exponential law (3.11)
is extrapolated from large R inwards to R = 0.
When Ken Freeman analyzed a sample of spiral gal-
axies, he found the remarkable result that the central
surface brightness fio of disks has a very low spread,
i.e., it is very similar for different galaxies (Freeman's
law, 1970). For Sa's to Sc's, a value of /x = 21.52±
0.39 B-mag/arcsec 2 is observed, and for Sd spirals and
later types, no = 22.61 ±0.47 B-mag/arcsec 2 . This re-
sult was critically discussed, for example with regard to
its possible dependence on selection effects. Their im-
portance is not implausible since the determination of
precise photometry of galaxies is definitely a lot eas-
ier for objects with a high surface brightness. After
accounting for such selection effects in the statistical
analysis of galaxy samples, Freeman's law was con-
firmed for "normal" spiral galaxies. However, galaxies
exist which have a significantly lower surface bright-
ness, the low surface brightness galaxies (LSBs). They
seem to form a separate class of galaxies whose study
is substantially more difficult compared to normal spi-
rals because of their low surface brightness. In fact,
the central surface brightness of LSBs is much lower
than the brightness of the night sky, so that search-
ing for these LSBs is problematic and requires very
accurate data reduction and subtraction of the sky
background.
3. The World of Galaxies
Whereas the bulge and the disk can be studied in
spirals even at fairly large distances, the stellar halo
has too low a surface brightness to be seen in distant
galaxies. However, our neighboring galaxy M31, the
Andromeda galaxy, can be studied in quite some detail.
In particular, the brightness profile of its stellar halo can
be studied more easily than that of the Milky Way. taking
advantage of our "outside" view. This galaxy should
be quite similar to our Galaxy in many respects; for
example, tidal streams from disrupted accreted galaxies
were also clearly detected in M3 1 .
A stellar halo of red giant branch stars was detected
in M31, which extends out to more than 150 kpc from
its center. The brightness profile of this stellar distri-
bution indicates that for radii r < 20 kpc it follows the
extrapolation from the brightness profile of the bulge,
i.e., a de Vaucouleurs profile. However, for larger radii it
exceeds this extrapolation, showing a power-law profile
which corresponds to a radial density profile of approx-
imately p oc r~ 3 , not unlike that observed in our Milky
Way. It thus seems that stellar halos form a generic prop-
erty of spirals. Unfortunately, the corresponding surface
brightness is so small that there is little hope of detect-
ing such a halo in other spirals for which individual stars
can no longer be resolved and classified.
The thick disk in other spirals can only be studied if
they are oriented edge-on. In these cases, a thick disk
can indeed be observed as a stellar population outside
the plane of the disk and well beyond the scale-height
of the thin disk. As is the case for the Milky Way,
the scale-height of a stellar population increases with
its age, increasing from young main-sequence stars to
old asymptotic giant branch stars. For luminous disk
galaxies, the thick disk does not contribute substantially
to the total luminosity; however, in lower-mass disk
galaxies with rotational velocities < 120 km/s, the thick
disk stars can contribute nearly half the luminosity and
may actually dominate the stellar mass. In this case,
the dominant stellar population of these galaxies is old,
despite the fact that they appear blue.
3.3.3 Rotation Curves and Dark Matter
The rotation curves of other spiral galaxies are easier to
measure than that of the Milky Way because we are able
to observe them "from outside". These
are achieved by utilizing the Doppler effect, where the
inclination of the disk, i.e., its orientation with respect
to the line-of-sight, has to be accounted for. The in-
clination angle is determined from the observed axis
ratio of the disk, assuming that disks are intrinsically
axially symmetric (except for the spiral arms). Mainly
the stars and Hi gas in the galaxies are used as lumi-
nous tracers, where the observable Hi disk is in general
significantly more extended than the stellar disk. There-
fore, the rotation curves measured from the 21 -cm line
typically extend to much larger radii than those from
optical stellar spectroscopy.
Like our Milky Way, other spirals also rotate con-
siderably faster in their outer regions than one would
expect from Kepler's law and the distribution of visible
matter (see Fig. 3.15).
The rotation curves of spirals do not decrease for
R > h r , as one would expect from the light distri-
bution, but are basically flat. We therefore conclude
that spirals are surrounded by a halo of dark mat-
ter. The density distribution of this dark halo can be
derived from the rotation curves.
Indeed, the density distribution of the dark matter can
be derived from the rotation curves. The force balance
between gravitation and centrifugal acceleration yields
the Kepler rotation law,
v 2 (R) = GM(R)/R ,
from which one directly obtains the mass M(R) within
a radius R. The rotation curve expected from the visible
matter distribution is 4
vl um (R) = GM ium (R)/R .
Mi um (R) can be determined by assuming a constant,
plausible value for the mass-to-light ratio of the lu-
minous matter. This value is obtained either from the
spectral light distribution of the stars, together with
knowledge of the properties of stellar populations, or by
fitting the innermost part of the rotation curve (where
'This consideration is strong!) simplified insofar as the given rela
nous are onh valid in this form for spherical mass distributions. The
rotational velocity produced b\ an oblate (disk shaped) mass distribu-
tion is more complicated to calculate: for instance, for an exponential
mass distribution in a disk, the maximum of i>i um occurs at - 2.2/?, .
with a Kepler decrease. v\ um oc /? -1 / 2 , at larger radii.
3.3 Spiral Galaxies
Distance from Nucleus (kpc)
Fig. 3.15. Examples of rotation curves of spiral galaxies. They
are all flat in the outer region and do not behave as expected
from Kepler's law if the galaxy consisted only of luminous
matter. Also striking is the fact thai Ihe amplitude oi the
rotation curve is higher for earl) i\ pes than for late types.
the mass contribution of dark matter can presumably
be neglected), assuming that M/L is independent of
radius for the stellar population. From this estimate
of the mass-to-light ratio, the discrepancy between
v 2 um and v 2 yields the distribution of the dark matter,
^dark = V 2 ~ V lm = GM^/R, OV
R r
i(fl)]
An example of this decomposition of the mass contri-
butions is shown in Fig. 3.16.
The corresponding density profiles of the dark matter
halos seem to be flat in the inner region, and decreas-
ing as R 2 at large radii. It is remarkable that p oc R~ 2
implies a mass profile M oc R, i.e., the mass of the halo
increases linearly with the radius for large R. As long as
the extent of the halo is undetermined the total mass of
a galaxy will be unknown. Since the observed rotation
curves are flat out to the largest radius for which 21 -cm
emission can still be observed, a lower limit for the ra-
dius of the dark halo can be obtained, 7?haio ^ 30/i _1 kpc.
To derive the density profile out to even larger radii,
other observable objects in an orbit around the galax-
ies are needed. Potential candidates for such luminous
tracers are satellite galaxies - companions of other spi-
rals, like the Magellanic Clouds are for the Milky Way.
(kpc)
Fig. 3.16. The flat rotation curves of spiral galaxies cannot be
explained by \ isible matter alone. The example of NGC 3198
demonstrates the rotation curve which would be expected from
the visible matter alone (curve labeled "disk"). To explain the
observed rotation curve, a dark matter component has to be
present (curve labeled "halo"). However, the decomposition
into disk and halo mass is not unambiguous because for it to
be so it would be necessary to know the mass-to-light ratio of
the disk. In the ease considered here, a "maximum disk"' was
assumed, i.e., it was assumed that the innermost part of the ro-
tation curve is produced solely by the visible matter in the disk
Because we cannot presume that these satellite galaxies
move on circular orbits around their parent galaxy, con-
clusions can be drawn based only on a statistical sample
of satellites. These analyses of the relative velocities of
satellite galaxies around spirals still give no indication
of an "edge" to the halo, leading to a lower limit for the
radius of /? ha to > 100 h~ x kpc.
For elliptical galaxies the mass estimate, and thus the
detection of a possible dark matter component, is sig-
nificantly more complicated, since the orbits of stars are
substantially more complex than in spirals. In particular,
the mass estimate from measuring the stellar velocity
dispersion via line widths depends on the anisotropy of
the stellar orbits, which is a priori unknown. Neverthe-
less, in recent years it has been unambiguously proven
that dark matter also exists in ellipticals. First, the
degeneracy between the anisotropy of the orbits and
the mass determination was broken by detailed kine-
matic analysis. Second, in some ellipticals hot gas was
detected from its X-ray emission. As we will see in
Sect. 6.3 in the context of clusters of galaxies, the tem-
perature of the gas allows an estimate of the depth of
3. The World of Galaxies
the potential well, and therefore the mass. Both methods
reveal lhat ellipticals are also surrounded by a dark halo.
The weak gravitational lens effect, which we will dis-
cuss in Sect. 6.5.2 in a different context, offers another
way to determine the masses of galaxies up to very large
radii. With this method we cannot study individual gal-
axies but only the mean mass properties of a galaxy
population. The results of these measurements confirm
the large size of dark halos in spirals and in ellipticals.
Correlations of Rotation Curves with Galaxy Prop-
erties. The form and amplitude of the rotation curves
of spirals are correlated with their luminosity and their
Hubble type. The larger the luminosity of a spiral, the
steeper the rise of v(R) in the central region, and the
larger the maximum rotation velocity u max . This latter
fact indicates that the mass of a galaxy increases with
luminosity, as expected. For the characteristic values of
the various Hubble types, one finds u max ~ 300 km/s
for Sa's, u max ~ 175 km/s for Sc's, whereas Irr's have
a much lower u max ~ 70 km/s. For equal luminosity,
Umax is higher for earlier types of spirals. However,
the shape (not the amplitude) of the rotation curves
of different Hubble types is similar, despite the fact that
they have a different brightness profile as seen, for in-
stance, from the varying bulge-to-disk ratio. This point
is another indicator that the rotation curves cannot be
explained by visible matter alone.
These results leave us with a number of obvious ques-
tions. What is the nature of the dark matter? What are
the density profiles of dark halos, how are they deter-
mined, and where is the "boundary" of a halo? Does the
fact that galaxies with v rot < 100 km/s have no promi-
nent spiral structure mean that a minimum halo mass
needs to be exceeded in order for spiral arms to form?
Some of these questions will be examined later, but
here we point out that the major fraction of the mass of
(spiral) galaxies consists of non-luminous matter. The
fact that we do not know what this matter consists of
leaves us with the question of whether this invisible
matter is a new, yet unknown, form of matter. Or is the
dark matter less exotic, normal (baryonic) matter that
is just not luminous for some reason (for example, be-
cause it did not form any stars)? We will see in Chap. 4
that the problem of dark matter is not limited to galax-
ies, but is also clearly present on a cosmological scale;
furthermore, the dark matter cannot be baryonic. A cur-
rently unknown form of matter is, therefore, revealing
itself in the rotation curves of spirals.
3.3.4 Stellar Populations and Gas Fraction
The color of spiral galaxies depends on their Hubble
type, with later types being bluer; e.g., one finds B —
V ~ 0.75 for Sa's, 0.64 for Sb's, 0.52 for Sc's, and 0.4
for Irr's. This means that the fraction of massive young
stars increases along the Hubble sequence towards later
spiral types. This conclusion is also in agreement with
the findings for the light distribution along spiral anus
where we clearly observe active star-formation regions
in the bright knots in the spiral arms of Sc's. Further-
more, this color sequence is also in agreement with the
decreasing bulge fraction towards later types.
The formation of stars requires gas, and the mass
fraction of gas is larger for later types, as can be mea-
sured, for instance, from the 2 1 -cm emission of Hi, from
Ha and from CO emission. Characteristic values for the
ratio (M gas /M tot ) are about 0.04 for Sa's, 0.08 for Sb's,
0.16 for Sc's, and 0.25 for Irr's. In addition, the fraction
of molecular gas relative to the total gas mass is smaller
for later Hubble types. The dust mass is less than 1% of
the gas mass.
Dust, in combination with hot stars, is the main
source of far-infrared (FIR) emission from galaxies.
Sc galaxies emit a larger fraction of FIR radiation than
Sa's, and barred spirals have stronger FIR emission than
normal spirals. The FIR emission arises due to dust be-
ing heated by the UV radiation of hot stars and then
reradiating this energy in the form of thermal emission.
A prominent color gradient is observed in spirals:
they are red in the center and bluer in the outer regions.
We can identify at least two reasons for this trend. The
first is a metallicity effect, as the metallicity is increasing
inwards and metal-rich stars are redder than metal-poor
ones, due to their higher opacity. Second, the color gra-
dient can be explained by star formation. Since the gas
fraction in the bulge is lower than in the disk, less star
formation takes place in the bulge, resulting in a stellar
population that is older and redder in general. Further-
more, it is found that the metallicity of spirals increases
with luminosity.
Abundance of Globular Clusters. The number of
globular clusters is higher in early types and in more
3.3 Spiral Galaxies
luminous galaxies. The specific abundance of globular
clusters in a galaxy is denned as their number, nor-
malized to a galaxy of absolute magnitude M v — — 15.
This can be done by scaling the observed number N t
of globular clusters in a galaxy of visual luminosity
L v or absolute magnitude M v , respectively, to that of
a hypothetical galaxy with M v — — 15:
Sn = N t -
= N t 10 04
(3.13)
If the number of globular clusters were proportional to
the luminosity (and thus roughly to the stellar mass) of
a galaxy, then this would imply Sn — const. However,
this is not the case: For Sa's and Sb's we find Sn ~ 1-2,
whereas Sn ~ 0.5 for Sc's. Sn is larger for ellipticals
and largest for cD galaxies.
3.3.5 Spiral Structure
The spiral arms are the bluest regions in spirals and they
contain young stars and Hll regions. For this reason,
the brightness contrast of spiral arms increases as the
wavelength of the (optical) observation decreases. In
particular, the spiral structure is very prominent in a blue
filter, as is shown impressively in Fig. 3.17.
Naturally, the question arises as to the nature of the
spiral arms. Probably the most obvious answer would be
that they are material structures of stars and gas, rotating
around the galaxy's center together with the rest of the
disk. However, this scenario cannot explain spiral arm
structure since, owing to the differential rotation, they
would wind up much more tightly than observed within
only a few rotation periods.
Rather, it is suspected that spiral arms are a wave
structure, the velocity of which does not coincide with
the physical velocity of the stars. Spiral arms are quasi-
stationary density waves, regions of higher density (pos-
sibly 10-20% higher than the local disk environment).
If the gas, on its orbit around the center of the galaxy
enters a region of higher density, it is compressed, and
this compression of molecular clouds results in an en-
hanced star-formation rate. This accounts for the blue
color of spiral arms. Since low-mass (thus red) stars live
longer, the brightness contrast of spiral arms is lower in
red light, whereas massive blue stars are born in the spi-
ral arms and soon after explode there as SNe. Indeed,
only few blue stars are found outside spiral arms.
In order to better understand density waves we may
consider, for example, the waves on the surface of a lake.
Peaks at different times consist of different water parti-
cles, and the velocity of the waves is by no means the
bulk velocity of the water.
3.3.6 Corona in Spirals?
Hot gas resulting from the evolution of supernova rem-
nants may expand out of the disk and thereby be ejected
to form a gaseous halo of a spiral galaxy. We might
<\\
• ■
- W :
\ 1 }
.
Fig. 3.17. The galaxy NGC 1300 in the B
filter (left panel) and in the I filter (right
panel). The spiral arms are much more
prominent in the blue than in the red. Also,
the tips of the bar are more pronounced
in the blue - an indicator of an enhanced
star-formation rate
tions are a very important tool for distance estimations,
as will be discussed in Sect. 3.6. Furthermore, these
scaling relations express relations between galaxy prop-
erties which any successful model of galaxy evolution
must be able to explain. Here we will describe these
scaling relations and discuss their physical origin.
3.4.1 The Tully-Fisher Relation
Using 21 -cm observations of spiral galaxies, in 1977
R. Brent Tully and J. Richard Fisher found that the
maximum rotation velocity of spirals is closely related
to their luminosity, following the relation
Fig. 3.18. The spiral galaxy NGC 4631. The optical (HST)
image of the galaxy is shown
are regions of very active sta
of massive stars eject hot gas
gas (at a temperature of T ~
shown as the blue diffuse
satellite. The image has i
red; the many luminous :
formation. The SN explosions
nto the halo of the galaxy. Tlik
10 6 K) emits X-ray radiation,
observed by the Chandra
of 2.'5
therefore suspect that such a "coronal" gas exists out-
side the galactic disk. While the existence of this coronal
gas has long been suspected, the detection of its X-ray
emission was first made possible with the ROS AT satel-
lite in the early 1990s. However, the limited angular
resolution of ROSAT rendered the distinction between
diffuse emission and clusters of discrete sources diffi-
cult. Finally, the Chandra observatory unambiguously
detected the coronal gas in a number of spiral galax-
ies. As an example, Fig. 3.18 shows the spiral galaxy
NGC 4631.
3.4 Scaling Relations
The kinematic properties of spirals and ellipticals
are closely related to their luminosity. As we shall
discuss below, spirals follow the Tully-Fisher rela-
tion (Sect. 3.4.1), whereas elliptical galaxies obey the
Faber-Jackson relation (Sect. 3.4.2) and are located in
the fundamental plane (Sect. 3.4.3). These scaling rela-
Locu
(3.14)
where the slope of the Tully-Fisher relation is about
a ~ 4. The larger the wavelength of the filter in which
the luminosity is measured, the smaller the dispersion
of the Tully-Fisher relation (see Fig. 3.19). This is to
be expected because radiation at larger wavelengths
is less affected by dust absorption and by the current
star-formation rate, which may vary to some extent be-
tween individual spirals. Furthermore, it is found that
the value of a increases with the wavelength of the fil-
ter; the Tully-Fisher relation is steeper in the red. The
dispersion of galaxies around the relation (3.14) in the
near infrared (e.g., in the H-band) is about 10%.
Because of this close correlation, the luminosity of
spirals can be estimated quite precisely by measur-
ing the rotational velocity. The determination of the
(maximum) rotational velocity is independent of the
galaxy's distance. By comparing the luminosity, as
determined from the Tully-Fisher relation, with the
measured flux one can then estimate the distance of
the galaxy - without utilizing the Hubble relation!
The measurement of u max is obtained either from
a spatially resolved rotation curve, by measuring v I0t (9),
which is possible for relatively nearby galaxies, or by
observing an integrated spectrum of the 21 -cm line of
Hi that has a Doppler width corresponding to about
2u m ax (see Fig. 3.20). The Tully-Fisher relation shown
in Fig. 3.19 was determined by measuring the width of
the 21 -cm line.
Explaining the Tully-Fisher Relation. The shapes of
ves of spirals are very similar to each
3.4 Scaling Relations
1.8 2.0 2.2 2.4 2.6 2.8 1.8 2.0 2.2 2.4 2.6 2.8 3.0
logWj,
Fig. 3.19. The Tully-Fisher relation for galaxies in the Lo-
cal Group (dots), in the Sculptor group (triangles), and in the
M81 group (squares). The absolute magnitude is plotted as
a function of the width of the 21-cm profile which indicates
the maximum rotation velocity (sec rig. 3.20). Filled symbols
i pn .i.i ilaxies lot which independent distance estimates
were obtained, cither from RR L\ rac stars. Ccphcids. or plan
etary nebulae. For galaxies represented by open symbols, the
average distance of the respective group is used. The solid line-
is a iit to similar data foi the Ursa Major cluster, together with
data of tho ! i fi lich in li idu 1 distan lim i
arc available (filled symbols). The larger dispersion around
the mean relation for the Sculptor group galaxies is due to the
group's extent along the line-of-sight
other, in particular with regard to their flat behavior in
the outer part. The flat rotation curve implies
_' 1
I I | I I I | I I I | I I I | |_
NGC7331 *;
600
';. :';
400
200
? ;
♦ 4
mIimImiImi I,"
400 600 800 1000 1200
Heliocentric velocity [km/s]
Fig. 3.20. 21 cm profile of the galaxy NGC 7331. The bold
dots indicate 20% and 50% of the maximum flux; these are of
relevance for the determination of the line width from which
the rotational velocity is derived
where the distance R from the center of the galaxy
refers to the flat part of the rotation curve. The exact
value is not important, though, if only v(R) « const. By
re- writing (3.15),
( M\
(3.16)
;s (/> =
(3.17)
and replacing R by the mean surface brightm
L/R 2 ,v/e obtain
This is the Tully-Fisher relation if M/L and (/> are the
same for all spirals. The latter is in fact suggested by
Freeman's law (Sect. 3.3.2). Since the shapes of rota-
tion curves for spirals seem to be very similar, the radial
dependence of the ratio of luminous to dark matter may
also be quite similar among spirals. Furthermore, since
the red or infrared mass-to-light ratios of a stellar pop-
ulation do not depend strongly on its age, the constancy
of M/L could also be valid if dark matter is included.
Although the line of argument presented above is far
from a proper derivation of the Tully-Fisher-relation,
it nevertheless makes the existence of such a scaling
relation plausible.
3. The World of Galaxies
log V c [km/s]
Fig. 3.21. Left panel: the mass contained ii
n of the rotational velocity V c for spirals. This stellar mass
is computed from the luminosity by multiplying it with a suit-
able stellar mass-to-light ratio which depends on the chosen
filter and which can be calculated from stellar population mod-
els. This is the "classical" Tully-Fisher relation. Squares and
circles denote galaxies for which V c was determined from the
21 -cm line width or from a spatially resolved rotation curve,
respectively. The colors of the symbols indicate the filter band
in which the luminosity was measured: H (red), K' (black), 1
(green). 6 (blue). Right panel: instead oi'lhe stellar mass, here
the sum of the stellar and gaseous mass is plotted. The gas mass
was derived from the flux in the 21-cm line, M gas = 1.4Mhi,
corrected for helium and metals. Molecular gas has no signif-
icant contribution to the baryonic mass. The line in both plots
is the Tully-Fisher relation with a slope of a = 4
Mass-to-Light Ratio of Spirals. We are unable to de-
termine the total mass of a spiral because the extent of
the dark halo is unknown. Thus we can measure M/L
only within a fixed radius. We shall define this radius as
R75, the radius at which the surface brightness attains
the value of 25 mag/arcsec 2 in the B-band; 5 then spirals
follow the relation
independently of their Hubble type. Within R25 one
finds M/L B = 6.2 for Sa's, 4.5 for Sb's, and 2.6 for Sc's.
This trend does not come as a surprise because late types
of spirals contain more young, blue and luminous stars.
.: had the surface hiightness docs
The Baryonic Tully-Fisher Relation. The above
"derivation" of the Tully-Fisher relation is based on the
assumption of a constant M/L value, where M is the to-
tal mass (i.e., including dark matter). Let us assume that
(i) the ratio of baryons to dark matter is constant, and
furthermore that (ii) the stellar populations in spirals are
similar, so that the ratio of stellar mass to luminosity is
a constant. Even under these assumptions we would ex-
pect the Tully-Fisher relation to be valid only if the gas
does not, or only marginally, contribute to the baryonic
mass. However, low-mass spirals contain a significant
fraction of gas, so we should expect that the Tully-
Fisher relation does not apply to these galaxies. Indeed,
it is found that spirals with a small u max < 100 km/s de-
viate significantly from the Tully-Fisher relation - see
Fig. 3.21(a).
Since the luminosity is approximately proportional
to the stellar mass, Lea M*, the Tully-Fisher relation is
a relation between t; max and M*. Adding the mass of the
3.4 Scaling Relations
gas, which can be determined from the strength of the
2 1 -cm line, to the stellar mass a much tighter correlation
is obtained, see Fig. 3.21(b). It reads
Mdis
= 2xl0 9 h~ 2 M Q ( " max V
°\100km/s/
(3.19)
and is valid over five orders of magnitude in disk mass
Mdi s k = M* + M gas . If no further baryons exist in spirals
(such as, e.g., MACHOs), this close relation means thai
the ratio of baryons and dark matter in spirals is constant
over a very wide mass range.
3.4.2 The Faber-Jackson Relation
A relation for elliptical galaxies, analogous to the Tully-
Fisher relation, was found by Sandra Faber and Roger
Jackson. They discovered that the velocity dispersion in
the center of ellipticals, ao, scales with luminosity (see
Fig. 3.22),
Lcxa
log(er ) = -0.1 M B + const
3.4.3 The Fundamental Plane
The Tully-Fisher and Faber-Jackson relations specify
a connection between the luminosity and a kinematic
property of galaxies. As we discussed previously, vari-
ous relations exist between the parameters of elliptical
galaxies. Thus one might wonder whether a relation ex-
ists between observables of elliptical galaxies for which
the dispersion is smaller than that of the Faber-Jackson
relation. Such a relation was indeed found and is known
as the fundamental plane.
To explain this relation, we will consider the vari-
ous relations between the parameters of ellipticals. In
Sect. 3.2.2 we saw that the effective radius of normal el-
lipticals is related to the luminosity (see Fig. 3.7). This
implies a relation between the surface brightness and
the effective radius,
R e oc (/)- a
(3.2 ii
where (7} e is the average surface brightness within the
effective radius, so that
(3.20)
"Deriving" the Faber-Jackson scaling relation is pos-
sible under the same assumptions as the Tully-Fisher
relation. However, the dispersion of ellipticals about
this relation is larger than that of spirals about the
Tully-Fisher relation.
= 2nR z e (7) e
From this, a relation between the luminosity and (7) e
results,
L oc R\ </) e oc (7); '
Fig. 3.22. The Faber-Jackson relation expresses a relation be-
tween the velocity dispersion and the luminosity of elliptical
galaxies. It can be derived from the virial theorem
<7} e o<L--
(3.23)
Hence, more luminous ellipticals have smaller surface
brightnesses, as is also shown in Fig. 3.7. By means
of the Faber-Jackson relation, L is related to oq, the
central velocity dispersion, and therefore, oq, (7) e , and
7? e are related to each other. The distribution of elliptical
galaxies in the three-dimensional parameter space (7? e ,
(7> e , Co) is located close to a plane defined by
7? e cxa L4 (7> e - -
(3.24)
Writing this relation in logarithmic form, we obtain
log 7? e = 0.34 (/z) e + 1.4 log CT + const , (3.25)
3. The World of Galaxies
where {fi) e is the average surface brightness within
R e , measured in mag/arcsec 2 . Equation (3.25) defines
a plane in this three-dimensional parameter space that
is known as the fundamental plane (FP). Different
projections of the fundamental plane are displayed in
Fig. 3.23.
How can this be Explained? The mass within R e can be
derived from the virial theorem, Ma(T 2 S e . Combining
this with (3.22) yields
Hence, the FP follows from the virial theorem provided
I — \ cc M or
oc V-
respectively ,
(3,27)
i.e., if the mass-to-light ratio of galaxies increases
slightly with mass. Like the Tully-Fisher relation, the
fundamental plane is an important tool for distance
estimations. It will be discussed more thoroughly later.
M </) e
which agrees with the FP in the form of (3.24) if
3.4.4 The D,-a Relation
Another scaling relation for ellipticals which is of sub-
stantial importance in practical applications is the D„-a
relation. D n is defined as that diameter of an ellipse
within which the average surface brightness /„ corre-
sponds to a value of 20.75 mag/arcsec 2 in the B-band.
If we now assume that all ellipticals have a self-similar
brightness profile, I(R) = I e f(R/R e ), with /(l) = 1,
then the luminosity within D n can be written as
6 7 8
loga + 0.26(u) e
Fig. 3.23. Projections of the fundamental
plane onto different two-parameter planes.
Upper left: the relation between radius and
m 1 in I 1 Ihi 1I1 ili 1 1
radius. Upper right: Faber-Jackson rela-
tion. Lower left: the relation between mean
surface brightness and velocity dispersion
shows the fundamental plane viewed from
above. Lower right: the fundamental plane
viewed from the side - the linear relation be-
tween radius and a combination of surface
brightness and velocity dispersion
3.5 Black Holes in the Centers of Galaxies
2 Dn/2
Mt) 7T = 27r/e / dRR f( R / R e)
D„/(2Re)
= 2nhRl j Ax x fix).
For a de Vaucouleurs profile we have approximately
f(x) oc x -1 2 in the relevant range of radius. Computing
the integral with this expression, we obtain
Replacing ^? e by the fundamental plane (3.24) then re-
sults in
D„oco- ( 5- 4 {7),: - 85 / e - 8 .
Since {/> e oc 7 e due to the assumed self-similar bright-
ness profile, we finally find
^OCOr'- 4 /^
(3.29)
This implies that D n is nearly independent of I e and
only depends on er - The D n -a relation (3.29) de-
scribes the properties of ellipticals considerably better
than the Faber-Jackson relation and, in contrast to the
fundamental plane, it is a relation between only two
observables. Empirically, we find that ellipticals follow
the normalized D„-a relation
-2.05
(3.30)
This result then instigates further questions: what dis-
tinguishes a "normal" galaxy from an AGN if both have
a SMBH in the nucleus? Is it the mass of the black
hole, the rate at which material is accreted onto it, or
the efficiency of the mechanism which is generating the
energy?
We will start with a concise discussion of how to
search for SMBHs in galaxies, then present some ex-
amples for the discovery of such SMBHs. Finally, we
will discuss the very tight relationship between the mass
of the SMBH and the properties of the stellar component
of a galaxy.
3.5.1 The Search for Supermassive Black Holes
We will start with the question of what a black hole
actually is. A technical answer is that a black hole
is the simplest solution of Einstein's theory of Gen-
eral Relativity which describes the gravitational field of
a point mass. Less technically - though sufficient for
our needs - we may say that a black hole is a point
mass, or a compact mass concentration, with an extent
smaller than its Schwarzschild radius rs (see below).
The Schwarzschild Radius. The first discussion of
black holes can be traced back to Laplace in 1795, who
considered the following: if one reduces the radius r of
a celestial body of mass M, the escape velocity v esc at
its surface will change,
2GM
and they scatter around this relation with a relative width
of about 15%.
3.5 Black Holes in the Centers
of Galaxies
As we have seen in Sect. 2.6.3, the Milky Way harbors
a black hole in its center. Furthermore, it is generally
accepted that the energy for the activity of AGNs is
generated by accretion onto a black hole (see Sect. 5.3).
Thus, the question arises as to whether all (or most)
galaxies contain a supermassive black hole (SMBH) in
their nuclei. We will pursue this question in this sec-
tion and show that SMBHs are very abundant indeed.
As a thought experiment, one can now see that for a suf-
ficiently small radius v esc will be equal to the speed of
light, c. This happens when the radius decreases to
- = 2.95xl0 5 cm —
\ M,
The radius rs is named the Schwarzschild radius, af-
ter Karl Schwarzschild who, in 1916, discovered the
point-mass solution for Einstein's field equations. For
our purpose we will define a black hole as a mass con-
centration with a radius smaller than r s . As we can
see, rs is very small: about 3 km for the Sun, and
r s ~ 10 12 cm for the SMBH in the Galactic center. At
a distance of D — Rq » 8 kpc, this corresponds to an
3. The World of Galaxies
angular radius of ~ 6 x 10" arcsec. Current observing
capabilities are still far from resolving scales of order
rs, but in the near future VLBI observations at very
short radio wavelengths may achieve sufficient angular
resolution to resolve the Schwarzschild radius for the
Galactic black hole. The largest observed velocities of
stars in the Galactic center, ~ 5000 km/s <^C c, indicate
that they are still well away from the Schwarzschild ra-
dius. However, in the case of the SMBH in our Galactic
center we can "look" much closer to the Schwarzschild
radius: with VLBI observations at wavelengths of 3 mm
the angular size of the compact radio source Sgr A* can
be constrained to be less than 0.3 mas, corresponding to
about 20rs. We will show in Sect. 5.3.3 that relativistic
effects are directly observed in AGNs and that veloci-
ties close to c do in fact occur there - which again is
a very direct indication of the existence of a SMBH.
If even for the closest SMBH, the one in the GC,
the Schwarzschild radius is significantly smaller than
the achievable angular resolution, how can we hope to
prove that SMBHs exist in other galaxies? Like in the
GC, this proof has to be found indirectly by detecting
a compact mass concentration incompatible with the
mass concentration of the stars observed.
The Radius of Influence. We consider a mass con-
centration of mass M. in the center of a galaxy where
the characteristic velocity dispersion of stars (or gas)
is a. We compare this velocity dispersion with the char-
acteristic velocity (e.g., the Kepler rotational velocity)
around a SMBH at a distance r, given by *jGM.jr.
From this it follows that, for distances smaller than
GM.
' U0 6 M o / V 100 km/s/
pc .
the SMBH will significantly affect the kinematics of
stars and gas in the galaxy. The corresponding angular
scale is
i(-£-)(^— Y 2 (—Y
\10 6 M Q ) V 100 km/s/ \lMpc/
where D is the distance of the galaxy. From this we im-
mediately conclude that our success in finding SMBHs
will depend heavily on the achievable angular resolu-
tion. The HST enabled scientists to make huge progress
in this field. The search for SMBHs promises to be suc-
cessful only in relatively nearby galaxies. In addition,
from (3.33) we can see that for increasing distance D the
mass M. has to increase for a SMBH to be detectable
at a given angular resolution.
Kinematic Evidence. The presence of a SMBH inside
r B H is revealed by an increase in the velocity dispersion
for r < r BH , which should then behave as a oc r" 1/2
for r < r B H- If the inner region of the galaxy rotates,
one expects, in addition, that the rotational velocity v mt
should also increase inwards ex r~ 1/2 .
Problems in Detecting These Signatures. The practi-
cal problems in observing a SMBH have already been
mentioned above. One problem is the angular resolu-
tion. To measure an increase in the velocities for small
radii, the angular resolution needs to be better than
0bh- Furthermore, projection effects play a role because
only the velocity dispersion of the projected stellar dis-
tribution, weighted by the luminosity of the stars, is
measured. Added to this, the kinematics of stars can be
rather complicated, so that the observed values for a
and f rot depend on the distribution of orbits and on the
geometry of the distribution.
Despite these difficulties, the detection of SMBHs
has been achieved in recent years, largely due to the
much improved angular resolution of optical telescopes
(like the HST) and to improved kinematic models.
3.5.2 Examples for SMBHs in Galaxies
Figure 3.24 shows an example for the kinematical
method discussed in the previous section. A long-slit
spectrum across the nucleus of the galaxy M84 clearly
shows that, near the nucleus, both the rotational velocity
and the velocity dispersion change; both increase dra-
matically towards the center. Figure 3.25 illustrates how
strongly the measurability of the kinematical evidence
for a SMBH depends on the achievable angular resolu-
tion of the observation. For this example of NGC 3115,
observing with the resolution offered by space-based
spectroscopy yields much higher measured velocities
than is possible from the ground. Particularly interest-
3.5 Black Holes in the Centers of Galaxies
ing is the observation of the rotation curve very close
to the center. Another impressive example is the central
region of M87, the central galaxy of the Virgo Cluster.
The increase of the rotation curve and the broadening
of the [Oll]-line (a spectral line of singly-ionized oxy-
gen) at k — 3727 A towards the center are displayed in
Fig. 3.26 and argue very convincingly for a SMBH with
M. % 3 x 10 9 M o .
The mapping of the Kepler rotation in the center of
the Seyfert galaxy NGC 4258 is especially spectacu-
lar. This galaxy contains water masers - very compact
sources whose position can be observed with very high
precision using VLBI techniques (Fig. 3.27). In this
case, the deviation from a Kepler rotation in the grav-
itational field of a point mass of M. ~ 3.5 x 10 7 M Q is
much less than 1%. The maser sources are embedded in
an accretion disk having a thickness of less than 0.3% of
its radius, of which also a warping is detected. Changes
in the radial velocities and the proper motions of these
maser sources have already been measured, so that the
model of a Kepler accretion disk has been confirmed in
detail.
All these observations are of course no proof of the
: of a SMBH in these galaxies because the
sources from which we obtain the kinematic evidence
are still too far away from the Schwarzschild radius. The
conclusion of the presence of SMBHs is rather that of
a missing alternative, as was already explained for the
case of the GC (Sect. 2.6.3). We have no other plausible
model for the mass concentrations detected. As for the
case of the SMBH in the Milky Way, an ultra-compact
star cluster might be postulated, but such a cluster would
not be stable over a long period of time. Based on the
existence of a SMBH in our Galaxy and in AGNs, the
SMBH hypothesis is the only plausible explanation for
these mass c
3.5.3 Correlation Between SMBH Mass
and Galaxy Properties
Currently, strong indications of SMBHs have been
found in about 35 normal galaxies, and their masses
have been estimated. This permits us to examine
whether, and in what way, M. is related to the properties
of the host galaxy. This leads us to the discovery of a re-
markable correlation; it is found that M, is correlated
with the absolute magnitude of the bulge component
(or the spheroidal component) of the galaxy in which
Fig. 3.24. An HST image of the nucleus of the galaxy M84 is
shown in the left-hand panel. M84 is a member of the Virgo
Cluster, about 15 Mpc away from us. The small rectangle de-
picts the position of the slit used by the STIS (Space Telescope
I 11 ( in i a ' 1 11 i in irument on-board the HST to ob-
tain a spectrum of the central region. This long-slit spectrum
is shown in the right-hand panel; the position along the slit
is plotted vertically, the wavelength of the light horizontally,
also illustrated by colors. Near the center of the galaxy the
wavelength suddenly changes because the rotational velocity
steeply increases inwards and then changes sign on the other
side of the center. This shows the Kepler rotation in the central
gra\ itational field of a SMBH, whose mass can be estimated
as M. ~ 3 x 1O 8 M
3. The World of Galaxies
. HST FOS {ap=0".21, a,=0".00)
•CFHTSIS (o,=0".24):
- CFHT Herzberg (o,=0".44)
Fig. 3.25. Rotational velocity (bottom) and velocity disper-
sion (top), as functions of the distance from the center along
the major axis of the galaxy NGC 3115. Colors of the sym-
bols mark observations with different instruments. Results
from CFHT data which have an angular resolution of 0'.'44 are
shown in blue. The SIS instrument at the CFHT uses active
optics to achieve roughly twice this angular resolution; corre-
sponding results are plotted in green. Finally, the red symbols
show the result from HST observations using the Faint Object
Spectrograph (FOS). As expected, with improved angular res-
olution an increase in the velocity dispersion is seen towards
the center. Even more dramatic is the impact of resolution on
measurements of the rotational velocity. Due to projection ef-
fects, the measured central velocity dispersion is smaller than
the real one; this effect can be corrected for. After correc-
tion, a central value of a ~ 600 km/s is found. This value is
much higher than the escape velocity from the central star
cluster if it were to consist solely of stars - it would dissolve
within ~ 2 x 10 4 years. Therefore, an additional compact
mass component of M. ~ 10 9 M Q must exist
0.5 0.0 0.5
Position, ;-:.'
3720 3750
Fig. 3.26. M87 has long been one of the
most promising candidates for harboring an
SMBH in its center. In this figure, the po-
sition of the slit is shown superimposed on
an Ha image of the galaxy (lower left) to-
gether with the spectrum of the [On] line
along this slit (bottom, center), and six spec-
tra corresponding to six different positions
along the slit, separated by 0'.' 14 each (lower
right). In the upper right panel the rotation
curve extracted from the data using a kine-
matical model is displayed. These results
show that a central mass concentration with
~ 3 x 10 9 M Q must be present, confined
to a region less than 3 pc across - indeed
le;i\ ing basically no alternative but a SMBH
3.6 Extragalactic Distance Determination
it is statistically highly significant, but the deviations
of the data points from this power law are considerably
larger than their error bars. An alternative way to express
this correlation is provided by the relation M/L on L° 25
found previously - see (3.27) - by which we can also
write M. on M° u ? .
An even better correlation exists between M. and the
velocity dispersion in the bulge component, as can be
seen in the right-hand panel of Fig. 3.28. This relation
is best described by
Fig. 3.27. The Seyfert galaxy NGC 4258 c
disk in its center in which several water masers are embed-
ded. In the top image, an artist's impression of (he hidden disk
and the jet is displayed, together with the line spectrum of the
maser sources. Their positions (center image) and velocities
have been mapped by VLBI observations. From these mea-
surements, the Kepler law for rotation in the gravitational field
of a point mass of M. = 25 x 10 6 M o in the center of this gal-
axy was verified. The best-fitting model of the central disk is
also plotted. The bottom image is a 20-cm map showing the
large-scale radio structure of the Seyfert galaxy
the SMBH is located (see Fig. 3.28, left). Here, the
bulge component is either the bulge of a spiral gal-
axy or an elliptical galaxy as a whole. This correlation
is described by
M. = 1.35 x 10 8 M Q ( — )
°\200km/s/
M. = 0.93 x 10 s M Q
(3.34)
where the exact value of the exponent is still subject to
discussion, and where a slightly higher value M. oc a 45
might better describe the data. The difference in the re-
sults obtained by different groups can partially be traced
back to different definitions of the velocity dispersion,
especially concerning the choice of the spatial region
across which it is measured. It is remarkable that the
deviations of the data points from the correlation (3.35)
are compatible with the error bars for the measurements
of M.. Thus, we have at present no indication of an
intrinsic dispersion of the M.-a relation.
In fact, there have been claims in the literature that
even globular clusters contain a black hole; however,
these claims are not undisputed. In addition, there may
be objects that appear like globular clusters, but are in
fact the stripped nucleus of a former dwarf galaxy. In
this case, the presence of a central black hole is not
unexpected, provided the scaling relation (3.35) holds
down to very low velocity dispersion.
To date, the physical origin of this very close relation
has not been understood in detail. The most obvious
apparent explanation - that in the vicinity of a SMBH
with a very large mass the stars are moving faster than
around a smaller-mass SMBH - is not conclusive: the
mass of the SMBH is significantly less than one percent
of the mass of the bulge component. We can therefore
disregard its contribution to the gravitational field in
which the stars are orbiting. Instead, this correlation has
to be linked to the fact that the spheroidal component
of a galaxy evolves together with the SMBH. A better
understanding of this relation can only be found from
models of galaxy evolution. We will continue with this
topic in Sect. 9.6.
Fig. 3.28. Correlation of SMBH mass M.
with the absolute magnitude Mb, bulge (left)
and the velocity dispersion <r e (right) in the
bulge component of the host galaxy. Circles
(squares, triangles) indicate measurements
that are based on stellar kinematics (gas
kinematics, maser disks)
8 2.0 2.2 2.4 2.6
loga e (kms- 1 )
3.6 Extragalactic Distance
Determination
In Sect. 2.2 we discussed methods for distance deter-
mination within our own Galaxy. We will now proceed
with the determination of distances to other galaxies.
It should be noted that the Hubble law (1.2) specifies
a relation between the redshift of an extragalactic ob-
ject and its distance. The redshift z is easily measured
from the shift in spectral lines. For this reason, the
Hubble law (and its generalization - see Sect. 4.3.3)
provides a simple method for determining distance.
However, to apply this law, the Hubble constant Ho
must first be known, i.e., the Hubble law must be cal-
ibrated. Therefore, in order to determine the Hubble
constant, distances have to be measured independently
from redshift.
Furthermore, it has to be kept in mind that besides
the general cosmic expansion, which is expressed in the
Hubble law, objects also show peculiar motion, like the
velocities of galaxies in clusters of galaxies or the mo-
tion of the Magellanic Clouds around our Milky Way.
These peculiar velocities are induced by gravitational
acceleration resulting from the locally inhomogeneous
mass distribution in the Universe. For instance, our
Galaxy is moving towards the Virgo Cluster of gal-
axies, a dense accumulation of galaxies, due to the
gravitational attraction caused by the cluster mass. The
measured redshift, and therefore the Doppler shift, is al-
ways a superposition of the cosmic expansion velocity
and peculiar velocities.
CMB Dipole Anisotropy. The peculiar velocity of
the Galaxy is very precisely known. The radiation of
the cosmic microwave background is not completely
isotropic but instead shows a dipole component. This
component originates in the velocity of the Solar Sys-
tem relative to the rest-frame in which the CMB appears
isotropic (see Fig. 1.17). Due to the Doppler effect, the
CMB appears hotter than average in the direction of our
motion and cooler in the opposite direction. Analyzing
this CMB dipole allows us to determine our peculiar
velocity, which yields the result that the Sun moves at
a velocity of (368 ± 2) km/s relative to the CMB rest-
frame. Furthermore, the Local Group of galaxies (see
Sect. 6.1) is moving at i>lg ^ 600 km/s relative to the
CMB rest-frame.
Distance Ladder. For the redshift of a source to be
dominated by the Hubble expansion, the cosmic ex-
pansion velocity v — cz — HqD has to be much larger
than typical peculiar velocities. This means that in order
to determine Hq we have to consider sources at large
distances for the peculiar velocities to be negligible
compared to H D.
Making a direct estimate of the distances of dis-
tant galaxies is very difficult. Traditionally one uses
a distance ladder, at first, the absolute distances to
nearby galaxies are measured directly. If methods to
measure relative distances (that is, distance ratios) with
sufficient precision are utilized, the distances to galax-
ies further away are then determined relative to those
nearby. In this way, by means of relative methods, dis-
tances are estimated for galaxies that are sufficiently far
away for their redshift to be dominated by the Hubble
flow.
3.6.1 Distance of the LAAC
The distance of the Large Magellanic Cloud (LMC) can
be estimated using various methods. For example, we
can resolve and observe individual stars in the LMC,
which forms the basis of the MACHO experiments (see
Sect. 2.5.2). Because the metallicity of the LMC is sig-
nificantly lower than that of the Milky Way, some of
the methods discussed in Sect. 2.2 are only applicable
after correcting for metallicity effects, e.g., the photo-
metric distance determination or the period-luminosity
relation for pulsating stars.
Perhaps the most precise method of determining
the distance to the LMC is a purely geometrical one.
The supernova SN 1987A that exploded in 1987 in the
LMC illuminates a nearly perfectly elliptical ring (see
Fig. 3.29). This ring consists of material that was once
ejected by the stellar winds of the progenitor star of
the supernova and that is now radiatively excited by
energetic photons from the supernova explosion. The
corresponding recombination radiation is thus emitted
only when photons from the SN hit this gas. Because
the observed ring is almost certainly intrinsically cir-
cular and the observed ellipticity is caused only by its
inclination with respect to the line-of- sight, the distance
to SN 1987 A can be derived from observations of the
ring. First, the inclination angle is determined from its
observed ellipticity. The gas in the ring is excited by
photons from the SN a time R/c after the original ex-
plosion, where R is the radius of the ring. We do not
observe the illumination of the ring instantaneously be-
cause light from the section of the ring closer to us
reaches us earlier than light from the more distant part.
Thus, its illumination was seen sequentially along the
ring. Combining the time delay in the illumination be-
tween the nearest and farthest part of the ring with its
inclination angle, we then obtain the physical diameter
of the ring. When this is compared to the measured an-
gular diameter of ~ 1'.'7, the ratio yields the distance to
SN 1987 A,
DsNi987A«51.8kpc±6%.
If we now assume the extent of the LMC along the line-
of-sight to be small, this distance can be identified with
Fig. 3.29. The ring around supernova 1987A in the LMC is
illuminated by photons from the explosion which induce the
radiation from the gas in the ring. It is inclined towards the
line of sight; thus it appears to be elliptical. Lighting up of
the ring was not instantaneous, due to the finite speed of light:
those sections of the ring closer to us lit up earlier than the
more distant parts. From the time shift in the onset of radia-
tion across the ring, its diameter can be derived. Combining
this with the measured angular diameter of the ring, the dis-
tance to SN 1987A - and thus the distance to the LMC - can
be determined
the distance to the LMC. The value is also compatible
with other distance estimates (e.g., as derived by using
photometric methods based on the properties of main-
sequence stars - see Sect. 2.2.4).
3.6.2 The Cepheid Distance
In Sect. 2.2.7, we discussed the period-luminosity re-
lation of pulsating stars. Due to their high luminosity,
Cepheids turn out to be particularly useful since they
can be observed out to large distances.
For the period-luminosity relation of the Cepheids to
be a good distance measure, it must first be calibrated.
This calibration has to be done with as large a sample
of Cepheids as possible at a known distance. Cepheids
in the LMC are well-suited for this purpose because
we believe we know the distance to the LMC quite pre-
cisely, see above. Also, due to the relatively small extent
of the LMC along the line-of-sight, all Cepheids in the
3. The World of Galaxies
LMC should be located at approximately the same dis-
tance. For this reason, the period-luminosity relation
is calibrated in the LMC. Due to the large number of
Cepheids available for this purpose (many of them have
been found in the microlensing surveys), the resulting
statistical errors are small. Uncertainties remain in the
form of systematic errors related to the metallicity de-
pendence of the period-luminosity relation; however,
these can be corrected for since the color of Cepheids
depends on the metallicity as well.
With the high angular resolution of the HST, individ-
ual Cepheids in galaxies are visible at distances up to
that of the Virgo cluster of galaxies. In fact, determining
the distance to Virgo as a central step in the determi-
nation of the Hubble constant was one of the major
scientific aims of the HST. In the Hubble Key Project,
the distances to numerous spiral galaxies in the Virgo
Cluster were determined by identifying Cepheids and
measuring their periods.
3.6.3 Secondary Distance Indicators
The Virgo Cluster, at a measured distance of about
16 Mpc, is not sufficiently far away from us to directly
determine the Hubble constant from its distance and
redshift, because peculiar velocities still contribute con-
siderably to the measured redshift at this distance. To
get to larger distances, a number of relative distance
indicators are used. They are all based on measuring
the distance ratio of galaxies. If the distance to one of
the two is known, the distance to the other is then ob-
tained from the ratio. By this procedure, distances to
more remote galaxies can be measured. Below, we will
review some of the most important secondary distance
indicators.
SN la. Supernovae of Type la are to good approximation
standard candles, as will be discussed more thoroughly
in Sect. 8.3.1. This means that the absolute magnitudes
of SNe la are all within a very narrow range. To mea-
sure the value of this absolute magnitude, distances must
be known for galaxies in which SN la explosions have
been observed and accurately measured. Therefore, the
Cepheid method was applied especially to such galax-
ies, in this way calibrating the brightness of SNe la.
SNe la are visible over very large distances, so that they
also permit distance estimates at such large redshifts
that the simple Hubble law (1.6) is no longer valid,
but needs to be generalized based on a cosmological
model (Sect. 4.3.3). As we will see later, these measure-
ments belong to the most important pillars on which our
standard model of cosmology rests.
Surface Brightness Fluctuations of Galaxies. Another
method of estimating distance ratios is surface bright-
ness fluctuations. It is based on the fact that the number
of bright stars per area element in a galaxy fluctuates -
purely by Poisson noise: If N stars are expected in an
area element, relative fluctuations of V~N/N — l/^/N
of the number of stars will occur. These are observed in
fluctuations of the local surface brightness. To demon-
strate that this effect can be used to estimate distances,
we consider a solid angle dco. The corresponding area
element dA — D 2 dco depends quadratically on the dis-
tance D of the galaxy; the larger the distance, the larger
the number of stars N in this solid angle, and the smaller
the relative fluctuations of the surface brightness. By
comparing the surface brightness fluctuations of differ-
ent galaxies, one can then estimate relative distances.
This method also has to be calibrated on the galaxies
for which Cepheid distances are available.
Planetary Nebulae. The brightness distribution of plan-
etary nebulae in a galaxy seems to have an upper limit
which is the nearly the same for each galaxy (see
Fig. 3.30). If a sufficient number of planetary nebulae
are observed and their brightnesses measured, it enables
us to determine their luminosity function from which
the maximum apparent magnitude is then derived. By
calibration on galaxies of known Cepheid distance, the
corresponding maximum absolute magnitude can be de-
termined, which then allows the determination of the
distance modulus for other galaxies, thus their distances.
Scaling Relations. The scaling relations for galaxies -
fundamental plane for ellipticals, Tully-Fisher relation
for spirals (see Sect. 3.4) - can be calibrated on local
groups of galaxies or on the Virgo Cluster, the dis-
tances of which have been determined from Cepheids.
Although the scatter of these scaling relations can be
15% for individual galaxies, the statistical fluctuations
are reduced when observing several galaxies at about
the same distance (such as in clusters and groups). This
3.7 Luminosity Function of Galaxies
Absolute 515007 magnitude
Fig. 3.30. Brightness distribution of planetary nebulae in An-
dromeda (M31), M81, three galaxies in the Leo I group, and
six galaxies in the Virgo Cluster. The plotted absolute mag-
nilud v i m< i in .1 in ill. emi ion lini ol d< ul I. ionized
oxygen at X = 5007 A in which a large fraction of the lu-
minosity of a planetary nebula is emitted. This characteristic
property is also used in the identification of such objects in
other galaxies. In all cases, the distribution is described by
a nearly identical luminosity function; it seems to be a univer-
sal function in galaxies. Therefore, the brightness distribution
of planetary nebulae can be used to estimate the distance
of a galaxy. In the fits shown, the data points marked by
open symbols were disregarded: at these magnitudes, the
distribution function is probably not complete
enables u
of galaxies.
The Hubble Constant. In particular, the ratio of dis-
tances to the Virgo and the Coma clusters of galaxies is
estimated by means of these various secondary distance
measures. Together with the distance to the Virgo Clus-
ter as determined from Cepheids, we can then derive
the distance to Coma. Its redshift (z ss 0.023) is large
enough for its peculiar velocity to make no significant
contribution to its redshift, so that it is dominated by
the Hubble expansion. By combining the various meth-
ods we obtain a distance to the Coma cluster of about
90 Mpc, resulting in a Hubble constant of
Hq--
72±8km/s/Mpc
(3.36)
The error given here denotes the statistical uncertainty
in the determination of Hq. Besides this uncertainty,
possible systematic errors of the same order of magni-
tude may exist. In particular, the distance to the LMC
plays a crucial role. As the lowest rung in the distance
latter, it has an effect on all further distance estimates.
We will see later (Sect. 8.7.1) that the Hubble constant
can also be measured by a completely different method,
based on tiny small-scale anisotropics of the cosmic
microwave background, and that this method results in
a value which is in impressively good agreement with
the one in (3.36).
3.7 Luminosity Function of Galaxies
Definition of the Luminosity Function. The luminos-
ity function specifies the way in which the members of
a class of objects are distributed with respect to their lu-
minosity. More precisely, the luminosity function is the
number density of objects (here galaxies) of a specific
luminosity. <P(M) dM is defined as the number den-
sity of galaxies with absolute magnitude in the interval
[M, M + dM]. The total density of galaxies is then
Accordingly, <P(L) dL is defined as the number density
of galaxies with a luminosity between L and L + dL. It
3. The World of Galaxies
should be noted here explicitly that both definitions of
the luminosity function are denoted by the same symbol,
although they represent different mathematical func-
tions, i.e., they describe different functional relations. It
is therefore important (and in most cases not difficult)
to deduce from the context which of these two functions
is being referred to.
Problems in Determining the Luminosity Function.
At first sight, the task of determining the luminosity
function of galaxies does not seem very difficult. The
history of this topic shows, however, that we encounter
a number of problems in practice. As a first step, the
determination of galaxy luminosities is required, for
which, besides measuring the flux, distance estimates
are also necessary. For very distant galaxies redshift is
a sufficiently reliable measure of distance, whereas for
nearby galaxies the methods discussed in Sect. 3.6 have
to be applied.
Another problem occurs for nearby galaxies, namely
the large-scale structure of the galaxy distribution. To
obtain a representative sample of galaxies, a suffi-
ciently large volume has to be surveyed because the
galaxy distribution is heavily structured on scales of
~ lOO/i" 1 Mpc. On the other hand, galaxies of partic-
ularly low luminosity can only be observed locally, so
the determination of <P(L) for small L always needs
to refer to local galaxies. Finally, one has to deal with
the so-called Malmquist bias; in a flux-limited sample
luminous galaxies will always be overrepresented be-
cause they are visible at larger distances (and therefore
are selected from a larger volume). A correction for this
effect is always necessary.
3.7.1 The Schechter Luminosity Function
The global galaxy distribution is well approximated by
the Schechter luminosity function
w K^)(zO" exp (- L/L *)
where L* is a characteristic luminosity above which the
distribution decreases exponentially, a is the slope of
the luminosity function for small L, and 0* specifies
the normalization of the distribution. A schematic plot
of this function is shown in Fig. 3.31.
Expressed in magnitudes, this function appears much
more complicated. Considering that an interval dL in
luminosity corresponds to an interval dM in abso-
lute magnitude, with dL/L — —0.4 In 10 dM, and using
<P(L) dL — <P(M) dM, i.e., the number of sources in
these intervals are of course the same, we obtain
dL
= (0.4 1nlO)<P*10 ,M
xexp|
-io u -'
(3.39)
(3.40)
As mentioned above, the determination of the parame-
ters entering the Schechter function is difficult; a set of
parameters in the blue band is
<P* = 1.6x 10~ 2 /z 3 Mpc~ 3 ,
M* =-19.7 + 51og/i, or
= 1.2xl0 10 ft-
= -1.07.
(3.41)
While the blue light of galaxies is strongly affected by
star formation, the luminosity function in the red bands
measures the typical stellar distribution. In the K-band,
we have
<P* = 1.6 x 10"
-0.9 .
- 2 h 3 Mpc" 3 ,
5 log/;,
The total number density of galaxies is formally infinite
if a < — 1, but the validity of the Schechter function
does of course not extend to arbitrarily small L. The
luminosity density
w
dL L 0(L) = <P* L* r(2 + a)
(3,43)
is finite for a > — 2. 6 The integral in (3.43), for a ~ — 1,
is dominated by L ~ L*, and n — <P* is thus a good
estimate for the mean density of L*-galaxies.
( 3.38) ''/"(., i is the Gamma function, defined by
! » in in c int. l\n + \) = n\. Wehave 7"(0.7) ~ 1.30,71(1) = 1,
71(1. 3) ~ 0.90. Since these values are all close to unity, l m ~ &*L*
is a good approximation for the luminosity density.
3.7 Luminosity Function of Galaxies
s of the galaxy luminosity function from
the Schechter form are common. There is also no obvi-
ous reason why such a simple relation for describing
the luminosity distribution of galaxies should exist.
Although the Schechter function seems to be a good
representation of the total distribution, each type of
galaxy has its own luminosity function, with each func-
tion having a form that strongly deviates from the
Schechter function - see Fig. 3.32. For instance, spi-
rals are relatively narrowly distributed in L, whereas
the distribution of ellipticals is much broader if we
account for the full L-range, from giant ellipticals to
dwarf ellipticals. E's dominate in particular at large L;
the low end of the luminosity function is likewise
dominated by dwarf ellipticals and Irr's. In addition,
the luminosity distribution of cluster and group gal-
axies differs from that of field galaxies. The fact that
the total luminosity function can be described by an
equation as simple as (3.38) is, at least partly, a coin-
cidence ("cosmic conspiracy") and cannot be modeled
easily.
3.7.2 The Bimodal Color Distribution of Galaxies
The classification of galaxies by morphology, given by
the Hubble classification scheme (Fig. 3.2), has the dis-
advantage that morphologies of galaxies are not easy to
quantify. Traditionally, this was done by visual inspec-
tion but of course this method bears some subjectivity of
the researcher doing it. Furthermore, this visual inspec-
tion is time consuming and cannot be performed on large
samples of galaxies. Various techniques were developed
to perform such a classification automatically, includ-
ing brightness profile fitting - a de Vaucouleurs profile
indicates an elliptical galaxy whereas an exponential
brightness profile corresponds to a spiral.
Even these methods cannot be applied to galaxy sam-
ples for which the angular resolution of the imaging is
not much better than the angular size of galaxies - since
then, no brightness profiles can be fitted. An alternative
to classify galaxies is provided by their color. We ex-
pect that early-type galaxies are red, whereas late-type
galaxies are considerably bluer. Colors are much eas-
o Composite cluster galaxy
luminosity distribution
• CD galaxies included
Absolute Magnitude M J( ,
(b)
7 ^
Slope a
tx
L* \
Fig. 3.31. Left panel: gaki.\\ luminosity function as obtained
from 1 3 clusters of galaxies. Tor the solid circles, cD galaxies
have also been included. Upper panel: a schematic plot of
the Schechter function
3. The World of Galaxies
-16 -14 -12
local field
Fig. 3.32. The luminosity function for different Hubble types
of field galaxies (top) and galaxies in the Virgo Cluster of
galaxies (bottom). Dashed curves denote extrapolation-;. In
contrast to Fig. 3.31, the more luminous galaxies are plot-
ted towards the left. The Schechter luminosity function of the
total galaxy distribution is compiled from the sum of the lumi
nosity distributions of individual galaxy types that all deviate
significantly from the Schechter function. One can see that
in clusters the majoi contribution at faint magnitudes comes
from the dwarf ellipticals (dEs), and that at the bright end
ellipticals and SO's contribute much more strongly to the lu-
minosity function than they do in the field. This trend is even
more prominent in regular clusters of galaxies
ier to measure than morphology, in particular for very
small galaxies. Therefore, one can study the luminosity
function of galaxies, classifying them by their color.
Using photometric measurements and spectroscopy
from the Sloan Digital Sky Survey (see Sect. 8.1.2), the
colors and absolute magnitudes of ~ 70 000 low-red-
shift galaxies has been studied; their density distribution
in a color-magnitude diagram are plotted in the left-
hand side of Fig. 3.33. From this figure we see imme-
diately that there are two density peaks of the galaxy
distribution in this diagram: one at high luminosities
and red color, the other at significantly fainter absolute
magnitudes and much bluer color. It appears that the
galaxies are distributed at and around these two den-
sity peaks, hence galaxies tend to be either luminous
and red, or less luminous and blue. We can also easily
see from this diagram that the luminosity function of
red galaxies is quite different from that of blue galaxies,
which is another indication for the fact that the sim-
ple Schechter luminosity function (3.38) for the whole
galaxy population most likely is a coincidence.
We can next consider the color distribution of galax-
ies at a fixed absolute magnitude M r . This is obtained
by plotting the galaxy number density along vertical
cuts through the left-hand side of Fig. 3.33. When this
is done for different M r , it turns out that the color dis-
tribution of galaxies is bimodal: over a broad range
in absolute magnitude, the color distribution has two
peaks, one at red, the other at blue u—r. Again, this
fact can be seen directly from Fig. 3.33. For each value
of M r , the color distribution of galaxies can be very well
fitted by the sum of two Gaussian functions. The cen-
tral colors of the two Gaussians is shown by the two
dashed curves in the left panel of Fig. 3.33. They be-
come redder the more luminous the galaxies are. This
luminosity-dependent reddening is considerably more
pronounced for the blue population than for the red
galaxies,.
To see how good this fit indeed is, the right-hand
side of Fig. 3.33 shows the galaxy density as obtained
from the two-Gaussian fits, with solid contours corre-
sponding to the red galaxies and dashed contours to
the blue ones. We thus conclude that the local galaxy
population can be described as a bimodal distribution
in u—r color, where the characteristic color depends
slightly on absolute magnitude. The galaxy distribu-
tion at bright absolute magnitudes is dominated by red
galaxies, whereas for less luminous galaxies the blue
population dominates. The luminosity function of both
populations can be described by Schechter functions;
however these two are quite different. The characteris-
tic luminosity is about one magnitude brighter for the
red. galaxies than for the blue ones, whereas the faint-end
slope a is significantly steeper for the blue galaxies. This
3.8 Galaxies as Gravitational Lenses
I - of galaxies
V survey /V ir]a) . corrected,
ontours on a log scale.
M r -51og(h w )
Fig. 3.33. The density of galaxies in color-magnitude space.
The color of ~ 70 000 galaxies with redshifts 0.01 < z < 0.08
from the Sloan Digital Sky Survey is measured by the rest-
frame u — r, i.e., after a (small) correction for their redshifi
was applied. The density contours, which were corrected for
selection effect'; like the Malnu|uisl bias, are logaiiihinicalK
spaced, with a factor of ~J2 between c
the left-hand panel, the measured distribution is shown. Ob-
viously, two peaks of the galaxy density are clearly visible,
one at a red color of u — r ~ 2.5 and an absolute magnitude
of M r ~ —21, the other at a bluer color of u — r ~ 1.3 and
significantly fainter magnitudes. The right-hand panel corre-
sponds to the modeled galaxy density, as is described in the
again is in agreement of what we just learned: for high
luminosities, the red galaxies clearly dominate, whereas
at small luminosities, the blue galaxies are much more
abundant.
The mass-to-light ratio of a red stellar population is
larger than that of a blue population, since the former no
longer contains massive luminous stars. The difference
in the peak absolute magnitude between the red and blue
galaxies therefore corresponds to an even larger differ-
ence in the stellar mass of these two populations. Red
galaxies in the local Universe have on average a much
higher stellar mass than blue galaxies. This fact is il-
lustrated by the two dotted lines in the right-hand panel
of Fig. 3.33 which correspond to lines of constant stel-
lar mass of ~ 2-3 x 10 10 M Q . This seems to indicate
a very characteristic mass scale for the galaxy distribu-
tion: most galaxies with a stellar mass larger than this
characteristic mass scale are red, whereas most of those
with a lower stellar mass are blue.
Obviously, these statistical properties of the galaxy
distribution must have an explanation in terms of the
evolution of galaxies; we will come back to this issue
in Chap. 9.
3.8 Galaxies as Gravitational Lenses
In Sect. 2.5 the gravitational lens effect was discussed,
where we concentrated on the deflection of light by
point masses. The lensing effect by stars leads to im-
age separations too small to be resolved by any existing
telescope. Since the separation angle is proportional to
the square root of the lens mass (2.79), the angular sepa-
ration of the images will be about a million times larger
if a galaxy acts as a gravitational lens. In this case it
should be observable, as was predicted in 1937 by Fritz
Zwicky. Indeed, multiple images of very distant sources
have been found, together with the galaxy responsible
for the image splitting. In this section we will first de-
scribe this effect by continuing the discussion we began
in Sect. 2.5.1. Examples of the lens effect and its various
applications will then be discussed.
3.8.1 The Gravitational Lensing Effect - Part II
The geometry of a typical gravitational lens system is
sketched in Fig. 2.21 and again in Fig. 3.34. The phys-
3. The World of Galaxies
« = £*•
We assume that the deflecting mass has a small extent
along the line-of- sight, as compared to the distances be-
tween observer and lens (D d ) and between lens and
source (D ds ), L <$C D d and L <$C D ds . All mass ele-
ments can then be assumed to be located at the same
distance D d . This physical situation is called a geometri-
cally ih in lens. If a galaxy acts as the lens, this condition
is certainly fulfilled - the extent of galaxies is typically
~ 100/i _1 kpc while the distances of lens and source
are typically ~ Gpc. We can therefore write (3.45) as
a superposition of Einstein angles of the form (2.71),
w-Zf «-»
II-&I 2 '
Fig. 3.34. As a reminder, another sketch of the lens geometry
ical description of such a lens system for an arbitrary
mass distribution of the deflector is obtained from the
following considerations.
If the gravitational field is weak (which is the case
in all situations considered here), the gravitational ef-
fects can be linearized. 7 Hence, the deflection angle of
a lens that consists of several mass components can
be described by a linear superposition of the deflection
angles of the individual components,
(3,45)
where £, is the projected position vector of the mass
element m,, and § describes the position of the light ray
in the lens plane, also called the impact vector.
For a continuous mass distribution we can imag-
ine subdividing the lens into mass elements of mass
dm = i7(|)d 2 £, where E(%) describes the surface mass
density of the lens at the position £, obtained by pro-
jecting the spatial (three-dimensional) mass density p
along the line-of-sight to the lens. With this definition
the deflection angle (3.46) can be transformed into an
integral.
(3.46)
4G f , ,
j-j
This deflection angle is then inserted into the lens
equation (2.75),
where £ = D<±0 describes the relation between the posi-
tion £ of the light ray in the lens plane and its apparent
direction 0. We define the scaled deflection angle as in
(2.76),
a( fi) = ^&(D d 0) ,
so that the lens equation (3.48) can be written in the
simple form (see Fig. 3.34)
P = o
-a(0)
' «{»')
te the scale
-0'
(3.49)
A more co
as follows,
«(*) =
where
hi*
d deflection is
(3.50)
\9-
-0'| 2 '
quantif) [he licld strength is to apph the viilal theorem: if a mass
distribution is in \irial equilibrium, then v 1 - <t>. and weak fields are
therefore characterized In r ; /<~ « I. Because the typical \clocitics
in galaxies are ~ 200km/s, for galaxies &/c 2 < 1(T 6 . The typical
velocities ol galaxies in a cluster of galaxies arc - 1000 knt/s. so that
in clusters 0/c 2 < 10 -5 . Thus the gra\ itational fields occurring are
weak in both cases.
is the dimensionless surfc
called critical surface mass density
(3.51)
density, and the so-
3.8 Galaxies as Gravitational Lenses
depends only on the distances to the lens and to the
source. Although I7 cr incorporates a combination of cos-
mological distances, it is of a rather "human" order of
magnitude,
\D s lGpcJ &
A source is visible at several positions on the sphere,
or multiply imaged, if the lens equation (3.49) has sev-
eral solutions for a given source position /?. A more
detailed analysis of the properties of this lens equation
yields the following general result:
If £ > S a in at least one point of the lens, then
source positions fi exist such that a source at ft has
multiple images. It immediately follows that k is
a good measure for the strength of the lens. A mass
distribution with k <K 1 at all points is a weak lens,
unable to produce multiple images, whereas one
with k > 1 for certain regions of is a strong lens.
denotes the distance of a point from the center of the
lens. In this case, the deflection angle is directed radially
inwards, and we obtain
where M(f) is the mass within radius f . Accordingly.
for the scaled deflection angle we have
a( e ) = r ^.-±. 2 jd9'e'K(e'), 0.55)
where, in the last step, m (0) was defined as the dimen-
sionless mass within 6. Since a and 9 are collinear, the
lens equation becomes one-dimensional because only
the radial coordinate needs to be considered,
For sources that are small compared to the character-
istic scales of the lens, the magnification fi of an image,
caused by the differential light deflection, is given by
(2.83), i.e.,
fi =
o
-1
The importance of the gravitational lens effect for extra-
galactic astronomy stems from the fact that gravitational
light deflection is independent of the nature and the state
of the deflecting matter. Therefore, it is equally sensi-
tive to both dark and baryonic matter and independent
of whether or not the matter distribution is in a state
of equilibrium. The lens effect is thus particularly suit-
able for probing matter distributions, without requiring
any further assumptions about the state of equilibrium
or the relation between dark and luminous matter.
3.8.2 Simple Models
Axially Symmetric Mass Distributions. The simplest
models for gravitational lenses are those which are axi-
ally symmetric, for which 17 (|) — I7(£), where £ = |£|
P = e-a(6) =
•u0)
13.56)
m(ff) =
An illustration of this one-dimensional lens mapping is
shown in Fig. 3.35.
Example: Point-Mass Lens. For a point mass M, the
dimensionless mass becomes
AGM D ds
' c 2 D d D s '
reproducing the lens equation from Sect. 2.5.1 for
a point-mass lens.
Example: Isothermal Sphere. We saw in Sect. 2.4.2
that the rotation curve of our Milky Way is flat for large
radii, and we know from Sect. 3.3.3 that the rotation
curves of other spiral galaxies are flat as well. This in-
dicates that the mass of a galaxy increases proportional
to r, thus p(r) ex r~ 2 , or more precisely,
P(r) =
2jiGr :
(3,57)
Here, a v is the one-dimensional velocity dispersion of
stars in the potential of the mass distribution if the
distribution of stellar orbits is isotropic. In principle,
a v is therefore measurable spectroscopically from the
line width. The mass distribution described by (3.57) is
called a singular isothermal sphere (SIS). Because this
mass model is of significant importance not only for the
analysis of the lens effect, we will discuss its properties
in a bit more detail.
3. The World of Galaxies
The density (3.57) diverges for r -> as p ex r~
so that the mass model cannot be applied up to tl
very center of a galaxy. However, the steep central ii
crease of the rotation curve shows that the core region
of the mass distribution, in which the density function
will deviate considerably from the r _2 -law, must be
small for galaxies. Furthermore, the mass diverges for
large r such that M(r) <x r. The mass profile thus has
to be cut off at some radius in order to get a finite
total mass. This cut-off radius is probably very large
(> 100 kpc for L* -galaxies) because the rotation curves
are flat to at least the outermost point at which they are
observable.
The SIS is an appropriate simple model for gravita-
tional lenses over a wide range in radius since it seems
to reproduce the basic properties of lens systems (such
as image separation) quite well. The surface mass den-
sity is obtained from the projection of (3.57) along the
line-of-sight,
27(£) = —?-
which yields the projected mass Af(£) within radius %
Fij». 335. Sketch oi'an axially symmetric lens. In the ton panel.
9 — a{6) is plotted as a function of the angular separation
from the center of the lens, together with the straight line
ft = 9. The three intersection points of the horizontal line at
fixed ft with the curve 9 - u{9) are the three solutions of
the lens equation. The bottom image indicates the positions
and sizes of the images on the observer's sky. Here, Q is the
un leased source (which is not \ isible itself in the case of light
de i lection, of course!), and A, Bl, B2 are the observed images
of the source. The sizes of the images, and thus their fluxes,
differ considerably : ihe inner image B2 is particularly weak in
the case depicted here. The flux of B2 relative to thai of image
A depends strongly on the core radius of the lens; it can be so
low as to render the third image unobservable. In the special
case of a singular isothermal sphere, the innermost image is
in fact absent
M(£) -
it J Vr £«') = ^. (3.5
With (3.54) the deflection angle can be obtained,
^)=4^)\
y(6) = 4it
vc/ \D S/
Thus the deflection angle for an SIS is constant and
equals G E , and it depends quadratically on a v . 6 E is
called the Einstein angle of the SIS. The characteristic
scale of the Einstein angle is
0e = 1-15
-) 2 (-)
from which we conclude that the angular scale of the
lens effect in galaxies is about an arcsecond for massive
galaxies. The lens equation (3.56) for an SIS is
3.8 Galaxies as Gravitational Lenses
where we took into account the fact that the deflection
angle is negative for 9 < since it is always directed
inwards.
Solution of the Lens Equation for the Singular Iso-
thermal Sphere. If \/}\ <9 E , two solutions of the lens
equation exist,
9 l = p+9 E .
(3.63)
Without loss of generality, we assume /? > 0; then
9\ > 9 E > and > 9 2 > —9 E : one image of the source
is located on either side of the lens center, and the
separation of the images is
A^, 1 -, 2 .2, E .2:'3(^^) 2 (^)
A,
"(3^64)
Thus, the angular separation of the images does not de-
pend on the position of the source. For massive galaxies
acting as lenses it is of the order of somewhat more than
one arcsecond. For p > 9 E only one image of the source
exists, at 9\ , meaning that it is located on the same side
of the center of the lens as the unlensed source.
For the magnification, we find
fi(ff)--
\0/0b\
II W-
(3.65)
If 9 « $e, ^ is very large. Such solutions of the lens
equation exist for |y0| <gc 9 E , so that sources close to the
center of the source plane may be highly magnified. If
/J = 0, the image of the source will be a ring of radius
9 — 9 E , a so-called Einstein ring.
More Realistic Models. Mass distributions occurring
in nature are not expected to be truly symmetric. The
ellipticity of the mass distribution or external shear
forces (caused, for example, by the tidal gravitational
field of neighboring galaxies) will disturb the symme-
try. The lensing properties of the galaxy will change
by this symmetry breaking. For example, more than
two images may be generated. Figure 3.36 illustrates
the lens properties of such elliptical mass distribu-
tions. One can see, for example, that pairs of images,
which are both heavily magnified, may be observed with
a separation significantly smaller than the Einstein ra-
dius of the lens. Nevertheless, the characteristic image
separation is still of the order of magnitude given by
(3.64).
3.8.3 Examples for Gravitational Lenses
Currently, about 70 gravitational lens systems are
known in which a galaxy acts as the lens. Some of them
were discovered serendipitously, but most were found
in systematic searches for lens systems. Amongst the
most important lens surveys are: (1) The HST Snapshot
Survey. The ~ 500 most luminous quasars have been
imaged with the HST, and six lens systems have been
identified. (2) JVAS. About 2000 bright radio sources
with a flat radio spectrum (these often contain com-
pact radio components, see Sect. 5.1.3) were scanned
for multiple components with the VLA. Six lens sys-
tems have been found. (3) CLASS. Like in JVAS, radio
sources with a flat spectrum were searched with the
VLA for multiple components, but the flux limit was
lower than in JVAS, which form a subset of the CLASS
sources. The survey contains 15 000 sources, of which,
to data, 22 have been identified as lenses. In this sec-
tion we will discuss some examples of identified lens
systems.
QSO 0957+561: The First Double Quasar. The first
lens system was discovered in 1979 by Walsh, Carswell
& Weymann when the optical identification of a ra-
dio source showed two point-like optical sources (see
Fig. 3.37). Both could be identified as quasars located
at the same redshift of z s — 1.41 and having very sim-
ilar spectra (see Fig. 3.38). Deep optical images of the
field show an elliptical galaxy situated between the two
quasar images, at a redshift of id — 0.36. The galaxy
is so massive and so close to image B of the source
that it has to produce a lens effect. However, the ob-
served image separation of A9 — 6" 1 is considerably
larger than expected from the lens effect by a single
galaxy (3.64). The explanation for this is that the lens
galaxy is located in a cluster of galaxies; the additional
lens effect of the cluster adds to that of the galaxy,
3. The World of Galaxies
Fig. 3.36. Geometry of an "elliptical" lens, whereby it is of lit-
tle importance whether the surface mass density £ is constant
on ellipses (i.e., the mass distribution has elliptical isodensity
contours) or whether an originally spherical mass distribution
is distorted b\ an external tidal field. On the right-hand side
in both panels, several different source positions in the source
plane arc displayed, each corresponding to a different color.
The origin in the source plane is chosen as the intersection
point of the line connecting the center of symmetry in the lens
and the observer with the source plane (see also Fig. 2.22).
Depending on the position of the source, one, three, or five
images ma\ appear in the lens plane (i.e.. the observer's sky);
the) are shown on the left hand side of each panel. The curves
in the lens plane are the critical curves, the location of all
points for which fjb ->■ oo. The curves in the source plane (i.e.,
on the right-hand side of each panel) are caustics, obtained
by mapping the critical curves onto the source plane using the
lens equation. Obviously, the number of images of a source
depends on the source location relative to the location of the
caustics. Strongly elongated images of a source occur close
to the critical curves
g the image separation to a large value. The lens
system QSO 0957+561 was observed in all wavelength
ranges, from the radio to the X-ray. The two images
of the quasar are very similar at all X, including the
VLBI structure (Fig. 3.38) - as would be expected since
the lens effect is independent of the wavelength, i.e.,
achromatic.
QSO PG1115+080. In 1980, the so-called triple quasar
was discovered, composed of three optical quasars at
a maximum angular separation of just below 3". Com-
ponent (A) is significantly brighter than the other two
images (B, C; see Fig. 3.39, left). In high-resolution im-
ages it was found that the brightest image is in fact
a double image: A is split into Al and A2. The angu-
lar separation of the two roughly equally bright images
is ~ 0'.'5, which is considerably smaller than all other
angular separations in this system. The four quasar im-
ages have a redshift of z s — 1.72, and the lens is located
at z d = 0.31. The image configuration is one of those
that are expected for an elliptical lens, see Fig. 3.36.
With the NIR camera NICMOS on-board HST,
not only were the quasar images and the lens gal-
axy observed, but also a nearly complete Einstein ring
(Fig. 3.39, right). The source of this ring is the host gal-
axy of the quasar (see Sect. 5.4.5) which is substantially
redder than the active galactic nucleus itself.
From the image configuration in such a quadruple
system, the mass of the lens within the images can be
estimated very accurately. The four images of the lens
system trace a circle around the center of the lens galaxy,
the radius of which can be identified with the Einstein
radius of the lens. From this, the mass of the lens within
the Einstein radius follows immediately because the
Einstein radius is obtained from the lens equation (3.56)
by setting ji — 0. Therefore, the Einstein radius is the
solution of the equation
This equation is best written as
| M(9 E ) = 7T(D d 9 E ) 2 i; cr ,
which is readily interpreted:
3.8 Galaxies as Gravitational Lenses
/S^
mL
16 L
06 ••/ bS.O
S G
_jg
Right Ascension (B1950)
Fig. 3.37. Top: optical images of the double quasar QSO
0957+561. The image on the left has a short exposure time;
here, the two point-like images A,B of the quasar arc clearly
visible. In contrast, the image on the right has a longer expo-
sure lime, show ing the lens galaxy Gl between the two quasar
images. Several other galaxies (G2-G5) are visible as well. The
lens galaxy is a member of a cluster of galaxies at za — 0.36.
Bottom: two radio maps of QSO 0957+561, observed with
the VLA at 6 cm (left) and 3.6 cm (right), respectively. The
Right Ascension (B1950)
nages of the quasar are denoted by A,B; G is the radio
i of the lens galaxy. The quasar has a radio jet, which
is a common property of many quasars (see Sect. 5.3.1). On
small angular scales, the jet can be observed by VLBI tech-
niques in both images (see Fig. 3.38). On large scales only
a single image of the jet exists, seen in image A; this property
should be compared with Fig. 3,36 where it was demonstrated
that the number of images of a source (component) depends
on its position in the source plane
3. The World of Galaxies
VLBI Observations of 0957 + 561
Fig. 3.38. Left: milliarcsecond structure of the two im-
ages of the quasar QSO 0957+561, a VLBI map at 13 cm
wavelength by Gorenstein et al. Both quasar images show
a core jet structure, and it is clearly seen that they are mirror
symmetric, as predicted by lens models, right: spectra of
the two quasar images QSO 0957+561A,B, observed by the
1000 1100 1200 1300 1400
Wavelength (A) in z QS0 = 1 .41 Rest Frame
Faint Object Camera (FOC) on-board HST. The similarity
of the spectra, in particular the identical redshift, is a clear
indicator of a common source of the two quasar images.
The broad Lya line, in the wings of which an Nv line is
visible, is virtually always the strongest e
quasars
The mass within 9 E of a lens follows from the fact
that the mean surface mass density within 6 E equals
the critical surface mass density E a . A more ac-
curate determination of lens masses is possible by
means of detailed lens models. For quadruple image
systems, the masses can be derived with a precision
of a few percent - these are the most precise mass
determinations in (extragalactic) astronomy.
QSO 2237+0305: The Einstein Cross. A spectroscopic
survey of galaxies found several unusual emission lines
in the nucleus of a nearby spiral galaxy which can-
not originate from this galaxy itself. Instead, they are
emitted by a background quasar at redshift z s — 1.7 situ-
ated exactly behind this spiral. High-resolution images
show four point sources situated around the nucleus
of this galaxy, with an image separation of Ad « 1'.'8
(Fig. 3.40). The spectroscopic analysis of these point
sources revealed that all four are images of the same
quasar (Fig. 3.41).
The images in this system are positioned nearly sym-
metrically around the lens center; this is also a typical
lens configuration which may be caused by an ellipti-
cal lens (see Fig. 3.36). The Einstein radius of this lens
is #e * 0C9, and we can determine the mass within this
radius with a precision of ~ 3%.
Einstein Rings. More examples of Einstein rings are
displayed in Figs. 3.42 and 3.43. The first of these
is a radio galaxy, with its two radio components be-
3.8 Galaxies as Gravitational Lenses
Fig. 3.39. A NIR image of QSO 1 1 15+080
is shown on the left, as observed with the
NICMOS camera on-board HST. The dou-
ble structure of image A (the left of the QSO
images) is clearly visible, although the im-
age separation of the two A components
is less than 0'.'5. The lens galaxy, located
in the "middle" of the QSO images, has
a much redder spectral energy distribution
than the quasar images. In the right-hand
panel, the quasar images and the lens gal-
axy have been subtracted. What remains is
a nearly closed ring: the light of the gakt.\>
which hosts the active galactic nucleus is
imaged into an Einstein ring
Fig. 3.40. Left: in the center of a nearby spiral galaxy, four
point-like sources were found whose spectra show strong
emission lines. This image from the CFHT clearly shows the
bar structure in the core of the lens galaxy. An HST/NICMOS
image of the center of QSO 2237+0305 is shown on the right.
The central source is not a fifth quasar image but rather the
bright nucleus of the lens galaxy
ing multiply imaged by a lens galaxy - one of the
two radio sources is imaged into four components, the
other mapped into a double image. In the NIR the ra-
dio galaxy is visible as a complete Einstein ring. This
example shows very clearly that the appearance of the
images of a source depends on the source size: to ob-
tain an Einstein ring a sufficiently extended source is
needed.
At radio wavelengths, the quasar MG 1654+13 con-
sists of a compact central source and two radio lobes.
As we will discuss in Sect. 5.1.3, this is a very typical
radio morphology for quasars. One of the two lobes has
a ring-shaped structure, which prior to this observation
had never been observed before. An optical image of the
field shows the optical quasar at the position of the com-
pact radio component and, in addition, a bright elliptical
galaxy right in the center of the ring-shaped radio lobe.
This galaxy has a significantly lower redshift than the
quasar and hence is the gravitational lens responsible
for imaging the lobe into an Einstein ring.
3. The World of Galaxies
C Ml]
I 1 sky
5000 5500
Wavelength [A]
Fig. 3.41. Spectra of the four images of the quasar 2237+0305,
observed with the CFHT. As is clearly visible, the spectral
properties of these four images are very similar; this is the
final proof that we are dealing with a lens system here. Mea-
suring the indi\ idual spectra of these four very closely spaced
sources is extremely difficult and can only be performed under
optimum observing conditions
3.8.4 Applications of the Lens Effect
Mass Determination. As mentioned previously, the
mass within a system of multiple images can be de-
termined directly, sometimes very precisely. Since the
length-scale in the lens plane (at given angular scale)
and S a depend on Ho, these mass estimates scale
with H . For instance, for QSO 2237+0305, a mass
within 0'.'9 of (1.08±0.02)/i" 1 x 10 I0 M o is derived.
An even more precise determination of the mass was
obtained for the lens galaxy of the Einstein ring in
the system MG 1654+13 (Fig. 3.43). The dependence
on the other cosmological parameters is comparatively
weak, especially at low redshifts of the source and the
lens. Most lens galaxies are early-type galaxies (el-
lipticals), and from the determination of their mass
it can be concluded that ellipticals also contain dark
matter.
Environmental Effects. Detailed lens models show
that the light deflection of most gravitational lenses
is affected by an external tidal field. This is due to
the fact that lens galaxies are often members of gal-
axy groups which contribute to the light deflection as
well. In some cases the members of the group have
been identified. Mass properties of the corresponding
group can be derived from the strength of this external
influence.
Determination of the Hubble Constant. The light
travel times along the different paths (according to the
multiple images) are not the same. On the one hand
the paths have different geometrical lengths, and on the
other hand the light rays traverse different depths of the
gravitational potential of the lens, resulting in a (gen-
eral relativistic) time dilation effect. The difference in
the light travel times At is measurable because lumi-
nosity variations of the source are observed at different
times in the individual images. At can be measured
from this difference in arrival time, called the time
delay.
•
if
0.2"
Fig. 3.42. The radio source 1938+666 is
seen to be multiply imaged (contours in the
right hand figure); here, the radio source
consists of two components, one of which is
imaged four-fold, the other two-fold. A NIR
image taken with the NICMOS camera on-
board the HST ! left hand figure, also shown
on the right in gray-scale) shows the lens
galaxy in the center of an Einstein ring that
originaios from the stellar light of the host
galaxy of the active galactic nucleus
3.9 Population Synthesis
16 h 52 rn 24 s
Right Ascension
Fig. 3.43. The quasar MG 1654+ 13 shows, in addition to the
compact radio core (Q), two radio lobes: (he northern lobe is
denoted by C, whereas the southern lobe is imaged into a ring.
An optical image is displayed in gray-scales, showing not
only the quasar at Q (z s = 1 -72) but also a massive foreground
galaxy at z c i = 0.25 that is responsible for the lensing of the
lobe into an Einstein ring. The mass of this galaxy within the
ring can be derived with a precision of ~ 1%
It is easy to see that At depends on the Hubble con-
stant, or in other words, on the size of the Universe. If
a universe is twice the size of our own, At would be
twice as large as well - see Fig. 3.44. Thus if the mass
distribution of the lens can be modeled sufficiently well,
by modeling the geometry of the image configuration,
then the Hubble constant can be derived from measur-
ing the difference in the light travel time. To date, At has
been measured in about 10 lens systems (see Fig. 3.45
for an example). Based on "plausible" lens models we
can derive values for the Hubble constant that are com-
patible with other measurements (see Sect. 3.6), but
which tend towards slightly smaller values of Ho than
that determined from the HST Key Project (3.36). The
main difficulty here is that the mass distribution in lens
Small H
Fig. 3.44. Lens geometry in two universes with different Hub-
ble constant. All observables are dimensionless - angular
separations, flux ratios, redshifts - except for the difference in
tire light travel time. Tins is larger in the universe at the hot
torn than in the one at the top; hence, At oc Hq ' . If the time
delay At can be measured, and if one has a good model for
the mass distribution of the lens, then the Hubble c<
be derived from measuring At
galaxies cannot unambiguously be derived from the
positions of the multiple images. Therefore, these de-
terminations of Hq are currently not considered to be
precision measurements. On the other hand, we can
draw interesting conclusions about the radial mass pro-
file of lens galaxies from A; if we assume H is known.
In Sect. 6.3.4 we will discuss the value of H deter-
minations from lens time delays in a slightly different
context.
The ISM in Lens Galaxies. Since the same source is
seen along different sight lines passing through the lens
galaxy, the comparison of the colors and spectra of the
individual images provides information on reddening
and on dust extinction in the ISM of the lens galaxy.
From such investigations it was shown that the extinc-
tion in ellipticals is in fact very low, as is to be expected
from the small amount of interstellar medium they con-
tain, whereas the extinction is considerably higher for
spirals. These analyses also enable us to study the re-
lation between extinction and reddening, and from this
to search for deviations from the Galactic reddening
law (2.21). In fact, the constant of proportionality R v is
different in other galaxies, indicating a different compo-
sition of the dust, e.g., with respect to the chemical com-
position and to the size distribution of the dust grains.
3. The World of Galaxies
I DecM I JanSS | Feb95| Mai35 | Apr95 I
leaf- g
Fig. 3.45. Left: optical light curves of the double quasar
0957+561 in two broad-band filters. The light curve of im-
age A is displayed in red and that of image B in blue, where
the latter is shifted in time by 417 days. With this shift, the two
light curves are made to coincide - this light travel time dif-
3.9 Population Synthesis
The light of normal galaxies originates from stars. Stel-
lar evolution is largely understood, and the spectral radi-
ation of stars can be calculated from the theory of stellar
atmospheres. If the distribution of the number density
of stars is known as a function of their mass, chemical
composition, and evolutionary stage, we can compute
the light emitted by them. The theory of population syn-
thesis aims at interpreting the spectrum of galaxies as
a superposition of stellar spectra. We have to take into
account the fact that the distribution of stars changes
over time; e.g., massive stars leave the main sequence af-
ter several 10 6 years, the number of luminous blue stars
thus decreases, which means that the spectral distribu-
tion of the population also changes in time. The spectral
energy distribution of a galaxy thus reflects its history of
star formation and stellar evolution. For this reason, sim-
ulating different star-formation histories and comparing
them with observed galaxy spectra provides important
clues to understanding the evolution of galaxies. In this
section, we will discuss some aspects of the theory
ference of 117 da) s is determined w ith an accuracy of ~ ±3
days. Right: radio light curves of QSO 0957+56 1A,B at 6 cm.
From these radio measurements At can also be measui ed, and
the corresponding value is compatible with that obtained from
optical data
of population synthesis; this subject is of tremendous
importance for our understanding of galaxy spectra.
3.9.1 Model Assumptions
The processes of star formation are not understood in
detail; for instance, it is currently impossible to compute
the mass spectrum of a group of stars that jointly formed
in a molecular cloud. Obviously, high-mass and low-
mass stars are born together and form young (open) star
clusters. The mass spectra of these stars are determined
empirically from observations.
The initial mass function (IMF) is defined as the ini-
tial mass distribution at the time of birth of the stars, such
that (him) dm specifies the fraction of stars in the mass
interval of width dm around m, where the distribution
is normalized,
3.9 Population Synthesis
The integration limits are not well defined. Typically,
one puts m L ~ O.1M because less massive stars do
not ignite their hydrogen (and are thus brown dwarfs),
and my ~ 100M Q , because more massive stars have not
been observed. Such very massive stars would be dif-
ficult to observe because of their very short lifetime;
furthermore, the theory of stellar structure tells us that
more massive stars can probably not form a stable con-
figuration due to excessive radiation pressure. The shape
of the IMF is also subject to uncertainties; in most cases,
the Salpeter-IMF is used,
4>(m) ex m" 2 - 35 , (3.67)
as obtained from investigating the stellar mass spectrum
in young star clusters. It is by no means clear whether
a universal IMF exists, or whether it depends on specific
conditions like metallicity, the mass of the galaxy, or
other parameters. The Salpeter IMF seems to be a good
description for stars with M > 1M , whereas the IMF
for less massive stars is less steep.
The star-formation rate is the gas mass that is
converted into stars per unit time,
dM
The metallicity Z of the ISM defines the metallicity
of the newborn stars, and the stellar properties in turn
depend on Z. During stellar evolution, metal-enriched
matter is ejected into the ISM by stellar winds, plan-
etary nebulae, and SNe, so that Z(t) is an increasing
function of time. This chemical enrichment must be
taken into account in population synthesis studies in
a self-consistent form.
Let Sx,z(t') be the emitted energy per wavelength and
time interval, normalized to an initial total mass of 1 M Q ,
emitted by a group of stars of initial metallicity Z and
age t' . The function Si.z(t-t')(t'), which describes this
emission at any point t in time, accounts for the different
evolutionary tracks of the stars in the Hertzsprung-
Russell diagram (HRD) - see Appendix B.2. It also
accounts for their initial metallicity (i.e., at time t — t'),
where the latter follows from the chemical evolution
of the ISM of the corresponding galaxy. Then the total
spectral luminosity of this galaxy at a time t is given by
i\U) -
I dt'ir(t-t')S KZ(t - t r ) (t'),
(3.68;
thus by the convolution of the star- formation rate with
the spectral energy distribution of the stellar popula-
tion. In particular, Fx (t) depends on the star-formation
history.
3.9.2 Evolutionary Tracks in the HRD;
Integrated Spectrum
In order to compute Sx,z(t-t')(t'), models for stellar
evolution and stellar atmospheres are needed. As a re-
minder, Fig. 3.46(a) displays the evolutionary tracks
in the HRD. Each track shows the position of a star
with specified mass in the HRD and is parametrized
by the time since its formation. Positions of equal time
in the HRD are called isochrones and are shown in
Fig. 3.46(b). As time proceeds, fewer and fewer massive
stars exist because they quickly leave the main sequence
and end up as supernovae or white dwarfs. The num-
ber density of stars along the isochrones depends on the
IMF. The spectrum Sx,z(t-t')(t') is then the sum over all
spectra of the stars on an isochrone - see Fig. 3.47(b).
In the beginning, the spectrum and luminosity of
a stellar population are dominated by the most massive
stars, which emit intense UV radiation. But after ~ 10 7
years, the flux below 1000 A is diminished significantly,
and after ~ 10 8 years, it hardly exists any more. At the
same time, the flux in the NIR increases because the
massive stars evolve into red supergiants.
For 10 8 yr < t < 10 9 yr, the emission in the NIR re-
mains high, whereas short-wavelength radiation is more
and more diminished. After ~ 10 9 yr, red giant stars
(RGB stars) account for most of the NIR production.
After ~ 3 x 10 9 yr, the UV radiation increases again
due to blue stars on the horizontal branch into which
stars evolve after the AGB phase, and due to white
dwarfs which are hot when they are born. Between an
age of 4 and 13 billion years, the spectrum of a stellar
population evolves fairly little.
Of particular importance is the spectral break located
at about 4000 A which becomes visible in the spectrum
after a few 10 7 years. This break is caused by a strongly
changing opacity of stellar atmospheres at this wave-
length, mainly due to strong transitions of singly ionized
cak 'iii m and the Balmer lines of hydrogen. This 4000 A-
break is one of the most important spectral properties of
galaxies; as we will discuss in Sect. 9.1.2, it allows us to
estimate the redshifts of early-type galaxies from their
3. The World of Galaxies
log (T eff /K)
Fig. 3.46. a) Evolutionary tracks in the HRD for stars of dif-
ferent masses, as indicated b\ (he numbers near the tracks (in
units of Mq). The ZAMS (zero age main sequence) is the
place of birth in the HRD; evolution moves stars away from
the main sequence. Depending on the mass, they explode as
a core-collapse SN (for M > 8M©) or end as a white dwarf
log (T ef /K)
(WD). Prior to this, they move along the red giant branch
( RGB i and the as\ mptotie giant branch (AGB). b) Isochrones
at different times, indicated in units of 10 9 years. The upper
main sequence is quickh depopulated by the rapid evolution
of massive stars, whereas the red giant branch is populated
Fig. 3.47. a) Comparison of the spectrum of a main-set] uenee
star with a blackbody spectrum of equal effective temperature.
The opacit) of the stellar atmosphere causes clear deviations
from the Planck spectrum in the UV/optical. b) Spectrum
of a stellar popuL
stantaneously born
l with solar metallicity that v
;o; t is given in units of 1
3.9 Population Synthesis
photometric properties - so-called photometric redshift
estimates.
3.9.3 Color Evolution
Detailed spectra of galaxies are often not available.
Instead we have photometric images in different broad-
band niters, since the observing time required for spec-
troscopy is substantially larger than for photometry. In
addition, modern wide-field cameras can obtain photo-
metric data of numerous galaxies simultaneously. From
the theory of population synthesis we can derive pho-
tometric magnitudes by multiplying model spectra with
the filter functions, i.e., the transmission curves of the
color filters used in observations, and then integrating
over wavelength (A. 25). Hence the spectral evolution
implies a color evolution, as is illustrated in Fig. 3.48(a).
For a young stellar population the color evolution is
rapid and the population becomes redder, again because
the hot blue stars have a higher mass and thus evolve
quickly in the HRD. For the same reason, the evolution
is faster in B — V than in V — K. It should be mentioned
that this color evolution is also observed in star clusters
of different ages. The mass-to-light ratio M/L also in-
creases with time because M remains constant while L
decreases.
As shown in Fig. 3.48(b), the blue light of a stel-
lar population is always dominated by main-sequence
stars, although at later stages a noticeable contribution
also comes from horizontal branch stars. The NIR ra-
diation is first dominated by stars burning helium in
their center (this class includes the supergiant phase of
massive stars), later by AGB stars, and after ~ 10 9 yr
by red giants. Main sequence stars never contribute
more than 20% of the light in the K-band. The fact
that M/L K varies only little with time implies that the
NIR luminosity is a good indicator for the total stellar
mass: the NIR mass-to-light ratio is much less depen-
dent on the age of the stellar population than that for
bluer filters.
age/Gyr
Fig. 3.48. a) For the same stellar population as in Fig. 3. 17(b),
the upper two graphs show the colors B — V and V — K as
a function of age. The lower two graphs show the mass-to-
light ratio M/L in two color bands in Solar units. The solid
curves show the total M/L (i.e., including the mass that is
log (age/yr)
later returned into the ISM), whereas the dashed curves show
the M/L of the stars itself, b) The fraction of B- (top) and
/T-luminosity (bottom) contributed by stars in their different
phases of stellar evolution (CHeB: core helium burning stars;
SGB: subgiant branch)
3. The World of Galaxies
3.9.4 Star Formation History and Galaxy Colors
Up to now, we have considered the evolution of a stellar
population of a common age (called an instantaneous
burst of star formation). However, star formation in
a galaxy takes place over a finite period of time.
We expect that the star-formation rate decreases over
time because more and more matter is bound in stars
and thus no longer available to form new stars. Since
the star-formation history of a galaxy is a priori un-
known, it needs to be parametrized in a suitable manner.
A "standard model" of an exponentially decreasing
star-formation rate was established for this,
1 exp[-(f-fr)/r]H(f-fr)
where r is the characteristic duration and tf the onset
of star formation. The last factor in (3.69) is the Heav-
iside step function, H(x) = 1 for x > 0, H(x) = for
x < 0. This Heaviside step function accounts for the fact
that i/f(f) = for t < tf. We may hope that this simple
model describes the basic aspects of a stellar popula-
tion. Results of this model are plotted in Fig. 3.49(a) in
a color-color diagram.
From the diagram we find that the colors of the pop-
ulation depend strongly on r. Specifically, galaxies do
not become very red if r is large because their star-
formation rate, and thus the fraction of massive blue
stars, does not decrease sufficiently. The colors of Sc
spirals, for example, are not compatible with a constant
star-formation rate - except if the total light of spirals is
strongly reddened by dust absorption (but there are good
reasons why this is not the case). To explain the colors
of early-type galaxies we need r < 4 x 10 9 yr. In gen-
eral, one deduces from these models that a substantial
evolution to redder colors occurs for t > x. Since the
luminosity of a stellar population in the blue spectral
range decreases quickly with the age of the population,
whereas increasing age affects the red luminosity much
less, we conclude:
The spectral distribution of galaxies is mainly de-
termined by the ratio of the star-formation rate
today to the mean star-formation rate in the past,
yr(today)/<yr}.
One of the achievements of this standard model is
that it explains the colors of present day galaxies, which
have an age of > 10 billion years. However, this model
is not unambiguous because other star-formation histo-
ries \j/(t) can be constructed with which the colors of
galaxies can be modeled as well.
3.9.5 Metallicity, Dust, and H11 Regions
Predictions of the model depend on the metallicity Z -
see Fig. 3.49(b). A small value of Z results in a bluer
color and a smaller M/L ratio. The age and metallicity
of a stellar population are degenerate in the sense that
an increase in the age by a factor X is nearly equivalent
to an increase of the metallicity by a factor 0.65X with
respect to the color of a population. The age estimate of
a population from color will therefore strongly depend
on the assumed value for Z. However, this degeneracy
can be broken by taking several colors, or information
from spectroscopy, into account.
Intrinsic dust absorption will also change the colors
of a population. This effect cannot be easily accounted
for in the models because it depends not only on the
properties of the dust but also on the geometric distribu-
tion of dust and stars. For example, it makes a difference
whether the dust in a galaxy is homogeneously dis-
tributed or concentrated in a thin disk. Empirically, it is
found that galaxies show strong extinction during their
active phase of star formation, whereas normal galaxies
are presumably not much affected by extinction, with
early-type galaxies (E/SO) affected the least.
Besides stellar light, the emission by Hll regions also
contributes to the light of galaxies. It is found, though,
that after ~ 10 7 yr the emission from gas nebulae only
marginally contributes to the broad-band colors of gal-
axies. However, this nebular radiation is the origin of
emission lines in the spectra of galaxies. Therefore,
emission lines are used as diagnostics for the star-
formation rate and the metallicity in a stellar population.
3.9.6 Summary
After this somewhat lengthy section, we shall summa-
rize the most important results of population synthesis
here:
3.9 Population Synthesis
V-K
Fig. 3.49. a) Evolution of colors between <t < 17 x 10 9 yr
for a stellar population with star-formation rate given by
(3.69), for five different values of the characteristic time-scale
r (r = oo is the limiting case for a constant star-formation
rate) -Galactic center see solid curves. The typical colors for
four diffcrcni morphological types of galaxies are plotted. For
each r, the evolution begins at the lower left, i.e., as a blue
population in both color indices. In the case of constant star
formation, the population never becomes redder than Irr's; to
Age /Gyr
achieve redder colors, r has to be smaller. The dashed line
connects points of t = 10 10 yr on the different curves. Here,
a Salpctcr IMF and Solar metallicity was assumed. The shift
in color obtained by doubling the metallicity is indicated by
an arrow, as well as that due to an extinction coefficient of
E(B — V) — 0.1; both effects w iil make galaxies appear red
der. b) The dependence of colors and M/L on the metallicit)
of the population
• A simple model of star- formation history reproduces
the colors of today's galaxies fairly well.
• (Most of) the stars in elliptical and SO galaxies are
old - the earlier the Hubble type, the older the stellar
population.
• Detailed models of population synthesis provide
information about the star-formation history, and
predictions by the models can be compared with
observations of galaxies at high redshift (and thus
smaller age).
We will frequently refer to results from population
synthesis in the following chapters. For example, we
will use them to interpret the colors of galaxies at
high redshifts and the different spatial distributions of
early-type and late-type galaxies (see Chap. 6). Also,
we will present a method of estimating the redshift of
galaxies from their broad-band colors (photometric red-
shifts). As a special case of this method, we will discuss
the efficient selection of galaxies at very high redshift
(Lyman-break galaxies, LBGs, see Chap. 9). Because
the color and luminosity of a galaxy are changing even
when no star formation is taking place, tracing back
such a passive evolution allows us to distinguish this
passive aging process from episodes of star formation
and other processes.
3.9.7 The Spectra of Galaxies
At the end of this section we shall consider the typical
spectra of different galaxy types. They are displayed
for six galaxies of different Hubble types in Fig. 3.50.
To make it easier to compare them, they are all plotted
in a single diagram where the logarithmic flux scale
is arbitrarily normalized (since this normalization does
not affect the shape of the spectra).
It is easy to recognize the general trends in these spec-
tra: the later the Hubble type, (1) the bluer the overall
spectral distribution, (2) the stronger the emission lines,
(3) the weaker the absorption lines, and (4) the smaller
the 4000- A break in the spectra. From the above discus-
sion, we would also expect these trends if the Hubble
sequence is considered an ordering of galaxy types
3. The World of Galaxies
Fig. 3.50. Spectra of gal-
axies of different types,
where the spectral flux is
plotted logarithmically in
arbitrary units. The spec-
tra are ordered according
to the Hubble sequence,
with early types at the bot-
tom and late-type spectra
at the top
5000 5500 6000 6500
Wavelength in restframe/A
according to the characteristic age of their stellar popu-
lation or according to their star- formation rate. Elliptical
and SO galaxies essentially have no star-formation ac-
tivity, which renders their spectral energy distribution
dominated by red stars. Furthermore, in these galaxies
there are no Hll regions where emission lines could be
generated. The old stellar population produces a pro-
nounced 4000-A break, which corresponds to a jump
by a factor of ~ 2 in the spectra of early-type galaxies.
It should be noted that the spectra of ellipticals and SO
galaxies are quite similar.
By contrast, Sc spirals and irregular galaxies have
a spectrum which is dominated by emission lines, where
the Balmer lines of hydrogen as well as nitrogen and
oxygen lines are most pronounced. The relative strength
of these emission lines are characteristic for Hll regions,
implying that most of this line emission is produced in
the ionized regions surrounding young stars. For irregu-
lar galaxies, the spectrum is nearly totally dominated by
the stellar continuum light of hot stars and the emission
lines from Hll regions, whereas clear contributions by
cooler stars can be identified in the spectra of Sc spiral
galaxies.
The spectra of Sa and Sb galaxies form a kind of
transition between those of early-type galaxies and Sc
galaxies. Their spectra can be described as a super-
position of an old stellar population generating a red
continuum and a young population with its blue con-
tinuum and its emission lines. This can be seen in
connection with the decreasing contribution of the
bulge to the galaxy luminosity towards later spiral
types.
The properties of the spectral light distribution of
different galaxy types, as briefly discussed here, is de-
scribed and interpreted in the framework of population
synthesis. This gives us a detailed understanding of
stellar populations as a function of the galaxy type. Ex-
tending these studies to spectra of high-redshift galaxies
allows us to draw conclusions about the evolutionary
history of their stellar populations.
3.10 Chemical Evolution of Galaxies
During its evolution, the chemical composition of a gal-
axy changes. Thus the observed metallicity yields
information about the galaxy's star-formation history.
We expect the metallicity Z to increase with star-
formation rate, integrated over the lifetime of the galaxy.
We will now discuss a simple model of the chemical evo-
3.10 Chemical Evolution of Galaxies
lution of a galaxy, which will provide insight into some
of the principal aspects.
We assume that at the formation epoch of the stellar
population of a galaxy, at time t = 0, no metals were
present; hence Z(0) = 0. Furthermore, the galaxy did
not contain any stars at the time of its birth, so that all
baryonic matter was in the form of gas. In addition, we
consider the galaxy as a closed system out of which no
matter can escape or be added later on by processes of
accretion or merger. Finally, we assume that the time-
scales of the stellar evolution processes that lead to the
metal enrichment of the galaxy are small compared to
the evolutionary time-scale of the galaxy. Under these
assumptions, we can now derive a relation between the
metallicity and the gas content of a galaxy.
Of the total mass of a newly formed stellar popu-
lation, part of it is returned to the ISM by supernova
explosions and stellar winds. We define this fraction as
R, so that the fraction a — (1 — R) of a newly-formed
stellar population remains enclosed in stars, i.e., it no
longer takes part in the further chemical evolution of the
ISM. The value of a depends on the IMF of the stellar
population and can be computed from models of pop-
ulation synthesis. Furthermore, let q be the ratio of the
mass in metals, which is produced by a stellar popula-
tion and then returned into the ISM, and the initial total
mass of the population. The yield y — q/a is defined
as the ratio of the mass in metals that is produced by
a stellar population and returned into the ISM, and the
mass that stays enclosed in the stellar population. The
yield can also be calculated from population synthesis
models. If ty{i) is the star-formation rate as a function
of time, then the mass of all stars formed in the history
of the galaxy is given by
-f dfi
and the total mass that remains enclosed in stars is s(t) =
aS(i). Since we have assumed a closed system for the
baryons, the sum of gas mass g(t) and stellar mass s(t)
is a constant, namely the baryon mass of the galaxy,
dt dt
g(t) + s(t) = M b
(3.70)
of the ISM and thus also that of its metals decreases.
On the other hand, metals are also returned into the
ISM by processes of stellar evolution. Under the above
assumption that the time-scales of stellar evolution are
small, this return occurs virtually instantaneously. The
metals returned to the ISM are composed of metals
that were already present at the formation of the stellar
population - a fraction R of these will be returned - and
newly formed metals. Together, the total mass of the
metals in the ISM obeys the evolution equation
d(gZ) _
d;
- = ijf(RZ + q)-Z
where the last term specifies the rate of the metals ex-
tracted from the ISM in the process of star formation
and the first term describes the return of metals to the
ISM by stellar evolution processes. Since dS/dt — i/r,
this can also be written as
d(gZ)
d.«>
= (R-l)Z + q =
The mass of the metals in the ISM is gZ; it changes
when stars are formed. Through this formation, the mass
Dividing this equation by a and using s — aS and the
definition of the yield, y — q/a, we obtain
d(gZ) dg dZ
— — = — Z + g— = y-Z. (3.71)
ds ds ds
From (3.70) it follows that dg/ds = -1 and dZ/d.s =
— dZ/dg, and so we obtain a simple equation for the
metallicity,
where \i % = g/M b is the fraction of baryons in the ISM,
and where we chose the integration constant such that at
the beginning, when /x g — 1 , the metallicity was Z = 0.
From this relation, we can now see that with decreasing
gas content in a galaxy, the metallicity will increase; in
our simple model this increase depends only on the
yield y. Since y can be calculated from population
synthesis models, (3.72) is a well-defined relation.
If (3.72) is compared with observations of galax-
ies, rather strong deviations from this relation are found
which are particularly prominent for low-mass galaxies.
While the assumption of an instantaneous evolution of
the ISM is fairly well justified, we know from structure
formation in the Universe (Chap. 7) that galaxies are
3. The World of Galaxies
by no means isolated systems: their mass continuously
changes through accretion and merging processes. In
addition, the kinetic energy transferred to the ISM by
supernova explosions causes an outflow of the ISM,
in particular in low-mass galaxies where the gas is
not strongly gravitationally bound. Therefore, the ob-
served deviations from relation (3.72) allow us to draw
conclusions about these processes.
Also, from observations in our Milky Way we find
indications that the model of the chemical evolution
sketched above is too simplified. This is known as the
G-dwarf problem. The model described above predicts
that about half of the F- and G-main-sequcncc stars
should have a metallicity of less than a quarter of the
Solar value. These stars have a long lifetime on the
main sequence, so that many of those observed today
should have been formed in the early stages of the Gal-
axy. Thus, in accordance with our model they should
have very low metallicity. However, a low metallicity
is in fact observed in only very few of these stars. The
discrepancy is far too large to be explained by selec-
tion effects. Rather, observations show that the chemical
evolution of our Galaxy must have been substantially
more complicated than described by our simple model.
4. Cosmology I: Homogeneous Isotropic World Models
We will now begin to consider the Universe as a whole.
Individual objects such as galaxies and stars will
no longer be the subject of discussion, but instead
we will turn our attention to the space and time in
which these objects are embedded. These considera-
tions will then lead to a world model, the model of our
cosmos.
This chapter will deal with aspects of homogeneous
cosmology. As we will see, the Universe can, to first
approximation, be considered as being homogeneous.
At first sight this fact obviously seems to contradict
observations because the world around us is highly
inhomogeneous and structured. Thus the assumption
of homogeneity is certainly not valid on small scales.
But observations are compatible with the assump-
tion that the Universe is homogeneous when averaged
over large spatial scales. Aspects of inhomogeneous
cosmology, and thus the formation and evolution of
structures in the Universe, will be considered later in
Chap. 7.
4.1 Introduction
and Fundamental Observations
Cosmology is a very special science indeed. To be able
to appreciate its peculiar role we should recall the typical
way of establishing knowledge in natural sciences. It
normally starts with the observation of some regular
patterns, for instance the observation that the height h
a stone falls through is related quadratically to the time t
it takes to fall, h = (g/2)t 2 . This relation is then also
found for other objects and observed at different places
on Earth. Therefore, this relation is formulated as the
"law" of free fall. The constant of proportionality g/2 in
this law is always the same. This law of physics is tested
by its prediction of how an object falls, and wherever
this prediction is tested it is confirmed - disregarding
the resistance of air in this simple example, of course.
Relations become physical laws if the predictions
they make are confirmed again and again; the validity
of such a law is considered more secure the more diverse
the tests have been. The law of free fall was tested only
on the surface of the Earth and it is only valid there
with this constant of proportionality. In contrast to this,
Newton's law of gravity contains the law of free fall as
a special case, but it also describes the free fall on the
surface of the Moon, and the motion of planets around
the Sun. If only a single stone were available, we would
not know whether the law of free fall is a property of this
particular stone or whether it is valid more generally.
In some ways, cosmology corresponds to the latter
example: we have only one single Universe available
for observation. Relations that are found in our cosmos
cannot be verified in other universes. Thus it is not
possible to consider any property of our Universe as
"'typical" - we have no statistics on which we could
base a statement like this. Despite this special situation,
enormous progress has been made in understanding our
Universe, as we will describe here and in subsequent
chapters.
Cosmological observations are difficult in general,
simply because the majority of the Universe (and with
it most of the sources it contains) is very far away from
us. Distant sources are very dim. This explains why our
knowledge of the Universe runs in parallel with the de-
velopment of large telescopes and sensitive detectors.
Much of today's knowledge of the distant Universe be-
came available only with the new generation of optical
telescopes of the 8-m class, as well as new and powerful
telescopes in other wavelength regimes.
The most important aspect of cosmological observa-
tions is the finite speed of light. We observe a source
at distance D in an evolutionary state at which it was
Af = (D/c) younger than today. Thus we can observe
the current state of the Universe only very locally. An-
other consequence of this effect, however, is of even
greater importance: due to the finite speed of light, it
is possible to look back into the past. At a distance of
10 billion light years we observe galaxies in an evolu-
tionary state when the Universe had only a third of its
current age. Although we cannot observe the past of our
own Milky Way, we can study that of other galaxies. If
we are able to identify among them the ones that will
form objects similar to our Galaxy in the course of cos-
1 Strict!} speaking, the constant of proportionality g depends slightly
I i 'nil ( i I I: Hoi
DOI: 10.1007/1 1614371_4 © Springer- Ve
is Isotropic World Models.
/ and Cosmology, pp. 141-174 (2006)
ig Berlin Heidelberg 2006
4. Cosmology I: Homogeneous Isotropic World Models
mic evolution, we will be able to learn a great deal about
the typical evolutionary history of such spirals.
The finite speed of light in a Euclidean space, in
which we are located at the origin r — today (t — to),
implies that we can only observe points in spacetime for
which \r\ — c(t — t); an arbitrary point (r, t) in space-
time is not observable. The set of points in spacetime
which satisfy the relation \r\ = c(to — t) is called our
backward light cone.
The fact that our astronomical observations are re-
stricted to sources which are located on our backward
light cone implies that our possibilities to observe the
Universe are fundamentally limited. If somewhere in
spacetime there would be a highly unusual event, we
will not be able to observe it unless it happens to lie on
our backward light cone. Only if the Universe has an
essentially "simple" structure we will be able to under-
stand it, by combining astronomical observations with
theoretical modeling. Luckily, our Universe seems to be
basically simple in this sense.
4.1.1 Fundamental Cosmological Observations
We will begin with a short list of key observations that
have proven to be of particular importance for cosmol-
ogy. Using these observational facts we will then be
able to draw a number of immediate conclusions; other
observations will be explained later in the context of
a cosmological model.
1. The sky is dark at night (Olbers' paradox).
2. Averaged over large angular scales, faint galaxies
(e.g., those with R > 20) are uniformly distributed
on the sky (see Fig. 4.1).
3. With the exception of a very few very nearby galax-
ies (e.g., Andromeda = M31), a redshift is observed
in the spectra of galaxies - most galaxies are mov-
ing away from us, and their escape velocity increases
linearly with distance (Hubble law; see Fig. 1.10).
4. In nearly all cosmic objects (e.g., gas nebulae,
main-sequence stars), the mass fraction of helium
is 25-30%.
5. The oldest star clusters in our Galaxy have an age of
~ 12 Gyr = 12 x 10 9 yr (see Fig. 4.2).
6. A microwave radiation (cosmic microwave back-
ground radiation, CMB) is observed, reaching us
from all directions. This radiation is isotropic except
Fig. 4.1. The APM survey: galaxy distribution in a ~ 100 x
50 degree 2 field around the South Galactic Pole. The intensi-
ties of the pixels are scaled with the number of galaxies per
pixel, i.e., the projected galaxy number density on the sphere.
The "holes" are regions around bright stars, globular clusters,
etc., that have not been surveyed
for very small, but immensely important, fluctuations
with relative amplitude ~ 10~ 5 .
. The spectrum of the CMB corresponds, within the
very small error bars that were obtained by the
measurements with COBE, to that of a perfect black-
body, i.e., a Planck radiation of a temperature of
T Q = 2.728 ± 0.004 K - see Fig. 4.3.
. The number counts of radio sources at high Gal-
actic latitude does not follow the simple law
N(> S) oc S~ 3/2 (see Fig. 4.4).
4.1.2 Simple Conclusions
We will next draw a number of simple conclusions from
the observational facts listed above. These will then
serve as a motivation and guideline for developing the
cosmological model. We will start with the assump-
tion of an infinite, Euclidean, static Universe, and show
that these assumptions are in direct contradiction to
observations (1) and (8).
Olbers' Paradox (1): We can show that the night sky
would be bright in such a universe - uncomfortably
bright, in fact. Let «* be the mean number density of
stars, constant in space and time according to the as-
sumptions, and let R* be their mean radius. A spherical
4.1 Introduction and Fundamental Observatio
V 10-
(a)
12-
M5 (NGC 5904)
14-
' *tW
16-
**w .
*'i? :
18-
tfr ••
20-
'm&. .::..
22-
F j ; : ;Jf^V'\'
Fig. 4.2. (a) Color-magnitude diagram of the globular clus-
ter M5. The different sections in this diagram arc labeled,
A: main sequence; B: red giant branch; C: point of helium
flash; D: horizontal branch; E: Schwarzschild gap in the hori-
zontal branch; F: white dwarfs, below the arrow. At the point
where the main sequence turns over to the red giant branch
(called the "turn-off point"), stars have a mass correspond-
ing to a main-sequence lifetime which is equal to the age of
the globular cluster (see Appendix B.3). Therefore, the age of
the cluster can be determined from the position of the turn-
off point by comparing it with models of stellar evolution.
Color (magnitudes)
(b) Isochrones, i.e., curves connecting the stellar evolution-
ary position in the color-magnitude diagram of stars of equal
age, are plotted for different ages and compared to the stars of
the globular cluster 47 Tucanae. Such analyses reveal that the
oldest globular clusters in our Milky Way are about 1 3 billion
years old, where different authors obtain slight!} differing re
suits - details of stellar evolution may play a role here. The
age thus obtained also depends on the distance of the clus-
ter. A revision of these distances by the HIPPARCOS satellite
led to a decrease of the estimated ages by about 2 billion
Cosmic Microwave Background Spectrum from COBE
Waves /centimeter
Fig. 4.3. CMB spectrum, plotted as intensity vs. frequency,
measured in waves per centimeter. The solid line shows
the expected spectrum of a blackbody of temperature T =
2.728 K. The error bars of the data, observed by the FIRAS
instrument on-board COBE, are so small that the data points
with error bars cannot be distinguished from the theoretical
4. Cosmology I: Homogeneous Isotropic World Models
-3-2-101234
log S v (mJy)
Fig. 4.4. Number counts of radio sources as a function of flux,
normalized by the Euclidean expectation N( S) oc S -5 / 2 , corre-
sponding to the integrated counts N(> S) <x S~ 3/2 . Counts are
displayed for three different frequencies; the) clear!) deviate
from the Euclidean expectation
shell of radius r and thickness dr around i
n* dV — Anr 2 dr n* stars. Each of these stars subtends
a solid angle of jzR 2 /r 2 on our sky, so the stars in the
shell cover a total solid angle of
Sun as observed from Earth is the same as seen by an
observer who is much closer to the Solar surface - the
sky would have a temperature of ~ 10 4 K; fortunately,
this is not the case!
Source Counts (8): Consider now a population of
sources with a luminosity function that is constant in
space and time, i.e., let n(> L) be the spatial num-
ber density of sources with luminosity larger than L.
A spherical shell of radius r and thickness dr around
us contains Aitr 2 drn(> L) sources with luminosity
larger than L. Because the observed flux S is re-
lated to the luminosity via L — Aitr 2 S, the number of
sources with flux > S in this spherical shell is given as
dN(> S) = Anr 2 dr n(> 4jt r 2 S), and the total number
of sources with flux > S results from integration over
the radii of the spherical shells,
dco = 47ir 2 dr «„ -^— = An 2 n* R% dr .
We see that this solid angle is independent of the radius r
of the spherical shell because the solid angle of a single
staroc r~ 2 just compensates the volume of the shell oc r 2 .
To compute the total solid angle of all stars in a static
Euclidean universe, (4.1) has to be integrated over all
distances r, but the integral
[dco , , f
o = dr — = 4jt 2 n* Rl dr
diverges. Formally, this means that the stars cover an
infinite solid angle, which of course makes no sense
physically. The reason for this divergence is that we
disregarded the effect of overlapping stellar disks on the
sphere. However, these considerations demonstrate that
the sky would be completely filled with stellar disks, i.e.,
from any direction, along any line-of-sight, light from
a star would reach us. Since the specific intensity I v is
independent of distance - the surface brightness of the
./d.
47rr 2 «(>47rr z S).
Changing the integration variable to L — An r 2 S, or
= JL/(4tiS), with dr = dL/(2^AirLS), yields
(4.1)
S' 3/2 I dL VIn(> L) . (4.2)
From this result we deduce that the source counts in
x 5" 3/2 , independent of
n contradiction to the
s - Olbers' paradox and
its - we conclude that
such a universe is N(> S) o
the luminosity function. This i;
observations.
From these two contradici
the non-Euclidean source c
at least one of the assumptions must be wrong. Our
Universe cannot be all three of Euclidean, infinite, and
static. The Hubble flow, i.e., the redshift of galaxies, in-
dicates that the assumption of a static Universe is wrong.
The age of globular clusters (5) requires that
the Universe is at least 12Gyr old because it can-
not be younger than the oldest objects it contains.
Interestingly, the age estimates for globular clusters
yield values which are very close to the Hubble
time Hq 1 = 9.78 h~ l Gyr. This similarity suggests that
4.2 An Expanding Univ
the Hubble expansion may be directly linked to the
evolution of the Universe.
The apparently isotropic distribution of galaxies
(2), when averaged over large scales, and the CMB
isotropy (6) suggest that the Universe around us is
isotropic on large angular scales. Therefore we will
first consider a world model that describes the Uni-
verse around us as isotropic. If we assume, in addition,
that our place in the cosmos is not privileged over any
other place, then the assumption of isotropy around
us implies that the Universe appears isotropic as seen
from any other place. The homogeneity of the Universe
follows immediately from the isotropy around every
location, as explained in Fig. 4.5. The combined as-
sumption of homogeneity and isotropy of the Universe
is also known as the cosmological principle. We will see
that a world model based on the cosmological principle
in fact provides an excellent description of numerous
observational facts.
However, homogeneity is in principle unobservable
because observations of distant objects show those at
an earlier epoch. If the Universe evolves in time, as
the aforementioned observations suggest, evolution-
ary effects cannot directly be separated from spatial
The assumption of homogeneity of course breaks
down on small scales. We observe structures in the Uni-
verse, like galaxies and clusters of galaxies, and even
accumulations of clusters of galaxies, so-called super-
clusters. Structures have been found in redshift surveys
that extend over ~ 100 h~ l Mpc. However, we have no
indication of the existence of structures in the Universe
with scales JJ> 100 Mpc. This length-scale can be com-
pared to a characteristic length of the Universe, which
is obtained from the Hubble constant. If H^ ' specifies
the characteristic age of the Universe, then light will
travel a distance c/Hq in this time. With this, one ob-
tains the Hubble radius as a characteristic length-scale
of the Universe (or more precisely, of the observable
Universe),
Mpc : Hubble length
The Hubble volume ~ R^ can contain a very large num-
ber of structures of size ~ 100 h~ x Mpc, so that it still
Fig. 4.5. Homogeneity follows from the isotropy around two
points. If the Universe is isotropic around observer B, the
densities atC, D indl m qual Drai inj pheres of different
radii around observer A, it is seen that the region within the
spherical shell around A has to be homogeneous. By varying
the radius of the shell, we can conclude the whole Universe
must be homogeneous
makes sense to assume an on-average homogeneous
Universe. In this homogeneous Universe we then have
density fluctuations that are identified with the observed
large-scale structures; these will be discussed in detail
in Chap. 7. To a first approximation we can neglect these
density perturbations in a description of the Universe as
a whole. We will therefore consider next world mod-
els that are based on the cosmological principle, i.e., in
which the Universe looks the same for all observers (or,
in other words, if observed from any point).
Homogeneous and isotropic world models are the
simplest cosmological solutions of the equations of
General Relativity (GR). We will examine how far such
simple models are compatible with observations. As we
shall see, the application of the cosmological principle
results in the observational facts which were mentioned
in Sect. 4. 1.1.
4.2 An Expanding Universe
Gravitation is the fundamental force in the Universe.
Only gravitational forces and electromagnetic forces
can act over large distance. Since cosmic matter is elec-
trically neutral on average, electromagnetic forces do
not play any significant role on large scales, so that
gravity has to be considered as the driving force in the
4. Cosmology I: Homogeneous Isotropic World Models
Universe. The laws of gravity are described by the the-
ory of General Relativity, formulated by A. Einstein
in 1915. It contains Newton's theory of gravitation as
a special case for weak gravitational fields and small
spatial scales. Newton's theory of gravitation has been
proven to be eminently successful, e.g., in describing the
motion of planets. Thus it is tempting to try to design
a cosmological model based on Newtonian gravity. We
will proceed to do that as a first step because not only is
this Newtonian cosmology very useful from a didactic
point of view, but one can also argue why the Newto-
nian cosmos correctly describes the major aspects of
a relativistic cosmology.
4.2.1 Newtonian Cosmology
The description of a gravitational system necessitates
the application of GR if the length-scales in the system
are comparable to the radius of curvature of spacetime;
this is certainly the case in our Universe. Even if we
cannot explain at this point what exactly the "curvature
radius of the Universe" is, it should be plausible that
it is of the same order of magnitude as the Hubble ra-
dius Ra. We will discuss this more thoroughly further
below. Despite this fact, one can expect that a Newtonian
description is essentially correct: in a homogeneous uni-
verse, any small spatial region is characteristic for the
whole universe. If the evolution of a small region in
space is known, we also know the history of the whole
universe, due to homogeneity. However, on small scales,
the Newtonian approach is justified. We will therefore,
based on the cosmological principle, first consider spa-
tially homogeneous and isotropic world models in the
framework of Newtonian gravity.
4.2.2 Kinematics of the Universe
Comoving Coordinates. We consider a homogeneous
sphere which may be radially expanding (or contract-
ing); however, we require that the density p(t) remains
spatially homogeneous. The density may vary in time
due to expansion or contraction. We choose a point
t — to in time and introduce a coordinate system x at
this instant with the origin coinciding with the center of
the sphere. A particle in the sphere which is located at
position jc at time to will be located at some other time t
at the position r(i) which results from the expansion
of the sphere. Since the expansion is radial or, in other
words, the velocity vector of a particle at position r(f)
is parallel to r, the direction of r(f) is constant. Because
r(to) = x, this means that
r(i) = a(t) x
(4.4)
The function a(t) can depend only on time. Although
requiring radial expansion alone could make a depend
on |*| as well, the requirement that the density remains
homogeneous implies that a must be spatially c
The function a(t) is called the cosmic scale fac
to r(t ) — x, it obeys
a(to) = 1 .
(4.5)
The value of to is arbitrary; we choose to = today. Par-
ticles (or observers) which move according to (4.4)
arc called comoving panicles (observers!, and x is llic
comoving coordinate. The world line (r, t) of a co-
moving observer is unambiguously determined by x,
(r,t) = [a(t)x,t].
Expansion Rate. The velocity of such a comoving
particle is obtained from the time derivative of its
position,
where in the last step we defined the expansion r,
H(i) :-
(4.7)
The choice of this notation is not accidental, since H
is closely related to the Hubble constant. To see this,
we consider the relative velocity vector of two comov-
ing particles at positions r and r+ Ar, which follows
directly from (4.6):
Ad = v(r+Ar, t) - v(r, t) = H(t) Ar .
(4.8)
Hence, the relative velocity is proportional to the sep-
aration vector, and the constant of proportionality H(t)
depends only on time but not on the position of the two
particles. Obviously, (4.8) is very similar to the Hubble
law
4.2 An Expanding Univ
in which v is the radial velocity of a source at distance D
from us. Therefore, setting t = to and Hq = H(to), (4.8)
is simply the Hubble law, in other words, (4.8) is a gen-
eralization of (4.9) for arbitrary time. It expresses the
fact that any observer expanding with the sphere will
observe an isotropic velocity field that follows the Hub-
ble law. Since we are observing an expansion today -
sources are moving away from us - we have Hq > 0,
and a(t Q ) > 0.
4.2.3 Dynamics of the Expansion
The above discussion describes the kinematics of the
expansion. However, to obtain the behavior of the func-
tion a(t) in time, and thus also the motion of comoving
observers and the time evolution of the density of the
sphere, it is necessary to consider the dynamics. The
evolution of the expansion rate is determined by self-
gravity of the sphere, from which it is expected that it
will cause a deceleration of the expansion.
It is important to note that this equation of motion does
not dependent on x. The dynamics of the expansion,
described by a(t), is determined solely by the matter
density.
"Conservation of Energy". Another way to describe
the dynamics of the expanding shell is based on the
law of energy conservation: the sum of kinetic and po-
tential energy is constant in time. This conservation of
energy is derived directly from (4.13). To do this, (4.13)
is multiplied by 2d, and the resulting equation can be
integrated with respect to time since d(d 2 )/dt — Ida.
andd(-l/a)/dt = d/a 2 :
here, Kc 2 is a constant of integration that will be inter-
preted later. After multiplication with x 2 /2, (4.14) can
Equation of Motion. We therefore consider a spherical
surface of radius x at time to and, accordingly, a ra-
dius r(t) = a(t)x at arbitrary time t. The mass Mix)
enclosed in this comoving surface is constant in time,
and is given by
Mix)
4.T
- Po x
4;r
p{t)r\t)
4:r
p(t)a\t)x
(4.10)
where p must be identified with the mass density of
the Universe today (t — to). The density is a function
of time and, due to mass conservation, it is inversely
proportional to the volume of the sphere,
p{i) = p a-\i). (4.11)
The gravitational acceleration of a particle on the spher-
ical surface is GM(x)/r 2 , directed towards the center.
This then yields the equation of motion of the particle,
GM(x) _ AttG pox
= dt 2 ~ ~~ 3~~ "r 2
or, after substituting r(t) — x a(t), an equation for a,
^) = -nr = —
(4.12)
r{t) = 4ttG po
3 a 2 {t)~
4jtG
2 r(t) KC 2 '
which is interpreted such that the kinetic + poten-
tial energy (per unit mass) of a particle is a constant
on the spherical surface. Thus (4.14) in fact describes
the conservation of energy. The latter equation also
immediately suggests an interpretation of the integra-
tion constant: K is proportional to the total energy of
a comoving particle, and thus the history of the expan-
sion depends on K. The sign of K characterizes the
qualitative behavior of the cosmic expansion history.
• If K < 0, the right-hand side of (4. 14) is always pos-
itive. Since da/dt > today, da/dt remains positive
for all times or, in other words, the Universe will
expand forever.
• If K = 0, the right-hand side of (4. 14) is always pos-
itive, i.e., da/dt > for all times, and the Universe
will also expand forever, but in a way that da/dt —>
for t -> 00 - the limiting expansion velocity for
t -> 00 is zero.
• If K > 0, the right-hand side of (4.14) vanishes if
a = a m . dx = (8ttGpo)/(3Kc 2 ). For this value of a,
da/dt — 0, and the expansion will come to a halt.
After that, the expansion will turn into a contraction,
and the Universe will re-collapse.
4. Cosmology I: Homogeneous Isotropic World Models
In the special case of K — 0, which separates eternally
expanding world models from those that will re-collapse
in the future, the Universe has a current density called
critical density which can be inferred from (4.14) by
setting t = to and Hq = d(to):
Per
.^l.SSxlO-^g/cm 3 .
(4.15)
Obviously, p cr is a characteristic density of the current
Universe. As in many situations in physics, it is useful
to express physical quantities in terms of dimensionless
parameters, for instance the real density of the Universe
today. We therefore define the density parameter
Qu :
_ A) .
(4.16)
where K > corresponds to Qq > 1, and K < cor-
responds to &o < 1. Thus, Qq is one of the central
cosmological parameters. Its determination was made
possible only quite recently, and we shall discuss this
in detail later. However, we should mention here that
matter which is visible as stars contributes only a small
fraction to the density of the Universe, Q* < 0.01. But,
as we already discussed in the context of rotation curves
of spiral galaxies, we find clear indications of the pres-
ence of dark matter which can in principle dominate the
value of Qq. We will see that this is indeed the case.
4.2.4 Modifications due to General Relativity
The Newtonian approach contains nearly all essential
aspects of homogeneous and isotropic world models,
otherwise we would not have discussed it in detail. Most
of the above equations are also valid in relativistic cos-
mology, although the interpretation needs to be altered.
In particular, the image of an expanding sphere needs
to be revised - this picture implies that a "center" of the
Universe exists. Such a picture implicitly contradicts the
cosmological principle in which no point is singled out
over others - our Universe neither has a center, nor is it
expanding away from a privileged point. However, the
image of a sphere does not show up in any of the relevant
equations : Eq. (4. 1 1 ) for the evolution of the density of
the Universe and Eqs. (4.13) and (4.14) for the evolu-
tion of the scale factor a(t) contain no quantities that
refer to a sphere.
General Relativity modifies the Newtonian model in
several respects:
• We know from the theory of Special Relativity that
mass and energy are equivalent, according to Ein-
stein's famous relation E — mc 2 . This implies that
it is not only the matter density that contributes
to the equations of motion. For example, a radia-
tion field like the CMB has an energy density and,
due to the equivalence above, this has to enter the
expansion equations. We will see below that such
a radiation field can be characterized as matter with
pressure. The pressure will then explicitly appear in
the equation of motion for a{t).
• The field equation of GR as originally formulated by
Einstein did not allow a solution which corresponds
to a homogeneous, isotropic, and static cosmos. But
since Einstein, like most of his contemporaries, be-
lieved the Universe to be static, he modified his
field equations by introducing an additional term,
the cosmological constant.
• The interpretation of the expansion is changed com-
pletely: it is not the particles or the observers that
are expanding away from each other, nor is the
Universe an expanding sphere. Instead, it is space
itself that expands. In particular, the redshift is no
Doppler redshift, but is itself a property of expanding
spacetimes. However, we may still visualize redshift
locally as being due to the Doppler effect without
making a substantial conceptual error.
In the following, we will explain the first two aspects in
more detail.
First Law of Thermodynamics. When air is com-
pressed, for instance when pumping up a tire, it heats up.
The temperature increases and accordingly so does the
thermal energy of the air. In the language of thermody-
namics, this fact is described by the first law: the change
in internal energy dU through an (adiabatic) change in
volume dV equals the work dU — — P dV, where P is
the pressure in the gas. From the equations of GR as ap-
plied to a homogeneous isotropic cosmos, a relation is
derived which reads
(4.17)
dt
(c 2 p« 3 ) =
dt
in full analogy to this law. Here, p c 2 is the energy den-
sity, i.e., for "normal" matter, p is the mass density,
and P is the pressure of the matter. If we now con-
sider a constant comoving volume element V x , then its
4.2 An Expanding Univ
physical volume V — a 3 (t) V x will change due to expan-
sion. Thus, a 3 = V/ V x is the volume, and c 2 p a 3 the
energy contained in the volume, each divided by V x .
Taken together, (4.17) corresponds to the first law of
thermodynamics in an expanding universe.
The Friedmann-Lemaitre Expansion Equations.
Next, we will present equations for the scale factor a(i)
which follow from GR for a homogeneous isotropic uni-
verse. Afterwards, we will derive these equations from
the relations stated above - as we shall see, the modifi-
cations by GR are in fact only minor, as expected from
the argument that a small section of a homogeneous
universe characterizes the cosmos as a whole. The field
equations of GR yield the equations of motion
(Si
8ttG
Kc 1
A
4jtG
3~~
( 3P \ A
where A is the aforementioned cosmological constant
introduced by Einstein. Compared to equations (4.13)
and (4.14), these two equations have been changed
in two places. First, the cosmological constant occurs
in both equations, and second, the equation of mo-
tion (4.19) now contains a pressure term. The pair of
equations (4.18) and (4.19) are called the Frieclmann
equations.
The Cosmological Constant. When Einstein intro-
duced the A -term into his equations, he did this solely
for the purpose of obtaining a static solution for the
resulting expansion equations. We can easily see that
(4.18) and (4.19), without the .A -term, have no solu-
tion for a = 0. However, if the yl-term is included, such
a solution can be found (which is irrelevant, however, as
we now know that the Universe is expanding). Einstein
had no real physical interpretation for this constant, and
after the expansion of the Universe was discovered he
discarded it again. But with the genie out of the bot-
tle, the cosmological constant remained in the minds of
cosmologists, and their attitude towards A has changed
frequently in the past 90 years. Around the turn of the
millennium, observations were made which strongly
suggest a non-vanishing cosmological constant, i.e., we
believe today that A ^ 0.
But the physical interpretation of the cosmological
constant has also been modified. In quantum mechanics
even completely empty space, the so-called vacuum,
may have a finite energy density, the vacuum energy
density. For physical measurements not involving grav-
ity, the value of this vacuum energy density is of no
relevance since those measurements are only sensitive
to energy differences. For example, the energy of a pho-
ton that is emitted in an atomic transition equals the
energy difference between the two corresponding states
in the atom. Thus the absolute energy of a state is mea-
surable only up to a constant. Only in gravity does the
absolute energy become important, because E = m c 2
implies that it corresponds to a mass.
It is now found that the cosmological constant is
equivalent to a finite vacuum energy density - the equa-
tions of GR, and thus also the expansion equations, are
not affected by this new interpretation. We will explain
this fact in the following.
4.2.5 The Components of Matter in the Universe
Starting from the equation of energy conservation
(4.14), we will now derive the relativistically correct
expansion equations (4.18) and (4.19). The only change
with respect to the Newtonian approach in Sect. 4.2.3
will be that we introduce other forms of matter. The es-
sential components of the Universe can be described as
pressure-free matter, radiation, and vacuum energy.
Pressure-Free Matter. The pressure in a gas is deter-
mined by the thermal motion of its constituents. At room
temperature, molecules in the air move at a speed com-
parable to the speed of sound, c s ~ 300 m/s. For such
a gas, P ~ pc 2 <$C pc 2 , so that its pressure is of course
gravitationally completely insignificant. In cosmology,
a substance with P <£ pc 2 is denoted as (pressure-free)
matter, also called cosmological dust. 2 We approximate
P m = 0, where the index "m" stands for matter. The
constituents of the (pressure-free) matter move with
velocities much smaller than c.
2 The notation "dust" should not be confused with the dust that is
1 i 1 1 1 1 1 1 1 11 Ii ii du 11 nolo I
denotes matter with P = 0.
4. Cosmology I: Homogeneous Isotropic World Models
Radiation. If this condition is no longer satisfied, thus
if the thermal velocities are no longer negligible com-
pared to the speed of light, then the pressure will also
no longer be small compared to pc 2 . In the limiting
case that the thermal velocity equals the speed of light,
we denote this component as "radiation". One example
of course is electromagnetic radiation, in particular the
CMB photons. Another example would be other parti-
cles of vanishing rest mass. Even particles of finite mass
can have a thermal velocity very close to c if the ther-
mal energy of the particles is much larger than the rest
mass energy, i.e., k^T » mc 2 . In these cases, the pres-
sure is related to the density via the equation of state for
radiation,
P t =-p t c 2 . (4.20)
Vacuum Energy. The equation of state for vacuum en-
ergy takes a very unusual form which results from the
first law of thermodynamics. Because the energy den-
sity p v of the vacuum is constant in space and time,
(4.17) immediately yields the relation
A =
-p v c .
Thus the vacuum energy has a negative pressure. This
unusual form of an equation of state can also be made
plausible as follows: consider the change of a volume V
that contains only vacuum. Since the internal energy is
U oc V, and thus a growth by dV implies an increase
in U, the first law dU — -PdV demands that P be
negative.
4.2.6 "Derivation" of the Expansion Equation
Beginning with the equation of energy conservation
(4.14), we are now able to derive the expansion equa-
tions (4.18) and (4.19). To achieve this, we differentiate
both sides of (4.14) with respect to t and obtain
2aa= — {pa 2 + 2aap) .
Next, we carry out the differentiation in (4.17), thereby
obtaining pa 3 + 3pa 2 d — —3Pa 2 d/c 2 . This relation is
then used to replace the term containing p in the
previous equation, yielding
a \nG ( 3P\
This derivation therefore reveals that the pressure term
in the equation of motion results from the combination
of energy conservation and the first law of thermody-
namics. However, we point out that the first law in
the form (4.17) is based explicitely on the equivalence
of mass and energy, resulting from Special Relativ-
ity. When assuming this equivalence, we indeed obtain
the Friedmann equations from Newtonian cosmology,
as expected from the discussion at the beginning of
Sect. 4.2.1.
Next we consider the three aforementioned compo-
nents of the cosmos and write the density and pressure
as the sum of dust, radiation, and vacuum energy,
P = Pm + Pi + Pv = Pm-
P=P r +P v ,
where p m+T combines the density in matter and radia-
tion. In the second equation, the pressureless nature of
matter, P m — 0, was used so that P m+r = P t . By inserting
the first of these equations into (4.14), we indeed obtain
the first Friedmann equation (4. 1 8) if the density p there
is identified with p m+r (the density in "normal matter"),
and if
(4.21!
8ttG
(4.23)
(4.22)
Furthermore, we insert the above decomposition of den-
sity and pressure into the equation of motion (4.22) and
immediately obtain (4.19) if we identify p and P with
p m+r and P m+I — P t , respectively. Hence, this approach
yields both Friedmann equations; the density and the
pressure in the Friedmann equations refer to normal
matter, i.e., all matter except the contribution by A.
Alternatively, the A -terms in the Friedmann equations
may be discarded if instead the vacuum energy density
and its pressure are explicitly included in P and p.
4.2.7 Discussion of the Expansion Equations
Following the "derivation" of the expansion equa-
tions, we will now discuss their consequences. First
we consider the density evolution of the various cos-
mic components resulting from (4.17). For pressure-free
matter, we immediately obtain p m oca~ 3 which is in
agreement with (4.11). Inserting the equation of state
(4.20) for radiation into (4.17) yields the behavior
. energy density is a constant in
4.2 An Expanding Univ
time. Hence
Pm(0 = Pm,0 a~\t) ■ Pr (t) = Pr, «" 4 » I
p v (0 = p v = const , (4.24)
where the index "0" indicates the current time, t — to.
The physical origin of the a~ A dependence of the ra-
diation density is seen as follows: as for matter, the
number density of photons changes oc a~ 3 because the
number of photons in a comoving volume is unchanged.
However, photons are redshifted by the cosmic expan-
sion. Their wavelength A changes proportional to a (see
Sect. 4.3.2). Since the energy of a photon is E — h? v
(h-p: Planck constant) and v — c/X, the energy of a pho-
ton changes as cC x due to cosmic expansion so that the
photon energy density changes oc aT A .
Analogous to (4.16), we define the dimensionless
density parameters for matter, radiation, and vacuum,
_ Pr,0 .
Pv
Q r ~ 4.2 x KT 5 h"
(4.26)
between matter and radiation density was different at
earlier epochs since p r evolves faster with a than p m ,
Pr(0 _ Pr,0 1 _ J2 r 1
Thus radiation and dust had the same energy density at
an epoch when the scaling factor was
Q m
= 4.2 x 10" 5 {Q m h 2 )
(4.25)
SO that S2 = &m + £?r + &A - 3
By now we know the current composition of the
Universe quite well. The matter density of galaxies (in-
cluding their dark halos) corresponds to fi m > 0.02,
depending on the - largely unknown - extent of their
dark halos. This value therefore provides a lower limit
for Q m . Studies of galaxy clusters, which will be dis-
cussed in Chap. 6, yield a lower limit of ,f2 m >0.1.
Finally, we will show in Chap. 8 that Q m ~ 0.3.
In comparison to matter, the radiation energy density
today is much smaller. It is dominated by photons of
the cosmic background radiation and by neutrinos from
the early Universe, as will be explained below. For the
density parameter of radiation we obtain
This value of the scaling factor and the corresponding
epoch in cosmic history play a very important role in
structure evolution in the Universe, as we will see in
Chap. 7.
With p = p m+r = p mfi a' 3 + p r>0 a' 4 and (4.25), the
expansion equation (4. 1 8) can be written as
H 2 (t) = (4.29)
? T 4 , 9 Kc 2 1
Hq \a- 4 (f)Q T + a~ \t)Q m -a~ 2 {t)^ + Q A \ .
Evaluating this equation at the present epoch, with
H{t{)) — Ho and a(fo) = 1, yields the value of the
integration constant K,
'"(?)'
so that today, the energy density of radiation in the
Universe can be neglected when compared to that of
matter. However, equations (4.24) reveal that the ratio
e used. Often the
(4.30)
Hence the constant K is obtained from the density
parameters of matter and vacuum and from the afore-
mentioned fact that Q T <§C £2 m , and has the dimension
of (length) -2 . In the context of GR, K is interpreted as
the curvature scalar of the Universe today, or more pre-
cisely, the homogeneous, isotropic three-dimensional
space at time t = t o has a curvature K. Depending on
the sign of K, we can distinguish the following cases:
• If K — 0, the three-dimensional space for any fixed
time t is Euclidean, i.e., fiat.
• If K > 0, 1/*J~K can be interpreted as the curvature
radius of the spherical 3-space - the two-dimensional
analogy would be the surface of a sphere. As already
Fig. 4.6. Two-dimensional analogies for the three possible
s of space. In a universe with positive curvature
> 0) the sum of the angles in a triangle is larger than
is smaller than
tgles is exactly
speculated in Sect. 4.2.1, the order of magnitude of
the curvature radius is c/H Q according to (4.30).
• If K < 0, the space is called hyperbolic - the two-
dimensional analogy would be the surface of a saddle
(see Fig. 4.6).
Hence GR provides a relation between the curvature
of space and the density of the Universe. In fact, this
is the central aspect of GR which links the geometry
of spacetime to its matter content. However, Einstein's
theory makes no statement about the topology of space-
time and, in particular, says nothing about the topology
of the Universe. 4 If the Universe has a simple topology,
it is finite in the case of K > 0, whereas it is infinite if
K < 0. However, in both cases it has no boundary (com-
pare: the surface of a sphere is a finite space without
boundaries).
With (4.29) and (4.30), we finally obtain the
expansion equation in the form
®*
= H
(0
= H {
[«"
\t)Q, + C
~ i {t)Q m
+ c
" 2 «(1-
Q m -Q;
) + n A ]
I l) ml i i Hilda I i I I 1 I ll i I III |ll
because Ihe sum of angles in a triangle on a cylinder is also ISO .
ButthesurfaceofacylimKi.il n ,, i ], lopof lil'l i nl from
a plane; in particular, closed straight lines do exist - walking on
a cylinder in a direction perpendicular to its axis, one will return to
Ihe starting point after a finite amount ot time.
4.3 Consequences
of the Friedmann Expansion
The cosmic expansion equations imply a number of im-
mediate consequences, some of which will be discussed
next. In particular, we will first demonstrate that the
early Universe must have evolved out of a very dense
and hot state called the Big Bang. We will then link
the scaling factor a to an observable, the redshift, and
explain what the term "distance" means in cosmology.
4.3.1 The Necessity of a Big Bang
The terms on the right-hand side of (4.31) each have
a different dependence on a:
• For very small a, the first term dominates and the
Universe is radiation dominated then.
• For slightly larger a > a eq , the dust (or matter) term
dominates.
• If K 7^ 0, the third term, also called the curvature
term, dominates for larger a.
• For very large a, the cosmological constant
dominates if it is different from zero.
The differential equation (4.31) in general cannot be
solved analytically. However, its numerical solution for
a(t) poses no problems. Nevertheless, we can analyze
the qualitative behavior of the function a(t) and thereby
understand the essential aspects of the expansion his-
tory. From the Hubble law, we conclude that d(to) > 0,
i.e., a is currently an increasing function of time. Equa-
4.3 Consequences of the Friedmann Expansion
tion (4.31) shows that a{i) > for all times, unless the
right-hand side of (4.31) vanishes for some value of a:
the sign of a can only switch when the right-hand side
of (4.31) is zero. If H 2 = for a value of a > 1, the
expansion will come to a halt and the Universe will rec-
ollapse afterwards. On the other hand, if H 2 — for
a value a — a m \ n with < a m [ n < 1, then the sign of a
switches at a m ; n . At this epoch, a collapsing Universe
changes into an expanding one.
Which of these alternatives describes our Uni-
verse depends on the density parameters. We find the
following classification (also see Fig. 4.7):
• If A = 0, then H 2 > for all a < 1, whereas the
behavior for a > 1 depends on Q m :
- if Q m < 1 (or K < 0, respectively), H 2 > for
all a: the Universe will expand for all times.
This behavior is expected from the Newtonian
approach because if K < 0, the kinetic energy in
any spherical shell is larger than the modulus of
the potential energy, i.e., the expansion velocity
exceeds the escape velocity and the expansion
will never come to a halt.
- If Q m > 1 (K > 0), H 2 will vanish for
a — a max = Q m /{Q m — 1). The Universe will
have its maximum expansion when the scale
factor is a max and will recollapse thereafter. In
Newtonian terms, the total energy of any spher-
ical shell is negative, so that it is gravitationally
bound.
• In the presence of a cosmological constant A > 0,
the discussion becomes more complicated:
- If Q m < 1 , the Universe will expand for all a > 1 .
- However, for Q m > 1 the future behavior of a(t)
depends on Q A : if Q A is sufficiently small,
a value a max exists at which the expansion comes
to a halt and reverses. In contrast, if Q A is large
enough the Universe will expand forever.
- \{Q A < 1, then// 2 >0 for alia < 1.
However, if Q A > 1, it is in principle possible
that H 2 = for an a — a m ; n < 1. Such mod-
els, in which a minimum value for a existed in
the past, can be excluded by observations (see
Sect. 4.3.2).
With the exception of the last case, which can be ex-
cluded, we come to the conclusion that a must have
attained the value a — at some point in the past, at
no Big Bang^^
■ /\=1
A=2
expands forever
-
\. closed
open^--^
1
Fig. 4.7. Classification of cosmological models. The siniighl
solid line connects flat models (i.e., those without spatial cur
vature, Q m + Q A = 1 ) and separates open ( K - 1 and closet!
(K > 0) models. The nearly horizontal curve separates mod-
els that will expand forever from those that will recollapse in
the distant future. Models in the upper left corner have an ex-
pansion histon where a has never been close to zero and thus
did not experience a Big Bang. In those models, a maximum
redshift for sources exists, which is indicated for two cases.
Since we know that Q m > 0.1, and sources at redshift > 6
have been observed, these models can be excluded
least formally. At this instant the "size of the Universe"
formally vanished. As a -> 0, both matter and radiation
densities diverge so that the density in this state must
have been singular. The epoch at which a — and the
evolution away from this state is called the Big Bang.
It is useful to define this epoch (a — 0) as the origin
of time, so that t is identified with the age of the Uni-
verse, the time since the Big Bang. As we will show,
the predictions of the Big Bang model are in impressive
agreement with observations. The expansion history for
the special case of a vanishing vacuum energy density
is sketched in Fig. 4.8 for three values of the curvature.
To characterize whether the current expansion of the
Universe is decelerated or accelerated, the deceleration
parameter
qo---
aa/a 2
(4.32)
s defined where the right-hand side has to be evaluated
it t = f . With (4.19) and (4.31) it follows that
qo = Q m /2-G A . (4.33)
Time
Fig, 4.8. The scale factor ait) as a function of cosmic time / for
three models with a vanishing cosmological constant, Q A = 0.
Closed models (K > 0) attain a maximum expansion and then
recollapse. In contrast, open models (K < 0) expand forever,
and the Einstein-de Sitter model of K = separates these two
cases. In all models, the scale factor tends towards zero in the
past; this time is called the Big Bang and defines the origin of
the time axis
If Q A = then q > 0, a < 0, i.e., the expansion de-
celerates, as expected due to gravity. However, if Q A
is sufficiently large the deceleration parameter may
become negative, corresponding to an accelerated ex-
pansion of the Universe. The reason for this behavior,
which certainly contradicts intuition, is seen in the vac-
uum energy. Only a negative pressure can cause an
accelerated expansion - more precisely, as seen from
(4.22), P < —pc 2 /3 is needed for a > 0. Indeed, we be-
lieve today that the Universe is currently undergoing an
accelerated expansion and thus that the cosmological
constant differs significantly from zero.
Age of the Universe. The age of the Universe at
a given scale factor a follows from dt = da(da/dt)~ l =
dtt/iciH). This relation can be integrated,
) — — I da [a 2 Q
Ha J
¥a 2 Q A
l Q m + {\-Q m -Q A )
(4.34)
where the contribution from radiation for a ^> a eq can
be neglected because it is relevant only for very small a
and thus only for a very small fraction of cosmic time.
To obtain the current age to of the Universe, (4.34)
is calculated for a — 1. For models of vanishing spa-
Fig. 4.9. Top panel: scale factor a(f) as a function of cosmic
time, here sealed as (t - to) Ho, foi an Einstein-de Sitter model
(Q m = l,Q A = Q\ dotted curve), an open universe (Q m = 0.3,
Q A = 0; dashed curve), and a flat universe of low density
(Q m = 0.3, Q A = 0.7; solid curve). At the current epoch,
t — to anda = 1. Bottom panel: aye of the universe in units of
the Hubble time f H = H ~ l for flat world models with K =
(Q m + Q A = 1; solid curve) and models with a vanishing
cosmological constant (dashed curve). We see that for a flat
universe with small Q m (thus large Q A = 1 — Q m ), to may be
considerably largei than //„
tial curvature K — and for those with A — 0, Fig. 4.9
displays t as a function of Q m .
The qualitative behavior of the cosmological models
is characterized by the density parameters Q m and Q A ,
whereas the Hubble constant Ho determines "only" the
overall length- or time-scale. Today, mainly two families
of models are considered:
• Models without a cosmological constant, A = 0. The
difficulties in deriving a "sensible" value for A from
particle physics is often taken as an argument for neg-
4.3 Consequences of the Friedmann Expansion
lecting the vacuum energy density. However, there
are very strong observational indications that in fact
A>0.
• Models with £2 m + G A = 1, i.e., £ = 0. Such flat
models are preferred by the so-called inflationary
models, which we will briefly discuss further below.
A special case is the Einstein-de Sitter model, Q m — 1,
Q A = 0. For this model, f = 2/(3H Q ) % 6.7 ft" 1 x
10 9 yr.
For many world models, to is larger than the age of
the oldest globular clusters, so they are compatible with
this age determination. The Einstein-de Sitter model,
however, is compatible with stellar ages only if Ho is
very small, considerably smaller than the value of Ho
derived from the HST Key Project discussed in Sect. 3.6.
Hence, this model is ruled out by observations.
It is believed that the values of the cosmological
parameters are now quite well known. We list them here
for later reference without any further discussion. Their
determination will be described in the course of this
chapter and in Chap. 8. The values are approximately
Q m
-0.3; Q A -0.7; ft ~ 0.7
(4.35)
4.3.2 Redshift
The Hubble law describes a relation between the red-
shift, or the radial component of the relative velocity,
and the distance of an object from us. Furthermore,
(4.6) specifies that any observer is experiencing a local
Hubble law with an expansion rate H(t) which depends
on the cosmic epoch. We will now derive a relation
between the redshift of a source, which is directly ob-
servable, and the cosmic time t or the scaling factor a(t),
respectively, at which the source emitted the light we
receive today.
To do this, we consider a light ray that reaches us to-
day. Along this light ray we imagine fictitious comoving
observers. The light ray is parametrized by the cosmic
time t, and is supposed to have been emitted by the
source at epoch r e . Two comoving observers along the
light ray with separation dr from each other see their
relative motion due to the cosmic expansion according
to (4.6), di> = H(t) dr, and they measure it as a redshift
of light, dX/X — dz — dv/c. It takes a time dr = dr/c for
the light to travel from one observer to the other. Fur-
thermore, from the definition of the Hubble parameter,
a = da/dt — Ha, we obtain the relation dr = da/(H a).
Combining these relations, we find
dX dv
= Hdt =
(4.36)
The relation dX/X = da/a is now easily integrated since
the equation dX/da — X/a obviously has the solution
X — Ca, where C is a constant. That constant is deter-
mined by the wavelength A obs of the light as observed
today (i.e., at a — 1), so that
X(a)=aX obs
(4.37)
(see Fig. 4.10). The wavelength at emission was there-
fore A. e = a(r e )A. bs- On the other hand, the redshift z
is defined as (1 + z) — X obs /X e . From this, we finally
obtain the relation
1+z
(4.38)
between the observable z and the scale factor a which is
linked via (4.34) to the cosmic time. The same relation
can also be derived by considering light rays in GR.
The relation between redshift and the scale factor is of
immense importance for cosmology because, for most
sources, redshift is the only distance information that
we are able to measure. If the scale factor is a mono-
tonic function of time, i.e., if the right-hand side of
(4.31) is different from zero for all a e [0, 1], then z is
also a monotonic function of r. In this case, which cor-
responds to the Universe we happen to live in, a, t, and
z are equally good measures of the distance of a source
from us.
Fig. 4.10. Due to cosmic expansion, photons are redshiiicd.
i.e., their wavelength, as measured by a comoving observer,
increases with the scale factor a. This sketch visualizes this
effect as a standing wave in an expanding box
4. Cosmology I: Homogeneous Isotropic World Models
Local Hubble Law. The Hubble law applies for nearby
sources: with (4.8) and v « zc it follows that
(4.39!
3000 Mpc '
where D is the distance of a source with redshift ;;.
This corresponds to a light travel time of At — D/c.
On the other hand, due to the definition of the Hubble
parameter, we have Aa — (1 — a) «s // At, where a is
the scale factor at time to — At, and we used a(to) — 1
and H(t ) = H . This implies D = (l-a)c/H . Uti-
lizing (4.39), we then find z = 1 — a, or a= 1 — z,
which agrees with (4.38) in linear approximation since
(1 + z) _1 = 1 - z + 0(z 2 ). Hence we conclude that the
general relation (4.38) contains the local Hubble law as
a special case.
Energy Density in Radiation. A further consequence
of (4.38) is the dependence of the energy density of radi-
ation on the scale parameter. As mentioned previously,
the number density of photons is oc a~ 3 if we assume
that photons are neither created nor destroyed. In other
words, the number of photons in a comoving volume el-
ement is conserved. According to (4.38), the frequency v
of a photon changes due to cosmic expansion. Since
the energy of a photon is oc v, E Y — h P v oc \/a, the
energy density of photons decreases, p r oc n E y oc aT A .
Therefore (4.38) implies (4.24).
Cosmic Microwave Background. Assuming that, at
some time t\ , the Universe contained a blackbody radi-
ation of temperature T\ , we can examine the evolution
of this photon population in time by means of relation
(4.38). We should recall that the Planck function B v
(A. 13) specifies the radiation energy of blackbody radi-
ation that passes through a unit area per unit time, per
unit area, per unit frequency interval, and per unit solid
angle. Using this definition, the number density dN v of
photons in the frequency interval between v and v + dv
is obtained as
diV v 4jt B v 8ttv 2 1
(4.40)
dv
chpv
-1
At a later time ?2 > h, the Universe has expanded by
a factor a(t2)/a(ti). An observer at ?2 therefore observes
the photons redshifted by a factor (1 + z) = a{t2)/a(t\),
i.e., a photon with frequency v at t\ will then be
measured to have frequency v' = v/(l +z). The orig-
inal frequency interval is transformed accordingly as
dv' = dv/(l +z). The number density of photons de-
creases with the third power of the scale factor, so
lhai iLV',, = dN v /(l + z) 3 . Combining these relations,
we obtain for the number density dN' v , of photons in the
frequency interval between v' and v' + dv'
dN' v , _ dN v /(l+zf
dv'
dv/(l + z)
8;n.' :
(a)-
L exp
where we used T 2 — Ti/(\ +z) in the last step. The
distribution (4.41) has the same form as (4.40) except
that the temperature is reduced by a factor (1 + z) _1 . If
a Planck distribution of photons had been established at
an earlier time, it will persist during cosmic expansion.
As we have seen above, the CMB is such a blackbody
radiation, with a current temperature of T = T CM ^ »
2.73 K. We will show in Sect. 4.4 that this radiation
originates in the early phase of the cosmos. Thus it is
meaningful to consider the temperature of the CMB as
the "temperature of the Universe" which is a function
of redshift,
(4,42)
T{z) = T {\ + z) = T Q a- i
i.e., the Universe was hotter in the past than it is today.
The energy density of the Planck spectrum is
= ass T =
( * 2 k B '
so p r behaves like (1 + z) 4 — a 4 in accordance with
(4.24).
Generally, it can be shown that the specific intensity
I v changes due to redshift according to
4 = —4 ■ (4.44)
y3 (v') 3
Here, /„ is the specific intensity today at frequency v
and I' v , is the specific intensity at redshift z at frequency
v' = (l + z)v.
4.3 Consequences of the Friedmann Expansion
Finally, it should be stressed again that (4.38) allows
all relations to be expressed as functions of a as well as
of z. For example, the age of the Universe as a function
of z is obtained by replacing the upper integration limit,
fl^(l+z)- 1 ,in(4.34).
The Necessity of a Big Bang. We discussed in
Sect. 4.3.1 that the scale factor must have attained the
value a = at some time in the past. One gap in our ar-
gument that inevitably led to the necessity of a Big Bang
still remains, namely the possibility that at sometime in
the past a — occurred, i.e., that the Universe under-
went a transition from a contracting to an expanding
state. This is possible only if Q A > 1 and if the mat-
ter density parameter is sufficiently small (see Fig. 4.7).
In this case, a attained a minimum value in the past.
This minimum value depends on both Q m and Q A .
For instance, for Q m > 0.1, the value is a m \ n > 0.3. But
a minimum value for a implies a maximum redshift
z max = l/a m j n — 1. However, since we have observed
quasars and galaxies with z > 6 and the density param-
eter is known to be £2 m > 0.1, such a model without
a Big Bang can be excluded.
4.3.3 Distances in Cosmology
In the previous sections, different distance measures
were discussed. Because of the monotonic behavior of
the corresponding functions, each of a, t, and z provide
the means to sort objects according to their distance. An
object at higher redshift zi is more distant than one at
Z\ < Z2 such that light from a source at zi may become
absorbed by gas in an object at redshift z\, but not vice
versa. The object at redshift z 1 is located between us and
the object at zi- The more distant a source is from us,
the longer the light takes to reach us, the earlier it was
emitted, the smaller a was at emission, and the larger z
is. Since z is the only observable of these parameters,
distances in extragalactic astronomy are nearly always
expressed in terms of redshift.
But how can a redshift be translated into a distance
that has the dimension of a length? Or, phrasing this
question differently, how many Megaparsecs away from
us is a source with redshift z = 2? The corresponding
answer is more complicated than the question suggests.
For very small redshifts, the local Hubble relation (4.39)
may be used, but this is valid only for z <5C 1 .
In Euclidean space, the separation between two
points is unambiguously defined, and several prescrip-
tions exist for measuring a distance. We will give two
examples here. A sphere of radius R situated at dis-
tance D subtends a solid angle of co = 7tR 2 /D 2 on our
sky. If the radius is known, D can be measured using this
relation. As a second example, we consider a source of
luminosity L at distance D which then has a measured
flux S = L/(AnD 2 ). Again, if the luminosity is known,
the distance can be computed from the observed flux. If
we use these two methods to determine, for example, the
distance to the Sun, we would of course obtain identical
results for the distance (within the range of accuracy),
since these two prescriptions for distance measurements
are defined to yield equal results.
In a non-Euclidean space like, for instance, our Uni-
verse this is no longer the case. The equivalence of
different distance measures is only ensured in Euclidean
space, and we have no reason to expect this equiva-
lence to also hold in a curved spacetime. In cosmology,
the same measuring prescriptions as in Euclidean space
are used for defining distances, but the different defini-
tions lead to different results. The two most important
definitions of distance are:
• Angular-diameter distance: As above, we consider
a source of radius R observed to cover a solid angle co.
The angular-diameter distance is defined as
1 Luminosity distance: We consider a source with lumi-
nosity L and flux S and define its luminosity distance
/ An S
These two distances agree locally, i.e., for z <$C 1; on
small scales, the curvature of spacetime is not notice-
able. In addition, they are unique functions of redshift.
They can be computed explicitly. However, to do this
some tools of GR are required. Since we have not dis-
cussed GR in this book, these tools are not available to
us here. The distance-redshift relations depend on the
cosmological parameters; Fig. 4.11 shows the angular-
diameter distance for different models. For A = 0, the
4. Cosmology I: Homogeneous Isotropic World Models
0.8
Q=0flat ___.„
0.6
,.--''"'
Q=0 open
0.4
/ ^^^rrvr^TTTr^ - Q=0.3 flat _
0.2
- jjT n=i
the angular-diameter distance on the Earth's surface, we
define D A (D) = L/<p = R sin(D/R), in analogy to the
definition (4.45). For values of D that are considerably
smaller than the curvature radius R of the sphere, we
therefore obtain that D A « D, whereas for larger D, D A
deviates considerably from D. In particular, D A is not
a monotonic function of D, rather it has a maximum at
D = nR/2.
There exists a general relation between angular-
diameter distance and luminosity distance,
redshift z
Fig. 4.11. Angular-diameter distance vs. redshift for different
cosmological models. Solid curves display models with no
vacuum energy; dashed curves show flat models with Q m +
Qa = 1. In both cases, results are plotted for Q m = 1, 0.3,
andO
famous Mattig relation applies,
Hflm-2)(V
In particular, D A is not necessarily a monotonic func-
tion of z. To better comprehend this, we consider the
geometry on the surface of a sphere. Two great circles
on Earth are supposed to intersect at the North Pole
enclosing an angle <p <§C 1 - they are therefore meridi-
ans. The separation L between these two great circles,
i.e., the length of the connecting line perpendicular to
both great circles, can be determined as a function of
the distance D from the North Pole, which is measured
as the distance along one of the two great circles. If
6 is the geographical latitude (0 — jt/2 at the North
Pole, = -n/2 at the South Pole), L = Rep cos 6 is
found, where R is the radius of the Earth. L vanishes
at the North Pole, attains its maximum at the equator
(where 6 — 0), and vanishes again at the South Pole;
this is because both meridians intersect there again.
Furthermore, D — R(tt/2 — 0), e.g., the distance to the
equator D — Riz/2 is a quarter of the Earth's circumfer-
ence. Solving the last relation for 6, the distance is then
given by L — R<p cos(jt/2 - D/R) = Rep sin(D/R). For
D L (z) = (l+z) 2 D A (z)
(4.48)
The reader might now ask which of these distances
is the correct one? Well, this question does not make
sense since there is no unique definition of the dis-
tance in a curved spacetime like our Universe. Instead,
the aforementioned measurement prescriptions must be
used. The choice of a distance definition depends on
the desired application of this distance. For example,
if we want to compute the linear diameter of a source
with observed angular diameter, the angular-diameter
distance must be employed because it is defined just
in this way. On the other hand, to derive the luminos-
ity of a source from its redshift and observed flux, the
luminosity distance needs to be applied. Due to the def-
inition of the angular-diameter distance (length/angular
diameter), those are the relevant distances that appear
in the gravitational lens equation (3.48). A statement
that a source is located "at a distance of 3 billion light
years" away from us is meaningless unless it is men-
tioned which type of distance is meant. Again, in the
low-redshift Universe (z <$C 1), the differences between
different distance definitions are very small, and thus
it is meaningful to state, for example, that the Coma
cluster of galaxies lies at a distance of ~ 90 Mpc.
In Fig. 4.12 a Hubble diagram extending to high red-
shifts is shown, where the brightest galaxies in clusters
of galaxies have been used as approximate standard
candles. With an assumed constant intrinsic luminosity
for these galaxies, the apparent magnitude is a measure
of their distance, where the luminosity distance D L (z)
must be applied to compute the flux as a function of
redshift.
Without derivation, we compile several expressions
that are required to compute distances in general
Friedmann-Lemaitre models. To do this, we need to
4.3 Consequences of the Friedmann Expansion
and thus can be computed for all redshifts and cosmo-
logical parameters by (in general numerical) integration
of (4.49). The luminosity distance then follows from
(4.48). The angular-diameter distance of a source at red-
shift zi, as measured by an observer at redshift z\ < zi,
reads
Fig. 4.12. A modern Hubble diagram: for several clusters of
galaxies, [he K band magnitude ol" the brightest cluster gal
axy is plotted versus the escape velocity, measured as redshi ft
z = Ak/X (symbols). If these galaxies all had the same lu-
minosity, [he apparent magnitude would be a measure of
distance. For low redshifts, the curves follow the linear Hubble
law (4.9), with z <*> v/c, whereas for higher redshifts modifica-
tions to this law are necessary. The solid curve corresponds to
con ti n J lumino u\ al all redshifts, whereas the two
other curves take evolutionary effects of the luminosity into ac-
count according to models of population synthesis (Sect. 3.9).
Two different epochs of star format Ion w ere assumed for these
galaxies. The diagram is based on a cosmological model with
a deceleration parameter of go = (see Eq. 4.33)
define the function
1/VK sin(VZx)
K>0
K =
K smh(v^Kx) K <
where K is the curvature scalar (4.30). The comoving
radial distance x of a source at redshift z can be com-
puted using dx = a" 1 dr — -a~ x cdt — -cda/(a 2 H).
Hence with (4.31)
(4.49)
da(c/H )
D A (zuz 2 ) =
1
-f K [x(Z2)-x(Zl)l .
(4.51)
This is the distance that is required in equations
of gravitational lens theory for Dd s . In particular,
D A (ZI,Z2)^D A (Z2)-D A (Z 1 ).
4.3.4 Special Case: The Einstein-de Sitter Model
As a final note in this section, we will briefly exam-
ine one particular cosmological model more closely,
namely the model with Q A = and vanishing curvature,
K = 0, and hence Q m — 1 . We disregard the radiation
component, which contributes to the expansion only at
very early times and thus for very small a. For a long
time, this Einstein-de Sitter (EdS) model was the pre-
ferred model among cosmologists because inflation (see
Sect. 4.5.3) predicts K = and because a finite value for
the cosmological constant was considered "unnatural".
In fact, as late as the mid-1990s, this model was termed
the "standard model". In the meantime we have learned
that A ^ 0; thus we are not living in an EdS universe.
But there is at least one good reason to examine this
model a bit more, since the expansion equations be-
come much simpler for these parameters and we can
formulate simple explicit expressions for the quantities
introduced above. These then yield estimates which for
other model parameters are only possible by means of
numerical integration.
The resulting expansion equation a — H a a~ 112 is
easily solved by making the ansatz a = (Ci)P which,
when inserted into the equation, yields the solution
a(t) =
(3 H t
(4.52)
J jaQ m + a 2 (l-i2 m -n A ) + a 4 S2 A '
The angular-diameter distance is then given as
Da(z) = — f K [*(*)] , (4.50)
Setting a = 1, we obtain the age of the Universe,
t — 2/ (3 Ho). The same result also follows immedi-
ately from (4.34) if the parameters of an EdS model are
inserted there. Using Hq ss 70kms _1 Mpc -1 results in
an age of about 10 Gyr, which is slightly too low to be
4. Cosmology I: Homogeneous Isotropic World Models
compatible with the age of the oldest star clusters. The
angular-diameter distance (4.45) in an EdS universe is
obtained by considering the Mattig relation (4.38) for
the case Q m — 1 :
H (l+Z)\ y/T+i)
where we used (4.48) to obtain the second relation from
the first.
4.3.5 Summary
We shall summarize the most important points of the
two preceeding lengthy sections:
• Observations are compatible with the fact that the
Universe around us is isotropic and homogeneous on
large scales. The cosmological principle postulates
this homogeneity and isotropy of the Universe.
• General Relativity allows homogeneous and
isotropic world models, the Friedmann-Lemaitre
models. In the language of GR, the cosmological
principle reads as follows: "A family of solutions
of Einstein's field equations exists such that a set
of comoving observers see the same history of the
Universe; for each of them, the Universe appears
isotropic."
• The shape of these Friedmann-Lemaitre world mod-
els is characterized by the density parameter Q m
and by the cosmological constant £2 A , the size by
the Hubble constant Hq. The cosmological parame-
ters determine the expansion rate of the Universe as
a function of time.
• The scale factor a(t) of the Universe is a monoton-
ically increasing function from the beginning of the
Universe until now; at earlier times the Universe was
smaller, denser, and hotter. There must have been an
instant when a -> 0, which is called the Big Bang.
The future of the expansion depends on £2 m and
Q A .
• The expansion of the Universe causes a redshift of
photons. The more distant a source is from us, the
more its photons are redshifted.
4.4 Thermal History of the Universe
Since T oc(l + z) the Universe was hotter at earlier
times. For example, at a redshift of z — 1 100 the tem-
perature was about T ~ 3000 K. And at an even higher
redshift, z — 10 9 , it was T ~ 3 x 10 9 K, hotter than in
a stellar interior. Thus we might expect energetic pro-
cesses like nuclear fusion to have taken place in the
early Universe.
In this section we shall describe the essential pro-
cesses in the early Universe. To do so we will assume
that the laws of physics have not changed over time. This
assumption is by no means trivial - we have no guaran-
tee whatsoever that the cross-sections in nuclear physics
were the same 13 billion years ago as they are today.
But if they have changed in the course of time the only
chance of detecting this is through cosmology. Based
on this assumption of time-invariant physical laws, we
will study the consequences of the Big Bang model de-
veloped in the previous section and then compare them
with observations. Only this comparison can serve as
a test of the success of the model. A few comments
should serve as preparation for the discussion in this
section.
1. Temperature and energy may be converted into
each other since k B T has the dimension of en-
ergy. We use the electron volt (eV) to measure
temperatures and energies, with the conversion
leV= 1.1605 x 10 4 £ B K.
2. Elementary particle physics is very well understood
for energies below ~ 1 GeV. For much higher en-
ergies our understanding of physics is a lot less
certain. Therefore, we will begin the consideration
of the thermal history of the cosmos at energies be-
low 1 GeV.
3. Statistical physics and thermodynamics of elemen-
tary particles are described by quantum mechanics.
A distinction has to be made between bosons,
which are particles of integer spin (like the pho-
ton), and fermions, particles of half-integer spin
(like, for instance, electrons, protons, or neutri-
nos).
4. If particles are in thermodynamical and chemical
equilibrium, their number density and their energy
distribution are specified solely by the temperature -
e.g., the Planck distribution (A. 13), and thus the en-
4.4 Thermal History of the Universe
ergy density of the radiation (4.43), is a function of T
only.
The necessary condition for establishing chemical
equilibrium is the possibility for particles to be be cre-
ated and destroyed, such as in electron-positron pair
production and annihilation.
4.4.1 Expansion in the Radiation-Dominated
Phase
As mentioned above (4.28), the energy density of radia-
tion dominates in the early Universe, at redshifts z J5> z eq
where
- 1 & 23 900 i'2 m lr
(4.54)
The radiation density behaves like p r oc T 4 , where the
constant of proportionality depends on the number of
species of relativistic particles (these are the ones for
which k B T JJ> mc 2 ). Since T oc \/a and thus p r oc a~ 4 ,
radiation then dominates in the expansion equation
(4. 1 8). The latter can be solved by apower law, a(t) oc t p ,
which after insertion into the expansion equation yields
j6= 1/2 and thus
Y 32jz-Gp '
in radiation-dominated phase
(4,55)
where the constant of proportionality depends again on
the number of relativistic particle species. Since the lat-
ter is known, assuming thermodynamical equilibrium,
the time dependence of the early expansion is uniquely
specified by (4.55). This is reasonable because for early
times neither the curvature term nor the cosmologi-
cal constant contribute significantly to the expansion
dynamics.
4.4.2 Decoupling of Neutrinos
We start our consideration of the Universe at a temper-
ature of T % 10 12 K which corresponds to ~ 100 MeV.
This energy can be compared to the rest mass of various
particles:
proton, m p
= 938.3 MeV/c 2 ,
neutron, m
„ = 939.6 MeV/c 2 ,
electron, »
e = 511keV/c 2 ,
muon, m M
= 140MeV/c 2 .
Protons and neutrons (i.e., the baryons) are too heavy
to be produced at the temperature considered. Thus all
baryons that exist today must have already been present
at this early time. Also, the production of muon 5 pairs,
according to the reaction y + y -> /x + + fi~ , is not ef-
ficient because the temperature, and thus the typical
photon energy, is not sufficiently high. Hence, at the
temperature considered the following relativistic parti-
cle species are present: electrons and positrons, photons
and neutrinos. These species contribute to the radiation
density p T . The mass of the neutrinos is not accu-
rately known, though we recently learned that they have
a small but finite rest mass. As will be explained in
Sect. 8.7, cosmology allows us to obtain a very strict
limit on the neutrino mass, which is currently below
1 eV. For the purpose of this discussion they may be
considered as massless.
In addition to relativistic particles, non-relativistic
particles also exist. These are the protons and neutrons,
and probably also the constituents of dark matter. We
assume that the latter consists of weakly interacting
massive particles (WIMPs), with rest mass larger than
~ 100 GeV because up to these energies no WIMP can-
didates have been found in terrestrial particle accelerator
laboratories. With this assumption, WIMPs are non-
relativistic at the energies considered. Thus, like the
baryons, they virtually do not contribute to the energy
density in the early Universe.
Apart from the WIMPs, all the aforementioned par-
ticle species are in equilibrium, e.g., by the following
» e* + y: Compton scattering,
** Y + Y'- pair-production and annihilation,
e + + e~ : neutrino-antineutrino scattering,
► v + e ± : neutrino-electron scattering.
W'lttons arc particles which bcha\c in many respects as electrons, ex-
cept thai thc\ are much hea\ ier. Furthermore, unions are unstable and
decay on a time-scale of ~ 2 x 10~ 6 s into an electron (or positron)
4. Cosmology I: Homogeneous Isotropic World Models
Reactions involving baryons will be discussed later.
The energy density at this epoch is
ln ^7T 2 far) 4
P = Pr = 10 ' 75 30^^'
which yields - see (4.55) -
r«0.3s
VIMeV/
(4.56!
Hence, about one second after the Big Bang the temper-
ature of the Universe was about 10 10 K. For the particles
to maintain equilibrium, the reactions above have to oc-
cur at a sufficient rate. The equilibrium state, specified
by the temperature, continuously changes, so that the
particle distribution needs to continually adjust to this
changing equilibrium. This is possible only if the mean
time between two reactions is much shorter than the
time-scale on which equilibrium conditions change. The
latter is given by the expansion. This means that the re-
action rates (the number of reactions per particle per unit
time) must be larger than the cosmic expansion rate H{t)
in order for the particles to maintain equilibrium.
The reaction rates r are proportional to the prod-
uct of the number density n of the particles and the
cross-section a of the corresponding reaction. Both
decrease with time: the number density decreases as
n oc a" 3 oc t~ 3/2 because of the expansion. Furthermore,
the cross-sections for weak interaction, which is respon-
sible for the reactions involving neutrinos, depend on
energy, approximately as a oc E 2 oc T 2 oc a~ 2 . Together
this yields r oc na oc aT s oc f~ 5/2 , whereas the expan-
sion rate decreases only as H oc t ~ ' . At sufficiently early
times, the reaction rates were larger than the expansion
rate, and thus particles remained in equilibrium. Later,
however, the reactions no longer took place fast enough
to maintain equilibrium. The time or temperature, re-
spectively, of this transition can be calculated from the
cross-section of weak interaction,
H
U-6xl0 10 K/
so that for T < 10 K neutrinos are no longer in equilib-
rium with the other particles. This process of decoupling
from the other particles is also called freeze-out; neutri-
nos freeze out at T ~ 10 10 K. At the time of freeze-out,
they had a thermal distribution with the same tem-
perature as the other particle species which stayed in
mutual equilibrium. From this time on neutrinos propa-
gate without further interactions, and so have kept their
thermal distribution up to the present day, with a temper-
ature decreasing as T oc I /a. This consideration predicts
that these neutrinos, which decoupled from the rest of
the matter about one second after the Big Bang, are
still around in the Universe today. They have a num-
ber density of 113 cm -3 per neutrino family and are
at a temperature of 1 .9 K (this value will be explained
in more detail below). However, these neutrinos are
currently undetectable because of their extremely low
cross-section.
The expansion behavior is unaffected by the neutrino
freeze-out and continues to proceed according to (4.56).
4.4.3 Pair Annihilation
At temperatures smaller than ~ 5 x 10 9 K, or k^T ~
500 keV, electron-positron pairs can no longer be pro-
duced efficiently since the number density of photons
with energies above the pair production threshold of
5 1 1 keV is becoming too small. However, the annihila-
tion e + + e~ -> y + Y continues to proceed and, due
to its large cross-section, the density of e + e~ -pairs
decreases rapidly.
Pair annihilation injects additional energy into the
photon gas, originally present as kinetic and rest mass
energy of the e + e~ pairs. This changes the energy dis-
tribution of photons, which continues to be a Planck
distribution but now with a modified temperature rela-
tive to that it would have had without annihilation. The
neutrinos, already decoupled at this time, do not bene-
fit from this additional energy. This means that after the
annihilation the photon temperature exceeds that of the
neutrinos. From the thermodynamics of this process,
the change in photon temperature is computed as
T (after annihilation)
/11\ 1/3
= I — T (before annihilation)
This temperature ratio is preserved afterwards, so that
have a temperature lower than that of the
4.4 Thermal History of the Universe
photons by (1 1/4) 1/3 ~ 1.4 - until the present epoch.
This result has already been mentioned and taken into
i the estimate of p t $ in (4.26); we find
After pair annihilation, the expansion law
t = 0.55 s
VlMeV/
applies. This means that, as a result of annihilation, the
constant in this relation changes compared to (4.56)
because the number of relativistic particles species has
decreased. Furthermore, the ratio r\ of baryon to photon
density remains constant after pair annihilation. The
former is characterized by the density parameter Q h —
Pb,o/£? CT in baryons (today), and the latter is determined
by 7b:
n : = (—\ = 2.74 x lrr 8 (n b h 2 )
Before pair annihilation there were about as many
electrons and positrons as there were photons. After
annihilation nearly all electrons were converted into
photons - but not entirely because there was a very
small excess of electrons over positrons to compensate
for the positive electrical charge density of the protons.
Therefore, the number density of electrons that survive
the pair annihilation is exactly the same as the number
density of protons, for the Universe to remain electri-
cally neutral. Thus, the ratio of electrons to photons is
also given by rj, or more precisely by about 0.8r), since
r\ includes both protons and neutrons.
(4.58)
Proton-to-Neutron Abundance Ratio. As already dis-
cussed, the baryons (or nucleons) do not play any role in
the expansion dynamics in the early Universe because of
their low density. The most important reactions through
which they maintain chemical equilibrium with the rest
of the particles are
p + e«*n+v ,
The latter is the decay of free neutrons, with a time-
scale for the decay of r n = 887 s. The first two reactions
maintain the equilibrium proton-to-neutron ratio as long
as the corresponding reaction rates are large compared
to the expansion rate. The equilibrium distribution is
specified by the Boltzmann factor,
-= ex P —
(4.60)
k B T )
where Am — m n — m p — 1 .293 MeV/c 2 is the mass dif-
ference between neutrons and protons. Hence, neutrons
are slightly heavier than protons; otherwise the neutron
decay would not be possible. After neutrino freeze-out
equilibrium reactions become rare because the above re-
actions are based on weak interactions, the same as those
which kept the neutrinos in chemical equilibrium. At
the time of neutrino decoupling, we have « n /w p « 1/3.
After this, protons and neutrons are no longer in equi-
librium, and their ratio is no longer described by (4.60).
Instead, it changes only by the decay of free neutrons
on the time-scale r n . To have neutrons survive at all un-
til the present day, they must quickly become bound in
atomic nuclei.
4.4.4 Primordial Nucleosynthesis
Protons and neutrons can fuse to form atomic nuclei
if the temperature and density of the plasma are suffi-
ciently high. In the interior of stars, these conditions for
nuclear fusion are provided. The high temperatures in
the early phases of the Universe suggest that atomic
nuclei may also have formed then. As we will dis-
cuss below, in the first few minutes after the Big Bang
some of the lightest atomic nuclei were formed. The
quantitative discussion of this primordial nucleosyn-
thesis (Big Bang nucleosynthesis, BBN) will explain
observation (4) of Sect. 4.1.1.
Deuterium Formation. The simplest compound nu-
cleus is that of deuterium (D), consisting of a proton
and a neutron and formed in the reaction
p + n^D+y .
The binding energy of D is E\, — 2.225 MeV. This en-
ergy is only slightly larger than m e c 2 and Am - all
these energies are of comparable size. The formation
of deuterium is based on strong interactions and there-
fore occurs very efficiently. However, at the time of
neutrino decoupling and pair annihilation, T is not
much smaller than E\,. This has an important conse-
quence: because photons are so much more abundant
4. Cosmology I: Homogeneous Isotropic World Models
than baryons, a sufficient number of highly energetic
photons, with E y > E^, exist in the Wien tail of the
Planck distribution to instantly destroy newly formed D
by photo-dissociation. Only when the temperature has
decreased considerably, k B T <£ E^, can the deuterium
abundance become appreciable. With the corresponding
balance equations we can calculate that the formation
rate exceeds the photo-dissociation rate of deuterium at
about r D *8x 10 8 K, corresponding to t ~ 3 min. Up
to then, a fraction of the neutrons has thus decayed,
yielding a neutron-proton ratio at T D of n n /« p « 1/7.
After that time, everything happens very rapidly. Ow-
ing to the strong interaction, virtually all neutrons first
become bound in D. Once the deuterium density has
become appreciable, helium (He 4 ) forms, which is a nu-
cleus with high binding energy (~ 28 MeV) which can
therefore not be destroyed by photo-dissociation. Ex-
cept for a small (but, as we will later see, very important)
remaining fraction, all deuterium is quickly transformed
into He 4 . For this reason, the dependence of helium for-
mation on the small binding energy of D is known as
the "bottleneck of nucleosynthesis".
Helium Abundance. The number density of helium nu-
clei can now be calculated since virtually all neutrons
present are bound in He 4 . First, « He = n n /2, since ev-
ery helium nucleus contains two neutrons. Second, the
number density of protons after the formation of helium
is «h = "p — «n> since He 4 contains an equal number of
protons and neutrons. From this, the mass fraction Y of
He 4 of the baryon density follows,
2(«n/»p) „
" l + (n„/n p r
where in the last step we used the above ratio of
«n/ w P ^ 1/7 at T D . This consideration thus leads to
the following:
About 1/4 of the baryonic mass in the Universe
should be in the form of He 4 . This is a robust pre-
diction of Big Bang models, and it is in excellent
agreement with observations.
Minutes: 1/60
Temperature (1 9 K)
Fig. 4.13. The evolution of abundances of the light elements
formed in BBN, as a function of temperature I low ei axis) and
cosmic time / ( upper axis). The decrease in neutron abundance
in the first ~ 3 min is due to neutron decay. The density of
deuterium increases steeply - linked to the steep decrease
in neutron density - and reaches a maximum at t ~ 3 min
because then its density becomes sufficiently large for efficient
formation of He 4 to set in. Only a few deuterium nuclei do
not find a reaction partner and remain, with a mass fraction of
~ 10~ 5 . Only a few other light nuclei are formed in the Big
Bang, mainly He 3 and Li 7
The helium content in the Universe may change later
by nuclear fusion in stars, which also forms heavier
nuclei ("metals"). Observations of fairly unprocessed
material (i.e., that which has a low metal content) reveal
that in fact Y « 0.25. Figure 4.13 shows the result of
a quantitative model of BBN where the mass fraction
of several species is plotted as a function of time or
temperature, respectively.
Dependence of the Primordial Abundances on the
Baryon Density. At the end of the first 3 min, the com-
position of the baryonic component of the Universe is
about as follows: 25% of the baryonic mass is bound in
helium nuclei, 75% in hydrogen nuclei (i.e., protons),
with traces of D, He 3 and Li 7 . Heavier nuclei cannot
form because no stable nucleus of mass number 5 or
8 exists and thus no new, stable nuclei can be formed
in collisions of two helium nuclei or of a proton with
a helium nucleus. Collisions between three nuclei are
far too rare to contribute to nucleosynthesis. The den-
4.4 Thermal History of the Universe
sity in He 4 and D depends on the baryon density in the
Universe, as can be seen in Fig. 4.14 and through the
following considerations:
• The larger the baryon density Q h , thus the larger the
baryon-to-photon ratio r\ (4.59), the earlier D can
form, i.e., the fewer neutrons have decayed, which
then results in a larger n n /n v ratio. From this and
(4.61) it follows that Y increases with increasing Q^.
• A similar argument is valid for the abundance of deu-
terium: the larger Q b is, the higher the baryon density
during the conversion of D into He 4 . Thus the conver-
sion will be more efficient and more complete. This
means that fewer deuterium nuclei remain without
a reaction partner for helium formation. Thus fewer
of them are left over in the end, so the fraction of D
will be lower.
Baryon Content of the Universe. From measurements
of the primordial abundances of He 4 and D and their
comparison with detailed models of nucleosynthesis in
the early Universe, r\ or Q b , respectively, can be deter-
mined (see Fig. 4.14). The abundance of deuterium is
a particularly sensitive measure for Q b . Measurements
of the relative strength of the Lya lines of H and D,
which have slightly different transition frequencies due
to the different masses of their nuclei, in QSO absorption
lines (see Sect. 5.6.3) yield D/H%3.4x lfr 5 . Since
the intergalactic gas producing these absorption lines is
very metal-poor and thus presumably barely affected by
nucleosynthesis in stars, its D/H ratio should be close
to the primordial value. Combining the quoted value of
D/H with the model curves shown in Fig. 4.14 we find
Q b h
«0.02
With a Hubble constant of H ~ 70 km s" ' Mpc" ' , thus
h ~ 0.7, we have Q b % 0.04. But since Q m > 0.1, this
result implies that baryons represent only a small frac-
tion of the matter in the Universe. The major fraction of
matter is non-baryonic dark matter.
WIMPs as Dark Matter. Big Bang nucleosynthesis
therefore provides a clear indication of the existence
of non-baryonic dark matter on cosmological scales.
Whereas our discussion of rotation curves of spiral gal-
axies could not fully exclude the possibility that the dark
Fraction of critical density
,01 0.02 0.05
(4.62)
Baryon density(10 ' 31 g cm 3 )
Fig. 4.14. BBN predictions of the primordial abundances of
light elements as a function of today's baryon density (pb,o.
lower axis) and the corresponding density parameter Q b where
h — 0.65 is assumed. The vertical extent of the rectangles
marks the measured values of the abundances (top: He 4 , cen-
ter: D, bottom: Li 7 ). The horizontal extent results from the
overlap of these intervals with curves computed from theoret-
ical models. The ranges in Q b that are allowed by these three
species do overlap, as is indicated by the vertical strip. The
deuterium measurements \iekl the most stringent constraints
for Q h
s of baryons, BBN shows that this cannot
be the case.
As mentioned previously, the most promising candi-
date for a dark matter constituent is an as yet unknown
elementary particle, a WIMP. In fact, from the above
considerations, constraints on the properties of such
a particle can be derived. If the WIMP is weakly inter-
acting, it will decouple in a similar way to the neutrinos.
If its mass m WIMP is smaller than the decoupling tem-
perature (T ~ 1 MeV), the WIMP was relativistic at
the epoch of freeze-out and thus its number density
is the same as that of the neutrinos, n — 113 cm -3 to-
4. Cosmology I: Homogeneous Isotropic World Models
day. Hence, \
parameter,
n compute the corresponding density
"'WIMP
"91.5eV '
(4.63)
provided m W iMP ^ 1 MeV. This equation is of course
also valid for neutrinos with m v <\ MeV. Since Q m < 2
certainly, we conclude from (4.63) that no stable weakly
interacting particle can exist in the mass range of
100 eV < m < 1 MeV. In particular, none of the three
neutrinos can have a mass in this range. Until recently,
these constraints on the mass of the fi- and T-neutrinos
were many orders of magnitude better than those from
laboratory experiments. Only measurements of neutrino
oscillations, in connection with laboratory constraints
on the p e -mass, provided better mass constraints. From
structure formation in the Universe, which will be dis-
cussed in Chap. 7, we now know that neutrinos cannot
make a dominant contribution to the dark matter, and
therefore an upper limit on their mass m v < 1 eV can be
obtained.
If the WIMP is heavier than 1 MeV, it decouples at
a time when it is already non-relativistic. Then the es-
timate of the number density, and with it the relation
(4.63), needs to be modified. In particular, if the WIMP
mass exceeds that of the Z-boson 6 (m z = 91 GeV), then
^wimp h 2 ~ (wwimp/ 1 TeV) 2 . This means that a WIMP
mass of several hundred GeV would provide a density
of J2wimp % ^m ~ 0.3. The next generation of particle
accelerators, in particular the Large Hadron Collider at
CERN which is supposed to start operations in 2007,
will most likely be able to detect such a particle in the
laboratory if it really exists. In fact, arguably the most
promising extension to the standard model of particle
physics - the model of supersymmetry - predicts a sta-
ble particle with a mass of several hundred GeV, the
neutralino.
In the analysis of BBN we implicitly assumed that
not more than three (relativistic, i.e., with m v < 1 MeV)
neutrino families exist. If N v > 3, the quantitative pre-
dictions of BBN will change. In this case, the expansion
6 The Z-boson is one of the exchange particles in weak i
the oilier luo are the charged \V= particles. The) were predicted
h\ the model oi electron eak interactions, and linalh being found in
particle accelerator-,., thus bcautiiulh supporting the \alidit\ ul this
electrowcak unification model which lies at the heart ol the standard
model of particle physics. The Z-boson plays about the s
weak interactions as the photon does in electromagnetic i
would occur faster (see Eq. 4.55) because p(T) would
be larger, leaving less time until the Universe has cooled
down to T D - thus, fewer neutrons would decay and the
resulting helium abundance would be higher. Even be-
fore 1990, it was concluded from BBN (with relatively
large uncertainties, however) that N v — 3. In 1990, the
value of N v — 3 was then confirmed in the laboratory
from Z-boson decay.
To circumvent the conclusion of a dominant fraction
of non-baryonic matter, inhomogeneous models of BBN
have been investigated, but these also yield values for J2 b
which are too low and therefore do not provide a viable
alternative.
4.4.5 Recombination
About 3 min after the Big Bang, BBN comes to an end.
At this time, the Universe has a temperature of roughly
T ~ 8 x 10 8 K and consists of photons, protons, helium
nuclei, traces of other light elements, and electrons.
In addition, there are neutrinos that dominate, together
with photons, the energy density and thus also the ex-
pansion rate, and there are (probably) WIMPs. Except
for the neutrinos and the WIMPs, all particle species
have the same temperature, which is established by in-
teractions of charged particles with the photons, which
resemble some kind of heat bath.
At z — z eq % 23 900 Q m h 2 , pressureless matter (i.e.,
the so-called dust) begins to dominate the energy
density in the Universe and thus the expansion rate.
The second term in (4.31) then becomes largest, i.e.,
H 2 sa Hq f2 m /a 3 . If a power-law ansatz for the scale
factor, aoctP, is inserted into the expansion equation,
we find that p — 2/3, and hence
a(t) =
(^S^H Q t
r
for a eq < a «; 1
(4.64)
This describes the expansion behavior until either the
curvature term or, if this is zero, the yl-term starts to
dominate.
After further cooling, the free electrons can combine
with the nuclei and form neutral atoms. This process is
called recombination, although this expression is mis-
leading: since the Universe was fully ionized until then,
4.4 Thermal History of the Universe
it is not a recombination but rather the (first) transition to
a neutral state - however the expression recombination
has now long been established. The recombination of
electrons and nuclei is in competition with the ionization
of neutral atoms by energetic photons (photoionization),
whereas collisional ionization can be disregarded com-
pletely since r\ - (4.59) - is so small. Because photons
are so much more numerous than electrons, cooling has
to proceed to well below the ionization temperature,
corresponding to the binding energy of an electron in
hydrogen, before neutral atoms become abundant. This
happens for the same reasons as apply in the context of
deuterium formation: there are plenty of ionizing pho-
tons in the Wien tail of the Planck distribution, even if
the temperature is well below the ionization tempera-
ture. The ionization energy of hydrogen is x = 13.6 eV,
corresponding to a temperature of T> 10 5 K, but T
has to first decrease to ~ 3000 K before the ionization
fraction
number density of free electrons
total number density of existing protons
(4.65)
falls considerably below 1, for the reason mentioned
above. At temperatures T> 10 4 K we have x«l, i.e.,
virtually all electrons are free. Only at z ~ 1300 does x
deviate significantly from unity.
The onset of recombination can be described by an
equilibrium consideration which leads to the so-called
Saha equation,
1-
""«'(£?) «*(£)•
which describes the ionization fraction x as a function
of temperature. However, once recombination occurs,
the assumption of thermodynamical equilibrium is no
longer justified. This can be seen from the following
consideration.
Any recombination directly to the ground state leads
to the emission of a photon with energy E y > x- How-
ever, these photons can ionize other, already recombined
(thus neutral), atoms. Because of the large cross-section
for photoionization, this happens very efficiently. Thus
for each recombination to the ground state, one neutral
atom will become ionized, yielding a vanishing net ef-
fect. But recombination can also happen in steps, first
into an excited state and then evolving into the ground
state by radiative transitions. Each of these recombina-
tions will yield a Lya photon in the transition from the
first excited state into the ground state. This Lya pho-
ton will then immediately excite another atom from the
ground state into the first excited state, which has an
ionization energy of only //4. This yields no net pro-
duction of atoms in the ground state. Since the density
of photons with E y > //4 is very much larger than of
those of E Y > x, the excited atoms are more easily ion-
ized, and this indeed happens. Stepwise recombination
thus also provides no route towards a lower ionization
fraction.
The processes described above cause a small dis-
tortion of the Planck spectrum due to recombination
radiation (in the range x ^ &b T) which affects re-
combination. One cannot get rid of these energetic
photons - in contrast to gas nebulae like Hll regions,
in which the Lya photons may escape due to the finite
geometry.
Ultimately, recombination takes place by means of
a very rare process, the two-photon decay of the first
excited level. This process is less probable than the di-
rect Lya transition by a factor of ~ 10 8 . However, it
leads to the emission of two photons, both of which
are not sufficiently energetic to excite an atom from the
ground state. This 2y -transition is therefore a net sink
for energetic photons. 7 Taking into account all relevant
processes and using a rate equation, which describes
the evolution of the distribution of particles and pho-
tons even in the absence of thermodynamic equilibrium,
gives for the ionization fraction in the relevant redshift
range 800 < z < 1200
The ionization fraction is thus a very strong function of
redshift since x changes from 1 (complete ionization)
The recombination of hydrogen - and also that of helium which
occurred at higher rcclshifis - perturbed the exact Planck shape of
the photon distribution, adding to it the L\ man alpha photons and
the photon pairs from the two-photon transition. This slight perturba-
tion in the CMB spectrum should in principle still be present today.
Unfortunately, it lies in a wavelength range i - 200 pm] where the
dust emission from the Galaxy is very strong: in addition, the wave-
length range coincides with the peals of the far infrared background
radiation (see Sect.9.3.1). Therefore, the detection of this spectral
distortion will be extremely difficult.
4. Cosmology I: Homogeneous Isotropic World Models
to x ~ 10~ 4 (where essentially all atoms are neutral)
within a relatively small redshift range. The recom-
bination process is not complete, however. A small
ionization fraction of x ~ 10~ 4 remains since the re-
combination rate for small x becomes smaller than the
expansion rate - some nuclei do not find an electron fast
enough before the density of the Universe becomes too
low. From (4.66), the optical depth for Thomson scat-
tering (scattering of photons by free electrons) can be
computed,
VlOOO/
which is virtually independent of cosmological pa-
rameters. Equation (4.67) implies that photons can
propagate from z ~ 1000 (the last-scattering surface)
until the present day essentially without any interac-
tion with matter - provided the wavelength is larger
than 1216 A. For photons of smaller wavelength, the
absorption cross-section of neutral atoms is large. Dis-
regarding these highly energetic photons here - their
energies are > 10 eV, compared to r rec ~ 0.3 eV, so
they are far out in the Wien tail of the Planck distribu-
tion - we conclude that the photons present in the early
Universe have been able to propagate without further in-
teractions until the present epoch. Before recombination
they followed a Planck spectrum. As was discussed in
Sect. 4.3.2, the distribution will remain a Planck spec-
trum with only its temperature changing. Thus these
photons from the early Universe should still be observ-
able today, redshifted into the microwave regime of the
electromagnetic spectrum.
Our consideration of the early Universe predicts
thermal radiation from the Big Bang, as was first
realized by George Gamow in 1946 - the cos-
mic microwave background. The CMB is therefore
a visible relic of the Big Bang.
The CMB was detected in 1965 by Arno Penzias
& Robert Wilson (see Fig. 4.15), who were awarded
the 1978 Nobel prize in physics for this very important
discovery. At the beginning of the 1990s, the COBE
satellite measured the spectrum of the CMB with a very
high precision - it is the most perfect blackbody ever
A MEASUREMENT OF EXCESS
TEMPERATURE
antenna (Crawford, Hogg, and Hunt 1961) at the Crawford Hill Laboratory, Holmdel,
New Jersey, at 4080 Mc/s have yielded a value about 3.5° K higher than expected. This
Fig. 4.15. The first lines of the article by Penzias & Wilson,
1965, ApJ, 142, 419
measured (see Fig. 4.3). From upper limits of deviations
from the Planck spectrum, very tight limits for possible
later energy injections into the photon gas, and thus on
energetic processes in the Universe, can be obtained. 8
We have only discussed the recombination of hy-
drogen. Since helium has a higher ionization energy
it recombines earlier than hydrogen. Although recom-
bination defines a rather sharp transition, (4.67) tells
us that we receive photons from a recombination layer
of finite thickness (Az ~ 60). This aspect will be of
importance later.
The gas in the intergalactic medium at lower redshift
is highly ionized. If this were not the case we would
not be able to observe any UV photons from sources
at high redshift ("Gunn-Peterson test", see Sect. 8.5.1).
Sources with redshifts z > 6 have been observed, and
we also observe photons with wavelengths shorter than
the Lya line of these objects. Thus at least at the epoch
corresponding to redshift z ~ 6, the Universe must have
been nearly fully ionized or else these photons would
have been absorbed by photoionization of neutral hy-
drogen. This means that at some time between z ~ 1000
and z ~ 6, a reionization of the intergalactic medium
must have occurred, presumably by a first generation
of stars or by the first AGNs. The results from the new
CMB satellite WMAP suggest a reionization at red-
shift z ~ 15; this will be discussed more thoroughly in
Sect. 8.7.
s For instance, there exists an X ra\ background i XRB ) w hicli is radi
ation thai appeared isotropic in earl\ measurements. For a long time,
n 11 ill ion lot i'ii i ii iii i ii i i! i
medium w itli temperature of A B /' 10 kcV emitting bremsstrahhmg
radiation, Bui such a hot intergalactic gas would modif\ the spectrum
of the CMB via the scattering of CMB photons to higher frequencies
b\ energetic electrons (hnerse Compton scattering). This explanation
for the source of the XRB was excluded by the COBE measure-
ni n i-ii in obsenations b_\ I i i Hit O V Ch idi
and X.Y1M .Newton, with their high angular resolution, we know to
day that the XRB is a superposition of radiation from discrete sources,
mostly AGNs.
4.5 Achievements and Problems of the Standard Model
4.4.6 Summary
We will summarize this somewhat long section as
follows:
• Our Universe originated from a very dense, very hot
state, the so-called Big Bang. Shortly afterwards, it
consisted of a mix of various elementary particles,
all interacting with each other.
• We are able to examine the history of the Universe in
detail, starting at an early epoch where it cooled down
by expansion such as to leave only those particle
species known to us (electrons, protons, neutrons,
neutrinos, and photons).
• Because of their weak interaction and the decreasing
density, the neutrinos experience only little interac-
tion at temperatures below ~ 10 10 K, the decoupling
temperature.
• At T ~ 5 x 10 9 K, electrons and positrons annihi-
late into photons. At this low temperature, pair
production ceases to take place.
• Protons and neutrons interact and form deuterium nu-
clei. As soon as T ~ 10 9 K, deuterium is no longer
efficiently destroyed by energetic photons. Further
nuclear reactions produce mainly helium nuclei.
About 25% of the mass in nucleons is transformed
into helium, and traces of lithium are produced, but
no heavier elements.
• At about T ~ 3000 K, some 400 000 years after
the Big Bang, the protons and helium nuclei com-
bine with the electrons, and the Universe becomes
essentially neutral (we say that it "recombines").
From then on, photons can travel without further
interactions. At recombination, the photons follow
a blackbody distribution (i.e., a thermal spectrum, or
a Planck distribution). By the ongoing cosmic ex-
pansion, the temperature of the spectral distribution
decreases, T oc (1 + z) _1 , though its Planck property
remains.
• After recombination, the matter in the Universe is
almost completely neutral. However, we know from
the observation of sources at very high redshift that
the intergalactic medium is essentially fully ionized
at z < 6. Before z > 6, the Universe must therefore
have experienced a phase of reionization. This effect
cannot be explained in the context of the strictly ho-
mogeneous world models; rather it must be examined
in the context of structure formation in the Universe
and the formation of the first stars and AGNs. These
aspects will be discussed in Sect. 9.4.
4.5 Achievements and Problems
of the Standard Model
To conclude this chapter, we will evaluate the cos-
mological model which has been presented. We will
review its achievements and successes, but also appar-
ent problems, and point out the route by which those
might be understood. As is always the case in natural
sciences, problems with an otherwise very success-
ful model are often the key to a new and deeper un-
derstanding.
4.5.1 Achievements
The standard model of the Friedmann-Lemaitre Uni-
verse described above has been extremely successful in
numerous ways:
• It predicts that gas which has not been sub-
ject to much chemical processing (i.e., metal-poor
gas) should have a helium content of ~ 25%.
This is in extraordinarily good agreement with
observations.
• It predicts that sources of lower redshift are closer
to us than sources of higher redshift. 9 Therefore,
modulo any peculiar velocities, the absorption of ra-
diation from sources at high redshift must happen at
smaller redshifts. Not a single counter-example has
been found yet.
• It predicts the existence of a microwave background,
which has indeed been found.
• It predicts the correct number of neutrino families,
which was confirmed in laboratory experiments of
the decay of the Z-boson.
Further achievements will be discussed in the context
of structure evolution in the Universe.
A good physical model is one that can also be falsi-
fied. In this respect, the Friedmann-Lemaitre Universe
is also an excellent model: a single observation could ei-
9 We ignore peculiar motions here which may cause an additional
(Doppler (redshift. These are typically < 1000 km/s and are thus
small compared to cosmological redshifts.
4. Cosmology I: Homogeneous Isotropic World Models
ther cause a lot of trouble for this model or even disprove
it. To wit, it would be incompatible with the model
1. if the helium content of a gas cloud or of a low-
metallicity star were significantly below 25%;
2. if it were found that one of the neutrinos has a rest
mass > 100 eV;
3. if the Wien part of the CMB had a smaller amplitude
compared to the Planck spectrum;
4. if a source with emission lines at z e were found to
show absorption lines at z- d J5> z e ;
5. if the cosmological parameters were such that t <
lOGyr.
On (1): While the helium content may increase by stel-
lar evolution due to fusion of hydrogen into helium,
only a small fraction of helium is burned in stars. In
this process, heavier elements are of course produced.
A gas cloud or a star with low metallicity therefore
cannot consist of material in which helium has been
destroyed; it must contain at least the helium abun-
dance from BBN. On (2): Such a neutrino would lead to
£2 m > 2, which is in strict contradiction to the derived
model parameters. On (3): Though it is possible to gen-
erate additional photons by energetic processes in the
Universe, thereby increasing the Wien part of the coad-
ded spectrum compared to that of a Planck function,
it is thermodynamically impossible to extract photons
from the Wien part. On (4): Such an observation would
question the role of redshift as a monotonic measure of
relative distances and thus remove one of the pillars of
the model. On (5): Our knowledge of stellar evolution
allows us to determine the age of the oldest stars with
a precision of better than ~ 20%. An age of the Uni-
verse below ~ 10 Gyr would be incompatible with the
age of the globular clusters - naturally, these have to be
younger than the age of the Universe, i.e., the time after
the Big Bang.
Although these predictions have been known for
more than 30 years, no observation has yet been made
which disproves the standard model. Indeed, at any
given time there have been astronomers who disagree
with the standard model. These astronomers have tried
to make a discovery, like the examples above, which
would pose great difficulties for the model. So far, they
have not succeeded; this does not mean that such re-
sulls cannot be found in the literature, but rather such
results did not withstand closer examination. The simple
opportunities to falsify the model and the lack of any
corresponding observation, together with the achieve-
ments listed above have made the Friedmann-Lemaitre
model the standard model of cosmology. Alternative
models have either been excluded by observation (such
as steady-state cosmology) or have been unable to make
any predictions. Currently, there is no serious alternative
to the standard model.
4.5.2 Problems of the Standard Model
Despite these achievements, there are some aspects of
the model which require further consideration. Here we
will describe two conceptual problems with the standard
model more thoroughly - the horizon problem and the
flatness problem.
Horizons. The finite speed of light implies that we
are only able to observe a finite part of the Universe,
namely those regions from which light can reach us
within a time to- Since 1 « 13.5 Gyr, our visible Uni-
verse has - roughly speaking - a radius of 13.5 billion
light years. More distant parts of the Universe are at the
present time unobservable for us. This means that there
exists a horizon beyond which we cannot see. Such hori-
zons do not only exist for us today: at an earlier time t,
the size of the horizon was about ct , hence smaller than
today. We will now describe this aspect quantitatively.
In a time interval dt, light travels a distance cdt,
which corresponds to a comoving distance interval dx —
cdt I a at scale factor a. From the Big Bang to a time t
(or redshift z) the light traverses a comoving distance of
f cdt
From a — da/dt we get dt — da/a = da/(aH), so that
If Zeq 3> Z ^> 0, the main contribution to the integral
comes from times (or values of a) in which the so-
called dust dominates the expansion rate H. Then with
4.5 Achievements and Problems of the Standard Model
(4.31) we find H(a) « // V^a~ 3/2 , and (4.68) yields
In earlier phases, z^> z eq , H is radiation-dominated,
H(a) % H Q jTT r /a 2 , and (4.68) becomes
(4.70)
The earlier the cosmic epoch, the smaller the comoving
horizon length, as was to be expected. In particular, we
will now consider the recombination epoch, z rec ~ 1000,
for which (4.69) applies (see Fig. 4.16). The comoving
length r H ,com corresponds to a physical proper length
r H prop = a r Hcom , and thus
TUpropfcec) = 2-£- ^ m ' /2 (1 + Z re c)~ 3/2 (4.71)
no
is the horizon length at recombination. We can then
calculate the angular size on the sky that this length
corresponds to,
where D A is the angular-diameter distance (4.45) to the
last scattering surface of the CMB. Using (4.47), we
find that in the case of Qa =
Fig. 4.16. The horizon problem: the region of space which
was in causal contact before recombination has a much
smaller radius than the spatial separation between two re-
gions from which we receive the CMB photons. Thus the
question arises how these two regions may "know" of each
other's temperature
The Horizon Problem: Since no signal can travel
faster than light, (4.72) means that CMB radia-
tion from two directions separated by more than
about one degree originates in regions that were
not in causal contact before recombination, i.e., the
time when the CMB photons interacted with mat-
ter the last time. Therefore, these two regions have
never been able to exchange information, for ex-
ample about their temperature. Nevertheless their
temperature is the same, as seen from the high de-
gree of isotropy of the CMB, which shows relative
fluctuations of only AT/T ~ 10" 5 !
This means that the horizon length at recombina
subtends an angle of about one degree on the sky.
Redshift-Dependent Density Parameter. We have de-
fined the density parameters Q m and Q A as the current
density divided by the critical mass density p CI today.
These definitions can be generalized. If we existed al
a different time, the densities and the Hubble constant
would have had different values and consequently we
would obtain different values for the density parame-
ters. Thus we define the total density parameter for an
4. Cosmology I: Homogeneous Isotropic World Models
arbitrary redshift
„ , ,_ Pm(z) + pAz) + p v
where the critical density p CI is also a function of
redshift.
Then by inserting (4.24) into (4.73), we find
Using (4.31), this yields
| l-fl (z) = F[l-flo(0)] | , (4.75)
where £2q(0) is the total density parameter today, and
"-(£»)*■
From (4.75) we can now draw two important conclu-
sions. Since F > for all a, the sign of Qq — 1 is
preserved and thus is the same at all times as today.
Since the sign of Qq — 1 is the same as that of the curva-
ture - see (4.30) - the sign of the curvature is preserved
in cosmic evolution: a flat Universe will be flat at all
times, a closed Universe with K > will always have
a positive curvature.
The second conclusion follows from the analysis of
the function F at early cosmic epochs, e.g., at z ^> z eq ,
thus in the radiation-dominated Universe. Back then,
with (4.31), we have
1
~ J2 r (l + z) 2 '
so that for very early times, F becomes very small. For
instance, at z ~ 10 10 , the epoch of the neutrino freeze-
out, F ~ 10~ 15 . Today, &q is of order unity but not
necessarily exactly 1. From observations, we know that
0.1 < J2o(0) < 2, where this is a very conservative es-
timate (from the more recent CMB measurements we
are able to constrain this interval to [0.97, 1.04]), so
that \l-f2 (0)\<l. Since F is so small at large red-
shifts, this mean?, that f2r,{z) must have been very, very
close to 1; for example at z ~ 10 10 it is required that
l^o-l|<10~ 15 .
The Flatness Problem: For the total density par-
ameter to be of order unity today, it must have been
extremely close to 1 at earlier times, which means
that a very precise "fine tuning" of this parameter
was necessary.
This aspect can be illustrated very well by another
physical example. If we throw an object up into the air,
it takes several seconds until it falls back to the ground.
The higher the initial velocity, the longer it takes to
hit the ground. To increase the time of flight we need
to increase the initial velocity, for instance by using
a cannon. In this way, the time of flight may be extended
to up to about a minute. Assume that we want the object
to be back only after one day; in this case we must
use a rocket. But we know that if the initial velocity of
a rocket exceeds the escape velocity i; esc ~ 1 1.2 km/s,
it will leave the gravitational field of the Earth and never
fall back. On the other hand, if the initial velocity is too
much below u eS c, the object will be back in significantly
less than a day. So the initial velocity must be very well
chosen for the object to return after being up for at least
a day.
The flatness problem is completely analogous to this.
If J2 had not been so extremely close to 1 at z ~ 10 10 ,
the Universe would have recollapsed long ago, or it
would have expanded significantly more than the Uni-
verse we live in. In either case, the consequences for
the evolution of life in the Universe would have been
catastrophic. In the first case, the total lifetime of the
Universe would have been much shorter than is needed
for the formation of the first stars and the first planetary
systems, so that in such a world no life could be formed.
In the second case, extreme expansion would have pre-
vented the formation of structure in the Universe. In
such a Universe no life could have evolved either.
This consideration can be interpreted as follows: we
live in a Universe which had, at a very early time,
a very precisely tuned density parameter, because only
in such a Universe can life evolve and astronomers
exist to examine the flatness of the Universe. In all
other conceivable universes this would not be possible.
This approach is meaningful only if a large number of
universes existed - in this case we should not be too
surprised about living in one of those where this initial
fine-tuning took place - in the other ones, we, and the
4.5 Achievements and Problems of the Standard Model
question about the cosmological parameters, would just
not exist. This approach is called the anthropic princi-
ple. It may either be seen as an "explanation" for the
flatness of our Universe, or as a capitulation - where we
give up attempting to solve the question of the origin of
the flatness of the Universe.
The example of the rocket given above is helpful
in understanding another aspect of cosmic expansion.
If the rocket is supposed to have a long time of flight
but not escape the gravitational field of the Earth, its
initial velocity must be very, very close to, but a tiny
little bit smaller than v esc . In other words, the absolute
value of the sum of kinetic and potential energy has
to be very much smaller than either of these two com-
ponents. This is also true for a large part of the initial
trajectory. Independent of the exact value of the time
of flight, the initial trajectory can be approximated by
the limiting case vq — u esc at which the total energy is
exactly zero. Transferred to the Hubble expansion, this
reads as follows: independent of the exact values of the
cosmological parameters, the curvature term can be dis-
regarded in the early phases of expansion (as we have
already seen above). This is because our Universe can
reach its current age only if at early times the modu-
lus of potential and kinetic energy were nearly exactly
equal, i.e., the curvature term in (4.14) must have been
a lot smaller than the other two terms.
4.5.3 Extension of the Standard Model: Inflation
We will consider the horizon and flatness problems from
a different, more technical point of view. Einstein's field
equations of GR, one solution of which has been de-
scribed as our world model, are a system of coupled
partial differential equations. As is always the case for
differential equations, their solutions are determined by
(1) the system of equations itself and (2) the initial
conditions. If the initial conditions at e.g., t = 1 s were
as they have been described, the two aforementioned
problems would not exist. But why are the conditions at
t = 1 s such that they allow a homogeneous, isotropic,
nearly flat model? The set of homogeneous and isotropic
solutions to the Einstein equation is of measure zero
(i.e., nearly all solutions of the Einstein equation are not
homogeneous and isotropic); thus these particular solu-
tions are very special. Taking the line of reasoning that
the initial conditions "just happened to be so" is not sat-
isfying because it does not explain anything. Besides the
anthropic principle, the answer to this question can only
be that processes must have taken place even earlier, due
to known or as yet unknown physics, which have pro-
duced these "initial conditions" at t = 1 s. The initial
conditions of the normal Friedmann-Lemaitre expan-
sion thus have a physical origin. Cosmologists believe
they have found such a physical reason: the inflationary
model.
Inflation. In the early 1980s, a model was developed
which was able to solve the flatness and horizon prob-
lems (and some others as well). As a motivation for
this model, we first recall that the physical laws and
properties of elementary particles are well known up to
energies of ~ 100 GeV because they were experimen-
tally tested in particle accelerators. For higher energies,
particles and their interactions are unknown. This means
that the history of the Universe, as sketched above, can
be considered secure only up to energies of 100 GeV.
The extrapolation to earlier times, up to the Big Bang,
is considerably less certain. From particle physics we
expect new phenomena to occur at an energy scale of
the Grand Unified Theories (GUTs), at about 10 14 GeV,
corresponding to t ~ 10~ 34 s.
In the inflationary scenario it is presumed that at
very early times the vacuum energy density was much
higher than today, so that Q A dominated the Hubble
expansion. Then from (4.18) we find that a/ a « *J Aj1.
This implies an exponential expansion of the Universe,
a{f) = C exp
m
(4.77)
Obviously, this exponential expansion (or inflationary
phase) cannot last forever. We assume that a phase tran-
sition took place in which the vacuum energy density is
transformed into normal matter and radiation (a process
called reheating), which ends the exponential expan-
sion and after which the normal Friedmann evolution of
the Universe begins. Figure 4.17 sketches the expansion
history of the Universe in an inflationary model.
Inflation Solves the Horizon Problem. During infla-
tion, H(a) — *JA/3 is constant so that the integral (4.68)
for the comoving horizon length formally diverges. This
implies that the horizon may become arbitrarily large
in the inflationary phase, depending on the duration of
Fig. 4,17, During an inilationan phase. indicated here by ihe
r, the Universe expands exponentially; see (4.77). This
phase comes to an end when a phase transition transforms
n energy into matter and radiation, after which the
e follows the normal Friedmann expansion
the exponential expansion. For illustration we consider
a very small region in space of size L < ct\ at a time
t[ ~ 10~ 34 s prior to inflation which is in causal con-
tact. Through inflation, it expands tremendously, e.g.,
by a factor ~ 10 40 ; the original L ~ 10~ 24 cm inflate to
about 10 16 cm by the end of the inflationary phase, at
tf ~ 10~ 32 s. By today, this spatial region will have ex-
panded by another factor of ~ 10 25 by following (for
t > tf) the normal cosmic expansion, to ~ 10 41 cm. This
scale is considerably larger than the size of the currently
visible Universe, c/Hq. According to this scenario, the
whole Universe visible today was in causal contact prior
to inflation, so that the homogeneity of the physical con-
ditions at recombination, and with it the nearly perfect
isotropy of the CMB, is provided by causal processes.
Inflation Solves the Flatness Problem as well. Due
to the tremendous expansion, any initial curvature is
straightened out (see Fig. 4.18). Formally this can be
seen as follows: during the inflationary phase we have
A
and since it is assumed that the inflationary phase lasts
long enough for the vacuum energy to be completely
Fig.4.18. Due to tremendou .-xpan ion during inflation, e
a Universe with initial curvature will appear to be a
Universe by the end of the inflationary phase
dominant, when it ends we then have Qq = 1 . Hence the
Universe is flat to an extremely good approximation.
The inflationary model of the very early Universe
predicts that today Qq — 1 is valid to very high pre-
cision; any other value of Qq would require another
fine-tuning. Thus the Universe is flat.
The physical details of the inflationary scenario are
not very well known. In particular it is not yet un-
derstood how the phase transition at the end of the
inflationary phase took place and why it did not occur
earlier. But the two achievements presented above (and
some others) make an inflationary phase appear a very
plausible scenario. As we will see below (Chap. 8), the
prediction of a flat Universe was recently accurately
tested and it was indeed confirmed. Furthermore, the
inflationary model provides a natural explanation for
the origin of density fluctuations in the Universe which
must have been present at very early epochs as the seeds
of structure formation. We will discuss these aspects
further in Chap. 7.
5. Active Galactic Nuclei
The light of normal galaxies in the optical and near in-
frared part of the spectrum is dominated by stars, with
small contributions by gas and dust. This is thermal ra-
diation since the emitting plasma in stellar atmospheres
is basically in thermodynamical equilibrium. To a first
approximation, the spectral properties of a star can be
described by a Planck spectrum whose temperature de-
pends on the stellar mass and the evolutionary state of
the star. As we have seen in Sect. 3.9, the spectrum of
galaxies can be described quite well as a superposi-
tion of stellar spectra. The temperature of stars varies
over a relatively narrow range. Only few stars are found
with T > 40 000 K, and those with T < 3000 K hardly
contribute to the spectrum of a galaxy, due to their
low luminosity. Therefore, as a rough approximation,
the light distribution of a galaxy can be described by
a superposition of Planck spectra from a temperature
range that covers about one decade. Since the Planck
spectrum has a very narrow energy distribution around
its maximum at h P v ~ 3k B T, the spectrum of a gal-
axy is basically confined to a range between ~ 4000 A
and ~ 20 000 A. If the galaxy is actively forming stars,
young hot stars extend this frequency range to higher
frequency, and the thermal radiation from dust, heated
by these new-born stars, extends the emission to the
far-infrared.
However, there are galaxies which have a much
broader energy distribution. Some of these show signif-
icant emission in the full range from radio wavelengths
to the X-ray and even Gamma range (see Fig. 3.3). This
emission originates mainly from a very small central re-
gion of such an active galaxy which is called the active
galactic nucleus (AGN). Active galaxies form a family
of many different types of AGN which differ in their
spectral properties, their luminosities and their ratio of
nuclear luminosity to that of the stellar light. The optical
spectra of three AGNs are presented in Fig. 5.1.
Some classes of AGNs, in particular the quasars,
belong to the most luminous sources in the Universe,
and they have been observed out to the highest mea-
sured redshifts (z ~ 6). The luminosity of quasars can
exceed the luminosity of normal galaxies by a factor
of a thousand. This luminosity originates from a very
small region in space, r < 1 pc. The optical/UV spectra
Ssylerll Hkn 290 Z.0.030B
^^w
Fig. 5.1. Optical spectra of three AGNs. The top panel displaj s
the spectrum of a quasar at redshift z ~ 2, which shows the
characteristic broad emission lines. The strongest are Lya of
hydrogen, and the Civ-line and CmJ-line of triple and double
ionized carbon, respectively (where the squared bracket means
thai this is a semi forbidden transition, as will be explained in
Sect , 5 .1.2). The middle panel shows the spectrum of a nearby
Seyfert galaxy of Type 1. Here both very broad emission
lines and narrow lines, in particular of double ionized oxygen,
are visible. In contrast, the spectrum in the bottom panel,
of a Seyfert galaxy of Type 2, shows only relative!} narrow
Peter Schneider. Active Galactic Nude
In: Peter Schneider. Extragalactic Astrc
DOI: 10.1007/1 1614371_5 © Springer-
omy and Cosmology, pp. 175 222 (2006)
5. Active Galactic Nuclei
■ Lya/NV
CIV
jJ
I Clll]
I 1909
CM]
2326
Mgll
2798
[Oil]
[NeV] H8
3426 410
Hi
4340
H(3 "
' f Si[V/OIV]
./ 1400
[Nelll]
3869
[OMI]
4959 f 5007
1000 2000 3000 4000 5000
Quasar Rest-frame Wavelength (A)
Fig.5.2. Combined spectrum of a sample of 718 individ-
ual QSOs, taken from the Large Bright Quasar Survey. This
i« n i irum h i considerably bell ignal to noi i
tio and a larger wavelength coverage than individual spectra.
It was combined from the individual quasar spectra by trans-
forming their wavelengths into the sources' rest-frames. The
most prominent lines are marked
of quasars are dominated by numerous strong and very
broad emission lines, some of them emitted by highly
ionized atoms (see Figs. 5.2 and 5.3). The processes in
AGNs are among the most energetic in astrophysics.
The enormous bandwidth of AGN spectra suggests that
the radiation is nonthermal. As we will discuss later,
AGNs host processes which produce highly energetic
particles and which are the origin of the nonthermal
radiation.
After an introduction in which we will briefly present
the history of the discovery of AGNs and their basic
properties, in Sect. 5.2 we will describe the most impor-
tant subgroups of the AGN family. In Sect. 5.3, we will
discuss several arguments which lead to the conclusion
that the energy source of an AGN originates in accre-
tion of matter onto a supermassive black hole (SMBH).
In particular, we will learn about the phenomenon
of superluminal motion, where apparent velocities of
source components are larger than the speed of light.
We will then consider the different components of an
AGN where radiation in different wavelength regions is
produced.
Of particular importance for understanding the phe-
nomenon of active galaxies are the unified models of
AGNs that will be discussed next. We will see that
the seemingly quite different appearances of AGNs can
all be explained by geometric or projection effects. Fi-
nally, we will consider AGNs as cosmological probes.
Due to their enormous luminosity they are observable
up to very high redshifts. These observations allow us
to draw conclusions about the properties of the early
Universe.
Fig. 5.3. An enlargement of the
composite QSO spectrum shown
in Fig.5.2. Here, weakei lines
are also visible. Also clearly vis-
ible is the break in the spectral
flux bluewards of the Lya line
which is caused by the Lya for-
est (Sect. 5.6.3), absorption by
intcrgalaclic hydrogen along the
line-of-sight. The dashed line
indicates (he average contin-
uum, whereas the dotted line
marks line complexes of singly
ionized iron that has such a high
line density that they blend into
a quasi continuum at the spectral
resolution shown here
5.1 Introduction
5.1 Introduction
5.1.1 Brief History of ACNs
As long ago as 1908, strong and broad emission lines
were discovered in the galaxy NGC 1068. However,
only the systematic analysis by Carl Seyfert in 1943
drew the focus of astronomers to this new class of
galaxies. The cores of these Seyfert galaxies have an
extremely high surface brightness, as demonstrated in
Fig. 5.4, and the spectrum of their central region is dom-
inated by emission lines of very high excitation. Some
of these lines are extremely broad (see Fig. 5.1). The
line width, when interpreted as Doppler broadening,
AX/X = Av/c, yields values of up to Av ~ 8500 km/s
for the full line width. The high excitation energy of
some of the line-emitting atoms shows that they must
have been excited by photons that are more energetic
than photons from young stars that are responsible for
the ionization of Hll regions. The hydrogen lines are of-
ten broader than other spectral lines. Most of the Seyfert
galaxies are spirals, but one cD galaxy is also found in
his original catalog.
In 1959, Lodewijk Woltjer argued that the extent
of the cores of Seyfert galaxies cannot be larger than
r < 100 pc because they appear point-like on optical
images, i.e., they are spatially not resolved. If the line-
emitting gas is gravitationally bound, the relation
GM ^ 2
between the central mass M(<r), the separation r of the
gas from the center, and the typical velocity v must be
Fig.5.4a-c. Three images of the Seyfert galaxy NGC4151,
with the exposure time increasing to the right. In short ex-
I 1 u id 1 mi in 11 point m! iih longet po in
displaying the galaxy
satisfied. The latter is obtained from the line width: typi-
cally v ~ 1000 km/s. Therefore, with r < 100 pc a mass
estimate is immediately obtained,
M ^ lol °(ioo^) M0 - (5 - 1}
Thus, either r~ 100 pc, which implies an enormous
mass concentration in the center of these galaxies, or
r is much smaller than the estimated upper limit, which
then implies an enormous energy density inside AGNs.
An important milestone in the history of AGNs
was made with the 3C and 3CR radio catalogs
which were completed around 1960. These are sur-
veys of the northern (<5 > -22°) sky at 158 MHz and
178 MHz, with a flux limit of 5 min = 9Jy (a Jansky
is the flux unit used by radio astronomers, where
1 Jy= 10" 23 ergs" 1 cm" 2 Hz" 1 ). Many of these 3C
sources could be identified with relatively nearby gal-
axies, but the low angular resolution of radio telescopes
at these low frequencies and the resulting large posi-
tional uncertainty of the respective sources rendered the
identification with optical counterparts very difficult. If
no striking nearby galaxy was found on optical photo-
plates within the positional uncertainty, the source was
at first marked as unidentified. 1
In 1963, Thomas Matthews and Allan Sandage
showed that 3C48 is a point-like ("stellar-like") source
of m — 16 mag. It has a complex optical spectrum con-
sisting of a blue continuum and strong, broad emission
lines which could not be assigned to any atomic tran-
sition, and thus could not be identified. In the same
year, Maarten Schmidt succeeded in identifying the ra-
dio source 3C273 with a point-like optical source which
also showed strong and broad emission lines at unusual
wavelengths. This was achieved by a lunar eclipse: the
Moon passed in front of the radio source and eclipsed it.
From the exact measurement of the time when the radio
emission was blocked and became visible again, the po-
sition of the radio source was pinned down accurately.
Schmidt could identify the emission lines of the source
with those of the Balmer series of hydrogen, but at an,
for that time, extremely high redshift of z — 0. 158. Pre-
suming the validity of the Hubble law and interpreting
'The complete optical identification 01 the 3CR catalog, which was
made possible h\ the enormous!) increased angular resolution of ill
terlerontetric radio observations and thus b\ a considerably intpnned
positional accuracy, was linalized only in the 1<-M()s some of these
luminous radio sources are very fainl optically.
5. Active Galactic Nuclei
the redshift as cosmological redshift, 3C273 is located
at the large distance of D ~ 500 h~ 1 Mpc. This huge dis-
tance of the source then implies an absolute magnitude
of M B = -25.3 + 5 log h, i.e., it is about ~ 100 times
brighter than normal (spiral) galaxies. Since the optical
source had not been resolved but appeared point-like,
this enormous luminosity must originate from a small
spatial region. With the improving determination of ra-
dio source positions, many such quasars (quasi-stellar
radio sources = quasars) were identified in quick suc-
cession, the redshifts of some being significantly higher
than that of 3C273.
5.1.2 Fundamental Properties of Quasars
In the following, we will review some of the most im-
portant properties of quasars. Although quasars are not
the only class of AGNs, we will at first concentrate on
them because they incorporate most of the properties of
the other types of AGNs.
As already mentioned, quasars were discovered by
identifying radio sources with point- like optical sources.
Quasars emit at all wavelengths, from the radio to the
X-ray domain of the spectrum. The flux of the source
varies at nearly all frequencies, where the variability
time-scale differs among the objects and also depends
on the wavelength. In general, it is found that the vari-
ability time-scale is smaller and its amplitude larger
when going to higher frequencies of the observed radi-
ation. The optical spectrum is very blue; most quasars
at redshifts z < 2 have U — B < —0.3 (for comparison:
only hot white dwarfs have a similarly blue color in-
dex). Besides this blue continuum, very broad emission
lines are characteristic of the optical spectrum. Some of
them correspond to transitions of very high ionization
energy (see Fig. 5.3).
The continuum spectrum of a quasar can often be
described, over a broad frequency range, by a power
law of the form
S v oc y~ c
(5.2s
where a is the spectral index, a — corresponds to
a flat spectrum, whereas a — 1 describes a spectrum in
which the same energy is emitted in every logarithmic
frequency interval. Finally, we shall point out again the
high redshift of many quasars.
5.1.3 Quasars as Radio Sources:
Synchrotron Radiation
The morphology of quasars in the radio regime depends
on the observed frequency and can often be very com-
plex, consisting of several extended source components
and one compact central one. In most cases, the ex-
tended source is observed as a double source in the
form of two radio lobes situated more or less symmet-
rically around the optical position of the quasar. These
lobes are frequently connected to the central core by
jets, which are thin emission structures probably related
to the energy transport from the core into the lobes. The
observed length-scales are often impressive, in that the
total extent of the radio source can reach values of up
to 1 Mpc. The position of the optical quasar coincides
with the compact radio source, which has an angular
extent of <$C 1" and is in some cases not resolvable even
with VLBI methods. Thus the extent of these sources
is < 1 mas, corresponding to r < 1 pc. This dynamical
range in the extent of quasars is thus extremely large.
Classification of Radio Sources. Extended radio
sources are often divided into two classes. Fanaroff-
Riley Type I (FR I) are brightest close to the core, and the
surface brightness decreases outwards. They typically
have a luminosity of L v ( 1 .4 GHz) < 1 32 erg s" l Hz' l .
In contrast, the surface brightness of Fanawjj Riley
Type II sources (FR II) increases outwards, and their lu-
minosity is in general higher than that of FR I sources,
L v (1.4 GHz) > 10 32 ergs" 1 Hz" 1 . One example for
each of the two classes is shown in Fig. 5.5. FRII ra-
dio sources often have jets; they are extended linear
structures that connect the compact core with a radio
lobe. Jets often show internal structure such as knots
and kinks. Their appearance indicates that they trans-
port energy from the core out into the radio lobe. One
of the most impressive examples of this is displayed in
Fig. 5.6.
The jets are not symmetric. Often only one jet is ob-
served, and in most sources where two jets are found
one of them (the "counter-jet") is much weaker than
the other. The relative intensity of core, jet, and ex-
tended components varies with frequency, for sources
as a whole and also within a source, because the com-
ponents have different spectral indices. For this reason,
radio catalogs of AGNs suffer from strong selection ef-
5.1 Introduction
Right Ascension (B1 950)
Fig. 5.5. Radio maps at X = 6 cm for two radio galaxies: the top
one is M84, an FRI radio source, the bottom one is 3C175, an
FRII source. The radiation from M84 in the radio is strongest
near the centei and decreases outwards, whereas in 3C175
the most prominent components are the two radio lobes. The
radio lobe on the right is connected to the compact core by
a long and very thin jet, whereas on the opposite side no jet
(counter-jet) is visible
fects. Catalogs that are sampled at low frequencies will
predominantly select sources that have a steep spec-
trum, i.e., in which the extended structures dominate,
whereas high-frequency samples will preferentially
contain core-dominated sources with a flat spectrum. 2
2 For this reason, radio simeys [or gra\ national lens systems, which
I n ii mi ned in ct eoneenti n soui I la
spectral index because these are dominated h_\ [he compact nucleus.
Multiple image systems arc [has more easily recognized as such.
00 s 16"32 m 00M6 h 24 n 00
•HPBW
6 h 56 m 00'16 h 48"W
,^^j%;
'TYrFvt.:
\.
82°20' \
50 100 150-^200
Offset from Core (arcsecT**^.
250
f
,
,
,
Offset from Core (arcsei
Fig. 5.6. The radio galaxy NGC 6251, with angular resolution
increasing towards the bottom. On large scales (and at low
frequencies), the l\\ o radio lobes dominate, while the core and
the jets are clearly prominent at higher frequencies. NGC 625 1
has a counter jet. but with signiiicanth lower luminosity than
the main jet. Even at the highest resolution obtained by VLBI,
structure can still be seen. The jets have a very small opening
angle and are therefore strongly collimated
Synchrotron Radiation. Over a broad range in wave-
lengths, the radio spectrum of AGNs follows a power
law ofthe form (5.2), with a ~ 0.7 for the extended com-
ponents and a ~ for the compact core components.
Radiation in the radio is often linearly polarized, where
the extended radio source may reach a degree of polar-
ization up to 30% or even more. The spectral form and
the high degree of polarization are interpreted such that
the radio emission is produced by synchrotron radiation
5. Active Galactic Nuclei
of relativistic electrons. Electrons in a magnetic field
propagate along a helical, i.e., corkscrew-shaped path,
so that they are continually accelerated by the Lorentz
force. Since accelerated charges emit electromagnetic
radiation, this motion of the electrons leads to the emis-
sion of synchrotron radiation. Because of its importance
for our understanding of the radio emission of AGNs, we
will review some aspects of synchrotron radiation next.
The radiation can be characterized as follows. If
an electron has energy E =
frequency of the emission is
3y 2 eB
~4.2xlOV
c 2 , the characteristic
(5.3)
where B denotes the magnetic field strength, e the
electron charge, and m e = 5 1 1 keV/c 2 the mass of the
electron. The Lorentz factor y, and thus the energy of
an electron, is related to its velocity v via
(5.4)
Jl-(v/c) 2
For frequencies considerably lower than v c , the spec-
trum of a single electron is ex v 1/3 , whereas it decreases
exponentially for larger frequencies. To a first ap-
proximation, the spectrum of a single electron can
be considered as quasi-monochromatic, i.e., the width
of the spectral distribution is small compared to the
characteristic emission frequency v c . The synchrotron
radiation of a single electron is linearly polarized, where
the polarization direction depends on the direction of
the magnetic field projected onto the sky. The degree
of polarization of the radiation from an ensemble of
electrons depends on the complexity of the magnetic
field. If the magnetic field is homogeneous in the spa-
tial region from which the radiation is measured, the
observed polarization may reach values of up to 75%.
However, if the spatial region that lies within the tele-
scope beam contains a complex magnetic field, with
the direction changing strongly within this region, the
polarizations partially cancel each other out and the
observed degree of linear polarization is significantly
reduced.
To produce radiation at cm wavelengths (v ~
10 GHz) in a magnetic field of strength B ~ 10~ 4 G, y ~
10 5 is required, i.e., the electrons need to be highly rel-
ativistic! To obtain particles at such high energies, very
efficient processes of particle acceleration must occur
in the inner regions of quasars. It should be mentioned
in this context that comic ray particles of considerably
higher energies are observed (see Sect. 2.3.4). The ma-
jority of cosmic rays are presumably produced in the
shock fronts of supernova remnants. Thus, it is supposed
that the energetic electrons in quasars (and other AGNs)
are also produced by "diffusive shock acceleration",
where here the shock fronts are not caused by supernova
explosions but rather by other hydrodynamical phenom-
ena. As we will see later, we find clear indications
in AGNs for outflow velocities that are considerably
higher than the speed of sound in the plasma, so that the
conditions for the formation of shock fronts are satisfied.
Synchrotron radiation will follow a power law if the
energy distribution of relativistic electrons also behaves
like a power law (see Fig. 5.7). If N(E) dE oc E~ s &E
represents the number density of electrons with energies
between E and E + dE, the power-law index of the re-
sulting radiation will be a — (s — l)/2, i.e., the slope in
the power law of the electrons defines the spectral shape
of the resulting synchrotron emission. In particular, an
index of a — 0.7 results for s — 2.4. An electron distri-
bution with N(E) ex E~ 2A is very similar to the energy
distribution of the cosmic rays in our Galaxy, which
may be another indicator for the same or at least a sim-
ilar mechanism being responsible for the generation of
this energy spectrum.
log(v) (arbitrary units)
Fig. 5.7. Electrons at a given energy emit a synchrotron spec-
irum which is indicated by the indi\ klual cun cs; the maximum
of the radiation is at v c (5.3), which depends on the electron en-
ergy. The superposition of many such spectra, corresponding
to an energy distribution of the electrons, results in a power-
law spectrum provided the energy distribution of the electrons
follows a power law
5.1 Introduction
The synchrotron spectrum is self-absorbed at low
frequencies, i.e., the optical depth for absorption due
to the synchrotron process is close to or larger than
unity. In this case, the spectrum becomes flatter and, for
small v, it may even rise. In the limiting case of a high
optical depth for self-absorption, we obtain S v oc v 2 - 5 for
v -> 0. The extended radio components are optically
thin at cm wavelength, so that a ~ 0.7, whereas the
compact core component is often optically thick and
thus self-absorbed, which yields a ~ 0, or even inverted
so that a < 0.
Through emission, the electrons lose energy. Thus,
the electrons cool and for only a limited time can they
radiate at the frequency described by (5.3). The power
emitted by an electron of Lorentz factor y, integrated
over all frequencies, is
d£ 4 e 4 B 2 y 2
P = -IT= 9^f' (5 - 5)
The characteristic time in which an electron loses its
energy is then obtained from its energy E — ym e c 2 and
its energy loss rate E — — P as
r C ooi=- = 2.4xlO^^ I )"7^ r -V 2 yr.
p Viov Vio~ 4 g/ '
(5.6)
For relatively low-frequency radio emission, this life-
time is longer than or comparable to the age of radio
sources. But as we will see later, high-frequency syn-
chrotron emission is also observed for which t c00 \ is con-
siderably shorter than the age of a source component.
The corresponding relativistic electrons can then only be
generated locally. This means that the processes of par-
ticle acceleration are not confined to the inner core of an
AGN, but also occur in the extended source components.
Since the characteristic frequency (5.3) of syn-
chrotron radiation depends on a combination of the
Lorentz factor y and the magnetic field B, we cannot
measure these two quantities independently. There-
fore, it is difficult to estimate the magnetic field of
a synchrotron source. In most cases, the (plausible) as-
sumption of an equipartition of the energy density in
the magnetic field and the relativistic particles is made,
i.e., one assumes that the energy density B 2 /(&jt) of the
magnetic field roughly agrees with the energy density
of the relativistic electrons. Such approximate equipar-
tition holds for the cosmic rays in our Galaxy and
its magnetic field. Another approach is to estimate
the magnetic field such that the total energy of rela-
tivistic electrons and magnetic field is minimized for
a given source luminosity. The resulting value for B ba-
sically agrees with that derived from the assumption of
equipartition.
5.1.4 Broad Emission Lines
The UV and optical spectra of quasars feature strong
and very broad emission lines. Typically, lines of the
Balmer series and Lya of hydrogen, and metal lines
of ions like Mgll, Cm, Civ 3 are observed - these are
found in virtually all quasar spectra. In addition, a large
number of other emission lines occur which are not seen
in every spectrum (Fig. 5.2 S.
To characterize the strength of an emission line, we
define the equivalent width of a line W* as
W k = I AX
S C (X) ~ S C (X Q ) '
(5.7,
where Si(X) is the total spectral flux, and S C (X) is the
spectral flux of the continuum radiation interpolated
across the wavelength range of the line. Fi; ne is the total
flux in the line and X its wavelength. Hence, W k is
the width of the wavelength interval over which the
continuum needs to be integrated to obtain the same
flux as measured in the line. Therefore, the equivalent
width is a measure of the strength of a line relative to
the continuum intensity.
The width of a line is characterized as follows: af-
ter subtracting the continuum, interpolated across the
wavelength range of the line, the width is measured
at half of the maximum line intensity. This width AX
is called the FWHM (full width at half maximum);
it may be specified either in A, or in km/s if the
line width is interpreted as Doppler broadening, with
AX/X = Av/c.
Broad emission lines in quasars often have a FWHM
of ~ 10 000 km/s, while narrower emission lines still
have widths of several 100 km/s. Thus the "narrow"
f dyn e (y)ym e c
is of an element are distinguished by Roman
numbers. A neutral atom is denoted b> "I". a singly ionized atom by
"II", and so on. So, Civ is three times ionized carbon.
5. Active Galactic Nuclei
emission lines are still broad compared to the typical
velocities in normal galaxies.
Redshift. Quasar surveys are always flux limited, i.e.,
one tries to find all quasars in a certain sky region with
a flux above a predefined threshold. Only with such
a selection criterion are the samples obtained of any sta-
tistical value. In addition, the selection of sources may
include further criteria such as color, variability, radio
or X-ray flux. For instance, radio surveys are defined by
S v > Su m at a specific wavelength. The optical identifi-
cation of such radio sources reveals that quasars have
a very broad redshift distribution. For decades, quasars
have been the only sources known at z > 3. Below we
will discuss different kinds of AGN surveys.
In the 1993 issue of the quasar catalog by Hewitt &
Burbidge, 7236 sources are listed. This catalog contains
a broad variety of different AGNs. Although it is sta-
tistically not well-defined, this catalog provides a good
indication of the width of the redshift and brightness
distribution of AGNs (see Fig. 5.8).
The luminosity function of quasars extends over
a very large range in luminosity, nearly three orders
of magnitude in L. It is steep at its bright end and
has a significantly flatter slope at lower luminosities
(see Sect. 5.6.2). We can compare this to the luminosity
function of galaxies which is described by a Schechter
function (see Sect. 3.7). While the faint end of the dis-
tribution is also described here by a relatively shallow
power law, the Schechter function decreases exponen-
tially for large L, whereas that of quasars decreases as
a power law. For this reason, one finds quasars whose
luminosity is much larger than the value of L where the
break in the luminosity function occurs.
5.2 AGN Zoology
Quasars are the most luminous members of the class
of AGNs. Seyfert galaxies are another type of AGN
and were mentioned previously. In fact, a wide range of
objects are subsumed under the name AGN, all of which
n
a)
200
000
A
800
ii ■
600
-
400
r
^
200
■ ^ ^
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Fig.S.S. The redshift (left) ;md brighlness distribution (right)
of QSOs in the 1993 Hewitt & Burbidge catalog. These dis-
tributions provide no proper statistical information, but they
clearly show the width of the distributions. The decrease in
abundances for z > 2.3 is a selection effect: many QSO sur-
12 131415 16 17 18 19 20 21 22 23 24
b) Apparent Magnitude
veys start with a color selection, typically U
z > 2.3, the strong Lya e:
and hence the quasar becomes redder ir
drops out of the color selection
-0.3. If
B -filter
color index and
have in common strong non-thermal emission in the
core of a galaxy (host galaxy). We will mention the most
important types of AGN in this section. It is important to
keep in mind that the frequency range in which sources
are studied affects the source classification. We shall
return to this point at the end of this section.
The classification of AGNs described below is very
confusing at first glance. Different classes refer to dif-
ferent appearances of AGNs but do not necessarily
correspond to the physical nature of these sources. As
we will discuss in Sect. 5.5, the appearance of an AGN
in the context of unified models depends very strongly
on the orientation of the source with respect to its line-
of-sight. We will then be able to better organize the
variety of classes.
5.2.1 Quasi-Stellar Objects
The unusually blue color of quasars suggested the
possibility of searching for them not only with radio
observations but also at optical wavelengths, namely
to look for point-like sources with a very blue U — B
color index. These photometric surveys were very suc-
cessful. In fact, many more such sources were found
than expected from radio counts. Most of these sources
are (nearly) invisible in the radio domain of the spec-
trum; such sources are called radio-quiet. Their optical
properties are virtually indistinguishable from those of
quasars. In particular, they have a blue optical energy
distribution (of course, since this was the search crite-
rion!), strong and broad emission lines, and in general
a high redshift.
Hence, apart from their radio properties, these
sources appear to be like quasars. Therefore they were
called radio-quiet quasars, or quasi-stellar objects,
QSOs. Today this terminology is no longer very com-
mon because the clear separation between sources with
and without radio emission is not considered valid any
more. Radio-quiet quasars also show radio emission if
they are observed at sufficiently high sensitivity. In mod-
ern terminology, the expression QSO encompasses both
the quasars and the radio-quiet QSOs. About 10 times
more radio-quiet QSOs than quasars are thought to exist.
The QSOs are the most luminous AGNs. Their core
luminosity can be as high as a thousand times that of
an L* -galaxy. Therefore they outshine their host galaxy
and appear point-like on optical images. For QSOs of
lower L, their host galaxies were identified and spatially
resolved with the HST (see Fig. 1.11). According to
our current understanding, AGNs are the active cores
of galaxies. These galaxies are supposed to be fairly
normal galaxies, except for their intense nuclear activity,
and we will discuss possible reasons for the onset of this
activity further below.
5.2.2 Seyfert Galaxies
Seyfert galaxies are the AGNs which were detected
first. Their luminosity is considerably lower than that
of QSOs. On optical images they are identified as
spiral galaxies which have an extraordinarily bright
core (Fig. 5.4) whose spectrum shows strong and broad
emission lines.
We distinguish Seyfert galaxies of Type 1 and Type 2:
Seyfert 1 galaxies have both very broad and also
narrower emission lines, where "narrow" still means
several hundred km/s and thus a significantly larger
width than characteristic velocities (like rotational ve-
locities) found in normal galaxies. Seyfert 2 galaxies
show only the narrower lines. Later, it was discovered
that intermediate variants exist - one now speaks of
Seyfert 1.5 and Seyfert 1.8 galaxies, for instance -
in which very broad lines exist but which are much
less prominent than they are in Seyfert 1 galaxies.
The archetype of a Seyfert 1 galaxy is NGC4151 (see
Fig. 5.4), while NGC 1068 is a typical Seyfert 2 galaxy.
The optical spectrum of Seyfert 1 galaxies is very
similar to that of QSOs. A smooth transition exists
between (radio-quiet) QSOs and Seyfert 1 galaxies. For-
mally, these two classes of AGNs are separated at an
absolute magnitude of M B = —21.5 + 5 log h . The sep-
aration of Seyfert 1 galaxies and QSOs is historical since
these two categories were introduced only because of
the different methods of discovering them. However,
except for the different core luminosity, no fundamen-
tal physical difference seems to exist. Often both classes
are combined under the name Type 1-AGNs.
5.2.3 Radio Galaxies
Radio galaxies are elliptical galaxies with an ac-
tive nucleus. They were the first sources that were
identified with optical counterparts in the early radio
5. Active Galactic Nuclei
surveys. Characteristic radio galaxies are Cygnus A and
Centaums A.
In a similar fashion to Seyfert galaxies, for radio
galaxies we also distinguish between those with and
without broad emission lines: broad-line radio galaxies
(BLRG) and narrow-line radio galaxies (NLRG), re-
spectively. In principle, the two types of radio galaxy can
be considered as radio-loud Seyfert 1 and Seyfert 2 gal-
axies but with a different morphology of the host galaxy.
A smooth transition between BLRG and quasars also
seems to exist, again separated by optical luminosity as
for Seyfert galaxies.
Besides the classification of radio galaxies into
BLRG and NLRG with respect to the optical spectrum,
they are distinguished according to their radio morphol-
ogy. As was discussed in Sect. 5.1.2, radio sources are
divided into FRI and FRII sources.
5.2.4 Optically Violently Variables
One subclass of QSOs is characterized by the very
strong and rapid variability of its optical radiation. The
flux of these sources, which are known as Optically
Violently Variables (OVVs), can vary by a significant
fraction on time-scales of days (see Fig. 5.9). Besides
this strong variability, OVVs also stand out because of
their relatively high polarization of optical light, typi-
1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 '
'-
ciy
_
^otf-b
C"
~
2>%8
-=
>■■■
"<r Mg II
:^.o
*^ o „Xw v °"' b ' ta " , *
«*
NGC 5548 emission lines
-
-mil
1 1 il 1 1 1 1 1 1 1 11 1 1
,-
50 100 150 200 250
Times (days)
1970 1975 1980 1985 1990 1995
a) Time (years) b)
Fig. 5.9. Quasars, BL Lac objects, and Seyfert galaxies all
show clear variability at many different wavelengths. In the
upper left pam I. tl ra light curve of the Seyfert l Luila.w
IRAS 13225-3809 is plotted (observed by ROSAT); on time-
scales of days, the source frequently varies by more than
a factor of 20. The radio light curve of BL Lacertae at X =
3.8 cm covering a period of 28 years is shown in the lower
left panel. Variations of such blazars arc observed in a number
of bursts, some overlapping (see, e.g., the burst in 1981). The
UV variability of NGC 5548, a Seyfert 1 galaxy, observed
by the IUE satellite is plotted for three wavelengths in the
100 150 200 250
Time (days)
lowci right panel. Variations at these frequencies appeal to be-
in phase, but the amplitude becomes larger towards smaller
wavelengths. Simultaneously, the line strengths of three broad
emission lines of this Seyfert 1 galaxy have been measured
and are plotted in the upper right panel. It is found thai lines
of high ionization potentials, like Civ, have higher variability
amplitudes than those of low ionization potentials, like Mgu.
From the relative temporal shift in the line variability and
the continuum flux, the size of the broad-line region can be
estimated - see Sect. 5.4.2
5.3 The Central Engine: A Black Hole
cally a few percent, whereas the polarization of normal
QSOs is below ~ 1%. OVVs are usually strong radio
emitters. Their radiation also varies in other wavelength
regions besides the optical, with shorter time-scales and
larger amplitudes as one moves to higher frequencies.
5.2.5 BL Lac Objects
The class of AGNs called BL Lac objects (or short:
BL Lacs) is named after its prototypical source BL
Lacertae. They are AGNs with very strongly varying
radiation, like the OVVs, but without strong emission
and absorption lines. As for OVVs, the optical radiation
of BL Lacs is highly polarized. Since no emission lines
are observed in the spectra of BL Lacs, the determi-
nation of their redshift is often difficult and sometimes
impossible. In some cases, absorption lines are detected
in the spectrum which are presumed to derive from the
host galaxy of the AGN and are then identified with the
redshift of the BL Lac.
The optical luminosity of some BL Lacs varies by
several magnitudes if observed over a sufficiently long
time period. Particularly remarkable is the fact that in
epochs of low luminosity, emission lines are sometimes
observed and then a BL Lac appears like an OVV
For this reason, OVVs and BL Lacs are collectively
called blazars. All known blazars are radio sources. Be-
sides the violent variability, blazars also show highly
energetic and strongly variable y-radiation (Fig. 5.10).
Table 5.1 summarizes the fundamental properties of the
different classes of AGNs.
Day of 11
Fig. 5.10. Variability of the blazar 3C279 in X-ray (bot-
tom) and in y -radiation at photon energies above 100 MeV
(top). On time-scales of a few days, the luminosity varies by
a factor ~ 10
5.3 The Central Engine: A Black Hole
We have previously mentioned that the energy produc-
tion in AGNs must be related to a supermassive black
hole (SMBH) in its center. We will present arguments
for this conclusion in this section. To do this, we will
first summarize some of the relevant observational lads
for AGNs.
• The extent of some radio sources in AGNs may reach
> 1 Mpc. From this length-scale a minimum lifetime
for the activity in the nucleus of these objects can
be derived, since even if the radio source expands
outwards from the core with the speed of light, the
age of such a source would be x > 10 7 yr.
• Luminous QSOs have a luminosity of up to L bol ~
10 47 erg/s. Assuming that the luminosity does not
Table 5.1. Overview of the classification of
ictivc galactic nuclei
Normal galaxy
Radio galaxy
Seyfert galaxy
Quasar
Blazar
Example
Milky Way
M87, Cygnus A
NGC4151
3C273
BL Lac, 3C279
Galaxy type
spiral
elliptical, irregular
spiral
irregular
elliptical?
L/Lq
< 10 4
10 6 - 10 s
10 s -10"
10" -10 14
10" -10 14
JWbh/Mo
3x I0 6
3 x 10 9
10 6 -10 9
10 6 - 10 9
to 6 - 10 9
Radio emission
weak
only«5%radio-lo
ud only « 5% radio-
oud strong, short-time
Radiation in optical
NIR
fully absorbed
old stars, continuum
broad emission lin
es broad emission li
les weak or no lines
X-ray emission
weak
strong
strong
strong
strong
Gamma emission
weak
weak
medium
strong
strong
Variability
unknown
months-years
hours-months
weeks-years
hours-years
5. Active Galactic Nuclei
change substantially over the lifetime of the source,
a total energy can be estimated from the luminosity
and the minimum age,
; 10 4
3rg/s x 10 7 yr ~ 3 x 10 61 erg ,
(5.8)
however, the assumption of an essentially constant
luminosity is not necessarily justified.
• The luminosity of some AGNs varies by more than
50% on time-scales of a day. From this variability
time-scale, an upper limit for the spatial extent of
the source can be determined, because the source
luminosity can change substantially only on such
time-scales where the source as a whole, or at least
a major part of the emitting region, is in causal con-
tact. Otherwise "one end" of the source does not
know that the "other end" is about to vary. This
yields a characteristic extent of the central source
of R < 1 lightday ~ 3 x 10 15 cm.
5.3.1 Why a Black Hole?
We will now combine the aforementioned observations
and derive from them that the basic energy production
in AGNs has to be of a gravitational nature. To do this,
we note that the most efficient "classical" method of en-
ergy production is nuclear fusion, as is taking place in
stars. We will therefore make the provisional assump-
tion (which will soon lead to a contradiction) that the
energy production in AGNs is based on thermonuclear
processes.
By burning hydrogen into iron - the nucleus with the
highest binding energy per nucleon - 8 MeV/nucleon
are released, or 0.008 m p c 2 per nucleon. The maxi-
mum efficiency of nuclear fusion is therefore e < 0.8%,
where e is defined as the mass fraction of "fuel" that is
converted into energy, according to
E = emc 2 | . (5.9)
To generate the energy of E — 3 x 10 61 erg by nuclear
fusion, a total mass m of fuel would be needed, where
m is given by
E ,, Q
m = — ~4x 10 42 g~2x 1O 9 M , (5.10)
where we used the energy estimate from (5.8). If the
energy of an AGN was produced by nuclear fusion,
burnt-out matter of mass m [more precisely, (1 — e)m]
must be present in the core of the AGN.
However, the Schwarzschild radius of this mass is
(see Sect. 3.5.1)
_ 2Gm _ 2GM e m
c 2 ~ c 2 M e
= 3 x 10 5 cm — ~ 6 x 10 14 cm ,
M
i.e., the Schwarzschild radius of the "nuclear cinder" is
of the same order of magnitude as the above estimate of
the extent of the central source. This argument demon-
strates that gravitational effects must play a crucial role -
the assumption of thermonuclear energy generation has
been disproven because its efficiency e is too low. The
only known mechanism yielding larger e is gravitational
energy production.
Through the infall of matter onto a central black hole,
potential energy is converted into kinetic energy. If it is
possible to convert part of this inward-directed kinetic
energy into internal energy (heat) and subsequently emit
this in the form of radiation, e can be larger than that of
thermonuclear processes. From the theory of accretion
onto black holes, a maximum efficiency of e ~ 6% for
accretion onto a non-rotating black hole (also called
a Schwarzschild hole) is derived. A black hole with
the maximum allowed angular momentum can have an
efficiency of e ~ 29%.
5.3.2 Accretion
Due to its broad astrophysical relevance beyond the
context of AGNs, we will consider the accretion process
in somewhat more detail.
The Principle of Accretion. Gas falling onto a compact
object loses its potential energy, which is first converted
into kinetic energy. If the infall is not prevented, the gas
will fall into the black hole without being able to radiate
this energy. In general one can expect that the gas has
finite angular momentum. Thus it cannot fall straight
onto the compact object, since this is prevented by the
angular momentum barrier. Through friction with other
gas particles and by the resulting momentum transfer,
the gas will assemble in a disk oriented perpendicular to
the direction of the angular momentum vector. The fric-
tional forces in the gas are expected to be much smaller
5.3 The Central Engine: A Black Hole
than the gravitational force. Hence the disk will locally
rotate with approximately the Kepler velocity. Since
a Kepler disk rotates differentially, in the sense as the
angular velocity depends on radius, the gas in the disk
will be heated by internal friction. In addition, the same
friction causes a slight deceleration of the rotational ve-
locity, whereby the gas will slowly move inwards. The
energy source for heating the gas in the disk is provided
by this inward motion - namely the conversion of poten-
tial energy into kinetic energy, which is then converted
into internal energy (heat) by friction.
According to the virial theorem, half of the potential
energy released is converted into kinetic energy; in the
situation considered here, this is the rotational energy
of the disk. The other half of the potential energy can
be converted into internal energy. We now present an
approximately quantitative description of this process,
specifically for accretion onto a black hole.
Temperature Profile of a Geometrically Thin, Opti-
cally Thick Accretion Disk. When a mass m falls from
radius r + Ar to r, the energy
_ GM.m GM.m _ GM.m Ar
~ r r + Ar r 7
is released. Here M. denotes the mass of the SMBH,
assumed to dominate the gravitational potential, so that
self-gravity of the disk can be neglected. Half of this en-
ergy is converted into heat, £heat = AE/2. If we assume
that this energy is emitted locally, the corresponding
luminosity is
GM.m
AL= — Ar , (5.11)
2r 2
where m denotes the accretion rate, which is the mass
that falls into the black hole per unit time. In the
stationary case, m is independent of radius since other-
wise matter would accumulate at some radii. Hence the
same amount of matter per unit time flows through any
cylindrical radius.
If the disk is optically thick, the local emission corre-
sponds to that of a black body. The ring between r and
r + Ar then emits a luminosity
AL = 2x2jrrAro SB T 4 (r) , (5.12)
where the factor 2 originates from the fact that the disk
has two sides. Combining (5.11) and (5.12) yields the
radial dependence of the disk temperature,
'm -
/ GM.m
\87Tcr SB r
A more accurate derivation explicitly considers the dis-
sipation by friction and accounts for the fact that part
of the generated energy is used for heating the gas,
where the corresponding thermal energy is also partially
advected inwards. Except for a numerical correction
factor, the same result is obtained,
(IGM.m
\Sita SB r i /
which is valid in the range r ^> r$. Scaling
Schwarzschild radius r$, we obtain
_ /3GM.m\
\8jT(7 SB rlJ
'£)"
By replacing r s with (3.31) in the first factor, this can
be written as
(5.14)
From this analysis, we can immediately draw a num-
ber of conclusions. The most surprising one may be
the independence of the temperature profile of the disk
from the detailed mechanism of the dissipation because
the equations do not explicitly contain the viscosity.
This fact allows us to obtain quantitative predictions
based on the model of a geometrically thin, optically
thick accretion disk. 4 The temperature in the disk in-
creases inwards oc r~ 3/4 , as expected. Therefore, the
total emission of the disk is, to a first approximation,
a superposition of black bodies consisting of rings with
The physical mechanism thai is responsible for the wscosily is un
know n. The molecular \ iseosity is far too small to be considered as the
primary process. Rather, the \ iseosity is probably produced by turbu-
leni Hows in the disk oi by magnetic fields, which become spun up by
differential rotation and thus amplified, so that these fields may act as
an elTccli\e friction. In addition, hydrodynamic instabilities may act
as a source of \ iseosity. Although she properties oi the accretion disk
presented here -luminosity and temperature prolile-arc independent
oi the specific mechanism of Ihe \ iseosity. oilier disk properties def-
initely depend on it. For example, the temporal hcha\ ior of a disk in
the presence ol a perturbation, which is responsible for the \ ariability
in some binary systems, depends on the magnitude of the viscosity.
which therefore can b timatcd fn l obsei lions oi ueli . i m
5. Active Galactic Nuclei
different radii at different temperatures. For this reason,
the resulting spectrum does not have a Planck shape but
instead shows a much broader energy distribution.
For any fixed ratio r/r$, the temperature increases
with the accretion rate. This again was expected: since
the local emission is oc T 4 and the locally dissipated en-
ergy is oc m, it must be T oc m l/A . Furthermore, at fixed
ratio r/r$, the temperature decreases with increasing
mass M. of the black hole. This implies that the maxi-
mum temperature attained in the disk is lower for more
massive black holes. This may be unexpected, but it is
explained by a decrease of the tidal forces, at fixed r/r$,
with increasing M. . In particular, it implies that the max-
imum temperature of the disk in an AGN is much lower
than in accretion disks around stellar sources. Accre-
tion disks around neutron stars and stellar-mass black
holes emit in the hard X-ray part of the spectrum and
are known as X-ray binaries. In contrast, the thermal ra-
diation of the disk of an AGN extends to the UV range
only (see below).
5.3.3 Superluminal Motion
Besides the generation of energy, another piece of ev-
idence for the existence of SMBHs in the centers of
AGNs results from observing relative motions of source
components at superluminal velocities. These observa-
tions of central radio components in AGNs are mainly
made using VLBI methods since they provide the high-
est available angular resolution. They measure a time
dependence of the angular separation of source compo-
nents, which often leads to values > c if the angular ve-
locity is translated into a transverse spatial velocity (Fig.
5.11). These superluminal motions caused some dis-
comfort upon their discovery. In particular, they at first
raised concerns that the redshift of QSOs may not orig-
inate from cosmic expansion. Only if the QSO redshifts
are interpreted as being of cosmological origin can they
be translated into a distance, which is needed to convert
the observed angular velocity into a spatial velocity.
We consider two source components (e.g., the radio
core and a component in the jet) which are observed
to have a time-dependent angular separation 9{t). If D
denotes the distance of the source, then the apparent
relative transverse velocity of the two components is
dr
(5.15!
Fig. 5.11. Apparent superluminal velocities of source compo-
nents in the radio jet of the source 3C120. VLB A observations
of this source are presented for 16 different epochs (indicated
by the numbers at the left of the corresponding radio map),
observed at 7 mm wavelength. The ellipse at the lower left
indicates the beam of the VLBA interferometer and thus the
angular resolution of these observations. At the distance of
3C120 of 140 Mpc, a milliarcsecond corresponds to a linear
scale of 0.70 pc. The four straight lines, denoted by 1, o, t, and
u, connect the same source components at different epochs.
The linear motion of these components is clearly visible. The
observed angular velocities of the components yield at
velocities in the range of 4.1c to 5c
5.3 The Central Engine: A Black Hole
where r — DO is the transverse separation of the two
components. The final expression in (5.15) shows that
D app is directly observable if the distance D is assumed
to be known.
Frequently, VLB I observations of compact radio
sources yield values for u app that are larger than c!
Characteristic values for sources with a dominant core
component are n app ~ 5c (see Fig. 5.1 1). But according
to the theory of Special Relativity, velocities > c do not
exist. Thus it is not surprising that the phenomenon of
superluminal motion engendered various kinds of ex-
planations upon its discovery. By now, superluminal
motion has also been seen in optical observations of
jets, as is displayed in Fig. 5.12.
One possible explanation is that the cosmological
interpretation of the redshifts may be wrong, because
for a sufficiently small D velocities smaller than the
speed of light would result from (5.15). However,
no plausible alternative explanations for the observed
redshifts of QSOs exist, and more than 40 years of
QSO observations have consistently confirmed that red-
e "a Y
6.0c 5.5c 6.1c
Fif». 5.12. Also at optical wavelengths, apparent superluminal
motion was observed. The figure shows the optical jet in M87,
based on HST images taken over a period of about four years.
The angular velocity of the components is up to 23 mas/yr.
Assuming a distance of M87 of D = 16Mpc, velocities of up
to ~ 6c are obtained for the components
shift is an excellent measure for their distances - see
Sect. 4.5.1.
However, relativity only demands that no signal may
propagate with velocities > c. It is easy to construct
a thought-experiment in which superluminal velocities
occur. For instance, consider a laser beam or a flashlight
that is rotating perpendicular to its axis of symmetry.
The corresponding light point on a screen changes its
position with a speed proportional to the angular veloc-
ity and to the distance of the screen from the light source.
If we make the latter sufficiently large, it is "easy" to
obtain a superluminal light point on the screen. But this
light point does not carry a signal along its track. There-
fore, the superluminal motions in compact radio sources
may be explained by such a screen effect, but what is
the screen and what is the laser beam?
The generally accepted explanation of apparent su-
perluminal motion combines very fast motions of source
components with the finite speed of light. For this, we
consider a source component moving at speed v at an
angle (f> with respect to the line-of-sight (see Fig. 5.13).
We arbitrarily choose the origin of time t — to be the
time at which the moving component is close to the core
component. At time t = t t , the source has a distance v t e
from the original position. The observed separation is
the transverse component of this distance,
Ar — vt e sin0 .
Since at time t e the source has a smaller distance from
Earth than at t — 0, the light will accordingly take
slightly less time to reach us. Photons emitted at times
t = and t = t e will reach us with a time difference of
v f e cos (j)
= t t (l-pcos<j>) ,
where we define
6.0c
as the velocity in units of the speed of light.
Equation (5.15) then yields the apparent velocity,
Ar vsind)
u apD = — = — . (5.17)
At 1 - /J cos </>
We can directly draw some conclusions from this
equation. The apparent velocity u app is a function of the
direction of motion relative to the line-of-sight and of
5. Active Galactic Nuclei
I Jet component
at f=0
To observer
Fig. 5.13. Explanation of superluminal motion: a source com-
ponent is moving at velocity v and at an angle <j> relative to
the line-of-sight. We consider the emission of photons at two
different times t — and t — t e . Photons emitted at t = t e will
reach us by At = t e (l — /3cos(/>) later than those emitted at
t = 0. The apparent separation of the two source components
then is Ar = vt e sin </>, yielding an apparent velocity on the
sky of u app = Ar/At = v sin 0/(1 - $ cose/.)
the true velocity of the component. For a given value
of v, the maximum velocity u app is obtained if
where the Lorentz factor y — (1 — fi 2 )~ x/2 was al-
ready defined in (5.4). The corresponding value for the
maximum apparent velocity is then
K P p) m ax = ^- ( 5 - 19 >
Since y may become arbitrarily large for values of
v -> c, the apparent velocity can be much larger than c,
Viewing angle, <t>
Fig. 5.14. Apparent velocity /J app = u app /c of a source com-
ponent moving with Lorentz factor y at an angle <j> with
respect to the line-of-sight, for four different values of y.
Over a wide range in 9, /J app > 1, thus apparent superluminal
motion occurs. The maximum values for /S app are obtained if
sin0 = l/y
even if the true velocity v is - as required by Special
Relativity - smaller than c. In Fig. 5.14, t; app is plotted
as a function of cp for different values of the Lorentz
factor y. To get t> app > c for an angle </>, we need
a 0.707 .
Hence, superluminal motion is a consequence of the
finiteness of the speed of light. Its occurrence implies
that source components in the radio jets of AGNs are
accelerated to velocities close to the speed of light.
In various astrophysical situations we find that the
outflow speeds are of the same order as the escape veloc-
ities from the corresponding sources. Examples are the
Solar wind, stellar winds in general, or the jets of neu-
tron stars, such as in the famous example of SS433 (in
which the jet velocity is 0.26 c). Therefore, if the outflow
velocity of the jets in AGNs is close c, the jets should
originate in a region where the escape velocity has
a comparable value. The only objects compact enough
to be plausible candidates for this are neutron stars and
black holes. And since the central mass in AGNs is con-
siderably larger than the maximum mass of a neutron
star, a SMBH is the only option left for the central ob-
ject. This argument, in addition, yields the conclusion
that jets in AGNs must be formed and accelerated very
close to the Schwarzschild radius of the SMBH.
The processes that lead to the formation of jets are
still subject to intensive research. Most likely magnetic
fields play a central role. Such fields may be anchored in
the accretion disk, and then spun up and thereby ampli-
fied. The wound-up field lines may then act as a kind of
spring, accelerating plasma outwards along the rotation
axis of the disk. In addition, it is possible that rotational
energy is extracted from a rotating black hole, a process
in which magnetic fields again play a key role. As is
always the case in astrophysics, detailed predictions in
situations where magnetic fields dominate the dynamics
of a system (like, e.g., in star formation) are extremely
difficult to obtain because the corresponding coupled
equations for the plasma and the magnetic field are very
hard to solve.
5.3.4 Further Arguments for SAABHs
A black hole is not only the simplest solution of the
equations of Einstein's General Relativity, it is also the
natural final state of a very compact mass distribution.
The occurrence of SMBHs is thus highly plausible from
a theoretical point of view. The evidence for the ex-
istence of SMBHs in the center of galaxies that has
been detected in recent years (see Sect. 3.5) provides
an additional argument for the presence of SMBHs in
AGNs.
Furthermore, we find that the direction of the jets on
a milliarcsecond scale, as observed by VLBI, is essen-
tially identical to the direction of jets on much larger
scales and to the direction of the corresponding radio
lobes. These lobes often have a huge distance from the
core, indicating a long lifetime of the source. Hence,
the central engine must have some long-term memory
because the outflow direction is stable over ~ 10 7 yr.
A rotating SMBH is an ideal gyroscope, with a direction
being defined by its angular momentum vector.
X-ray observations of an iron line of rest energy
h P v = 6.35 keV in Seyfert galaxies clearly indicate that
the emission must be produced in the inner region of an
accretion disk, within only a few Schwarzschild radii of
a SMBH. An example for this is given in Fig. 5.15. The
shape of the line is caused by a combination of a strong
Doppler effect due to high rotation velocities in the disk
and by the strong gravitational field of the black hole,
as is explained in Fig. 5.16.
Energy (keV)
Fig. 5.15. The spectral form of the broad iron line in the
Seyfert 1 galaxy MCG-6-30-15 as observed with the ASCA
satellite. If the material emitting the line were at rest we would
observe a narrow line at hpv = 6.35 keV. We see that the
line is (a) broad, (b) strongly asymmetric, and (c) shifted to
smaller energies. A model for the shape of the line, based on
a disk around a black hole lhal is emitting in the radius range-
rs 5 r < 20r s , is sketched in Fig. 5.16
This iron line is not only detected in individual
AGNs, but also in the average spectrum of an ensem-
ble of AGNs. In a deep (~ 7.7 x 10 5 s) XMM-Newton
exposure of the Lockman hole, a region of very low
column density of Galactic hydrogen, a large number
of AGNs were identified and spectroscopically verified.
The X-ray spectrum of these AGNs in the energy ranges
of 0.2 to 3 keV and of 8 to 20 keV (each in the AGN
rest-frame) was modeled by a power law plus intrinsic
absorption. The ratio of the measured spectrum of each
individual AGN and the fitted model spectrum was then
averaged over the AGN population, after transforming
the spectra into the rest-frame of the individual sources.
As shown in Fig. 5.17, this ratio clearly shows the pres-
ence of a strong and broad emission line. The shape of
this average emission line can be very well modeled by
emission from an accretion disk around a black hole
where the radiation originates from a region lying be-
tween ~ 3 and ~ 400 Schwarzschild radii. The strength
of the iron line indicates a high metallicity of the gas in
these AGNs.
5. Active Galactic Nuclei
Transverse Doppler shift
Fig. 5.16. The profile of the broad iron line is caused by
a combination of Doppler shift, relativistic beaming, and
gravitational redshift. On the left, the observed energy of
the line as a function of position on a rotating disk is in-
dicated by colors. Here, the energy in the right part of the
disk which is moving towards us is blueshifted, whereas the
left part of the disk emits redshifted radiation. Besides this
Doppler effect, all radiation is redshifted because the pho-
tons must escape from the deep potential well. The smaller
the radius of the emitting region, the larger this gravitational
redshift. The line profile we would obtain from a ring-shaped
section of the disk (dashed ellipses) is plotted in the pan-
els on the right. The uppermost panel shows the shape of
the line we would obtain if no relativistic effects occurred
besides the non-relativistic Doppler effect. Below, the line
profile is plotted taking the relativistic Doppler effect and
beaming (see Eq. 5.31) into account. This line profile is
shifted towards smaller energies by gravitational redshift so
that, in combination, the line profile shown at the bottom
results
el energy (keV) (b)
Fig. 5.17. The ratio of the X-ray spectrum of AGNs and a fitted
power law averaged over 53 Type 1 AGNs (left panel) and 41
Type 2 AGNs (right panel). The gray and black data points
are from two different detectors on-board the XMM-Newton
observatory. In both AGN samples, a broad relativistic iron
-iVt/'lj
0.5 1 2 5 10
channel energy (keV)
line is visible; in the Type 2 AGNs, an additional n
line component at 6.4 keV can be identified. The line stt
indicates that the average iron abundance in these sout
about three times the Solar value
5.3 The Central Engine: A Black Hole
5.3.5 A First Mass Estimate for the SMBH:
The Eddington Luminosity
Radiation Force. As we have seen, the primary en-
ergy production in AGNs occurs through accretion of
matter onto a SMBH, where the largest part of the
energy is produced in the innermost region, close to
the Schwarzschild radius. The energy produced in the
central region then propagates outwards and can inter-
act with Mailing matter by absorption or scattering.
Through this interaction of outward-directed radiation
with matter, the momentum of the radiation is trans-
ferred to the matter, i.e., the infalling matter experiences
an outwards-directed radiation force. In order for matter
to fall onto the SMBH at all, this radiation force needs
to be smaller than the gravitational force. This condition
can be translated into a minimum mass of the SMBH,
required for its gravity to dominate the total force at
a given luminosity.
We consider a fully ionized gas, so that the interaction
of radiation with this infalling plasma is basically due
to scattering of photons by free electrons. This is called
Thomson scattering. The mean radiation force on an
electron at radius r is then
F m & ■■
(5.20)
where
8tt / e 2 \ 2 „ ,
(TT= T\nTc 1 ) = 6 - 65xl ° cm (5 - 21)
denotes the Thomson cross-section (in cgs units). This
cross-section is independent of photon frequency. 5 To
1 photon 1 if an el 1 n 1 1 1 1 11 1 II
Thomson scattering. 'To a lirst approximation, the energy ol the plio
ton is unchanged in this process, only its direction is different alter
scattering. This is not really hue though. Due to the fact that a pho-
ton with energy E,, carries a momentum I 1. scatterin ill impi
a recoil on the electron. After the scattering evenl the electron will
thus have a non-zero velocity and a corresponding kinetic energy.
Owing to energy eonscnation the photon energy alter scattering is
therefore slightly smaller than before. This energy loss of the photon
1 II I 1 1 1 I I
appreciable, this scatl ring pi ll n 'II I C ompton scattering.
If the electron is not at rest, the scattering can also lead to net energy
transfer to the photon, such as it happens when low- frequency pho
ions propagate through a hot gas (as we w ill discuss in Sect. 0.3. I ) or
through a distribution of relati\istic electrons. In this case one calls
it ih n rsc ( impl hi effect 1 he ph; i< ol ill th 11 1 is th
same, only their kinematics are different.
derive (5.20), we note that the flux S = L/(4nr 2 ) is the
radiation energy which flows through a unit area at dis-
tance r from the central source per unit time. Then S/c
is the momentum of photons flowing through this unit
area per time, or the radiation pressure, because the mo-
mentum of a photon is given by its energy divided by
the speed of light. Thus the momentum transfer to an
electron per unit time, or the radiation force, is given
by a-iS/c. From (5.20), we can see that the radiation
force has the same dependence on radius as the gravita-
tional force, oc r~ 2 , so that the ratio of the two forces is
independent of radius.
Eddington Luminosity. For matter to be able to fall in -
the condition for energy production - the radiation force
must be smaller than the gravitational force. For each
electron there is a proton, and these two kinds of parti-
cles are electromagnetically coupled. The gravitational
force per electron-proton pair is given by
GM.m p
where we have neglected the mass of the electron since
it is nearly a factor of 2000 smaller than the proton
mass m p . Hence, the condition
for the dominance of gravity can be w
a T L GM.m p
Ait r 2 c r 2
where we have defined the Eddington luminosity L e dd
of a black hole of mass M.. Since cr T is independent of
photon frequency, the luminosity referred to above is
the bolometric luminosity.
5. Active Galactic Nuclei
1 to occur at all, we need L < L edd .
Remembering that the Eddington luminosity is propor-
tional to M. we can turn the above argument around: if
a luminosity L is observed, we conclude L edd > L, or
M. > M edd := — — - — L
4nGcm p
)m
v 10 46 erg/s
(5.24)
Therefore, a lower limit for the mass of the SMBH can
be derived from the luminosity. For luminous AGNs,
like QSOs, typical masses are M. > 10 8 M Q , while
Seyfert galaxies have lower limits of M. > 1O 6 M .
Hence, the SMBH in our Galaxy could in principle
provide a Seyfert galaxy with the necessary energy.
In the above definition of the Eddington luminos-
ity we have implicitly assumed that the emission of
radiation is isotropic. In principle, the above argu-
ment of a maximum luminosity can be avoided, and
thus luminosities exceeding the Eddington luminosity
can be obtained, if the emission is highly anisotropic.
A geometrical concept for this would be, for example,
accretion through a disk in the equatorial plane and the
emission of a major part of the radiation along the po-
lar axes (see Fig. 5.18). Models of this kind have indeed
been constructed. It was shown that the Eddington limit
may be exceeded by this, but not by a large factor. How-
ever, the possibility of anisotropic emission has another
very important consequence. To derive a value for the
luminosity from the observed flux of a source, the rela-
tion L = 4jtD^ S is applied, which is explicitly based on
the assumption of isotropic emission. But if this emis-
sion is anisotropic and thus depends on the direction to
the observer, the true luminosity may differ consider-
ably from that which is derived under the assumption of
isotropic emission. Later we will discuss the evidence
for anisotropic emission in more detail.
Eddington Accretion Rate. If the conversion of in-
falling mass into energy takes place with an efficiency e,
the accretion rate can be determined,
*_^.<U,I( ')(&). (5.25)
ec 2 e Vl0 46 erg/s/ \lyrj
Since the maximum efficiency is of order e ~ 0.1, this
implies accretion rates of typically several Solar masses
per year for very luminous QSOs. If L is measured in
units of the Eddington luminosity, we obtain with ( 5.23 )
/ 1.3x 10 38 erg/s \,
" Ledd V ec 2 )\M Q J ~ L eM
—C-
where in the last step the Eddington
been defined,
(5.26)
rate has
Fig. 5.18. A sketch of the innermost region of ;
disk. Because of high temperatures in this region, radiation
pressure can dominate the gas pressure inside the disk: this
leads to an inflation into a thick disk. Radiation from the thick
part of the disk can then hit the thin parts and be parlialh
reflected. 'This reflection is a plausible explanation of the X ray
spectra of AGNs
Growth Rate of the SMBH Mass. The Eddington ac-
cretion rate is the maximum accretion rate if isotropic
emission is assumed, and it depends on the assumed ef-
ficiency e. We can now estimate a characteristic time in
which the mass of the SMBH will significantly increase,
M.
5 x 10 s yr ,
(5.28)
i.e., even with efficient energy production (e ~ 0. 1), the
mass of a SMBH can increase greatly on cosmologi-
cally short time-scales by accretion. However, this is
not the only mechanism which can produce SMBHs of
large mass. They can also be formed through the merger
of two black holes, each of smaller mass, as would be
expected after the merger of two galaxies if both part-
ners hosted a SMBH in its center. This aspect will be
discussed more extensively later.
5.4 Components of an AGN
In contrast to stars, which have a simple geometry, we
expect several source components in AGNs with differ-
ent, sometimes very complex geometric configurations
to produce the various components of the spectrum;
this is sketched in Fig. 5.19. Accretion disks and jets
in AGNs are clear indicators for a significant deviation
from spherical symmetry in these sources. The rela-
tion between source components and the corresponding
spectral components is not always obvious. How-
ever, combining theoretical arguments with detailed
observations has led to quite satisfactory models.
5.4.1 The IR, Optical, and UV Continuum
In Sect. 5.3.2 we considered an accretion disk with
a characteristic temperature, following from (5.14), of
T(r) « 6.3 x 10 s K
'(£
-3/4
(5.29)
/ M.
X \10 8 M C
The thermal emission of an accretion disk with this
radial temperature profile produces a broad spectrum
with its maximum in the UV. The continuum spectrum
of QSOs indeed shows an obvious increase towards
UV wavelengths, up to the limit of observable wave-
lengths, k > 1000 A. (This is the observed wavelength;
QSOs at high redshifts can be observed at significantly
shorter wavelengths in the QSO rest-frame.) At wave-
lengths k < 912 A, photoelectric absorption by neutral
hydrogen in the ISM of the Galaxy sets in, so that
the Milky Way is opaque for this radiation. Only at
considerably higher frequencies, namely in the soft
X-ray band (h P v > 0.2 keV), does the extragalactic sky
become observable again.
If the UV radiation of a QSO originates mainly from
an accretion disk, which can be assumed because of
the observed increase of the spectrum towards the UV,
the question arises whether the thermal emission of the
disk is also visible in the X-ray domain. In this case,
the spectrum in the range hidden from observation, at
13 eV < h P v < 0.2 keV, could be interpolated by such
an accretion disk spectrum. This seems indeed to be
the case. The X-ray spectrum of QSOs often shows
10 11 12 13 14 15 16 17 18 19
log v (Hz)
Fig. 5.19. Sketch of the characteristic spectral behavior of
a QSO. We distinguish between radio-loud (dashed curve)
and radio-quiet (solid curve) QSOs. Plotted is vS v (in arbi-
trary units), so that Hal sections in the spectrum correspond
to equal energy per logarithmic frequency interval. The most
prominent feature is the big blue bump, a broad maximum
in the UV up to the soft X-ray domain of the spectrum. Be-
sides this maximum, a less prominent secondary maximum is
found in the IR. The spectrum increases inwards higher ener-
gies in the X-ray domain of the spectrum - typically ~ 10%
of the total energy is emitted as X-rays
a very simple spectral shape in the form of a power
law, S v ex v~ a , where a ~ 0.7 is a characteristic value.
However, the spectrum follows this power law only
at energies down to ~ 0.5 keV. At lower energies, the
spectral flux is higher than predicted by the extrapo-
lation of the power-law spectrum observed at higher
energies. One interpretation of this finding is that the
(non-thermal) source of the X-ray emission produces
a simple power law, and the additional flux at lower
X-ray energies is thermal emission from the accretion
disk (see Fig. 5.19).
Presumably, these two spectral properties - the in-
crease of the spectrum towards the UV and the radiation
excess in the soft X-ray - have the same origin, being
two wings of a broad maximum in the energy distri-
bution, which itself is located in the spectral range
unobservable for us. This maximum is called the big
blue hump (BBB). A description of the BBB is possi-
ble using detailed models of accretion disks (Fig. 5.20).
For this modeling, however, the assumption of a local
Fig. 5.20. Spectrum of the QSO PKS0405-123 at z = 0.57
(data points with error bars) from the NIR and the optical
up to the UV spectral region, plus a model for this spec-
trum (solid curve). The latter combines various components:
i I ) the radiation from an accretion disk that causes the big blue
bump and whose spectrum is also shown for three indi\ idual
radius ranges, (2) the Balmer continuum, and (3) an under
lying power law which may have its origin in synchrotron
Planck spectrum at all radii of the disk is too simple
because the structure of the accretion disk is more com-
plicated. The spectral properties of an accretion disk
have to be modeled by an "atmosphere" for each radius,
similar to that in stars.
Besides the BBB, an additional maximum exists in
the MIR (IR-bump). This can be described by thermal
emission of warm dust (T < 2000 K). Later in this chap-
ter we will discuss other observations which provide
additional evidence for this dust component.
The optical continuum of blazars is different from
that of Seyfert galaxies and QSOs. It often features
a spectral pattern that follows, to very good approx-
imation, a power law and is strongly variable and
polarized. This indicates that the radiation is predom-
inantly non-thermal. The origin of this radiation thus
probably does not lie in an accretion disk. Rather, the
radiation presumably has its origin in the relativistic
jets which we already discussed for the radio domain,
with their synchrotron radiation extending up to optical
wavelengths. This assumption was strongly supported
by many sources where (HST) observations discovered
optical emission fromjets (see Fig. 5. 12 and Sect. 5.5.4).
5.4.2 The Broad Emission Lines
Characteristics of the Broad Line Region. One of the
most surprising characteristics of AGNs is the presence
of very broad emission lines. Interpreted as Doppler
velocities, the corresponding width of the velocity dis-
tribution of the components in the emitting region is
of order Av < lOOOOkm/s (or Ak/k < 0.03). These
lines cannot be due to thermal line broadening be-
cause that would imply k B T ~ m p (Av) 2 /2 ~ 1 MeV,
or T ~ 10 10 K - no emission lines would be produced
at such high temperatures because all atoms would be
fully ionized (plus the fact that at such temperatures
a plasma would efficiently produce e + e _ -pairs, and
the corresponding annihilation line at 5 1 1 keV should
be observable in Gamma radiation). Therefore, the ob-
served line width is interpreted as Doppler broadening.
The gas emitting these lines then has large-scale veloc-
ities of order ~ lOOOOkm/s. Velocities this high are
indicators of the presence of a strong gravitational field,
as would occur in the vicinity of a SMBH. If the emis-
sion of the lines occurs in gas at a distance r from
a SMBH, we expect characteristic velocities of
. \-l/2
GM.
so for velociti
of
s of v ~ c/30, we obtain a radial distance
Hence, the Doppler broadening of the broad emission
lines can be produced by Kepler rotation at radii of
about 1000rs. Although this estimate is based on the
assumption of a rotational motion, the infall velocity
for free fall does not differ by more than a factor \fl
from this rotational velocity. Thus the kinematic state of
the emitting gas is of no major relevance for this rough
estimate if only gravity is responsible for the occurrence
of high velocities.
The region in which the broad emission lines are pro-
duced is called the broad-line region (BLR). The density
of the gas in the BLR can be estimated from the lines that
are observed. To see this, it must be pointed out that al-
lowed and semi-forbidden transitions are found among
the broad lines. Examples of the former are Lya, Mgll,
and Civ, whereas Cm] and Niv] are semi-forbidden
5.4 Components of an ACN
transitions. However, no forbidden transitions are ob-
served among the broad lines. The classification into
allowed, semi-forbidden, and forbidden transitions is
done by means of quantum mechanical transition prob-
abilities, or the resulting mean time for a spontaneous
radiational transition. Allowed transitions correspond
to electric dipole radiation, which has a large transi-
tion probability, and the lifetime of the excited state
is then typically only 10~ 8 s. For forbidden transitions,
the time-scales are considerably larger, typically 1 s, be-
cause their quantum mechanical transition probability
is substantially lower. Semi-forbidden transitions have
a lifetime between these two values. To mark the dif-
ferent kinds of transitions, a double square bracket is
used for forbidden transitions, like in [Om], while semi-
forbidden lines are marked by a single square bracket,
like in Cm].
An excited atom can transit into its ground state (or
another lower-lying state) either by spontaneous emis-
sion of a photon or by losing energy through collisions
with other atoms. The probability for a radiational tran-
sition is defined by the atomic parameters, whereas
the collisional de-excitation depends on the gas den-
sity. If the density of the gas is high, the mean time
between two collisions is much shorter than the aver-
age lifetime of forbidden or semi-forbidden radiational
Iransilions. Therefore the corresponding line photons
are not observed. 6 The absence of forbidden lines is
then used to derive a lower limit for the gas density,
and the occurrence of semi-forbidden lines yields an
upper bound for the density. To minimize the depen-
dence of this argument on the chemical composition
of the gas, transitions of the same element are prefer-
entially used for these estimates. However, this is not
always possible. From the presence of the Cm] line and
the non-existence of the [Om] line in the BLR, com-
bined with model calculations, a density estimate of
n e ~ 3 x 10 9 cm -3 is obtained.
Furthermore, from the ionization stages of the line-
emitting elements, a temperature can be estimated,
typically yielding T ~ 20 000 K. Detailed photoioniza-
tion models for the BLR are very successful and are
able to reproduce details of line ratios very well.
''To make forbidden Iransilions visible, die pas density needs lo be
very low. Densities this low cannol be produced in the laboratory.
Forbidden lines are in i'ael nol observed in laboratory spectra; they
are "forbidden".
From the density of the gas and its temperature,
the emission measure can then be calculated (i.e., the
number of line photons per volume element). From the
observed line strength and the distance to the AGN, the
total number of emitted line photons can be calculated,
and by dividing through the emission measure, the vol-
ume of the line-emitting gas can be determined. This
estimated volume of the gas is much smaller than the
total volume (~ r 3 ) of the BLR. We therefore conclude
that the BLR is not homogeneously filled with gas;
rather, the gas has a very small filling factor. The gas in
which the broad lines originate fills only ~ 10~ 7 of the
total volume of the BLR; hence, it must be concentrated
in clouds.
Geometrical Picture of the BLR. From the previous
considerations, a picture of the BLR emerges in which it
contains gas clouds with a characteristic particle density
of « e ~ 10 9 cm" 3 . In these clouds, heating and cooling
processes take place. Probably the most important cool-
ing process is the observed emission in the form of
broad emission lines. Heating of the gas is provided
by energetic continuum radiation from the AGN which
photoionizes the gas, similar to processes in Galactic
gas clouds. The difference between the energy of a pho-
ton and the ionization energy yields the energy of the
released electron, which is then thermalized by colli-
sions and leads to gas heating. In a stationary state, the
heating rate equals the cooling rate, and this equilibrium
condition defines the temperature the clouds will attain.
The comparison of continuum radiation and line
emission yields the fraction of ionizing continuum pho-
tons which are absorbed by the BLR clouds; a value of
about 10% is obtained. Since the clouds are optically
thick to ionizing radiation, the fraction of absorbed con-
tinuum photons is also the fraction of the solid angle
subtended by the clouds, as seen from the central contin-
uum source. From the filling factor and this solid angle,
the characteristic size of the clouds can be estimated.
from which we obtain typical values of ~ 10 11 cm. In
addition, based on these argument, the number of clouds
in the BLR can be estimated. This yields a typical value
of~10 10 .
The characteristic velocity of the clouds corresponds
to the line width, hence several thousand km/s. How-
ever, the kinematics of the clouds are unknown. We do
not know whether they are rotating around the SMBH,
5. Active Galactic Nuclei
whether they are infalling or streaming outwards, or
whether their motion is rather chaotic. It is also pos-
sible that different regions within the BLR exist with
different kinematic properties.
Reverberation Mapping. A direct method to examine
the extent of the BLR is provided by reverberation map-
ping. This observational technique utilizes the fact that
heating and ionization of the gas in the BLR are both
caused by the central continuum source of the AGN.
Since the UV radiation of AGNs varies, we expect cor-
responding variations of the physical conditions in the
BLR. In this picture, a decreasing continuum flux should
then lead to a lower line flux, as is demonstrated in Fig.
5.21. Due to the finite extent of the BLR, the observed
variability in the lines will be delayed in time compared
to the ionizing continuum. This delay At can be identi-
fied with the light travel time across the BLR, At ~ r/c.
In other words, the BLR feels the variation in the contin-
uum source only after a delay of At. From the observed
correlated variabilities of continuum and line emission,
At can be determined for different line transitions, and
so the corresponding values of r can be estimated.
Such analyses of reverberation mapping are ex-
tremely time-consuming and complex because one
needs to continuously monitor the continuum light and,
simultaneously, the line fluxes of an AGN over a long
period. The relevant time-scales are typically months
for Seyfert 1 galaxies (see Fig. 5.22). To perform such
measurements, coordinated campaigns involving many
observatories are necessary because the light curves
have to be observed without any gaps, and one should
not depend on the local weather conditions at any ob-
servatory. From the results of such campaigns and the
correlation of the light curves in the UV continuum
and the different line fluxes (Fig. 5.23), the picture
of an inhomogeneous BLR is obtained which extends
over a large range in r and which consists of differ-
ent "layers". The extent of the BLR scales with the
luminosity of the AGN. Its ionization structure varies
with r; the higher the ionization energy of a transi-
tion, the smaller the corresponding radius r. For the
JD 7582-7594
JD 7642-7650
■
"1
i ■ > m
.-/ u "
1
5
**-
l
. , j , , , i . ,
i ' ■
■
\
CIVX1549
;
1
600
;
s
, '
+
■
400
&
»
-.
2
.•'.*
-+-
'-
...I... i i
. i .
, -
Fig. 5.21. In the left-hand
panel, the UV spectrum
of the Seyfert 1 galaxy
NGC5548 is plotted for
two different epochs in
which the source radiated
strongly and weakly, re-
spectively. It can clearly
be seen that not only
does the continuum ra-
diation of the source vary
but also the strength of
the emission lines. The
right-hand panels show
the flux of the continuum
at~ 1300 A, the Crv line
at k = 1549 A, and the
Hen line at X =1640 A, as
a function of the near-UV
flux at different epochs
during an eight-month
observational c
with the IUE
5.4 Components of an ACN
ft V t* V
W F ° 1
A/Na
-n . i |t. ..(i i ■ ■ | n i n i i i ■ f
■A *' ^ >.
i i
7500 7550 7600 7650 7700 7750
J. D.-2 440000
W
A
*UjW
Fig. 5.22. Light curve of
NGC 5548 over a period
of 8 months at different
wavelengths. In the left-
hand panels, from top to
bottom, the continuum at
1 = 1350 A, X =1840 A,
and X = 2670 A, the
broad and strong emis-
sion lines Lya and Civ,
as well as the optical
light curve are plotted.
The right-hand panels
shows the weaker lines
Nv at A. = 1240 A, Siiv
atA.= 1402A,Hell+Om]
at X = 1640 A, Cm] at
X = 1909 A, and Mgll at
X = 2798 A
7500 7550 7600 7650 7700 7750
J.D.-2440000
Seyfert 1 galaxy NGC 5548, one obtains At ~ 12 d for
Lya, about At ~ 26 d for Cm], and about 50 d for Mgll.
This may not come as a surprise because the ionizing
flux increases for smaller r. Furthermore, the relative
flux variations in lines of higher ionization energy are
larger, as can also be seen in Fig. 5.9. This picture is also
consistent with the different width of the various lines,
in that lines of higher ionization energy are broader. In
addition, lines of higher ionization energy have a mean
redshift systematically shifted bluewards compared to
narrower emission lines. This hints at a radial out-
ward motion of the clouds together with an intrinsic
absorption of part of the BLR radiation by interven-
ing absorbing material. For a simple picture, but not the
5. Active Galactic Nuclei
Fig. 5.23. The different
light curves from Fig. 5.22
are correlated with the con-
tinuum flux at X = 1350 A.
The autocorrelation func-
tion is shown by the
solid line in the central
panels, the others are cross-
correlation functions. We
can see that the maximum
of the correlation is shifted
towards positive times -
flux are not simultaneously
followed by the emission
lines but appear only after
a delay. This delay corre-
sponds to the light travel
time from the center of the
AGN to the clouds of the
BLR where the lines are
emitted. The smaller the
ionization level of the re-
spective ion, the longer the
delay. For example, we ob-
tain a delay of 12 days for
Lya, 26 days for Cm], and
about 50 days for Mgn,
where the latter value could
not be measured exactly
because the relative flux
variations of this line are
small and thus the corre-
lation function does not
show a very prominent
only plausible one, an outflow motion of the clouds in
the BLR can be assumed, where we see those clouds
that are, from our point of view, located behind the ac-
cretion disk partly absorbed by the disk material. The
received line radiation is therefore dominated by the
clouds that are in front of the disk and moving towards
us, so that it is systematically blueshifted.
The nature of the clouds in the BLR is unknown.
Their small extent and high temperature imply that they
should vaporize on very small time-scales unless they
are somehow stabilized. Therefore these clouds need to
be either permanently replenished or they have to be
stabilized, either by external pressure, e.g., from a very
hot but thin medium in the BLR in between the clouds,
5.4 Components of an ACN
by magnetic fields, or even gravitationally. One possi-
bility is that the clouds are the extended atmospheres
of stars; this would, however, imply a very high (too
high?) total mass of the BLR.
5.4.3 Narrow Emission Lines
Besides the broad emission lines that occur in QSOs,
Seyfert 1 galaxies, and broad-line radio galaxies, most
AGNs (with exception of the BL Lacs) show narrow
emission lines. Their typical width is ~ 400 km/s. This
is considerably narrower than lines of the BLR, but
still significantly broader than characteristic velocities
in normal galaxies. In analogy to the BLR, the region in
which these lines are produced is known as the narrow
line region (NLR). The strongest line from the NLR
is, besides Lya and ClV, the forbidden [Olll] line at
X = 5007 A. The existence of forbidden lines implies
that the gas density in the NLR is significantly lower
than in the BLR. From estimates analogous to those
for the BLR, the characteristic properties of the NLR
are determined. It should be noted that no reverberation
mapping can be applied, since the extent of the NLR
is ~ 100 pc. Because of this large extent, no variability
of the narrow line intensities is expected on time-scales
accessible to observation, and none has been found.
The line ratios of allowed and forbidden lines yield
« e ~ 10 3 cm" 3 for the typical density of the gas in which
the lines originate. The characteristic temperature of the
gas is likewise obtained from line ratios, T ~ 16 000 K,
which is slightly lower than in the BLR. The filling
factor here is also significantly smaller than one, about
10~ 2 . Hence, the geometrical picture of clouds in the
NLR also emerges. Like in the BLR, the properties of
the NLR are not homogeneous but vary with r.
Since the extent of the NLR in Seyfert galaxies is of
the order of r ~ 100 pc, it can be spatially resolved for
nearby Seyfert galaxies. The morphology of the NLR is
very interesting: it is not spherical, but appears as two
cone-shaped regions (Fig. 5.24). It seems as if the ion-
ization of the NLR by the continuum radiation of the
AGN is not isotropic, but instead depends strongly on
the direction.
5.4.4 X-Ray Emission
The most energetic radiation of an AGN is expected to
be produced in the immediate vicinity of the SMBH.
Ground View
Fig. 5.24. Image of the Seyfert galaxy NGC5728. Left:
a large-scale image showing the disk galaxy; right: an HST
image of its central region taken through a filter with a small
bandwidth (narrow-band filter) centered on a nai
line. This image shows the spatially resolved NLR. We <
HST View
see that it is not spherical but consists of two cones. From this,
it is concluded that the ionizing radiation of the AGN is not
isotropic, but is emitted in two preferred directions perpendic-
ular to the disk of the Galaxy (and thus probably perpendiculai
to the central accretion disk)
5. Active Galactic Nuclei
Therefore, the X-ray emission of AGNs is of special in-
terest for probing the innermost regions of these objects,
as we have already seen from the relativistic iron line
shown in Fig. 5.15. In fact, the variability on very short
time-scales (see Fig. 5.9) is a clear indicator of a small
extent of the X-ray source.
To a first approximation, the X-ray spectrum is char-
acterized by a power law, S v oc v~ a , with slope a ~ 0.7.
At energies h P v > 10 keV, the spectrum exceeds the ex-
trapolation of this power law, i.e., it becomes flatter.
Towards lower X-ray energies, the spectrum seems to
be steeper than the power law, which presumably re-
sults from the blue part of the BBB, as was mentioned
previously.
Besides this continuum radiation, emission and ab-
sorption lines are also found in the X-ray domain, the
strongest lines being those of highly ionized iron. The
improved sensitivity and spectral resolution of the X-ray
telescopes Chandra and XMM-Newton compared to
earlier X-ray observatories have greatly advanced the
X-ray spectroscopy of AGNs. Figure 5.25 shows an
example of the quality of these spectra.
The X-ray emission of Seyfert 1 and Seyfert 2 gal-
axies is very different. In the energy range of the
ROSAT X-ray satellite (0.1 keV < h P v < 2.4 keV), sig-
nificantly more Seyfert 1 galaxies were discovered than
Seyfert 2 galaxies. The origin of this was later un-
covered by Chandra and XMM-Newton. In contrast to
ROSAT, these two satellites are sensitive up to energies
of h P v ~ 10 keV and they have found large numbers of
Seyfert 2 galaxies. However, their spectrum differs from
that of Seyfert 1 galaxies because it is cut off towards
lower X-ray energies. The spectrum indicates the pres-
ence of an absorber with a hydrogen column density
of > 10 22 cm -2 and in some cases even orders of mag-
nitude higher. This fact will be used in the context of
unified models (Sect. 5.5) of AGNs.
5.4.5 The Host Galaxy
As the term "active galactic nuclei" already implies,
AGNs are considered the central engine of otherwise
quite normal galaxies. This nuclear activity is nourished
by accretion of matter onto a SMBH. Since it seems that
all galaxies (at least those with a spheroidal component)
harbor a SMBH, the question of activity is rather one of
accretion rate. What does it take to turn on a Seyfert gal-
axy, and why are most SMBHs virtually inactive? And
by what mechanism is matter brought into the vicinity
of the SMBH to serve as fuel?
For a long time it was not clear as to whether QSOs
are also hosted in a galaxy. Their high luminosity ren-
ders it difficult to identify the surrounding galaxy on
images taken from the ground, with their resolution
being limited by seeing to ~ 1". In the 1980s, the sur-
rounding galaxies of some QSOs were imaged for the
first time, but only with the HST did it become pos-
IRAS 13349+2438
j* '
XMM-Newton/RGS
l-jri--
5 Ms °l° -
> >>>> ll
s
" " >
z . a . 9 °. -=
* - *
Till
ll I I \
,UiJ
§ E=5? "j
iii
RM IjlLnliil
xg*S£ I
1 1
wavelength (A)
Fig. 5.25. X-ray spectrum of the quasar
IRAS 13349+2438 (z = 0.108), observed
by the XMM satellite. Various absorption
lines are marked
5.4 Components of an ACN
sible to obtain detailed images of QSO host galaxies
(see Fig. 5.26) and thus to include them in the class
of galactic nuclei. In these investigations, it was also
found that the host galaxies of QSOs are often heavily
disturbed, e.g., by tidal interaction with other galax-
ies or even by merging processes. These disturbances
of the gravitational potential are considered essential
for the gas to overcome the angular momentum bar-
rier and to flow towards the center of the galaxy. At
the same time, such disturbances seem to increase the
star-formation rate enormously, because starburst gal-
axies are also often characterized by disturbances and
interactions. A close connection seems to exist between
AGN activity and starbursts. Optical and NIR images of
QSOs (see Fig. 5.26) cannot unambiguously answer the
question of whether QSO hosts are spirals or ellipticals.
Today it seems established that the hosts of
low-redshift QSOs are predominantly massive and
bulge-dominated galaxies. This finding is in good
agreement with the fact that the black hole mass in "nor-
mal" galaxies scales with the mass of the spheroidal
component of the galaxies. It was recently found
that higher-redshift QSOs are also hosted by mas-
sive elliptical galaxies. Furthermore, the host galaxies
of radio-loud QSOs seem to be systematically more
luminous than that of radio-quiet QSOs.
■■■ •
4
•
•
■■■
Fig. 5.26. HST images of QSOs. In all cases the host galaxy
can clearly be identified, with the QSO itself being visible as a
(central) point source in these images. Top left: PG 0052+25 1
is located in the center of an apparently normal spiral galaxy.
Lower left: PHL 909 seems to be located in the center of a nor-
mal elliptical galaxy. Top center: the QSO IRAS 04505-2958
is obviously part of a collision of two galaxies and may be
provided with "fuel" by material ripped from the galaxies by
tidal forces. Surrounding the QSO core, a region of active
star formation is visible. PG 1012+008 (lower center) is also
part of a pair of merging galaxies. Top right: the host galaxj
of QSO 0316-346 seems to be about to capture a tidal tail.
Lower right: the QSO IRAS 13218+0552 seems to be located
in a galaxy which just went through a merger process
5. Active Galactic Nuclei
Binary QSOs. The connection between the activity of
galaxies and the presence of close neighbors is also seen
from the clustering properties of QSOs. In surveys for
gravitational lens systems, pairs of QSO images have
been detected which have angular separations of a few
arcseconds and very similar redshifts, but sufficiently
different spectra to exclude them being gravitational!}
lensed images of the same source. The number of bi-
nary QSOs thus found is considerably larger than the
expectation from the large-scale correlation function of
QSOs. This conclusion was further strengthened by an
extensive analysis from the QSOs in the Sloan Digital
Sky Survey (see Sect. 8.1.2). The correlation function of
QSOs at separations below ~ 30h~ ' kpc exceeds that of
the extrapolation of the correlation function from larger
scales by a factor of 10 or more. Hence it seems that
the small-scale clustering of QSOs is very much en-
hanced, say compared to normal galaxies, which could
be due to the triggering of activity by the proximity
of the neighbor: in this case, both galaxies attain a per-
turbed gravitational potential and start to become active.
5.4.6 The Black Hole Mass in AGNs
We now return to the determination of the mass of the
central black hole in AGNs. In Sect. 5.3.5, a lower limit
on the mass was derived, based on the fact that the
luminosity of an AGN cannot exceed the Eddington
luminosity. However, this estimate cannot be very pre-
cise, for at least two reasons. The first is related to the
anisotropic appearance of an AGN. The observed flux
can be translated into a luminosity only on the assump-
tion that the emission from the AGN is isotropic, and
we have discussed several reasons why this assump-
tion is not justified in many cases. Second, we do not
have a clear idea what the ratio of AGN luminosity to
its Eddington luminosity is. It is clear that this ratio
can vary a lot between different black holes. For exam-
ple, the black hole at the center of our Galaxy could
power a luminosity of several 10 44 erg/s if radiating
with the Eddington luminosity - and we know that the
true luminosity is several order of magnitudes below
this value.
M. from Reverberation Mapping. A far more accu-
rate method for estimating the black hole mass in AGNs
comes from reverberation mapping which we described
in Sect. 5.4.2. The principal quantity that is derived from
this technique is the size r of the BLR for a given atomic
line or for a given ionization state of a chemical ele-
ment. Furthermore, the relative line width Ak/k can be
measured, and can be related to the characteristic ve-
locity dispersion a in the BLR, a — c Ak/X. Assuming
that the gas is virialized, or moving on Keplerian or-
bits around the black hole, the mass of the latter can be
estimated to be
M. «
ra l /G ,
(5.30)
where the difference between random motion and cir-
cular orbits corresponds to a factor of order 2 in this
estimate. Thus, once reverberation mapping has been
conducted, the black hole mass can be estimated with
very reasonable accuracy.
However, this is a fairly expensive observing tech-
nique, requiring the photometric and spectroscopic
monitoring of sources over long periods of time, and
it can therefore be applied only to relatively small sam-
ples of sources. Furthermore, this technique is restricted
to low-luminosity AGNs, since the size of the BLR, and
thus the time delay and the necessary length of the mon-
itoring campaign, increases with the black hole mass.
We might therefore want to look at alternative methods
for estimating M..
M, from Scaling Relations. When applied to a set
of nearby Seyfert 1 galaxies, for which reverberation
mapping has been carried out, one finds that the black
hole mass in these AGNs satisfies the same relation
(3.35) between M. and the velocity dispersion a e of
the bulge as has been obtained for inactive galaxies.
This scaling relation then yields a useful estimate of
the black hole mass from the stellar velocity dispersion.
Unfortunately, even this method cannot be applied to
a broad range of AGNs, since the velocity dispersion of
stars cannot be measured in AGNs which are either too
luminous - since then the nuclear emission outshines
the stellar light, rendering spectroscopy of the latter
impossible - or too distant, so that a spatial separation
of nuclear light from stellar light is no longer possible.
However, another scaling relation was found which
turns out to be very useful and which can be extended
to luminous and high-redshift sources. The size of the
BLR correlates strongly with the continuum luminos-
5.4 Components of an ACN
ity of an AGN. This behavior can be understood by the
following argument. As we have seen, the BLR covers
a broad range in radii around the center, and the physical
conditions in the BLR are "layered": ions of higher ion-
ization energy are closer to the continuum source than
those with lower ionization energy. The gas in the BLR
is subject to photoionization, and hence the distribution
of ionization states will depend on the flux of energetic
photons. This flux is oc L/r 2 , thus it depends on the lu-
minosity L of the ionizing radiation and the distance r
for the central source. For a given ionization state, and
thus for a given broad emission line, the value of the
ionization parameter E — L/r 2 should be very similar
in all AGNs. Hence, this argument yields r oc L 1/2 . In
fact, direct estimates from sources where the radius was
determined with reverberation mapping confirmed such
a relation, which might be slightly steeper, r oc L~° 6 .
The value of E can be obtained for those sources for
which reverberation mapping yields a determination of
the size r. These sources are then used to calibrate the
L-r relation for a given line transition. Once this is done,
(5.30) can be applied again, with the radius now deter-
mined from the calibrated value of E and the continuum
luminosity of the AGN. This method can be extended
to high redshifts, if the value of E can be determined
for emission lines which are located in the optical win-
dow at a given redshift, where the Mgll and Civ lines
are the most important transitions.
The Eddington Efficiency. Once an estimate for M.
is obtained, the Eddington luminosity can be calculated
and compared with the observed luminosity. The ratio of
these two, 6 Edd = L/L Edd , is called the Eddington effi-
ciency. If one can ignore strongly beamed emission, e Edd
should be smaller than unity. For the estimate of € Edd ,
the observed luminosity in the optical band needs to be
translated into a bolometric luminosity, which can be
done with the help of the average spectral energy dis-
tribution of AGNs of a given class. All of these steps
involve statistical errors of a factor ~ 2 in any indi-
vidual object, but when averaged over an ensemble of
sources, they should yield approximately the correct
mean values.
We find that e Edd varies between a few percent to
nearly unity among QSOs. Hence, once a black hole
becomes sufficiently active as to radiate like a QSO, its
luminosity approaches the Eddington luminosity. There
might be a trend that radio-loud QSOs have a somewhat
larger e Edd , but these correlations are controversial and
might be based on selection effects. The fact that e Edd is
confined to a fairly narrow interval implies that the lu-
minosity of a QSO can be used to estimate M., just by
settingM. = e Edd M Edd (L).Thismassestimatehasasta-
tistical uncertainty of about a factor of ~ 3 in individual
sources.
The Galactic Black Hole. The Eddington efficiency
of the SMBH in the Galactic center is many orders of
magnitude smaller than unity; in fact, with its total lu-
minosity of 5 x 10 36 erg/s, € Edd ~ 10~ 8 . Such a small
value indicates that the SMBH in our Galaxy is starved;
the accretion rate must be very small. However, one can
estimate a minimum mass rate with which the SMBH in
the Galactic center is fed, by considering the mass-loss
rate of the stars near the Galactic center. This amounts
to ~ 10~ 4 M /yr, enough material to power an accre-
tion flow with L ~ 10 _2 L Edd . The fact that the observed
luminosity is so much smaller than this value leads to
two implications. The first of these is that there must
be other modes of accretion which are far less efficient
than that of the geometrically thin, optically thick ac-
cretion disk described in Sect. 5.3.2. Such models for
accretion flows were indeed developed. In these mod-
els, the generated internal energy (heat) is not radiated
away locally, but instead advected with the flow towards
the black hole. The second conclusion is that the cen-
tral mass concentration must indeed be a black hole -
a black hole is the only object which does not have
a surface. If, for example, one would postulate a hypo-
thetical object with M ~ 3 x 10 6 M Q which has a hard
surface (like a scaled-up version of a neutron star), the
accreted material would fall onto the surface, and its ki-
netic and inner energy would be deposited there. Hence,
this surface would heat up and radiate thermally. Since
we have strict upper limits on the radius of the object,
coming from mm-VLBI observations, we can estimate
the minimum luminosity such a source would have. This
estimate is again several orders of magnitude larger than
the observed luminosity from Sgr A*, firmly ruling out
the existence of such a solid surface.
The observed flaring activity of Sgr A* (see
Sect. 2.6.4) yields further information about the prop-
erties of the Galactic SMBH. In particular, the
quasi-periodicity of ~ 17 min most likely must be iden-
5. Active Galactic Nuclei
tified with a source component orbiting the SMBH.
From the theory of black holes it follows that objects
can have stable orbits around a black hole only if the
orbital radius is larger than some threshold. For a black
hole without rotation, this last stable orbit has a radius
of 3r s , whereas it can be smaller for spinning black
holes. Since we know the mass of the SMBH in our
Galaxy, and thus its Schwarzschild radius rs , we can cal-
culate the orbital period for this last stable orbit. This
turns out to be larger than 17 min for a non-rotating
black hole. In fact, assuming that the material which
emits the flared radiation orbits the black hole at or
near the last stable orbit, one concludes that the SMBH
in SgrA* spins at about half the maximally allowed
rate.
Recently, flaring activity from other low-luminosity
AGNs has been detected. Since their corresponding
black hole mass is estimated to be larger than that
of SgrA*, the time-scale of variability is accordingly
longer.
Black Hole Mass Scaling Relations at High Red-
shifts. As we have seen in Sect. 3.5.3, the black hole
mass in normal, nearby galaxies is correlated with the
bulge (or spheroidal) luminosity. As this component
of galaxies consists of an old stellar population, its
luminosity is very closely related to its stellar mass.
Estimating the black hole mass from the continuum lu-
minosity of the QSOs, and observing the spheroidal
luminosity of their host galaxies (which requires the
high angular resolution of HST), we can now investigate
whether such a scaling relation already existed at earlier
epochs, i.e., at high redshifts. When the evolution of the
stellar population is taken into account in determining
the mass of the stellar spheroidal component - stars at
high redshift are necessarily younger than the old stel-
lar population in local ellipticals or bulges - essentially
the same relation between M. and the spheroidal stellar
mass is obtained for z < 2 QSOs as for local galaxies.
This means that, whatever causes the close correlation
between these two quantities in the local Universe, these
processes must have already occurred in the early Uni-
verse. Needless to say, this observational result places
strong constraints on the joint evolution of galaxies and
their central supermassive black holes. At even higher
redshifts, there are indications that the ratio of black
hole mass and stellar mass was larger than today.
Black Hole Demography. Given that supermassive
black holes grow by accretion, 7 and that this accre-
tion is related to the energy release in AGNs, one might
ask whether the total mass density of black holes at
the present epoch is compatible with the integrated
AGN luminosity. In other words, can the mass density
of black holes be accounted for by the total accre-
tion luminosity over cosmic time, as seen in the AGN
population?
The first of these numbers is obtained from the
scaling relation between SMBH mass and the prop-
erties of the spheroidal components in galaxies, as
discussed in Sect. 3.5.3. This yields a value of the
spatial mass density of SMBHs in the mass range
10 6 < M./M e < 5 x 10 9 of ~ 4 x 1O 5 M /Mpc 3 , with
about a 30% uncertainty. About a quarter of this mass is
contributed by SMBHs in the bulges of late-type galax-
ies; hence, the total SMBH mass density is dominated
by ellipticals.
The overall accreted mass is obtained from the
redshift-dependent luminosity function of AGNs (see
Sect. 5.6.2), by assuming an efficiency e of the con-
version of mass into energy. Indeed, the local mass
density of SMBHs is matched if the accretion efficiency
is e ~ 0.10, as is expected from standard accretion
disk models. It therefore seems that the population of
SMBHs located in normal galaxies at the present epoch
have undergone an active phase in their past, causing
their mass growth. However, it may be that the effi-
ciency here is underestimated, as some fraction of the
energy released during the accretion process is con-
verted into kinetic energy, as seen by powerful jets in
AGN. This fraction is largely undetermined at present,
but may not be negligible. In this case, the true e needs
to be higher than 0.1, which is only possible for black
holes which rotate rapidly. In fact, the observed profile
of the iron emission line from AGNs indicates black
hole rotation.
A more detailed comparison between the SMBH and
AGN populations reveals that the characteristic Edding-
ton efficiency is e E dd ~ 0.3. With this value, combined
with (5.28), one can estimate the mean time-scale over
which a typical SMBH was active in the past, yield-
"''1 'he population ol supermassi\e black holes can also he changed by
merging proces-.es. i.e.. as [he result of merging black holes when
heir h las mei I r. in h in I il bl ' hole
mass is largely conserved, modulo some general relativistic effects.
5.5 Family Relations of ACNs
ing f act ~ 2 x 10 8 yr. Hence, the SMBH of a currer
day massive galaxy was active during about 2% of il
lifetime.
5.5 Family Relations of AGNs
5.5.1 Unified Models
In Sect. 5.2, different types of AGNs were listed. We
saw that many of their properties are common to all
types, but also that there are considerable differences.
Why are some AGNs seen as broad-line radio galaxies,
others as BL Lac objects? The obvious question arises
as to whether the different classes of AGNs consist of
rather similar objects which differ in their appearance
due to geometric or light propagation effects, or whether
more fundamental differences exist. In this section we
will discuss differences and similarities of the various
classes of AGNs and show that they presumably all
derive from the same physical model.
Common Properties. Common to all AGNs is a SMBH
in the center of the host galaxy, the supposed cen-
tral engine, and also an accretion disk that is feeding
the black hole. This suggests that a classification can
be based on M. and the accretion rate m, or perhaps
more relevantly the ratio m/m eM . M. defines the max-
imum (isotropic) luminosity of the SMBH in terms
of the Eddington luminosity, and the ratio m/m e dd
describes the accretion rate relative to its maximum
value. Furthermore, the observed properties, in par-
ticular the seemingly smooth transition between the
different classes, suggest that radio-quiet quasars and
Seyfert 1 galaxies basically differ only in their cen-
tral luminosity. From this, we would then deduce that
they have a similar value of m/m e dd but differ in M..
An analogous argument may be valid for the transition
from BLRGs to radio-loud quasars.
The difference between these two classes may be
due to the nature of the host galaxy. Radio galaxies (and
maybe radio-loud quasars?) are situated in elliptical gal-
axies, Seyfert nuclei (and maybe radio-quiet quasars?)
in spirals. A correlation between the luminosity of the
AGN and that of the host galaxy also seems to exist.
This is to be expected if the luminosity of the AGN is
strongly correlated with the respective Eddington lumi-
nosity, because of the correlation between the SMBH
mass in normal galaxies and the properties of the gal-
axy (Sect. 3.5.3). Another question is how to fit blazars
and Seyfert 2 galaxies into this scheme.
Anisotropic Emission. In the context of the SMBH plus
accretion disk model, another parameter exists that will
affect the observed characteristics of an AGN, namely
the angle between the rotation axis of the disk and the
direction from which we observe the AGN. We should
mention that in fact there are many indications that the
radiation of an AGN is not isotropic and thus its appear-
ance is dependent on this direction. Among these are the
observed ionization cones in the NLR (see Fig. 5.24)
and the morphology of the radio emission, as the ra-
dio lobes define a preferred direction. Furthermore, our
discussion of superluminal motion has shown that the
observed superluminal velocities are possible only if the
direction of motion of the source component is close to
the direction of the line-of-sight. The X-ray spectrum
of many AGNs shows intrinsic (photoelectric) absorp-
tion caused by high column density gas, where this
effect is mainly observed in Seyfert 2 galaxies. Because
of these clear indications it seems obvious to exam-
ine the dependence of the appearance of an AGN on the
viewing direction. For example, the observed difference
between Seyfert 1 and Seyfert 2 galaxies may simply
be due to a different orientation of the AGN relative to
the line-of-sight.
Broad Emission Lines in Polarized Light. In fact,
another observation of anisotropic emission provides
a key to understanding the relation between AGN types,
which supports the above idea. The galaxy NGC 1068
has no visible broad emission lines and is therefore
classified as a Seyfert 2 galaxy. Indeed, it is consid-
ered an archetype of this kind of AGN. However, the
optical spectrum of NGC 1068 in polarized light shows
broad emission lines (Fig. 5.27) such as one would find
in a Seyfert 1 galaxy. Obviously the galaxy must have
a BLR, but it is only visible in polarized light. The
photons that are emitted by the BLR are initially unpo-
larized. Polarization may be induced through scattering
of the light, however, where the direction perpendicu-
lar to the directions of incoming and scattered photons
define a preferred direction, which then defines the
polarization direction.
5. Active Galactic Nuclei
rrrrp
I I I II I I M I I
TTrrjTTTT|Trf
^y^y^_
4000 4500 5000 5500 6000
Wavelength (A)
Fig. 5.27. Spectrum of the Seyfert 2 galaxy NGC 1068. The
top panel displays the total flux which, besides the contin-
uum, also show s narrow emission lines, in particular |OmJ at
A. = 5007 A and k = 4959 A. However, in polarized light (bot-
tom panel!, broad emission lines (like H/i and Hy) typical of
a Seyferl I galaxy are also visible. Therefore, it is concluded
that the BLR becomes visible in light polarized via scattering;
the BLR is thus visible only indirectly
The interpretation of this observation (see Fig. 5.28)
now is that NGC 1068 has a BLR but our direct view of
it is obscured by absorbing material. However, this ab-
sorber does not fully engulf the BLR in all directions but
only within a solid angle of < An as seen from the cen-
tral core. If photons from the BLR are scattered by dust
or electrons in a way that we are able to observe the scat-
tered radiation, then the BLR would be visible in this
scattered light. Direct light from the AGN completely
outshines the scattered light, which is the reason why we
cannot identify the latter in the total flux. By scattering,
however, this radiation is also polarized. Thus in obser-
vations made in polarized light, the (unpolarized) direct
radiation is suppressed and the BLR becomes visible in
the scattered light.
This interpretation is additionally supported by
a strong correlation of the spatial distribution of the
polarization and the color of the radiation in NGC 1068
(see Fig. 5.29). We can conclude from this that the differ-
ences between Seyfert 1 and Seyfert 2 galaxies originate
in the orientation of the accretion disk and thus of the
absorbing material relative to the line-of-sight.
From the abundance ratio of Seyfert 1 to Seyfert 2
galaxies (which is about 1:2), the fraction of solid angle
in which the view to the BLR is obscured, as seen from
the AGN, can be estimated. This ratio then tells us that
about 2/3 of the solid angle is covered by an absorber.
Such a blocking of light may be caused by dust. It is
assumed that the dust is located in the plane of the
accretion disk in the form of a thick torus (see Fig. 5.28
and Fig. 5.30 for a view of this geometry).
Search for Type 2 QSOs. If the difference between
Seyfert galaxies of Type 1 and Type 2 is caused merely
by their orientation, and if likewise the difference be-
tween Seyfert 1 galaxies and QSOs is basically one
of absolute luminosity, then the question arises as to
whether a luminous analog for Seyfert 2 galaxies exists,
a kind of Type 2 QSO. Until a few years ago such Type 2
QSOs had not been observed, from which it was con-
cluded that either no dust torus is present in QSOs due
to the high luminosity (and therefore no Type 2 QSOs
exist) or that Type 2 QSOs are not easy to identify.
This question has finally now been settled: the cur-
rent X-ray satellites Chandra and XMM-Newton have
identified the population of Type 2 QSOs. Due to the
high column density of hydrogen which is distributed in
the torus together with the dust, low-energy X-ray radia-
tion is almost completely absorbed by the photoelectric
effect if the line-of-sight to the center of these sources
passes through the obscuring torus. These sources were
therefore not visible for ROSAT (E < 2.4 keV), but the
energy ranges of Chandra and XMM-Newton finally al-
lowed the X-ray detection and identification of these
Type 2 QSOs.
Another candidate for Type 2 QSOs are the ultra-
luminous infrared galaxies (ULIRGs), in which extreme
IR-luminosity is emitted by large amounts of warm dust
which is heated either by very strong star formation
or by an AGN. Since ULIRGs have total luminosi-
5.5 Family Relations of ACNs
-:v\V; ; ' ;:; "
eel. g|S % !
VrlQ'^J*-
.I- * i ■>'<",
» - . # j ^
X~ r ?'*,-=. ■
ni i in
. '~\ \% Ij Torus
Nar-r^Jitae^TY
Disk
Broad Line
^ .."£.'*;/
, 'j&'SUtfp.
'".'./- :.■■'... -
Fig. 5.28. Sketch of our current understanding of the unifi-
cation of AGN types. The accretion disk is surrounded by
a thick torus containing dust which thus obscures the view to
the center of the AGN. When looking from a direction neai
the plane of the disk, a direct view of the continuum source
and the BLR is blocked, whereas it is directly visible from
directions closer to the symmetry axis of the disk. The dif-
ference between Seyfert 1 (and BLRG) and Seyfcri 2 (and
NLRG) is therefore merely a matter of orientation relative to
the line-of-sight. If an AGN is seen exactly along the jet axis,
it appears as a blazar
Contours = UV/Red Color
Vectors = X4260 Polarization
Fig. 5.29. The contours show the color of the optical e:
in the Seyfert 2 galaxy NGC 1068, namely the flux ratio in
the U- and R-bands. The sticks indicate the strength and
orientation of the polarization in B-band light. The center of
the galaxy is located at Act = = AS. At its bluest (center
left), the polarization of the optical emission is strongest and is
perpendicular to the direction to the center of the galaxy; this
is the direction of polarization expected for local scattering by
electrons. Hence, where the scattering is strongest, iiie largest
fraction of direct light from the AGN is also observed, and the
optical spectrum of AGNs is considcrabh bluer than the stellar
light from galaxies
ties comparable to QSOs, the latter interpretation is
possible. In fact, distinguishing between the two possi-
bilities is not easy for individual ULIRGs, and in many
sources indicators of both strong star formation and
non-thermal emission (e.g., in the form of X-ray emis-
sion) are found. This discovery indicates that in many
5. Active Galactic Nuclei
Ground-Based Optical/Radio-lmage
HST Image of a Gas and Dust Disk
Fig. 5.30. The elliptical
galaxy NGC4261. The
left-hand panel shows
an optical image of this
galaxy together with the
radio emission (shown in
orange). An HST image
showing the innermost
region of the galaxy is
shown on the right. The
jet is virtually perpendic-
ular to the central disk
of gas and dust, which
is in agreement with the
theoretical picture in the
context of a unification
model
1 .7 arcsec
400 Light-Years
objects, the processes of strong star formation and accre-
tion onto a SMBH are linked. For both processes, large
amounts of gas are necessary, and the fact that both star-
burst galaxies and AGNs are often found in interacting
galaxies, where the disturbance in the gravitational field
provides the conditions for a gas flow into the center of
the galaxy, suggests a link between the two phenomena.
Next we will examine how blazars fit into this uni-
fied scheme. A first clue comes from the fact that all
blazars are radio sources. Furthermore, in our inter-
pretation of superluminal motion (Sect. 5.3.3) we saw
that the appearance and apparent velocity of the cen-
tral source components depend on the orientation of the
source with respect to us, and that it requires relativistic
velocities of the source components. To obtain an in-
terpretation of the blazar phenomenon that fits into the
above scheme, we first need to discuss an effect that
results from Special Relativity.
5.5.2 Beaming
Due to relativistic motion of the source components
relative to us, another effect occurs, known as beaming.
Due to beaming, the relation between source luminosity
and observed flux from a moving source depends on its
velocity with respect to the observer. One aspect of this
phenomenon is the Doppler shift in frequency space:
the measured flux at a given frequency is different from
that of a non-moving source because the measured fre-
quency corresponds to a Doppler-shifted frequency in
the rest-frame of the source. Another effect described by
Special Relativity is that a moving source which emits
isotropically in its rest-frame has an anisotropic emis-
sion pattern, with the angular distribution depending on
its velocity. The radiation is emitted preferentially in the
direction of the velocity vector of the source (thus, in the
forward direction), so that a source will appear brighter
if it is moving towards the observer. In Sect. 4.3.2, we
already mentioned the relation (4.44) between the radi-
ation intensity in the rest-frame of a source and in the
system of the observer. Due to the strong Doppler shift,
this implies that a source moving towards us appears
brighter by a factor
£>+ =
1
(5.31)
-/Jcos^)/
t, where a is the spectral index. Fur-
Vxd-
than the source ai
thermore, yS = v/c, <p is the angle between the velocity
vector of the source component and the line-of-sight
to the source, and the Lorentz factor y = (1 — p 2 )~ l/1
has already been defined in Sect. 5.3.3. Even at weakly
5.5 Family Relations of ACNs
relativistic velocities (fi ~ 0.9) this can already be a con-
siderable factor, i.e., the radiation from the relativistic
jet may appear highly amplified. Another consequence
of beaming is that if a second jet exists which is moving
away from us (the so-called counter-jet), its radiation
will be weakened by a factor
£>_ =
1
Vk(1 + £cos</>)/
(5.32)
relative to the stationary source. Obviously, £>_ can
be obtained from £) + by replacing by (j) + n, since
the counter-jet is moving in the opposite direction. In
particular, the flux ratio of jet and counter-jet is
£>+
n + /?cos0 \
U-0COS0/
(5,33)
and this factor may easily be a hundred or more
(Fig. 5.31). The large flux ratio (5.33) for relativistic jets
is the canonical explanation for VLBI jets being virtu-
ally always only one-sided. This effect is also denoted
as "Doppler favoritism" - the jet pointing towards us is
observed preferentially because of the beaming effect
and the resulting amplification of its flux.
\ ' '
' I ' I ' I ' I ' I ' I ■ I
7=15
-
5
■
S
7-
«
7-5
CD
y=2
l
3
2
"~""~-- ^^
_
-
1
^^=^^
-
10 20 30 40 50 60 70 80 90
Viewing angle, c&
Fig. 5.31. The logarithm of the flux ratio of jet and counter-
jet (5.33) is plotted as a function of the angle <f> for different
values of the Lorentz factor y. Even at relatively small values
of y, this ratio is large if <j> is close to 0, but even at <p ~ 30°
111 ratio i iill ippicci bl< Hence the plot shows the Doppler
favoritism and explains why, in most compact radio AGNs,
one jet is visible but the counter-jet is not
Beaming and the Blazar Phenomenon. If we observe
a source from a direction very close to the jet axis and
if the jet is relativistic, its radiation can outshine all
other radiation from the AGN because <£>+ can become
very large in this case. Especially if the beamed radia-
tion extends into the optical/UV part of the spectrum,
the line emission may also become invisible relative
to the jet emission, and the source will appear to us
as a BL Lac object. If the line radiation is not out-
shined completely, the source may appear as an OVV.
The synchrotron nature of the optical light is also the
explanation for the optical polarization of blazars since
synchrotron emission can be polarized, in contrast to
thermal emission.
The strong beaming factor also provides an explana-
tion for the rapid variability of blazars. If the velocity
of the emitting component is close to the speed of light,
P < 1, even small changes in the jet velocity or its di-
rection may noticeably change the Doppler factor <©+.
Such small changes in the direction are expected be-
cause there is no reason to expect a smooth outflow of
material along the jet at constant velocity. In addition,
we argued that, very probably, magnetic fields play an
important role in the generation and collimation of jets.
These magnetic fields are toroidally spun-up, and emit-
ting plasma can, at least partially, follow the field lines
along helical orbits (see Fig. 5.32).
Hence beaming can explain the dominance of radi-
ation from the jet components if the gas is relativistic.
and also the absence or relative weakness of emission
lines. At the same time, it provides a plausible scenario
for the strong variability of blazars. The relative strength
of the core emission and the extended radio emission
depends heavily on the viewing direction. In blazars,
a dominance of the core emission is expected, which is
exactly what we observe.
5.5.3 Beaming on Large Scales
A consequence of this model is that the jets on kpc
scales, which are mainly observed by the VLA, also
need to be at least semi-relativistic: kiloparsec-scale
jets are in most cases also one-sided, and they are al-
ways on the same side of the core as the VLBI jet on
pc scales. Thus, if the one-sidedness of the VLBI jet is
caused by beaming and the corresponding Doppler fa-
Fig. 5.32. Illustration of the relativistic jet model. The ac-
celeration of the jet to velocities close to the speed of light
is probably caused by a combination of very strong gra\ ila
tional fields in the vicinity of the SMBH and strong magnetic
fields which are rotating rapidly because they are anchored in
the accretion disk. Shock fronts within the jet lead to acceler-
ation processes of relativistic electrons, which then strongly
radiate and become visible as "blobs" in the jets. By rotation
of the accretion disk in which the magnetic field lines are an-
chored, the field lines obtain a characteristic helical shape. It
is supposed that this process is responsible for the focusing
(collimation) of the jet
voritism of an otherwise intrinsically symmetric source,
the one-sidedness of large-scale jets should have the
same explanation, implying relativistic velocities for
them as well. These do not need to be as close to c as
those of the components that show superluminal mo-
tion, but their velocity should also be at least a few
tenths of the speed of light. In addition, it follows that
the kiloparsec-scale jet is moving towards us and is
therefore closer to us than the core of the AGN; for the
counter-jet we have the opposite case. This prediction
can be tested empirically, and it was confirmed in polar-
ization measurements. Radiation from the counter-jet
crosses the ISM of the host galaxy, where it experiences
additional Faraday rotation (see Sect. 2.3.4). It is in fact
observed that the Faraday rotation of counter-jets is sys-
tematically larger than that of jets. This can be explained
by the fact that the counter-jet is located behind the host
galaxy and we are thus observing it through the gas of
that galaxy.
5.5.4 Jets at Higher Frequencies
Optical Jets. In Sect. 5.1.2, we discussed the radio
emission of jets, and Sect. 5.3.3 described how their
relativistic motion is detected from their structural
changes, i.e., superluminal motion. However, jets are
not only observable at radio frequencies; they also
emit at much shorter wavelengths. Indeed, the first two
jets were detected in optical observations, namely in
QSO 3C273 (Fig. 5.33) and in the radio galaxy M87
(Fig. 5.34), as a linear source structure pointing radi-
ally away from the core of the respective galaxy. With
the commissioning of the VLA (Fig. 1.21) as a sensitive
and high-resolution radio interferometer, the discovery
and examination of hundreds of jets at radio frequencies
became possible.
The HST, with its unique angular resolution, has de-
tected numerous jets in the optical (see also Fig. 5.12).
They are situated on the same side of the correspond-
ing AGNs as the main radio jet. Optical counterparts of
radio counter-jets have not been detected thus far. Op-
tical jets are always shorter, narrower, and show more
structure than the corresponding radio jets. The spec-
trum of optical jets follows a power law (5.2) similar to
that in the radio domain, with an index a that describes,
in general, a slightly steeper spectrum. In some cases,
linear polarization in the optical jet radiation of ~ 10%
was also detected. If we also take into account that the
positions of the knots in the optical and in the radio jets
agree very well, we inevitably come to the conclusion
that the optical radiation is also synchrotron emission.
This conclusion is further supported by a nearly con-
stant flux ratio of radio and optical radiation along
the jets.
As was mentioned in Sect. 5.1.3, the relativistic elec-
trons that produce the synchrotron radiation lose energy
by emission. In many cases, the cooling time (5.6)
of the electrons responsible for the radio emission is
longer than the time of flow of the material from the
central core along the jet, in particular if the flow is
(semi-)relativistic. It is thus possible that relativistic
electrons are produced or accelerated in the immedi-
ate vicinity of the AGN and are then transported away
by the jet. This is not the case for those electrons produc-
ing the optical synchrotron radiation, however, because
the cooling time for emission at optical wavelengths is
only f cool ~ 10 3 (S/10 -4 G) yr. Even if the relativistic
5.5 Family Relations of ACNs
Fig. 5.33. Jets are visible not only in the radio domain but ii
some cases also at other wavelengths. Left: an HST image o
the quasar 3C273 is shown, with the point-like quasar in th(
center and (displaj ed in blue I jet-shaped optical ei
spatially coincides with the radio jet (displayed in red). Right:
an X-ray image of this quasar taken by the Chandra satellite.
The jet is also visible at very high energies
VLBA
Radio
Fig. 5.34. Top left: a radio map of M87, the central galaxy
in the Virgo Cluster of galaxies. Top right: an HST image
of the region shown in the inset of the left-hand panel. The
radio jet is also visible at optical wavelengths. The lower
image shows a VLBI map of the region around the galaxy
core; the jet is formed within a few 10 17 cm from the core
of the galaxy, which presumably contains a black hole of
M. ~ 3 x 10 9 M o . Very close to the center the opening angle
of the jet is significantly larger than further out. This indicates
that the jet only becomes collimated at a larger distance
5. Active Galactic Nuclei
electrons are transported in a (semi-)relativistic jet, they
cannot travel more than a distance of ~ 1 kpc before los-
ing their energy. The observed length of optical jets is
much larger, though. For this reason, the corresponding
electrons cannot be originating in the AGN itself but in-
stead must be produced locally in the jet. The knots in
the jets, which are probably shock fronts in the outflow,
are thought to represent the location of the accelera-
tion of relativistic particles. Quantitative estimates of
the cooling time are hampered by the unknown beam-
ing factor (5.31). Since optical jets are all one-sided,
and in most cases observed in radio sources with a flat
spectrum, a very large beaming factor is generally as-
sumed. Transforming back into the rest-frame of the
electrons yields a lower frequency and a lower luminos-
ity. Since the latter is utilized for estimating the strength
of the magnetic fields (by assuming equipartition of
energy, for instance), this also changes the estimated
cooling time.
X-Ray Radiation of Jets. The Chandra satellite dis-
covered that many of the jets which had been identified
in the radio are also visible in X-ray light (Fig. 5.35).
This came as a real surprise. This discovery and the
strong correlation of the spatial distribution of radio,
optical, and X-ray emission imply that they must all
originate from the same regions in the jets, i.e., that the
origins of the emission must be linked to each other. As
we have discussed, radio and optical radiation originate
from synchrotron emission, the emission by relativistic
electrons moving in a magnetic field. The same elec-
trons that are responsible for the radio emission can
also produce X-ray photons by inverse Compton scat-
tering. In this process, low-energy photons are scattered
to much higher energies by collisions with relativistic
electrons - a photon of frequency v may have a fre-
quency v' «s y 2 v after being scattered by an electron
of energy ym e c 2 . Since the characteristic Lorentz fac-
tors of electrons causing the synchrotron radiation of
radio jets may reach values of y ~ 10 4 , these elec-
trons may scatter, by inverse Compton scattering, radio
photons into the X-ray domain of the spectrum. This
effect is also called synchrotron self-Compton radia-
tion. Alternatively, relativistic electrons can also scatter
optical photons from the AGN, for which less ener-
getic electrons are required. The omnipresent CMB
may also be considered as a photon source for the in-
verse Compton effect, and in many cases the observed
X-ray radiation is probably Compton-scattered CMB
radiation.
The inverse Compton model cannot, however, be
applied to all X-ray jets without serious problems oc-
curring. For instance, variability in X-ray emission was
observed in the knots of M87, indicating a very short
cooling time for the electrons. Since the electrons must
have a much larger Lorentz factor y if the radiation, at
Fig. 5.35. X-ray images of AGN jets. Left: a Chandra image of population of relativistic electrons. Right: a Chandra image of
the jet in the QSO PKS 1 127- 145, with overlaid contours of the active galaxy Centaurus A. Here the jet is visible, as well
radio emission (1 .4 cm, VLA). The direction of the jet and its as a large number of compact sources interpreted to be X-ray
substructure are very similar at both wavelengths, suggesting binaries
an interpretation in which the radiation is caused by the same
5.6 ACNs and Cosmology
a given frequency, originates from synchrotron emis-
sion instead of by inverse Compton scattering, their
cooling time t coo \ (5.6) would be much shorter as well.
In such sources, which are typically FR I radio sources,
the synchrotron process itself probably accounts for the
X-ray emission. This implies, on the one hand, very
short cooling time-scales and therefore the increased
necessity for a local acceleration of the electrons. On
the other hand, the required energies for the elec-
trons are very high, ~ 100 TeV. It is currently unclear
which acceleration processes may account for these
high energies.
Detecting radio jets at X-ray frequencies seems to be
a frequent phenomenon: about half of the flat-spectrum
radio QSO with jet-like extended radio emission also
show an X-ray jet. All of those are one-sided, although
the corresponding radio images often show lobes oppo-
site the X-ray jets, reinforcing the necessity for Doppler
favoritism also in the X-ray waveband.
Finally, it should be mentioned that our attempts at
finding a unification scheme for the different classes of
AGNs have been quite successful. The scheme of unifi-
cation is generally accepted, even though some aspects
are still subject to discussion. One particular model is
sketched in Fig. 5.36.
5.6 AGNs and Cosmology
AGNs, and QSOs in particular, are visible out to very
high redshifts. Since their discovery in 1963, QSOs
have held the redshift record nearly without interrup-
tion. Only in recent years have QSOs and galaxies been
taking turns in holding the record. Today, several hun-
dred QSOs are known with z > 4, and the number of
those with z > 5 continues to grow since a criterion was
found to identify these objects. This leads to the possi-
bility that QSOs could be used as cosmological probes,
and thus to the question of what we can learn about the
Universe from QSOs. For example, one of the most ex-
citing questions is how does the QSO population evolve
with redshift - was the abundance of QSO at high red-
shifts, i.e., at early epochs of the cosmos, similar to that
today, or does it evolve over time?
5.6.1 The K-Correction
To answer this question, we must know the luminosity
function of QSOs, along with its redshift dependence.
As we did for galaxies, we define the luminosity
function @(L,z)dL as the spatial number density of
Optical
emission line properties
1
Type 2
(narrow lines)
Type 1 Type
(broad lines) (unusual)
Sy2
Sy1
radio-quiet
Type-2-QSO
E
IR-Quasar?
QSO BAL QSO?
1
radio-loud
/FR 1
NLRG <
N FRII
BLRG /BLLac
QQpn Blazars \
bbHU X (FSRQ)
FSRQ I
Decreasing angle to line of sight
Fig. 5.36. This table presents a unification scheme for AGNs
via the angular momentum of the central black hole and the
orientation of the accretion disk with respect to the line-of-
sight. The closer the direction of the jet is to the line-of-
sight, the more the jet component dominates. Furthermore,
the relative strength of the radio emission in this particular
unified scheme is linked to the angular momentum of the
black hole. The classification pattern shown here is only one
of several possibilities, but the dependence of the AGN class
is generally considered to be accepted
5. Active Galactic Nuclei
QSOs with luminosity between L and L + AL. nor-
mally refers to a comoving volume element, so that
a non-evolving QSO population would correspond to
a z-independent <P. One of the problems in determin-
ing is related to the question of which kind of
luminosity is meant here. For a given observed fre-
quency band, the corresponding rest-frame radiation of
the sources depends on their redshift. For optical obser-
vations, the measured flux of nearby QSOs corresponds
to the rest-frame optical luminosity, whereas it corre-
sponds to the UV luminosity for higher-redshift QSOs.
In principle, using the bolometric luminosity would be
a possible solution; however, this is not feasible since
it is very difficult to measure the bolometric luminosity
(if at all possible) due to the very broad spectral distri-
bution of AGNs. Observations at all frequencies, from
the radio to the gamma domain, would be required, and
obviously, such observations can only be obtained for
selected individual sources.
Of course, the same problem occurs for all sources
at high redshift. In comparing the luminosity of gal-
axies at high redshift with that of nearby galaxies, for
instance, it must always be taken into account that, at
given observed wavelength, different spectral ranges in
the galaxies' rest-frames are measured. This means in
order to investigate the optical emission of galaxies at
Z ~ 1, observations in the NIR region of the spectrum
are necessary.
Frequently the only possibility is to use the lumi-
nosity in some spectral band and to compensate for
the above effect as well as possible by performing ob-
servations in several bands. For instance, one picks as
a reference the blue filter which has its maximum effi-
ciency at ~ 4500 A and measures the blue luminosity
for nearby objects in this filter, whereas for objects at
redshift z ~ 1 the intrinsic blue luminosity is obtained
by observing with the /-band filter, and for even larger
redshifts observations need to be extended into the near-
IR. The observational problems with this strategy, and
the corresponding corrections for the different sensitiv-
ity profiles of the filters, must not be underestimated
and are always a source of systematic uncertainties. An
alternative is to perform the observation in only one
(or a few) filters and to approximately correct for the
redshift effect.
In Sect. 4.3.3, we defined various distance measures
in cosmology. In particular, the relation S — L/{AjiD\)
between the observed flux S and the luminosity L
of a source defines the luminosity distance D L . Here
both the flux and the luminosity refer to bolometric
quantities, i.e., flux and luminosity integrated over all
frequencies. Due to the redshift, the measured spec-
tral flux S v is related to the spectral luminosity L v i at
a frequency v' — v(l + z), where one finds
AnDt
e this relation in a slightly different form.
(5.34)
S v =
XirDi
-(1 + z)
(5.35)
where the first factor is of the same form as in the relation
between the bolometric quantities while the second fac-
tor corrects for the spectral shift. This factor is denoted
the K-correction. It obviously depends on the spectrum
of the source, i.e., to determine the K-correction for
a source its spectrum needs to be known. Furthermore,
this I actor depends on the filter used. Since in opti-
cal astronomy magnitudes are used as a measure for
brightness, (5.35) is usually written in the form
m int = m ohs + K(z)
with K(z) = -2.5 log |^(1 + z) . (5-36)
where m int is the magnitude that would be measured in
the absence of redshift, and m obs describes the bright-
ness actually observed. The K-correction is not only
relevant for QSOs but for all objects at high redshift, in
particular also for galaxies.
5.6.2 The Luminosity Function of Quasars
By counting QSOs, we obtain the number density
N( > S) of QSOs with a flux larger than S. We find a rela-
tion of roughly N(> S) oc S~ 2 for large fluxes, whereas
the source counts are considerably flatter for smaller
fluxes. The flux at which the transition from steep counts
to flatter ones occurs corresponds to an apparent mag-
nitude of about B ~ 19.5. Up to this magnitude, about
10 QSOs per square degree are found.
From QSO number counts, combined with mea-
surements of QSO redshifts, the luminosity function
0(L,z) can be determined. As already defined above,
5.6 AGNs and Cosmology
<P(L, z) dL is the number density in a comoving vol-
ume element of QSOs at redshift z with a luminosity
between L and L + dL.
Two fundamental problems exist in determining the
luminosity function. The first is related to the above
discussion of wavelength shift due to cosmological red-
shift: a fixed wavelength range in which the brightness
is observed corresponds to different wavelength inter-
vals in the intrinsic QSO spectra, depending on their
redshift. We need to correct for this effect if the number
density of QSOs above a given luminosity in a cer-
tain frequency interval is to be compared for local and
distant QSOs. One way to achieve this is by assuming
a universal spectral shape for QSOs; over a limited spec-
tral range (e.g., in the optical and the UV ranges), this
assumption is indeed quite well satisfied. This univer-
sal spectrum is obtained by averaging over the spectra
of a larger number of QSOs (Fig. 5.2). By this means,
a useful K-correction of QSOs as a function of redshift
can then be derived.
The second difficulty in determining <P(L, z) is to
construct QSO samples that are "complete". Since
QSOs are point-like they cannot be distinguished from
stars by morphology on optical images, but rather only
by their color properties and subsequent spectroscopy.
However, with the star density being much higher than
that of QSOs, this selection of QSO candidates by color
criteria, and subsequent spectroscopic verification, is
very time-consuming. Only more recent surveys, which
image large areas of the sky in several filters, were
sufficiently successful in their color selection and sub-
sequent spectroscopic verification, so that very large
QSO samples could be compiled. An enormous increase
in statistically well-defined QSO samples was achieved
by two large surveys with the 2dF spectrograph and the
Sloan Digital Sky Survey. We will discuss this in the
context of galaxy redshift surveys in Sect. 8.1.2.
The luminosity function that results from such
analyses is typically parametrized as
i.e., for fixed z, <P is a double power law in L. At
L J?> L*(z), the first term in the square brackets in (5.37)
dominates if a > /}, yielding <$ oc L~ a . On the other
hand, the second term dominates for L <$; L*(z), so that
<P oc L~^. Typical values for the exponents are a » 3.9,
» 1.5. The characteristic luminosity L*(z) where the
/.-dependence changes, strongly depends on redshift.
A good fit to the data for z < 2 is achieved by
L*(z) = L* (l + z) k ,
(5.38)
with k s» 3.45, where the value of k depends on the
assumed density parameters Q m and Q A . This approx-
imation is valid for z < 2, whereas for larger redshifts
L*(z) seems to vary less with z. The normalization con-
stant is determined to be <P* «s 5.2 x 10 3 h 3 Gpc -3 , and
Lq corresponds to roughly M B — — 20.9 + 5 log h. The
luminosity function as determined from an extensive
QSO survey is plotted in Fig. 5.37.
With this form of the luminosity function, a number
of conclusions can be drawn. The luminosity function
of QSOs is considerably broader than that of galaxies,
which we found to decrease exponentially for large L.
The strong dependence of the characteristic luminos-
ity L*(z) on redshift clearly shows a very significant
cosmological evolution of the QSO luminosity func-
Fig. 5.37. The luminosity function of QSOs in six redshift
intervals in the range 0.4 <z<2.1, determined from the 2dF
QSO Redshift Survey by spectroscopy of more than 23 000
QSOs. The dotted curves represent the best fit to the data that
is achieved by a double power law as in (5.37) where the data
have been corrected for the selection function of the survey.
I hi iiii i i in QSO d> iv " \ nli ni« i :a inu n .1 hili i < i< ni;
visible. The dashed line denotes the formal separation between
Seyfert galaxies and QSOs
5. Active Galactic Nuclei
tion. For example, at z ~ 2, L*(z) is about 50 times
larger than today. Furthermore, for high luminosities,
ex [L*(z)] < *~ 1 L~ a . This means that the spatial num-
ber density of luminous QSOs was more than 1 000 times
larger at z ~ 2 than it is today (see Fig. 5.38). Another
way of seeing this is that the low-redshift luminosity
function in Fig. 5.37 does not extend to the very bright
luminosities for which the luminosity function at high
redshifts was measured. The reason for this is that the
number density of very luminous QSOs at low redshifts
is so small that essentially none of them are contained
in the survey volume from which the results in Fig. 5.37
were derived.
For redshifts z > 3, the evolution of the QSO pop-
ulation seems to turn around, i.e., the spatial density
apparently decreases again. The exact value of z at
which the QSO density attains its maximum is still
somewhat uncertain because of the difficulties in ob-
taining a complete sample of QSOs at high redshift.
Since a redshift z ~ 3 corresponds to an epoch where
the Universe had only about 20% of its current age (the
exact value depends on the cosmological parameters),
a kind of "QSO epoch" seems to have occurred, in the
sense that the QSO population seem to have quickly
formed and then largely became extinct again.
Redshift
2 4 6 8 10 12
Cosmic Time (Gyr)
Fig. 5.38. The relative spatial density of QSOs as a function
of the age of the Universe. We can see that the QSO density
has a well-defined narrow maximum which corresponds to
a redshift of about z ~ 2.5; towards even larger redshifts, the
density seems to decrease again. This plot suggests the notion
of a "QSO epoch"
There are several possible interpretations of the QSO
luminosity function and its redshift dependence. One of
them is that the luminosity of any one QSO varies in
time, parallel to the evolution of L* (z). Most likely this
interpretation is wrong because it implies that a lumi-
nous QSO will always remain luminous. Although the
efficiency of energy conversion into radiation is much
higher for accretion than for thermonuclear burning, an
extremely high mass would nevertheless accumulate in
this case. This would then be present as the mass of the
SMBH in local QSOs. 8 However, estimates of M. in
QSOs rarely yield values larger than ~ 3 x 10 9 M Q .
However, it is by no means clear that a given source
will be a QSO throughout its lifetime: a source may be
active as a QSO for a limited time, and later appear as
a normal galaxy again. For instance, it is possible that
virtually any massive galaxy hosts a potential AGN.
This supposition is clearly supported by the fact that
apparently all massive galaxies harbor a central SMBH.
If the SMBH is fed by accreting matter, this galaxy
will then host an AGN. However, if no more mass is
provided, the nucleus will cease to radiate and the galaxy
will no longer be active. Our Milky Way may serve
as an example of this effect, since although the mass
of the SMBH in the center of the Galaxy would be
sufficient to power an AGN luminosity of more than
10 44 erg/s considering its Eddington luminosity (5.23),
the observed luminosity is lower by many orders of
magnitude.
AGNs are often found in the vicinity of other galax-
ies. One possible interpretation is that the neighboring
galaxy disturbs the gravitational field of the QSO's host,
such that it allows its interstellar medium to flow into
the center of the host galaxy where it accretes onto the
central black hole - and "the monster starts to shine". If
this is the case, the luminosity function (5.37) does not
provide information about individual AGNs, but only
about the population as a whole.
Interpreting the redshift evolution then becomes ob-
vious. The increase in QSO density with redshift in the
scenario described above originates from the fact that
at earlier times in the Universe interactions between
galaxies and merger processes were significantly more
frequent than today. On the other hand, the decrease
^Compare the mas-, estimate in Sett. t.3. I where, instead of 10 7 yr,
the lifetime to be inserted here is the age of the Universe, ~ 10 10 yr.
5.6 ACNs and Cosmology
at very high z is to be expected because the SMBHs
in the center of galaxies first need to form, and this ob-
viously happens in the first ~ 10 9 years after the Big
Bang.
5.6.3 Quasar Absorption Lines
The optical/UV spectra of quasars are characterized by
strong emission lines. In addition, they also show ab-
sorption lines, which we have not mentioned thus far.
Depending on the redshift of the QSOs, the wavelength
range of the spectrum, and the spectral resolution, QSO
spectra may contain a large variety of absorption lines.
In principle, several possible explanations exist. They
may be caused by absorbing material in the AGN itself
or in its host galaxy, so they have an intrinsic origin.
Alternatively, they may arise during the long journey
between the QSO and us due to intervening gas along
the line-of-sight. We will see that different kinds of ab-
sorption lines exist, and that both of these possibilities
indeed occur. The analysis of those absorption lines
which do not have their origin in the QSO itself pro-
vides information about the gas in the Universe. For
this purpose, a QSO is basically a very distant bright
light source used for probing the intervening gas.
This gas can be either in intergalactic space or is cor-
related with foreground galaxies. In the former case,
we expect that this gas is metal-poor and thus consists
mainly of hydrogen and helium. Furthermore, in order
to cause absorption, the intergalactic medium must not
be fully ionized, but needs to contain a fraction of neu-
tral hydrogen. Gas located closer to galaxies may be
expected to also contain appreciable amounts of metals
which can give rise to absorption lines.
The identification of a spectral line with a spe-
cific line transition and a corresponding redshift is,
in general, possible only if at least two lines occur
at the same redshift. For this reason, doublet transi-
tions are particularly valuable, such as those of Mgll
(A = 2795 A and A = 2802 A), and Civ (k = 1548 A
and X = 1551 A). The spectrum of virtually any QSO at
high (emission line) redshift z em shows narrow absorp-
tion lines by Civ and Mgll at absorption line redshifts
Zabs < Zem- If the spectral coverage extends to shorter
wavelengths than the observed Lya emission line of
the QSO, numerous narrow absorption lines exist at
^obs < ^obsCLya) = (1 + Zem) 1216 A. The set of these
absorption lines is denoted as the Lyman-a forest. In
about 15% of all QSOs, very broad absorption lines
are found, the width of which may even considerably
exceed that of the broad emission lines.
Classification of QSO Absorption Lines. The different
absorption lines in QSOs are distinguished by classes
according to their wavelength and width.
• Metal systems: In general these are narrow absorp-
tion lines, of which Mgll and Civ most frequently
occur (and which are the easiest to identify). How-
ever, in addition, a number of lines of other elements
exist (Fig. 5.39). The redshift of these absorption
lines is < z a bs < z em ; therefore they are caused by
intervening matter along the line-of-sight and are
not associated with the QSO. Normally a metal sys-
:s of many different lines of different ions,
I 1.6 ■
cr_
I
MC 1331 +17
k
« - T /
frf f
I
f idp
^kI/ifT
3600 3800
Wavelength (A)
Fig. 5.39. Spectrum of the QSO 1331+17
at Zem = 2.081 observed by the Multi-
Mirror Telescope in Arizona. In the
spectrum, a whole series of absorption
lines can be seen which have all been
identified with gas at Zabs = 1.776. The
corresponding Lya line at X w 3400 A is
very broad; it belongs to the damped Lya
Fig. 5.40. Keck spectrum of the Lyman-cc forest towards QSO variety of narrow absorption lines of neutral hydrogen in
1422+231, aQSOatz = 3.62. As an aside, this is a quadruply- the intergalactic medium is visible. The statistical analysis
imaged lensed QSO. The wavelength resolution is about of these lines pro\ ides information on the gas distribution in
7 km/s. On the blue side of the Lya emission line, a large the Universe (see Sect. 8.5)
all at the same redshift. From the line strength, the
column density of the absorbing ions can be de-
rived. For an assumed chemical composition and
degree of ionization of the gas, the corresponding
column density of hydrogen can then be deter-
mined. Estimates for such metal systems yield typical
values of 10 17 cm -2 < N H < 10 21 cm~ 2 , where the
lower limit depends on the sensitivity of the spectral
observation,
i Associated metal systems: These systems have char-
acteristics very similar to those of the aforementioned
intervening metal systems, but their redshift is
Zabs ~ Zem- Since such systems are over-abundant
compared to a statistical z-distribution of the metal
systems, these systems are interpreted as belonging
to the QSO itself. Thus the absorber is physically as-
sociated with the QSO and may be due, for example,
to absorption in the QSO host galaxy or in a galaxy
associated with it.
i Lya forest: The large set of lines at X < (1 +
z em ) 1216 A, as shown in Fig. 5.40, is interpreted to
be Lya absorption by hydrogen along the line-of-
sight to the QSO. The statistical properties of these
lines are essentially the same for all QSOs and seem
to depend only on the redshift of the Lya lines, but
not on z em - This interpretation is confirmed by the
fact that for nearly any line in the Lya forest, the cor-
responding Ly/? line is found if the quality and the
wavelength range of the observed spectra permit this.
The Lya forest is further subdivided, according to the
strength of the absorption, into narrow lines, Lyman-
limit systems, and damped Lya systems. Narrow Lya
lines are caused by absorbing gas of neutral hydrogen
column densities of Na < 10 17 cm -2 . Lyman-limit
systems derive their name from the fact that at col-
umn densities of TVh ^ 10 17 cm -2 , neutral hydrogen
almost totally absorbs all radiation at A. < 912 A
(in the hydrogen rest-frame), where photons ion-
4500 5000 5500
Wavelength (A)
Fig. 5.41. A Lyman-limit sys-
tem along the line-of-sight
towards the QSO 2000-330
is absorbing virtually all radia-
tion at wavelengths A. < 912 A
in the rest-frame of the ab-
sorber, here redshifted to about
4150A
ize hydrogen (Fig. 5.41). If such a system is located
at ziimit in the spectrum of a QSO, the spectrum
at X < (1 + ziimit) 912 A is almost completely sup-
pressed. Damped Lya systems occur if the column
density of neutral hydrogen is /V H > 2 x 10 20 cm -2 .
In this case, the absorption line becomes very broad
due to the extended damping wings of the Voigt
profile. 9
» Broad absorption lines: For about 15% of the QSOs,
very broad absorption lines are found in the spectrum
at redshifts slightly below z em (Fig. 5.42). The lines
show a profile which is typical for sources with out-
flowing material, as seen, for instance, in stars with
stellar winds. However, in contrast to the latter, the
Doppler width of the lines in the broad absorption
line (BAL) QSOs is a significant fraction of the speed
of light.
Interpretation. The metal systems with a redshift sig-
nificantly smaller than z em originate either in overdense
regions in intergalactic space or they are associated with
galaxies (or more specifically, galaxy halos) located
along the line-of-sight. In fact, Mgll systems always
seem to be correlated with a galaxy at the same redshift
as the absorbing gas. From the statistics of the angu-
Wavelength (A)
Fig. 5.42. Spectra of three BAL-QSOs, QSOs with broad ab-
sorption lines. On the blue side of every strong emission line
very broad absoi ption is visible, such as can be caused by out-
flowing material. Such line shapes, at much lower width, of
course, are also found in the spectra of stars with strong stellar
"The Voigt prolile <j>{v) of a line, which specilics the spectral encrg)
distribution of the photons around the central frequenc\ n, of the lino,
is the com elation of the intrinsic line profile, described b\ a Lorentz
£/
i ' i» t , M
(v-vo-vov/c) 2 + (r/4it) 2 '
where the integral extends ox er the \ clocih component along the line-
of sight. In these equations, /"is the intrinsic line w idth which results
irom the natural line width (related to the lifetime of the atomic states)
and pressure broadening, m is the mass of the atom, which defines,
together with the temperature T of the gas. the Maxwellian velocity
distribution, i! the natural line width is small compared to the thermal
width, the Doppler profile dominates m the center of the line, that is
for frequencies close to r (l . The line profile is then well approximated
by a Gaussian. In the wings of the line, the Lorentz profile dominates.
For the wings of the line, where 0(v) is small, to become observable
the optical depth needs to be high. This is the case in damped Lya
systems.
lar separations of these associated galaxies to the QSO
sight-line and from their redshifts, we obtain a charac-
teristic extent of the gaseous halos of such galaxies of
~ 25/i" 1 kpc. For Civ systems, the extent seems to be
even larger, ~ 40h~ l kpc.
The Lya forest is caused by the diffuse intergalactic
distribution of gas. In Sect. 8.5, we will discuss models
of the Lya forest and its relevance for cosmology more
thoroughly (see also Fig. 5.43).
Broad absorption lines originate from material in
the AGN itself, as follows immediately from their red-
shift and their enormous width. Since the redshift of the
broad absorption lines is slightly lower than that of the
corresponding emission lines, the absorbing gas must
be moving towards us. The idea is that this is material
flowing out at a very high velocity. BAL-QSOs (broad
absorption line QSOs) are virtually always radio-quiet.
The role of BAL-QSOs in the AGN family is unclear.
A plausible interpretation is that the BAL property also
depends on the orientation of the QSO. In this case, any
5. Active Galactic Nuclei
y^ft^^
820
840
920
Wavelength [nm]
Fig. 5.43. A VLT spectrum of the QSO SDSS 1030+0524 at
z = 6.28, currently one of the highest known QSO redshifts.
The blue side of the Lyce emission line and the adjacent contin-
uum arc almost completely devoured by the dense Lya forest
QSO would be a BAL if observed from the direction
into which the absorbing material streams out.
Discussion. Most absorption lines in QSO spectra are
not physically related to the AGN phenomenon. Rather,
they provide us with an opportunity to probe the matter
along the line-of-sight to the QSO. The Lya forest will
be discussed in relation to this aspect in Sect. 8.5. Fur-
thermore, absorption line spectroscopy of QSOs carried
out with UV satellites has proven the existence of very
hot gas in the halo of our Milky Way. Such UV spec-
troscopy provides one of the very few opportunities to
analyze the intergalactic medium if its temperature is of
the order of ~ 10 6 K - gas at this temperature is very
difficult to detect since it emits in the extreme UV which
is unobservable from our location inside the Milky Way,
and since almost all atoms are fully ionized and there-
fore cause no absorption. Only absorption lines from
very highly ionized metals (such as the five times ion-
ized oxygen) can still be observed. Since the majority
of the baryons should be found in this hot gas phase
today, this test is of great interest for cosmology.
6. Clusters and Groups of Galaxies
Galaxies are not uniformly distributed in space, but
instead show a tendency to gather together in galaxy
groups and clusters of galaxies. This effect can be
clearly recognized in the projection of bright galax-
ies on the sky (see Figs. 6.1 and 6.2). The Milky Way
Fig. 6.1. The distribution ol galaxi . m the Northern sky, as
compiled in the Lick catalog. This catalog contains the galaxy
number counts for "pixels'* of 10' x 10' each. It is clearly seen
thai Ihe distribution of galaxies on lire sphere is fai from being
homogeneous. Instead it is distinct!} structured
Fig. 6.2. The distribution of all galaxies brighter than B < 14.:
on the sphere, plotted in Galactic coordinates. The Zone o
Avoidance is clearly seen as the region near the Galactic plam
itself is a member of a group, called the Local Group
(Sect. 6.1), which implies that we are living in a locally
overdense region of the Universe.
The transition between groups and clusters of gal-
axies is smooth. The distinction is made by the
number of their member galaxies. Roughly speaking,
an accumulation of galaxies is called a group if it
consists of N < 50 members within a sphere of diame-
ter D < 1.5/i" 1 Mpc. Clusters have N > 50 members
and diameters D> 1.5/i -1 Mpc. A formal definition
of a cluster is presented further below. An example of
a group and a cluster of galaxies is displayed in Fig. 6.3.
Clusters of galaxies are the most massive gravitation-
ally bound structures in the Universe. Typical values
for the mass are M > 3 x 10 14 M Q for massive clusters,
whereas for groups M~3x 10 13 M Q is characteris-
tic, with the total mass range of groups and clusters
extending over 10 12 M o < M < 1O 15 M .
Originally, clusters of galaxies were characterized as
such by the observed spatial concentration of galaxies.
Today we know that, although the galaxies determine
the optical appearance of a cluster, the mass contained
in galaxies contributes only a small fraction to the total
mass of a cluster. Through advances in X-ray astron-
omy, it was discovered that galaxy clusters are intense
sources of X-ray radiation which is emitted by a hot
gas (r~3x 10 7 K) located between the galaxies. This
intergalactic gas {inlraclusler medium. ICM) contains
more baryons than the stars seen in the member galax-
ies. From the dynamics of galaxies, from the properties
of the X-ray emission of the clusters, and from the grav-
itational lens effect we deduce the existence of dark
matter in galaxy clusters, dominating the cluster mass
like it does for galaxies.
Clusters of galaxies play a very important role in
observational cosmology. They are the most massive
bound and relaxed (i.e., in a state of approximate
dynamical equilibrium) structures in the Universe, as
mentioned before, and therefore mark the most promi-
nent density peaks of the large-scale structure in the
Universe. Their cosmological evolution is therefore di-
rectly related to the growth of cosmic structures. Due
to their high galaxy density, clusters and groups are
also ideal laboratories for studying interactions between
Peter Schneider. Clusters and ( Ironps uf Galaxies
DOI 1(1 100 i I I , priniri
Cosmology, pp. 223 275 (2006)
Fig. 6.3. Left: HCG40,
a compact group of gal-
axies, observed with
the Subaru telescope
on Mauna-Kea. Right:
the cluster of galaxies
CI 0053-37, observed
with the WFI at the
ESO/MPG 2.2-m tele-
galaxies and their effect on the galaxy population. For
instance, the fact that elliptical galaxies are preferen-
tially found in clusters indicates the impact of the local
galaxy density on the morphology and evolution of
galaxies.
6.1 The Local Group
The galaxy group of which the Milky Way is a mem-
ber is called the Local Group. Within a distance of
~ 1 Mpc around our Galaxy, about 35 galaxies are cur-
rently known; they are listed in Table 6.1. A sketch of
their spatial distribution is given in Fig. 6.4.
6.1.1 Phenomenology
The Milky Way (MW), M3 1 (Andromeda), and M33 are
the three spiral galaxies in the Local Group, and they
are also its most luminous members. The Andromeda
galaxy is located at a distance of 770 kpc from us. The
Local Group member next in luminosity is the Large
Magellanic Cloud (LMC, see Fig. 6.5), which is orbiting
around the Milky Way, together with the Small Magel-
lanic Cloud (SMC), at a distance of ~ 50 kpc (~ 60 kpc,
respectively, for the SMC). Both are satellite galaxies
of the Milky Way and belong to the class of irregular
galaxies (like about 11 other Local Group members).
The other members of the Local Group are dwarf gal-
axies, which are very small and faint. Because of their
low luminosity and their low surface brightness, many
of the known members of the Local Group have been
detected only in recent years. For example, the Antlia
galaxy, a dwarf spheroidal galaxy, was found in 1997.
Its luminosity is about 10 4 times smaller than that of the
Milky Way.
Many of the dwarf galaxies are grouped around the
Galaxy or around M31; these are known as satellite
Fig. 6.4. Schematic distribution of galaxies in the I
Group, with the Milky Way at the center of the figure
6.1 The Local Crc
Table 6.1. Members of the Local Group. Listed are the
of the galaxy, its morphological type, the absolute B
magnitude, its position on the sphere in both right a
sion/declination and in Galactic coordinates, its distance from
the Sun, and its radial velocity. A sketch of the spatial
configuration is displayed in Fig. 6.4
Galaxy
Type
M B
RA/Dec.
e,b
D(kpc)
v r (km/.)
Milky Way
Sbc HI
-20.0
1830-30
0,0
8
LMC
Ir III-IV
-18.5
0524-60
280, -33
50
270
SMC
Ir IV-V
-17.1
0051-73
303, -44
63
163
SgrI
dSph?
1856-30
6,-14
20
140
Fornax
dEO
0237-34
237,-65
138
55
Sculptor Dwarf
dSph
-9.8
0057-33
286, -84
88
110
Leo I
dSph
-11.9
1005+12
226, +49
790
168
Leo II
dSph
-10.1
1110 + 22
220, +67
205
90
Ursa Minor
dSph
-8.9
1508 + 67
105, +45
69
-209
Draco
dSph
-9.4
1719 + 58
86, +35
79
-281
Carina
dSph
-9.4
0640-50
260, -22
Sextans
dSph
-9.5
1010-01
243, +42
86
230
M31
Sb HI
-21.2
0040 + 41
121,-22
770
-297
M32=NGC221
dE2
-16.5
0039 + 40
121,-22
730
-200
M110=NGC205
dE5p
-16.4
0037 + 41
121,-21
730
-239
NGC 185
dE3p
-15.6
0036 + 48
121,-14
620
-202
NGC 147
dE5
-15.1
0030 + 48
120,-14
755
-193
And I
dSph
-11.8
0043 + 37
122, -25
790
—
And II
dSph
-11.8
0113 + 33
129, -29
680
—
And III
dSph
-10.2
0032+36
119,-26
760
—
Cas = And VII
dSph
2326 + 50
109, -09
690
—
Peg=DD0 216
dlr/dSph
-12.9
2328+14
94, -43
760
—
Peg II = And VI
dSph
-11.3
2351+24
106, -36
775
LGS3
dlr/dSph
-9.8
0101+21
126,-41
620
-277
M33
Sc II III
-18.9
0131+30
134,-31
850
-179
NGC 6822
dlr IV-V
-16.0
1942-15
025,-18
500
-57
1C 1613
dlrV
-15.3
0102 + 01
130, -60
715
-234
Sagittarius
dlrV
1927-17
21, +16
1060
-79
WLM
dlr IV-V
-14.4
2359-15
76, -74
945
-116
1C10
dlr IV
-16.0
0017 + 59
119,-03
660
-344
DDO210,Aqr
dlr/dSph
-10.9
2044-13
34,-31
950
-137
Phoenix Dwarf
dlr/dSph
-9.8
272, 68
405
56
Tucana
dSph
-9.6
2241 - 64
323, -48
870
—
Leo A = DDO 69
dlrV
-11.7
0959 + 30
196, 52
800
—
Cetus Dwarf
dSph
-10.1
0026-11
101,-72
775
-
galaxies. Distributed around the Milky Way are the
LMC, the SMC, and nine dwarf galaxies, several of
them in the so-called Magellanic Stream (see Fig. 6.6),
a long, extended band of neutral hydrogen which was
stripped from the Magellanic Clouds about 2 x 10 8 yr
ago by tidal interactions with the Milky Way. The Mag-
ellanic Stream contains about 2 x 10 8 M Q of neutral
hydrogen.
The spatial distribution of satellite galaxies around
the Milky Way shows a pronounced peculiarity, in that
these 1 1 satellites form a highly flattened system. These
satellites appear to lie essentially in a plane which is
oriented perpendicular to the Galactic plane. The satel-
lites around M31 also seem to be distributed in an
anisotropic way around their host. In fact, satellites gal-
axies around spirals seem to be preferentially located
near the short axes of the projected light distribution,
which has been termed the Holmberg effect, although
the statistical significance of this alignment has been
questioned.
6.1.2 Mass Estimate
We will present a simple estimate of the mass of the
Local Group, from which we will find that it is consid-
erably more massive than one would conclude from the
observed luminosity of the associated galaxies.
Fig. 6.5. An image of the Large Magellanic (loud (LMC),
taken with the CTIO 4-m telescope
M3 1 is one of the very few galaxies with a blueshifted
spectrum. Hence, Andromeda and the Milky Way are
approaching each other at a relative velocity of v «
120 km/s. This value results from the velocity of M31
relative to the Sun of v ss 300 km/s, and from the motion
of the Sun around the Galactic center. Together with the
distance to M3 1 of D ~ 770 kpc, we conclude that both
galaxies will collide on a time-scale of ~ 6 x 10 9 yr (if
we disregard the transverse component of the relative
velocity).
The luminosity of the Local Group is dominated by
the Milky Way and by M31, which together produce
about 90% of the total luminosity. If the mass density
follows the light distribution, the dynamics of the Local
Group should also be dominated by these two galaxies.
Therefore, one can try to estimate the mass of the two
galaxies from their relative motion, and with this also
the mass of the Local Group.
In the early phases of the Universe, the Galaxy and
M3 1 were close together and both took part in the Hub-
ble expansion. By their mutual gravitational attraction,
their relative motion was decelerated until it came to
a halt - at a time f max at which the two galaxies had
their maximum separation r max from each other. From
this time on, they have been moving towards each other.
The relative velocity v(t) and the separation r(i) follow
from the conservation of energy,
Galactic Longitude
Fig. 6.6. HI map of a large region in the sky containing the
Magellanic Clouds. This map is part of a large survey of HI,
observed through its 21 -cm line emission, that was performed
with the Parkes telescope in Australia, and which maps about
a quarter of the Southern sky with a pixel size of 5' and a veloc-
ity resolution of ~ 1 km/s. The emission from gas at Galactic
velocities has been removed in this map. Besides the HI emis-
sion by the Magellanic Clouds themselves, gas between them
is visible, the Magellanic Bridge and the Magellanic Stream,
the latter connected to the Magellanic Clouds by an "interface
region". Gas is also found in the direction of the orbital mo-
tion of the Magellanic Clouds around the Milky Way, forming
the "leading arm"
GM
(6.1)
determined by considering (6. 1 ) a
separation, when r = r max and v ■■
the time of maximum
= 0. With this,
where M is the sum of the masses of the Milky Way and
M23 1 , and C is an integration constant. The latter can be
6.1 The Local Cro
follows immediately. Since i
ferential equation for r(t),
It can be solved using the initial condition r — at
t — 0. For our purpose, an approximate consideration
is sufficient. Solving the equation for df we obtain, by
integration, a relation between r max and f max ,
-dr/dt, (6.1) is a dif- 6.1.3 Other Components of the Local Group
-h-J:
'2GM^\/r-\/r m . dx
(6.2)
Since the differential equation is symmetric with
respect to changing u -» — v, the collision will hap-
pen at 2? max . Estimating the time from today to the
collision, by assuming the relative velocity to be
constant during this time, then yields r(to)/v(to) =
D/v — 770 kpc/120 km/s, and one obtains 2f max s» t +
D/v, or
D
(6,3;
age of the Universe. Hence,
where to is the c
together with (6.2) this yields
v 2 GM GM GM
Now by inserting the values r(f ) = D and v = v(to),
we obtain the mass M,
M~3x 1O 12 M
(6.5)
where we have assumed t «s 14 x 10 9 yr. This mass
is much larger than the mass of the two galax-
ies as observed in stars and gas. The mass estimate
yields a mass-to-light ratio for the Local Group of
M/L ~70Mq/L q . This is therefore another indica-
tion of the presence of dark matter because we can see
only about 5% of the estimated mass in the Milky Way
and Andromeda. Another mass estimate follows from
the kinematics of the Magellanic Stream, which also
yields M/L > 80M o /L o .
One of the most interesting galaxies in the Local Group
is the Sagittarius dwarf galaxy which was only dis-
covered in 1994. Since it is located in the direction of
the Galactic bulge, it is barely visible on optical im-
ages, if at all, as an overdensity of stars. Furthermore,
it has a very low surface brightness. It was discovered
in an analysis of stellar kinematics in the direction of
the bulge, in which a coherent group of stars was found
with a velocity distinctly different from that of bulge
stars. In addition, the stars belonging to this overdensity
have a much lower metallicity, reflected in their col-
ors. The Sagittarius dwarf galaxy is located close to the
Galactic plane, at a distance of about 16kpc from the
Galactic center and nearly in the direct extension of our
line-of-sight to the GC. This proximity implies that it
must be experiencing strong tidal gravitational forces
on its orbit around the Milky Way; over the course of
time, these will have the effect that the Sagittarius dwarf
galaxy will be slowly disrupted. In fact, in recent years
a relatively narrow band of stars has been found around
the Milky Way. These stars are located along the orbit of
the Sagittarius galaxy. Their chemical composition sup-
ports the interpretation that they are stars stripped from
the Sagittarius dwarf galaxy by tidal forces. In addition,
globular clusters have been identified which presum-
ably once belonged to the Sagittarius dwarf galaxy, but
which have also been removed from it by tidal forces
and are now part of the globular cluster population in
the Galactic halo.
Compact high-velocity clouds (CHVCs) are high-
velocity clouds (see Sect. 2.3.6) with an angular
diameter of < 1°. The distance of these clouds is difficult
to determine, since they do not seem to contain any stars,
and hence the methods of distance determination based
on stellar properties cannot be applied. In those cases
where the spectrum of a background object shows an
absorption line at the same radial velocity as determined
for the cloud from measurements of the 21 -cm line, an
upper limit for the cloud distance is obtained; namely
the distance of the object whose spectrum displays the
absorption line.
Indirect arguments sometimes yield rather large es-
timates, of several hundred kpc, for the distance of the
CHVCs. If their distance is indeed this large, the ro-
tation curves of CHVCs, i.e., their differential infall
6. Clusters and Groups of Galaxies
velocities, suggest high masses for the clouds. In this
model, CHVCs would contain a large fraction of dark
matter, M ~ 10 7 M Q , and hence much more dark mat-
ter than their neutral hydrogen mass. CHVCs would
then be additional members of the Local Group, having
a mass not very different from that of dwarf galaxies,
but in which star formation was suppressed for some
reason so that they contain no, or only very few, stars.
This model of CHVCs is controversial, however, and
its verification or falsification would be of considerable
interest for cosmology, as we will discuss later. If a con-
centration of CHVCs exists around the Milky Way at
distances like the ones assumed in this model, a similar
concentration should also exist around our sister galaxy
M3 1 . Currently, an intensive search for these systems
is in progress. While HVCs have been found around
M31, the search for CHVCs has been without success
thus far. Therefore, one concludes a relatively low char-
acteristic Galacto-centric distance for Galactic CHVCs
of ~ 50 kpc. In this case, they would not be high-mass
objects.
The Neighborhood of the Local Group. The Local
Group is indeed a concentration of galaxies: while it
contains about 35 members within ~ 1 Mpc, the next
neighboring galaxies are found only in the Sculptor
Group, which contains about six members and is located
at a distance of D ~ 1.8 Mpc. The next galaxy group
after this is the M81 group of about eight galaxies at
D ~ 3 . 1 Mpc, the two most prominent galaxies of which
are displayed in Fig. 6.7.
The other nearby associations of galaxies within
10 Mpc from us shall also be mentioned: the Centaurus
group with 17 members and D ~ 3.5 Mpc, the M101
group with five members and D ~ 7.7 Mpc, the M66
and M96 group with together 10 members located at
D ~ 9.4 Mpc, and the NGC 1023 group with six mem-
bers at D — 9.6 Mpc. The numbers given here are those
of currently known galaxies. Dwarf galaxies like Sagit-
tarius would be very difficult to detect at the distances
of these groups.
Most galaxies are members of a group. Many more
dwarf galaxies exist than luminous galaxies, and dwarf
galaxies are located preferentially in the vicinity of
larger galaxies. Some members of the Local Group are
so under-luminous that they would hardly be observable
outside the Local Group.
Fig. 6.7. M8 1 (left) and M82 (right), two galaxies of the M8 1
group, about 3.1 Mpc away. These two galaxies are moving
around each other, and the gravitational interaction taking
place may be the reason for the violent star formation in M82.
M82 is an archetypical starburst galaxy
One large concentration of galaxies was already
known in the eighteenth century (W. Herschel) - the
\ Irvo Cluster. Its galaxies extend over a region of about
10° x 10° in the sky, and its distance is D ~ 16 Mpc.
The Virgo Cluster consists of about 250 large galaxies
and more than 2000 smaller ones. In the classification
scheme of galaxy clusters, Virgo is considered an irreg-
ular cluster. The closest regular massive galaxy cluster
is the Coma cluster (see Fig. 1 . 14), at a distance of about
D~ 90 Mpc.
6.2 Galaxies in Clusters and Groups
6.2.1 The Abell Catalog
George Abell compiled a catalog of galaxy clusters,
published in 1958, in which he identified regions in the
sky that show an overdensity of galaxies. This iden-
tification was performed by eye on photoplates from
the Palomar Observatory Sky Survey (POSS), a pho-
tographic atlas of the Northern (S > —30°) sky. 1 He
i) ]»«).
'The POSS, or more preciseh the first Palomar Sky Survey, c
of 879 pairs of photoplates observed in two color bands, and
the Northern sky at declinations > —30°. It was complete
The coverage of the southern part of the sky was completed in 1980
in the ESO/SERC Southern Sky Surveys, where this survey is about
two magnitudes deeper (B £ 23. R „ 22) than POSS. The photo-
nhttes from bolli miiac\s ha\e been digitized, forming the Digitized
6.2 Galaxies in Clusters and Groups
omitted the Galactic disk region because the observa-
tion of galaxies is considerably more problematic there,
due to extinction and the high stellar density (see also
Fig. 6.2).
Abell's Criteria and his Catalog. The criteria Abell
applied for the identification of clusters refer to an
overdensity of galaxies within a specified solid angle.
According to these criteria, a cluster contains > 50 gal-
axies in a magnitude interval 1113 <m <m^+2, where
m3 is the apparent magnitude of the third brightest
galaxy in the cluster. 2 These galaxies must be located
within a circle of angular radius
1.'7
0a = — (6.6)
where z is the estimated redshift. The latter is deter-
mined by the assumption that the luminosity of the tenth
brightest galaxy in a cluster is the same for all clusters.
A calibration of this distance estimate is performed on
clusters of known redshift. 6 A is called the Abell ra-
dius of a cluster, and corresponds to a physical radius
ofR A ^ 1.5A _1 Mpc.
The so-determined redshift should be within the
range 0.02 < z < 0.2 for the selection of Abell clus-
ters. The lower limit is chosen such that a cluster can
be found on a single POSS photoplate (~ 6° x 6°) and
does not extend over several plates, which would make
the search more difficult, e.g., because the photographic
sensitivity may differ for individual plates. The upper
redshift bound is chosen due to the sensitivity limit of
the photoplates.
The Abell catalog contains 1682 clusters which all
fulfill the above criteria. In addition, it lists 1030 clus-
ters that have been found in the search, but which do not
Sky Survey (DSS) that covers the full sky. Sections from the DSS can
be obtained directly via the Internet, with the full DSS having a data
volume ol some 600GB. Currently, the second Palontar Sky Survey
(POSS ll! is in progress, which will be about one magnitude deeper
compared to the lirst one and will contain data from three (instead
of two) color lifers, litis will probably he the last photographic atlas
of lite sky because, v. ith the development ol large CO.) cameras, w e
will soon be able 10 perform such surveys digitally The most promi-
nent example of this is the Sloan Digital Sky Survey, which we will
discuss in a different context in Sect. 8.1.2.
Ih i 1 i 1 1 11 1I1 11 1 111 I that the lumi
nosity of the brightest galaxy may vary considerably among clusters,
hven more important is the fact that there is a finite probability for
the brightest galaxy in a sky region under consideration to not belong
to the cluster, but to be located at some smaller distance from us.
fulfill all of the criteria (most of these contain between
30 and 49 galaxies). An extension of the catalog to the
Southern sky was published by Abell, Corwin & Olowin
in 1989. This ACO catalog contains 4076 clusters, in-
cluding the members of the original catalog. Another
important catalog of galaxy clusters is the Zwicky cat-
alog (1961-68), which contains more clusters, but for
which the applied selection criteria are considered less
reliable.
Problems in the Optical Search for Clusters. The
selection of galaxy clusters from an overdensity of gal-
axies on the sphere is not without problems, in particular
if these catalogs are to be used for statistical purposes.
An ideal catalog ought to fulfill two criteria: first it
should be complete, in the sense that all objects which
fulfill the selection criteria are contained in the catalog.
Second it should be reliable, i.e., it should not contain
any objects that do not belong in the catalog because
they do not fulfill the criteria (so-called false positives).
The Abell catalog is neither complete, nor is it reliable.
We will briefly discuss why completeness and reliabil-
ity cannot be expected in a catalog compiled in this
way.
A galaxy cluster is a three-dimensional object,
whereas galaxy counts on images are necessarily based
on the projection of galaxy positions onto the sky.
Therefore, projection effects are inevitable. Random
overdensities on the sphere caused by line-of-sight pro-
jection may easily be classified as clusters. The reverse
effect is likewise possible: due to fluctuations in the
number density of foreground galaxies, a cluster at high
redshift may be classified as an insignificant fluctuation
- and thus remain undiscovered.
Of course, not all members of a cluster classified as
such are in fact galaxies in the cluster, as here projec-
tion effects also play an important role. Furthermore,
the redshift estimate is relatively coarse. In the mean-
time, spectroscopic analyses have been performed for
many of the Abell clusters, and it has been found that
Abell's redshift estimates have an error of about 30% -
surprisingly accurate, considering the coarseness of his
assumptions.
The Abell catalog is based on visual inspection of
photographic plates. It is therefore partly subjective.
Today, the Abell criteria can be applied to digitized im-
ages in an objective manner, using automated searches.
6. Clusters and Groups of Galaxies
From these, it has been found that the results are not
much different. The visual search must have been per-
formed with great care and has to be recognized as
a great accomplishment. For this reason, and in spite of
the potential problems discussed above, the Abell and
the ACO catalogs are still frequently used.
The clusters in the catalog are ordered by right as-
cension and are numbered. For example, Abell 851 is
the 851st entry in the catalog, also denoted as A851.
With a redshift of z = 0.41, A851 is the most distant
Abell cluster.
class, so only very few clusters exist with a very I urge
number of cluster galaxies. As a reminder, the region of
the sky from where the Abell clusters were detected is
about 2/3 of the total sphere. Thus, only a few very rich
clusters do indeed exist (at redshift < 0.2).
The subdivision into six distance classes is based on
the apparent magnitude of the tenth brightest galaxy,
in accordance with the redshift estimate for the cluster.
Hence, the distance class provides a coarse measure of
the distance.
Abell Classes. The Abell and ACO catalogs divide
clusters into so-called richness and distance classes.
Table 6.2 lists the criteria for the richness classes, while
Table 6.3 lists those for the distance classes.
There are six richness classes, denoted from to 5,
according to the number of cluster member galaxies.
Richness class contains between 30 and 49 members
and therefore does not belong to the cluster catalog
proper. One can see from Table 6.2 that the number
of clusters rapidly decreases with increasing richness
Table 6.2. Definition of Abell's richness classes. N is t
number of cluster galaxies with magnitudes between r
and m3 + 2 inside the Abell radius (6.6), where m->, is t
brightness of the third brightest cluster galaxy.
Rid
nessch
ssfi
N
N
mbe
■ in Abell's
catalog
(0)
1
3
5
(30-49)
50-79
130-199
200-299
(> 1000)
1224
Table 6.3. Definition of Abell's distance classes, t
magnitude of the tenth brightest cluster galaxy.
Distanc
no
Est
Number in Abell's
catalog with R > 1
1
13.
-14.0
0.0283
9
14.
-14.8
0.0400
2
3
14.
-15.6
0.0577
33
4
15.
-16.4
0.0787
60
6
17.
-18.0
0.198
921
6.2.2 Luminosity Function of Cluster Galaxies
The luminosity function of galaxies in a cluster is de-
fined as in Sect. 3.7 for the total galaxy population. In
many clusters, the Schechter luminosity function (3.38)
represents a very good fit to the data if the brightest
galaxy is disregarded in each cluster (see Fig. 3.32 for
the Virgo Cluster of galaxies). The slope a at the faint
end is not easy to determine, since projection effects
become increasingly important for fainter galaxies. The
value of a seems to vary between clusters, but it is not
entirely clear whether this result may also be affected
by projection effects in different clusters of differing
strength. Thus, no final conclusion has been reached as
to whether the luminosity function has a steep increase
at L <$; L* or not, i.e., whether many more faint gal-
axies exist than luminous ~ L* -galaxies (compare the
galaxy content in the Local Group, Sect. 6.1.1, where
even in our close neighborhood it is difficult to ob-
tain a complete census of the galaxy population). L* is
very similar for many clusters, which is the reason
why the distance estimate by apparent brightness of
cluster members is quite reliable. However, a num-
ber of clusters exists with a clearly deviating value
ofL*.
Many clusters contain cD galaxies at their centers;
these differ from large ellipticals in several respects.
They have a very extended stellar envelope, whose size
may exceed R ~ 100 kpc and whose surface brightness
profile is much broader than that of a de Vaucouleurs
profile (see Fig. 3.8). cD galaxies are found only in the
centers of clusters or groups, thus only in regions of
strongly enhanced galaxy density. Many cD galaxies
have multiple cores, which is a rather rare phenomenon
among the other cluster members.
6.2 Galaxies in Clusters and Groups
6.2.3 Morphological Classification of Clusters
Clusters are also classified by the morphology of their
galaxy distribution. Several classifications are used, one
of which is displayed in Fig. 6.8. Since this is a descrip-
tion of the visual impression of the galaxy distribution,
the exact class of a cluster is not of great interest. How-
ever, a rough classification can provide an idea of the
state of a cluster, i.e., whether it is currently in dynami-
cal equilibrium or whether it has been heavily disturbed
by a merger process with another cluster. Therefore, one
distinguishes in particular between regular and irregu-
lar clusters, and also those which are intermediate; the
transition between classes is of course continuous. Reg-
ular dusters are compact whereas, in contrast, irregular
clusters are "open" (Zwicky's classification criteria).
This morphological classification indeed points at
physical differences between clusters, as correlations
between morphology and other properties of galaxy
clusters show. For example, it is found that regular
clusters are completely dominated by early-type galax-
ies, whereas irregular clusters have a fraction of spirals
nearly as large as in the general distribution of field
galaxies. Very often, regular clusters are dominated by
a cD galaxy at the center, and their central galaxy density
is very high. In contrast, irregular clusters are signifi-
cantly less dense in the center. Irregular clusters often
show strong substructure, which is rarely found in reg-
ular clusters. Furthermore, regular clusters have a high
richness, whereas irregular clusters have fewer cluster
members. To summarize, regular clusters can be said to
be in a relaxed state, whereas irregular clusters are still
in the process of evolution.
6.2.4 Spatial Distribution of Galaxies
Most regular clusters show a centrally condensed num-
ber density distribution of cluster galaxies, i.e., the
galaxy density increases strongly towards the center.
If the cluster is not very elliptical, this density dis-
tribution can be assumed, to a first approximation, as
being spherically symmetric. Only the projected den-
sity distribution N(R) is observable. This is related to
the three-dimensional number density n{r) through
(6.7)
where in the second step a simple transformation of
the integration variable from the line-of-sight coordi-
nate z to the three-dimensional radius r — ^/R 2 + z 2
was made.
Of course, no function N(R) can be observed, but
only points (the positions of the galaxies) that are
distributed in a certain way. If the number density
of galaxies is sufficiently large, N(R) is obtained by
smoothing the point distribution. Alternatively, one
considers parametrized forms of N(R) and fits the pa-
rameters to the observed galaxy positions. In most cases,
the second approach is taken because its results are more
robust. A parametrized distribution needs to contain at
least five parameters to be able to describe at least the
basic characteristics of a cluster. Two of these parame-
ters describe the position of the cluster center on the sky.
One parameter is used to describe the amplitude of the
density, for which, e.g., the central density N Q — N(Q)
••;*••
'Wis
Fig. 6.8. Rough morphological classifica
tion of clusters by Rood & Sastry: cDs are
those which are dominated by a central cD
galaxy, Bs contain a pair of bright galaxies
in the center. Ls are clusters with a nearly
linear alignment of the dominant galax-
ies, Cs have a single core of galaxies, Fs
are clusters with an oblate galaxy distribu-
tion, and Is are clusters with an irregular
distribution
6. Clusters and Groups of Galaxies
may be used. A forth parameter is a characteristic scale
of a cluster, often taken to be the core radius r c , de-
fined such that at R — r c , the projected density has
decreased to half the central value, N(r c ) — No/2. Fi-
nally, one parameter is needed to describe "where the
cluster ends"; the Abell radius is a first approximation
for such a parameter.
Parametrized cluster models can be divided into those
which are physically motivated, and those which are
of a purely mathematical nature. One example for the
latter is the de Vaucouleurs profile which is not de-
rived from dynamical models. Next, we will consider
a class of distributions that are based on a dynamical
model.
Isothermal Distributions. These models are based on
the assumption that the velocity distribution of the
massive particles (this may be both galaxies in the
cluster or dark matter particles) of a cluster is lo-
cally described by a Maxwell distribution, i.e., they
are thermalized. As shown from spectroscopic analy-
ses of the distribution of the radial velocities of cluster
galaxies, this is not a bad assumption. Assuming, in
addition, that the mass profile of the cluster follows
that of the galaxies (or vice versa), and that the tem-
perature (or equivalently the velocity dispersion) of the
distribution does not depend on the radius (so that one
has an isothermal distribution of galaxies), then one
obtains a one-parameter set of models, the so-called
isothermal spheres. These can be described physically
as follows.
In dynamical equilibrium, the pressure gradient must
be equal to the gravitational acceleration, so that
dP
GM(r)
where p(r) denotes the density of the distribution, e.g.,
the density of galaxies. By p(r) — (m) n(r), this mass
density is related to the number density n(r), where (m)
is the average particle mass. M(r) — 4jt f Q r dr' r' 2 p(r')
is the mass of the cluster within a radius r. By
differentiation of (6.8), we obtain
2 dP\
)+4jrGr z p = 0.
(6.9)
The relation between pressure and density is P = nk B T.
On the other hand, the temperature is related to the
velocity dispersion of the particles,
. < m > /.
(6.10)
where (u 2 ) is the mean squared velocity, i.e., the velocity
dispersion, provided the average velocity vector is set
to zero. The latter assumption means that the cluster
does not rotate, or contract or expand. If T (or (t> 2 )) is
independent of r, then
dP k B Tdp {v 2 )dp 2 dp
d^Md^Xd^ ^' (6 - H)
where a 2 is the one-dimensional velocity dispersion,
e.g., the velocity dispersion along the line-of-sight,
which can be measured from the redshift of the cluster
galaxies. If the velocity distribution corresponds to an
isotropic (Maxwell) distribution, the one-dimensional
velocity dispersion is exactly 1/3 times the three-
dimensional velocity dispersion, because of (i> 2 ) = a 2 +
a 2 + a 2 , or
With (6.9), it then follows that
dr \ p dr J
Singular Isothermal Sphere. In general, the dif-
ferential equation (6.13) for p(r) cannot be solved
analytically. Physically reasonable boundary conditions
are p(0) = po, the central density, and (dp/dr)\ r= o — 0,
for the density profile to be flat at the center. One particu-
lar analytical solution of the differential equation exists,
however: By substitution, we can easily show that
p(r) =
InGr-
(6.14)
solves (6.13). This density distribution is called singu-
lar isothermal sphere; we have encountered it before, in
the discussion of gravitational lens models in Sect. 3.8.2.
This distribution has a diverging density as r — »■ and
an infinite total mass Mir) oc r. It is remarkable that this
6.2 Galaxies in Clusters and Groups
density distribution is just what is needed to explain the
flat rotation curves of galaxies at large radii.
Numerical solutions of (6.13) with the initial condi-
tions specified above (thus, with a flat core) reveal that
the central density and the core radius are related to each
other by
9g 2 v
' AnGr}
Hence, these physical solutions of (6.13) avoid the infi-
nite density of the singular isothermal sphere. However,
these solutions also decrease outwards with p oc r~ 2 ,
so they have a diverging mass as well. The origin
of this mass divergence is easily understood because
these isothermal distributions are based on the as-
sumption that the velocity distribution is isothermal,
thus Maxwellian with a spatially constant temperature.
A Maxwell distribution has wings, hence it (formally)
contains particles with arbitrarily high velocities. Since
the distribution is assumed stationary, such particles
must not escape, so their velocity must be lower than the
escape velocity from the gravitational well of the cluster.
But for a Maxwell distribution this is only achievable
for an infinite total mass.
King Models. To remove the problem of the diverging
total mass, self-gravitating dynamical models with an
upper cut-off in the velocity distribution of their con-
stituent particles are introduced. These are called King
models and cannot be expressed analytically. However,
an analytical approximation exists for the central region
of these mass profiles,
p(f) = po 1 + I -
Using (6.7), we obtain from this the projected surface
mass density
^
The analytical fit (6.16) of the King profile also has
a diverging total mass, but this divergence is "only"
logarithmic.
These analytical models for the density distribu-
tion of galaxies in clusters are only approximations,
of course, because the galaxy distribution in clusters is
often heavily structured. Furthermore, these dynamical
models are applicable to a galaxy distribution only if
the galaxy number density follows the matter density.
However, one finds that the distribution of galaxies in
a cluster often depends on the galaxy type. The fraction
of early-type galaxies (Es and SOs) is often largest near
the center. Therefore, one should consider the possibil-
ity that the distribution of galaxies in a cluster may be
different from that of the total matter. A typical value
for the core radius is about r c ~ 0.25/z -1 Mpc.
6.2.5 Dynamical Mass of Clusters
The above argument relates the velocity distribution
of cluster galaxies to the mass profile of the cluster,
and from this we obtain physical models for the den-
sity distribution. This implies the possibility of deriving
the mass, or the mass profile, respectively, of a clus-
ter from the observed velocities of cluster galaxies. We
will briefly present this method of mass determination
here. For this, we consider the dynamical time-scale of
clusters, defined as the time a typical galaxy needs to
: the cluster once,
A'a
- 1.5/r
i'yr
(6.18)
where a (one-dimensional) velocity dispersion er„ ~
1000 km/s was assumed. The dynamical time-scale is
shorter than the age of the Universe. One therefore con-
cludes that clusters of galaxies are gravitationally bound
systems. If this were not the case they would dissolve on
a timescale ? cross . Since f cross <§C t one assumes a virial
equilibrium, hence that the virial theorem applies, so
that in a time-average sense,
2£ kin + £ po
where
= 0,
(6 J<»
with £q — 2/Oor,
--jl>
pot 2 /__
6. Clusters and Groups of Galaxies
are the kinetic and the potential energy of the cluster
galaxies, m, is the mass of the j-th galaxy, u,- is the
absolute value of its velocity, and r t j is the spatial sep-
aration between the i-th and the y'-th galaxy. The factor
1/2 in the definition of £"p t occurs since each pair of
galaxies occurs twice in the sum.
We define the total mass of the cluster,
the velocity dispersion, weighted by mass,
i v 2\. = Ly m . v 2
and the gravitational radius,
^= 2 We^)
With this, we obtain
for the kinetic and potential energy. Applying the virial
theorem (6.19) yields the mass i
Transition to Projected Quantities. The above deriva-
tion uses the three-dimensional separations r t of the
galaxies from the cluster center, which are, however,
not observable. To be able to apply these equations to
observations, they need to be transformed to projected
separations. If the galaxy positions and the directions of
their velocity vectors are uncorrelated, as it is the case,
e.g., for an isotropic velocity distribution, then
(u 2 ) = 3a„ 2 , r G =?-R G
R G = 2M> \J2^
where Ry denotes the projected separation between the
galaxies i and j. The parameters a v and R G are direct
observables; thus, the total mass of the cluster c
determined. One obtains
,l.lxlO 15 M (-
") 2 (— )
s/ VIMpc/
(6.21)
(6.27)
We explicitly point out that this mass estimate no longer
depends on the masses m, of the individual galaxies -
rather the galaxies are now test particles in the gravita-
tional potential. With a v ~ 1000 km/s and R c ~ 1 Mpc
as typical values for rich clusters of galaxies, one obtains
a characteristic mass of ~ 10 15 M Q for rich clusters.
The "Missing Mass" Problem in Clusters of Gal-
axies. With M and the number N of galaxies, one
can now derive a characteristic mass m — M/N for
the luminous galaxies. This mass is found to be very
high, m ~ 10 13 M Q . Alternatively, M can be compared
with the total optical luminosity of the cluster galaxies,
Ltot ~ 10 12 -10 13 L Q , and hence the mass-to-light ratio
can be calculated; typically
(£)
t M -
This value exceeds the M/L ratio of early-type galax-
ies by at least a factor of 10. Realizing this discrepancy,
Fritz Zwicky concluded as early as 1933, from an anal-
ysis of the Coma cluster, that clusters of galaxies must
contain considerably more mass than is visible in gal-
axies - the dawn of the missing mass problem. As we
will see further below, this problem has by now been
firmly established, since other methods for the mass
determination of clusters also yield comparable values
and indicate that a major fraction of the mass in gal-
axy clusters consists of (non-baryonic) dark mailer. The
slurs visible in galaxies contribute less than about 5Vr
to the total mass in clusters of galaxies.
6.2.6 Additional Remarks on Cluster Dynamics
Given the above line of argument, the question of course
arises as to whether the application of the virial theo-
6.2 Galaxies in Clusters and Groups
rem is still justified if the main fraction of mass is not
contained in galaxies. The derivation remains valid in
this form as long as the spatial distribution of galaxies
follows the total mass distribution. The dynamical mass
determination can be affected by an anisotropic velocity
distribution of the cluster galaxies and by the possibly
non-spherical cluster mass distribution. In both cases,
projection effects, which are dealt with relatively easily
in the spherically-symmetric case, obviously become
more complicated. This is also one of the reasons for
the necessity to consider alternative methods of mass
determination.
Two-body collisions of galaxies in clusters are of
no importance dynamically, as is easily seen from the
corresponding relaxation time-scale (3.3),
N
frelax- Across— ,
which is much larger than the age of the Universe. The
motion of galaxies is therefore governed by the collec-
tive gravitational potential of the cluster. The velocity
dispersion is approximately the same for the differ-
ent types of galaxies, and also only a weak tendency
exists for a dependence of a v on galaxy luminosity, re-
stricted to the brightest ones (see below in Sect. 6.2.9).
From this, we conclude that the galaxies in a cluster
are not "thermalized" because this would mean that
they all have the same mean kinetic energy, implying
a v oc m~ 1/2 . Furthermore, the independence of a v from
L implies that collisions of galaxies with each other are
not dynamically relevant; rather, the velocity distribu-
tion of galaxies is defined by collective processes during
cluster formation.
Violent Relaxation. One of the most important of the
aforementioned processes is known as violent relax-
ation. This process very quickly establishes a virial
equilibrium in the course of the gravitational collapse
of a mass concentration. The reason for it are the
small-scale density inhomogeneities within the collaps-
ing matter distribution which generate, via Poisson's
equation, corresponding fluctuations in the gravitational
field. These then scatter the infalling particles and, by
this, the density inhomogeneities are further amplified.
The fluctuations of the gravitational field act on the
matter like scattering centers. In addition, these field
fluctuations change over time, yielding an effective ex-
change of energy between the particles. In a statistical
average, all galaxies obtain the same velocity distri-
bution by this process. As confirmed by numerical
simulations, this process takes place on a time-scale
°f f cross. i- e -> roughly as quickly as the collapse itself.
Dynamical Friction. Another important process for the
dynamics of galaxies in a cluster is dynamical friction.
The simplest picture of dynamical friction is obtained
by considering the following. If a massive particle of
mass m moves through a statistically homogeneous dis-
tribution of massive particles, the gravitational force on
this particle vanishes due to homogeneity. But since the
particle itself has a mass, it will attract other massive
particles and thus cause the distribution to become in-
homogeneous. As the particle moves, the surrounding
"background" particles will react to its gravitational
field and slowly start moving towards the direction
of the particle trajectory. Due to the inertia of mat-
ter, the resulting density inhomogeneity will be such
that an overdensity of mass will be established along
the track of the particle, where the density will be
higher on the side opposite to the direction of mo-
tion (thus, behind the particle) than in the forward
direction (see Fig. 6.9). By this process, a gravita-
tional field will form that causes an acceleration of
the particle against the direction of motion, so that the
particle will be slowed down. Because this "polariza-
tion" of the medium is caused by the gravity of the
particle, which is proportional to its mass, the decel-
eration will also be proportional to m. Furthermore,
a fast-moving particle will cause less polarization in the
medium than a slow-moving one because each mass
element in the medium is experiencing the gravita-
tional attraction of the particle for a shorter time, thus
the medium becomes less polarized. In addition, the
particle is on average farther away from the density
accumulation on its backward track, and thus will expe-
rience a smaller acceleration if it is faster. Combining
these arguments, one obtains for the dependence of this
dynamical friction
ib
(6.29)
where p is the mass density in the medium. Applied
to clusters of galaxies, this means that the most mas-
6. Clusters and Groups of Galaxies
1&&
Fig. 6.9. The principle of dynamical friction. The gravitational
field of amassh e particle {here indicated by the large symbol)
accelerates the surrounding mattei towards its track. Through
this, an overdensity establishes on the backward side of its
orbit, the gravitational force ot which decelerates the particle
sive galaxies will experience the strongest dynamical
friction, so that they are subject to a significant deceler-
ation through which they move deeper into the potential
well. The most massive cluster galaxies should there-
fore be concentrated around the cluster center, so that
a spatial separation of galaxy populations with respect
to their masses occurs (mass segregation). If dynamical
friction acts over a sufficiently long time, the massive
cluster galaxies in the center may merge into a single
one. This is one possible explanation for the formation
of cD galaxies.
Dynamical friction also plays an important role in
other dynamical processes in astrophysics. For example,
the Magellanic Clouds experience dynamical friction
on their orbit around the Milky Way and thereby lose
kinetic energy. Consequently, their orbit will become
smaller over the course of time and, in a distant future,
these two satellite galaxies will merge with our Gal-
axy. In fact, dynamical friction is of vital importance in
galaxy merger processes which occur in the evolution
of the galaxy population, a subject we will return to in
Sect. 9.6.
6.2.7 Intergalactic Stars in Clusters of Galaxies
The space between the galaxies in a cluster is filled
with hot gas, as visible from X-ray observations. In re-
cent years, it has been found that besides hot gas there
are also stars in between the galaxies. The detection of
such an intergalactic stellar population comes as a sur-
prise at first sight, because our understanding of star
formation implies that they can only form in the dense
centers of molecular clouds. Hence, one expects that
stars cannot form in intergalactic space. This is not nec-
essarily implied by the presence of intergalactic stars,
however, since they can also be stripped from galax-
ies in the course of gravitational interactions between
galaxies in the cluster, and so form an intergalactic pop-
ulation. The fate of these stars is thus comparable to
that of the interstellar medium, which is metal-enriched
by the processes of stellar evolution in galaxies before
it is removed from these galaxies and becomes part of
the intergalactic medium in clusters; otherwise, the sub-
stantial metallicity of the ICM could not be explained.
The observation of diffuse optical light in clusters
of galaxies and, related to this, the detection of the
intracluster stellar population, is extremely difficult. Al-
though first indications have already been found with
photographic plate measurements, the surface bright-
ness of this cluster component is so low that even
with CCD detectors the observation is extraordinarily
challenging. To quantify this, we note that the sur-
face brightness of this diffuse light component is about
30 mag arcsec" 2 at a distance of several hundred kpc
from the cluster center. This value needs to be com-
pared with the brightness of the night sky, which is about
2 1 mag arcsec -2 in the V-band. One therefore needs to
correct for the effects of the night sky to better than
a tenth of a percent for the intergalactic stellar compo-
nent to become visible in a cluster. Furthermore, cluster
galaxies and objects in the foreground and background
need to be masked out in the images, in order to measure
the radial profile of this diffuse component. This is pos-
sible only up to a certain limiting magnitude, of course,
up to which individual objects can be identified. The ex-
istence of weaker sources has to be accounted for with
statistical methods, which in turn use the luminosity
function of galaxies.
The diffuse light component is best investigated
in a statistical superposition of the images of several
galaxy clusters. Statistical fluctuations in the sky back-
ground and uncertainties in the flatfield 3 determination
"the flat held of an image (or. more precise!}, of the system consisting
of telescope, tiller, and detector) is defined as the image old uniform!}
illuminated iield. so that in the ideal ease each pixel ot the detector
produces the same output signal. This is not the case in reality, how
e\er. as the sensitivity differs for individual pixels. For this reason,
the flatfield measures the sensitivity distribution of the pixels, w Inch
i 111 n 1 urn d 1 1 m th im 111 il;
6.2 Galaxies in Clusters and Groups
are in this case averaged out. In these analyses an r~ 1/4 -
law is found for the light distribution in the inner region
of clusters, i.e., the (de Vaucouleurs) brightness profile
of the central galaxy is measured. For radii larger than
about ~ 50 kpc, the brightness profile exceeds the ex-
Irapolation of the de Vaucouleurs profile, and has been
detected out to very large distances from the cluster cen-
ter. This fact needs to be considered in the context of the
existence of cD galaxies, which are defined by exactly
this light excess. The separation between the diffuse
light component and the extended light profile of a cD
galaxy is not easily performed, but at large distances
from the cluster center, one can exclude the possibility
that the corresponding stars are gravitationally bound to
the central cluster galaxy.
Besides this diffuse component, individual stars and
planetary nebulae have been detected in some neigh-
boring galaxy clusters which cannot be assigned to any
cluster galaxy. The diffuse cluster component accounts
for about 10% of the total optical light in a cluster.
Therefore, models of galaxy evolution in clusters should
provide an explanation for these observations.
Hydrogen Clouds in the Virgo Cluster, and a Dark
Galaxy? The interaction of galaxies in clusters can-
not only lead to the stripping of stars from galaxies,
but in the case of gas-rich galaxies its ISM can be
(partly) removed. Indeed, in the Virgo Cluster several
large clouds of gas have been found, through their Hi
21 -cm emission, which are not centered on optically
luminous galaxies. For one of them, with a neutral hy-
drogen mass of ~ 10 8 M o , the rotational velocity has
been measured, yielding the result that this cloud is
dominated by dark matter - an optically dark galaxy.
Combining the rotational velocity with the size of the
Hi distribution, a lower bound of the dynamical mass
of ~ 10 11 Af© is inferred - the lack of any visible coun-
terpart in the optical then yields a lower bound on the
mass-to-light ratio of about 500 in Solar units. How-
ever, this conclusion is based on the assumption that the
gas is in dynamical equilibrium. Rather, if it has been
recently stripped from a galaxy, an equilibrium state
may not have been established, and therefore the mass
estimate from the measured velocity field may be in er-
ror. In other cases, the Hi clouds can be identified with
the galaxy from which they originated, owing to the
tidal Sails connecting the cloud with the corresponding
galaxy. In any case, the possibility of having identified
a "dark galaxy" of this large mass is exceedingly excit-
ing, as such objects are not expected from our current
understanding of galaxy formation.
6.2.8 Galaxy Groups
Accumulations of galaxies that do not satisfy Abel-
l's criteria are in most cases galaxy groups. Hence,
groups are the continuation of clusters towards fewer
member galaxies and are therefore presumably of
lower mass, lower velocity dispersion, and smaller ex-
tent. The distinction between groups and clusters is
at least partially arbitrary. It was defined by Abell
mainly to be not too heavily affected by projection
effects in the identification of clusters. Groups are of
course more difficult to detect, since the overdensity
criterion for them is more sensitive to projection ef-
fects by foreground and background galaxies than for
clusters.
A special class of groups are the compact groups,
assemblies of (in most cases, few) galaxies with very
small projected separations. The best known examples
for compact groups are Stephan's Quintet and Seyfert's
Sextet (see Fig. 6.10). In 1982, a catalog of 100 compact
groups (Hickson Compact Groups, HCGs) was pub-
lished, where a group consists of four or more bright
members. These were also selected on POSS photo-
plates, again solely by an overdensity criterion. The
median redshift of the HCGs is about z = 0.03. Exam-
ples of optical images of HCGs are given in Figs. 6.3
and 1.16.
Follow-up spectroscopic studies of the HCGs have
verified that 92 of them have at least three galaxies
with conforming redshifts, defined such that the corre-
sponding recession velocities lie within 1000 km/s of
the median velocity of group members. Of course, the
similarity in redshift does not necessarily imply that
these groups form a gravitationally bound and relaxed
system. For instance, the galaxies could be tracers of an
overdense structure which we happen to view from a di-
rection where the galaxies are projected near each other
on the sky. However, more than 40% of the galaxies
in HCGs show evidence of interactions, indicating that
these galaxies have near neighbors in three-dimensional
Fig. 6.10. Left: Stephan's Quintet, also known as Hickson fact galaxies belonging to the group; the spiral galaxy (e) is
Compact Group 92, is a very dense accumulation of galaxies located at significantly higher distance. Another object orig-
with a diameter of about 80kpc. Right: Seyfert's Sextet, an inally classified as a galaxy is no galaxy but instead a tidal
apparent accumulation of six galaxies located very close to- tail that was ejected in tidal interactions of galaxies in the
gether on the sphere. Only four of the galaxies (a)-(d) are in group
space. Furthermore, about three quarters of HCGs
with four or more member galaxies show extended
X-ray emission, most likely coming from intragroup
hot gas, providing additional evidence for the presence
of a common gravitational potential well.
More recently, galaxy groups have been selected
from spectroscopic surveys. For these, a three-dimen-
sional overdensity criterion can be applied, which
considerably reduces projection effects and which also
allows the detection of groups in regions of larger
mean projected galaxy number density. The velocity
dispersion in groups is significantly smaller than that in
clusters; typical values are er„ ~ 300 km/s.
Compact groups have a lifetime which is much
shorter than the age of the Universe. The dynami-
cal timescale is fdyn ~ R/&v ~ 0.02 Hq l , thus small
compared to to ~ Hq ' . By dynamical friction (see
Sect. 6.2.6), galaxies in groups lose kinetic (orbital)
energy and move closer to the dynamical center where
interactions and mergers with other group galaxies take
place, as also seen by the high fraction of member gal-
axies with morphological signs of interactions. Since
the lifetime of compact groups is shorter than the age of
the Universe, they must have formed not too long ago.
If we do not happen to live in a special epoch of cos-
mic history, such groups must therefore still be forming
today. From dynamical studies, one finds that - as in
clusters - the total mass of groups is significantly larger
than the sum of the mass visible in galaxies; a typical
mass-to-light ratio is M/L ~ 50ft (in Solar units), which
is comparable to that of the Local Group.
As in clusters, the fraction of group members which
are spirals is lower than the fraction of spirals among
field (i.e., isolated) galaxies, and the relative abundance
of spiral galaxies decreases with increasing a v of the
group. Furthermore, galaxy groups are X-ray emitters,
so they likewise contain hot intergalactic gas, albeit at
lower temperatures and lower metallicities than clusters.
There are by now good indications that (compact)
galaxy groups contain a diffuse optical light component,
as is the case for galaxy clusters, and that the fraction
of optical emission due to the diffuse component varies
strongly between individual groups. Since the origin of
the intragroup stellar component is most likely related
to the history of galaxy interactions and tidal stripping
by the group potential, the relative contribution of the
intragroup light may contain valuable information about
the evolution of groups.
6.2 Galaxies in Clusters and Groups
6.2.9 The Morphology-Density Relation
As mentioned several times before, the mixture of gal-
axy types in clusters differs from that of isolated (field)
galaxies. Whereas about 70% of the field galaxies are
spirals, clusters are dominated by early-type galaxies, in
particular in their inner regions. Furthermore, the frac-
tion of spirals in a cluster depends on the distance to
the center and increases for larger r. Obviously, the lo-
cal density has an effect on the morphological mix of
galaxies.
More generally, one may ask whether the mixture of
the galaxy population depends on the local galaxy den-
sity. While earlier studies of this effect were frequently
constrained to galaxies within and around clusters,
new extensive redshift surveys like the 2dFGRS and
the SDSS (see Sect. 8.1.2) allow us to systematically
investigate this question with very large and care-
fully selected samples of galaxies. The morphological
classification of such large samples is performed by au-
tomated software tools, which basically measure the
light concentration in the galaxies. A comparison of
galaxies classified this way with visual classifications
shows very good agreement.
Results from the Sloan Digital Sky Survey. As an ex-
ample of such an investigation, results from the Sloan
Digital Sky Survey are shown in Fig. 6.1 1. The galaxies
have been morphologically classified, based on SDSS
photometry, and separated into four classes, correspond-
ing to elliptical galaxies, SO galaxies, and early (Sa) and
late (Sc) types of spiral. In this analysis, only galaxies
have been included for which the redshift was spec-
troscopically measured. Therefore, the spatial galaxy
density can be estimated. However, one needs to take
into account the fact that the measured redshift is a su-
perposition of the cosmic expansion and the peculiar
velocity of a galaxy. The peculiar velocity may have
rather large values (~ 1000 km/s), in particular in clus-
ters of galaxies. For this reason, for each galaxy in the
sample the surface number density of galaxies which
have a redshift within ±1000 km/s of the target galaxy
has been determined. The left panel in Fig. 6.1 1 shows
the fraction of the different galaxy classes as a function
of this local galaxy density. A very clear dependence, in
particular of the fraction of late-type spirals, on the local
density can be seen: in regions of higher galaxy den-
sity Sc spirals contribute less than 10% of the galaxies,
whereas their fraction is about 30% in low-density re-
^_^ ' :
—
arly-type
_
ate Disc
0.4
c 0.3
' J
1
it 0.2
/i\-
tfU.
j-**-*"!--^ :
0.1
Galaxy Density (Mpc~ a )
Fig. 6.11. The number fraction of galaxies of different mor-
phologies is plotted as a function of the local galaxy density
(lefi pain I ind 1 11 ilaxies in clusters as a function of the
distance from the cluster center, scaled by the corresponding
vnial radius (right panel). Galaxies have been divided into
four different classes. "Early-types" contain mainly ellipti-
R/R uiria|
cals, "intermediates" are mainly SO galaxies, "early and late
disks" are predominantly Sa and Sc spirals, respectively. In
both representations, a clear dependence of the galaxy mix on
the density or on the distance from the cluster center, respec-
tively, is visible. In the histograms at the top of each panel,
the number of galaxies in the various bins is plotted
6. Clusters and Groups of Galaxies
gions. Combined, the fraction of spirals decreases from
~ 65% in the field to about 35% in regions of high gal-
axy density. In contrast, the fraction of ellipticals and
SO galaxies increases towards higher densities, with the
increase being strongest for ellipticals.
In the right-hand panel of Fig. 6.11, the mixture of
galaxy morphologies is plotted as a function of the dis-
tance to the center of the nearest cluster, where the
distance has been scaled by the virial radius of the corre-
sponding cluster. As expected, a very strong dependence
of the fraction of ellipticals and spirals on this distance
is seen. Sc spirals contribute a mere 5% of galaxies in
the vicinity of cluster centers, whereas the fraction of
ellipticals and SO galaxies strongly increases inwards.
The two diagrams in Fig. 6. 1 1 are of course not mu-
tually independent: a region of high galaxy density is
very likely to be located in the vicinity of a cluster cen-
ter, and the opposite is valid accordingly. Therefore,
it is not immediately clear whether the mix of galaxy
morphologies depends primarily on the respective den-
sity of the environment of the galaxies, or whether it is
caused by morphological transformations in the inner
regions of galaxy clusters.
The morphology-density relation is also seen in gal-
axy groups. The fraction of late-type galaxies decreases
with increasing group mass. Furthermore, the fraction of
early-type galaxies increases with decreasing distance
from the group center, as is also the case in clusters.
Alternative Consideration: The Color-Density Rela-
tion. We pointed out in Sect. 3.7.2 that galaxies at fixed
luminosity seem to have a bimodal color distribution
(see Fig. 3.33). Using the same data set as that used for
Fig. 3.33, the fraction of galaxies that are contained in
the red population can be studied as a function of the
local galaxy density. The result of this study is shown
in the left-hand panel of Fig. 6.12, where the fraction
of galaxies belonging to the red population is plotted
against the local density of galaxies, measured in terms
of the fifth-nearest neighboring galaxy within a redshift
of ±1000 km/s. The fraction of red galaxies increases
towards higher local number density, and the relative in-
crease is stronger for the less luminous galaxies. If we
identify the red galaxies with the early-type galaxies in
Fig. 6.1 1, these two results are in qualitative agreement.
Surprisingly, the fraction of galaxies in the red sample
seems to be a function of a combination of the local
galaxy density and the luminosity of the galaxy, as i
shown in the right-hand panel of Fig. 6.12.
Interpretation. A closer examination of Fig. 6.11 may
provide a clue as to what physical processes are respon-
sible for the dependence of the morphological mix on
the local number density. We consider first the right-
hand panel of Fig. 6.11. Three different regimes in
radius can be identified: for R > R V { T , the fraction of
the different galaxy types remains basically constant.
In the intermediate regime, 0.3 < R/R v i T < 1, the frac-
tion of SO galaxies strongly increases inwards, whereas
the fraction of late-type spirals decreases accordingly.
This result is compatible with the interpretation that
in the outer regions of galaxy clusters spirals lose gas
(for instance, by their motion through the intergalactic
medium), and these galaxies then transform into passive
SO galaxies. Below R < 03R vk , the fraction of SO gal-
axies decreases strongly, and the fraction of ellipticals
increases substantially.
In fact, the ratio of the number densities of SO galax-
ies and ellipticals, for R < 0.37? vir , strongly decreases as
R decreases. This may hint at a morphological transfor-
mation in which SO galaxies are turned into ellipticals,
probably by mergers. Such gas-free mergers, also called
"dry mergers", may be the preferred explanation for the
generation of elliptical galaxies. One of the nice prop-
erties of dry mergers is that such a merging process
would not be accompanied by a burst of star forma-
tion, unlike the case of gas-rich collisions of galaxies.
The existence of a population of newly-born stars in
ellipticals would be difficult to reconcile with the gen-
erally old stellar population actually observed in these
galaxies.
Considering now the dependence on local galaxy
density (the left-hand panel of Fig. 6.1 1), a similar be-
havior of the morphological mix of galaxies is observed:
there seem to exist two characteristic values for the
galaxy density where the relative fractions of galaxy
morphologies change noticeably. Interestingly, the rela-
tion between morphology and density seems to evolve
only marginally between z — 0.5 and the local Universe.
One clue as to the origin of the morphological
transformation of galaxies in clusters, as a function of
distance from the cluster center, comes from the obser-
vation that the velocity dispersion of very bright cluster
galaxies seems to be significantly smaller than that of
6.3 X-Ray Radiation from Clusters of Galaxies
22.0<-M r <23.0 a
21.0<-M r <22.0 □
- 20.0<-M r <21.0 o
19.0<-M f <20.0 a
18.0<-M r <19.0 n
\ ...
3fP ,:l: -
- 17.0<-M r <18.0 o
H
h s^
'
'■pgp
(b) :
the red distri-
function of E$, an
of the local galaxy number density based (
log(I 5 /Mpc- 3 +L f /L_ 202 )
confirmed neighbor galaxy within ±1000 km/s. Different
symbols correspond to different luminosity bins, as indicated,
e red fraction is plotted against a combination
the projected distance of the fifth-nearest spectroscopically of the local galaxy density £5 and the luminosity of the galaxy
less luminous ones. Assuming that the mass-to-light ra-
tio does not vary substantially among cluster members,
this then indicates that the most massive galaxies have
smaller velocity dispersions. One way to achieve this
trend in the course of cluster evolution is by dynamical
interactions between cluster galaxies. Such interactions
tend to "thermalize" the velocity distribution of galax-
ies, so that the mean kinetic energy of galaxies tends to
become similar. This then causes more massive galax-
ies to become slower on average. If this interpretation
holds, then the morphology-density relation may be at-
tributed to these dynamical interactions, rather than to
the (so-called ram-pressure) stripping of the interstellar
medium as the galaxies move through the intracluster
medium.
E+A Galaxies. Galaxy clusters contain a class of gal-
axies which is denned in terms of spectral properties.
These galaxies show strong Balmer line absorption in
their spectra, characteristic of A stars, but no [Oil] or
Ha emission lines. The latter indicates that these galax-
ies are not undergoing strong star formation at present,
whereas the former shows that there was an episode of
star formation within the past ~ 1 Gyr, about as long
ago as the main- sequence lifetime of A stars. These
galaxies have been termed E+A galaxies since their
spectra appears like a superposition of that of A-stars
and that of otherwise normal elliptical galaxies. They
are interpreted as being post-starburst galaxies. Since
they were first seen in clusters, the interpretation of the
origin of E+A galaxies was originally centered on the
cluster environment - for example star-forming galax-
ies falling into a cluster and having their interstellar
medium removed by tidal forces caused by the clus-
ter potential well and/or stripping as the galaxies move
through the intracluster medium. However, E+A gal-
axies were later also found in different environments,
making the above interpretation largely obsolete. By in-
vestigating the spatial correlation of these galaxies with
other galaxies shows that the phenomenon is not associ-
ated with the large-scale environment. An overdensity
of neighboring galaxies can be seen only out to scales of
~ 100 kpc. If the sudden turn-off of the star- formation
activity is indeed caused by an external perturbation,
it is therefore likely that it is caused by the dynami-
cal interaction of close neighboring galaxies. Indeed,
about 30% of E+A galaxies are found to have morpho-
logical signatures of perturbations, such as tidal tails,
supporting the interaction hypothesis.
In fact, the spiral galaxies in clusters seem to dif-
fer statistically from those of field spirals, in that the
fraction of disk galaxies with absorption-line spectra,
and thus no ongoing star formation, seems to be larger
in clusters than in the field by a factor ~ 4, indicating
6. Clusters and Groups of Galaxies
that the cluster environment has a marked impact o
star-formation ability of these galaxies.
6.3 X-Ray Radiation
from Clusters of Galaxies
One of the most important discoveries of the UHURU
X-ray satellite, launched in 1970, was the detection
of X-ray radiation from massive clusters of galaxies.
With the later Einstein X-ray satellite and more recently
ROSAT, X-ray emission was also detected from lower-
mass clusters and groups. Three examples for the X-ray
emission of galaxy clusters are displayed in Figs. 6.13-
6.15. Figure 6.13 shows the Coma cluster of galaxies,
observed with two different X-ray observatories. Al-
though Coma was considered to be a fully relaxed
cluster, distinct substructure is visible in its X-ray
radiation. The cluster RXJ 1347-1145 (Fig. 6.14) is
regarded as the most luminous cluster in the X-ray do-
main. A large mass estimate of this cluster also follows
from the analysis of the gravitationally lensed arcs (see
Sect. 6.5) that are visible in Fig. 6.14; the cover of this
book shows a more recent image of this cluster, taken
with the ACS camera on-board HST, where a large num-
ber of arcs can be readily detected. Finally, Fig. 6.15
shows a superposition of the X-ray emission and an op-
tical image of the cluster MS 1 054—03 , which is situated
at z — 0.83 and to which we will refer as an example
frequently below.
6.3.1 General Properties of the X-Ray Radiation
Clusters of galaxies are the brightest extragalactic X-ray
sources besides AGNs. Their characteristic luminosity
is L x ~ 10 43 up to ~ 10 45 erg/s for the most massive
clusters. This X-ray emission from clusters is spatially
extended, so it does not originate in individual galax-
ies. The spatial region from which we can detect this
radiation can have a size of 1 Mpc or even larger. Fur-
thermore, the X-ray radiation from clusters does not
vary on timescales over which it has been observed
(< 30 yr). Variations would also not be expected if the
radiation originates from an extended region.
Continuum Radiation. The spectral energy dis-
tribution of the X-rays leads to the conclusion
that the emission process is optically thin thermal
bremsstrahlung (free-free radiation) from a hot gas.
This radiation is produced by the acceleration of elec-
Fig. 6.13. X-ray images of the Coma cluster, taken with the
ROSAT-PSPC (left) and XMM-EPIC (right). The image size
in the left panel is 2.7° x 2S ' . A rental kablc feature is the sec-
ondary maximum in the X-ray emission at the lower right of
the cluster center which shows that even Coma, long consid-
ered to be a regular cluster, is not completely in an equilibrium
state, but is dynamically evolving, presumably by the at
of a galaxy group
6.3 X-Ray Radiation from Clusters of Galaxies
•RXJ1347-1145
* R05AT HRI
•■'-*:•"-'
*■ z
Fig. 6.14. RXJ 1347- 1 145 is the most luminous galaxy clus-
ter in the X-ray domain. A color-coded ROSAT/HR1 image
of this cluster, which shows the distribution of the intergalac
tic gas, is superposed on an optical image of the cluster. The
two arrows indicate giant arcs, images of background galaxies
which are strongly distorted by the gravitational lens effect
trons in the Coulomb field of protons and atomic nuclei.
Since an accelerated electrically charged particle emits
radiation, such scattering processes between electrons
and protons in an ionized gas yields emission of pho-
tons. From the spectral properties of this radiation, the
gas temperature in galaxy clusters can be determined,
which is, for clusters with mass between ~ 10 I4 M o
and ~ 1O 15 M , in the range of 10 7 -10 8 K, or 1-10 keV,
respectively.
The emissivity of bremsstrahlung is described by
32jtZ 2 e 6 n t
? v/k * T g a (T,v),
(6.30)
where e denotes the elementary charge, n e and «; the
number density of electrons and ions, respectively, Z the
charge of the ions, and m e the electron mass. The func-
tion gg is called Gaunt factor; it is a quantum mechanical
correction factor of order 1, or, more precisely,
(9k B T\
gs « — = In , .
*fn \4hpvJ
Hence, the spectrum described by (6.30) is flat
for hpv <$C k%T, and exponentially decreasing for
h?v > k^T, as is displayed in Fig. 6.16.
HST * WFPC2
Ground + X-ray
PRC98-26 'August 19, 1998
STScI -OPO
M. Donahue (STScI) and NASA
Fig. 6.15. The cluster of
galaxies MS 1054-03 is,
at z = 0.83, the highest-
redshift cluster in the
Einstein Medium Sensi-
tivity Survey, which was
compiled from observa-
tions with the Einstein
satellite (see Sect. 6.3.5).
On the right, an HST
image of the cluster is
shown, while on the
left is an optical im-
age, obtained with the
2.2-m telescope of the
University of Hawaii, su-
perposed (in blue) with
the X-ray emission of the
cluster measured with the
ROSAT-HRI
6. Clusters and Groups of Galaxies
The temperature of the gas in massive clusters
is typically T ~ 5 x 10 7 K, or k B T ~ 5 keV - X-ray
astronomers usually specify temperatures and fre-
quencies in keV (see Appendix C). For a thermal
plasma with Solar abundances, the total bremsstrahlung
emission js
(6.31)
The energy resolution and the angular resolution of
X-ray satellites prior to the Chandra and XMM-Newton
observatories, which were both launched in 1999, did
not permit detailed analyses of the spatial dependence
of the gas temperature. Therefore, it is often assumed
in modeling the X-ray emission from clusters that T is
spatially constant. However, observations by these two
recent satellites show that in many cases this assumption
is not well justified because clear temperature gradients
are observed.
Line Emission. The assumption that the X-ray emission
originates from a hot, diffuse gas (intracluster medium,
ICM) was confirmed by the discovery of line emission
in the X-ray spectrum of clusters. The most prominent
line in massive clusters is located at energies just be-
low 7 keV: it is the Lyman-a line of 25-fold ionized iron
(thus, of an iron nucleus with only a single electron).
Slightly less ionized iron has a strong transition at some-
what lower energies of E ~ 6.4 keV. Later, other lines
were also discovered in the X-ray spectrum of clusters.
As a rule, the hotter the gas is, thus the more completely
ionized it is, the weaker the line emission. The X-ray
emission of clusters with relatively low temperatures,
fc B T < 2 keV, is sometimes dominated by line emission
from highly ionized atoms (C, N, O, Ne, Mg, Si, S, Ar,
Ca; see Fig. 6.16). The emissivity of a thermal plasma
with Solar abundance and temperatures in the range
10 5 K<T<4x 10 7 K can roughly be approximated by
e s» 6.2 x 10~ 19 ( | ( " e ,) erg cm" 3 s" 1 .
Equation (6.32) accounts for free-free emission as
well as line emission. Compared to (6.31), one finds
a different dependence on temperature: while the to-
la! emissivity for bremsstrahlung is oc T 1/2 , it increases
again towards lower temperatures where the line emis-
sion becomes more important. It should be noted in
particular that the emissivity depends quadratically on
the density of the plasma, since both bremsstrahlung
and the collisional excitation responsible for line emis-
sion are two-body processes. Thus in order to estimate
the mass of the hot gas from its X-ray luminosity, the
spatial distribution of the gas needs to be known. For ex-
ample, if the gas in a cluster is locally inhomogeneous,
the value of (w^) which determines the X-ray emissiv-
ity may deviate significantly from (n e ) 2 . As we will see
later, clusters of galaxies satisfy a number of scaling re-
lations, and one relation between the gas mass and the
X-ray luminosity is found empirically, from which the
gas mass can be estimated.
Morphology of the X-Ray Emission. From the mor-
phology of their X-ray emission, one can roughly
distinguish between regular and irregular clusters, as
is also done in the classification of the galaxy distribu-
tion. In Fig. 6.17, X-ray surface brightness contours are
superposed on optical images of four galaxy clusters
or groups, respectively, covering a wide range of clus-
ter mass and X-ray temperature. Regular clusters show
a smooth brightness distribution, centered on the optical
center of the cluster, and an outwardly decreasing sur-
face brightness. Typically, regular clusters have a high
X-ray luminosity L x and high temperatures. In contrast,
irregular clusters may have several brightness maxima,
often centered on cluster galaxies or subgroups of clus-
ter galaxies. Some of the irregular clusters show a high
temperature as well, which is interpreted as a conse-
quence of merger processes between clusters, in which
the gas is heated by shock fronts. The trend emerges that
in clusters with a larger fraction of spirals, L x and T are
lower. Irregular clusters also have a lower central galaxy
density compared to regular clusters. Clusters of galax-
ies with a dominating central galaxy often show a strong
central peak in X-ray emission. The X-ray emission
often deviates from axial symmetry, so that the assump-
tion of clusters being spherically symmetric is not well
founded in these cases.
6.3 X-Ray Radiation from Clusters of Galaxies
Bremsstrahlung (ff-emission)
k B T e =1 keV, 3keV, 9 keV; N H =0crrT 2
Fig. 6.16. X-ray emission of a hot plasma. In the top panel,
the bremsstrahlung spectrum is shown, for three different gas
temperatures; the radiation of hotter gas extends to higher
photon energies, and above E ~ k%T the spectrum is expo-
nentially cut off. In the central panel, atomic transitions and
recombination radiation are also taken into account. These
additional radiation mechanisms become more important to-
wards smaller T, as can be seen from the T = 1 keV curve. In
the bottom panel, photo-absorption is included, with different
column densities in hydrogen and a metallicity of 0.4 in So-
lar units. This absorption produces a cut-off in the spectrum
towards lower energies
Energy [keV]
ff+fb+bb-emission
k B T e =1 keV, 3keV, 9 keV; A = 0.4; N H =0cr
Energy [keV]
ff+fb+bb-emission
k B T e = 3 keV, A= 0.4; N H = 0, 3 x 1 0' 20 , 1 21
Fig. 6.17. Surface brightness contours of the X-ray emis-
sion for four different groups or clusters of galaxies. Upper
left: the galaxy group NGC 5044, at redshift z = 0.009, with
an X-ray temperature of T «* 1 .07 keV and a virial mass of
M 20 o ^0.32/i" 1 x 10 14 M o .Upperright: the group MKW4, at
z = 0.02, with T « 1.71 keV and M 2 oo ^ 0.5h~ l x 1O 14 M .
Lower left: the cluster of galaxies A 0754, atz = 0.053, with
r^9.5keVandM 20 o sa 13.1/j" 1 x 10 14 M Q .Lowerright:the
cluster of galaxies A 3667, at z = 0.056, with T « 7.0 keV and
M200 w 5.6/i -1 x 1O 14 M . The X-ray data were obtained by
ROSAT, and the optical images were taken from the Digitized
Sky Survey. These clusters are part of the HIFLUGCS Survey,
which we will discuss more thoroughly in Sect. 6.3.5
6.3.2 Models of the XRay Emission
Hydrostatic Assumption. To draw conclusions about
the properties of the intergalactic (intracluster) medium
from the observed X-ray radiation and about the distri-
bution of mass in the cluster, the gas distribution needs
to be modeled. For this, we first consider the speed of
sound in the cluster gas,
- 1000 km s
average molecular n
of a gas particle in u:
(m)
/z := .
nk B T
/ P% V p &
where P denotes the gas pressure, p g the gas density,
and n the number density of gas particles. Then, the
ss is defined as the average mass
ts of the proton mass,
(6.33)
so that p g — n{m)= n^m p . For a gas of fully ionized
hydrogen, one gets /x = 1/2 because in this case one has
one proton and one electron per ~proton mass. Since the
cluster gas also contains helium and heavier elements,
one obtains p, ~ 0.63. The sound-crossing time for the
cluster is
fsc = ^^7xl0 8 yr,
6.3 X-Ray Radiation from Clusters of Galaxies
and is thus, for a cluster with T ~ 10 K, significantly
shorter than the lifetime of the cluster, which can be
approximated roughly by the age of the Universe. Since
the sound-crossing time defines the time-scale on which
deviations from the pressure equilibrium are evened out,
the gas can be in hydrostatic equilibrium. In this case,
the equation
VP = -p g ^
(634)
applies, with <P denoting the gravitational potential.
Equation (6.34) describes how the gravitational force
is balanced by the pressure force. In the spherically
symmetric case in which all quantities depend only on
the radius r, we obtain
1 AP _
Pi dr
d<Z> _
~~d7~
GM{r)
(635)
where M{r) is the mass enclosed within radius r. Here,
M(r) is the total enclosed mass, i.e., not just the gas
mass, because the potential <t> is determined by the to-
tal mass. By inserting P = nk^T = p g kBT/(pun p ) into
(6.35), we obtain
ksTr 2 .
G/im p '
d In
dr
dlnT\
This equation is of central importance for the X-ray as-
tronomy of galaxy clusters because it shows that we can
derive the mass profile M(r) from the radial profiles of
p g and T. Thus, if one can measure the density and tem-
perature profiles, the mass of the cluster, and hence the
total density, can be determined as a function of radius.
However, these measurements are not without diffi-
culties. p g (r) and T(r) need to be determined from the
X-ray luminosity and the spectral temperature, using
the bremsstrahlung emissivity (6.30). Obviously, they
can be observed only in projection in the form of the
surface brightness
J JT^pJ
enough to measure both p g (r) and T(r) with suffi-
cient accuracy, except for the nearest clusters. For this
reason, the mass determination is often performed by
employing additional, simplifying assumptions.
Isothermal Gas Distribution. From the radial profile
of I(R), e(r) can be derived by inversion of (6.37).
Since the spectral bremsstrahlung emissivity depends
only weakly on T for h P v <$C k B T, due to (6.30). Hie
radial profile of the gas density p g can be derived
from e(r). The X-ray satellite ROSAT was sensitive
to radiation of 0. 1 keV < E < 2.4 keV, so that the X-ray
photons detected by it are typically from the regime
where h P v <$C k B T.
Assuming that the gas temperature is spatially con-
stant, T(r) — T g , (6.36) simplifies, and the mass profile
of the cluster can be determined from the density profile
of the gas.
The /3-Model. A commonly used method consists of fit-
ting the X-ray data by a so-called /J-model. This model
is based on the assumption that the density profile of
the total matter (dark and luminous) is described by an
isothermal distribution, i.e., it is assumed that the tem-
perature of the gas is independent of radius, and at the
same time that the mass distribution in the cluster is de-
scribed by the isothermal model that has been discussed
in Sect. 6.2.4. With (6.8) and (6.11), we then obtain for
the total density p{r)
dlnp
1 GM
(6,38)
On the other hand, in the isothermal case (6.36) reduces
dlnpg
/xOTp GM
~k^T~ g ^ r '
The comparison of (6.38) and (6.39) then shows that
d In Pg/dr oc d In p/dr, or
(6.37) p g (r) ex [p(r)f with
k B T g
from which the emissivity, and thus density and temper-
ature, need to be derived by de-projection. Furthermore,
the angular and energy resolution of X-ray telescopes
prior to XMM-Newton and Chandra were not high
must apply; thus the gas density follows the total density
to some power. Here, the index fi depends on the ratio of
the dynamical temperature, measured by a v , and the gas
temperature. Now, using the King approximation for an
6. Clusters and Groups of Galaxies
isothermal mass distribution - see (6.16) - as a model
for the mass distribution, we obtain
Pg(r) = P g o
where p gu is the central gas density. The brightness
profile of the X-ray emission in this model is then,
according to (6.37),
The X-ray emission of many clusters is well de-
scribed by this profile, 4 yielding values for r c of 0. 1 to
0.3ft -1 Mpc and a value for the index fi = fa « 0.65.
Alternatively, yS can be measured, with the definition
given in (6.40), from the gas temperature T g and the
velocity dispersion of the galaxies a v , which yields
typical values of /J = p\ pec ss 1. Such a value would
also be expected if the mass and gas distributions were
both isothermal. In this case, they should have the same
temperature, which was presumably determined by the
formation of the cluster.
The fact that the two values for /J determined above
differ from each other (the so-called ^-discrepancy ) is
as yet not well understood. The measured values for fa
often depend on the angular range over which the bright-
ness profile is fitted; the larger this range, the larger fa
becomes, and thus the smaller the discrepancy. Further-
more, temperature measurements of clusters are often
not very accurate because it is the emission-weighted
temperature which is measured, which is, due to the
quadratic dependence of the emissivity on p g , domi-
nated by the regions with the highest gas density. The
fact that the innermost regions of clusters where the
gas density is highest tend to have a temperature be-
low the bulk temperature of the cluster may lead to
an underestimation of "the" cluster temperature. In ad-
dition, the near independence of the spectral form of
ejj from T for h P v <^k s T renders the measurement of
4 We point out that the pair of equations (6.41) and (6.42) is valid
independent!) of the \alidit\ of die assumptions from which (6. ! 1 )
was obtained. If the obsened X ra\ emission is very well described
h\ (0.12). the gas densih profile (6. 11) can be obtained from it.
independent!) of the \ahdit\ of the assumptions made before
T difficult. Only with Chandra and XMM-Newton can
the X-ray emission also be mapped at energies of up
to E < 10 keV, which results in considerably improved
temperature measurements.
Such investigations have revealed that the gas is not
really isothermal. Typically, the temperature decreases
towards the center and towards the edge, while it is
rather constant over a larger range at intermediate radii.
Many clusters are found, however, in which the temper-
ature distribution is by no means radially symmetric, but
shows distinct substructure. Finally, as another possible
explanation for the ^-discrepancy, it should be men-
tioned that the velocity distribution of those galaxies
from which a v is measured may be anisotropic.
Besides all the uncertainty as to the validity of the
/J-model, we also need to mention that numerical sim-
ulations of galaxy clusters, which take dark matter and
gas into account, have repeatedly come to the conclu-
sion that the mass determination of clusters, utilizing
the /S-model, should achieve an accuracy of better than
~ 20%, although different gas dynamical simulations
have arrived at distinctly different results.
Dark Matter in Clusters from X-Ray Observations.
Based on measurements of their X-ray emission, a mass
estimate can be performed for galaxy clusters. It is
found, in agreement with the dynamical method, that
clusters contain much more mass than is visible in gal-
axies. The total mass of the intergalactic medium is
clearly too low to account for the missing mass; its gas
mass is only ~ 15% of the total mass of a cluster.
The mass of clusters of galaxies consists of ~ 3%
contribution from stars in galaxies and ~ 15% from
intergalactic gas, whereas the remaining ~ 80%
consists of dark matter which therefore dominates
the mass of the clusters.
6.3.3 Cooling Flows
In examining the intergalactic medium, we have as-
sumed hydrostatic equilibrium, but we have disregarded
the fact that the gas cools by its emission, thus it will
lose internal energy. For this reason, once established,
a hydrostatic equilibrium cannot be maintained over ar-
bitrarily long times. To decide whether this gas cooling
is important for the dynamics of the system, the cool-
ing time-scale needs to be considered. This cooling time
turns out to be very long,
^"•Mio^fGfe)"
where u — (3/2)nk s T g is the energy density of the gas
and « e the electron density. Hence, the cooling time is
longer than the Hubble time nearly everywhere in the
cluster, which allows a hydrostatic equilibrium to be
established. In the centers of clusters, however, the den-
sity may be sufficiently large to yield f coo i < t o ~ Hq l .
Here, the gas can cool quite efficiently, by which its
pressure decreases. This then implies that, at least close
to the center, the hydrostatic equilibrium can no longer
be maintained. To re-establish pressure equilibrium, gas
needs to flow inwards and is thus compressed. Hence,
an inward-directed mass flow should establish itself.
The corresponding density increase will further accel-
erate the cooling process. Since the emissivity (6.32)
of a relatively cool gas increases with decreasing tem-
perature, this process should then very quickly lead to
a strong compression and cooling of the gas in the cen-
ters of dense clusters. In parallel to this increase in
density, the X-ray emission will strongly increase, be-
cause e s on n\. As a result of this process, a radial density
and temperature distribution should be established with
a nearly unchanged pressure distribution. In Fig. 6.18,
the cooler gas in the center of the Centaurus cluster is
clearly visible.
These so-called cooling flows have indeed been ob-
served in the centers of massive clusters, in the form of
a sharp central peak in I(R). However, we need to stress
that, as yet, no inwardsy?ow.s have been measured. Such
a measurement would be very difficult, though, due to
the small expected velocities. The amount of cooling gas
can be considerable, with models predicting values of
up to several 100M o /yr. However, after spectroscopic
observations by XMM-Newton became available, we
have learned that these very high cooling rates implied
by the models were significantly overestimated.
Fig. 6.18. Chandra image of the Centaurus cluster; the size of
the field is 3' x 3'. Owing to the excellent angular resolution of
the Chandra satellite, the complexity of the morphology in the
X-ray emission of clusters can be analyzed. Colors indicate
photon energies, from low to high in red, yellow, green, and
blue. The relatively cool inner region might be the result of
a cooling flow
The Fate of the Cooling Gas. The gas cooling in this
way will accumulate in the center of the cluster, but
despite the expected high mass of cold gas, no clear
evidence has been found for it. In clusters harboring
a cD galaxy, the cooled gas may, over a Hubble time,
contribute a considerable fraction of the mass of this gal-
axy. Hence, the question arises whether cD galaxies may
have formed by accretion in cooling flows. In this sce-
nario, the gas would be transformed into stars in the cD
galaxy. However, the star-formation rate in these central
galaxies is much lower than the rate by which cluster
gas cools, according to the "old" cooling flow models.
The sensitivity and spectral resolution achieved with
XMM-Newton have strongly modified our view of cool-
ing flows. In the standard model of cooling flows, the
gas cools from the cluster temperature down to temper-
atures significantly below 1 keV. In this process many
atomic lines are emitted, produced by various ioniza-
tion stages, e.g., of iron, which change with decreasing
temperature. Figure 6. 19 (top panel) shows the expected
spectrum of a cooling flow in which the gas cools down
6. Clusters and Groups of Galaxies
Isobaric Multiphase Cooling Flow Model
Fig. 6.19. In the top panel, a model spectrum of
a cooling flow is shown, in which the gas cools
down from 8 keV to T g = 0. The strong lines
of FeXVII can be seen. In the central panel, the
spectrum of Abell 1835 is superposed on the
model spectrum; clear discrepancies are visi-
ble, especially the absence of strong emission
lines from FeXVII. If the gas is not allowed to
cool down to temperatures below 3 keV (bot-
tom panel), the agreement with observation
improves visibly
Rest Wavelength (A)
from the cluster temperature of T g « 8 keV to T g = 0,
where a chemical composition of 1/3 Solar abundance
is assumed. In the central panel, this theoretical spec-
trum is compared with the spectrum of the cluster Abell
1835, where very distinct discrepancies become visi-
ble. In the bottom panel, the model was modified such
that the gas cools down only to T g = 3 keV; this model
clearly matches the observed spectrum better.
Hence, cooler gas in the inner regions of clusters has
now been directly detected spectroscopically. However,
the temperature measurements from X-ray spectroscopy
are significantly different from what one would expect.
The above arguments imply that drastic cooling should
take place in the gas, because the process of compres-
sion and cooling will accelerate for ever decreasing T g .
Therefore, one expects to find gas at all temperatures
lower than the temperature of the cluster. But this seems
not to be the case: whereas gas at T g > 1 keV is found,
no gas seems to be present at even smaller temperatures,
although the cooling flow models predict the existence
of such gas. A minimum temperature seems to exist, be-
low which the gas cannot cool, or the amount of gas that
cools to T g = is considerably smaller than expected
from the cooling flow model. This lower mass rate of
gas that cools down completely would then also be com-
patible with the observed low star-formation rates in the
central galaxies of clusters. In fact, a correlation be-
tween the cooling rate of gas as determined from XMM
observations and the regions of star formation in clusters
has been found.
6.3 X-Ray Radiation from Clusters of Galaxies
What Prevents Massive Cooling Flows? One way to
explain the clearly suppressed cooling rates in cooling
flows is by noting that many clusters of galaxies harbor
an active galaxy in their center, the activity of which,
e.g., in form of (radio-)jets, may affect the ICM. For in-
stance, energy could be transferred from the jet to the
ICM, by which the ICM is heated. This heating might
then prevent the temperature from dropping to arbitrar-
ily small values. This hypothesis is supported by the fact
that many clusters are known in which the ICM is clearly
affected by the central AGN - see Fig. 6.20 for one of the
first examples where this effect has been seen. Plasma
from the jet seems to locally displace the X-ray emitting
gas. By friction and mixing in the interface region be-
tween the jet and the ICM, the latter is certainly heated.
It is unclear, though, whether this explanation is valid for
every cluster, because not every cluster in which a very
cool ICM is expected also contains an observed AGN.
On the other hand, this is not necessarily an argument
against the hypothesis of AGNs as heating sources, since
AGNs often have a limited time of activity and may be
switching on and off, depending on the accretion rate.
Thus, the gas in a cluster may very well be heated by an
AGN even if it is currently (at the time of observation)
inactive. Another example of apparently underdense
regions in the X-ray gas of a group is shown in Fig. 6.21.
The Bullet Cluster. Clusters of galaxies are indeed
excellent laboratories for hydrodynamic and plasma-
physical processes on large scales. In them, shock fronts,
for instance in merging clusters, cooling fronts (which
are also called "contact discontinuities" in hydrody-
namics), and the propagation of sound waves can be
observed. A particularly good example is the galaxy
cluster IE 0657-56 displayed in Fig. 6.22, the "bullet
cluster". To the right of the cluster center, strong and rel-
atively compact X-ray emission (the "bullet") is visible,
while further to the right of it one sees an arc-shaped dis-
continuity in surface brightness. From the temperature
distribution on both sides of the discontinuity one infers
Fig. 6.20. A ROSAT-HRI-image of the central region of the
Perseus cluster, with its central galaxy NGC 1275. The latter
is the center of the emission in both the radio and the X-
ray, here displayed as contours and color-coded, respectively.
Clearly identifiable is the effect of the radio-jets on the X-ray
emission - at the location of the radio lobes the X-ray ei
is strongly suppressed
Fig. 6.21. Galaxy groups are also X-ray emitters, albeit weaker
than clusters of galaxies. Moreover, the temperature of the
ICM is lower than in clusters. This 4' x 4' Chandra image
shows HCG 62. Note the complexity of the X-ray emission
and the two symmetrically aligned regions that seem to be
\ irtually devoid of hot ICM - possibly holes blown free by
jets from the central galaxy of this group (NGC 4761)
6. Clusters and Groups of Galaxies
Fig. 6.22. The cluster of galaxies IE 0657-56 is a perfect
example of a merging cluster. On the left, a Chandra im-
age of the cluster, while the right-hand image shows the
superposition of the X-ray contours on an optical R-band im-
age taken with the ESO NTT. The most remarkable feature
in the X-ray map is the compact region to the right (west-
wards) of the cluster center (from which the cluster derives
its name the "bullet cluster"), and the sharp tr
surface brightness further to the right of it. An analysis of
the brightness profile and of the X-ray temperature distribu-
tion shows that this must be a shock front moving at about
2.5 times the speed of sound, or v ~ 3500 km/s, through the
gas. To the right of this shock front, a group of galaxies is
visible
that it is a shock front. The strength of the shock im-
plies that the "bullet" is moving at about v ~ 3500 km/s
(from left to right in the figure) through the intergalac-
tic medium of the cluster. The interpretation of this
observation is that we are witnessing the merger of two
clusters, where one less massive cluster has passed, from
left to right in Fig. 6.22, through a more massive one.
The "bullet" in this picture is understood to be gas from
the central region of the less massive cluster, which is
still rather compact. This interpretation is impressively
supported by the group of galaxies to the right of the
shock front, which are probably the former member gal-
axies of the less massive cluster. As this cluster crosses
through the more massive one, its galaxies and dark
matter are moving collisionlessly, whereas the gas is
decelerated by friction with the gas in the massive clus-
ter: the galaxies and the dark matter are thus able to
move faster through the cluster than the gas, which is
lagging behind.
Indeed, a weak lensing analysis of this cluster (see
Sect. 6.5.2) shows that the mass of the small cluster
component is centered on the associated group of gal-
axies, not on the intracluster gas component causing
the X-ray emission near the bullet. This result provides
additional strong evidence for the interpretation given
above, showing that galaxies and dark matter together
behave as a collisionless gas as they cross the cluster,
whereas the gas itself is held back by friction.
6.3.4 The Sunyaev-Zeldovich Effect
Electrons in the hot gas of the intracluster medium can
scatter photons of the cosmic microwave background.
The optical depth and thus the scattering probability
for this Compton scattering is relatively low, but the
effect is nevertheless observable and, in addition, is of
great importance for the analysis of clusters, as we will
now see.
A photon moving through a cluster of galaxies to-
wards us will have a different direction after scattering
and thus will not reach us. But since the cosmic back-
ground radiation is isotropic, for any CMB photon that
is scattered out of the line-of- sight, another photon ex-
ists - statistically - that is scattered into it, so that the
6.3 X-Ray Radiation from Clusters of Galaxies
total number of photons reaching us is preserved. How-
ever, the energy of the photons changes slightly through
scattering by the hot electrons, in a way that they have an
(on average) higher frequency after scattering. Hence,
by Compton scattering, energy is on average transferred
from the electrons to the photons (see Fig. 6.23).
As a consequence, this scattering leads to a re-
duced number of photons at lower energies, relative
to the Planck spectrum, and higher energy photons be-
ing added. This effect is called the Sunyaev-Zeldovich
effect (SZ effect). It was predicted in 1970 and has now
been observed in a large number of clusters.
The CMB spectrum, measured in the direction
of a galaxy cluster, deviates from a Planck spec-
trum; the degree of this deviation depends on the
temperature of the cluster gas and on its density.
In the Ray leigh- Jeans domain of the CMB spectrum,
thus at wavelengths larger than about 1 mm, photons are
effectively removed by the SZ effect. For the change in
specific intensity in the RJ part, one obtains
Al Kl
-2v
Wavelength (mm)
(6.44)
Frequency (GHz)
Fig. 6.23. The influence of the Sunyaev-Zeldovich effect on
llic cosmic background radiation. The dashed curve represents
the Planck distribution of the unperturbed CMB spectrum, the
solid curve shows the spectrum after the radiation has passed
through a cloud of hot electrons. The magnitude of this effect,
for clarity, has been very much exaggerated in this sketch
(6.45)
is the Compton-y parameter and crj the Thomson
cross-section for electron scattering. Obviously, v is
proportional to the optical depth with respect to Comp-
ton scattering, given as an integral over n e er T along the
line-of- sight. Furthermore, y is proportional to the gas
temperature, because that defines the average energy
transfer per scattering event. Overall, y is proportional
to the integral over the gas pressure P — nk B T along
the line-of- sight through the cluster.
Observations of the SZ effect provide another possi-
bility for analyzing the gas in clusters. For instance, if
one can spatially resolve the SZ effect, which is possible
today with interferometric methods (see Fig. 6.24), one
obtains information about the spatial density and tem-
perature distribution. Here it is of crucial importance
that the dependence on temperature and gas density is
different from that in X-ray emission. Because of the
quadratic dependence of the X-ray emissivity on n e ,
the X-ray luminosity depends not only on the total gas
mass, but also on its spatial distribution. Small-scale
clumps in the gas, for instance, would strongly affect
the X-ray emission. In contrast, the SZ effect is linear
in gas density and therefore considerably less sensitive
with respect to inhomogeneities in the ICM.
The next generation of radio telescopes, which will
operate in the mm-domain of the spectrum, will perform
SZ surveys and search for clusters of galaxies by the SZ
effect. The outcome of these surveys is expected to be
a particularly useful sample of galaxy clusters because
this selection criterion for clusters does not depend on
the detailed gas distribution. Equations (6.44) and (6.45)
also show that the SZ effect is independent of the cluster
redshift, as long as the change in CMB temperature is
spatially resolved. For this reason, it is expected that
SZ surveys will identify a large number of high-redshift
clusters.
Distance Determination. For a long time, the SZ effect
was mainly considered a tool for measuring distances to
clusters of galaxies, and from this the Hubble constant.
We will now schematically show how the SZ effect,
6. Clusters and Groups of Galaxies
Fig. 6.24. Sunyaev-Zeldovich maps of three clusters of galax-
ies at 0.37 < z < 0.55. Plotted is the temperature difference of
the measured CMB relative to the average CMB temperature
(or, at fixed frequency, the difference in radiation intensities).
The black ellipse in each image specifies the instrument's
beam size. For each of the clusters shown here, the spatial
dependence of the SZ effect is clearly visible. Since the SZ
effect is proportional to the electron density, the mass frac-
tion of baryons in clusters can be measured if one additionally
knows the total mass of the cluster from dynamical methods
or from the X-ray temperature. The analysis of the clusters
shown here yields for the mass fraction of the intergalactic
gas / g fs 0.08 h~ l
in combination with the X-ray emission, allows us to
determine the distance to a cluster. The change in the
CMB intensity has the dependence
A RJ
- oc n e L T g ,
where L is the extent of the cluster along the line-of-
sight. To obtain this relation, we replace the /-integration
in (6.45) by a multiplication with L, which yields the
correct functional dependence. On the other hand, the
surface brightness of the X-ray radiation behaves as
Ix oc Ln\ .
Combining these two relations, we are now able to elim-
inate n e . Since T g is measurable by means of the X-ray
spectrum, the dependence
remains. Now assuming that the cluster is spherical, its
extent L along the line-of-sight equals its transverse
extent R — 6D A , where 9 denotes its angular extent and
D A the angular-diameter distance (4.45) to the cluster.
With this assumption, we obtain
Da
M/, RJ
l
(d.4(i)
Hence, the angular-diameter distance can be determined
from the measured SZ effect, the X-ray temperature of
the ICM, and the surface brightness in the X-ray domain.
Of course, this method is more complicated in practice
than demonstrated here, but it is applied to the distance
determination of clusters. In particular, the assumption
of the same extent of the cluster along the line-of-sight
as its transverse size is not well justified for any individ-
ual cluster, but one expects this assumption to be valid
on average for a sample of clusters. Hence, the SZ effect
is another method of distance determination, indepen-
dent of the redshift of the cluster, and therefore suitable
for determining the Hubble constant.
Discussion. The natural question arises whether this
method, in view of the assumptions it is based on, can
compete with the determination of the Hubble constant
via the distance ladder and Cepheids. The same ques-
tion also needs to be asked for the determination of Hq
by means of the time delay in gravitational lens systems,
which we discussed in Sect. 3.8.4. In both cases, the an-
swer to this question is the same: presumably neither
of the two methods will provide a determination of the
Hubble constant with an accuracy comparable to that
achieved by the Hubble Key Project (Sect. 3.6.3) and
from the angular fluctuations in the CMB (see Sect. 8.6).
Nevertheless, both methods are of great value for cos-
mology: first, the distance ladder has quite a number of
rungs. If only one of these contains an as yet undetected
severe systematic error, it could affect the resulting
value for H . Second, the Hubble Key Project mea-
sured the expansion rate in the local Universe, typically
within ~ 100 Mpc (the distance to the Coma cluster).
As we will see later, the Universe contains inhomo-
6.3 X-Ray Radiation from Clusters of Galaxies
geneities on these length-scales. Thus, it may well be
that we live in a slightly overdense or underdense region
of the Universe, where the Hubble constant deviates
from the global value. In contrast to this, both the SZ
effect and the lensing method measure the Hubble con-
stant on truly cosmic scales, and both methods do so
in only a single step - there is no distance ladder in-
volved. For these reasons, these two methods are of great
importance in additionally confirming our Hq measure-
ments. Another aspect adds to this, which must not
be underestimated: even if the same or a similar value
results from these measurements as the one from the
Hubble Key Project, we still have learned an impor-
tant fact, namely that the local Hubble constant agrees
with the one measured on cosmological scales - this
is one of the predictions of our cosmological model,
which can thus be tested in an impressive way. Indeed,
both methods have been applied to quite a number of
lens systems and luminous clusters showing an SZ ef-
fect, respectively, and they yield values for H which
are slightly smaller than, but compatible within the er-
ror bars with the value of H obtained from the Hubble
Key Project.
6.3.5 X-Ray Catalogs of Clusters
Originally, clusters of galaxies were selected by over-
densities of galaxies on the sphere using optical
methods. As we have seen, projection effects may play
a crucial role in this, in the form of coincidental over-
densities in the projected galaxy distribution, which do
not correspond to spatial overdensities. In addition, one
has the superposition of foreground and background
galaxies, which renders the selection more difficult the
farther the clusters are away from us.
A more reliable way of selecting clusters is by their
X-ray emission, since the hot X-ray gas signifies a deep
potential well, thus a real three-dimensional overdensity
of matter, so that projection effects become virtually
negligible. The X-ray emission is oc n\, which again
renders projection effects improbable. In addition, the
X-ray emission, its temperature in particular, seems to
be a very good measure for the cluster mass, as we
will discuss further below. Whereas the selection of
clusters is not based on their temperature, but on the
X-ray luminosity, we shall see that L x is also a good
indicator for the mass of a cluster (see Sect. 6.4).
The first cosmologically interesting X-ray catalog of
galaxy clusters was the EMSS (Extended Medium Sen-
sitivity Survey) catalog. It was constructed from archival
images taken by the Einstein observatory which were
scrutinized for X-ray sources other than the primary
target in the field-of-view of the respective observation.
These were compiled and then further investigated us-
ing optical methods, i.e., photometry and spectroscopy.
The EMSS catalog contains 835 sources, most of them
AGNs, but it also contains 104 clusters of galaxies.
Among these are six clusters at redshift > 0.5; the most
distant is MS 1054-03 at z = 0.83 (see Fig. 6.15). Since
the Einstein images all have different exposure times,
the EMSS is not a strictly flux-limited catalog. But with
the flux limit known for each exposure, the luminosity
function of clusters can be derived from this.
The same method as was used to compile the EMSS
was applied to ROSAT archival images by various
groups, leading to several catalogs of X-ray- selected
clusters. The selection criteria of these different groups,
and therefore of the different catalogs, differ. Since
ROSAT was more sensitive than the Einstein observa-
tory, these catalogs contain a larger number of clusters,
and also ones at higher redshift (Fig. 6.25). Furthermore,
ROSAT performed a survey of the full sky, the ROSAT
All Sky Survey (RASS). The RASS contains about 10 5
sources distributed over the whole sky. The identifica-
tion of extended sources in the RASS (in contrast to
non-extended sources - about five times more AGNs
than clusters are expected) yielded a catalog of clusters
as well which, owing to the relatively short exposure
times in the RASS, contains the brightest clusters. The
exposure time in the RASS is not uniform over the sky
since the applied observing strategy led to particularly
long exposures for the regions around the Northern and
Southern ecliptic pole (see Fig. 6.26).
One of the cluster catalogs that were ex-
tracted from the RASS data is the HIFLUGCS
catalog. It consists of the 63 X-ray-brightest clus-
ters and is a strictly flux-limited survey, with
/ x (0.1-2.4keV) > 2.0 x 10" 11 ergs" 1 cm" 1 ; it ex-
cludes the Galactic plane, \b\ > 20°, as well as other
regions around the Magellanic clouds and the Virgo
Cluster of galaxies in order to avoid large column den-
sities of Galactic gas which lead to absorption, as well as
Galactic and other nearby X-ray sources. The extended
HIFLUGCS survey contains, in addition, several other
Fig. 6.25. Left: Chandra
image of a 6' x 6' -field
with two clusters of galax-
ies at high redshift. Right:
a 2' x 2' -field centered on
one of the clusters pre-
sented on the left (RX
J0849+4452), in B, I, and
K, overlaid with the X-ray
brightness cc
clusters for which good measurements of the brightness
profile and the X-ray temperature are available.
From the luminosity function of X-ray clusters,
a mass function can be constructed, using the rela-
tion between L x and the cluster mass that will be
discussed in the following section. As we will explain
in more detail in Sect. 8.2, this cluster mass function is
an important probe for cosmological parameters.
6.4 Scaling Relations
for Clusters of Galaxies
Our examination of galaxies revealed the existence of
various scaling relations, for example the Tully-Fisher
relation. These have proven to be very useful not only for
the distance determination of galaxies, but also because
any successful model of galaxy evolution needs be able
to explain these empirical scaling relations. Therefore,
it is of great interest to examine whether clusters of
galaxies also fulfill any such scaling relation. As we
will see, the X-ray properties of clusters play a central
role in this.
6.4.1 Mass-Temperature Relation
It is expected that the larger the spatial extent, velocity
dispersion of galaxies, temperature of the X-ray gas,
and luminosity of a cluster are, the more massive it is.
In fact, from theoretical considerations one can deduce
the existence of relations between these parameters. The
X-ray temperature T specifies the thermal energy per
gas particle, which should be proportional to the binding
energy for a cluster in virial equilibrium,
M
Tex — .
Since this relation is based on the virial theorem,
r should be chosen to be the radius within which the
matter of the cluster is virialized. This value for r is
called the virial radius r vlT . From theoretical consider-
ations of cluster formation (see Chap. 7), one finds that
the virial radius is defined such that within a sphere of
radius r vir , the average mass density of the cluster is
about A c ss 200 times as high as the critical density p cr
of the Universe. The mass within r v ; r is called the virial
mass M v ; r which is, according to this definition,
4/(
^c Per 4
Combining the two above relations, one obtains
T oc — ^ oc r 2 - <x M 2 (?
This relation can now be tested on observations by us-
ing a sample of galaxy clusters with known temperature
and with mass determined by the methods discussed in
Sect. 6.3.2. An example of this is displayed in Fig. 6.27,
in which the mass is plotted versus temperature for clus-
ters from the extended HIFLUGCS sample. Since it is
6.4 Scaling Relations for Clusters of Galaxies
Fig. 6.26. The top panel shows the total exposure time in the
ROS AT All Sky Survey as a function of sky position. Near the
ecliptic poles the exposure time is longest, as a consequence of
the applied observing strategy. Because of the "South Atlantic
Anomaly" (a region of enhanced cosmic ray flux over the
South Atlantic Ocean, off the coast of Brazil, caused by the
shape of the Earth's magnetosphere), the exposure time is
generally higher in the North than in the South. The X ra\ sky,
as observed in the RASS, is shown in the lower panel. The
colors indicate the shape of the spectral energy distribution,
where blue indicates sources with a harder spectrum
"clusters" of low mass and temperature) are located be-
low the power-law fit that is obtained from higher mass
clusters. If one confines the sample to clusters with
M > 5 x 10 13 M o , the best fit is described by
M 5OO = 3.57xlO-M (^) L
(6,49»
with an uncertainty of slightly more than 10%. This rela-
tion is very similar to the one deduced from theoretical
considerations, M oc T L5 . With only small variations
in the parameters, the relation (6.49) is obtained both
from a cluster sample in which the mass was de-
termined based on an isothermal ,6-model, and from
a cluster sample in which the measured radial tempera-
ture profile T(r) was utilized in the mass determination
(see Eq. 6.36). Constraining the sample to clusters
with temperatures above 3 keV, one obtains a slope
of 1.48 ±0.1, in excellent agreement with theoretical
expectations. Considerably steeper mass-temperature
relations result from the inclusion of galaxy groups
into the sample, from which we conclude that they
do not follow the scaling argument sketched above in
detail.
The X-ray temperature of galaxy clusters appar-
ently provides a very precise measure for their
virial mass, better than the velocity dispersion (see
below).
With the current X-ray observatories, it will be pos-
sible to test these mass-temperature relations with even
higher accuracy; the first preliminary results confirm the
above result, and the improved accuracy of future ob-
servations will lead to an even smaller dispersion of the
data points around the power law.
easier to determine the mass for small radii than the
virial mass itself, the mass M 50 o within the radius rsoo,
the radius within which the average density is 500
times the critical density, has been plotted here. The
measured values clearly show a very strong correla-
tion, and best-fit straight lines describing power laws of
the form M = AT 01 are also shown in the figure. The
exact values of the two fit parameters depend on the
choice of the cluster sample; the right-hand panel of
Fig. 6.27 shows in particular that galaxy groups (thus,
6.4.2 Mass-Velocity Dispersion Relation
The velocity dispersion of the galaxies in a cluster also
can be related to the mass: from (6.25) we find
3r vir o-^
m . — vir » (& sm
., it then follows that
M Vu otol
kT , keV
Fig. 6,27. For ihc clusters of galaxies from the extended HI
FLUGCS sample (see Sect. 6.3.5), the mass within a mean
overdensity of 500 is plotted as a function of X-ray tem-
perature, where a dimensionless Hubble constant of h — 0.5
has been assumed. In the left-hand panel, the mass was de-
termined b> applying an isothermal /3-model, while in the
right-hand panel, the radial temperature profile T(r) was
used to determine the mass, by means of (6.36). Most of
the temperature measurements are from observations by the
ASCA satellite. The solid and dash-dotted curves in the
kT , keV
left-hand panel show the best fit to the data, where for the
latter only the clusters from the original HIFLUGCS sample
were used. In the right-hand panel, the dotted line is a fit
to all the data in the plot, while the solid line takes into
account only clusters with a mass > 5 x 10 13 M Q . In both
panels, (he upper dotted line shows the mass-temperature
relation that was obtained from a simulation using sim-
plified gas dynamics - the slope agrees with that found
from the observations, but the amplitude is significantly too
high
This relation can now be tested using clusters for which
the mass has been determined using the X-ray method,
and for which the velocity dispersion of the cluster
galaxies has been measured. Alternatively, the relation
T oc er 2 can be tested. One finds that these relations are
essentially satisfied for the observed clusters. However,
the relation between a v and M is not as tight as the M-T
relation. Furthermore, numerous clusters exist which
strongly deviate from this relation. These are clusters
of galaxies that are not relaxed, as can be deduced from
the velocity distribution of the cluster galaxies (which
strongly deviates from a Maxwell distribution in these
cases) or from a bimodal or even more complex gal-
axy distribution in the cluster. These outliers need to
be identified, and removed, if one intends to apply the
scaling relation between mass and velocity dispersion.
6.4.3 Mass-Luminosity Relation
The total X-ray luminosity that is emitted via brems-
strahlung is proportional to the squared gas density and
the gas volume, hence it should behave as
L x a p 2 T 1/2 r 3 ir oc p 2 T 1/2 M vir . (6.52)
Estimating the gas density through p g ~ M g r~ 3 =
/ g M v i r r~ 3 , where / g = Mg/M v ; r denotes the gas frac-
tion with respect to the total mass of the cluster, and
using (6.48), we obtain
L x oc/ 2 M v 4 / 3 . (6.53)
This relation needs to be modified if the X-ray lu-
minosity is measured within a fixed energy interval.
Particularly for observations with ROSAT, which could
only measure low-energy photons (below 2.4 keV), the
received photons typically had E y < k B T, so that the
measured X-ray luminosity becomes independent of T.
Hence, one expects a modified scaling relation between
the X-ray luminosity measured by ROSAT L<2.4 keV and
the mass of the cluster,
£<2.4keVOC/ 2 M vir .
(6.54)
This scaling relation can also be tested empirically,
as shown in Fig. 6.28, where the X-ray luminosity in the
energy range of the ROSAT satellite is plotted against
the virial mass. One can immediately see that clusters of
galaxies indeed show a strong correlation between lumi-
nosity and mass, but with a clearly larger scatter than in
the mass-temperature relation. 5 Therefore, the temper-
5 It should be noted, though, that the determination of Lx and M
arc independent of each ether, whereas in the mass determination the
temperature is an explicit parameter so that the measurements ot these
two parameters are correlated.
6.4 Scaling Relations for Clusters of Galaxies
Fig. 6.28. Foi Ihc galaxy clusters in the extended HIFLUGCS
sample, ihc X ra} luminosit) in the energy range of the
ROSAT satellite is plotted versus the mass of the cluster.
The solid points show the clusters of the HIFLUGCS sam-
ple proper. For the full sample and for the main HIFLUGCS
sample, a best-fit power law is indicated by the solid line and
dashed line, respectively
ature of the intergalactic gas is a better mass indicator
than the X-ray luminosity or the velocity dispersion of
the cluster galaxies.
However, determining the slope of the relation from
the data approximately yields L<2.4kev oc M L5 , instead
of the expected behavior (L <2 .4 k e v c< M 1X> ). Obviously,
the above scaling arguments are not valid with the as-
sumption of a constant gas fraction. This discrepancy
between theoretical expectations and observations has
been found in several samples of galaxy clusters and is
considered well established. An explanation is found in
models where the intergalactic gas has not only been
heated by gravitational infall into the potential well of
the cluster. Other sources of heating may have been
present or still are. For cooler, less massive clusters,
this additional heating should have a larger effect than
for the very massive ones, which could also explain the
deviation of low-M clusters from the mass-luminosity
relation of massive clusters visible in Fig. 6.28. As has
already been argued in the discussion of cooling flows
in Sect. 6.3.3, an AGN in the inner regions of the cluster
may provide such a heating. The heating and additional
kinetic energy provided by supernovae in cluster gal-
axies is also considered a potential source of additional
heating of the intergalactic gas. It is obvious that solv-
ing this mystery will provide us with better insights into
the formation and evolution of the gas component in
clusters of galaxies.
Despite this discrepancy between the simple models
and the observations, Fig. 6.28 shows a clear correlation
between mass and luminosity, which can thus empiri-
cally be used after having been calibrated. Although
the temperature is the preferred measure for a cluster's
mass, one will in many cases resort to the relation be-
tween mass and X-ray luminosity because determining
the luminosity (in a fixed energy range) is consider-
ably simpler than measuring the temperature, for which
significantly longer exposure times are required.
6.4.4 Near-Infrared Luminosity as Mass Indicator
Whereas the optical luminosity of galaxies depends not
only on the mass of the stars but also on the star-
formation history, the NIR light is much less dependent
on the latter. As we have discussed before, the NIR lu-
minosity is thus quite a reliable measure of the total
mass in stars. For this reason, we would expect that the
NIR luminosity of a cluster is very strongly correlated
with its total stellar mass. Furthermore, if the latter is
closely related to the total cluster mass, as would be the
case if the stellar mass is a fixed fraction of the cluster's
total mass, the NIR luminosity can be used to estimate
the masses of clusters.
The Two-Micron All Sky Survey (2MASS) pro-
vides the first opportunity to perform such an analysis
on a large sample of galaxy clusters. One selects
clusters of galaxies for which masses have been de-
termined by X-ray methods, and then measures the
K-band luminosities of the galaxies within the clus-
ter. Figure 6.29 presents the resulting mass-luminosity
diagram within r 500 for 93 galaxy clusters and groups,
where the mass was derived from the clusters' X-ray
temperatures (plotted on the top axis) by means of
(6.49). A surprisingly close relation between these two
parameters is seen, which can be described by a power
law of the form
-=3.95
M50
(6.55)
1O 12 L ' V2xl0 14 M Q ;
where a Hubble constant of h — 0.7 is assumed. The
dispersion of individual clusters around this power law
is about 32%, where at least part of this scatter origi-
nates in uncertainties in the mass determination - thus,
6. Clusters and Groups of Galaxies
,(h 7O - 1 1O 14 M )
IV W
Fig. 6.29. The correlation between K-band luminosity and the
mass of galaxy clusters, measured within the radius inside
which the mean density is 500 times the critical density of
the Universe. The cluster mass has been determined by the
relation (6.49) between mass and temperature
the intrinsic scatter is even lower. This result is of great
potential importance for future studies of galaxy clus-
ters, and it renders the NIR luminosity a competitive
method for the determination of cluster masses, which
is of great interest in view of the next generation of
NIR wide-field instruments (like VISTA on Paranal, for
instance).
6.5 Clusters of Galaxies
as Gravitational Lenses
6.5.1 Luminous Arcs
In 1986, two groups independently discovered unusu-
ally stretched, arc-shaped sources in two clusters of
galaxies at high redshift (see Figs. 6.30 and 6.31). The
nature of these sources was unknown at first; they were
named arcs, or giant luminous arcs, which did not imply
any interpretation originally. Different hypotheses for
the origin of these arcs were formulated, like for instance
emission by shock fronts in the ICM, originating from
Fig. 6.30. The cluster of galaxies A 370 at redshift z = 0.375 is
one of the first two clusters in which giant luminous arcs were
found in 1986. In this HST image, the arc is clearly \ isible;
it is about 20" long, tangentially oriented with respect to the
center of the cluster which is located roughly halfway between
the two brighi lustei il us and curved towards the center
of the cluster. Only with HST images was it realized how
thin these arcs are. In this image, several other lens effects
are visible as well, for example a background galaxy that
is imaged three-fold. The arc is the image of a galaxy at
z s = 0.724
explosive events. All these scenarios were disproven
when the spectroscopy of the arc in the cluster Abell 370
showed that the source is at a much higher redshift than
the cluster itself. Thus, the arc is a background source,
subject to the gravitational lens effect (see Sect. 3.8)
of the cluster. By differential light deflection, the light
beam of the source can be distorted in such a way that
highly elongated arc-shaped images are produced.
The discovery that clusters of galaxies may act as
strong gravitational lenses came as a surprise at that
time. Based on the knowledge about the mass distribu-
6.5 Clusters of Galaxies as Gravitational Lenses
Fig. 6.31. The cluster of galaxies CI 2244-02 at redshift
z — 0.33 is the second cluster in which an arc was discov-
ered. Spectroscopic analysis of this arc revealed the redshift
of the corresponding source to be z s = 2.24 - at the time of
discovery in 1987, it was the first normal galaxy detected at
a redshift > 2. This image was observed with the near-IR cam
era ISAAC at the VLT Above the arc, one can see another
slrongh elongated source which, is probabh associated with
a galaxy at very high redshift as well
tion of clusters, derived from X-ray observations before
ROSAT, it was estimated that the central surface mass
density of clusters is not sufficiently high for strong
effects of gravitational light deflection to occur. This
incorrect estimate of the central surface mass density
in clusters originated from analyses utilizing the fi-
model which, as briefly discussed above, starts with
some heavily simplifying assumptions. 6
Hence, arcs are strongly distorted and highly magni-
fied images of galaxies at high redshift. In some massive
clusters several arcs were discovered and the unique an-
gular resolution of the HST played a crucial role in such
observations. Some of these arcs are so thin that their
'Another lesson that can be learned from the discovery of the arcs
is one regarding the psycholog) of researchers. After the first ohser
vations of arcs were published, several astronomers took a second
lock at their own images of these two clusters sad clearl) detected
the arcs in them, the reason win tins phenomenon, which had been
ohserxed much earlier, was not published before can be explained
by the fact that researchers were not completely sure about u hether
these sources were real. A certain tendency prevails in not recogniz
ins phenomena that occur une.\pcctedl\ in data as rcadilv as results
which are expected. However, there are also those researchers who
beha\c in exactly the opposite manner and even interpret phenomena
expected from theors in some unusual way.
width is unresolved even by the HST, indicating an ex-
treme length-to-width ratio. For many arcs, additional
images of the same source were discovered, sometimes
called "counter arcs". The identification of multiple
images is performed either by optical spectroscopy
(which is difficult in general, because one arc is highly
magnified while the other images of the same source
are considerably less strongly magnified and therefore
much fainter in general, and also because spectroscopy
of faint sources is very time-consuming), by multi-
color photometry (all images of the same source should
have the same color), or by common morphological
properties.
Lens Models. Once again, the simplest mass model
for a galaxy cluster as a lens is the singular isothermal
sphere (SIS). This lens model was discussed previously
in Sect. 3.8.2. Its characteristic angular scale is specified
by the Einstein radius (3.60), or
E = 28!'8( T
") 2 (~) ■
(6.56)
1000 km/s/
Very high magnifications and distortions of images can
occur only very close to the Einstein radius. This im-
mediately yields an initial mass estimate of a cluster, by
assuming that the Einstein radius is about the same as the
angular separation of the arc from the center of the clus-
ter. The projected mass within the Einstein radius can
then be derived, using (3.66). Since clusters of galaxies
are, in general, not spherically symmetric and may show
significant substructure, so that the separation of the arc
from the cluster's center may deviate significantly from
the Einstein radius, this mass estimate is not very accu-
rate in general; the uncertainty is estimated to be ~ 30%.
Models with asymmetric mass distributions predict a va-
riety of possible morphologies for the arcs and the posi-
tions of multiple images, as is demonstrated in Fig. 6.32
for an elliptical lens. If several arcs are discovered in
a cluster, or several images of the source of an arc, we
can investigate detailed mass models for such a cluster.
The accuracy of these models depends on the number
and positions of the observed lensed images; e.g., on
how many arcs and how many multiple image systems
are available for modeling. The resulting mass models
are not unambiguous, but they are robust. Clusters that
contain many lensed images have very well-determined
mass properties, for instance the mass and the mass
6. Clusters and Groups of Galaxies
<2)
«f
^O
^
Fig. 6.32. Distortions by the lens effect of an elliptical poten-
tial, as a function of the source position. The first panel shows
the source itself. The second panel displays ten positions of
the source in the source plane (numbered from 1 to 10) rela-
tive to the center of the lens; the solid curves show the inner
and outer caustics. The remaining panels ( numbered from 1 to
10) show the inner and outer critical curves and the resulting
profile within the radii at which arcs are found, or the
ellipticity of the mass distribution and its substructure.
Figure 6.33 shows two clusters of galaxies which
contain several arcs. For a long time, A 22 18 was the
classic example of the existence of numerous arcs in
a single galaxy cluster. Then after the installation of
the ACS camera on-board HST in 2002, a spectacu-
lar image of the cluster A 1689 was obtained in which
more than 100 arcs and multiple images were identified.
Several sections of this image are shown in Fig. 6.34.
For clusters of galaxies with such a rich inventory of
lens phenomena, very detailed mass models can be
constructed.
Such mass models have predictive power, allowing
an iterative modeling process. An initial simple mass
model is fitted to the most prominent lensed images
in the observation, i.e., either giant arcs or clearly rec-
ognizable multiple images. In general, this model then
predicts further images of the source producing the arc.
Close to these predicted positions, these additional im-
ages are then searched for, utilizing the morphology of
the light distribution and the color. If this initial model
describes the overall mass distribution quite well, such
images are found. The exact positions of the new im-
ages provide further constraints on the lens model which
is then refined accordingly. Again, the new model will
predict further multiple image systems, and so on. By
this procedure, very detailed models can sometimes be
obtained. Since the lens properties of a cluster depend
on the distance or the redshift of the source, the redshift
of lensed sources can be predicted from the identifica-
tion of multiple image systems in clusters if a detailed
mass model is available. These predictions can then be
verified by spectroscopic analysis, and the success of
this method gives us some confidence in the accuracy
of the lens models.
Results. We can summarize the most important results
of the examination of clusters using arcs and multi-
ple images as follows: the mass of galaxy clusters is
indeed much larger than the mass of their luminous
matter. The lensing method yields a mass which is in
very good agreement with mass estimates from the X-
ray method or from dynamical methods. However, the
core radius of clusters, i.e., the scale on which the mass
profile flattens inwards, is significantly smaller than de-
termined from X-ray observations. A typical value is
r c ~ 30/j -1 kpc, in contrast to ~ I50h~ l kpc from the
X-ray method. This difference leads to a discrepancy
in the mass determination between the two methods on
scales below ~ 200ft -1 kpc. We emphasize that, at least
in principle, the mass determination based on arcs and
multiple images is substantially more accurate because
it does not require any assumptions about the symmetry
of the mass distribution, about hydrostatic equilibrium
of the X-ray gas, or about an isothermal temperature
distribution. On the other hand, the lens effect mea-
sures the mass in cylinders because the lens equation
contains only the projected mass distribution, whereas
the X-ray method determines the mass inside spheres.
The conversion between the two methods introduces
i particular for clusters which deviate
6.5 Clusters of Galaxies as Gravitational Lenses
Fig. 6.33. Top image: the cluster of galax-
ies A 2218 (z d = 0.175) contains one of the
most spectacular arc systems. The majority
of the galaxies visible in the image are as-
sociated with the cluster and the redshifts
of many of the strongly distorted arcs have
now been measured. Bottom inuiiie: the
cluster of galaxies CI 0024+17 (z = 0.39)
contains a rich system of arcs. The arcs ap-
pear bluish, stretched in a direction which
is tangential to the cluster center. The three
arcs to the left of the cluster center, and the
arc to the right of it and closer to the center,
are images of the same background galaxy
which has a redshift of z = 1.62. Another
image of the same source was found close
to the cluster center. Also note the identi-
cal ("pretzel"-shaped) morphology of the
images
significantly from spherical symmetry. Overestimating
the core radius was the main reason why the discov-
ery of the arcs was a surprise because clusters with core
radii like the ones determined from the early X-ray mea-
surements would in fact not act as strong gravitational
lenses. Hence, the mere existence of arcs shows that the
core radius must be small.
A closer analysis of galaxy clusters with cooling
flows shows that, in these clusters, the mass profile es-
timated from X-ray observations is compatible with the
observed arcs. Such clusters are considered dynamically
relaxed, so that for them the assumption of a hydro-
static equilibrium is well justified. The X-ray analysis
has to account explicitly for the existence of a cool-
ing flow, though, and the accordingly modified X-ray
emission profile is more sophisticated than the simple
/J-model. Clusters without cooling flows are distinctly
more complex dynamically. Besides the discrepancy in
mass determination, lensing and X-ray methods can lead
to different estimates of the center of mass in such un-
relaxed clusters, which may indicate that the gas has
not had enough time since the last strong interaction or
merging process to settle into an equilibrium state.
The mass distribution in clusters often shows signif-
icant substructure. Clusters of galaxies in which arcs
are observed are often not relaxed. These clusters still
undergo dynamical evolution - they are young systems
with an age not much larger than f cross , or systems whose
Fig. 6.34. The cluster of galaxies Abell 1689
has the richest system of arcs and multiple
images found to date. In a deep ACS ex-
posure of this cluster more than a hundred
such lensed images were detected. Six sec-
tions of this ACS image are shown in which
various arcs are visible, some with an ex-
treme length-to-width ratio, indicating very
high magnification factors
equilibrium was disturbed by a fairly recent merger pro-
cess. For such clusters, the X-ray method is not well
founded because the assumptions about symmetry and
equilibrium are not satisfied. The distribution of arcs in
the cluster A2218 (Fig. 6.33) clearly indicates a non-
spherical mass distribution. Indeed, this cluster seems
to consist of at least two massive components around
which the arcs are curved, indicating that the cluster
is currently undergoing a strong merging event. This
is further supported by measurements of the temper-
ature distribution of the intracluster gas, which shows
a strong peak in the center, where the temperature is
about a factor of 2 higher than in its surrounding region.
From lens models, we find that for clusters with a cen-
tral cD galaxy, the orientation of the mass distribution
follows that of the cD galaxy quite closely. We conclude
from this result that the evolution of the cD galaxy must
be closely linked to the evolution of the cluster, e.g., by
accretion of a cooling flow onto the cD galaxy. Often,
the shape of the mass distribution very well resembles
the galaxy distribution and the X-ray emission.
The investigation of galaxy clusters with the grav-
itational lens method provides a third, completely
independent method of determining cluster masses.
It confirms that the mass of galaxy clusters signifi-
cantly exceeds that of the visible matter in stars and
in the intracluster gas. We conclude from this re-
sult that clusters of galaxies are dominated by dark
matter.
6.5.2 The Weak Gravitational Lens Effect
The Principle of the Weak Lensing Effect. In Sect. 3.8
we saw that gravitational light deflection does not only
deflect light beams as a whole, but also that the size
and shape of light beams are distorted by differential
light deflection. This differential light deflection leads,
e.g., to sources appearing brighter than they would be
without the lens effect. The giant arcs discussed above
are a very good example of these distortions and the
corresponding magnifications.
If some background sources exist which are distorted
in such an extreme way as to become visible as gi-
ant luminous arcs, then it appears plausible that many
more background galaxies should exist which are less
strongly distorted. Typically, these are located at larger
angular separations from the cluster center, where the
lens effect is weaker than at the location of the luminous
arcs. Their distortion then is so weak that it cannot be
identified in an individual galaxy image. The reason for
this is that the intrinsic light distribution of galaxies is
not circular; rather, the observed image shape is a super-
position of the intrinsic shape and the gravitational lens
distortion. The intrinsic ellipticity of galaxies is con-
siderably larger than the shear, in general, and acts as
a kind of noise in the measurement of the lensing ef-
fect. However, the distortion of adjacent galaxy images
should be similar since the gravitational field their light
beams are traversing is similar. By averaging over many
6.5 Clusters of Galaxies as Gravitational Lenses
such galaxy images, the distortion can then be measured
(see Fig. 6.35) because no preferred direction exists in
the intrinsic random orientation of galaxies. Since the
results from the Hubble Deep Field (Fig. 1 .27) became
available, if not before, we have known that the sky is
densely covered by small, faint galaxies. In deep op-
lical images, one should therefore find a high number
density of such galaxies located in the background of
a galaxy cluster. Their measured shapes can be used for
investigating the weak lensing effect of the cluster.
The distortion, obtained by averaging over image
ellipticities, reflects the contribution of the tidal forces to
the local gravitational field of the cluster. In this context,
it is denoted as shear. It is given by the projection of
the tidal contribution to the gravitational field along the
Line-of-sight. The shear results from the derivative of
Fig. 6.35. The principle of the weak gravitational lensing ef-
fect is illustrated here with a simulation. Due to the tidal
component of the gravitational field in a cluster, the shape
of the images (ellipses) of background galaxies get distorted
and, as for arcs, the galaxy images will be aligned, on aver-
age, tangentially to the cluster center. By local averaging over
Ihe ellipticities of galaxy image'-, a local estimate of the lidal
gravitational field can be obtained (the direction of the sticks
indicates the orientation of the tidal field, and their length is
proportional to its strength). From this estimated tidal field,
the projected mass distribution can then be reconstructed
the deflection angle, where the deflection angle (3.47)
depends linearly on the surface mass density of the lens.
Hence, it is possible to reconstruct the surface mass
density of galaxy clusters in a completely parameter-
free way using the measured shear: it can be used to
map the (dark) matter in a cluster.
Observations. Since shear measurements are based on
averaging over image ellipticities of distant galaxies,
this method of weak gravitational lensing requires op-
tical images with as high a galaxy density as possible.
This implies that the exposures need to be very deep
to reach very faint magnitudes. But since very faint
galaxies are also very distant and, as a consequence,
have small angular extent, the observations need to
be carried out under very good observing conditions,
to be able to accurately measure the shape of gal-
axy images without them being smeared into circular
images by atmospheric turbulence, i.e., the seeing. Typ-
ically, to apply this method images from 4-m class
telescopes are used, with exposure times of one to
three hours. This way, we reach a density of about
30 galaxies per square arcminute (thus, 10 5 per square
degree) of which shapes are sufficiently well measur-
able. This corresponds to a limiting magnitude of about
R~25. The seeing during the exposure should not be
larger than ~ 0"8 to still be able to correct for seeing
effects.
Systematic observations of the weak lensing effect
only became feasible in recent years with the devel-
opment of wide-field cameras. 7 This, together with the
improvement of the dome seeing at many telescopes and
the development of dedicated software for data analysis,
rendered quantitative observational studies with weak
lensing possible; the best telescopes at the best obser-
vatories regularly accomplish seeing below 1", and the
dedicated software is specifically designed for measur-
ing the shapes of extremely faint galaxy images and for
correcting for the effects of seeing and anisotropy of the
point-spread function.
7 Prominent examples for such cameras are, for instance, the
~ 12000 x 8000-pixel camera CFH12k, mounted on the Canada-
France-Hawaii Telescope (CFHT), or the Wide-Field Imager (WFI),
a ~ (8000) 2 -pixel camera at the ESO/MPG 2.2-m telescope on La
Silla. In 2003. [he lirst square-degree camera was installed at the
CFHT, Megacam, with ~ (18000) 2 pixels. Another square-degree
camera, OmegaCAM, is due to start operations at ESO's newly built
VLT Survey Telescope (VST) in 2007.
Fig. 6.36. Left: the tidal (or shear) field of the cluster
CI 0024+17 is indicated by sticks whose length and direction
represent the strength and orientation of the tidal gravitational
field. Right: the surface mass density is shown, reconstructed
by means of the weak ^mutational lens effect. The bright
galaxies in the cluster are seen to follow the (dark) matter dis-
tribution; the orientation of the isodensity contours is the same
of the light in the center of the cluster
Mass Reconstruction of Galaxy Clusters. By means
of this method, the reconstruction of the mass density of
a large number of clusters became possible. The most
important results of these investigations are as follows:
the center of the mass distribution corresponds to the
optical center of the cluster (see Fig. 6.36). If X-ray
information is available, the mass distribution is, in gen-
eral, found to be centered on the X-ray maximum. The
shape of the mass distribution - e.g., its ellipticity and
orientation - is in most cases very similar to the dis-
tribution of bright cluster galaxies. The comparison of
the mass profile determined by this method and that de-
termined from X-ray data agree well, typically within
a factor of ~ 1.5 (see Fig. 6.37 for an example). Through
the weak lensing effect, substructure in the mass dis-
tribution is also detected in some clusters (Fig. 6.38)
which does not in all cases reflect the distribution of
cluster galaxies. However, in general a good correspon-
dence between light and mass exists (Fig. 6.39). From
these lensing studies, we obtain a mass-to-light ratio for
clusters that agrees with that found from X-ray anal-
yses, about M/L ~ 250/i in Solar units. Clusters of
galaxies that strongly deviate from this average value
do exist, however. Two independent analyses for the
cluster MS 1224+20 resulted in a mass-to-light ratio of
6 [arcseconds]
Fig. 6.37. Radial mass profile of the galaxy cluster Abell 22 1 8.
The data points with error bars are mass estimates from the
w eak lensing effect, the solid and dashed curves are isothermal
sphere models assuming different velocity dispersions. The
cross denotes the mass estimated from luminous arcs, and the
triangle depicts the mass obtained from the central cD gaiaw
6.5 Clusters of Galaxies as Gravitational Lenses
;;~^
-^
-
/ ' /^---Z 1ZJ
-/,.///,, ,
\
' / \ I / I \ '
-J
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. A^
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■
i
sis
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-
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Fig. 6.38. Analysis of the cluster of galaxies MS 1054-03 by
llic weak lensing cfi'ccl. In ihe upper left panel, aground based
image is shown with a field size of 7'5 x 7.'5. In this image,
about 2400 faint objects are detected, the majority of which
i ilaxi I high i I'i'i in n Ii in i i i Hii in in in
the galaxies, the tidal field of the cluster can be reconstructed,
and from this ihe projected mass distribution L'(0). presented
in the lower left panel; the latter is indicated by the black
contours, while the white contours represent the smoothed
light distribution of the cluster galaxies. A mosaic of HST
-200 -100 100 200
images allows the ellipticity measurement of a significantly
larger number of galaxies, and with better accuracy. The tidal
field resulting from these measurements is displayed in the
upper right panel, with the reconstructed surface mass density
shown in the lower right panel. One can clearly see that the
cluster is strongly structured, with three density maxima which
correspond to regions \\ ith bright clustct galaxies. Tins cluster
seems to be currently in the process of formation through
a merger of smaller entities
6. Clusters and Croups of Galaxies
Fig. 6.39. The cluster of galaxies C10939+4713 (A851) is the
cluster with the highest redshift in the Abell catalog. The
HS 1 image in the upper left panel was obtained shortly after
the refurbishment of the HST in 1994; in this image, North
is down, whereas it is up in the bottom images. The mass
distribution of the cluster was reconstructed from this image
and is shown in the upper right panel, both as the level surface
and by the contours on top. We see that the distributions
of bright galaxies and of (dark) matter are very similar: (licit'
respective centers are aligned, a secondary maximum exists in
both the light and the matter distribution, as does the prominent
minimum in which no bright galaxies are visible either. This
cluster also shows strong lensing effects, which can be seen
from the image at the bottom: a triple image system at z ^ 3 .98
and an arc with z = 3.98 were confirmed spectroscopically
6.6 Evolutionary Effects
M/L s» 800/j in Solar units, more than twice the value
normally found in clusters.
The similarity of the mass and galaxy distributions
is not necessarily expected because the lens effect
measures the total mass distribution, and therefore
mainly the dark matter in a cluster of galaxies. The simi-
lar distributions then imply that the galaxies in a cluster
seem to basically follow the distribution of the dark
matter, although there are some exceptions.
The Search for Clusters of Galaxies with Weak Grav-
itational Lensing. The weak lensing effect can not only
be used to map the matter distribution of known clus-
ters, but it can also be used to search for clusters. Mass
concentrations generate a tangential shear field in their
vicinity, which can specifically be searched for. The ad-
vantage of this method is that it detects clusters based
solely on their mass properties, in contrast to all other
methods which rely on the emission of electromagnetic
radiation, whether in the form of optical light from clus-
ter galaxies or as X-ray emission from a hot intracluster
medium. In particular, if clusters with atypically low
gas or galaxy content exist, they could be detected in
this way.
With this method, quite a number of galaxy clus-
ters have been detected already - see Fig. 6.40. Further
candidates exist, in that from the shear signal a signif-
icant mass concentration is indicated but it cannot be
identified with any concentration of galaxies on opti-
cal images. The clarification of the nature of these lens
signals is of great importance: if in fact matter con-
centrations do exist which correspond to the mass of
a cluster but which do not contain luminous galaxies,
then our understanding of galaxy evolution needs to be
revised. However, we cannot exclude the possibility that
these statistically significant signals are statistical out-
liers, or result from projection effects - remember, lens-
ing probes the line-of-sight integrated matter density.
Together with the search for galaxy clusters by means
of the SZ effect (Sect. 6.3.4), the weak lensing effect
provides an interesting alternative for the detection of
mass concentrations compared to traditionally methods.
VLT l-band Image: 36mn exposure
Dark matter reconstruction
H9BH
3a
m
^^
§f#|§
"-_■"'
CLy^
'Vri
Fig. 6.40. Top left: a VLT/FORS1 image,
taken as part of a survey of "empty fields".
Top right: the mass reconstruction, as was
obtained from the optical data by employ-
ing the weak lensing effect. Clearly visible
is a peak in the mass distribution; the opti-
cal image shows a concentration of galaxies
in this region. Hence, in this field a cluster
of galaxies was detected for the first time by
its lens properties. Bottom: as above, here
a galaxy cluster was also detected through
its lensing effect. On the left, an optical
wide-field image is shown, obtained by the
Big Throughput Camera, and the mass re-
construction is displayed on the right. The
location of the peak in the latter coincides
with a concentration of galaxies. Spectro-
scopic measurements yield that these form
a cluster of galaxies at z = 0.276
6. Clusters and Groups of Galaxies
6.6 Evolutionary Effects
Today, we are able to discover and analyze clusters of
galaxies at redshifts z ~ 1 and higher; thus the question
arises whether these clusters have the same properties as
local clusters. At z ~ 1 the age of the Universe is only
about half of that of the current Universe. One might
therefore expect an evolution of cluster properties.
Luminosity Function. First, we shall consider the
comoving number density of clusters as a function
of redshift or, more precisely, the evolution of the
luminosity function of clusters with z. As Fig. 6.41
demonstrates, such evolutionary effects are not very
pronounced, and only at the highest luminosities or the
most massive clusters, respectively, does an evolution
become visible. This reveals itself by the fact thai at high
redshift, clusters of very high luminosity or very high
mass are less abundant than they are today. The inter-
pretation and the relevance of this fact will be discussed
later (see Sect. 8.2.1).
Butcher-Oemler Effect. We saw in Chap. 3 that early-
type galaxies are predominantly found in clusters and
groups, whereas spirals are mostly field galaxies. For ex-
ample, a massive cluster like Coma contains only 10%
spirals, the other luminous galaxies are ellipticals or SO
galaxies (see also Sect. 6.2.9). Besides these morpho-
logical differences, the colors of galaxies are very useful
for a characterization: early-type galaxies (ellipticals
and SO galaxies) have little ongoing star formation and
therefore consist mainly of old, thus low-mass and cool
stars. Hence they are red, whereas spirals feature active
star formation and are therefore distinctly bluer. The
fraction of blue galaxies in nearby clusters is very low.
Butcher and Oemler found that this changes if one
examines clusters of galaxies at higher redshifts: these
contain a larger fraction of blue galaxies, thus of spi-
rals (see Fig. 6.42). This means that the mixture of
galaxies changes over time. In clusters, spirals must
become scarcer with increasing cosmic time, e.g., by
transforming into early-type galaxies.
A possible and plausible explanation is that spirals
lose their interstellar gas. Since they move through the
intergalactic gas (which emits the X-ray radiation) at
high velocities, the ISM in the galaxies may be torn
away and mix with the ICM. This is plausible be-
cause the ICM also has a high metallicity. These metals
can only originate in a stellar population, thus in the
enriched material in the ISM of galaxies. Later, we
will discuss some further evidence for transformations
between galaxy types.
L x (0.5-2.0 keV) erg s" 1
Fig. 6.41. X-ray luminosity function of galaxy clusters, as was
obtained i'l'om a region around the North Ecliptic Pole (NEP).
the region with the longest exposure time in the ROSAT All
Sky Survey (see Fig. 6.26). Plotted is dA?/dL x , the (comov-
in») number density per luminosity interval, for clusters with
0.02 < z S 0.3 (left panel) and 0.3 < z < 0.85 (right panel),
NEP: z= [0.30-0.85]
- BCSXLF
• RASS1BSXLF
" REFLEX XLF
L x (0. 5-2.0 keV) erg s 1
respectively. The luminosity was derived from the flux in the
photon energy range from 0.5 keV to 2 keV. The three differ
enl curves specif) the local luminosity function of clusters as
found in other cluster surveys at lower redshifts. We sec that
evolutionary effects in the luminosity function are relatively
small and become visible only at high Lx
Redshift
Fig. 6.42. Butcher-Oemler effect: in the upper panel, the frac-
tion of blue galaxies f\, in a sample of 195 galaxy clusters is
plotted as a function of cluster redshift, where open (filled I cir
cies indicate photometric (spectroscopic) redshift data foi the
clusters. The lower panel shows a selection of clusters w ith
spectroscopically determined redshifts and well-defined red
cluster sequence. For the determination of /;,. foreground and
background galaxies need to be statistically subtracted using
control fields, which may also rcsul tin ncgatn e values for ft,.
A clear increase in /b with redshift is visible, and a line of
regression yields f b = 1.34 Z - 0.03
Color-Magnitude Diagram. Plotting the color of clus-
ter galaxies versus their magnitude, one finds a very
well-defined, nearly horizontal sequence (Fig. 6.43).
This red cluster sequence (RCS) is populated by the
early-type galaxies in the cluster.
The scatter of early-type galaxies around this se-
quence is very small, which suggests that all early-type
galaxies in a cluster have nearly the same color, only
weakly depending on luminosity. Even more surprising
is the fact that the color-magnitude diagrams of differ-
ent clusters at the same redshift define a very similar
red cluster sequence: cluster galaxies with the same
redshift and luminosity have virtually the same color.
Abell 2390, observed with the HST. Star symbols represent
earl) type galaxies, identified by their morphology, while di-
amonds denote other galaxies in the field. The red cluster
sequence is clearly visible
Comparing the red sequences of clusters at different
redshifts, one finds that the sequence of cluster galax-
ies is redder the higher the redshift is. In fact, the red
cluster sequence is so precisely characterized that, from
the color-magnitude diagram of a cluster alone, its red-
shift can be estimated, whereby a typical accuracy of
Az ~ 0.1 is achieved. The accuracy of this estimated
redshift strongly depends on the choice of the color
filters. Since the most prominent spectral feature of
early-type galaxies is the 4000-A break, the redshift
is estimated best if this break is located right between
two of the color bands used.
This well-defined red cluster sequence is of crucial
importance for our understanding of the evolution of
galaxies. We know from Sect. 3.9 that the composition
of a stellar population depends on the mass spectrum
at its birth (the initial mass function, IMF) and on its
age: the older a population is, the redder it becomes.
The fact that cluster galaxies at the same redshift all
have roughly the same color indicates that their stellar
populations have very similar ages. However, the only
age that is singled out is the age of the Universe itself.
In fact, the color of cluster galaxies is compatible with
their stellar populations being roughly the same age as
the Universe at that particular redshift. This also pro-
vides an explanation for why the red cluster sequence is
6. Clusters and Groups of Galaxies
0.4
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2000 2500 3000
Rest Wavelength (A)
80 100
Rest
Fig. 6.44. The radio galaxy LBDS 53W091 has a redshift of
z = 1.552, and it features a very red color (R-K& 5.8).
Optical spectroscopy of the galaxy provides us with the spec-
tral light distribution of the UV emission in the galaxy's
rest frame. The LTV lighl of a stellar population is almost com-
pletely due to stars on the upper main sequence - see Fig. 3.38.
In the upper left panel, the spectrum of LBDS 53W091 is com-
pared to those of different F stars; one can see that F6 stars
match the spectral distribution of the galaxy nearly perfectly.
In the bottom panel, synthetic spectra from population syn-
thesis calculations are compared to the observed spectrum.
3000 3500
Wavelength (A)
A population with an age of about 4 Gyr represents the best
fit to the observed spectrum; this is also comparable to the
lifetime of F6 stars: the most luminous (still existing) stars
dominate the light distribution of a stellar population in the
UV. In combination, this reveals that this galaxy at z — 1 .552 is
at least 3 Gyr old. Phrased differently, the age of the Universe
at z = 1 .55 must be at least 3 Gyr. In the upper right panel,
Ihe age of the Universe at z — 1 .55 is displayed as a function
of Ho and Qa (for Q m + Qa = 1)- Hence, this single galaxy
provides significant constraints on cosmological parameters
6.6 Evolutionary Effects
Fig. 6.45. The cluster of galaxies
MS 1054-03, observed with the HST, is the
most distant cluster in the EMSS X-ray sur-
vey (z = 0.83). The reddish galaxies in the
image on the left form a nearly linear struc-
ture. This cluster is far from being spherical,
as we have also seen from its weak lens-
ing results (Fig. 6.38) - it is not relaxed.
The smaller images on the right show blow-
ups of selected cluster fields where mergers
of galaxies become visible: in this cluster,
the merging of galaxies is directly observ-
able. At least six of the nine merging pairs
found in this cluster have been shown to be
gravitationally bound systems
■ :. V„ : :
A' '
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.,
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25' I 59 ,
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■ 'b:\
■
26W
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4
15 s 22'35'MO'
Right Ascension (2000)
Fig. 6.46. The cluster of galaxies XMMU J2235 .2-2557 was
discovered in the field-of-view of an XMM-Newton image for
which a different source was the original target. The image on
the left shows the X-ray contours, superposed on an R-band
image, while the image on the right shows the central section,
here superposed on a K-band image. Galaxies in the field
Right Ascent
follow a red cluster sequence if the color is measured in R — z-
The symbols denote galaxies at redshift 1.37 < z < 1.40. The
strong X-ray source to the upper right of the cluster center is
a Sc\ fori uaki\\ al lowci Tctkhifi. As of 2005. ihi\ cluster is the
most distant X-ray selected cluster known, with a temperature
of ~ 6 keV and a velocity dispersion of a ~ 750 km/s
shifted towards bluer colors at higher redshifts - there, magnitude diagram of early-type galaxies in clusters
the age of the Universe was smaller, and thus the stel- is not flat, in that more luminous galaxies are redder,
lar population was younger. This effect is of particular follows from the dependence of galaxy colors on the
importance at high redshifts. The fact that the color- metallicity of their stellar populations. The higher the
6. Clusters and Groups of Galaxies
Kuremeli D >tar i < usier ol' ' i;il i- ■ n Mind Radio Galaxj TOJI338-1942
(VLTKUEYEN + FORS2)
I 2.0 ^KlWtt^W^
Wavelength (nm)
s ( 1 1 l i i.l l , (1 1 l. in l In l r in i h >iln><, ili v II II" h I'H.
(VLTKUEYEN + F0RS2) jl
Fig. 6.47. The most distant known group of galaxies. The
region around the radio galaxy TN J1338-1942 (z = 4.1)
was scanned for galaxies at the same redshift; 20 such galaxies
were found with the VLT, marked by circles in the left image.
For 10 of these galaxies, the spectra are shown on the right;
in all of them, the Lya emission line is clearly visible. Hence,
groups of galaxies were already formed in an early stage of
the Universe
luminosity of a galaxy, and thus its stellar mass, the
higher its metallicity.
Indeed, from the colors of cluster galaxies it is pos-
sible to derive very strict upper limits on their star
formation in recent times. The color of cluster galaxies
at high redshifts even provides interesting constraints
on cosmological parameters - only those models are
acceptable which have an age of the Universe, at the re-
spective redshift, larger than the estimated age of the
stellar population. One interesting example of this is
presented in Fig. 6.44.
Therefore, we conclude from these observations that
the stars in cluster galaxies formed at very early times
in the Universe. But this does not necessarily mean
that the galaxies themselves are also this old, because
galaxies can be transformed into each other by merger
processes (see Fig. 6.45). This changes the morphology
of galaxies, but may leave the stellar populations largely
unchanged.
Clusters of Galaxies at Very High Redshift. The
search for clusters at high redshift is of great cosmolog-
ical interest. As will be demonstrated in Sect. 7.5.2, the
expected number density of clusters as a function of z
strongly depends on the cosmological model. Hence,
this search offers an opportunity to constrain cosmo-
logical parameters by the statistics of galaxy clusters.
The search for clusters in the optical (thus, by gal-
axy overdensities) becomes increasingly difficult at
high z because of projection effects. Nevertheless, sev-
eral groups have managed to detect clusters at z ~ 0.8
with this technique. In particular, the overdensity of
galaxies in three-dimensional space can be analyzed
if, besides the angular coordinates on the sphere, the
galaxy colors are also taken into account. Because of
the red cluster sequence, the overdensity is much more
prominent in this space than in the sky projection alone.
Projection effects play a considerably smaller role
in X-ray searches for clusters. Using sensitive X-ray
6.6 Evolutionary Effects
satellites like ROSAT, some clusters with z ~ 1.2 have
been found (see Fig. 6.25). The new X-ray satellites
Chandra and XMM-Newton are even more sensitive.
Therefore, one expects them to be able to find clusters
at even higher redshifts; one example for a cluster at z =
1.393 is shown in Fig. 6.46. This example demonstrates
combining deep X-ray images with observations in the
optical and the NIR is an efficient method of compiling
samples of distant clusters.
Through optical methods, it is also possible to iden-
tify galaxy concentrations at very high redshift. One
approach is to assume that luminous AGNs at high
redshift are found preferentially in regions of high over-
density, which is also expected from models of galaxy
formation. With the redshift of the AGN known, the red-
shift at which one should search for an overdensity of
galaxies near the AGN is defined. Those searches have
proven to be quite successful; for instance, they are per-
formed using narrow-band filter photometry, with the
filter centered on the redshifted Lya line, tuned to the
redshift of the AGN. Candidates need to be verified
spectroscopically afterwards. One example of a strong
galaxy concentration at z = 4.1 is presented in Fig. 6.47.
The identification of a strong spatial concentration of
galaxies is not sufficient to have identified a cluster
of galaxies though, because it is by no means clear
whether one has found a gravitationally bound sys-
tem of galaxies (and the corresponding dark matter).
Rather, such galaxy concentrations are considered to
be the predecessors of galaxy clusters which will only
evolve into bound systems during later cosmological
evolution.
7. Cosmology II: Inhomogeneities in the Universe
7.1 Introduction
In Chap. 4, we discussed homogeneous world models
and introduced the standard model of cosmology. It is
based on the cosmological principle, the assumption of
a (spatially) homogeneous and isotropic Universe. Of
course, the assumption of homogeneity is justified only
on large scales because observations show us that our
Universe is inhomogeneous on small scales - otherwise
no galaxies or stars would exist.
The distribution of galaxies on the sky is not uni-
form or random (see Fig. 6.1), rather they form clusters
and groups of galaxies. Also clusters of galaxies
are not distributed uniformly, but their positions are
correlated, grouped together in superclusters. The three-
dimensional distribution of galaxies, obtained from
redshift surveys, shows an interesting large-scale struc-
ture, as can be seen in Fig. 7.1 which shows the spatial
distribution of galaxies in the two-degree Field Galaxy
Redshift Survey (2dFGRS).
Even larger structures have been discovered. The
Great Wall is a galaxy structure with an extent of
~ 100/i -1 Mpc, which was found in a redshift survey
of galaxies (Fig. 7.2). Such surveys also led to the dis-
covery of the so-called voids, nearly spherical regions
which contain virtually no (bright) galaxies, and which
have a diameter of typically 50ft -1 Mpc. The discovery
of these large-scale inhomogeneities raises the question
of whether even larger structures might exist in the Uni-
verse, or more precisely: does a scale exist, averaged
over which the Universe appears homogeneous? The
existence of such a scale is a requirement for the homo-
geneous world models to provide a realistic description
of the mean behavior of the Universe.
To date, no evidence of structures with linear di-
mension > IOO/j -1 Mpc have been found, as can also
be seen from Fig. 7.1. Hence, the Universe seems to
be basically homogeneous if averaged over scales of
R ~ 200ft" 1 Mpc. This "homogeneity scale" needs to
be compared to the Hubble radius R H = c/H Q « 3000
h~ x Mpc. This implies R <$C c/H , so that after averag-
ing, l(c/H {) )/R] 3 ~ (15) 3 ~ 3000 independent volume
elements exist per Hubble volume. This justifies the
approximation of a homogeneous world model when
considering the mean history of the Universe.
On small scales, the Universe is inhomogeneous.
Evidence for this is the galaxy distribution projected
on the sky, the three-dimensional galaxy distribution
determined by redshift surveys, and the existence of
clusters of galaxies, superclusters, "Great Walls", and
voids. In addition, the anisotropy of the cosmic mi-
crowave background (CMB), with relative fluctuations
of AT/T ~ 10~ 5 , indicates that the Universe already
contained small inhomogeneities at redshift z ~ 1000,
which we will discuss more thoroughly in Sect. 8.6.
H
V8*
Fig. 7.1. The distribution of galaxies in the
complete 2dF Galaxy Redshift Survey. In
the radial direction, the escape velocity, or
redshift, is plotted, and the polar angle is the
right ascension. In this survey, more than
350 000 spectra were taken; plotted here is
the distribution of more than 200 000 galax-
ies with reliable rcdshifi measurements. The
data from the complete survey are publicly
available
Peter Schneider, Co\moln-j\ II: Inhnmnvcneitiex in the Universe.
In: Peter Schneider. Extragalactic Astronomy and Cosmology, pp. 277-307 (2006)
DOI: 10.1007/11614371_7 © Springer- Verlag Berlin Heidelberg 2006
7. Cosmology II: Inhomogeneities in the Univ
CfA2 First 6 Slices
5.5<6<42.5
< m B < 1 5.5 ' Copyright SAO 1
Fig. 7.2. The Great Wall: in a redshift survey of galaxies with
radial velocities of cz < 15 000 km/s, a galaxy structure was
disco\ ered which is located at a redshift of cz ~ 6000 km/s,
extending in right ascension between 9 h < a < 16 h . Plotted
arc galaxies with declination 8.5° < S < 42°
In this chapter, we will examine the evolution of such
density inhomogeneities and their description.
7.2 Gravitational Instability
7.2.1 Overview
The smallness of the CMB anisotropy (AT/T ~ 10~ 5 ;
see Sect. 8.6) suggests that the density inhomogeneities
at redshift z ~ 1000 - this is the epoch where most of the
CMB photons interacted with matter for the last time -
must have had very small amplitudes. Today, the ampli-
tudes of the density inhomogeneities are considerably
larger; for example, a massive cluster of galaxies con-
tains within a radius of ~ 1.5/i _1 Mpc more than 200
times more mass than an average sphere of this radius
in the Universe. Thus, these are no longer small density
fluctuations.
Obviously, the Universe became more inhomoge-
neous in the course of its evolution; as we will see,
density perturbations grow over time. One defines the
relative density contrast
z ~ 1000, \S\ «: 1. The dynamics of the cosmic Hubble
expansion is controlled by the gravitational field of the
average matter density p(t), whereas the density fluc-
tuations Ap(r, t) — p(r, t) — p(t) generate an additional
gravitational field.
We shall here be interested only in very weak grav-
itational fields, for which the Newtonian description
of gravity can be applied. Since the Poisson equation,
which specifies the relation between matter density and
the gravitational potential, is linear, the effects of the
homogeneous matter distribution and of density fluctu-
ations can be considered separately. The gravitational
field of the total matter distribution is then the sum of
the average matter distribution and that of the density
fluctuations.
We consider a region in which Ap > 0, hence S > 0,
so that the gravitational field in this region is stronger
than the cosmic average. An overdense region produces
a stronger gravitational field than that corresponding
to the mean Hubble expansion. By this additional
self-gravity, the overdense region will expand more
slowly than the average Hubble expansion. Because of
the delayed expansion, the density in this region will
also decrease more slowly than in the cosmic mean,
p(t) = (H-z)Vo = a~ 3 (i)po, and hence the density
contrast in this region will increase. As a consequence,
the relative density will increase, which again produces
an even stronger gravitational field, and so on. It is
obvious that this situation is unstable. Of course, the
argument also works the other way round: in an under-
dense region with S < 0, the gravitational field generated
is weaker than in the cosmic mean, therefore the self-
gravity is weaker than that which corresponds to the
Hubble expansion. This implies that the expansion is de-
celerated less than in the cosmic mean, the underdense
region expands faster than the Hubble expansion, and
thus the local density will decrease more quickly than
the mean density of the Universe. In this way, the den-
sity contrast decreases, i.e., <5 becomes more negative
over the course of time.
Sir.
p(r, t) - p(t)
Pit)
(7.1)
where p(t) denotes the mean cosmic matter density
in the Universe at time t. From the definition of S,
one can immediately see that S > — 1, because p > 0.
The smallness of the CMB anisotropy suggests that at
Density fluctuations grow over time due to their
self-gravity; overdense regions increase their den-
sity contrast over the course of time, while
underdense regions decrease their density contrast.
In both cases, \8\ increases. Hence, this effect of
7.2 Gravitational Instability
gravitational instability leads to an increase of
density fluctuations over the course of time. The
evolution of structure in the Universe is described
by the model of gravitational instability.
Equations of Motion. The behavior of this fluid
described by the continuity equation
dp
'Strictly speaking, the cosmic dust cannot be described as a fluid
because the mailer is assumed to lie eollisionless. This means thai no
interactions occur between the particles, except lor sanitation. Two
iiows of such dust call thus penetrate each other This situation can
be compared to that of a fluid whose molecules are interacting by
collisions. Through these collisions, the velocity distribution of the
molecules will, at each position, assume an approximate Maxwell
distribution, with a well defined average velocity that corresponds lo
the How velocity at this point. Such an unambiguous velocity does not
exist lor Just in general. However, at carl} limes, when deviations from
the Hubble flow arc still very small, no multiple flow , are expected,
so that in this case, the velocity held is unambiguously defined.
fV-0oi0 = 0,
(7.21
The evolution of structure in the Universe can be
understood in the framework of this model. In this chap-
ter we will describe structure formation quantitatively.
This includes the analysis of the evolution of density
perturbations over time, as well as a statistical descrip-
tion of such density fluctuations. We will then see that
the evolution of inhomogeneities is directly observable,
and that the Universe was less inhomogeneous at high
redshift than it is today. Since the evolution of pertur-
bations depends on the cosmological model, we need
to examine whether this evolution can be used to obtain
an estimate of cosmological parameters. In Chap. 8, we
will give an affirmative answer to this question. Finally,
we will briefly discuss the origin of density fluctuations.
7.2.2 Linear Perturbation Theory
We first will examine the growth of density pertur-
bations. For this discussion, we will concentrate on
length-scales that are substantially smaller than the
Hubble radius. On these scales, structure growth can
be described in the framework of the Newtonian theory
of gravity. The effects of spacetime curvature and thus
of General Relativity need to be accounted for only for
density perturbations on length-scales comparable to, or
larger than the Hubble radius. In addition, we assume
for simplicity that the matter in the Universe consists
only of dust (i.e., pressure-free matter), with density
p(r, t). The dust will be described in the fluid approx-
imation, where the velocity field of this fluid shall be
denoted by v(r, t). 1
which expresses the fact that matter is conserved: the
density decreases if the fluid has a diverging veloc-
ity field (thus, if particles are moving away from each
other). In contrast, a converging velocity field will
lead to an increase in density. Furthermore, the Euler
equation applies,
(7.3)
which describes the conservation of momentum and
the behavior of the fluid under the influence of forces.
The left-hand side of (7.3) is the time derivative of the
velocity as would be measured by an observer moving
with the flow, because dv/dt is the derivative at a fixed
point in space, whereas the total left-hand side of (7.3)
is the time derivative of the velocity measured along the
flow lines. The latter is affected by the pressure gradient
and the gravitational field <P , the latter satisfying the
Poisson equation
V 2 = AnGp .
(7.4)
Since we are only considering dust, the pressure van-
ishes, P = 0. These three equations for the description
of a self-gravitating fluid can in general not be solved
analytically. However, we will show that a special,
cosmologically relevant exact solution can be found,
and that by linearization of the system of equations
approximate solutions can be constructed for \S\ <JC 1.
Hubble Expansion. The special exact solution is the
flow that we have already encountered in Chap. 4: the
homogeneous expanding cosmos. By substituting into
the above equations it is immediately shown that
v(r, t) = H(t)r
is a solution of the equations if p is homogeneous and
satisfies (4. 1 1), and if the Friedmann equation (4.13) for
the scale factor applies.
As long as the density contrast 1 5 1 <$C 1 , the deviations
of the velocity field from the Hubble expansion will be
small. We expect that in this case, physically relevant
solution of the above equations are those which deviate
only slightly from the homogeneous case.
7. Cosmology II: Inhomogeneities in the Univ
a \a I
It is convenient to consider the problem in comoving
coordinates; hence we define, as in (4.4),
r = a{t) x .
In a homogeneous cosmos, x is a constant for every
matter particle, and its spatial position r changes only
due to the Hubble expansion. Likewise, the velocity
field is written in the form
(7.5)
where u (x, t) is a function of the comoving coordinate x.
In (7.5), the first term represents the homogeneous Hub-
ble expansion, whereas the second term describes the
deviations from this homogeneous expansion. For this
reason, u is called the peculiar velocity.
Transforming the Fluid Equations to Comoving Co-
ordinates. We will now show how the above equations
read in comoving coordinates. For this, we first note
that the partial derivative d/dt in (7.2) means a time
derivative at fixed r. If the equations are to be written in
comoving coordinates, this partial time derivative needs
to be transformed into one where x is kept fixed. For
example,
\i)i
i)i !
= { Jt ) x PA X ,t)--x-V xPx (x, t ),
(7.6)
where V x is the gradient with respect to comov-
ing coordinates, and where we define the function
p x (x, t) = p(ax, t). Note that p x (x, t) and p(x, t) both
describe the same physical density field, but that p
and p x are different mathematical functions of their
arguments. After these transformations, (7.2) becomes
dp 3d
Accordingly, the gravitational potential <P is written as
f0(x,f); (7.9)
the first term is the Newtonian potential for a homoge-
neous density field, and (j> satisfies the Poisson equation
for the density inhomogeneities,
V 2 (l)(x, t) = 4jrGa 2 (t)p(t)S(x, t)
3H 2 S2 m
= W* ( *'°' (? - 10)
where in the last step we used p oc cT 3 and the definition
of the density parameter Q m . Then, the Euler equation
(7.3) becomes
9« h-V a 1 1
— + H+-H = -— VP--V0, (7.11)
dt a a pa a
where (4.13) has been utilized.
Linearization. In the homogeneous case, S = 0, u = 0,
= 0, p — p, an d (7.7) then implies p + 3Hp = 0,
which also follows immediately from (4.17) in the case
of P = 0. Now we will look for approximate solutions
of the above set of equations which describe only small
deviations from this homogeneous solution. For this
reason, in these equations we only consider first-order
terms in the small parameters S and u, i.e., we disregard
terms that contain uS or are quadratic in the velocity u.
After this linearization, we can eliminate the peculiar
velocity u and the gravitational potential (f> from the
equations 2 and then obtain a second-order differential
equation for the density contrast S,
-PH
-V • (pu) = ,
(7.7)
where from now on all spatial derivatives are to be
considered with respect to x. For notational simplicity
we from now on set p = p x and S = S(x,t), and note that
the partial time derivative is to be understood to mean at
fixed x. Writing p — 75(1 + S) and using p oc a~ 3 , (7.7)
reads in comoving coordinates
--V-[(1+S)k] = 0.
(7.8)
d 2 S 2d 8S
;;/-
a dt
= 4nGpS .
(7.12)
It is remarkable that neither does this equation c
derivatives with respect to spatial coordinates, nor do
the coefficients in the equation depend on x. Therefore,
(7.12) has solutions of the form
S(x, t) = D(t) S(x) ,
2 For this, the linearized form of (7.8), 38/ dt + a~'V u = 0, is dif-
ni ii u Ih res] i in nd u i H I ii i il i ii
d1" the linearized form of equation (7.11) for the pressure- free case.
du/dt+Hu = -a~' Vtf>. Finally, the Laplacian of <j> is replaced b\
the Poisson equation (7.10).
7.2 Gravitational Instability
i.e., the spatial and temporal dependences factorize in
these solutions. Here, S(x) is an arbitrary function of
the spatial coordinate, and D(t) satisfies the equation
.. 2d .
D+—D-4jrGp(t)D = 0. (7.13)
The Growth Factor. The differential equation (7.13)
has two linearly independent solutions. One can show
that one of them increases with time, whereas the other
decreases. If, at some early time, both functional de-
pendences were present, the increasing solution will
dominate at later times, whereas the solution decreas-
ing with t will become irrelevant. Therefore, we will
consider only the increasing solution, which is denoted
by D + (t), and normalize it such that D + (f ) = 1. Then,
the density contrast becomes
S(x,t) = D+(t)S (x).
(7.14)
This mathematical consideration allows us to draw im-
mediately a number of conclusions. First, the solution
(7.14) indicates that in linear perturbation theory the
spatial shape of the density fluctuations is frozen in co-
moving coordinates, only their amplitude increases. The
growth factor D + if) of the amplitude follows a simple
differential equation that is easily solvable for any cos-
mological model. In fact, one can show that for arbitrary
values of the density parameter in matter and vacuum
energy, the growth factor has the form
D+(a)
H(a) C
#o J [Q m /a' + Q A a' 2 -
(i2 m + Q A
-Df 2
where the factor of proportionality is determined from
the condition D + (to) — 1 .
In accordance with D + (t Q ) — 1, S (x) would be the
distribution of density fluctuations today if the evolution
was indeed linear until the present epoch. Therefore,
8()(x) is denoted as the linearly extrapolated density fluc-
tuation field. However, the linear approximation breaks
down if | S | is no longer <§C 1. In this case, the terms
that have been neglected in the above derivations are no
longer small and have to be included. The problem then
becomes considerably more difficult and defies analyt-
ic Instead one needs, in general, to rely on
numerical procedures for analyzing the growth of den-
sity perturbations. Furthermore, it shall be noted once
again that, for large density perturbations, the fluid ap-
proximation is no longer valid, and that up to now we
have assumed the Universe to be matter dominated. At
early times, i.e., for z > z eq (see Eq. 4.54), this assump-
tion becomes invalid, so that the above equations need
to be modified for these early epochs.
Example: Einstein-de Sitter Model. In the special
case of a universe with Q m — 1, Q A — 0, (7.13) can be
solved explicitly. In this case, a(t) — (f/f ) 2/3 , so that
\a) 3t H y 8ttG W
furthermore, in this model to Ho — 2/3, so that (7.13)
reduces to
A . 9
(7.15)
This equation is easily solved by making the ansatz
Doct q ; this ansatz is suggested because (7.15) is
equidimensional in t, i.e., each term has the dimen-
sion D/(time) 2 . Inserting into (7.15) yields a quadratic
equation for q,
q(q-l) + -q-
= 0,
with solutions q — 2/3 and q = —\. The latter corre-
sponds to fluctuations decreasing with time and will be
disregarded in the following. So, for the Einstein-de
Sitter model, the increasing solution
\ 2/3
D+(t) =
\ t !
= a(i) ,
(7.16)
is found, i.e., in this case the growth factor equals the
scale factor. For different cosmological parameters this
is not the case, but the qualitative behavior is quite sim-
ilar, which is demonstrated in Fig. 7.3 for three models.
In particular, fluctuations were able to grow by a factor
~ 1000 from the epoch of recombination at z ~ 1000,
from which the CMB photons originate, to the present
day.
Evidence for Dark Matter on Cosmic Scales. At the
present epoch, S ^> 1 certainly on scales of clusters of
galaxies (~2Mpc), and 6 ~ 1 on scales of superclusters
7. Cosmology II: Inhomogeneities in the Univ
r''— '" d =l'" v =6''''-
% a d =o.3a v =o ■
0.8
A\--- O d =0.3 a v =o.7-
0.6
=" \\\
0.4
- \v;; .. :
0.2
0.2 0.4 0.6 0.8
1
Fig. 7.3. Growth factor D + for three different cosmological
models, as a function of the scale factor a (left panel) and
of redshift (right panel). It is clearly visible how quickly
D + decreases with increasing redshift in the EdS model, in
comparison to the models of lower density
(~ 10 Mpc). Hence, because of the law of linear struc-
ture growth (7.14) and the behavior of D + (t) shown
in Fig. 7.3, we would expect S > 10~ 3 at z = 1000
for these structures to be able to grow to non-linear
structures at the current epoch. For this reason, we
should also expect CMB fluctuations to be of compara-
ble magnitude, AT/T > 10~ 3 . The observed fluctuation
amplitude is AT/T ~ 10~ 5 , however. The correspond-
ing density fluctuations therefore cannot have grown
sufficiently strongly up to today to form non-linear
structures.
This contradiction can be resolved by the dominance
of dark matter. Since photons interact with baryonic
matter only, the CMB anisotropics basically provide
(at least on angular scales below ~ 1°) information on
the density contrast of baryons. Dark matter may have
had a higher density contrast at recombination and may
have formed potential wells, into which the baryons
then "fall" after recombination.
7.3 Description of Density Fluctuations
We will now examine the question of how to describe
an inhomogeneous universe quantitatively, i.e., how to
quantify the structures it contains. This task sounds eas-
ier at first sight than it is in reality. One has to realize
that the aim of such a theoretical description cannot be
to describe the complete function S(x, t) for a partic-
ular universe. No model of the Universe will be able
to describe, for instance, the matter distribution in the
vicinity of the Milky Way in detail. No model based on
the laws of physics alone will be able to predict that at
a distance of ~ 800 kpc from the Galaxy a second mas-
sive spiral galaxy is located, because this specific feature
of our local Universe depends on the specific initial con-
ditions of the matter distribution in the early Universe.
We can at best hope to predict the statistical properties
of the mass distribution, such as, for example, the aver-
age number density of clusters of galaxies above a given
mass, or the probability of a massive galaxy being found
within 800 kpc of another one. Likewise, numerical sim-
ulations of the Universe (see below) cannot reproduce
our Universe; instead, they are at best able to gener-
ate cosmological models that have the same statistical
properties as our Universe.
It is quite obvious that a very large number of statis-
tical properties exist for the density field, all of which
we can examine and which we hope can be explained
quantitatively by the correct model of structure forma-
tion in the Universe. To make any progress at all, the
statistical properties need to be sorted or classified. How
can the statistical properties of a density field best be
described?
Two universes are considered equivalent if their den-
sity fields S have the same statistical properties. One may
then imagine considering a large (statistical) ensemble
of universes whose density fields all have the same sta-
tistical properties, but for which the individual functions
<5(jc) are all different. This statistical ensemble is called
a random field, and any individual distribution with the
respective statistical properties is called a realization of
the random field.
An example may clarify these concepts. We consider
the waves on the surface of a large lake. The statisti-
cal properties of these waves - such as how many of
them there are with a certain wavelength, and how their
amplitudes are distributed - depend on the shape of the
lake, its depth, and the strength and direction of the
wind blowing over its surface. If we assume that the
wind properties are not changing with time, the statis-
tical properties of the water surface are constant over
time. Of course, this does not mean that the ampli-
tude of the surface as a function of position is o
7.3 Description of Density Fluctuations
Rather, it means that two photographs of the surface
that are taken at different times are statistically indistin-
guishable: the distribution of the wave amplitudes will
be the same, and there is no way of deciding which of
the snapshots was taken first. Knowing the surface to-
pography and the wind properties sufficiently well, one
is able to compute the distribution of the wave ampli-
tudes, but there is no way to predict the amplitude of the
surface of the lake as a function of position at a partic-
ular time. Each snapshot of the lake is a realization of
the random field, which in turn is characterized by the
statistical properties of the waves.
7.3.1 Correlation Functions
Galaxies are not randomly distributed in space, but
rather they gather in groups, clusters, or even larger
structures. Phrased differently, this means that the
probability of finding a galaxy at location jc is not in-
dependent of whether there is a galaxy in the vicinity
of x. It is more probable to find a galaxy in the vicin-
ity of another one than at an arbitrary location. This
phenomenon is described such that one considers two
points x and y, and two volume elements dV around
these points. If n is the average number density of gal-
axies, the probability of finding a galaxy in the volume
element dV around jc is then
Pi =ndV ,
independent of x if we assume that the Universe is statis-
tically homogeneous. We choose dV such that Pi <§C 1,
so that the probability of finding two or more galaxies
in this volume element is negligible.
The probability of finding a galaxy in the volume
element dV at location jc and at the same time finding
a galaxy in the volume element d V at location y is then
= (ndV) 2 [l + ? g (x,jO] •
(7.17)
If the distribution of galaxies was uncorrected, the
probability P 2 would simply be the product of the prob-
abilities of finding a galaxy at each of the locations
x and j in a volume element dV, so P2 — P\. But
since the distribution is correlated, the relation does not
apply in this simple form; rather, it needs to be modi-
fied, as was done in (7.17). Equation (7.17) defines the
two-point correlation function (or simply "correlation
function") of galaxies % g (x, y).
By analogy to this, the correlation function for the
total matter density can be defined as
(p(x) P (y)) = p 2 {[1 +S(x)] [1 +S(y)])
= p 2 (l + {S(x)S(y)))
=:75 2 [1+£(jc,j)] , (7.18)
because the mean (or expectation) value (S(x)) = for
all locations x.
In the above equations, angular brackets denote av-
eraging over an ensemble of distributions that all have
identical statistical properties. In our example of the
lake, the correlation function of the wave amplitudes al
positions x and y, for instance, would be determined by
taking a large number of snapshots of its surface and
then averaging the product of the amplitudes at these
two locations over all these realizations.
Since the Universe is considered statistically homo-
geneous, § can only depend on the difference x — y and
not on x and y individually. Furthermore, £ can only
depend on the separation r — \x — y | , and not on the
direction of the separation vector x — y because of the
assumed statistical isotropy of the Universe. Therefore,
£ = £(r) is simply a function of the separation between
two points.
For a homogeneous random field, the ensemble av-
erage can be replaced by spatial averaging, i.e., the
correlation function can be determined by averaging
over the density products for a large number of pairs
of points with given separation r. The equivalence of
ensemble average and spatial average is called the er-
godicity of the random field. Only by this can the
correlation function (and all other statistical properties)
in our Universe be measured at all, because we are able
to observe only a single - namely our - realization of the
hypothetical ensemble. From the measured correlations
between galaxy positions, as determined from spectro-
scopic redshift surveys of galaxies (see Sect. 8.1.2), one
finds the approximate relation
§ g (r)
■(3
(7.19)
for galaxies of luminosity ~ L* (see Fig. 7.4), where
rrj ~ 5h~ l Mpc denotes the correlation length, and
7. Cosmology II: Inhomogeneities in the Univ
Fig. 7.4. The correlation function £ g of
galaxies, as it was determined from the
Las Campanas Redshift Survey. In the top
panel, | g is shown for small and intermedi-
ate separations, whereas the bottom panel
shows it for large separations. Dashed and
dotted lines indicate the northern and south-
ern part, respectively, of the survey, and
the solid triangles denote the correlation
function obtained from combining both.
A power law with slope y = 1.52 is plotted
for comparison (bold solid curve)
where the slope is about y — 1-8- This relation
is approximately valid over a range of separations
2/j- 1 Mpc<r <30h~ l Mnc.
Hence, the correlation function provides a means to
characterize the structure of the matter distribution in the
Universe. Besides this two-point correlation function,
correlations of higher order may also be defined, leading
to general n -point correlation functions. These are more
difficult to determine from observation, though. It can
be shown that the statistical properties of a random field
are fully specified by the set of all n -point correlations.
7.3.2 The Power Spectrum
An alternative (and equivalent) description of the sta-
tistical properties of a random field, and thus of the
structure of the Universe, is the power spectrum P(k).
Roughly speaking, the power spectrum P(k) describes
the level of structure as a function of the length-scale
L ~ 2jt/k; the larger P(k), the larger the amplitude
of the fluctuations on a length-scale litjk. Here, k is
a wave number. Phrased differently, the density fluc-
tuations are decomposed into a sum of plane waves
of the form S(x) = J2 a k cos(x-k), with a wave vec-
tor k and an amplitude a*. The power spectrum P(k)
then describes the distribution of amplitudes with equal
k — \k\. Technically speaking, this is a Fourier decom-
position. Referring back to the example of waves on the
surface of a lake, one finds that a characteristic wave-
length L c exists, which depends, among other factors,
on the wind speed. In this case, the power spectrum will
have a prominent maximum at k — 2n/L c .
The power spectrum P(k) and the correlation func-
tion are related through a Fourier transform; formally,
one has 3
P(k) = lit
h
, sin kr
- $(r) ,
(7.20)
3 This may not look like a "standard" homier transform on first sight.
However, the relation between P(k) and f(r) is given by a three
dimensional Fourier transform. Since the correlation function depends
only on the separation r— \r\, the two integrals over the angular
coordinates can he performed explicitly, leading to the lorn: of (7.20).
7.4 Evolution of Density Fluctuations
i.e., the integral over the correlation function with
a weight factor depending on k ~ 2it/L. This relation
can also be inverted, and thus §(r) can be computed
from P(k).
In general, knowing the power spectrum is not
sufficient to unambiguously describe the statistical
properties of any random field - in the same way as
the correlation function f(r) on ly provides an incom-
plete characterization. However, random fields do exist,
so-called Gaussian random fields, which are uniquely
characterized by P(k). Such Gaussian random fields
play an important role in cosmology because it is as-
sumed that at very early epochs, the density field obeyed
Gaussian statistics.
7.4 Evolution of Density Fluctuations
P(k) and §(r) both depend on cosmological time or red-
shift because the density field in the Universe evolves
over time. Therefore, the dependence on t is explicitly
written P(k, t) and §(r, t). Note that P(k, t) is linearly
related to §(r, t), according to (7.20), and § in turn de-
pends quadratically on the density contrast S. If we
interpret jc as a comoving separation vector, from (7.14)
we then know the time dependence of the density fluc-
tuations, 5(jc, t) — D + (t)So(x). Thus, within the scope
of the validity of (7. 14),
£(*,*) = D\(t) &x,to),
(7.21)
P(k, t) = D z + (t) P(k, t ) =: D%(t) P (k) , (7.22)
where k is a comoving wave number. We shall stress
once again that these relations are valid only in the
framework of Newtonian, linear perturbation theory in
the matter dominated era of the Universe, to which we
had restricted ourselves in Sect. 7.2.2. Equation (7.22)
states that the knowledge of Po(k) is sufficient to obtain
the power spectrum P(k, t) at any time, again within the
framework of linear perturbation theory.
7.4.1 The Initial Power Spectrum
The Harrison-Zeldovich Spectrum. Initially it may
seem as if Po(k) is a function that can be chosen arbi-
trarily, but one objective of cosmology is to calculate
this power spectrum and to compare it to observations.
More than thirty years ago, arguments were already
developed to specify the functional form of the initial
power spectrum.
At early times, the expansion of the Universe follows
a power law, a(t) on t 1/2 in the radiation-dominated era.
At that time, no natural length-scale existed in the Uni-
verse to which one might compare a wavelength. The
only mathematical function that depends on a length
but does not contain any characteristic scale is a power
law; 4 hence for very early times one should expect
P(k) ex k ns . (7.23)
Many years ago, Harrison, Zeldovich, Peebles and oth-
ers argued, based on scaling relations, that it should
be ra s = 1. For this reason, the spectrum (7.23) with
n s = 1 is called Harrison Zeldovich spectrum. With
such a spectrum, we may choose a time t\ after the
inflationary epoch and write
P(k, ti) = D 2 + (ti)Ak ns , (7.24)
where A is a normalization constant that cannot be deter-
mined from theory but has to be fixed by observations.
Assuming the validity of (7.22),
P (k) = Ak n *
would then apply.
The Transfer Function. This relation above needs
to be modified for several reasons. In linear pertur-
bation theory, which led to S(x,t) — D+(t) 8q(x), we
assumed the validity of Newtonian dynamics, consid-
ered only the matter-dominated epoch of the Universe,
and disregarded any pressure terms. The evolution of
perturbations in the radiation-dominated cosmos pro-
ceeds differently though, also depending on the scale
of the perturbations in comparison to the length of the
horizon, so that a correction term of the form
P (k) = Ak" s T 2 (k)
(7.25)
You can com ince \ourself ot this b\ Irving to lind another t\pe of
function of a scale that docs not involve a characteristic length: e.g..
sin.\ docs not work, if \ is a length, since the sine ol a length is not
delined: one thus needs something like sint i/.v h >. hence introducing
a length scale. The same arguments apply to other functions, such
1 1 I 1 1 I m 111 pon nti I 1 Iso 1 111 III I l
power laws. e.g.. ,\.v" -- B.\ j " delines a characteristic scale, naincK
that value of x where the two terms become equal.
7. Cosmology II: Inhomogeneities in the Univ
needs to be introduced. T(k) is called the transfer func-
tion; it can be computed for any cosmological model if
the matter content of the Universe is specified. In par-
ticular, T(k) depends on the nature of dark matter. One
distinguishes between cold dark matter (CDM) and hot
dink matter (HDM). These two kinds of dark matter
differ in the thermal velocities of their constituents at
time t eq , when radiation and matter had equal density.
The particles of CDM were non-relativistic at this time,
whereas those of HDM had velocities of order c. If
dark matter consists of weakly interacting elementary
particles, the difference between CDM and HDM de-
pends on the mass m of the particles. Assuming that the
"temperature" of the dark matter particles is close to
the temperature of the Universe, then a particle mass m
satisfying the relation
mc 2 » k B T(t eq ) -k B x 2.73 K (1 +z eq )
= k B x 2.73 K x 23 900 Q m h 2 ~ 6Q m h 2 eV
indicates CDM, whereas HDM is characterized by the
opposite inequality, i.e., mc 2 <$C k B T(t eq ); for instance,
neutrinos belong to HDM. The important distinction be-
tween HDM and CDM follows from the considerations
below.
If density fluctuations become too large on a certain
scale, linear perturbation theory breaks down and (7.25)
is no longer valid. Then the true current power spectrum
P(k, t ) will deviate from Pq{K). Nevertheless, in this
case it is still useful to examine Po(k) - it is then called
the linearly extrapolated /tower spectrum.
7.4.2 Growth of Density Perturbations
Within the framework of linear Newtonian perturbation
theory in the "cosmic fluid", S(x, t) — D + (t) &o(x) ap-
plies. Modifications to this behavior are necessary for
several reasons:
• If dark matter consists of relativistic particles, these
are not gravitationally bound in the potential well of
a density concentration. In this case, they are able to
move freely and to escape from the potential well,
which in the end leads to its dissolution if these par-
ticles dominate the matter overdensity. From this, it
follows immediately that for HDM small-scale den-
sity perturbations cannot form. For CDM this effect
of free-steaming does not occur.
• At redshifts z> z eq , radiation dominates the density
of the Universe. Since the expansion law a(t) is then
distinctly different from that in the matter-dominated
phase, the growth rate for density fluctuations will
also change.
• As discussed in Sect. 4.5.2, a horizon exists with
comoving scale r H>C om(0- Physical interactions can
take place only on scales smaller than r H , C om(0- F° r
fluctuations of length-scales L ~ 2it/k > r H , C om(0>
Newtonian perturbation theory will cease to be valid,
and one needs to apply linear perturbation theory in
the framework of the General Relativity.
CDM and HDM. The first of the above points immedi-
ately implies that a clear difference must exist between
HDM and CDM models as regards structure formation
and evolution. In HDM models, small-scale fluctua-
tions are washed out by free-streaming of relativistic
particles, i.e., the power spectrum is completely sup-
pressed for large k, which is expressed by the transfer
function T(k) decreasing exponentially for large k. In
the context of such a theory, very large structures will
form first, and galaxies can form only later by frag-
mentation of large structures. However, this formation
scenario is in clear contradiction with observations. For
example, we observe galaxies and QSOs at z ~ 6 so that
small-scale structure is already present at times when
the Universe had less than 10% of its current age. In
addition, the observed correlation function of galaxies,
both in the local Universe (see Fig. 7.4) and at higher
redshift, is incompatible with cosmological models in
which the dark matter is composed mainly of HDM.
Hot dark matter leads to structure formation that
does not agree with observation. Therefore we
can exclude HDM as the dominant constituent of
dark matter. For this reason, it is now commonly
assumed that the dark matter is "cold". The achieve-
ments of the CDM scenario in the comparison
between model predictions and observations fully
justify this assumption.
We shall elaborate on the last statement in quite some
detail in Chap. 8.
In linear perturbation theory, fluctuations grow on all
scales, or for all wave numbers, independent of each
7.4 Evolution of Density Fluctuations
other. This applies not only in the Newtonian case, but
also remains valid in the framework of General Rel-
ativity as long as the fluctuation amplitudes are small.
Therefore, the behavior on any (comoving) length-scale
can be investigated independently of the other scales. At
very early times, perturbations with a comoving scale L
are larger than the (comoving) horizon, and only for
z < z e nter(L) does the horizon become larger than the
considered scale L. Here, z en ter(£) is defined as the
redshift at which the (comoving) horizon equals the
(comoving) length-scale L,
r H ,com(Zenter(£)) = L .
(7.26)
It is common to say that at z en ter(L) the perturbation
under consideration "enters the horizon", whereas actu-
ally the process is the opposite - the horizon outgrows
the perturbation. Relativistic perturbation theory shows
that density fluctuations of scale L grow as long as
L > fH.com, namely oc a 2 if radiation dominates (thus, if
z > z eq ), or oc a if matter dominates (thus, if z < z eq ).
Free-streaming particles or pressure gradients cannot
impede the growth on scales larger than the hori-
zon length because, according to the definition of the
horizon, physical interactions - which pressure or free-
steaming particles would be - cannot extend to scales
larger than the horizon size.
Qualitative Behavior of the Transfer Function. The
behavior of the growth of a density perturbation on
a scale L for z < z en i e r(L) depends on z enter itself.
If a perturbation enters the horizon in the radiation-
dominated phase, Zeq £ z en tei(L), the fluctuation cannot
grow during the epoch z eq <z< z en ter (L)- In this pe-
riod, the energy density in the Universe is dominated
by radiation, and the resulting expansion rate pre-
vents an efficient perturbation growth. At later epochs,
when z < Zeq, the growth of density perturbation con-
tinues. If Zenter(i) £ Zeq, thus if the perturbation enters
the horizon during the matter-dominated epoch of
the Universe, these perturbations will grow as de-
scribed in Sect. 7.2.2, with S oc D + (t). This implies that
a length-scale Lq is singled out, namely the one for
which
Zeq = Ze„ te r(i ) , (7.27)
1
" V2H0 V(l + Zeq)^m
-~ l2(Q m h 2 y 1 Mpc,
h
(7.28)
where the expression for r H , C om(z) generalizes (4.69),
and where (4.54) has been used for z eq .
Density fluctuations with L > Lq enter the horizon
after matter started to dominate the energy density of the
Universe; hence their growth is not impeded by a phase
of radiation-dominance. In contrast, density fluctuations
with L < Lq enter the horizon at a time when radiation
dominates. These then cannot grow further as long as
z > z eq , and only in the matter-dominated epoch will
their amplitudes proceed to grow again. Their relative
growth up to the present time has therefore grown by
a smaller factor than that of fluctuations with L > Lq
(see Fig. 7.5). The quantitative consideration of these
effects allows us to compute the transfer function. In
general, this needs to be done numerically, but very
log (a)
Fig. 7.5. A density perturbation that enters the horizon dur-
ing the radiation-dominated epoch of the Universe ceases to
grow until matter starts to dominate the energy content of the
Universe. In comparison to a perturbation that enters the hori
zon later, during the mailer dominated epoch, the amplitude of
in ni ill i | null lion i up] i din a i loi m i )
which explains Ihe qualitative behavior (7.29) of the transfer
function
7. Cosmology II: Inhomogeneities in the Univ
good approximations e;
treated analytically,
T(k)K lforfc« 1/L ,
T(k) « (kL Q y 2 fork » 1/L ;
but the important point is:
t. Two limiting cases are easily
In the framework of the CDM model, the trans-
fer function can be computed, and thus, by means
of (7.19), also the power spectrum of the density
fluctuations as a function of length-scale and red-
shift. The amplitude of the power spectrum has to
be obtained from observations.
The Shape Parameter. The transfer function depends
on the combination kLo, which is the inverse of the
ratio of the length-scale under consideration (~ litjk)
and the horizon scale at the epoch of equality, and thus
on k{Q m h 2 )~ l . Since distances determined from red-
shift are measured in units of h~ l Mpc, the shape of
the transfer function, and thus also that of the power
spectrum, depends on r = Q m h. r is called the shape
parameter of the power spectrum. It is sometimes used
as a free parameter instead of being identified with Q m h.
A detailed analysis shows that r depends also on Q b ,
but since Q b < 0.05 is small, according to primordial
nucleosynthesis (see Sect. 4.4.4), this effect is relatively
small and often neglected.
If the galaxy distribution follows the distribution of
dark matter, the former can be used to determine the cor-
relation function or the power spectrum. Both from the
distribution of galaxies projected onto the sphere (angu-
lar correlation function) and from its three-dimensional
distribution (which is determined from redshift sur-
veys), values in the range r ~ 0.15-0.25 are found.
From T(k) « 1 for kL «; 1, and with (7.24), we find
that P(k) oc k for kLo <?C 1. This behavior is compatible
with the CMB anisotropy measurements by COBE on
large scales, as we will discuss in detail in Chap. 8.
In Fig. 7.6, the power spectrum is plotted for sev-
eral cosmological models that have different density
parameter, shape parameter, and normalization of the
power spectrum. The thin curves show P(k) as derived
from linear perturbation theory, and the bold curves
display the power spectrum with non-linear structure
100 1000 10 4 10 5 10 6
(c/H ) k
Fig. 7.6. The current power spectrum of density fluctuations
for CDM models. The wave number A is given in units of //o/c.
and (Hq/c) 3 P(k) is dimensionless. The various curves have
different cosmological parameters: EdS: Q m = 1, Q A = 0;
OCDM: Q m = 0.3, Q A = 0; ytCDM: Q m = 0.3, Q A = 0.7.
The values in parentheses specify (ag, F), where o% is the
normalization of the power spectrum (which will be discussed
below), and where F is the shape parameter. The thin curves
correspond to the power spectrum Po(k) linearly extrapolated
to the present day, and the bold curves take the non linear
evolution into account
evolution taken into account. The power spectra dis-
played all have a characteristic wave number at which
the slope of P(k) changes. It is specified by ~ 2jt/L ,
with the characteristic length Lq being defined in (7.28).
Besides pure CDM and HDM models (the latter be-
ing excluded by observation), one can consider models
which are dominated by CDM, but which have a (small)
contribution by HDM; these are called mixed dark mat-
ter (MDM) models. Such a contribution has indeed now
become part of the standard model, due to the detected
finite rest mass of neutrinos which implies < Q v <$C 1 .
With this contribution, T(k) is changed in such a way
that small scales (i.e., large k) are slightly suppressed in
the power spectrum. We will see later that by observ-
ing the power spectrum we can constrain the rest mass
of neutrinos very well, and cosmological observations
provide, in fact, by far the most stringent mass limits
for neutrinos.
Density Distribution of Baryons. The evolution of
density fluctuations of baryons differs from that of dark
matter. The reason for this is essentially the interaction
7.5 Non-Linear Structure Evolution
of baryons with photons: although matter dominates
the Universe for z < z eq , the density of baryons re-
mains smaller than that of radiation for a long time,
until after recombination begins. Since photons and
baryons interact with each other by photon scattering
on free electrons, which again are tightly coupled elec-
tromagnetically to protons and helium nuclei, and since
radiation cannot fall into the potential wells of dark
matter, baryons are hindered from doing so as well.
Hence, the baryons are subject to radiation pressure.
For this reason, the density distribution of baryons is
initially much smoother than that of dark matter. Only
after recombination does the interaction of baryons with
photons cease to exist, and the baryons can fall into the
potential wells of dark matter, i.e., some time later the
distribution of baryons will closely resemble that of the
dark matter.
The linear theory of the evolution of density fluc-
tuations will break down at the latest when \S\ ~ 1;
the above equations for the power spectrum P(k, t) are
therefore valid only if the respective fluctuations are
small. However, very accurate fitting formulae now ex-
ist for P(k, t) which are also valid in the non-linear
regime. For some cosmological models, the non-linear
power spectrum is displayed in Fig. 7.6.
7.5 Non-Linear Structure Evolution
Linear perturbation theory has a limited range of ap-
plicability; in particular, the evolution of structures like
clusters of galaxies cannot be treated within the frame-
work of linear perturbation theory. One might imagine
that one can evolve the system of equations (7.2)-(7.4)
to higher orders in the small variables S and \u\, and so
consider a non-linear perturbation theory. In fact, a quite
extensive literature exists on this topic in which such
calculations have been performed. It is worth mention-
ing, though, that while this higher-order perturbation
theory indeed allows us to follow density fluctuations
to slightly larger values of | S |, the achievements of this
theory do not, in general, justify the large mathematical
effort. In addition, the fluid approximation is no longer
valid if gravitationally bound systems form because, as
mentioned earlier, multiple steams of matter will occur
in this case.
However, for some interesting limiting cases, ana-
lytical descriptions exist which are able to represent
the non-linear evolution of the mass distribution in the
Universe. We shall now investigate a special and very
important case of such a non-linear model. In gen-
eral, studying the non-linear structure evolution requires
the use of numerical methods. Therefore, we will also
discuss some aspects of such numerical simulations.
7.5.1 Model of Spherical Collapse
We consider a spherical region in an expanding Uni-
verse, with its density p(i) enhanced compared to the
mean cosmic density 75(f),
p(t) = [l + S(l)]p(l).
(7,30)
where we use the density contrast S as defined in (7.1).
For reasons of simplicity we assume that the density
within the sphere is homogeneous although, as we will
later see, this is not really a restriction. The density
perturbation is assumed to be small for small t, so that
it will grow linearly at first, S(t) ex D + (t), as long as
5 <$C 1. If we consider a time t\ which is sufficiently
early such that S(ti) «: 1, then 5(f ; ) = S D + (t[), where
<5o is the density contrast linearly extrapolated to the
present day. It should be mentioned once again that
S ^ S(to), because the latter is determined by the non-
linear evolution.
Let R com be the initial comoving radius of the over-
dense sphere; as long as 8 <$C 1 , the comoving radius will
change only marginally. The mass within this sphere is
M =
An
4tt
4, m A>(l + <5i)«— 4,,
(7.31)
because the physical radius is R — aR com , and
p = po/a 3 . This means that a unique relation exists be-
tween the initial comoving radius and the mass of this
sphere, independent of the choice of t{ and <5 , if only
we choose <5(fj) = <5 D + (h) «: 1.
Due to the enhanced gravitational force, the sphere
will expand slightly more slowly than the Universe as
a whole, which again will lead to an increase in its den-
sity contrast. This then decelerates the expansion rate
even further, relative to the cosmic expansion rate. In-
deed, the equations of motion for the radius of the sphere
are identical to the Friedmann equations for the cosmic
expansion, only with the sphere having an effective Q m
7. Cosmology II: Inhomogeneities in the Univ
different from that of the mean Universe. If the initial
density is sufficiently large, the expansion of the sphere
will come to a halt, i.e., R(t) will reach a maximum;
after this, the sphere will recollapse.
If ?max is the time of maximum expansion, then the
sphere will, theoretically, collapse to a single point at
time f co n = 2f max . The relation f co u = 2f max follows from
the time reversal symmetry of the equation of motion:
the time to the maximum expansion is equal to the
time from that point back to complete collapse. 5 The
question of whether the expansion of the sphere will
come to a halt depends on the density contrast S(t[)
or Sq - compare the discussion of the expansion of the
Universe in Sect. 4.3.1- and on the model for the cosmic
background.
Special Case: The Einstein-de Sitter Model. In the
special case of Q m = 1 and Qa = 0, this behavior can
easily be quantified analytically; we thus treat this case
separately. In this cosmological model, any sphere with
So > is a "closed universe" and will therefore recol-
lapse at some time. For the collapse to take place before
t\, 5(fj) or So needs to exceed a threshold value. For in-
stance, for a collapse at f C oii < to, a linearly extrapolated
overdensity of
20
- (12tt) 2/3 ~ 1.69
(7.32)
is required. More generally, one finds that So > S c (1 + z)
is needed for the collapse to occur before redshift z.
Violent Relaxation and Virial Equilibrium. Of
course, the sphere will not really collapse to a single
point. This would only be the case if the sphere was per-
fectly homogeneous and if the particles in the sphere
moved along perfectly radial orbits. In reality, small-
scale density and gravitational fluctuations will exist
within such a sphere. These then lead to deviations of
the particles' tracks from perfectly radial orbits, an ef-
fect that is more important the higher the density of
the sphere becomes. The particles will scatter on these
fluctuations in the gravitational field and will virialize;
this process of violent relaxation has already been de-
scribed in Sect. 6.2.6 and occurs on short time-scales -
roughly the d\ namical time-scale, i.e., the time it takes
the particles to fully cross the sphere. In this case, the
virialization is essentially complete at t m \\. After that,
the sphere will be in virial equilibrium, and its average
density will be 6
(p} = (l+5 vir )p(f coU ),
where (1 +<5 vir ) ~ 178^ m ' 6 . (7.33)
This relation forms the basis for the statement that the
virialized region, e.g., of a cluster, is a sphere with an
average density ~ 200 times the critical density p CI of
the Universe at the epoch of collapse. Another conclu-
sion from this consideration is that a massive galaxy
cluster with a virial radius of 1.5/j _1 Mpc must have
formed from the collapse of a region that originally had
a comoving radius of about six times this size, roughly
10/i _1 Mpc. Such a virialized mass concentration of
dark matter is called a dark mailer halo.
Up to now, we have considered the collapse of a ho-
mogeneous sphere. From the above arguments one can
easily convince oneself that the model is still valid if
the sphere has a radial density gradient, e.g., if the den-
sity decreases outwards. In this case, the initial density
contrast will also decrease as a function of radius. The
inner regions of such a sphere will then collapse faster
than the outer ones; a halo of lower mass will form first,
and only later, when the outer regions have also col-
lapsed, will a halo with higher mass form. From this it
follows that halos of low initial mass will grow in mass
by further accretion of matter.
The spherical collapse model is a simple model for
the non-linear evolution of a density perturbation in the
Universe. Despite being simplistic, it represents the fun-
damental principles of gravitational collapse and yields
approximate relations, e.g., for the collapse time and
mean density inside the virialized region, as they are
found from numerical simulations.
' For the same reason that it takes a stone thrown up into the air the
same time to reach its peak altitude as to fall hack to the ground from
''This result is obtained from conservation of energy and from the
\irial theorem. The total energ\ £„„ of the sphere is a constant. At
the time of maximum expansion, it is given solely by the gravitational
binding energy of the system since then the expansion velocity, and
thus the kinetic energy, vanishes. On the other hand, the virial theorem
implies that in \irial equilibrium £u„ — — E pol /2, and by combining
this witli the conserxation of energy £ u „ — £ki n + £pot one ' s men
able to compute £,„, in equilibrium and hence the radius and density
of the collapsed sphere.
7-5-2 Number Density of Dark Matter Halos
Press-Schechter Model. The model of spherical col-
lapse allows us to approximately compute the number
density of dark matter halos as a function of their mass
and redshift; this model is called the Press-Schechter
model.
We consider a field of density fluctuations 8q{x), fea-
turing fluctuations on all scales according to the power
spectrum Po(k). Assume that we smooth this field with
a comoving smoothing length R, by convolving it with
a filter function of this scale . In our example of the waves
on a lake, we could examine a picture of its surface taken
through a pane of milk-glass, by which all the contours
on small scales would be blurred. Then, let S R (x) be
the smoothed density field, linearly extrapolated to the
present day. This field does not contain any fluctuations
on scales < R, because these have been smoothed out.
Each maximum in 8 R (x) corresponds to a peak with
characteristic scale > R and, according to (7.31), each
of these maxima corresponds to a mass peak of mass
M ~ (4jtR 3 /3)p . If the amplitude S R of the density
peak is sufficiently large, a sphere of (comoving) ra-
dius R around the peak will decouple from the linear
growth of density fluctuations and will begin to grow
non-linearly. Its expansion will come to a halt, and then
it will recollapse. This process is similar to that in the
spherical collapse model and can be described approx-
imately by this model. The density contrast required
for the collapse, S R > 5 m j n , can be computed for any
cosmological model and for any redshift.
If the statistical properties of Sq(x) are Gaussian -
which is expected for a variety of reasons - the statisti-
cal properties of the fluctuation field <5 are completely
defined by the power spectrum P(k). Then the num-
ber density of density maxima with S R > 5 m j n can be
computed, and hence the (comoving) number density
n(M, z) of relaxed dark matter halos in the Universe as
a function of mass M and redshift z can be determined.
The Mass Spectrum. The most important results of
the Press-Schechter model are easily explained (see
Fig. 7.7). The number density of halos of mass M
depends of course on the amplitude of the density fluc-
tuation So - i.e., on the normalization of the power
spectrum Po(k). Hence, the normalization of Po(k)
-1 -0.5
log l0 (M/(10 15 h- 1 M o ))
Fig. 7.7. Number density of dark matter halos with mass > M,
computed from the Press-Schechter model. The comoving
number density is shown for three difl'crcnl redshifts, z —
i upper eur\ es j. „ — 0.33. and z = 0.5 (lower curves), for three
different cosmological models: an Einstein-de Sitter model
(solid lines), a low-density open model with Q m = 0.3 and
Qa = (dotted lines), and a flat universe of low density with
Q m = 1 — Q A = 0.3 (dashed lines). The normalization of the
density fluctuation field has been chosen such that the number
density of halos with M > 10 14 /i -1 M© at z = in all models
agrees with the local number dens i I ol il i 1 In I i oti
the dramatic redshift evolution in the EdS model
can be determined by comparing the prediction of the
Press-Schechter model with the observed number den-
sity of galaxy clusters, as we will discuss further in
Sect. 8.2.1 below. The corresponding result is called the
"cluster-normalized power spectrum".
Furthermore, we find that n(M,z) is a decreasing
function of halo mass M. This follows immediately
from the previous argument, since a larger M requires
a larger smoothing length R, together with the fact that
the number density of mass peaks of a given ampli-
tude (5 m ; n decreases with increasing smoothing length.
For large M, n(M, z) decreases exponentially because
sufficiently high peaks become very rare for large
smoothing lengths. Therefore, very few clusters with
mass >2x 10 15 M Q exist. From Fig. 7.7, we can see
that the number density of clusters with M > 10 15 M Q
today is about 10~ 7 Mpc~ 3 , so the average separation
between two such clusters is larger than 100 Mpc, which
7. Cosmology II: Inhomogeneities in the Univ
is compatible with the observation that the most nearby
massive cluster (Coma) is about 90 Mpc away from us.
The density contrast <5 m i n required for a collapse be-
fore redshift z is a function of z, as we have seen above.
In particular, for the Einstein-de Sitter model we have
<5 min ~ 1.69(1 + z). In general, <5 min = S c /D + (z), where
<5 C and D + (z) each depend on the cosmological model.
This means that the redshift dependence of 5 m ; n depends
on the cosmological model and is basically described by
the growth factor D + (z). Since D + (z) is, at fixed z (we
recall that, by definition, D + (0) — 1), larger for smaller
C2 m (see Fig. 7.3), the ratio of the number density of
halos at redshift z to the one in the current Universe,
n(M, z)/n(M, 0), is larger the smaller Q m is. For clus-
ter masses (M ~ 1O 15 M ), the evolution of this ratio in
the Einstein-de Sitter model is dramatic, whereas it is
less strong in open and in flat, Tl-dominated universes
(see Fig. 7.7).
By comparing the number density of galaxy clusters
at high redshift with the current abundance, we can
Fig. 7.8. Expected (comoving) number density oi'galax) clus-
ters with mass > 8 x 10 14 /! _1 M Q within a (comoving) radius
of R < 1.5/i _1 Mpc, for flat cosmological models and different
values of the density parameter Q m . The normalization of the
power spectrum in the models has been chosen such that the
current cluster number dcnsit\ is approximately reproduced.
The points with error bars show results from observations of
ilaxy cli ii i it ili. ii in in nil - ilthot 'I ih i rroi I it ■,
at high redshift are very large, a high-density i
to be excluded
thus obtain constraints on Q m , and in some sense also
on Q A . Even a few very massive clusters at z > 0.5
are sufficient to exclude the Einstein-de Sitter model
by this argument. As a matter of fact, the existence of
the cluster MS 1054-03 (Fig. 6.15) alone, the mass of
which was determined by dynamical methods, from its
X-ray emission, and by the lens effect, is already nearly
sufficient to falsify the Einstein-de Sitter model (see
Fig. 7.8). However, at least one problem exists in the
application of this method, namely making a sufficiently
accurate mass determination for distant clusters and, in
addition, determining whether they are relaxed and thus
accounted for in the Press-Schechter model. Also the
completeness of the local cluster sample is a potential
problem.
A Special Case. To get a more specific impression of the
Press-Schechter mass spectrum, we consider the spe-
cial case where the power spectrum Po(k) is described
by a power law, Po(k) ex k n . From Fig. 7.6, we can see
that this provides quite a good description over a large
range of k if one concentrates on scales either clearly
above or far below the maximum of Pq. The length-
scale at which P has its maximum is specified roughly
by (7.28). As we can also see from Fig. 7.6, the non-
linear evolution that the Press-Schechter model refers
to is relevant only for scales considerably smaller than
this maximum, rendering the power law a valid approx-
imation, with n ~ — 1.5. In this case, the mass function
can be written in closed form,
n(M.z)
Pct^m
M
\ y/2
V^r" M 2 \M*(z)J
x exp I -
(7.34)
\M*(z)J J '
where y = l+n/3 ~ 0.5, and where M*{z) is the
z-dependent mass-scale above which the mass spec-
trum is exponentially cut off. For masses considerably
smaller than M* (z), the Press-Schechter mass spectrum
is basically a power law in M. The characteristic mass-
scale M* (z) depends on the normalization of the power
spectrum and on the growth factor,
M*(z) = M* [D + (z)] 2/y = Ml (l+zy 2/Y , (7.35)
where the final expression applies to an Einstein-de Sit-
ter universe. Hence, the characteristic mass-scale grows
over time, and it describes the mass-scale on which
7.5 Non-Linear Structure Evolution
the mass distribution in the Universe is just becoming
non-linear for a particular redshift. This mass-scale at
the current epoch, Mq , depends on the normalization of
the power spectrum; it approximately separates groups
from clusters of galaxies, and explains the fact that
clusters are (exponentially) less abundant than groups.
Furthermore, the Press-Schechter model describes
a very general property of structure formation in a CDM
model, namely that low-mass structures - like galaxy-
mass dark matter halos - form at early times, whereas
large mass accumulations evolve only later. The ex-
planation for this is found in the shape of the power
spectrum P(k) as described in (7.25) together with the
asymptotic form (7.29) of the transfer function T(k).
A model like this is also called a hierarchical structure
formation or a "bottom-up" scenario. In such a model,
small structures that form early later merge to form large
structures.
Comparison with Numerical Simulations. The Press-
Schechter model is a very simple model, based on
assumptions that are not really justified in detail. Never-
theless, its predictions are in astounding agreement with
the number density of halos determined from simula-
tions, and this model, published in 1974, has for nearly
25 years predicted the halo density with an accuracy
that was difficult to achieve in numerical simulations.
Only since the mid-1990s have the precision and statis-
tics of numerical simulations of structure formation
reached a level on which significant discrepancies with
the Press-Schechter model become clearly noticeable.
However, the analytical description has also been im-
proved; instead of a spherical collapse, the more realistic
ellipsoidal collapse has been investigated, by which
the number density of halos is modified relative to the
Press-Schechter model. This advanced model is found
to be in very good agreement with the numerical results,
as demonstrated in Fig. 7.9, so that today we have a good
description of n(M, z) that very accurately resembles
the results from numerical simulations.
7.5.3 Numerical Simulations
of Structure Formation
Analytical considerations - such as, for instance, lin-
ear perturbation theory or the spherical collapse model
- are only capable of describing limiting cases of struc-
10"'
n
10~ 2
X \7 = 3.06 \ 'X |
!„•
\z-5.72 > \ \
'-.
5 1(T 4
"A \ \ ^
?■
10" 6
V\__ \ \
L
,■„"
10 m 10" 10 12 10 1a 10 14 10 15 10 ,G
M [fi" 1 M e ]
Fig. 7.9. The mass spectrum of dark matter halos is plotted
for five different redshifts (data points with error bars), as
determined in the Millennium simulation (which we will dis-
cuss more extensively below - see Fig. 7.12). The solid curves
describe an approximation for the mass spectrum, which has
been obtained from different simulations, and which obviously
provides an excellent description of the simulation results. For
z — and z — 10, the prediction of the Press-Schechter model
is indicated by the dotted curves, underestimating the abun-
dance of vcr\ massive halos and overestimating the density of
lower-mass halos. The vertical dotted line indicates the lowest
halo mass which can still be resolved in these simulations
ture formation. In general, gravitational dynamics is
too complicated to be analytically examined in de-
tail. For this reason, experiments to simulate structure
formation by means of numerical methods have been
performed for some time already. The results of these
simulations, when compared to observations, have con-
tributed very substantially to establishing the standard
model of cosmology, because only through them did it
become possible to quantitatively distinguish the pre-
dictions of this model from those of other models. Of
course, the enormous development in computer hard-
ware rendered corresponding progress in simulations
possible; in addition, the continuous improvement of
numerical algorithms has allowed steadily improved
spatial resolution of the simulations.
Since the Universe is dominated by dark matter, it is
often sufficient to compute the behavior of this dark mat-
ter and thus to consider solely gravitational interactions.
Only in recent years has computing power increased
to a level where hydrodynamic processes can also ap-
proximately be taken into account, so that the baryonic
7. Cosmology II: Inhomogeneities in the Univ
component of the Universe can be traced as well. In
addition, radiative transfer can be included in such sim-
ulations, hence the influence of radiation on the heating
and cooling of the baryonic component can also be
examined.
The Principle of Simulations. Representative Dark
Matter Particles. We will now give a brief description
of the principle of such simulations, where we confine
ourselves to dark matter. Of course, no individual par-
ticles of dark matter are traced in the simulations: since
it presumably consists of elementary particles, which
therefore have a high number density, one would only be
able to simulate an extremely small, microscopic section
of the Universe. Rather, one examines the behavior of
dark matter in the expanding Universe by representing
its particles by bodies of mass M, and by then assuming
that these "macroscopic particles" behave like the dark
matter particles in a volume V — M/p. Effectively, this
corresponds to the assumption that dark matter consists
of particles of mass M. Since this assumption cannot be
valid in detail, we will later need to modify the resulting
equations.
Choice of Simulation Volume. The next point one
needs to realize right from the start is that one can-
not simulate the full spatial volume of the Universe
(which may be infinite) but only a representative sec-
tion of it. Typically, a comoving cube with side length L
is chosen. For this section to be representative, the lin-
ear extent L should be larger than the largest observed
structures in the Universe. Otherwise, the effects of the
large-scale structure would be neglected. For example,
hardly any structure is found in the Universe on scales
> 200 h~ x Mpc, so that L — 200 h~ x Mpc is a reason-
able value for the comoving size of the cube. Since the
numerical effort scales with the number of grid points at
which the gravitational force is computed, and which is
limited by the computer's speed and memory, the choice
of L also immediately implies the length-scale of the nu-
merical resolution. Furthermore, the total mass within
the numerical volume is oc Q m L 3 , so that for a given
maximum number of particles, the minimum mass that
can be resolved in the simulation is also known.
Periodic Boundary Conditions. Since particles close
to the boundaries of the cube also feel gravitational
forces from matter outside the cube, one cannot simply
assume the region outside the cube to be empty. We need
to make assumptions about the matter distribution out-
side the numerical volume. Since one assumes that the
Universe is essentially homogeneous on scales > L, the
cube is extended periodically - for instance, a particle
leaving the cube at its upper boundary will immediately
re-enter the cube from the lower side. The mass distri-
bution (and with it also the force field) is periodic in
these simulations, with a period of L. This assumption
of periodicity has an effect on the results for the mass
distribution on scales comparable to L; the quantitative
analysis of the results from these simulations should
therefore be confined to scales < L/2.
Softening Length. With the above assumptions, the
equation of motion for all particles can now be set up.
The force on the i-th particle is
' frf |r ; -r ; | 3
thus the sum of forces exerted by all the other parti-
cles, where these are periodically extended. This aspect
may appear at first sight more difficult than it actually
is, as we will explain below. In particular, the force law
(7.36) also describes strong collisions of particles, e.g.,
where a particle changes its velocity direction by ~ 90°
in a collision if it comes close enough to another parti-
cle. Of course, this effect is a consequence of replacing
the dark matter constituents by macroscopic "particles"
of mass M. As we have seen in Sect. 3.2.4, the typical re-
laxation time-scale for a system is oc N/ In N, and since
the mass in the numerical volume is denned by L, one
has N oc \/M. Reducing the particles' mass and increas-
ing N accordingly, the abundance of strong collisions
would decrease, but computer power and memory is
then a limiting factor. Thus to correct for the artefact of
strong collisions, the force law is modified for small sep-
arations such that strong collisions no longer occur. The
length-scale below which the force equation is modified
("softened") and deviates from oc 1/r 2 is called soften-
ing length and is chosen to be about the mean separation
of two particles of mass M - the smaller M, the smaller
the softening length. This then also defines a limit for
the spatial resolution in the simulation: scales below
or comparable to the softening length are not resolved,
7.5 Non-Linear Structure Evolution
and the behavior on these small scales is affected by
numerical artefacts.
Computation of the Force Field. The computation of
the force acting on individual particles by summation,
as in (7.36), is not feasible in practice, as can be seen
as follows. Assume the simulation to trace 10 8 parti-
cles, ihcn in total 10 16 terms need to be calculated using
(7.36) - for each time step. Even on the most power-
ful computers this is not feasible today. To handle this
problem, one evaluates the force in an approximate way.
One first notes that the force experienced by the i-th par-
ticle, exerted by the j'-th particle, is not very sensitive to
small variations in the separation vector r, — r,, as long
as these variations are much smaller than the separa-
tion itself. Except for the nearest particles, the force on
the i-th particle can then be computed by introducing
a grid into the cube and shifting the particles in the sim-
ulation to the closest grid point. 7 With this, a discrete
mass distribution on a regular grid is obtained. The force
field of this mass distribution can then be computed by
means of a Fast Fourier Transform (FFT), a fast and
very efficient algorithm. However, the introduction of
the grid establishes a lower limit to the spatial force
resolution; this is often chosen such that it agrees with
the softening length. Because the size of the grid cells
also defines the spatial resolution of the force field, it
is chosen to be roughly the mean separation between
two particles, so that the number of grid points is typi-
cally of the same order as the number of particles. This is
called the PM (particle-mesh) method. To achieve better
spatial resolution, the interaction of closely neighbor-
ing particles is considered separately. Of course, this
force component first needs to be removed from the
force field as computed by FFT. This kind of calcu-
lation of the force is called the P 3 M (particle-particle
particle-mesh) method.
Initial Conditions and Evolution. The initial condi-
tions for the simulation are set at very high redshift.
The particles are then distributed such that the power
spectrum of the resulting mass distribution resembles
a Gaussian random field with the theoretical (linear)
power spectrum P(k, z) of the cosmological model. The
In practice, the mass ol a particle is distributed to all 8 neighboring
' its, with the relative proportion ol the mass depending on the
of the panicle to each oi these grid points.
grid
equations of motion for the particles with the force field
described above are then integrated in time. The choice
of the time step is a critical issue in this integration,
as can be seen from the fact that the force on parti-
cles with relatively close neighbors will change more
quickly than that on rather isolated particles. Hence, the
time step is either chosen such that it is short enough for
the former particles - which requires substantial com-
putation time - or the time step is varied for different
particles individually, which is clearly the more efficient
strategy. For different times in the evolution, the parti-
cle positions and velocities are stored; these results are
then available for subsequent analysis.
Examples of Simulations. The size of the simulations,
measured by the number of particles considered, has
increased enormously in recent years with the cor-
responding increase in computing capacities and the
development of efficient algorithms. In modern simu-
lations, 512 3 or even more particles are traced. One
example of such a simulation is presented in Fig. 7.10,
where the structure evolution was computed for four dif-
ferent cosmological models. The parameters for these
simulations and the initial conditions (i.e., the initial re-
alization of the random field) were chosen such that the
resulting density distributions for the current epoch (at
z — 0) are as similar as possible; by this, the depen-
dence of the redshift evolution of the density field on
the cosmological parameters can be recognized clearly.
Comparing simulations like these with observations has
contributed substantially to our realizing that the mat-
ter density in our Universe is considerably smaller than
the critical density.
Massive clusters of galaxies have a very low number
density, which can be seen from the fact that the mas-
sive cluster closest to us (Coma) is about 90 Mpc away.
This is directly related to the exponential decrease of
the abundance of dark matter halos with mass, as de-
scribed by the Press-Schechter model (see Sect. 7.5.2).
In simulations such as that shown in Fig. 7.10, the sim-
ulated volume is still too small to derive statistically
meaningful results on such sparse mass concentrations.
This difficulty has been one of the reasons for simulat-
ing considerably larger volumes. The Hubble Volume
Simulations (see Fig. 7.1 1) use a cube with a side length
of 3000/i~' Mpc, not much less than the currently visi-
ble Universe. This simulation is particularly well-suited
7. Cosmology II: Inhomogeneities in the Univ
ACDM
SCDM
tCDM
OCDM
Fig. 7.10. Simulations of the dark matter distribution in the
Universe for four different cosmological models: Q m = 0.3,
Q A =0.1 (ACDM), Q m =\.0, ^ = 0.0 (SCDM and
rCDM), and Q m = 0.3, Q A = 0. (OCDM). The two Einstein-
de Sitter models differ in their shape parameter F which
specifies the shape of the power spectrum P(k). For each
of the models, the mass distribution is presented for three
different redshifts, z — 3, z — 1, and today, z = 0. Whereas
the current mass distribution is quite similar in all four
models (the model parameters were chosen as such), they
clearh differ at high redshift. We can see, for instance,
that significantly less structure has formed at high redshift
in the SCDM model compared to the other models. From
the analysis of the matter distribution at high redshift, one
can therefore distinguish between the different models. In
these simulations by the VIRGO Consortium, 256 3 parti-
cles were traced; the side length of the simulated volume
is ~ 240/i -'Mpc
to studying the statistical properties of very massive
structures, like, e.g., the distribution of galaxy clusters.
On the other hand, this large volume, together with
the limited total number of particles that can be fol-
lowed, means that the mass and spatial resolution of
this simulation are insufficient for studying galaxies.
To date (2006), by far the largest simulation is the
Millennium simulation, carried out for a cosmological
model with J2 m =0.25, Q a — 0.75, a power spectrum
normalization of erg =0.9, and a Hubble constant of
h=0J3. A cube of side length 500/j -1 Mpc was con-
sidered, in which (2160) 3 «sl0 10 particles with a mass
of 8.6 x 10 8 /z _1 M each were traced. With this choice
of parameters, one can spatially resolve the halos of
galaxies. At the same time, the volume is large enough
for the simulation to contain a large number of massive
7.5 Non-Linear Structure Evolution
The Hubble Volume Simulation
=0.7, h=0.7,
o e =0.9{ACDM)
3000 x 3000 x 30 h~ 3 Mpc 3
P 3 M: ^=35, s=100/r 1 kpc
1000 3 particles, 1024 3 mesh
T3E(Garching) - 51 2cpus
1500 Mpc/h
Fig. 7.11. The Hubble Volume Simulations: simulated is a box
of volume (3000/j -1 Mpc) 3 , containing 10 9 particles, where
a ACDM model with Q m = 0.3 and Q A = 0.7 was cho-
sen. Displayed is the projection of the density distribution
of a 30/i _1 Mpc thick slice of the cube. Simulations like this
can be used to analyze the statistical properties of the mass
distribution in the Universe on large scales. The sector in
the lower left corner represents roughly the size of the CfA
redshift survey (see Fig. 7.2)
clusters whose evolutionary history can be followed.
The spatial resolution of the simulation is ~ 5h~ x kpc,
yielding a linear dynamic range of ~ 10 5 . The result-
ing mass distribution at z=0 is displayed in Fig. 7.12
in slices of 15/z -1 Mpc thickness each, where the linear
scale changes by a factor of four from one slice to the
next. The images zoom in to a region around a massive
cluster that becomes visible with its rich substructure
in the uppermost slice, as well as filaments of the mat-
ter distribution, at the intersections of which massive
halos form. The mass distribution in the Millennium
simulation is of great interest for numerous different in-
vestigations. We will discuss some of its results further
in Chap. 9.
Analysis of Numerical Results. The analysis of the
numerical results is nearly as intricate as the simula-
tion itself because the positions and velocities of ~ 10 9
particles alone do not provide any new insights. The
output of the simulation needs to be analyzed with re-
spect to specific questions. Obviously, the (non-linear)
power spectrum P(k, z) of the matter distribution can be
computed from the spatial distribution of particles; the
corresponding results have led to the construction of the
analytic fit formulae presented in Fig. 7.6. Furthermore,
one can search for voids in the resulting particle dis-
tribution, which can then be compared to the observed
abundance and typical size of voids.
One of the main applications is the search for col-
lapsed mass concentrations (i.e., dark matter halos), and
their number density can be compared to predictions
from the Press-Schechter model and to observations.
From this, it has been found that the Press-Schechter
mass function represents the basic aspects of the mass
spectrum astonishingly well, but even more accurate
formulae for the mass spectrum of halos have been con-
structed from the simulations (see Fig. 7.9). However,
the identification of a halo and the determination of its
mass from the positions and velocities of the particles
is by no means trivial, and various methods for this
are applied. For instance, we can concentrate on spatial
overdensities of particles and define a halo as a spher-
ical region, within which the average density is just
200 times the critical density - this definition of a halo
is suggested by the spherical collapse model. Alterna-
tively, those particles which are gravitationally bound,
as can be obtained from the particle velocities, can be
assigned to a halo.
The direct link between the results from dark matter
simulations and the observed properties of the Uni-
verse requires an understanding of the relation between
dark matter and luminous matter. Dark matter halos in
simulations cannot be compared to the observed gal-
axy distribution without further assumptions, e.g., on
the mass-to-light ratio. We will return to these aspects
later.
Fig. 7.12. Distribution of
matter in slices of thickness
15/z -1 Mpc each, com-
puted in the Millennium
simulation. This simula-
tion took about a month,
running on 512 CPU pro-
cessors. The output of the
simulation, i.e., the posi-
tion and velocities of all
10 10 particles at 64 times
steps, has a data volume of
~ 27 TB. The region shown
in the two lower slices is
larger than the simulated
box which has a sidelength
of 500h~ l Mpc; neverthe-
less, the matter distribution
shows no periodicity in the
figure as the slice was cut at
a skewed angle to the box
7.5.4 Profile of Dark Matter Halos
As already mentioned above, dark matter halos can be
identified in mass distributions generated by numerical
simulations. Besides the abundance of halos as a func-
tion of their mass and redshift, their radial mass profile
can also be analyzed if individual halos are represented
by a sufficient number of dark matter particles. The
ability to obtain halo mass profiles depends on the mass
resolution of a simulation. A surprising result has been
obtained from these studies, namely that halos seem to
show a universal density profile. We will briefly discuss
this result in the following.
If we define a halo as described above, i.e., as a spher-
ical region within which the average density is ~ 200
times the critical density at the respective redshift, the
mass M of the halo is related to its (virial) radius r2oo by
_4tt ,
3 r ' z
Since the critical density at redshift z is specified by
p CI ( z ) = 3// 2 (z)/(8ttG), we can write this as
100r 2 3 00 H 2 (z)
M =
(7.37)
so that at each redshift, a unique relation exists between
the halo mass and its radius. We can also define the
virial velocity V 2 oo °f a halo as the circular velocity at
the virial radius,
VL= — ■ (7-38)
>-200
7.5 Non-Linear Structure Evolution
Combining (7.37) and (7.38), we can express the halo
mass and virial radius as a function of the virial velocity,
M
y 200
: 10GH(z) '
lOH(z)
(7,39)
Since the Hubble function H(z) increases with redshift,
the virial radius at fixed virial velocity decreases with
redshift. From (7.37) we also see that r 2 oo decreases
with redshift at fixed halo mass. Hence, halos at a given
mass (or given virial velocity) are more compact at
higher redshift than they are today.
The NFW Profile. The density profile of halos averaged
over spherical shells seems to have a universal func-
tional form, which was first reported by Julio Navarro,
Carlos Frenk & Simon White in a series of articles in
the mid-1990s. This NFW-profile is described by
Ps
where p s is the amplitude of the density profile, and
r s specifies a characteristic radius. For r <£r s we find
p oc r~ l , whereas for r » r s , the profile follows p oc r~ 3 .
Therefore, r s is the radius at which the slope of the den-
sity profile changes (see Fig. 7.13). p s can be expressed
in terms of r s , since, according to the definition of r 2 no,
p = 200p CI (z) = t- / 4nr 2 dr p(r)
47Tr 200 •>
f dxx 2
where in the last step the integration variable was
changed to x — r/r2oo, and the concentration index
(7.40) was defined. The larger the value of c, the more strongly
the mass is concentrated towards the inner regions.
1 2 3 4 12 3
Fig. 7.13. For eight different cosmological
simulations, the density profile is shown for
the most massive and Ihc least massive halo.
each as a function of the radius, together
with the best fitting density profile (7.40).
The cosmological models represent an EdS
model (here denoted by SCDM), a ylCDM
model, and different models with power
spectra lhal are assumed to be power laws
locally, P(k) oc k" . The arrows indicate the
softening length in the gravitational force
for the respecth e halos; thus, the major part
of the profiles is numerically well resolved
7. Cosmology II: Inhomogeneities in the Univ
Equation (7.41) implies that p s can be expressed ii
terms of p a (z) and c, and performing the integration ii
(7.41) yields
3 ^ crv 'ln(l+c)-c/(l+c) ■
Since M is determined by r 2 oo, the NFW profile is
parametrized by 7-200 (or by the mass of the halo) and by
the concentration c that describes the shape of the distri-
bution. Simulations show that the concentration index c
is strongly correlated with the mass and the redshift of
the halo; one finds approximately
M -
(H
where M* is the non-linear mass scale already men-
tioned in Sect. 7.5.2. This result can also be obtained
from analytical scaling arguments, under the assump-
tion of the existence of a universal density profile. In
Fig. 7.14, the density profile of dark halos is plotted as
a function of the scaled radius r/r2oo, where the simi-
larity in the profile shapes for the different simulations
becomes clearly visible, as well as the dependence of
the concentration index on the halo mass. The range
over which the density distribution of numerically sim-
ulated halos is described by the profile (7.40) is limited
by the virial radius r2oo, whereas in the central region of
halos the numerical resolution of the simulations is too
low to test (7.40) for very small r. The latter comment
concerns the inner ~ 1% of the halo mass.
Generalization. No good analytical argument has yet
been found for the existence of such a universal density
profile, in particular not for the specific functional form
of the NFW profile. As a matter of fact, other numerical
simulations find a slightly different density profile that
can be expressed as
p (X
1
(r/r s )«(l + r/r s ) 3 -
with a ~ 1.5, whereas the NFW profile is characterized
by a — 1. The reason for the difference between differ-
ent simulations has not conclusively been established,
but probably the density profile in the innermost region
Fig. 7.14. The density profiles from Fig.
7.13, but now the density is scaled by the
critical density, and die radius scaled by
.i'Nio. Solid (dashed) curves correspond to
halos of low (high) mass - thus, halos of
low mass are relatively denser close to the
center, and they have a higher
7.5 Non-Linear Structure Evolution
(which is difficult to resolve numerically) is more com-
plicated than a power law. For large radii, the different
research groups agree on the shape of the profile oc r~ 3 .
Comparison with Observations. The comparison of
these theoretical profiles with an observed density dis-
tribution is by no means simple because the density
profile of dark matter is of course not directly observ-
able. For instance, in normal spiral galaxies, p(r) is
dominated by baryonic matter at small radii. For exam-
ple, in the Milky Way, roughly half of the matter within
^?o consists of stars and gas, so that only little informa-
tion is provided on p DM in the central region. In general,
it is assumed that galaxies with very low surface bright-
ness (LSBs) are dominated by dark matter well into the
center. The rotation curves of LSB galaxies are appar-
ency not in agreement with the expectations from the
NFW model (Fig. 7.15); in particular, they provide no
evidence of a cusp in the central density distribution
(p -> oo for r -> 0).
Part of this discrepancy may perhaps be explained
by the finite angular resolution of the 21 -cm line
of the rotation curves; however, the dis-
crepancy remains if higher-resolution rotation curves
are measured using optical long-slit and integral-field
spectroscopy. As an additional point, the kinematics of
these galaxies may be more complicated, and in some
cases their dynamical center is difficult to determine.
The orbits of stars and gas in these galaxies may show
a more complex behavior than expected from a smooth
density profile. The mass distribution in the (inner parts
of a) dark matter halo is neither smooth nor axially
symmetric, and stars and gas do not move on circular
orbits in a thin plane of symmetry. Instead, simulations
show that the pressure support of the gas, together with
non-circular motions and projection effects systemati-
cally underestimate the rotational velocity in the center
of dark matter halos, thereby creating the impression
of a constant density core. Nevertheless, the observed
rotation curves of LSB galaxies may prove to be a ma-
jor problem for the CDM model - hence, this potential
discrepancy must be resolved.
An additional complication is the fact that not only is
baryonic matter present in the inner regions of galaxies
Fig. 7.15. The rotation curves in the NFW
density profiles from Fig. 7.13, in units of
the rotational velocity at r2oo • All curves ini-
tially increase, reach a maximum, and then
decrease again; over a fairly wide range
in radius, the rotation curves are approx-
imately flat. The solid curves are taken
directly from the simulation, while dashed
curves indicate the rotation curves expected
from the NFW profile. The dotted curve in
each panel presents a fit to the low-mass
halo data with the so-called Hernquist pro-
file, a mass distribution frequently used in
modeling - it fits the rotation curve very
well in the inner part of the halo, but fails
beyond ~ 0.1i?200- In these scaled units,
halos of low mass have a relatively higher
n rotational velocity
7. Cosmology II: Inhomogeneities in the Univ
(and clusters), thus contributing to the density, but also
these baryons have modified the density profile of dark
matter halos in the course of cosmic evolution. Baryons
are dissipative, they can cool, form a disk, and accrete
inwards. The change in the resulting density distribution
of baryons by dissipative processes cause a change of the
gravitational potential over time, to which dark matter
also reacts. The dark matter profile in real galaxies is
thus modified compared to pure dark matter simulations.
Despite these difficulties, it has been found that the
X-ray data of many clusters are compatible with an
NFW profile. Analyses based on the weak lensing effect
also show that an NFW mass profile provides a very
good description for shear data. In Fig. 7.16 it is shown
that the radial profile of the galaxy density in clusters
on average follows an NFW profile, where the mean
concentration index is c % 3, i.e., smaller than expected
for the mass profile of clusters. One interpretation of this
result is that the galaxy distribution in clusters is less
strongly concentrated than the density of dark matter.
7.5.5 The Substructure Problem
As we will discuss in detail in the next chapter, the
CDM model of cosmology has proven to be enormously
successful in describing and predicting cosmological
observations. Because this model has achieved this suc-
cess and is therefore considered the standard model,
results that apparently do not fit into the standard model
are of particular interest. The rotation curves of LSB
galaxies mentioned above are one such result. Either
one finds a good reason for this apparent discrep-
ancy between observation and the predictions of the
CDM model or, otherwise, results of this kind indi-
cate the necessity to introduce extensions to the CDM
model. In the former case, the model would have over-
come another hurdle in demonstrating its validity and
would be confirmed even further, whereas in the latter
case, new insights would be gained into the physics of
cosmology.
Sub-Halos of Galaxies and Clusters of Galaxies.
Besides the rotation curves of LSB galaxies, there is
another observation that does not seem to fit into the
picture of the CDM model at first sight. Numerical
simulations of structure formation show that a halo of
mass M contains numerous halos of much lower mass,
Fig. 7.16. The galaxy distribution averaged over 93 nearby
clusters of galaxies, as a function of the projected distance
to the cluster center. Galaxies have been selected in the NIR,
and cluster masses, and thus f-200, have been determined from
X ray Jala. Plotted is the projected number density of cluster
galaxies, averaged over the various clusters, versus the scaled
radius r/r2oo- In the top panel the galaxy sample is split into
luminous and less luminous galaxies, while in the bottom
panel the cluster sample is split according to the cluster mass.
The solid curves show a fit of the projected NFW profile,
which turns out to be an excellent description in all cases. The
concentration index is, with c «* 3, roughly the same in all
eases, and smaller than expected for the mass profile of clusters
so-called sub-halos. For instance, a halo with the mass
of a galaxy cluster contains hundreds or even thousands
of halos with masses that are orders of magnitude lower.
7.5 Non-Linear Structure Evolution
Indeed, this can be expected because clusters of galaxies
contain substructure, visible in the form of the cluster
galaxies. In the upper part of Fig. 7.17, the simulation of
a cluster and its substructure is displayed. Indeed, this
mass distribution looks just like the mass distribution
expected in a cluster of galaxies, with the main clus-
ter halo and its distribution of member galaxies. The
lower part of Fig. 7.17 shows the simulation of a halo
with mass ~ 2 x 10 12 M Q , which corresponds to a mas-
sive galaxy. As one can easily see, its mass distribution
shows a large number of sub-halos as well. In fact, the
two mass distributions are nearly indistinguishable, ex-
Fig. 7.17. Density distribution of two simulated dark mat-
ter halos. In the top image, the halo has a virial mass of
5 x 10 14 M Q , corresponding to a cluster of galaxies. The halo
in the bottom image has a mass of 2 x 10 12 M Q , representing
a massive galaxy. In both cases, the presence of substructure
in ihc mass distribution can be seen. It can. be identified with
individual cluster galaxies in the case of the galaxy cluster.
The substructure in a galaxy can not be identified easily with
any observable source population; one may expect that these
are satellite galaxies, but obscn ations show that these are eon
sidcrably less abundant than the substructure seen here. Apart
from the length-scale (and thus also the mass-scale), both
halos appear very similar from a qualitative point of view
Fig. 7.18. Number density of sub halos as a function of their
iii i 1 1 i mi i i ] i d b tin i hi i ondin ! i i I rian
rotational velocity ig. measured in units of the corresponding
rotational velocity of the main halo. The curves show this
number density of sub-halos with rotational velocity > v c for
a halo of either cluster mass or galaxy mass. The observed
numbers of sub-halos (i.e.. of galaxies) in the Virgo Cluster
are plotted as open circles with error bars, and the numbei
oi i Hit ili i in ih hli \ i\ as filled cncles. One
can see that the simulations describe the abundance oi cluster
galaxies quite well, but around the Galaxy significantly fewer
satellite galaxies exist than predicted by a CDM model
cept for their scaling in the total mass. 8 The presence
of substructure over a very wide range in mass is a di-
rect consequence of hierarchical structure formation,
in which objects of higher mass each contain smaller
structures that have been formed earlier in the cosmic
evolution.
Whereas this substructure in clusters is easily iden-
tified with the cluster member galaxies, the question
arises as to what the sub-halos in galaxies can possibly
correspond to. These show a broad mass spectrum, as
displayed in Fig. 7.18. Some of these sub-halos are rec-
' The reason for this is found in the property of the power spectrum
oi density fluctuations that has been discussed in Sect. /.S.2. namch
that l'(k) can he approximated b\ a power law over a w ide rang.: in k.
Such a power law features no characteristic scale. For this reason,
the properties oi halos of high and low mass are scale invariant, as is
clearly visible in Fig. 7.17.
7. Cosmology II: Inhomogeneities in the Univ
ognized in our Milky Way, namely the known satellite
galaxies like, e.g., the Magellanic Clouds. In a simi-
lar way, the satellite galaxies of the Andromeda galaxy
may also be identified with sub-halos. However, as we
have seen in Sect. 6.1, fewer than 40 members of the
Local Group are known - whereas the numerical sim-
ulations predict hundreds of satellite galaxies for the
Galaxy. This apparent deficit in the number of observed
sub-halos is considered to be another potential problem
of CDM models.
However, one always needs to remember that the
simulations only predict the distribution of mass, and
not that of light (which is accessible to observation).
One possibility of resolving this apparent discrepancy
centers on the interpretation that these sub-halos do in
fact exist, but that most of them do not, or only weakly,
emit radiation. What appears as a cheap excuse at first
sight is indeed already part of the models of the forma-
tion and evolution of galaxies. As will be discussed in
Sect. 9.6.3 in more detail, it is difficult to form a con-
siderable stellar population in halos of masses below
~ 1O 9 M . Most halos below this mass threshold will
therefore be hardly detectable because of their low lu-
minosity. In this picture, sub-halos in galaxies would in
fact be present, as predicted by the CDM models, but
most of these would be "dark".
Evidence for the Presence of CDM Substructure in
Galaxies. A direct indication of the presence of sub-
structure in the mass distribution of galaxies indeed
exists, which originates from gravitational lens systems.
As we have seen in Sect. 3.8, the image configuration of
multiple quasars can be described by simple mass mod-
els for the gravitational lens. Concentrating on those
systems with four images of a source, for which the po-
sition of the lens is also observed (e.g., with the HST),
a simple mass model for the lens has fewer free pa-
rameters than the coordinates of the observed quasar
images that need to be fitted. Despite of this, it is pos-
sible, with very few exceptions, to describe the angular
positions of the images with such a model very ac-
curately. This result is not trivial, because for some
lens systems which were observed using VLBI tech-
niques, the image positions are known with a precision
of better than 10~ 4 arcsec, with an image separation
of the order of 1". This result demonstrates that the
mass distribution of lens galaxies is, on scales of the
image separation, quite well described by simple mass
models.
Besides the image positions, such lens models also
predict the magnifications /x of the individual images.
Therefore, the ratio of the magnifications of two im-
ages should agree with the flux ratio of these images
of the background source. The surprising result from
the analysis of lens systems is that, although the image
positions of (nearly) all quadruply imaged systems are
very precisely reproduced by a simple mass model, in
not a single one of these systems does the mass model
reproduce the flux ratios of the images !
Perhaps the simplest explanation for these results is
that the simple mass models used for the lens are not
correct and other kinds of lens models should be used.
However, this explanation can be excluded for many of
the observed systems. Some of these systems contain
Fig. 7.19. 8.5-GHz map of the lens system 2045+265. The
source at z s = 1-28 is imaged four- fold (components A-D) by
a lens galaxy at z s = 0.867, while component E represents
emission from the lens, as is evident from its different radio
pei in ni I mm tin • en nil pro] f th i ition I It n
mapping, one can show that any "smooth" mass model of the
lens predicts the flux of B to be roughly the same as the sum
of the fluxes of components A and C. Obviously, this rule is
strongly \ iolated in llris lens system, because B is weaker Irian
both A and C. This result can only be explained by small-scale
le mass distribution of the lens galaxy
7.5 Non-Linear Structure Evolution
two or three images of the source that are positioned
very closely together, for which one therefore knows
that they are located close to a critical curve. In such
a case, the magnification can be estimated quite well
analytically; in particular, it no longer depends on the
exact form of the lens model employed. Hence, the exis-
tence of such "universal properties" of the lens mapping
excludes the existence of simple (i.e., "smooth") mass
models capable of describing the observed flux ratios.
One example of this is presented in Fig. 7.19.
The natural explanation for these flux discrepancies
is the fact that a lensing galaxy does not only have
a smooth large-scale mass profile, but that there is also
small-scale substructure in its density. In the case of
spiral galaxies, this may be the spiral arms, which can
be seen as a small-scale perturbation in an otherwise
smooth mass profile. However, most lens galaxies are
ellipticals. The sub-halos that are predicted by the CDM
model may then represent the substructure in their mass
distribution. For a further discussion of this model, we
first should mention that a small-scale perturbation of
the mass profile only slightly changes the deflection an-
gle caused by the lens, whereas the magnification [i
may be modified much more strongly. As a matter
of fact, by means of simulations, it has been demon-
strated that lens galaxies containing sub-halos of about
the same abundance as postulated by the CDM model
give rise to a statistical distribution of discrepancies in
the flux ratios which is very similar to that found in the
observed lens systems. Furthermore, these simulations
show that, on average, a particular image of the source
is clearly demagnified compared to the predictions by
simple, smooth lens models, again in agreement with
the observational results. And finally, in the case that
a relatively massive sub-halo is located close to one of
the images, the image position should also be slightly
shifted, compared to the smooth mass model. This ef-
fect was in fact directly detected in two lens systems: in
these cases, a sub-halo exists in the lens galaxy which is
massive enough to form stars, and which therefore can
be observed. Its effect on the magnification and the im-
age position can then be inferred from the lens model
(see Fig. 7.20).
For these reasons, it is probable that galaxies contain
sub-halos, as predicted by the CDM model, but most
sub-halos, in particular those with low mass, <
Fig. 7.20. On the right, an H-band image of the lens sys-
tem MG 2016+1 12 is shown, consisting of a lens galaxy in
the center and four images of the background source, the
two southernmost of which are nearly merged in this image.
On the left, VLBI maps of these components are presenied;
the radio source consists of a compact core and a jet com-
ponent, clearly visible in images A and B. The VLBI map
of component C reveals that is it in fact a double image
of the source, in which the core and jet components each
are visible twice. Any smooth mass model for the lens gal-
axy predicts that the separation C12-C11 should roughly be
the same as that between C13-C2, which obviously contra-
dicts the observation. In this case, the substructure in the
mass distribution is even visible: if one includes the weak
emission south of component C, which is visible in the im-
age on the right, into the lens model as a mass component,
the separation of the components in image C can be well
modeled
7. Cosmology II: Inhomogeneities in the Univ
only few stars and are therefore not visible. One conse-
quence of this explanation is that the low-mass satellite
galaxies that are seen in our Local Group should be
dominated by dark matter. Given the faintness and low
surface brightness of these galaxies, obtaining kinemat-
ical information for them is very difficult and requires
large telescopes for spectroscopy of individual stars in
these objects. The results of such investigations indicate
that the dwarf galaxies in the Local Group are indeed
dark matter dominated, with a mass-to-light ratio of
~ 100 in Solar units. However, this conclusion is based
on the assumption that the stars in these systems are in
dynamical equilibrium, an assumption which is difficult
to test.
7.6 Peculiar Velocities
As mentioned on several occasions before, cosmic
sources do not exactly follow the Hubble expansion, but
have an additional peculiar velocity. Deviations from
the Hubble flow are caused by local gravitational fields,
and such fields are in turn generated by local density
fluctuations. These inevitably lead to an acceleration,
which affects the matter and generates peculiar veloc-
ities. In numerical simulations, the peculiar velocities
of individual particles are followed in the computations
automatically. In this brief section, we will investigate
the large-scale peculiar velocities as they are derived
from linear perturbation theory.
Since the spatial dependence of the density contrast 8
is constant in time, 8(x, t) — <5 (jc) D+(t) (see Eq. 7.14),
the acceleration vector g has a constant direction in
the framework of linear perturbation theory. Hence, one
obtains the peculiar velocity in the form
u(x) =
-f(V m )g(x),
3H Q m
where we defined the function
a{t) AD + _ d log D+
D + (t) da dloga
/(Am) : =
For t — t , the function f{Q m ) can be expressed
by a very simple and very accurate approximation,
/(J2 m ) ss i2^f. This was first discovered for the case
where Q A — 0, but it was later found that a cosmologi-
cal constant has only a marginal effect on this relation.
Introducing corrections arising from Q A , one obtains
the slightly more accurate approximation
Q A
Q m
/^r+^l + ^j- (7-45)
From the smallness of the last term, one can see that the
correction for A is marginal indeed, because of which
one sets / = Q { ^ 6 in most cases.
On the other hand, g(x) is the gradient of the grav-
itational potential, goc—V(p. This implies that u(x)
is a gradient field, i.e., a scalar function \/f(x) exists
such that u = V\fr, where the gradient is taken with
respect to the comoving spatial coordinate x. There-
fore, V • g oc — V 2 </> oc — <5, so that also V • u oc —8; here,
the Poisson equation (7.10) has been utilized. Taken
together, these results yield for today
V-u(x) = -H Q™8 (x).
(7.46)
u(x)~ I dtg(x,t) ,
i.e., parallel to g(x). Quantitatively, we obtain for to-
day, thus for t = to,a relation between the velocity and
acceleration field:
We would like to derive this result in somewhat more
detail and begin with the linearized form of Eq. (7.8),
where the gradient is, here and in the following, alw ays
taken with respect to comoving coordinates. The fact
that 8(x, t) factorizes (see Eq. 7.14) immediately yields
35 _ D+
Combining this equation with (7.47) and, as above,
defining u — V\j/ leads to
Cl D+
a H(a) f(Q m ) 8 *
M D+ da
a H(a) Q^ 6 8 ,
where we used the previously defined function f(£2 m ).
This Poisson equation for jjr can be solved, and by com-
puting the gradient the peculiar velocity field can be
7.7 Origin of the Density Fluctuations
u(x,t) = -p-aH(a) [ d 3 yS(y,t) T
4tt J |.
(7.49)
Equation (7.49) shows that the velocity field can be
derived from the density field. If the density field in the
Universe were observable, one would obtain a direct
prediction for the corresponding velocity field from the
above relations. This depends on the matter density Q m ,
so that from a comparison with the observed velocity
field, one could estimate the value for Q m . We will come
back to this in Sect. 8.1.6.
7.7 Origin of the Density Fluctuations
We have seen in Sect. 4.5.3 that the horizon and the
flatness problem in the normal Friedmann-Lemaitre
evolution of the Universe can be solved by postulating
an early phase of very rapid - exponential - expansion
of the cosmos. In this inflationary phase of the Uni-
verse, any initial curvature of space is smoothed away
by the tremendous expansion. Furthermore, the expo-
nential expansion enables the complete currently visible
Universe to have been in causal contact prior to the in-
flationary phase. These two aspects of the inflationary
model are so attractive that today most cosmologists
consider inflation as part of the standard model, even if
the physics of inflation is as yet not understood in detail.
The inflationary model has another property that is
considered very promising. Through the huge expan-
sion of the Universe, microscopic scales are blown
up to macroscopic dimensions. The large-scale struc-
ture in the current Universe corresponds to microscopic
scales prior to and during the inflationary phase. From
quantum mechanics, we know that the matter distribu-
tion cannot be fully homogeneous, but it is subject to
quantum fluctuations, expressed, e.g., by Heisenberg's
uncertainty relation. By inflation, these small quantum
fluctuations are expanded to large-scale density fluc-
tuations. For this reason, the inflationary model also
provides a natural explanation for the presence of initial
density fluctuations.
In fact, one can study these effects quantitatively and
attempt to calculate the initial power spectrum of these
fluctuations. The result of such investigations will de-
pend slightly on the details of the inflationary model
they are based on. However, these models agree in their
prediction that the initial power spectrum should have
a form very similar to the Harrison-Zeldovich fluctua-
tion spectrum, except that the spectral index n s of the
primordial power spectrum should be slightly smaller
than the Harrison-Zeldovich value of n s — 1. Thus, the
model of inflation can be directly tested by measuring
the power spectrum and, as we shall see in Chap. 8, the
power-law slope n s indeed seems to be slightly flatter
that unity, as expected from inflation.
The various inflationary models also differ in their
predictions of the relative strength of the fluctuations of
spacetime, which should be present after inflation. Such
fluctuations are not directly linked to density fluctua-
tions, but they are a consequence of General Relativity,
according to which spacetime itself is also a dynami-
cal parameter. One consequence of this is the existence
of gravitational waves. Although no gravitational waves
have been directly detected until now, the analysis of the
double pulsar PSR J 19 15+ 1606 proves the existence of
such waves. 9 Primordial gravitational waves provide
an opportunity to empirically distinguish between the
various models of inflation. These gravitational waves
leave a "footprint" in the polarization of the cosmic
microwave background that is measurable in principle.
A satellite mission to perform these measurements is
currently being discussed.
9 The double pulsar PSRJ1915+1606 was discovered in 1974. From
the orbital motion of the pukar and its companion star. gnu national
waxes arc emitted, according to General Rclalivil; . Through this, the
sclent loses kinetic (orbital) energy, so thai the size of the orbit de-
creases o\er time. Since pulsars represent excellent clocks, and we
can measure time w ith extreme!) high precision, this change in the or
bital motion can be observed with very high accuracy and compared
with predictions from General Relaioitv The fantastic agreement of
theory and observation is considered a delinite proof of the existence
of gravitational waves. For the discovery of the double pulsar and
the detailed analysis of this system. Russell llulse and Joseph Taylor
were awarded the Nobel Prize in Physics in 1993. In 2003, a dou-
ble neutron star binary was discovered where pulsed radiation front
both components cam be observed. This fact, together with the small
orbital period of 2. I it imply ing a small separation of the two stars,
makes tins an e\en better laboratory for study ing strong- held gra\ in.
8. Cosmology III: The Cosmological Parameters
In Chaps. 4 and 7, we described the fundamental aspects
of the standard model of cosmology. Together with the
knowledge of galaxies, clusters of galaxies, and AGNs
that we have gained in the other chapters, we are now
ready to discuss the determination of the various cos-
mological parameters. In the course of this discussion,
we will describe a number of methods, each of which is
in itself useful for estimating cosmological parameters,
and we will present the corresponding results from these
methods. The most important aspect of this chapter is
that we now have more than one independent estimate
for each cosmological parameter, so that the determina-
tion of these parameters is highly redundant. This very
aspect is considerably more important than the precise
values of the parameters themselves, because it provides
a test for the consistency of the cosmological model.
We will give an example in order to make this
point clear. In Sect. 4.4.4, we discussed how the cos-
mic baryon density can be determined from primordial
nucleosynthesis and the observed ratio of deuterium to
hydrogen in the Universe. Thus, this determination is
based on the correctness of our picture of the thermal
history of the early Universe, and on the validity of the
laws of nuclear physics shortly after the Big Bang. As
we will see later, the baryon density can also be derived
from the angular fluctuations in the cosmic background
radiation, for which the structure formation in a CDM
model, discussed in the previous chapter, is needed as
a foundation. If our standard model of cosmology was
inconsistent, there would be no reason for these two
values of the baryon density to agree - as they do in
a remarkable way. Therefore, in addition to obtaining
a more precise value of £2^ from this comparison than
from each of the individual methods alone, the agree-
ment is also a strong indication of the validity of the
standard model.
We will begin in Sect. 8.1 with the observation of the
large-scale distribution of matter, the large-scale struc-
ture (LSS). It is impossible to observe the large-scale
structure of the matter distribution itself; rather, only the
spatial distribution of visible galaxies can be measured.
Assuming that the galaxy distribution follows, at least
approximately (which we will specify later), that of the
dark matter, the power spectrum of the density fluctu-
ations can be estimated from that of the galaxies. As
we discussed in the previous chapter, the power spec-
trum in turn depends on the cosmological parameters.
In Sect. 8.2, we will summarize some aspects of clus-
ters of galaxies which are relevant for the determination
of the cosmological parameters.
In Sect. 8.3, Type la supernovae will be used as
cosmological tools, and we will discuss their Hubble
diagram. Since SN la are considered to be standard can-
dles, their Hubble diagram provides information on the
density parameters Q m and Q A . These observations
provided the first clear indication, around 1998, that
the cosmological constant differs from zero. We will
then analyze the lensing effect of the LSS in Sect. 8.4,
by means of which information about the statistical
properties of the LSS of matter is obtained directly,
without the necessity for any assumptions on the rela-
tion between matter and galaxies. As a matter of fact,
this galaxy-mass relation can be directly inferred from
the lens effect. In Sect. 8.5, we will turn to the proper-
ties of the intergalactic medium and, in particular, we
will introduce the Lyman-a forest in QSO spectra as
a cosmological probe.
Finally, we will discuss the anisotropy of the cosmic
microwave background in Sect. 8.6. Through observa-
tions of the cosmic microwave background and their
analysis, a vast amount of very accurate information
about the cosmological parameters are obtained. In par-
ticular, we will report on the recent and exciting results
concerning CMB anisotropics, and will combine these
findings with the results obtained by other methods.
This combination yields a set of parameters for the
cosmological model which is able to describe nearly
all observations of cosmological relevance in a self-
consistent manner, and which today defines the standard
model of cosmology.
8.1 Redshift Surveys of Galaxies
8.1.1 Introduction
The inhomogeneous large-scale distribution of matter
that was described in Chap. 7 is not observable directly
Peter Schneider. Cosmolosrx 111: The Cosmological Parameters.
In: Peter Schneider. Extragalactic Astronomy and Cosmology, pp. 309-354 (2006)
DOI: 10.1007/1 1614371_8 © Springer- Verlag Berlin Heidelberg 2006
8. Cosmology III: The Cosmological Parameters
because it consists predominantly of dark matter. If it is
assumed that the distribution of galaxies traces the un-
derlying distribution of dark matter fairly, the properties
of the LSS of matter could be studied by observing the
galaxy distribution in the Universe. Quite a few good
reasons exist for this assumption not to be completely
implausible. For instance, we observe a high galaxy
density in clusters of galaxies, and with the methods
discussed in Chap. 6, we are able to verify that clusters
indeed represent strong mass concentrations. Qualita-
tively, this assumption therefore seems to be justified.
We will later modify it slightly.
In any case, the distribution of galaxies on the sphere
appears inhomogeneous and features large-scale struc-
ture. Since galaxies have evolved from the general
cosmic density field, they should contain information
about the latter. It is consequently of great interest
to examine and quantify the properties of the galaxy
distribution.
In principle, two possible ways exist to accomplish
this study of the galaxy distribution. With photometric
sky surveys, the two-dimensional distribution of galax-
ies on the sphere can be mapped. To also determine the
third spatial coordinate, it is necessary to measure the
redshift of the galaxies using spectroscopy, deriving the
distance from the Hubble law (1.6). It is obvious that
we can learn considerably more about the statistical
properties of the galaxy distribution from their three-
dimensional distribution; hence, redshift surveys are of
particular interest.
The graphical representation of the spatial galaxy po-
sitions is accomplished with so-called wedge diagrams.
They represent a sector of a circle, with the Milky Way
at its center. The radial coordinate is proportional to
z (or cz - by this, the distance is measured in km/s),
and the polar angle of the diagram represents an angu-
lar coordinate in the sky (e.g., right ascension), where
an interval in the second angular coordinate is selected
in which the galaxies are located. An example for such
a wedge diagram is shown in Fig. 8.1.
8.1.2 Redshift Surveys
Performing redshift surveys is a very time-consuming
task compared to making photometric sky maps,
because recording a spectrum requires much more ob-
CfA2
Max radius 12000
0<h< 12000 (km/s)
m E sl5.5
Copyright 2001 SAO
equatorial coordinates.
Fig. 8.1. The Cf A redshift survey,
Along its radial axis, this wedge diagram shows the escape
velocity cz up to 12 000 km/s. and the polar angle specifics
ihc right ascension of a galaxy. The Great Wall extends from
9 h to 15 h . The overdensity at l h and cz = 4000 km/s is the
Pisces-Perseus supercluster
serving time than the mere determination of the apparent
magnitude of a source. Hence, the history of redshift
surveys, like that of many other fields in astronomy,
is driven by the development of telescopes and instru-
ments. The introduction of CCDs in astronomy in the
early 1980s provided a substantial increase in sensi-
tivity and accuracy of optical detectors, and enabled
us to carry out redshift surveys of galaxies in the
nearby Universe containing several thousand galaxies
(see Fig. 8.1). Using a single slit in the spectrograph
implied that in each observation the spectra of only
one or very few galaxies could be recorded simultane-
ously. The situation changed with the introduction of
spectrographs with high multiplexity which were de-
signed specifically to perform redshift surveys. With
them, the spectra of many objects (up to a thousand)
in the field-of-view of the instrument can be observed
simultaneously.
The Strategy of Redshift Surveys. Such a survey is ba-
sically defined by two criteria. The first is its geometry:
8.1 Redshift Surveys of Galaxies
a region of the sky is chosen in which the survey is per-
formed. Second, those objects in this region need to be
selected for which spectra should be obtained. In most
cases, for practical reasons the objects are selected ac-
cording to their brightness, i.e., spectra are taken of all
galaxies above a certain brightness threshold. The latter
defines the number density of galaxies in the survey, as
well as the required exposure time. To apply the second
criterion, a photometric catalog of sources is required
as a starting point. The criteria may be refined further
in some cases. For instance, a minimum angular extent
of objects may be chosen to avoid the inclusion of stars.
The spectrograph may set constraints on the selection
of objects; e.g., a multi-object spectrograph is often un-
able to observe two sources that are too close together
on the sky.
Examples of Redshift Surveys. In the 1980s, the Cen-
ter for Astrophysics (CfA) Survey was carried out which
measured the redshifts of more than 14000 galaxies in
the local Universe (Fig. 8.1). The largest distances of
these galaxies correspond to about cz ~ 15 OOOkm/s.
One of the most spectacular results from this survey
was the discovery of the "Great Wall", a huge structure
in the galaxy distribution (see also Fig. 7.2).
In the Las Campanas Redshift Survey (LCRS), car-
ried out in the first half of the 1990s, the redshifts of
more than 26 000 galaxies were measured. They are
located in six narrow strips of 80° length and 1.5°
width each. With distances of up to ~ 60 000 km/s,
this survey is considerably deeper than the CfA Red-
shift Survey. The distribution of galaxies is displayed in
Fig. 8.2, from which we can recognize the typical bub-
ble or honeycomb structure. Galaxies are distributed
along filaments, which are surrounding large regions in
which virtually no galaxies exist - the aforementioned
voids. The galaxy distribution shows a structure which is
qualitatively very similar to the dark matter distribution
generated in numerical simulations (see, e.g., Fig. 7.12).
In addition, we see from the galaxy distribution that no
structures exist with scales comparable to the extent of
the survey. Thus, the LCRS has probed a scale larger
than that where significant structures of the mass dis-
tribution are found. The survey volume of the LCRS
therefore covers a representative section of the Universe.
A different kind of redshift survey became possi-
ble through the sky survey carried out with the IRAS
satellite (see Sect. 1.3.2). In these redshift surveys of
IRAS galaxies, the selection of objects for which spectra
were obtained was based on the 60 |xm flux mea-
sured by IRAS in its (near) all-sky survey. Various
redshift surveys are based on this selection, differing
in the flux limit applied; for example the 2Jy sur-
vey (hence, S 6 o nm > 2 ly), or the 1.2 Jy survey. The
QDOT and PSCz surveys both have a limiting flux of
^60 [un > 0.6 Jy, where QDOT observed spectra for one
out of six randomly chosen galaxies from the IRAS
sample, while PSCz is virtually complete and contains
~ 15 500 redshifts. One of the advantages of the IRAS
Fig. 8.2. The Las Campanas Redshiii Survey consists of three
fields each at the North and South Galactic Pole. Each of these
fields is a strip 1 .5° wide and 80° long. Overall, the survey con-
tains about 26 000 galaxies, and the median of their redshiii
is about 0.1. The six strips show the distribution of galaxies
on Hie sphere, and the wedge diagram indicates, for galaxies
with measured redshift, the right ascension versus distance
from the Milky Way, measured in units of 1000 kms -1
8. Cosmology III: The Cosmological Parameters
surveys is that the FIR flux is nearly unaffected by Gal-
actic absorption, an effect that needs to be corrected
for when galaxies are selected from optical photometry.
Furthermore, the PSCz is an "all-sky" survey, contain-
ing the galaxy distribution in a sphere around us, so that
we obtain a complete picture of the local galaxy dis-
tribution. However, one needs to be aware of the fact
that in selecting galaxies via their FIR emission one
is thus selecting a particular type of galaxy, predomi-
nantly those which have a high dust content and active
star formation which heats the dust.
The Canada-France Redshift Survey (CFRS) ob-
tained spectroscopy of faint galaxies with 17.5 < I <
22.5, with a median redshift of about 0.5. The resulting
catalog contains 948 objects, 591 of which are galaxies.
This survey was performed by a multi-object spectro-
graph at the CFHT (see Sect. 1.3.3) which was able to
take the spectra of up to 100 objects simultaneously.
For the first time, due to its faint limiting magnitude it
enabled us to study the evolution of (optically-selected)
galaxies, for example by means of their luminosity func-
tion and their star-formation rate, and to investigate the
redshift dependence of the galaxy correlation function -
and thus to see the evolution of the large-scale structure.
Currently, two large spectroscopic surveys with faint
limiting magnitudes are being carried out. Both of them
use high multiplex spectrographs mounted on 10-m
class telescopes: the VIMOS instrument on the VLT
and the DEIMOS instrument on Keck. The target of
both surveys, the VIMOS VLT Deep Survey (VVDS)
and the DEEP2 survey, is to obtain spectra of several
tens of thousands of galaxies with z ~ 1, thus extend-
ing the CFRS by more than an order of magnitude in
sample size and by ~ 1.5 magnitudes in depth.
The 2dF Survey and the Sloan Digital Sky Survey.
The scientific results from the first redshift surveys mo-
tivated the production of considerably more extended
surveys. By averaging over substantially larger volumes
in the Universe, it was expected that the statistics on the
galaxy distribution could be significantly improved. In
addition, the analysis of the galaxy distribution at higher
redshift would also enable a measurement of the evolu-
tion in the galaxy distribution. Two very extensive red-
shift surveys were performed with these main objectives
in mind: the two-degree Field Galaxy Redshift Survey
(2dFGRS) and the Sloan Digital Sky Survey (SDSS).
The 2dFGRS was carried out using a spectrograph
specially designed for this project, which was mounted
at the 4-m Anglo Australian Telescope. Using optical
fibers to transmit the light of the observed objects from
the focal plane to the spectrograph, up to 400 spectra
could be observed simultaneously over a usable field
with a diameter of 2°. The positioning of the individual
fibers on the location of the pre-selected objects was
performed by a robot. The redshift survey covered two
large connected regions in the sky, of 75° x 15° and
75° x7.5°, plus 100 additional, randomly distributed
fields. This survey geometry was chosen so as to yield
the optimal cosmological information about the galaxy
distribution, that is, the most precise measurement of the
correlation function at relevant scales. The photometric
input catalog was the APM galaxy catalog which had
been compiled from digitized photographic plates. The
limiting magnitude of the galaxies for which spectra
were obtained is approximately B < 19.5, where this
value is corrected for Galactic extinction. The 2dFGRS
has been completed, and it contains redshifts for more
than 230 000 galaxies (see Fig. 7.1). The spectra and
redshifts are publicly available. The scientific yield from
this large data set is already very impressive, as we will
show further below.
For the SDSS, a dedicated telescope was built,
equipped with two instruments. The first is a camera
with 30 CCDs which has scanned nearly a quarter of
the sky in five photometric bands, generating by far the
largest photometric sky survey with CCDs. The amount
of data collected in this survey is enormous, and its
storage and reduction required a tremendous effort. For
this photometric part of the Sloan Survey, a new pho-
tometric system was developed, with its five filters (u,
g, r, i, z) chosen such that their transmission curves
overlap as little as possible (see Appendix A.4). The
selection of targets for spectroscopy was carried out us-
ing this photometric information. As in the 2dF Survey,
the multi-object spectrograph used optical fibers, and in
this case these had to be manually installed in holes that
had been punched into a metal plate. With about 640 si-
multaneously observed spectra, the strategy was similar
to that for the 2dFGRS. The aim of the spectroscopic
survey was to obtain about a million galaxy spectra.
The data products of the SDSS have been made pub-
licly available at regular intervals, and currently (2006)
about half of the survey has been published. For the
8.1 Redshift Surveys of Galaxies
SDSS, the scientific yield has also been very high al-
ready, and not just based on the redshift survey. In fact,
the photometric data has been used in a large variety of
other Galactic and extragalactic projects.
Both the 2dF survey and the SDSS also recorded,
besides the spectra of galaxies, those of QSOs which
were selected based on their optical colors; this yielded
by far the most extensive QSO surveys.
\n) v n N V '
where S y is the density contrast of matter, averaged over
the volume V. Under the assumption of linear biasing,
we can then infer the statistical properties of matter from
those of the galaxy distribution. A physical model for
biasing is sketched in Fig. 8.3.
8.1.3 Determination of the Power Spectrum
We will now return to the question of whether the dis-
tribution of (dark) matter in the Universe can be derived
from the observed distribution of galaxies. If galaxies
trace the distribution of dark matter fairly, the power
spectrum of dark matter can be determined from the
galaxy distribution. However, since the formation and
evolution of galaxies is as yet not understood suffi-
ciently well to allow us to quantitatively predict the
relation between galaxies and dark matter (at least
not without introducing a number of model assump-
tions), this assumption is not justified a priori. For
instance, it may be that there is a threshold in the
local density of dark matter, below which the forma-
tion of galaxies does not occur or is at least strongly
suppressed.
The connection between dark matter and galaxies is
parametrized by the so-called linear bias factor b. It is
defined by
An
Ap
where n is the average density of the galaxy population
considered, and An — n — n is the deviation of the local
number density of galaxies from their average density.
Hence, the bias factor is the ratio of the relative overden-
sities of galaxies to dark matter. Such a linear relation is
not strictly justified from theory. However, it is a plau-
sible ansatz on scales where the density field is linear.
In principle, the bias factor b may depend on the gal-
axy type, on redshift, and on the length-scale that is
considered.
The definition given in (8.1) must be understood in
a statistical sense. In a volume V, we expect on aver-
age N = nV galaxies, whereas the observed number of
Normalization of the Power Spectrum. In Sect. 7.4.2,
we demonstrated that the power spectrum of the den-
sity fluctuations can be predicted in the framework
of a CDM model, except for its normalization which
has to be measured empirically. A convenient way for
its parametrization is through the parameter og. This
parameter is motivated by the following observation.
Analyzing spheres of radius R — 8h~ l Mpc in the lo-
cal Universe, it is found that optically-selected galaxies
have, on this scale, a fluctuation amplitude of about 1 ,
(8.2,
where the averaging is performed over different spheres
of identical radius R — %h~ l Mpc. Accordingly, we
define the dispersion of the matter density contrast,
averaged over spheres of radius R = %h~ l Mpc as
-n
Using the definition of the bias factor (
obtain
(8.3)
.1), we then
8.4)
Because of this simple relation, it has become common
practice to use er 8 as a parameter for the normalization
of the power spectrum. 1 If b = 1, thus if galaxies trace
the matter distribution fairly, then one has org «s 1 . If b
is not too different from unity, we see that the density
fluctuations on a scale of ~ %h~ l Mpc are becoming
non-linear at the present epoch, in the sense that S ~ 1 .
On larger scales, the evolution of the density contrast
can approximately be described by linear perturbation
theory.
'More precisely, one considers o\ the normalization of the power
spectrum linearly extrapolated to the present day l'o(k). so thai the
relation (8.4) needs to be modified slightly.
8. Cosmology III: The Cosmological Parameters
Fig. 8.3. The sketch represents a particular model oi biasing.
Let the one-dimensional density profile of matter be speci-
fied by the solid curve, which results from a superposition of
a large scale (represented by the dashed curve) and a small-
scale fluctuation. Assuming thai galaxies can form only at
locations where the density field exceeds a certain threshold -
plotted as a straight line - the galaxies in this density profile
will be localized at the positions indicated by the arrows. Ob-
viously, the locations of the galaxies are highly correlated;
they only form near the peaks of the large-scale fluctuation. In
this picture, ihe correlation of galaxies on small scales is much
stronger than the correlation of the underlying density field
Shape of the Power Spectrum. If one assumes that
b does not depend on the length-scale considered, the
shape of the dark matter power spectrum can be deter-
mined from the power spectrum P g (k) of the galaxies.
whereas its amplitude depends on b. As we have seen in
Sect. 7.4.2, the shape of P(k) is described by the shape
parameter r — h Q m in the framework of CDM models.
The comparison of the shape of the power spectrum
of galaxies with that of CDM models yields r ~ 0.25
(see Fig. 8.4). Since r — hQ m , this result indicates
a Universe of low density (unless h is unreasonably
low).
In the 2dFGRS, the power spectrum of galaxies
was measured with a much higher accuracy than had
previously been possible. Since a constant b can be
expected, at best, in the linear domain, i.e., on scales
above ~ \§h~ l Mpc, only such linear scales are used
in the comparison with the power spectra from CDM
models. As the density parameter Q m seems to be rela-
tively small, the baryonic density plays a noticeable role
in the transfer function (see Eq. 7.25) which depends
on £?b as well as on r. The measurement accuracy of
the galaxy distribution in the 2dFGRS is high enough
to be sensitive to this dependence. In Fig. 8.5, the mea-
sured power spectrum of galaxies from the 2dFGRS
is shown, together with predictions from CDM mod-
els for different shape parameters r. Two families of
model curves are drawn: one where the baryon density
is set to zero, and the other for the value of Q b which
results from the analysis of primordial nucleosynthesis
(see Sect. 4.4.4).
Considering models in which Q b is a free parameter,
there are two domains in parameter space for which
good fits of the power spectrum of the galaxy distribu-
tion are obtained (see Fig. 8.5). One of the two domains
is characterized by a very high baryon fraction of the
matter density, and by a very large value for Q m h. These
parameter values are incompatible with virtually every
other determination of cosmological parameters. On the
other hand, a good fit to the shape of the power spectrum
is obtained by
r = Q m h = 0.18 ±0.02, Q h /Q m
= 0.17 ±0.06.
(8.5)
As is seen in Fig. 8.5, and as we will show further be-
low, these values for the parameters are in very good
agreement with those obtained from other cosmological
observations.
Comparing the power spectra of two different types
of galaxies, we should observe that they are propor-
tional to each other and to the power spectrum of dark
matter. Their amplitudes, however, may differ if their
bias factors are different. The comparison of red and
blue galaxies in the 2dFGRS shows that these indeed
have a very similar shape, supporting the assumption of
a linear biasing on large scales. However, the bias factor
8.1 Redshift Surveys of Galaxies
ff'
Sfi
A?-'"
* Abell
* Radio
* Abell x IRAS
«CfA
a APM/Stromlo
* Radio x IRAS
•IRAS
« APM (angular)
k/h Mpc -1
Fig. 8.4. Left: the power spectrum of galaxies is displayed,
as determined from different galaxy surveys, where A 2 (k) <x
/e P(.k) is a dimensionless description of the power spectrum.
Right: model spectra for A(k) are plotted, where F varies
0.01
k/h Mpc" 1
from 0.5 (uppermost curve) to 0.2; the data from the various
surveys have been suitably averaged. We see that a value of
r ~ 0.25 for the shape parameter fits the observations quite
k/h Mpc 1
Fig. 8.5. Left: power spectrum of the galaxy distribution as
measured in the 2dFGRS (points with error bars), here rep-
resented as A 2 (k) ock 3 P(k). The curves show power spectra
from CDM models with different shape parameter r = O m h,
and two values of Q^\ one as obtained from primordial nu-
cleosynthesis (BBN, solid curves), and the other for models
without baryons (dashed curves). The Hubble constant h—Q.l
and the slope n s = 1 of the primordial power spectrum were
assumed. A very good fit to the observational data is obtained
for r ss 0.2 (from Peacock, 2003, astro-ph/0309240). Right:
confidence contours in the Q m h-Q\, / ^ m -plane. Two regions
in parameter space are seen to provide good fits to the data.
The upper region is incompatible with many other cosmologi
cal data. In contrast, the lower left domain in parameter space
is in remarkable agreement with measurements from BBN
(see Sect. 4.4.4), with the baryon fraction in clusters of galax-
ies (see Sect. 8.2.3), and with CMB anisotropy rr
(see Sect. 8.6)
8. Cosmology III: The Cosmological Parameters
for red galaxies is larger by about a factor 1 .4 than for
blue galaxies. This result is not completely unexpected,
because red galaxies are located preferentially in clus-
ters of galaxies, whereas blue galaxies are rarely found
in massive clusters. Hence, red galaxies seem to fol-
low the density concentrations of the dark matter much
more closely than blue galaxies.
8.1.4 Effect of Peculiar Velocities
Redshift Space. The relative velocities of galaxies in
the Universe are not only due to the Hubble expansion
but, in addition, galaxies have peculiar velocities. The
peculiar velocity of the Milky Way is measurable from
the CMB dipole (see Fig. 1.17). Owing to these pecu-
liar velocities, the observed redshift of a source is the
superposition of the cosmic expansion velocity and its
peculiar velocity v along the line-of-sight,
--H D + v
(8.6!
The measurement of the other two spatial coordinates
(the angular position on the sky) is not affected by
the peculiar velocity. The peculiar velocity therefore
causes a distortion of galaxy positions in wedge dia-
grams, yielding a shift in the radial direction relative to
their true positions. Since, in general, only the redshift
is measurable and not the true distance D, the observed
three-dimensional position of a source is specified by
the angular coordinates and the redshift distance
ity dispersion, the galaxies span a broad range in s,
which is easily identified in a wedge diagram as a highly
stretched structure pointing towards us, as can be seen
in Fig. 7.2.
Mass concentrations of smaller density have the op-
posite effect: galaxies that are closer to us than the center
of this overdensity move towards the concentration,
hence away from us. Therefore their redshift distance s
is larger than their true distance D. Conversely, the pecu-
liar velocity of galaxies behind the mass concentration
is pointing towards us, so their s is smaller than their true
distance. If we now consider galaxies that are located
on a spherical shell around this mass concentration, this
sphere in physical space becomes an oblate ellipsoid
with symmetry axis along the line-of-sight in redshift
space. This effect is illustrated in Fig. 8.6.
Hence, the distortion between physical space and
redshift space is caused by peculiar velocities which
manifest themselves in the transformation (8.7) of the
radial coordinate in space (thus, the one along the line-
of-sight). Due to this effect, the correlation function of
galaxies is not isotropic in redshift space. The reason for
this is the relation between the density field and the cor-
responding peculiar velocity field. Specializing (7.49)
to the current epoch and using the relation (8.1) between
the density fields of matter and of galaxies, we obtain
! S/^
y-x
where we defined the parameter
Ho
--D +
H
(8.7)
The space that is spanned by these three coordinates
is called redshift space. In particular, we expect that
the correlation function of galaxies is not isotropic in
redshift space.
Galaxy Distribution in Redshift Space. The best
known example of this effect is the "Fingers of God". To
understand their origin, we consider galaxies in a clus-
ter. They are situated in a small region in space, all at
roughly the same distance D and within a small solid
angle on the sphere. However, due to the high veloc-
This relation between the density field of galaxies and
the peculiar velocity is valid in the framework of lin-
ear perturbation theory under the assumption of linear
biasing. The anisotropy of the correlation function is
now caused by this correlation between u(x) and S s (x),
and the degree of anisotropy depends on the param-
eter p. Since the correlation function is anisotropic, this
likewise applies to the power spectrum.
Cosmological Constraints. Indeed, the anisotropy of
the correlation function can be measured, as is shown in
Fig. 8.7 for the 2dFGRS (where in this figure, the usual
8.1 Redshift Surveys of Galaxies
"Finger of God"
convention of denoting the transverse separation as a
and that along the line-of-sight in redshift space as it is
followed). Clearly visible is the oblateness of the curves
of equal correlation strength along the line-of-sight for
separations > 10ft -1 Mpc, for which the density field is
still linear, whereas for smaller separation the finger-of-
god effect emerges. This oblateness at large separations
depends directly on ji, due to (8.8), so that (3 can be de-
Fig. 8.6. The influence of peculiar velocities on
the location of galaxies in redshift space. The
upper left panel shows the positions of galaxies
(points) in redshift space, which are in reality
located on spherical shells. Galaxies connected
by curves have the same separation from the
center of a spherically-symmetric overdensity
(such as a galaxy cluster) in real space. The
explanation for the distortion in redshift space
is given in the lower panel. On large scales,
galaxies are falling into the cluster, so that
galaxies closer to us have a peculiar velocity
directed away from us. Thus, in redshift space
they appear to be more distant than they in
fact are. The inner virialized region of the
cluster generates a "Finger of God", shown by
the highly elongated ellipses in redshift space
directed toward the observer. Here, galaxies
from a small spatial region are spread out
in redshift space due to the large velocity
dispersion yielding large radial patterns in
corresponding wedge diagrams. In the upper
right panel, the same effect is shown for the
case where the cluster is situated close to us
(small circle in lower center)
termined from this anisotropy. 2 However, one needs to
take into account the fact that galaxies are not strictly
-In fact, one can decompose the correlation function into nniltipole
components, such as the monopole (which is the isotropic part of
the correlation function) qnadrnpole (describing the oblateness), etc.
'1'hc ratio ol the quadriipole and monopole components in the lineai
regime is independent of the under!} ing pou cr specs ram. and depends
8. Cosmology III: The Cosmological Parameters
a/rTMpc
Fig. 8.7. The 2-point correlation functioi I measured from
the 2dFGRS, plotted as a function of the transverse separa-
tion a and the radial separation n in redshift space. Solid
contours connect values of constant £ g . The dashed curves
show the same correlation function, determined from a cos-
mological simulation that accounts for small-scale velocities.
1 h obi u. n "i i, di u ibution foi lis cpai ilion .ti ■• t ih
Fingers of God are clearly visible
following the cosmic velocity field. Due to small-scale
gravitational interactions they have a velocity disper-
sion o-p around the velocity field as predicted by linear
theory. A quantitative interpretation of the anisotropy of
the correlation function needs to account for this effect,
which causes an additional smearing of galaxy positions
in redshift space along the line-of- sight. Therefore, the
derived value of fl is related to or p . It is possible to deter-
mine both quantities simultaneously, by comparing the
observed correlation function with models for differ-
ent values of fi and a v . From this analysis, confidence
regions in the /S-er p -plane are obtained, which feature
a distinct minimum in the corresponding x 2 function
and by which both parameters can be estimated simul-
taneously. For the best estimate of these values, the
2dFGRS yielded
8.1.5 Angular Correlations of Galaxies
Measuring the correlation function or the power spec-
trum is not only possible with extensive redshift surveys
of galaxies, which have become available only relatively
recently. In fact, the correlation properties of galaxies
can also be determined from their angular positions on
the sphere. The three-dimensional correlation of gal-
axies in space implies that their angular positions are
likewise correlated. These angular correlations are eas-
ily visible in the projection of bright galaxies onto the
sphere (see Fig. 6.2).
The angular correlation function w(9) is defined in
analogy with the three-dimensional correlation func-
tion |(r) (see Sect. 7.3.1). Considering two solid angle
element dco at 6 X and 2 , the probability of finding a gal-
axy at 9i is P\—n dco, where n denotes the average
density of galaxies on the sphere (with well-defined
properties like, for instance, a minimum magnitude
limit). The probability of finding a galaxy near 9\ and
another one near 2 is then
P 2 = (ndco) z [1 + W (\0 l -O 2 \)] ,
(8.11)
where we utilize the statistical homogeneity and
isotropy of the galaxy distribution, by which the corre-
lation function w depends only on the absolute angular
separation. The angular correlation function w(9) is
of course very closely related to the three-dimensional
correlation function £ g of galaxies. Furthermore, w(0)
depends on the redshift distribution of the galaxies con-
sidered; the broader this distribution is, the fewer pairs
of galaxies are found at a given angular separation which
are also located close to each other in three-dimensional
space, and hence are correlated. This means that the
broader the redshift distribution of galaxies, the smaller
the expected angular correlation.
The relation between w (6) and £ g (r) is given by the
Limber equation, which can, in its simplest form, be
written as
w(0)
=/ d W
d(Az)
(8.12)
g LW(z)0] 2 +(^) {AzA .
P = 0.51 ±0.05;
s 520 km/s .
where Da(z) is the angular diameter distance (4.45),
(8.10) p(z) describes the redshift distribution of galax-
8.1 Redshift Surveys of Galaxies
ies, and dD specifies the physical distance interval
corresponding to a redshift interval dz,
AD =
:iH
dz (l + z)H(z)
Long before extensive redshift surveys were per-
formed, the correlation w(0) had been measured. Since
it is linearly related to § g , and since § g in turn is related
to the power spectrum of the matter fluctuations and to
the bias factor, the measured angular correlation func-
tion could be compared to cosmological models. For
some time, such analyses have hinted at a small value
for the shape parameter r — £2 m h of about 1/4 (see
Fig. 8.4), which is incompatible with an Einstein-de
Sitter model. Figure 8.8 shows w(6) for four magni-
tude intervals measured from the SDSS. We see that
w(9) follows a power law over a wide angular range,
which we would also expect from (8.12) and from the
fact that f (r) follows a power law. 3 In addition, the fig-
ure shows that w(0) becomes smaller the fainter the
galaxies are, because fainter galaxies have a higher
redshift on average and they define a broader redshift
distribution.
0.0001
6 (degrees)
Fig. 8.8. The angular correlation function w(8) in the four
magnitude intervals 18 < r* < 19, 19 < r* < 20, 20 < r* <
21, and 21 < r* < 22, as measured from the first photometric
data of the SDSS, together with a power law fit to the data in
the angular range V < 6 < 30'; the slope in all cases is very
close to # -0 - 7
8.1.6 Cosmic Peculiar Velocities
The relation (8.8) between the density field of galaxies
and the peculiar velocity can also be used in a different
context. To see this, we assume that the distance of gal-
axies can be determined independently of their redshift.
In the relatively local Universe this is possible by using
secondary distance measures (such as, e.g., the scaling
relations for galaxies that were discussed in Sect. 3.4).
With the distance known, we are then able to determine
the radial component of the peculiar velocity by means
of the redshift,
v = cz-H D.
To measure values of v of order ~ 500 km/s, D needs
to be determined with a relative accuracy of
angular correlation functioi
w that a power law f (r)
With the distance measurements being accurate to about
10%, the distance to which this method can be applied
is limited to cz/Hq ~ 100 Mpc, corresponding to an
expansion velocity cz ~ 6000 km/s. Thus, the peculiar
velocity field can be determined only relatively locally.
In order to measure D, one typically uses the Tully-
Fisher relation for spirals, and the fundamental plane or
the D n -a relation for ellipticals. In most cases, these
measurements are carried out for groups of galaxies
which then all have roughly the same distance; in this
way, the measurement accuracy of their common (or
average) distance is improved.
Equation (8.8) now allows us to predict the peculiar
velocity field from the measured density field of gal-
axies, which can then be compared with the measured
peculiar velocities - where this relation depends on fi
(see Eq. 8.9). Therefore, we can estimate /3 from this
comparison. The inverse of this method is also possible:
to derive the density distribution from the peculiar ve-
locity field, and then to compare this with the observed
8. Cosmology III: The Cosmological Parameters
Potent Mass Density
Fig.8.9. The peculiar velocity field (top panel) and the de-
rived density field (bottom panel) in our neighborhood. The
distances here are specified as expansion velocities in units
of 1000 km/s. The mass concentration on the left, towards
which the velocity vectors are pointing, is the Great Attractor
(see text), and on the right is the Pisces-Perseus supercluster.
By comparing this reconstructed mass distribution with the
di-.iril.Hii ion of galaxies, p can be determined. Early analyses
of this kind resulted in relatively large values of ft, whereas
more recent results show p ~ 0.5. Since the bias facto, ma}
be different for the various types of galaxies, ft may depend
on the type of observed galaxies as well. For instance, one
finds that IRAS galaxies have a lower p, thus also a lower b,
than galaxies which have been selected optically
galaxy distribution. 4 Such a comparison is displayed in
Fig. 8.9.
Measurements of the peculiar velocity field in the
mid-1980s led to the conclusion that an unseen mass
concentration, i.e., one that could not, at that time, be
identified with a large concentration of galaxies, was
having a significant effect on the local velocity field.
This mass concentration (which was termed the "Great
Attractor") was located roughly in the direction of the
Galactic Center, which is the reason why it was not
directly observable.
X-ray cluster samples are much less affected by Gal-
actic absorption than optically selected clusters, and
therefore provide a much clearer view of the mass dis-
tribution surrounding the Local Group, including the
Zone of Avoidance. In recent years, based on such
X-ray selected clusters, the simple picture of the Great
Attractor has been modified. In fact, at the proposed
distance to the Great Attractor of ~ 80 Mpc, the matter
density seems to be considerably smaller than origi-
nally thought. However, behind the Great Attractor there
seems to be a significant overdensity of clusters at larger
distances (see Fig. 8.10).
The Velocity Dipole of the Galaxy Distribution. A re-
lated aspect of these studies is the question of whether
the observed peculiar velocity of the Local Group, as
determined from the dipole anisotropy of the CMB,
can be traced back to the matter distribution around
us. We would expect to find a related dipole in the
matter distribution which caused an acceleration of
the Local Group to the observed value of the pecu-
liar velocity of 627 ±22 km/s towards the direction
£ = 273° ± 3, b = 29° ± 3° in Galactic coordinates (this
value is obtained from the direct measurement of the
dipole velocity in the rest-frame of the Sun, to which the
motion of the Sun relative to the Local Group rest-frame
is added).
In principle, this question can be answered from pho-
tometric galaxy surveys alone. We found a relation (8.8)
between the fractional galaxy overdensity <5 g and the pe-
At first sight Ihis seems to be impossible, since only the radial com
ponent oi the peculiar velocity can be measured - proper motions
of galaxies arc far too small to be observable. However, in the lin-
ear regime we can assume the velocity held to be a gradient field,
u = Vi/r; see Sect. 7.6. The velocity potential i/r can be obtained by
integrating the peculiar velocity. i//U') — i//(0) - /„' dl-u. where the
integral is taken over a curve connecting the observer at Oto a point jr.
Choosing nidial c in I ia I colli) i I i uliai
velocity enters the integral. Therefore, this component is sufficient,
in principle, lo construct the velocity potential •'■> . and therefore tiie
three dimensional velocity held.
8.2 Cosmological Para meters from Clusters of Galaxies
Fig. 8.10. An optical image taken in the direction of the Great
Attractor. This image has a side length of half a degree and
was observed by the WFI at the ESO/MPG 2.2-m telescope
on La Silla. The direction of this pointing is only ~ 7° away
from the Galactic disk. For this reason, the stellar densit) in
the image is extremely high (about 200 000 stars can be found
iii this image) and, due to extinction in the disk of the Milky
Way, much fewer faint galaxies at high redshift are found
in this image than in comparable images at high Galactic
latitude. Nevertheless, a large number of galaxies are visible
(greenish), belonging to ahuge cluster of galaxies (ACO 3627,
at a distance of about 80 Mpc), which is presumably the main
contributor to the Great Attractor
culiar velocity u(x) which we can specialize to the point
of origin x — 0. This relation is based on the assumption
of linear biasing. A galaxy at distance D contributes to
the peculiar velocity by an amount oc m/D 2 , where m
is its mass. If we assume that the mass-to-light ratio of
galaxies are all the same, then m oc L and the contribu-
tion of this galaxy to u is oc L/D 2 oc S. Hence, under
these simplifying assumptions the contribution of a gal-
axy to the peculiar velocity depends only on its observed
flux.
To apply this simple idea to real data, we need an
all-sky map of the galaxy distribution. This is difficult
to obtain, due to the presence of extinction towards the
Galactic plane. However, if the galaxy distribution is
mapped at infrared wavelengths, these effects are min-
imized. It is therefore not surprising that most of the
studies on the dipole distribution of galaxies concen-
trate on infrared surveys. The IRAS source catalog still
provides one of the major catalogs for such an anal-
ysis. More recently, the Two-Micron All Sky Survey
(2MASS) catalog provided an all-sky map in the near-IR
which can be used as well. The NIR also has the ad-
vantage that the luminosity at these wavelengths traces
the mass of the stellar population of a galaxy quite well,
in contrast to shorter wavelength for which the mass-
to-light ratio among galaxies varies much more. The
results of these studies is that the dipole of the galaxy
distribution lies within ~ 20° of the CMB dipole. This
is quite a satisfactory result, if we consider the number
of assumptions that are made in this method. The am-
plitude of the expected velocity depends on the factor
P — Q^ 6 /b. Thus, by comparing the predicted velocity
from the galaxy distribution with the observed dipole
of the CMB this factor can be determined, yielding
£ = 0.49 ±0.04.
Supplementing the photometric surveys with red-
shifts allows the determination of the distance out to
which the galaxy distribution has a marked effect on
the Local Group velocity, by adding up the contribu-
tions of galaxies within a maximum distance from the
Local Group. Although the detailed results from differ-
ent groups vary slightly, the characteristic distance turns
out to be ~ 150/j -1 Mpc, i.e., larger than the distance to
the putative Great Attractor. In fact, earlier results sug-
gested a considerably smaller distance, which was one
of the reasons for postulating the presence of the Great
Attractor.
8.2 Cosmological Parameters
from Clusters of Galaxies
Being the most massive and largest gravitationally
bound and relaxed objects in the Universe, clus-
ters of galaxies are of special value for cosmology.
In this section, we will explain various methods by
which cosmological parameters have been derived from
observations of galaxy clusters.
8. Cosmology III: The Cosmological Parameters
8.2.1 Number Density
In Sect. 7.5.2, we demonstrated that it is possible, for
a given cosmological model, to calculate the number
density of halos as a function of mass and redshift. This
finding suggests that we should now compare the ob-
served number density of galaxy clusters with these the-
oretical results and draw conclusions from this. We saw
in Chap. 6 that the selection of clusters by their X-ray
emission is currently viewed as the most reliable method
of finding clusters. Hence, we use the X-ray cluster cat-
alogs described in Sect. 6.3.5 for a comparison of the
halo number density with model predictions.
In order to perform this comparison, the masses of
clusters need to be determined. We discussed various
methods of cluster mass determination in Chap. 6. Since
a very detailed mass determination is possible only for
individual clusters, but not for a large sample (which
is required for a statistical comparison), one usually
applies the scaling relations discussed in Sect. 6.4. In
particular, the relation (6.52) between X-ray tempera-
ture, X-ray luminosity, and virial mass plays a central
role. The scaling relations are then calibrated on clusters
for which detailed mass estimates have been performed.
The comparison of the number density of observed
clusters to the halo density in cosmological models can
be performed either in the local Universe or as a func-
tion of redshift. In the former case, one obtains the
normalization of the power spectrum from this compar-
ison, hence erg, for a given matter density parameter Q m .
More precisely, the number density of halos depends on
the combination cr 8 i?^ 5 , where the exact value of the
exponent of Q m depends on the mass range of the ha-
los that are considered. The analysis of cluster catalogs
like the ones compiled from the ROSAT All Sky Survey
(RASS) yields a value of about
<H<2^
~ 0.5 .
(8.13)
where the uncertainty in this value mainly comes from
the calibration of the scaling relations.
The degeneracy between Q m and <r 8 can be broken by
considering the redshift evolution of the number density
of clusters. As we have seen in Sect. 7.2.2, the growth
factor D + of the density perturbations depends on the
cosmological parameters. For a low-density universe,
the growth factor D + at high redshift is consider-
ably larger than in an Einstein-de Sitter universe (see
Fig. 7.3). Hence, the expected number density of clus-
ters at high redshift is considerably smaller in an EdS
model than in one of low density, for a fixed local
number density of clusters. Indeed, in an EdS universe
virtually no clusters of high mass are expected at z > 0.5
(see Fig. 7.7), whereas the evolution of the halo number
density is significantly weaker for cosmological mod-
els with small Q m . The fact that very massive clusters
have been discovered at redshift z > 0.5 is therefore in-
compatible with a cosmological model of high matter
density.
8.2.2 Mass-to-Light Ratio
On average, the mass-to-light ratio of cosmic objects
seems to be an increasing function of their mass. In
Chap. 3 we saw that M/L is smaller for spirals than
for ellipticals, and furthermore that for ellipticals M/L
increases with mass. In Chap. 6, we argued that galaxy
groups like the Local Group have M/L ~ 100/z, and
that for galaxy clusters M/L is several hundreds, where
all these values are quoted in Solar units. We conclude
from this sequence that M/L increases with the length-
or mass-scale of objects. Going to even larger scales -
superclusters, for instance - M/L seems not to increase
any further, rather it seems to approach a saturation
value (see Fig. 8.11).
Thus, if we assume the M/L ratio of clusters to be
characteristic of the average M/L ratio in the Universe,
the average density of the Universe po can be calculated
from the measured luminosity density JL and the M/L
ratio for clusters,
Here, L and £ refer to a fixed frequency interval, e.g.,
to radiation in the B-band; £. can be measured, for
instance, by determining the local luminosity function
of galaxies, yielding
_ (M/L) B
1200 ft
Since several methods of determining cluster masses
now exist (see Chap. 6), and since L is measurable as
well, (8.14) can be applied to clusters in order to esti-
mate f2 m . Typically, this results in Q m ~ 0.2, a value for
Q m which is slightly smaller than that obtained by other
methods. However, this method is presumably less reli-
(8.14)
8.2 Cosmological Para meters from Clusters of Galaxies
Universe closed
W Peculiar
Scale motion
100 kpc 10Mpc
Length scale
Fig. 8.11. The mass-to-light ratio M/L seems to be a function
of the length- or mass-scale of cosmic objects. The luminous
region in spirals has M/L ~ 3 (all values in Solar units of
Mq/Lq), whereas that of ellipticals has M/L ~ 10. How-
evei. galaxies have a dark mailer halo, so that the true mass
of galaxies, and thus their M/L, is much larger than that
which is measured in their visible region. Masses can also be
estimated from the dynamics of galaxy pairs, typically yield-
ing M/L ~ 50 for galaxies, including their dark halo. Galaxy
a roups and clusters have an even higher M/L ratio, hence thc\
are particularly strongly dominated by dark mailer reaching
M/L ~ 250. If the M/L ratio in clusters corresponds to the
average M/L in the Universe, it is possible to determine the
matter density in the Universe from the luminosity density,
and to obtain a value of Q m ~ 0.2. Only some early investi-
gations of large-scale peculiar motions in the Universe have
indicated even higher M/L, but these values seem not to be
confirmed by more re
able than the other ones described in this section: X is
not easily determined (e.g., the normalization of the
Schechter luminosity function has been revised consid-
erably in recent years, and its accuracy is not better than
~ 20%), and the M/L ratio in clusters is not necessarily
representative. For instance, the evolution of galaxies in
a cluster is different from that of a "mean galaxy".
8.2.3 Baryon Content
As discussed in Chap. 6, clusters of galaxies largely
consist of dark matter. Only about 15% of their mass is
baryonic, the major part of which is contributed by hot
intergalactic gas, visible through its X-ray emission.
Within the accuracy of the measurements, the baryon
content of clusters does not seem to vary between dif-
ferent clusters, rather it seems to have a uniform value.
The existence of a universal baryon fraction is to be
expected, since it is difficult to imagine how for struc-
tures as large as clusters the mixture of baryons and
dark matter would strongly differ from the cosmic av-
erage. A massive cluster with a current virial radius of
~ 1 .5 Mpc has formed by the gravitational contraction
of a comoving volume with a linear extent of about
~ 10 Mpc. Effects like feedback from supernova explo-
sions or other outflow phenomena, that are occurring in
galaxies and which may reduce their baryon mass, are
not effective in galaxy clusters due to their size.
Assuming the baryon fraction fi, in clusters to be
representative of the Universe, the density parameter
of the Universe can be determined, because the cos-
mic baryon density is presumed to be known from
primordial nucleosynthesis (Sect. 4.4.4). This yields
&m ;>
i'2 b >-
,/h
«0.3.
(8.15«
8.2.4 The LSS of Clusters of Galaxies
Under the assumption that the galaxy distribution fol-
lows that of dark matter, the galaxy distribution enables
8. Cosmology III: The Cosmological Parameters
us to draw conclusions about the statistical properties
of the dark matter distribution, e.g., its power spectrum.
At least on large scales, where structure evolution still
proceeds almost linearly today, this assumption seems
to be justified if an additional bias factor is allowed for.
Hence, it is obvious to also examine the large-scale dis-
tribution of galaxy clusters, which should follow the
distribution of dark matter on linear scales as well,
although probably with a different bias factor.
The ROSAT All Sky Survey (see Sect. 6.3.5) al-
lowed the compilation of a homogeneous sample of
galaxy clusters with which the analysis of the large-
scale distribution of clusters became possible for the
first time. Figure 8.12 shows that the power spectrum
of clusters has the same shape as that of galaxies, how-
ever with a considerably larger normalization. The ratio
of the two power spectra displayed in this figure is
based on different bias factors for galaxies and clus-
ters, ^dusters % 2.6b s . For this reason the power spectrum
I"" ' .reflexgclst'"' '
Fit (Par)
:
10 6
f f
; Durham/UKST GAL
Fit(CDM)
i
10 5
-J
>
5T
10 4
1000
100
0.01 0.05 0.1 0.5
k [h/Mpc]
Fig. 8.12. The power spectrum of galaxies (open symbols) and
of galaxy clusters from the REFLEX survey (filled symbols).
The two power spectra have basically the same shape, but
they differ by a multiplicative factor. This factor specifies the
square of the ratio of the bias factors of optically selected
galaxies and of X-ray clusters, respectively. Particularly on
large scales, mapping the powei spectrum from clusters is of
substantial importance
for clusters has an amplitude that is larger by a factor of
about (2.6) 2 than that for galaxies. Since clusters of gal-
axies are much less abundant than galaxies, the density
maxima of the dark matter corresponding to the former
need to have a higher threshold than those of galaxies.
which will, in the biasing model illustrated in Fig. 8.3,
result in stronger correlations.
The analysis of the power spectrum by means of
clusters is interesting, particularly on large scales, yield-
ing an additional data point for the shape parameter
r = £2 m h. Together with the cluster abundance, their
correlation properties yield values of Q m « 0.34 and
erg « 0.71.
8.3 High-Redshift Supernovae
and the Cosmological Constant
8.3.1 Are SN la Standard Candles?
As mentioned in Sect. 2.3.2, Type la supernovae are sup-
posed to be the result of explosion processes of white
dwarfs which cross a critical mass threshold by ac-
cretion of additional matter. This threshold should be
identical for all SNe la, making it at least plausible that
they all have the same luminosity. If this were the case,
they would be ideal for standard candles: owing to their
high luminosity, they can be detected and examined
even at large distances.
However, it turns out that SNe la are not really stan-
dard candles, since their maximum luminosity varies
from object to object with a dispersion of about 0.4 mag
in the blue band light. This is visible in the top panel
of Fig. 8.13. If SNe la were standard candles, the data
points would all be located on a straight line, as de-
scribed by the Hubble law. Clearly, deviations from the
Hubble law can be seen, which are significantly larger
than the photometric measurement errors.
It turns out that there is a strong correlation be-
tween the luminosity and the shape of the light curve
of SNe la. Those of higher maximum luminosity show
a slower decline in the light curve, as measured from
its maximum. Furthermore, the observed flux is possi-
bly affected by extinction in the host galaxy, in addition
to the extinction in the Milky Way. With the resulting
reddening of the spectral distribution, this effect can be
8.3 High-Redshift Supernovae and the Cosmological Constant
DISTANCE MODULUS
Fig. 8.13. The Hubble diagram for relatively nearby SNe la.
Plotted is the measured expansion velocity a. as a function of
Ihe distance modulus for the individual supernovae. In the top
panel, ii is assumed thai all sources have die same luminosity.
If this was correct, all data points should be aligned along
the straight line, as follows from the Hubble law. Obviously,
in iii mil i 1 1 ill I i in | i i 1 ill ii i mi ii
have been corrected by means of the so-called MLCS method
in which the shape of the light curve and the colors of the SN
are used to "standardize" the luminosity (see text for more
explanations). By this the deviations from the Hubble law
become dramatically smaller - the dispersion is reduced from
0.42 mag to 0. 15 mag
derived from the observed colors of the SN. The com-
bined analysis of these effects provides a possibility
for deducing an empirical correction to the maximum
luminosity from the observed light curves in several fil-
ters, accounting both for the relation of the width of the
curve to the observed luminosity and for the extinction.
This correction was calibrated on a sample of SNe la
for which the distance to the host galaxies is very accu-
rately known. With this correction applied, the SNe la
follow the Hubble law much more closely, as can be
seen in the bottom panel of Fig. 8.13. A scatter of only
a = 0.15 mag around the Hubble relation remains. Fig-
ure 8.14 demonstrates the effect of this correction on
the light curves of several SNe la which initially ap-
pear to have very different maximum luminosities and
widths. After correction they become nearly identical.
The left panel of Fig. 8.14 suggests that the light curves
of SN la can basically be described by a one-parameter
family of functions, and that this parameter can be de-
duced from the shape, in particular the width, of the
light curves.
With this correction, SNe la become standardized
candles, i.e., by observing the light curves in several
bands their "corrected" maximum luminosity can be de-
termined. Since the observed flux of a source depends
on its luminosity and its luminosity distance D L , and the
latter also depends, besides redshift, on the cosmologi-
cal model, SNe la can be used for the determination of
cosmological parameters by measuring the luminosity
distance as a function of redshift. To apply this method,
it is necessary to detect and observe SNe la at apprecia-
ble redshifts, where deviations from the linear Hubble
law become visible.
8.3.2 Observing SNe la at High Redshifts
An efficient strategy for the discovery of supernovae
at large distances has been developed, and two large
international teams have performed extensive searches
for SNe la at high redshifts in recent years. Two pho-
tometric images of the same field, observed about four
weeks apart, are compared and searched for sources
which are not visible in the image taken first but which
are seen in the later one. Of these candidates, spec-
tra are then immediately taken to verify the nature of
the source as a SNIa and to determine its redshift.
Subsequently, these sources become subject to exten-
sive photometric monitoring in order to obtain precise
light curves with a time coverage (sampling rate) as
complete as possible. For this observation strategy to
be feasible, the availability of observing time for both
spectroscopy and subsequent photometry needs to be
secured well before the search for candidates begins.
Hence, this kind of survey requires a very well-planned
strategy and coordination involving several telescopes.
Since SNe la at high redshift are very faint, the new 8-m
8. Cosmology III: The Cosmological Parameters
-20
B Band
_ ~ 19
J/j^~~^(&%. as measured
§. -18
X^%^
T -17
-16
; ^^.;-i ;
Calan/Tololo SNe la
Fig. 8.14. Left panel: B-band light curves of different SNe la.
One sees that the shape of the light curves and the max-
imum luminosity of the SNe la differ substantially among
the sample. A transformation was found empirically with panel
ingle parameter described by the width of the light
/e. By means of this transformation, the different light
n all be made congruent, as displayed in the right
class telescopes need to be used for the spectroscopic
observations.
Both teams were very successful in detecting dis-
tant SNe la. In their first large campaigns, the results of
which were published in 1998, they detected and an-
alyzed sources out to redshifts of z < 0.8. Since then,
further SNe la have been found, some with redshifts
> 1 . Substantial advances have also been made by ob-
serving with the HST. Among other achievements, the
HST detected a SNIa at redshift z = 1.7. Of special
relevance is that the conclusions of both teams are in ex-
traordinary agreement. Since they use slightly different
methods in the correction of the maximum luminos-
ity, this agreement serves as a significant test of the
systematic uncertainties intrinsic to this method.
8.3.3 Results
As a first result, we mention that the width of the light
curve is larger for SNe la at higher redshift than it is for
local objects. This is expected because, due to redshift,
the observed width evolves by a factor (1+z). This
dependence has been convincingly confirmed, showing
in a direct way the transformation of the intrinsic to the
observed time interval as a function of redshift.
Plotting the observed magnitudes in a Hubble di-
agram, one can look for the set of cosmological
parameters which best describes the dependence of ob-
served magnitudes m Q \, s on redshift, as is illustrated in
Fig. 8.15.
Comparing the maximum magnitude of the mea-
sured SNe la, or their distance modulus respectively,
with that which would be expected for an empty uni-
verse (Q m = = &a), one obtains a truly surprising
'■ Hf,T Discovorw)
0.5
1.0
1.5
2.0
Fig. 8.15. Distance modulus of nearby and distant SNe la,
determined from the corrected maximum flux of the source.
Diamond symbols represent supernovae that were detected
from the ground, circles those that were found by the HST.
Particularrj remarkable is the small scatter of the data points
around the curve that corresponds to a cosmological model
with Q m = 0.29, Q A = 0.7 1
8.3 High-Redshift Supernovae and the Cosmological Constant
Fig. 8.16. Difference between the maximum brightn
of SNela and that expected in an empty
(Q m = = Q A ). Diamond symbols represent events that \\ ere
detected from the ground, circles the ones discovered by the
HST. In the top panel, the individual SNela are presented,
whereas in the bottom panel they are averaged in redshift
bins. An empty universe would correspond to the dotted
straight line, A (m — M) = 0. The dashed curve corresponds to
a cosmological model with Q m = 0.27, Q A = 0.73. Further-
more, model curves for universes with constant acceleration
are drawn; these models, which are not well-motivated from
physics, and models including "gray dust" (in which the ex-
tinction is assumed to be independent of the wavelength), can
be excluded
result (see Fig. 8.16). Considering at first only the su-
pernovae with z < 1 , one finds that these are fainter than
predicted even for an empty universe. It should be men-
tioned that, according to (4.13), such an empty universe
would expand at constant rate, a — 0. The luminosity
distance in such a universe is therefore larger than in
any other universe with a vanishing cosmological con-
stant. The luminosity distance can only be increased by
assuming that the Universe expanded more slowly in the
past than it does today, hence that the expansion has ac-
celerated over time. From (4.19) it follows that such an
accelerated expansion is possible only if Q A > 0. This
result, first published in 1998, meant a turnaround in
our physical world view because, until then, we were
convinced that the cosmological constant was zero.
More recently, this result has been confirmed by ever
more detailed investigations. In particular, the sample
of SNe la was enlarged and (by employing the HST) ex-
tended to higher redshifts. From this, it was shown that
for z > 1 the trend is reversed and SN la become brighter
than they would be in an empty universe (see Fig. 8.16).
At these high redshifts the matter density dominates
the Universe, proceeding as (1 + z) 3 in contrast to the
constant vacuum energy.
The corresponding constraints on the density param-
eters Q m and Q A are plotted in Fig. 8.17, in comparison
to those that were obtained in 1998. As becomes clear
from the confidence contours, the SNIa data are not
compatible with a universe without a cosmological
constant. An Einstein-de Sitter model is definitely
excluded, but also a model with Q m = 0.3 (a value
derived from galaxy redshift surveys) and Q A — is
incompatible with these data.
We conclude from these results that a non-vanishing
dark energy component exists in the Universe, caus-
ing an accelerated expansion through its negative
pressure. The simplest form of this dark energy is
the vacuum energy or the cosmological constant.
8. Cosmology III: The Cosmological Parameters
,:'.:!::'::!::
&
<&
I ::': :! ..' .! I ' ' ' ' I ' ' ' ' I .
2
a
1
-1
;/><// J
X;"
, . , ,
\ c Recollapses q a 40 •
I , . . , I , . ,\l , . . , I '
Fig. 8.17. From the measured magnitudes of SNIa and the
corresponding!) implied values for the luminosity distances,
confidence regions in the Q^-Qa plane are plotted here. The
solid contours result from the 157 SNela that are also plotted
in Fig. 8.16, whereas the dotted contours represent the results
from 1998. Dashed lines represent cosmological models with
the same deceleration parameter c/ ()
Other forms of dark energy, such as that with a modi-
fied equation of state P — w pc 2 , with — 1/3 > w > —1,
where w — — 1 corresponds to a cosmological constant,
are currently the subject of intense discussion. Con-
straints on w will, in the foreseeable future, only be
possible through astronomical observations, and they
will help us to shed some light on the physical nature of
dark energy.
8.3.4 Discussion
The discovery of the Hubble diagram of SNe la bein
incompatible with a universe having a vanishing vac
uum energy came as a surprise. It was the first evidence
of the existence of dark energy. The cosmological con-
stant, first introduced by Einstein, then later discarded
again, seems to indeed have a non-vanishing value.
This far-reaching conclusion, with its consequences
for fundamental physics, obviously needs to be critically
examined. Which options do we have to explain the ob-
servations without demanding an accelerated expansion
of the Universe?
Evolutionary Effects. The above analysis is based on
the implicit assumption that, on average, SNe la all lia\ e
the same maximum (corrected) luminosity, independent
of their redshift. As for other kinds of sources for which
a Hubble diagram can be constructed and from which
cosmological parameters can be derived, the major diffi-
culty lies in distinguishing the effects of spacetime cur-
vature from evolutionary effects. A z -dependent evolu-
tion of SNe la, in such a way that they become less lumi-
nous with increasing redshift, could have a similar effect
on a Hubble diagram as would an accelerated expansion.
At first sight, such an evolution seems improbable
since, according to our current understanding, the ex-
plosion of a white dwarf close to the Chandrasekhar
mass limit is responsible for these events, and this mass
threshold solely depends on fundamental physical con-
stants. On the other hand, the exact mass at which the
explosion will be triggered may well depend on the
chemical composition of the white dwarf, and this in
turn may depend on redshift. Although it is presum-
ably impossible to prove that such evolutionary effects
are not involved or that their effect is at least smaller
than cosmological effects, one can search for differ-
ences between SNe la at low and at high z. For instance,
it has been impressively demonstrated that the spectra of
high-redshift SNe la are very similar to those of nearby
ones. Hence, no evidence for evolutionary effects has
been found from these spectral studies. Furthermore,
the time until the maximum is reached is independent
of z, if one accounts for the time dilation (1 +z).
Extinction. The correction to the luminosity for ex-
tinction in the host galaxy and in the Milky Way is
determined from reddening. The relation between ex-
tinction and reddening depends on the properties of
the dust - if these evolve with z the correction may
become systematically wrong. To test for this possibil-
8.4 Cosmic Shear
ity one can separately investigate SNe la that occur in
early-type galaxies, in which only little dust exists, and
compare these to events in spiral galaxies. In this test,
no systematic differences are found, neither in events at
high redshift nor in nearby SNe la.
One possibility that has been discussed is the ex-
istence of "gray dust": dust that causes an absorption
independent of wavelength. In such a case extinction
would not reveal itself by reddening. However, this
hypothesis lacks any theoretical explanation for the
physical nature of the dust particles. In addition, the
observation of SNe la at z > 1 shows that the evo-
lution of their magnitude at maximum is compatible
with a A -universe. In contrast, in a scenario involv-
ing "gray dust", a monotonic decrease of the brightness
with redshift would be expected, relative to an empty
universe.
Although it cannot be completely ruled out that the
results from SN la investigations are affected by system-
atic effects that mimic a cosmological effect, all tests
that have been performed for such systematics have been
negative. For this reason, the results are a very strong
indication of a universe with finite vacuum energy den-
sity. The confirmation of this conclusion by the CMB
anisotropies (see Fig. 8.6) is indeed impressive.
8.4 Cosmic Shear
On traversing the inhomogeneous matter distribution in
the Universe, light beams are deflected and distorted,
where the distortion is caused by the tidal gravita-
tional field of the inhomogeneously distributed matter.
As was already discussed in the context of the recon-
struction of the matter distribution in galaxy clusters
(see Sect. 6.5.2), by measuring the shapes of images of
distant galaxies this tidal field can be mapped. From
probing the tidal field, conclusions can be drawn about
Fig. 8.18. As light beams propagate through the Universe they
are affected by the inhomogeneous matter distribution; they
are deflected, and the shape and size of their cross-section
changes. This effect is displayed schematically here - light
beams from sources at the far side of the cube are propa-
gating through the large- scale distribution of matter in the
Universe, and we observe the distorted images of the sources.
In particular, the image of a circular source is elliptical to
a first approximation. Since the distribution of matter is highl)
structured on large scale--, the image distortion caused by light
deflection is coherent: the distortion of two neighboring light
beams is very similar, so that the observed ellipticities of
neighboring galaxies are correlated. From a statistical anal) sis
of the shapes of galaxy images, conclusions about the statis-
tical properties of the matter distribution in the Universe can
be drawn. Hence, the ell i, n itiesofim I listant sources
are closely related to the (projected) mailer distribution, as
displayed schematically in the right panel
8. Cosmology III: The Cosmological Parameters
the matter distribution. This effect, called cosmic shear,
is sketched in Fig. 8.18. In contrast to the case of a gal-
axy cluster in which the tidal field is rather strong, the
large-scale distribution of matter causes a very much
weaker tidal field: a typical value for this shear is about
1 % on angular scales of a few arcminutes, meaning that
the image of an intrinsically circular source attains an
axis ratio of 0.99:1.
The shear field results from the projection of the
three-dimensional tidal field along the line-of- sight.
Hence, we are able to obtain information about the
statistical properties of the density inhomogeneities in
the Universe, by a statistical analysis of the image
shapes of distant galaxies. For instance, the two-point
correlation function of the image ellipticities can be
measured. This is linked to the power spectrum P(k)
of the matter distribution. Thus, by comparing mea-
surements of cosmic shear with cosmological models
we obtain constraints on the cosmological parameters,
without the need to make any assumptions about the
0.001
777T- :.:::- ',:
■ MvWM+ (VLT)
~f- !"«":«"»'-« \
• vWME+(CFH-1)
0,0008
■
I
- KWL (CFHT)
• BRE (WHT)
\ WTK+ (CTIO)
\. vWMR+ (CFH-2)"
" 4Jtft
%
nIMTt
T \ :
0.0004
(arcmin)
Parameters: (Q M , Q A h, r, <r 3 ); <z s >=0,8
Fig. 8.19. Earl\ measurements of cosmic shear. Plotted is the
shear dispersion, measured from the ellipticities of faint and
small galaxy images on deep CCD exposures, as a function
of angular scale. Data from different teams are represented by
different symbols. For instance, MvWM+ resulted from a VLT
project, vWMR+ from a large survey (VIRMOS-Descartes) at
the CFHT. For this latter project, the images of about 450 000
galaxies have been analyzed; the corresponding error bars
from this survey are significantly smaller than those of the
earlier surveys. The curves indicate cosmic shear predictions
in different cosmological models, where the curves are labeled
by the cosmological parameters Q m , Qa, h, F and a$,
relation between luminous n
r (galaxies) and dark
Since the size of the effect is expected to be very
small, systematic effects like the anisotropy of the
point-spread function or distortions in the telescope op-
tics need to be understood very well, and they need
to be corrected for in the measurements. In principle,
the problems are the same as in the mass reconstruc-
tion of galaxy clusters with the weak lensing effect
(Sect. 6.5.2), but they are substantially more difficult
to deal with since the measurable signal is considerably
smaller.
In March 2000, four research groups published,
quasi-simultaneously, the first measurements of cosmic
shear, and in the fall of 2000 another measurement was
obtained from VLT observations. Since then, several
teams worldwide have successfully performed mea-
surements of cosmic shear, for which a large number
of different telescopes have been used, including the
HST. The development of wide-field cameras and of
special software for data analysis are mainly responsi-
ble for these achievements. Some of the early results are
compiled in Fig. 8.19.
By comparison of these measurement results with
theoretical models, constraints on cosmological param-
eters are obtained; one example of this is presented
in Fig. 8.20. Currently, a major source of uncertainty
in this cosmological interpretation is our insufficient
knowledge of the redshift distribution of the faint gal-
axies that are used for the measurements. In the coming
years this uncertainty will be greatly reduced, as exten-
sive redshift surveys of faint galaxies will be conducted
with the next generation of multi-object spectrographs
at 10-m class telescopes.
The most significant result that has been obtained
from cosmic shear so far is a derivation of a combina-
tion of the matter density Q m and the normalization a%
of the power spectrum of density fluctuations, which
can also be seen in Fig. 8.20. The near-degeneracy of
these two parameters has roughly the same functional
form as for the number density of galaxy clusters, since
with both methods we probe the matter distribution on
similar physical length-scales. For an assumed value
of C2 m = 0.3, ct 8 can thus be constrained. Although
the values obtained by different groups differ slightly,
they are compatible with <r 8 « 0.8 within the range of
uncertainty.
8.5 Origin of the Lyman-a Forest
6.5 deg 2 , 450 000 galaxies 1=24.
z=0.8-Open CDM, A=0, 1=0.21
Fig. 8.20. From the analysis of the data in Fig. 8.19 and com-
parison with model predictions, constraints on cosmological
parameters can be derived. Here, confidence contours in the
^m-o"8 parameter plane are shown, where Q A = was as-
sumed. The effect of Q A on the prediction of the shear is
relatively small. The data suggest a Universe of low density.
Presently the largest uncertainty in the quanlilalne analysis
of shear data is the insufficiently known redshift distribution
of the faint galaxies
8.5 Origin of the Lyman-a Forest
We have seen in Sect. 5.6.3 that in the spectrum of
any QSO a large number of absorption lines at wave-
lengths shorter than the Lya emission line of the QSO
are found. The major fraction of these absorption lines
originate from the Lya transition of neutral hydrogen
located along the line-of-sight to the source. Since
the absorption is found in the form of a line spec-
trum, the absorbing hydrogen cannot be distributed
homogeneously. A homogeneous intergalactic medium
containing neutral hydrogen would be visible in contin-
uum absorption. In this section, we will first examine
this continuum absorption. We will then summarize
some observational results on the Lya forest and explain
why studying this provides us with valuable information
about the cosmological parameters.
8.5.1 The Homogeneous Intergalactic Medium
We first ask whether part of the baryons in the Uni-
verse may be contained in a homogeneous intergalactic
medium. This question can be answered by means
of the Gunn-Peterson test. Neutral hydrogen absorbs
photons at a rest wavelength of X = X hya = 1216 A.
Photons from a QSO at redshift zqso attain this wave-
length X Lya somewhere along the line-of-sight between
us and the QSO, if they are emitted by the QSO at
^-Lya (l + zqso) < X < Xiy a . However, if the wave-
length at emission is > X Lya , the radiation can nowhere
on its way to us be absorbed by neutral hydrogen. Hence,
a jump in the observed continuum radiation should
occur between the red and the blue side of the Lya
emission line of the QSO: this is the Gunn-Peterson
effect. The optical depth for absorption is, for models
with Q A = 0, given by
= 4.14x I0 w h~
« ffl (z)/cm ■
(8.16)
(l+Z)Vl + ^mZ
where n H i (z) is the density of neutral hydrogen at the ab-
sorption redshift z, with (1 + z) — X/X Lya < (1 + Zqso)-
Such a jump in the continuum radiation of QSOs
across their Lya emission line, with an amplitude
5'(blue)/5'(red) = e~ T , has not been observed for QSOs
at z < 5. Tight limits for the optical depth were ob-
tained by detailed spectroscopic observation, yielding
r < 0.05 for z < 3 and r < 0. 1 for z ~ 5. At even higher
redshift observations become increasingly difficult, be-
cause the Lya forest then becomes so dense that hardly
any continuum radiation is visible between the indi-
vidual absorption lines (see, e.g., Fig. 5.40 for a QSO at
zq S0 = 3.62). From the upper limit for the optical depth,
one obtains bounds for the density of neutral hydrogen,
«Hi(comoving) <2x 10~ 13 h cm -3
or J2 ffl <2x lO^/r 1 .
From this we conclude that hardly any homogeneously
distributed baryonic matter exists in the intergalactic
medium, or that hydrogen in the intergalactic medium
is virtually fully ionized. However, from primordial nu-
cleosynthesis we know the average density of hydrogen
- it is much higher than the above limits - so that hydro-
gen must be present in essentially fully ionized form.
We will discuss in Sect. 9.4 how this reionization of the
intergalactic medium presumably happened.
8. Cosmology III: The Cosmological Parameters
In recent years, QSOs at redshifts > 6 have been
discovered, not least by careful color selection in data
from the Sloan Digital Sky Survey (see Sect. 8.1.2). The
spectrum of one of these QSOs is displayed in Fig. 5.42.
For this QSO, we can see that virtually no radiation
bluewards of the Lya emission line is detected. After
this discovery, it was speculated whether the redshift had
been identified at which the Universe was reionized.
The situation is more complicated, though. First, the
Lya forest is so dense at these redshifts that lines blend
together, making it very difficult to draw conclusions
about a homogeneous absorption. Second, in spectra of
QSOs at even higher redshift, radiation bluewards of the
Lya emission line has been found. As we will soon see,
the reionization of the Universe probably took place at
a redshift significantly higher than z ~ 6.
8.5.2 Phenomenology of the Lyman-a Forest
Neutral hydrogen in the IGM is being observed in the
Lya forest. For the observation of this Lya forest, spec-
tra of QSOs with high spectral resolution are required
because the typical width of the lines is very small, cor-
responding to a velocity dispersion of ~ 20 km/s. To
obtain spectra of high resolution and of good signal-to-
noise ratio, very bright QSOs are selected. In this field,
enormous progress has been made since the emergence
of 10 m-class telescopes.
As mentioned before, the line density in the Lya
forest is a strong function of the absorption redshift. The
number density of Lya absorption lines with equivalent
width (in the rest-frame of the absorber) W > 0.32 A at
z > 2 is found to follow
dN
dF
-k(l + z) Y
(8.17)
with y ~ 2.5 and k ~ 4, which implies a strong redshift
evolution. At lower redshift, where the Lya forest is
located in the UV part of the spectrum and therefore
is considerably more difficult to observe (only by UV-
sensitive satellites like the IUE, FUSE, and the HST),
the evolution is slower and the number density deviates
from the power law given above.
From the line strength and width, the Hi column
density Nm of a line can be measured. The number
density of lines as a function of Nm
dN m
- « N>
(8.18)
with ft ~ 1.6. This power law approximately describes
the distribution over a wide range of column densi-
ties, 10 12 cm' 2 < N m < 10 22 cm" 2 , including Ly-limit
systems and damped Lya systems.
The temperature of the absorbing gas can be esti-
mated from the line width as well, by identifying the
width with the thermal line broadening. As typical val-
ues, one obtains ~ 10 4 K to 2 x 10 4 K which, however,
are somewhat model-dependent.
The Proximity Effect. The statistical properties of the
Lya forest depend only on the redshift of the absorption
lines, and not on the redshift of the QSO in the spec-
trum of which they are measured. This is as expected
if the absorption is not physically linked to the QSO,
and this observational fact is one of the most important
indicators for an intergalactic origin of the absorption.
However, there is one effect in the statistics of Lya
absorption lines which is directly linked to the QSO.
One finds that the number density of Lya absorption
lines at those redshifts which are only slightly smaller
than the emission line redshift of the QSO itself, is
lower than the mean absorption line density at this red-
shift (averaged over many different QSO lines-of-sight).
This effect indicates that the QSO has some effect on the
absorption lines, if only in its immediate vicinity; for
this reason, it is named the proximity effect. An expla-
nation of this effect follows directly from considering
the ionization stages of hydrogen. The gas is ionized
by energetic photons which originate from hot stars and
AGNs and which form an ionizing background. On the
other hand, ionized hydrogen can recombine. The de-
gree of ionization results from the equilibrium between
these two processes.
The number of photoionizations of hydrogen atoms
per volume element and unit time is proportional to the
density of neutral hydrogen atoms and given by
= Ah «hi ,
(8.19)
where Fm, the photoionization rate, is proportional
to the density of ionizing photons. The correspond-
ing number of recombinations per volume and time is
8.5 Origin of the Lyman-Q- Forest
proportional to the density of free protons and electrons,
n lec = an p n e , (8.20)
where the recombination coefficient a depends on the
gas temperature. The Gunn-Peterson test tells us that
the intergalactic medium is essentially fully ionized, and
thus «m <K n p = « e ^ «b (we disregard the contribution
of helium in this consideration). We then obtain for
the density of neutral hydrogen in an equilibrium of
ionization and recombination
«m=^-«p. (8.21)
J HI
This results shows that n m is inversely proportional to
the number density of ionizing photons. However, the
intergalactic medium in the vicinity of the QSO does not
only experience the ionizing background radiation field
but, in addition, the energetic radiation from the QSO
itself. Therefore, the degree of ionization of hydrogen
in the immediate vicinity of the QSO is higher, and
consequently less Lya absorption can take place there.
Since the contribution of the QSO to the ionizing
radiation depends on the distance of the gas from the
QSO (oc r -2 ), and since the spectrum and ionizing flux
of the QSO is observable, examining the proximity ef-
fect provides an estimate of the intensity of the ionizing
background radiation as a function of redshift. This
value can then be compared to the total ionizing ra-
diation which is emitted by QSOs and young stellar
populations at the respective redshift. This comparison,
in which the luminosity function of AGNs and the star-
formation rate in the Universe are taken into account,
yields good agreement, thus confirming our model for
the proximity effect.
8.5.3 Models of the Lyman-a Forest
Since the discovery of the Lya forest, various models
have been developed in order to explain its nature. Since
about the mid-1990s, one model has been established
that is directly linked to the evolution of large-scale
structure in the Universe.
The "Old" Model of the Lyman-a Forest. Prior to this
time, models were designed in which the Lya forest was
caused by quasi-static hydrogen clouds. These clouds
(Lya clouds) were postulated and were initially seen as
a natural picture given the discrete nature of the absorp-
tion lines. From the statistics of the number density of
lines, the cloud properties (such as radius and density)
could then be constrained. If the line width represented
a thermal velocity distribution of the atoms, the temper-
ature and, together with the radius, also the mass of the
clouds could be derived (e.g., by utilizing the density
profile of an isothermal sphere). The conclusion from
these arguments was that such clouds would evaporate
immediately unless they were gravitationally bound in
a dark matter halo (mini-halo model), or confined by
the pressure of a hot intergalactic medium. 5
The New Picture of the Lyman-ct Forest. For about
a decade now, a new paradigm has existed for the
nature of the Lya forest. Its establishment became pos-
sible through advances in hydrodynamic cosmological
simulations.
We discussed structure formation in Chap. 7, where
we concentrated mainly on dark matter. After recombi-
nation at z ~ 1100 when the Universe became neutral
and therefore the baryonic matter no longer experienced
pressure by the photons, baryons were, just like dark
matter, only subject to gravitational forces. Hence the
behavior of baryons and dark matter became very sim-
ilar up to the time when baryons began to experience
significant pressure forces by heating (e.g., due to pho-
toionization) and compression. The spatial distribution
of baryons in the intergalactic medium thus followed
that of dark matter, as is also confirmed by numerical
simulations. In these simulations, the intensity of ioniz-
ing radiation is accounted for - it is estimated, e.g., from
the proximity effect. Figure 8.21 shows the column den-
sity distribution of neutral hydrogen which results from
such a simulation. It shows a structure similar to the
distribution of dark matter, however with a higher den-
sity contrast due to the quadratic dependence of the Hi
density on the baryon density - see (8.21).
From the distribution of neutral gas simulated this
way, synthetic absorption line spectra can then be com-
puted. For these, the temperature of the gas and its
peculiar velocity are used, the latter resulting from
the simulation as well. Such a synthetic spectrum is
"The I in 1 1 umption 1 ludcd at last b> tl 1 ( il'l m 1 Lire
mails ofthe CMB spectrum, because such a hoi intergalactic medium
would cause deviations of the CMB spectrum from its Planck shape.
by Compton scattering of the CMB photons.
Fig. 8.21. Column density of neutral hydrogen, computed in
a joint simulation of dark mallei- and gas. The size of ihe cube
displayed here is 10/i _1 Mpc (comoving). By computing the
Lya absorption of photons crossing a simulated cube like this,
simulated spectra of the Lya forest are obtained, which can
then be compared statistical!} with observed spectra
displayed in Fig. 8.22, together with a measured Lya
spectrum. These two spectra are, from a statistical point
of view, virtually identical, i.e., their density of lines, the
width and optical depth distributions, and their correla-
tion properties are equal. For this reason, the evolution
of cosmic structure provides a natural explanation for
the Lya forest, without the necessity of additional free
parameters or assumptions. In this model, the evolution
of dN/dz is driven mainly by the Hubble expansion and
the resulting change in the degree of ionization in the
intergalactic medium.
Besides the correlation properties of the Lya lines
in an individual QSO spectrum, we can also consider
the correlation between absorption line spectra of QSOs
which have a small angular separation on the sky. In this
case, the corresponding light rays are close together,
probing neighboring spatial regions of the intergalactic
medium. If the neutral hydrogen is correlated on scales
larger than the transverse separation of the two lines-
of-sight towards the QSOs, correlated Lya absorption
lines should be observable in the two spectra. As a mat-
ter of fact, it is found that the absorption line spectra
Wavelength [A]
Fig. 8.22. One of the spectra is a section of the Lya forest
towards the QSO 1422+231 (see also Fig. 5.40), the other is
a simulated spectrum: both are statistical!} so similar that it is
impossible to distinguish them - which one is which?
of QSOs show correlations, provided that the angular
separation is sufficiently small. The correlation lengths
derived from these studies are > 100 A -1 kpc, in agree-
ment with the results from numerical simulations. In
particular, the lines-of-sight corresponding to different
images of multiple-imaged QSOs in gravitational lens
systems are very close together, so that the correlation
of the absorption lines in these spectra can be very well
verified.
Where are the Baryons Located? As another result
of these investigations it is found that at 2 < z < 4 the
majority (~ 85%) of baryonic matter is contained in
the Lya forest, mainly in systems with column densi-
ties of 10 14 cm" 2 <%<3x 10 15 cm" 2 . Thus, at these
high redshifts we observe nearly the full inventory of
baryons. At lower redshift, this is no longer the case.
Indeed, only a fraction of the baryons can be observed
in the local Universe, for instance in stars or in the in-
tergalactic gas in clusters of galaxies. From theoretical
arguments, we expect that the majority of baryons to-
day should be found in the form of intergalactic gas,
for example in galaxy groups and large-scale filaments
that are seen in simulations of structure formation. This
gas is expected to have a temperature between ~ 10 5 K
and ~ 10 7 K and is therefore very difficult to detect; it
is called the warm-hot intergalactic medium. At these
temperatures, the gas is essentially fully ionized so that
8.5 Origin of the Lyman-Q- Forest
Fig. 8.23. Optical depth for Lya absorp-
tion versus gas density, obtained from
a cosmological simulation. Each data point
represents a line-of-sight through a gas dis-
tribution like the one presented in Fig. 8.21.
For the panel on the right, peculiar mo-
tion of the gas -.a as neglected: in iiiis ease.
the points follow the relation (8.22) very
accurately. With the peculiar motions and
thermal line broadening taken into account
(left panel), the points also follow this
relation on average
log Pg/p g
ICQ Pg/Pg
it cannot be detected in absorption line spectra. How-
ever, the temperature and density are too low to expect
significant X-ray emission from this gas. 6
8.5.4 The Lya Forest as Cosmological Tool
The aforementioned simulations of the Lya forest pre-
dict that most of the lines originate in regions of
the intergalactic medium where the gas density is
p g < 10p g . Hence, the density of the absorbing gas is
relatively low, compared, e.g., to the average gas den-
sity in a galaxy. The temperature of the gas causing
the absorption is about ~ 10 4 K. At these densities and
temperatures, pressure forces are small compared to
gravitational forces, so that the gas follows the density
distribution of dark matter very closely. From the ab-
sorption line statistics, it is therefore possible to derive
the statistical properties of the dark matter distribution.
More precisely, the two-point correlation function of the
Lya lines reflects the spectrum of density fluctuations
in the Universe, and hence it can be used to measure the
power spectrum P(k).
We will consider some aspects of this method in
more detail. The temperature of the intergalactic gas
is not homogeneous because gas heats up by com-
pression. Thus at a fixed redshift dense gas is hotter
than the average baryon temperature 7b. As long as
the compression proceeds adiabatically, T basically de-
pends on the density, T = TQ(p g /p g ) a , where T and
the exponent a depend on the ionization history and
on the spectrum of the ionizing photons. Typical val-
ues are 4000 K < T < 10 000 K and 0.3 < a < 0.6.
The density of neutral hydrogen is specified by (8.21),
"hi oc pgT~ 0J / r m , where the temperature dependence
of the recombination rate has been taken into account.
Since the temperature depends on the density, one
obtains for the optical depth of Lya absorption
''Although hydrogen is not delectable in this intergalaelic medium due
to its complete ionization, hues from metal ions at a high ionization
stage can be obscned in U V absorption line spectra, for instance the
lines of 0\ 1. the lb e times ionized oxygen. To derive a baryon density
from ohserxations oi diese lines, assumptions about the temperature
of the gas and about its tnetallicit} are required. The latesl result'-.
which have mainly been obtained using the UV satellite FUSE, are
compatible with the idea that today the major fraction ol baryons is
contained in this warm hot intergalactic medium.
•00'
where P — 2 — O.la^ 1.6, with the prefactor depending
on the observed redshift, the ionization rate /hi, and the
average temperature 7b.
In Fig. 8.23, the distribution of optical depth and gas
density at redshift z — 3 is plotted, obtained from a hy-
drodynamical simulation. As is seen from the right-hand
panel, the distribution follows the relation (8.22) very
closely, which means that a major fraction of the gas
has not been heated by shock fronts, but rather by adi-
abatic compression. Even with peculiar motion of the
gas and thermal broadening taken into account, as is the
case in the panel on the left, the average distribution still
follows the analytical relation very closely.
8. Cosmology III: The Cosmological Parameters
blc
From the observed distribution of t, it is thus possi-
draw conclusions about the distribution of the
gas overdensity p g /p g . As argued above, the latter
is basically the same as the corresponding overden-
sity of dark matter. From an absorption line spectrum,
r(A.) can be determined (wavelength-)pixel by pixel,
where X corresponds, according to X — (1+z) 1216 A,
to a distance along the line-of-sight, at least if peculiar
velocities are disregarded. From r(X), the overdensity
as a function of this distance follows with (8.22), and
thus a one-dimensional cut through the density fluc-
tuations is obtained. The correlation properties of this
density are determined by the power spectrum of the
matter distribution, which can be measured in this way.
This probe of the density fluctuations is applied at
redshifts 2 < z < 4, where, on the one hand, the Lya for-
est is in the optical region of the observed spectrum, and
on the other hand, the forest is not too dense for this anal-
ysis to be feasible. This technique therefore probes the
large-scale structure at significantly earlier epochs than
is the case for the other cosmological probes described
earlier. At such earlier epochs the density fluctuations
are linear down to smaller scales than they are today.
For this reason, the Lya forest method yields invalu-
able information about the power spectrum on smaller
scales than can be probed with, say, galaxy redshift sur-
veys. We shall come back to the use of this method in
combination with the CMB anisotropies in Sect. 8.7.
8.6 Angular Fluctuations of the Cosmic
Microwave Background
The cosmic microwave background consists of photons
that last interacted with matter at z ~ 1000. Since the
Universe must have already been inhomogeneous at this
time, in order for the structures present in the Universe
today to be able to form, it is expected that these spatial
inhomogeneities are reflected in a (small) anisotropy of
the CMB: the angular distribution of the CMB temper-
ature reflects the matter inhomogeneities at the redshift
of decoupling of radiation and matter.
Since the discovery of the CMB in 1965, such
anisotropies have been searched for. Under the assump-
tion that the matter in the Universe only consists of
baryons, the expectation was that we would find relative
fluctuations in the CMB temperature of AT/T ~ 10~ 3
on scales of a few arcminutes. This expectation is based
on the theory of gravitational instability for structure
growth: to account for the density fluctuations ob-
served today, one needs relative density fluctuations
at z ~ 1000 of order 10~ 3 . Despite increasingly more
sensitive observations, these fluctuations were not de-
tected. The upper limits resulting from these searches
for anisotropies provided one of the arguments that, in
the mid-1980s, caused the idea of the existence of dark
matter on cosmic scales to increasingly enter the minds
of cosmologists. As we will see soon, in a Universe
which is dominated by dark matter the expected CMB
fluctuations on small angular scales are considerably
smaller than in a purely baryonic Universe. Only with
the COBE satellite were temperature fluctuations in the
CMB finally observed in 1992 (Fig. 1.17). Over the last
few years, sensitive and significant measurements of
the CMB anisotropy have also been carried out using
balloons and ground-based telescopes.
We will first describe the physics of CMB an-
isotropies, before turning to the observational results
and their interpretation. As we will see, the CMB
anisotropies depend on nearly all cosmological pa-
rameters, such as Q m , £2b, £2 A , £2rdm, Ho, the
normalization or 8 , the primordial slope n s , and the shape
parameter r of the power spectrum. Therefore, from
an accurate mapping of the angular distribution of the
CMB and by comparison with theoretical expectations,
all these parameters can, in principle, be determined.
8.6.1 Origin of the Anisotropy: Overview
The CMB anisotropies reflect the conditions in the
Universe at the epoch of recombination, thus at
Z ~ 1000. Temperature fluctuations originating at this
time are called primary anisotropics. Later, as the
CMB photons propagate through the Universe, they
may experience a number of distortions along their way
which, again, may change their temperature distribu-
tion on the sky. These effects then lead to secondary
anisotropies.
The most basic mechanisms causing primary
anisotropies are the following:
• Inhomogeneities in the gravitational potential cause
photons which originate in regions of higher den-
8.6 Angular Fluctuations of the Cosmic Microwave Background
sity to climb out of a potential well. As a result of
this, they loose energy and are redshifted (gravita-
tional redshift). This effect is partly compensated for
by the fact that, besides the gravitational redshift,
a gravitational time delay also occurs: a photon that
originates in an overdense region will be scattered at
a slightly earlier time, and thus at a slightly higher
temperature of the Universe, compared to a photon
from a region of average density. Both effects always
occur side by side. They are combined under the term
Sachs-Wolfe effect. Its separation into two processes
is necessary only in a simplified description; a gen-
eral relativistic treatment of the Sachs- Wolfe effect
jointly yields both processes.
» We have seen that density fluctuations are always
related to peculiar velocities of matter. Hence, the
electrons that scatter the CMB photons for the
last time do not follow the pure Hubble expan-
sion but have an additional velocity that is closely
linked to the density fluctuations (compare Sect. 7.6).
This results in a Doppler effect: if photons are
scattered by gas receding from us with a speed
larger than that corresponding to the Hubble expan-
sion, these photons experience an additional redshift
which reduces the temperature measured in that
direction.
» In regions of a higher dark matter density, the baryon
density is also enhanced. On scales larger than
the horizon scale at recombination (see Sect. 4.5.2),
the distribution of baryons follows that of the
dark matter. On smaller scales, the pressure of the
baryon-photon fluid is effective because, prior to re-
combination, these two components had been closely
coupled by Thomson scattering. Baryons are adiabat-
ically compressed and thus get hotter in regions of
higher baryon density, hence their temperature - and
with it the temperature of the photons coupled to
them - is also larger.
• The coupling of baryons and photons is not perfect
since, owing to the finite mean free path of photons,
the two components are decoupled on small spatial
scales. This implies that on small length-scales, the
temperature fluctuations can be smeared out by the
diffusion of photons. This process is known as Silk
damping, and it implies that on angular scales be-
low about ~ 5', only very small primary fluctuations
Obviously, the first three of these effects are closely cou-
pled to each other. In particular, on scales > r H ,com(Zrec)
the first two effects can partially compensate each other.
Although the energy density of matter is, at recombina-
tion, higher than that of the radiation (see Eq. 4.54), the
energy density in the baryon-photon fluid is dominated
by radiation, so that it is considered a relativistic fluid.
Its speed of sound is thus c s « *JP/p « c/>/3. The high
pressure of this fluid causes oscillations to occur. The
gravitational potential of the dark matter is the driving
force, and pressure the restoring force. These oscilla-
tions, which can only occur on scales below the sound
horizon at recombination, then lead to adiabatic com-
pression and peculiar velocities of the baryons, hence
to anisotropies in the background radiation.
Secondary anisotropies result, among other things,
from the following effects:
• Thomson scattering of CMB photons. Since the Uni-
verse is currently transparent for optical photons
(since we are able to observe objects at z > 6), it must
have been reionized between z ~ 1000 and z ~ 6,
presumably by radiation from the very first gener-
ation of stars and/or by the first QSOs. After this
reionization, free electrons are available again, which
may then scatter the CMB photons. Since Thom-
son scattering is essentially isotropic, the direction
of a photon after scattering is nearly independent
of its incoming direction. This means that scattered
photons no longer carry information about the CMB
temperature fluctuations. Hence, the scattered pho-
tons form an isotropic radiation component whose
temperature is the average CMB temperature. The
main effect resulting from this scattering is a reduc-
tion of the measured temperature anisotropies, by the
fraction of photons which experience such scattering.
• Photons propagating towards us are traversing a Uni-
verse in which structure formation takes place. Due to
this evolution of the large-scale structure, the grav-
itational potential is changing over time. If it was
time-independent, photons would enter and leave
a potential well with their frequency being unaf-
fected, compared to photons that are propagating in
a homogeneous Universe: the blueshift they expe-
rience when falling into a potential well is exactly
balanced by the redshift they attain when climbing
out. However, this "conservation" of photon energy
8. Cosmology III: The Cosmological Parameters
no longer occurs if the potential is varying with time.
One can show that for an Einstein-de Sitter model,
the peculiar gravitational potential </> (7.10) is con-
stant over time, 7 and hence, the propagation in the
evolving Universe yields no net frequency shift. For
other cosmological models this effect does occur; it
is called the integrated Sachs-Wolfe effect.
• The gravitational deflection of CMB photons, caused
by the gravitational field of the cosmic density fluctu-
ations, leads to a change in the photon direction. This
means that two lines-of-sight separated by an angle 6
at the observer have a physical separation at recom-
bination which may be different from D A (z KC )@, due
to the gravitational light deflection. Because of this,
the correlation function of the temperature fluctua-
tions is slightly smeared out. This effect is relevant
on small angular scales.
• The Sunyaev-Zeldovich effect, which we discussed
in Sect. 6.3 .4 in the context of galaxy clusters, also af-
fects the temperature distribution of the CMB. Some
of the photons propagating along lines-of-sight pass-
ing through clusters of galaxies are scattered by
the hot gas in these clusters, yielding a tempera-
ture change in these directions. We recall that in
the direction of clusters the measured intensity of
the CMB radiation is reduced at low frequencies,
whereas it is increased at high frequencies. Hence,
the SZ effect can be identified in the CMB data if
measurements are conducted over a sufficiently large
frequency range.
8.6.2 Description of the Cosmic Microwave
Background Anisotropy
Correlation Function and Power Spectrum. In order
to characterize the statistical properties of the angular
distribution of the CMB temperature, the two-point cor-
relation function of the temperature on the sphere can
be employed, in the same way as it is used for describ-
ing the density fluctuations or the angular correlation
function of galaxies. To do this, the relative temperature
fluctuations T(n) = [T(n) - T ] /T are defined, where
n is a unit vector describing the direction on the sphere,
7 This is seen with (7.10) due to the dependence p ex a~ 3 and S oc
D+ = a for an EdS model.
and T is the average temperature of the CMB. The cor-
relation function of the temperature fluctuations is then
defined as
C{6) = {T{n)T{ri)) , (8.23)
where the average extends over all pairs of directions n
and n' with angular separation 6. As for the description
of the density fluctuations in the Universe, for the CMB
it is also common to consider the power spectrum of
the temperature fluctuations, instead of the correlation
function.
We recall (see Sect. 7.3.2) that the power spectrum
P(k ) of the density fluctuations is defined as the Fourier
transform of the correlation function. However, exactly
the same definition cannot be applied to the CMB.
The difference here is that the density fluctuations S(x)
are defined on a flat space (approximately, at the rele-
vant length-scales). In this space, the individual Fourier
modes (plane waves) are orthogonal, which enables
a decomposition of the field S(x) into Fourier modes
in an unambiguous way. In contrast to this, the tem-
perature fluctuations T are defined on the sphere. The
analog to the Fourier modes in a flat space are spher-
ical harmonics on the sphere, a complete orthogonal
set of functions into which T(n) can be expanded. 8 On
small angular scales, where a sphere can be considered
locally flat, spherical harmonics approximately behave
like plane waves. The power spectrum of temperature
fluctuations, in most cases written as £(l+ 1)C, . then
describes the amplitude of the fluctuations on an angu-
lar scale 6 ~ itjt = 180°/£. I = 1 describes the dipole
anisotropy, I = 2 the quadrupole anisotropy, and so on.
Line-of-Sight Projection. The CMB temperature fluc-
tuations on the sphere result from projection, i.e.,
the integration along the line-of-sight of the three-
dimensional temperature fluctuations which have been
discussed above. This integration also needs to account
for the secondary effects, those in the propagation of
photons from z ~ 1000 to us. Overall, this is a rel-
atively complicated task that, moreover, requires the
explicit consideration of some aspects of General Rel-
ativity. The necessity for this can clearly be seen by
s Spherical h
matical physics, for in
the hydrogen atom or
problems in physics.
n many problems in mathc
e in the quantum mechanical treatment of
e generally, in all spherically symmetric
8.6 Angular Fluctuations of the Cosmic Microwave Background
considering the fact that two directions which are sepa-
rated by more than ~ 1° have a spatial separation at
recombination which is larger than the horizon size
at that time - so spacetime curvature explicitly plays
a role. Fortunately, the physical phenomena that need
to be accounted for are (nearly) all of a linear nature.
This means that, although the corresponding system
of coupled equations is complicated, it can neverthe-
less easily be solved, since the solution of a system of
linear equations is not a difficult mathematical prob-
lem. Generally accessible software packages exist (i.e.,
CMBFAST), which compute the power spectrum Q for
any combination of cosmological parameters.
8.6.3 The Fluctuation Spectrum
Horizon Scale. To explain the basic features of CMB
fluctuations, we first point out that a characteristic
length-scale exists at z rec , namely the horizon length.
It is specified by (4.71). For cosmological models with
£2 a = 0, the horizon spans an angle of - see (4.72) -
frirec
n.i°y/s2 m
This angle is modified for models with a cosmological
constant; if the Universe is flat (£2 m + Q A — 1), one
finds
Sound Horizon and Acoustic Peaks. On angular scales
< #H,rec, fluctuations are observed that were inside the
horizon prior to recombination, hence physical effects
may act on these scales. As already mentioned, the fluid
of baryons and photons is dominated by the energy den-
sity of the photons. Their pressure prevents the baryons
from falling into the potential wells of dark matter. In-
stead, this fluid oscillates. Since the energy density is
dominated by photons, i.e., by relativistic particles, this
fluid is relativistic and its sound speed is c s «» c/V3.
Therefore, the maximum wavelength at which a wave
may establish a full oscillation prior to recombination is
A ma x^WC s = r H (frec)/V / 3. (8.25)
This length-scale is called the sound horizon. It cor-
responds to an angular scale of 6\ » 9u,iec/V3 ~ 1°,
or l\ ~ 200 for a flat cosmological model with Q m +
Qa — 1- By the Doppler effect and by adiabatic
compression, these oscillations generate temperature
fluctuations that should be visible in the temperature
fluctuation spectrum Q . Hence, £ (£ + 1 ) Q should have
a maximum at l\ ~ 200; additional maxima are ex-
pected at integer multiples of l\. These maxima in
the angular fluctuation spectrum are termed acoustic
peaks (or Doppler peaks); their ^-values and their am-
plitudes are the most important cosmological means of
diagnostics on the CMB anisotropies.
with a very weak dependence on the matter density,
about (x £2~ 0A . As we will demonstrate in the following,
this angular scale of the horizon is directly observable.
Fluctuations on Large Scales. On scales ^> 0H >rec the
Sachs- Wolfe effect dominates, since oscillations in the
baryon-photon fluid can occur only on scales below
the horizon length. For this reason, the CMB angular
spectrum directly reflects the fluctuation spectrum P(k)
of matter. In particular, for a Harrison-Zeldovich
spectrum, P(k) oc k one expects that
t(l + Y)C t ^ const for t «;
0H,re
-100,
and the amplitude of the fluctuations immediately yields
the amplitude of P(k). This flat behavior of the fluctu-
ation spectrum for n s — 1 is modified by the integrated
Sachs-Wolfe effect.
Silk Damping. Since recombination is not instanta-
neous but extends over a finite range in redshift, CMB
photons are last scattered within a shell of finite thick-
ness. Considering a length-scale that is much smaller
than the thickness of this shell, several maxima and
minima of T are located within this shell along a line-
of-sight. For this reason, the temperature fluctuations
on these small scales are averaged out in the integration
along the line-of-sight. The thickness of the recombi-
nation shell is roughly equal to the diffusion length
of the photons, therefore this effect is relevant on the
same length-scales as the aforementioned Silk damp-
ing. This means that on scales < 5' (£> 2500), one
expects a damping of the anisotropy spectrum and, as
a consequence, only very small (primary) temperature
fluctuations on such small scales.
Model Dependence of the Fluctuation Spectrum.
Figure 8.24 shows the power spectra of CMB flue-
8. Cosmology III: The Cosmological Parameters
100 ( a ) Curvature
Fig. 8.24. Dependence of the CMB
fluctuation spectrum on cosmological pa-
rameters. Plotted is the square root of
the power per logarithmic interval in I,
A T = Jl(l + l)Ce/(2n) T . These power
spectra were obtained from an accu-
rate calculation, taking into account all
the processes previously discussed in
the framework of perturbation theory in
General Relativity. In all cases, the refer-
ence model is defined by Q m + Q A = 1,
Q A = 0.65, Q h h 2 = 0.02, Q m h 2 = 0.147,
and a slope in the primordial density fluctu-
ation spectrum of n s = 1, corresponding to
the Harrison-Zeldovich spectrum. In each
of the four panels, one of these parametei s
is varied, and the other three remain fixed.
The various dependences are discussed in
detail in the main text
tuations where, starting from some reference model,
individual cosmological parameters are varied. First we
note that the spectrum is basically characterized by
three distinct regions in £ (or in the angular scale). For
I < 100, £{l + 1)Q is a relatively flat function if - as
in the figure - a Harrison-Zeldovich spectrum is as-
sumed. In the range £ > 100, local maxima and minima
can be seen that originate from the acoustic oscillations.
For £ > 2000, the amplitude of the power spectrum is
strongly decreasing due to Silk damping.
Figure 8.24(a) shows the dependence of the power
spectrum on the curvature of the Universe, thus on
J2 tot = Q m + C2 A . We see that the curvature has two
fundamental effects on the spectrum: first, the locations
of the minima and maxima of the Doppler peaks are
shifted, and second, the spectral shape at £ < 100 de-
pends strongly on Q tot . The latter is a consequence of
the integrated Sachs-Wolfe effect because the more the
world model is curved, the stronger the time variations
of the gravitational potential cj). The shift in the acous-
tic peaks is essentially a consequence of the change in
the geometry of the Universe: the size of the sound
horizon depends only weakly on the curvature, but
the angular diameter distance D A (z rec ) is a very sen-
sitive function of this curvature, so that the angular
scale that corresponds to the sound horizon changes
accordingly.
The dependence on the cosmological constant for
flat models is displayed in Fig. 8.24(b). Here one can
see that the effect of Q A on the locations of the acous-
tic peaks is comparatively small, so that these basically
depend on the curvature of the Universe. The most im-
portant influence of Q A is seen for small £. For Q A = 0,
the integrated Sachs- Wolfe effect vanishes and the
power spectrum is flat (for n s = 1), whereas larger Q A
always produce a strong integrated Sachs-Wolfe effect.
The influence of the baryon density is presented in
Fig. 8.24(c). An increase in the baryon density causes
the amplitude of the first Doppler peak to rise, whereas
that of the second peak decreases. In general, the am-
plitudes of the odd-numbered Doppler peaks increase,
and those of the even-numbered peaks decrease with
increasing Q\,\r . Furthermore, the damping of fluctua-
tions sets in at smaller £ (hence, larger angular scales)
8.6 Angular Fluctuations of the Cosmic Microwave Background
Fig. 8.25. The uppermost curve in each of the two panels
shows the spectrum of primary temperature iluclualions for the
same reference model as used in Fig. 8.24, whereas the other
curves represent the effect of secondarx anisotropics. On large
angular settles (small ( .). the integrated Sachs-Wolfe effect
dm in nates, whereas the effects of grax national light deflection
i lensing) and of the Sunyaev-Zeldovich effect ( SZ) dominate
at huge (. On intermediate angular scales, the scattering of
photons by free electrons which arc present in the intergalactic
gas after rcionization (curve labeled "suppression" ) is the most
efficient secondary process. Other secondary effects which arc
included in these plots are considerably smaller than the ones
mentioned above and arc thus of little interest here
if £2^ is reduced, since in this case the mean free path of
photons increases, and so the fluctuations are smeared
out over larger scales. Finally, Fig. 8.24(d) demonstrates
the dependence of the temperature fluctuations on the
density parameter Q m h 2 . Changes in this parameter re-
sult in both a shift in the locations of the Doppler peaks
and in changes of their amplitudes.
From this discussion, it becomes obvious that the
CMB temperature fluctuations can provide an enor-
mous amount of information about the cosmological
parameters. Thus, from an accurate measurement of
the fluctuation spectrum, very tight constraints on these
parameters can be obtained.
Secondary Anisotropics. In Fig. 8.25, the secondary
effects in the CMB anisotropies are displayed and com-
pared to the reference model used above. Besides the
already extensively discussed integrated Sachs-Wolfe
effect, the influence of free electrons after reioniza-
tion of the Universe has to be mentioned in particular.
Scattering of CMB photons on these electrons essen-
tially reduces the fluctuation amplitude on all scales,
by a factor e~ r , where x is the optical depth with re-
spect to Thomson scattering. The latter depends on the
reionization redshift, since the earlier the Universe was
reionized, the larger r is. Also visible in Fig. 8.25 is
the fact that, on small angular scales, gravitational light
deflection and the Sunyaev-Zeldovich effect become
dominant. The identification of the latter is possible
by its characteristic frequency dependence, whereas
distinguishing the lens effect from other sources of
anisotropies is not directly possible.
8.6.4 Observations of the Cosmic Microwave
Background Anisotropy
To understand why so much time lies between the
discovery of the CMB in 1965 and the first measure-
ment of CMB fluctuations in 1992, we note that these
fluctuations have a relative amplitude of ~ 2 x 10~ 5 .
The smallness of this effect means that in order to
observe it very high precisions is required. The main
difficulty with ground-based measurements is emis-
sion by the atmosphere. To avoid this, or at least
to minimize it, satellite experiments or balloon-based
observations are strongly preferred. Hence, it is not sur-
prising that the COBE satellite was the first to detect
CMB fluctuations. 9 Besides mapping the temperature
distribution on the sphere (see Fig. 1.17) at an angular
resolution of ~ 7°, COBE also found that the CMB is
the most perfect blackbody that has ever been mea-
sured. The power spectrum for I < 20 measured by
COBE was almost flat, and therefore compatible with
the Harrison-Zeldovich spectrum.
"'With the exception of the dipole anisotropy. caused b\ the peculiar
\elocit_\ of the Sun. which lias an amplitude of ~ 10~ 3 ; this was
identified earlier
8. Cosmology III: The Cosmological Parameters
Galactic Foreground. The measured temperature dis-
tribution of the microwave radiation is a superposition
of the CMB and of emission from Galactic (and extra-
galactic) sources. In the vicinity of the Galactic disk,
this foreground emission dominates, which is clearly
visible in Fig. 1.17, whereas it seems to be consid-
erably weaker at higher Galactic latitudes. However,
due to its different spectral behavior, the foreground
emission can be identified and subtracted. We note
that the Galactic foreground basically consists of three
components: synchrotron radiation from relativistic
electrons in the Galaxy, thermal radiation by dust, and
bremsstrahlung from hot gas. The synchrotron compo-
nent defines a spectrum of about I v oc v~° 8 , whereas the
dust is much warmer than 3 K and thus shows a spec-
tral distribution of about /„ oc v 3 5 in the spectral range
of interest for CMB measurements. Bremsstrahlung
has a flat spectrum in the relevant spectral region,
I v ss const. This can be compared to the spectrum of the
CMB, which has a form I v oc v 2 in the Rayleigh-Jeans
region.
There are two ways to extract the foreground emis-
sion from the measured intensity distribution. First, by
observing at several frequencies the spectrum of the mi-
crowave radiation can be examined at any position, and
the three aforementioned foreground components can
be identified by their spectral signature and subtracted.
As a second option, external datasets may be taken into
account. At larger wavelengths, the synchrotron radia-
tion is significantly more intense and dominates. From
a sky map at radio frequencies, the distribution of syn-
chrotron radiation can be obtained and its intensity at
the frequencies used in the CMB measurements can be
extrapolated. In a similar way, the infrared emission
from dust, as measured, e.g., by the IRAS satellite (see
Fig. 2.11), can be used to estimate the dust emission
of the Galaxy in the microwave domain. Finally, one
expects that gas that is emitting bremsstrahlung also
shows strong Balmer emission of hydrogen, so that the
bremsstrahlung pattern can be predicted from an Ha
map of the sky. Both options, the determination of the
foregrounds from multifrequency data in the CMB ex-
periment and the inclusion of external data, are utilized
in order to obtain a map of the CMB which is as free
from foreground emission as possible - which indeed
seems to have been accomplished in the bottom panel
of Fig. 1.17.
N X
^N %
CMBAnisotr
Jp¥
3^
-
Fig. 8.26. The antenna temperature (oc /„ v~ z ) of the CMB and
of the three foreground components discussed in the text, as
a function of frequency. The five frequency bands of WMAP
are marked. The dashed curves specify the average antenna
temperature of the foreground radiation in the 77%' and 859?
of the sky, respectively, in which the CMB analysis was con-
ducted. We see that the three high-frequency channels are not
dominated by foreground ei
The optimal frequency for measuring the CMB
anisotropies is where the foreground emission has
a minimum; this is the case at about 70 GHz (see
Fig. 8.26). Unfortunately, this frequency lies in a spec-
tral region that is difficult to access from the ground.
From COBE to WMAP. In the years after the COBE
mission, different experiments performed measure-
ments of the anisotropy from the ground, focusing
mainly on smaller angular scales. In around 1997,
evidence was accumulating for the presence of the
first Doppler peak, but the error bars of individual
experimental results were too large at that time to
clearly localize this peak. The breakthrough was then
achieved in March 2000, when two groups published
their CMB anisotropy results: BOOMERANG and
MAXIMA. Both are balloon-based experiments, each
observing a large region of the sky at different frequen-
cies. In Fig. 8.27, the maps from the BOOMERANG
experiment are presented. Both experiments have unam-
biguously measured the first Doppler peak, localizing
it at £ % 200. From this, it was concluded that we live
in a nearly flat Universe - the quantitative analysis of
the data yielded Q m + Q A «s 1 ± 0. 1 . Furthermore, clear
8.6 Angular Fluctuations of the Cosmic Microwave Background
-300 -200 -100
100 200 300
r\
J' : --t
- :■ ■ s^^gfgfc-
^^S|^
-|M .
W^\
-™- 1S o G h, ' / : ' ;
...
\ * r '
J - *
g?3s*
r *vt
V ^
■' IMS
S&xH"^
m*P\
Fig. 8.27. In 2000, two groups published the results of their
CMB observations, BOOMERANG and MAXIMA. This
figure shows the BOOMERANG data. On the left, the tem-
perature distributions at 90 GHz, 150 GHz, and 240 GHz are
displa) ed, while the low ci right panel shows that at 400 GHz.
The three small circles in each panel denote the location of
known strong point sources. The two upper panels on the right
show the differences of temperature maps obtained at two dif-
ferent frequencies, e.g., the temperature map obtained with
the 90-GHz data minus that obtained from the 150-GHz data.
These difference maps feature considerably smaller fluctua-
tions than the individual maps. This is compatible with the
idea that the major fraction of the radiation originates in the
CMB and not, e.g., in Galactic radiation which has a different
spectral distribution and would thus be more prominent in the
difference maps. Only the region within the dashed rectangle
was used in the original analysis of the temperature fluctu-
ations, in order to avoid boundary effects. The fluclualion
spectrum computed from the difference maps is compatible
with pure noise
indications of the presence of the second Doppler peak
were found.
In April 2001, refined CMB anisotropy mea-
surements from three experiments were released,
BOOMERANG, MAXIMA, and DASI. For the former
two, the observational data were the same as in the year
before, but improved analysis methods were applied;
in particular, a better instrumental calibration was ob-
tained. The resulting temperature fluctuation spectrum
is presented in Fig. 8.28, demonstrating that it was now
possible to determine the locations of the first three
Doppler peaks.
The status of measurements of the CMB anisotropy
as of the end of 2002 is shown in Fig. 8.29. In the left-
hand panel, the results of numerous experiments are
plotted individually. The panel on the right shows the
weighted mean of these experiments. Although it might
not be suspected at first sight, the results of all ex-
periments shown on the left are compatible with each
other. With that we mean that the individual measure-
8. Cosmology III: The Cosmological Parameters
5 200 ° ii -IIjJjI 1 }
6000
strong H D
weak £i M =1 & LSS
O DMR
- 4000
T \
o
:± 2000
/ \
200 400
200 400
Fig. 8.28. Power spectrum of the CMB angular fluctuations,
measured with the BOOMERANG experiment. These re-
sults were published in 2002, based on the same data as
the previously released results, but using an improved analy-
sis. Plotted are the coefficients 1(1 + l)C t /(2jt) as a function
of wave number or the multipole order I ~ 180° /0, respec-
tively. The first three peaks can clearly be distinguished; they
originate from oscillations in the photon-baryon fluid at the
time of recombination. In the panel on the right, the fluctu-
ation spectra of several cosmological models which provide
good fits to the CMB data are plotted. The model denoted
"weak" (solid curve) uses the constraints 0.45 < h < 0.90,
t > 10 Gyr, andithas Q A = 0.51, Q m = 0.51, Q h h 2 = 0.022,
h — 0.56, and accordingly to = 15.2 Gyr. The short dashed
curve ("strong H ") uses a stronger constraint h = 0.71 ±
0.08, and yields Q A = 0.62, Q m = 0.40, Q b h 2 = 0.022,
h = 0.65, and accordingly Jo = 13.7 Gyr
ments, given their error bars, are statistically compatible
with the power spectrum that results from the weighted
moan.
With the optimally averaged power spectrum, we
can now determine the cosmological model which best
describes these data. Under the assumption of a flat
model, we obtain Q A = 0.71 ± 0. 1 1 and a baryon den-
sity of Q b h 2 = 0.023 ±0.003, in excellent agreement
with the value obtained from primordial nucleosyn-
thesis (see Eq. 4.62). Furthermore, the spectral index
of the primordial density fluctuations is constrained to
n s = 0.99 ±0.06, which is very close to the Harrison-
Zeldovich value of 1 . In addition, the Hubble constant
is estimated to be h — 0.71 ±0.13, again in extraordi-
narily good agreement with the value obtained from
local investigations using the distance ladder, which is
a completely independent measurement. These agree-
ments are truly impressive if one recalls the assumptions
our cosmological model is based upon.
Baryonic Oscillations in the Galaxy Distribution. As
an aside, though a very interesting one, it should be
mentioned here that the baryonic oscillations which
are responsible for generating the acoustic peaks in the
CMB anisotropy spectrum have now also been observed
in the large-scale distribution of galaxies. To understand
how this can be the case, we consider what happens after
recombination. Imagine that recombination happened
instantaneously; then right at that moment there are den-
sity fluctuations in the dark matter component as well as
the acoustic oscillations in the baryons. The photons can
stream freely, due to the absence of free electrons, and
the sudden drop of pressure in the baryon component
reduces the sound speed from c/~j3 essentially to zero.
We said before that the baryons can then fall into the
potential wells of the dark matter. However, since the
cosmic baryon density is only about six times smaller
than that of the dark matter, the baryonic density fluc-
tuations at recombination are not completely negligible
8.6 Angular Fluctuations of the Cosmic Microwave Background
AIAC KT0CO97_ O
* QMASK c
B0OM9? O
2 10 40 100 200 400 600 800 1000 1200 1400 1600
Multipole t
Fig. 8.29. This figure summarizes the status of the CMB
anisotropy measurements as of the end of 2002. Left: the re-
sults from a large number of individual experiments are shown.
Right: the "best" spectrum of the fluctuations is plotted, ob-
Multipole I
tained by a weighted mean of the individual results where the
corresponding error bars have been taken into account for the
weighting. The red curve shows the fluctuation spectrum of
the best-fitting cosmological model
compared to those of the dark matter. Therefore, they
form their own potential wells, and part of the dark
matter will fall into them. After some time, baryons and
dark matter have about the same spatial distribution,
which is described by the linear evolution of the density
field, where the initial condition is a superposition of
the dark matter fluctuations at recombination plus that
of the baryonic oscillations. Whereas the corresponding
density contrast of the latter is small compared to that of
the dark matter, it has the unique feature that it carries
a well-defined length-scale, namely the sound horizon
at recombination. The matter correlation function in the
local Universe should therefore contain a feature at just
this length-scale. If galaxies trace the underlying mat-
ter distribution, this length-scale should then be visible
in the galaxy correlation function.
In 2005, this feature in the galaxy correlation function
was indeed observed, using the 2dFGRS and the SDSS
redshift surveys. In these correlations, we thus see the
same features as displayed by the acoustic oscillations
in the CMB, but at much lower redshift. The reason why
this discovery is of great importance is seen from the fact
that the baryonic oscillations define a specific length-
scale. The ratio of this length-scale to the observed
angular scale yields the angular diameter distance to the
redshift specified by the sample of galaxies considered.
Thus, we have a "standard rod" in the Universe by which
we can directly measure the distance-redshift relation,
which in turn depends on the cosmological parameters,
yielding a new and very valuable probe for cosmology.
The fact that the baryonic oscillations cause a feature
in the galaxy correlation function at a separation of
~ 100 Mpc immediately implies that very large redshift
surveys are needed to measure this effect, encompassing
very large volumes of the Universe.
8.6.5 WAAAP: Precision Measurements
of the Cosmic Microwave
Background Anisotropy
In June 2001, the Wilkinson Microwave Anisotropy
Probe satellite was launched, named in honor of David
Wilkinson, one of the pioneers of CMB research.
WMAP is, after COBE, only the second experiment
to obtain an all-sky map in the microwave regime.
Compared to COBE, WMAP observes over a wider
8. Cosmology III: The Cosmological Parameters
verification of the COBE measurements. In Fig. 8.30,
sky maps by COBE and by WMAP are displayed. The
dramatically improved angular resolution of the WMAP
map is obvious. In addition, it can clearly be seen that
both maps are very similar if one compares them at
a common angular resolution. This comparison can be
performed quantitatively by "blurring" the WMAP map
to the COBE resolution using a smoothing algorithm.
Since WMAP is not observing at exactly the same fre-
quencies as COBE, it is necessary to interpolate between
two frequencies in the WMAP maps to match the fre-
quency of the COBE map. The comparison then shows
that, when accounting for the noise, the two maps are
completely identical, with the exception of a single lo-
cation in the Galactic disk. This can be explained, e.g.,
by a deviation of the spectral behavior of this source
from the 2.73 K blackbody spectrum that was implicitly
assumed for the aforementioned interpolation between
two WMAP frequencies. The confirmation of the COBE
measurements is indeed highly impressive.
Fig. 8.30. Comparison of the CMB anisotropy i
by COBE (top) and WMAP (bottom), after subtraction of
the dipole originating from the motion of the Sun relative to
the CMB rest-frame. The enormously improved angular res-
olution of WMAP is easily seen. Although these maps were
recorded at different frequencies, the similarity in the tem-
perature distribution is clearly visible and could be confirmed
quantitatively. From this, the COBE results have, for the first
time, been confirmed independently
frequency range, using five (instead of three) frequen-
cies; it has a much improved angular resolution (which
is frequency-dependent; about 20', compared to ~ 7°
for COBE), and in addition, WMAP is able to mea-
sure the polarization of the CMB. Results from the
first year of observation with WMAP were published in
2003. These excellent results confirmed our cosmologi-
cal world model in such a way that we are now justified
in calling it the standard model of cosmology. The most
important results from WMAP will be discussed in the
following.
Comparison to COBE. Since WMAP is the first satel-
lite after COBE to map the full sky in the relevant
frequency range, its first year results allowed the first
Cosmic Variance. Before we continue discussing the
WMAP results we need to explain the concept of
cosmic variance. The angular fluctuation spectrum of
CMB anisotropics is quantified by the multipole coef-
ficients C(. For instance, C\ describes the strength of
the dipole. The dipole has three components; these can
be described, for example, by an amplitude and two
angles which specify a direction on the sphere. Accord-
ingly, the quadrupole has five independent components,
and in general, Q is defined by (21+ 1) independent
components.
Cosmological models of the CMB anisotropics pre-
dict the expectation value of the amplitude of the
individual components Q. In order to compare mea-
surements of the CMB with these models one needs
to understand that we will never measure the expec-
tation value, but instead we measure only the mean
value of the components contributing to the Q on our
microwave sky. Since the quadrupole has only five inde-
pendent components, the expected statistical deviation
of the average from the expectation value is C2/V5. In
general, the statistical deviation of the average of Q
from the expectation value is
AC t =
(8.26)
8.6 Angular Fluctuations of the Cosmic Mi'
;t to many other situations, in which the statis-
tical uncertainties can be reduced by analyzing a larger
sample, this is not possible in the case of the CMB : there
is only one microwave sky that we can observe. Hence,
we cannot compile a sample of microwave maps, but
instead depend on the one map of our sky. Observers
at another location in the Universe will see a different
CMB sky, and thus will measure different values Q,
since their CMB sky corresponds to a different real-
ization of the random field which is specified by the
power spectrum P(k) of the density fluctuations. This
means that (8.26) is a fundamental limit to the sta-
tistical accuracy, which cannot be overcome by any
improvements in instrumentation. This effect is called
cosmic variance. The precision of the WMAP mea-
surements is, for all I < 350, better than the cosmic
variance (8.26). Therefore, the fluctuation spectrum for
I < 350 measured by WMAP is "definite", i.e., further
improvements of the accuracy in this angular range will
not provide additional cosmological information (how-
ever, in future measurements one may test for potential
systematic effects).
The Fluctuation Spectrum. Since WMAP observes at
five different frequencies, the Galactic foreground ra-
diation can, in principle, be separated from the CMB
due to the different spectral behavior. Alternatively, ex-
ternal datasets may be utilized for this, as described in
Sect. 8.6.4. This second method is preferred because,
by using multifrequency data in the foreground sub-
traction, the noise properties of the resulting CMB map
would get very complicated. The sky regions in which
the foreground emission is particularly strong - mainly
in the Galactic disk - are disregarded in the determina-
tion of Ci. Furthermore, known point sources are also
masked in the map.
The resulting fluctuation spectrum is presented in
Fig. 8.31. In this figure, instead of plotting the individ-
ual Q, the fluctuation amplitudes have been averaged
in £-bins. The solid curve indicates the expected fluctu-
ation spectrum in a ylCDM-Universe whose parameters
are quantitatively discussed further below. The gray
region surrounding the model spectrum specifies the
width of the cosmic variance, according to (8.26) and
modified with respect to the applied binning.
The first conclusion is that the measured fluctuation
spectrum agrees with the model extraordinarily well.
Virtually no statistically significant deviations of the
data points from the model are found. Smaller deviations
which are visible are expected to occur as statistical
outliers. The agreement of the data with the model is in
fact spectacular: despite its enormous potential for new
discoveries, WMAP "only" confirmed what had already
been concluded from earlier measurements. Hence, the
results from WMAP confirmed the cosmological model
in an impressive way and, at the same time, considerably
improved the accuracy of the parameter values.
W 100 200 400
Multipole moment (f)
Fig. 8.31. As the central result from the first-year WMAP
measurements, the top panel shows the fluctuation spectrum of
the CMB temperature (TT), whereas the bottom panel displa} s
the power spectrum of the correlation between the temperature
distribution and polarization amplitude i Tit ). Besides the data
points from WMAP, which are plotted here in £-bins, the
results from two other CMB experiments (CBI and ACBAR)
are also plotted, at larger ('. The curve in each panel shows the
best-fitting ACDM model, and the gray region surrounding
it indicates the cosmic variance. The large amplitude of the
point in the TE spectrum at small I indicates an unexpectedly
high polarization on lai ingular seal which su i n
early reionization of the Universe
8. Cosmology III: The Cosmological Parameters
The only data point which deviates substantially from
the model is that of the quadrupole, I = 2. In the COBE
measurements, the amplitude of the quadrupole was
also smaller than expected, as can be seen in Fig. 8.29.
If one assigns physical significance to this deviation,
this discrepancy may provide the key to possible ex-
tensions of the standard model of cosmology. Indeed,
shortly after publication of the WMAP results, a num-
ber of papers were published in which an explanation
for the low quadrupole amplitude was sought. Another
kind of explanation may be found in the fact that for the
analysis of the angular spectrum about 20% of the sky
was disregarded, mainly the Galactic disk. The fore-
ground emission is concentrated towards the disk, and
we cannot rule out the possibility that it has a measur-
able impact on those regions of the sphere that have not
been disregarded. This influence would affect the spec-
trum mainly at low I, Anomalies in the orientation of
the low-order multipoles have in fact been found in the
data. Currently, there is probably no reason to assume
that the low quadrupole amplitude is of cosmological
relevance. If, on the other hand, future analysis of the
data can rule out a substantial foreground contribution
from the Galaxy (or even from the Solar System), the
low quadrupole amplitude may be a smoking gun for
modifications of the standard model.
Polarization of the CMB. The cosmic background ra-
diation is blackbody radiation and should therefore be
unpolarized. Nevertheless, polarization measurements
of the CMB have been conducted which revealed a fi-
nite polarization. This effect shall be explained in the
following.
The scattering of photons on free electrons not only
changes the direction of the photons, but also produces
a linear polarization of the scattered radiation. The di-
rection of this polarization is perpendicular to the plane
spanned by the incoming and the scattered photons.
Consider now a region of space with free electrons. Pho-
tons from this direction have either propagated from
the epoch of recombination to us without experienc-
ing any scattering, or they have been scattered into our
direction by the free electrons. Through this scatter-
ing, the radiation is, in principle, polarized. Roughly
speaking, photons that have entered this region from
the "right" or the "left" are polarized in north-south di-
rection, and photons Mailing from "above" or "below"
show a polarization in east-west direction after scatter-
ing. If the CMB, as seen from the scattering electrons,
was isotropic, an equal number of photons would en-
ter from right and left as from above and below, so that
the net polarization would vanish. However, the scat-
tering electrons see a slightly anisotropic CMB sky, in
much the same way as we observe it; therefore, the
net-polarization will not completely vanish.
This picture implies that the CMB radiation may
be polarized. The degree of polarization depends on
the probability of a CMB photon having been scattered
since recombination, thus on the optical depth with re-
spect to Thomson scattering. Since the optical depth
depends on the redshift at which the Universe was reion-
ized, this redshift can be estimated from the degree of
polarization.
In the lower part of Fig. 8.31, the power spectrum
of the correlation between the temperature distribution
and the polarization is plotted. One finds a surprisingly
large value of this cross-power for small I. This mea-
surement is probably the most unexpected discovery
in the WMAP data from the first year of observation,
because it requires a very early reionization of the Uni-
verse, Zi on ~ 15, hence much earlier than derived from,
e.g., the spectra of QSOs at z > 6.
The Future of CMB Measurements. Before dis-
cussing the cosmological parameters that result from
the WMAP data, we will briefly outline the prospects
of CMB measurements in the years after 2005. On the
one hand, WMAP will continue to carry out measure-
ments for several years, improving the accuracy of the
measurements and, in particular, testing the results from
the first year. The power spectrum of the polarization
itself, to data (February 2006) has not yet been pub-
lished, so that we can expect new insights (or another
confirmation of the standard model) from that as well,
in particular regarding the reionization redshift. As for
COBE, the WMAP data will also be a rich source of
research for many years.
Balloon and ground-based observations will con-
duct CMB measurements on small angular scales
and so extend the results from WMAP towards
larger I. For example, the experiments DASI, CBI,
and BOOMERANG have measured polarization fluctu-
ations of the CMB, as well as temperature-polarization
cross-correlations. As their measurements extend to
8.7 Cosmological Parameters
smaller angular scales than WMAP, the damping tail
in the angular power spectrum was detected by these
experiments. In particular, the results from the 2003
flight of BOOMERANG have confirmed the large scat-
tering optical depth found by WMAP and thus the high
redshift of reionization.
The progress in such measurements has already been
enormous in recent years, and outstanding results can
be expected for the near future. Finally, the ESA satel-
lite Planck is due to be launched in 2008, observing
in a frequency range between 30 and 850 GHz and at
an angular resolution of about 5'. Like WMAP, Planck
will also map the full sky. Besides measuring the CMB,
this mission will produce very interesting astrophysical
results; it is expected, for instance, that the Planck satel-
lite will discover about 10 4 clusters of galaxies by the
Sunyaev-Zeldovich effect.
8.7 Cosmological Parameters
For a long time, the determination of the cosmologi-
cal parameters has been one of the prime challenges in
cosmology, and numerous different methods were de-
veloped and applied to determine Hq , £2 m ,£2 A ,etc. Until
a few years ago, these different methods yielded re-
sults with relatively large error margins, some of which
did not even overlap. In recent years, the situation has
fundamentally changed, as already discussed in the pre-
vious sections. The measurements by WMAP form the
current highlight in the determination of the cosmolog-
ical parameters, and thus we begin this section with
a presentation of these results.
8.7.1 Cosmological Parameters with WMAP
The precise measurement of the first Doppler peak,
together with consideration of the full angular power
spectrum, provides very tight constraints on the devi-
ation of the cosmological model from a model with
vanishing curvature. In Fig. 8.32, the confidence regions
in the Q m -Q A plane are given, determined either from
the CMB data alone (WMAP, combined with measure-
ments at small angular scales, which in the following is
called WMAPext), or by combining these with SNela
data and/or the value of H as determined from the Hub-
ble Key Project (see Sect. 3.6.2). As in earlier CMB
measurements presented in Sect. 8.6.4, the results from
WMAP also show that the deviation of Q m + Q A from
unity is very small.
For this reason, we consider the other cosmologi-
cal parameters under the assumption of a flat Universe,
Q m + £2 A = \. The WMAP team analyzed a six-
dimensional cosmological parameter space, spanned by
the amplitude A of the density fluctuations (this am-
plitude is directly linked to er 8 , but it is measured on
considerably larger scales than 8 h~ l Mpc because the
CMB data probe the power spectrum on such large
scales), by the slope n s of the spectrum of primordial
fluctuations (with n s — 1 for a Harrison-Zeldovich spec-
trum), by the optical depth x with respect to Thomson
scattering after the reionization of the Universe, by the
scaled Hubble constant h, and by the density parame-
ters Q m h 2 and Q\,h 2 . Table 8.1 lists the best-fit values
for these parameters, where four different combinations
of data were used: WMAP alone, WMAP in combi-
nation with measurements of the CMB fluctuations on
smaller scales (WMAPext), WMAPext in combination
with the results from the 2dFGRS, and finally the com-
bination of WMAPext, the 2dFGRS, and Lya forest
results.
Considering first the CMB results alone, we find that
the value of n s is very close to unity, hence the primor-
dial fluctuation spectrum must have a slope very similar
to, but slightly smaller than the Harrison-Zeldovich
spectrum - in agreement with the predictions from
inflationary models (see Sect. 7.7). The value of the
Hubble constant, h = 0.73 ± 0.05, is in excellent agree-
ment with that determined from the Hubble Key Project.
The derived baryon density Q\,h 2 is also in outstand-
ing agreement with the value obtained from primordial
nucleosynthesis. Combining the values for Q m h : and
h stated in the table, we obtain a value for r — Q m h
which is in very good agreement with that found from
the galaxy distribution in the 2dFGRS (see Eq. 8.5). We
should recall once again that we are dealing with com-
pletely independent methods for the determination of
these parameters.
The measurement of the integrated Sachs- Wolfe ef-
fect in the fluctuation spectrum is a verification of the
value for Q A being different from zero, fully indepen-
dent of the supernovae results. As a matter of fact, the
8. Cosmology III: The Cosmological Parameters
1.0
0.8
*>w
0.6
\
0.4
0.2
WMAP drily.
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 8.32. Ict and 2ct confidence regions (dark gray and light
graj areas, respectively) in the Q m -Q A plane. In the upper
left panel, only the WMAP data were used. In the upper right
panel, the WMAP data were combined with CMB measure-
ments on smaller angular scales (WMAPext). In the lower
i It panel In WMAP i lata were combined with the deter-
' the Hubble constant from the HST Key Project,
and the confidence region which is obtained from SN la mea
suremcnls is included onh for comparison. In the lower right
panel, die SX la dala arc included in addition. The dashed Hue
indicates models of vanishing ci
Table 8.1. The six basic parameters determined from the
WMAP data, where a flat cosmological model (Q m + Q A = 1)
is assumed. A is the amplitude and n s the slope of the primor-
dial power spectrum, and r is the optical depth with respect
to Thomson scattering after reionization. >rj iT /v is a statistical
measure for the agreement of the best-fit model and the data.
The different columns list the best-fit parameters obtained by
using the WMAP data alone, WMAP in combination with
CMB measurements on small angular scales (WMAPext),
WMAPext with additional inclusion of the power spectrum
from the 2dFGRS (WMAPext+2dFGRS), and finally the ad-
ditional inclusion of the power spectrum from the Lycf forest
(WMAPext+2dFGRS+LyoO (from Spergel et al., 2003, ApJS,
148, 175)
WMAP
WMAPext
WMAPext+2dFGRS
WMAPext+2dFGRS+Ly<*
A
0.9 ±0.1
0.8 ±0.1
0.8±0.1
0.75+°°*
n
0.99 ±0.04
0.97 ±0.03
0.97 ±0.03
0.96 ±0.02
0.166+5S 0'^
°- 143+ 0062
0.148+°,°,^
°- 117+ 0053
h
0.72 ±0.05
0.73 ±0.05
0.73 ±0.03
0.72 ±0.03
Q m h 2
0.14 ±0.02
0.13±0.01
0.1 34 ±0.006
0.1 33 ±0.006
n h h 2
0.024 ±0.001
0.023 ±0.001
0.023 ±0.001
0.0226 ±0.0008
XeV"
1429/1341
1440/1352
1468/1381
8.7 Cosmological Parameters
1
IB
J
a
0.1
0.01
m
: Intergalactic
: hydrogen
- clumping
E3
Gravitational
lensing
H
Cluster
abundance
i \ i Cosmic
\ background "
0.001
SDSS
galaxy
•
\ 1
.0001
., ^
Fig. 8.33. The power spectrum of density
fluctuations in the Universe, as determined
by different methods. Here, A 2 (k) <x k 3 P(k)
is plotted. Note that small length-scales
(or large k, respectively) are towards the
left in the plot. Going from large to small
scales, the results presented here are ob-
tained from CMB temperature fluctuations,
from the abundance of galaxy clusters, from
the large-scale dKiribuiion of galaxies, from
cosmic shear, and from the statistical prop-
erties of the Lya forest. One can see that the
power spectrum of a /1CDM model is able
to describe all these data over many orders
of magnitude in scale
00 1000 10000
ale (millions of lightyears)
physical origin of this effect can be proven directly be-
cause the integrated Sachs-Wolfe effect is produced at
relatively low redshifts (where the influence of a cos-
mological constant is noticeable), as a result of the time
evolution of the gravitational potential. Therefore, it
should be directly correlated with the large-scale matter
overdensities which are observable in the distribution
of galaxies and clusters of galaxies, assuming a bias
model. For example, one can correlate the CMB temper-
ature map with luminous elliptical galaxies, as they are
observed photometrically in the Sloan Digital Sky Sur-
vey over very large regions on the sky (see Sect. 9.1.2).
The significant correlation signal found in this anal) sis
yields very strong evidence for the temperature fluctu-
ations having originated from the Sachs-Wolfe effect
on large angular scales, hence providing a direct proof
of Q A £ 0.
A big surprise in the WMAP results is the large value
of t, which is derived in particular from the TE power
spectrum. This value for t implies that the Universe was
reionized at a fairly high redshift of z ~ 15.
The combination of CMB results with those from the
large-scale distribution of galaxies and the statistics of
the Lya forest allows us to measure the power spectrum
at smaller length-scales, as shown in Fig. 8.33. This
combination therefore provides stronger constraints on
the cosmological parameters.
We can see from Table 8. 1 that with this combination
the error margins of some parameters can indeed be re-
duced, compared to considering the WMAP data alone;
in particular, this is the case for £2 m h 2 . It is important to
note that by combining the different data sets, the values
of the parameters change only within the range of their
error bars as determined from the CMB data, which
means that the different datasets are compatible with
each other (and with the flat ylCDM model). With these
primary parameters known, further parameters may now
be derived; these are listed in Table 8.2.
The combined data yield, as a best value for the total
density of the Universe,
Q m + Q A = 1.02 ±0.02,
(8.27)
in outstanding agreement with the prediction from the
inflationary model. Furthermore, the fraction of hot dark
matter can be constrained, for which the small-scale
observations (here from the Lya forest) are of particular
importance, since HDM reduces the power on small
8. Cosmology III: The Cosmological Parameters
Table 8.2. Cosmological parameters, as derived from the
CMB data (WMAP) and the combination of these with the
data from 2dFGRS and the Lya forest. Here, Zi on is the red-
shift of the reionization of the Universe (where it is assumed
that the reionization was homogeneous, instantaneous and
complete), z KC is the redshift of the recombination (this is
the redshift at which the z -distribution of the last scattering
of CMB photons has its maximum), z e q is the redshift where
matter and radiation had the same energy density, n\, is the
number density of baryons today, and ;; is the number density
ratio of baryons to photons (from Spergel et al., 2003, ApJS,
148, 175)
WMAP WMAPext+2dFGRS+Lya
0.72 ±0.05 C
• 71+ 003
4< e
0.9±0.1 C
0.44±0.10 (
.84 ±0.04
.38+g"
fib
0.047 ±0.006 (
.044± 0.004
Q m
0.29 ±0.07 (
.27 ±0.04
I
13.4±0.3Gyr
17±5
3.7±0.2Gyr
7±4
Zrec
1088+}
089 ±1
Zeq
3454+^2
233+™
»b
(2.7±0.1)xl0- 7 cm- 3
2.5±0.1)x 10- 7 cm- 3
"
(6.5t° l) x 10-'°
6.i«i) x lo-i"
scales. We obtain
Q v h 2 < 0.0076 ,
(8.28)
with a 2ct significance, which implies a strict upper limit
for the neutrino mass of m v < 0.23 eV, where (4.63) was
used. This limit is significantly tighter than that which
is currently achievable in laboratory measurements. Be-
sides the Lya forest, cosmic shear can also be utilized
for measurements on small scales, as is demonstrated in
Fig. 8.34.
8.7.2 Cosmic Harmony
With the exception of the high optical depth r, the
WMAP results to data have not brought big surprises.
However, this fact in itself is surprising: given the high
sensitivity and angular resolution of this satellite, it
could well have been possible that the measured fluc-
tuation spectrum showed discrepancies with respect to
our standard model. Remarkably, this does not seem to
be the case.
Hence we are in a situation in which the basic cosmo-
logical parameters are not only known with an accuracy
Rat 4
BBN
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Fig. 8.34. This figure illustrates the complementarity of the
CMB data with those from cosmic shear. The individual con-
fidence regions of both methods (blue for the CMB. orange
for cosmic shear) in the Q m -o% plane are nearly orthogonal,
so that a combination of both methods leads to a significantly
smaller region (green) of allowed parameter pairs
that had been unimaginable only a few years ago, but
also each of these individual values has been mea-
sured by more than one independent method, confirming
the self-consistency of the model in an impressive
manner.
• Hubble Constant. H has been determined with the
Hubble Key Project, by means of the distance lad-
der, particularly using Cepheids. The resulting value
is in outstanding agreement with that derived from
CMB anisotropies. Other estimates of Ho yield com-
parable values. Although the determination of H by
means of the time delay measurement for galaxy-
scale gravitational lenses, and by means of the SZ
effect typically yield somewhat smaller values, these
are still compatible with the values from the Hubble
Key Project and the CMB measurements within the
range of the expected statistical errors and systematic
effects which are difficult to control.
• Contribution of Baryons to the Total Density. The
ratio Q\,/Q m has been determined from the baryon
fraction in clusters of galaxies, from redshift sur-
veys, and from the CMB fluctuations, all yielding
^ b /.Q m «s0.15.
• Baryon Density. The value for Q\,h 2 determined
from primordial nucleosynthesis combined with
8.7 Cosmological Parameters
s of the deuterium abundance in Lya
systems has also impressively been confirmed by the
WMAP results.
• Matter Density. Assuming the value of Hq to be
known, Q m has been determined from the distribu-
tion of galaxies in redshift surveys, from the CMB,
and from the evolution of the number density of
galaxy clusters.
• Vacuum Energy. The very tight limits on the cur-
vature of the Universe obtained from the CMB
, and the implied tight limits on the
n of Q m + Q A from unity, allows us to de-
Q A from the measurement of Q m and the
integrated Sachs-Wolfe effect. These values are in
excellent agreement with the SNIa measurements,
as shown in Fig. 8.35.
• Normalization of the Power Spectrum. Since the
CMB fluctuations measure the power spectrum at
large length-scales, the normalization obtained from
these measurements can be compared to the value
of erg only if the shape of Pik) is very well known.
However, with the shape of P(k) being tightly con-
strained within the framework of CDM models by
the accurate determination of the other parameters,
the CMB measurements yield a value for org that
is in very good agreement with that obtained from
the abundance of galaxy clusters and from cosmic
shear (see Fig. 8.34). Furthermore, these values are
compatible with those derived from the peculiar ve-
locity field of galaxies. However, the uncertainties
in erg for the individual methods are about ±10%
each, so that <r g , at the present time, is per-
haps the least accurately determined cosmological
parameter.
• Age of the Universe. The age of the Universe de-
rived from the WMAP data, t « 13.4 x 10 9 yr, is
compatible with the age of globular clusters and of
the oldest white dwarfs in our Galaxy.
The observational results described in this chapter
opened an era of precision cosmology. On the one hand,
the accuracy of the individual cosmological parameters
will doubtlessly be improved in the coming years by new
observational results; on the other hand, the interest of
cosmology will increasingly shift towards observations
of the early Universe. Studies of the evolution of cos-
mic structure, of the formation of galaxies and clusters,
X
expands fc
Fig. 8.35. This figure shows the allowed regions of the param-
eter pair Q m and Q A , as derived from the CMB anisotropy,
SN la measurements, and the z -evolution of the abundance of
galaxj clusters. Since the individual confidence regions have
substantially different orientations in this parameter plane,
their combination provides much better constraints on these
parameters than each method by itself. The smallness of the
indh idual confidence regions and the fact that they are over-
lapping is an impressive demonstration of the self-consistency
of our cosmological model
and of the history of the reionization of the Universe
will increasingly become the focus of cosmological
research.
Another central objective of future cosmological re-
search will remain the investigation of dark matter and
of dark energy. For the foreseeable future, the latter in
particular will be accessible only through astronomical
observations. Due to the enormous importance of a non-
vanishing dark energy density for fundamental physics,
studying its properties will be at the center of interest of
more than just astrophysicists. It is expected that a sue-
8. Cosmology III: The Cosmological Parameters
cessful theory describing dark energy will necessitate
a significant breakthrough in our general understanding
of fundamental physics.
The search for the constituents of dark matter will
keep physicists busy in the coming years. Experiments
at future particle accelerators (e.g., the LHC at CERN)
and the direct search, in underground laboratories, for
particles which may account for candidates of dark mat-
ter, are promising. In any case, dark matter (if it indeed
consists of elementary particles) will open up a new
field in particle physics. For these reasons, the interests
of cosmology and particle physics are increasingly con-
verging - in particular since the Universe is the largest
and cheapest laboratory for particle physics.
9. The Universe at High Redshift
In the previous chapter we explained by what means
the cosmological parameters may be determined, and
what progress has been achieved in recent years. This
might have given the impression that, with the deter-
mination of the values for Q m , Q A , etc., cosmology is
nearing its conclusion. As a matter of fact, for several
decades cosmologists have considered the determina-
tion of the density parameter and the expansion rate of
the Universe their prime task, and now this goal has
seemingly largely been achieved. However, from this
point on, the future evolution of the field of cosmology
will probably proceed in two directions. First, we will
try to uncover the nature of dark energy and to gain new
insights into fundamental physics along the way. Sec-
ond, astrophysical cosmology is much more than the
mere determination of a few parameters. We want to
understand how the Universe evolved from a very prim-
itive initial state into what we are observing around us
today - galaxies of different morphologies, the large-
scale structure of their distribution, clusters of galaxies,
and active galaxies. We seek to study the formation of
stars and of metals, and also the processes that reionized
the intergalactic medium.
The boundary conditions for studying these pro-
cesses are now very well defined. A few years ago,
the cosmological parameters in models of galaxy evo-
lution, for instance, could vary freely because they had
not been determined sufficiently well at that time. Today,
a successful model needs to come up with predictions
compatible with observations, but using the parameters
of the standard model. There is little freedom left in
designing such models. In other words, the stage on
which the formation and evolution of objects and struc-
ture takes place is prepared, and now the cosmic play
can begin.
Progress in recent years, with developments in in-
strumentation having played a vital role, has allowed us
to examine the Universe at very high redshift. An obvi-
ous indication of this progress is the increasingly high
maximum redshift of sources that can be observed; as
an example, Fig. 9.1 presents the spectrum of a QSO at
redshift z = 6.43. Today, we know quite a few galaxies
at redshift z > 6, i.e., we observe these objects at a time
when the Universe had less than 10% of its current age.
Besides larger telescopes, which enabled these deep im-
ages of the Universe, gaining access to new wavelength
domains is of particular importance for our studies of the
distant Universe. This can be seen, for example, from
the fact that the optical radiation of a source at redshift
z ~ 1 is shifted into the NIR. Because of this, near-
infrared astronomy is about as important for galaxies
at z > 1 as optical astronomy is for the local Universe.
Furthermore, the development of submillimeter astron-
omy has provided us with a view of sources that are
J1 1481 6.64+5251 50.3 z=6.43 Keck/ESI
<Mhw>MU»
Wavelength (A)
Fig. 9.1. Spectrum of a QSO at the high red
shift of z = 6.43. Like many other QSOs
at very high redshift, this source was dis-
covered with the Sloan Digital Sky Survey.
The spectrum was obtained with the Keck
telescope. The redshifted Lya line is clearly
visible, its blue side "eaten" away by in-
tergalactic absorption. Almost all radiation
bluewards of the Lya line is absorbed, with
only the emission from the Ly/S line still
getting through. For A. < 7200 A the spectral
flux is compatible with zero; intergalactic
absorption is too strong here
Peter Schneider. The Universe al High
DOI: 10.1007/11614371 JUO Springer-
id Cosmology, pp. 355^105 (2006)
9. The Universe at High Redshift
nearly completely hidden to the optical eye because of
strong dust absorption.
In this chapter, we will attempt to provide an im-
pression of astronomy of the distant Universe, and shed
light on some interesting aspects that are of particu-
lar importance for our understanding of the evolution of
the Universe. This field of research is currently develop-
ing very rapidly, so we will simply address some of the
main topics in this field today. We begin in Sect. 9. 1 with
a discussion of methods to specifically search for high-
redshift galaxies, and we will then focus on a method by
which galaxy redshifts can be determined solely from
photometric information in several bands (thus, from the
color of these objects). This method can be applied to
deep sky images observed by HST, and we will present
some of the results of these HST surveys. Finally, we
will emphasize the importance of gravitational lenses as
"natural telescopes", which, due to their magnification,
provide us with a deeper view into the Universe.
Gaining access to new wavelength domains paves
the way for the discovery of new kinds of sources;
in Sect. 9.2 we will present galaxy populations which
have been identified by submillimeter and NIR obser-
vations, and whose relation to the other known types
of galaxies is yet to be uncovered. In Sect. 9.3 we will
show that, besides the CMB, background radiation also
exists at other wavelengths, but whose nature is consid-
erably different from that of the CMB. The question of
when and by what processes the Universe was reion-
ized will be discussed in Sect. 9.4. Then, in Sect. 9.5,
we will focus on the history of cosmic star formation,
and show that at redshift z > 1 the Universe was much
more active than it is today - in fact, most of the stars
which are observed in the Universe today were already
formed in the first half of cosmic history. This empir-
ical discovery is one of the aspects that one attempts
to explain in the framework of models of galaxy for-
mation and evolution. In Sect. 9.6 we will highlight
some aspects of these models and their link to observa-
tions. Finally, we will discuss the sources of gamma-ray
bursts. These are explosive events which, for a very
short time, appear brighter than all other sources of
gamma rays on the sky put together. For about 25 years
the nature of these sources was totally unknown; even
their distance estimates were spread over at least seven
orders of magnitude. Only since 1997 has it been
known that these sources are of extragalactic origin.
9.1 Galaxies at High Redshift
In this section we will first consider the question of how
distant galaxies can be found, and how to identify them
as such. The properties of these high-redshift galaxies
can then be compared with those of galaxies in the
local Universe, which were described in Chap. 3. The
question then arises as to whether galaxies at high ::. and
thus in the early Universe, look like local galaxies, or
whether their properties are completely different. One
might, for instance, expect that the mass and luminosity
of galaxies are evolving with redshift. Examining the
galaxy population as a function of redshift, one can trace
the history of global cosmic star formation and analyze
when most of the stars visible today have formed, and
how the density of galaxies changes as a function of
redshift. We will investigate some of these questions in
this and the following sections.
9.1.1 Lyman-Break Galaxies (LBGs)
How to Find High-Redshift Galaxies? Until about
1995 only a few galaxies with z > 1 had been known;
most of them were radio galaxies discovered by op-
tical identification of radio sources. The most distant
normal galaxy with z > 2 then was the source of the
giant luminous arc in the galaxy cluster CI 2244— 02
(see Fig. 6.31). Very distant galaxies are faint, and so
the question arises of how galaxies at high z can be
detected at all.
The most obvious answer to this question may per-
haps be by spectroscopy of a sample of faint galaxies.
This method is not feasible though, since galaxies with
R < 22 have redshifts z < 0.5, and spectra of galaxies
with R > 22 are only observable with 4-m telescopes
and with a very large investment of observing time.
Also, the problem of finding a needle in a haystack
arises: most galaxies with R < 24.5 have redshifts z < 2
(a fact that was not known before 1995), so how can
we detect the small fraction of galaxies with larger
redshifts?
Narrow-Band Photometry. A more systematic method
that has been applied is narrow-band photometry. Since
hydrogen is the most abundant element in the Universe,
one expects that some fraction of galaxies feature a Lya
emission line (as do all QSOs). By comparing two sky
images, one taken with a narrow-band filter centered
on a wavelength A, the other with a broader filter also
centered roughly on A, this line emission can be searched
for specifically. If a galaxy atz«A./(1216A) — 1 has
a strong Lya emission line, it should be particularly
bright in the narrow-band image in comparison to the
broad-band image, relative to other sources. This search
for Lya emission line galaxies had been almost without
success until the mid-1990s. Among other reasons, one
did not know what to expect, e.g., how faint galaxies at
z ~ 3 are and how strong their Lya line would be.
The Lyman-Break Method. The breakthrough was
obtained with a method that became known as the
Lyman-break method. Since hydrogen is so abundant
and its ionization cross-section so large, one can expect
that photons with A < 912 A are very heavily absorbed
by neutral hydrogen in its ground state. Therefore, pho-
tons with A < 912 A have a low probability of escaping
from a galaxy without being absorbed.
Intergalactic absorption also contributes. In
Sect. 5.6.3 we saw that each QSO spectrum features
a Lya forest and Lyman-limit absorption. The inter-
galactic gas absorbs a large fraction of photons emitted
by a high-redshift source at A < 1216 A, and virtually
all photons with a rest-frame wavelength X < 912 A.
As also discussed in Sect. 8.5.2, the strength of this ab-
sorption increases with increasing redshift. Combining
these facts, we conclude that spectra of high-redshift
galaxies should display a distinct feature - a "break" -
at X = 1216 A. Furthermore, radiation with X < 912 A
should be strongly suppressed by intergalactic absorp-
tion, as well as by absorption in the interstellar medium
of the galaxies themselves, so that only a very small
fraction of these ionizing photons will reach us.
From this, a strategy for the detection of galaxies
at z > 3 emerges. We consider three broad-band fil-
ters with central wavelengths X\ < A 2 < A3, where their
spectral ranges are chosen to not (or only marginally)
overlap. If Ai < (1+z) 912 A < Xi, a galaxy containing
young stars should appear relatively blue as measured
with the filters X 2 and A 3 , and be virtually invisible in
the A i -filter: because of the absorption, it will drop out
of the A i -filter (see Fig. 9.2). For this reason, galaxies
that have been detected in this way are called Lyman-
3000
7000
4000 5000 6000
Wavelength (A)
Fig. 9.2. Principle of the Lyman-break method. The histogram
shows the synthetic spectrum of a galaxy at z = 3. 15. gener
ated by models of population synthesis; the spectrum belongs
to a QSO at slightly higher redshift. Clearly, the decline of the
spectrum at A < 912(1 +z) A is noticeable. The three dotted
curves arc the transmission curves of three broad band fillers,
chosen such that one of them (U n ) blocks all photons with
wavelengths above the Lyman break. The color oi'this galax)
would then be blue in G — ,Ji, and very red in U n — G
break galaxies (LBG) or drop-o
is displayed in Fig. 9.3.
i. An example of this
Large Samples of LBGs. The method was first ap-
plied systematically in 1996, using the filters specified
in Fig. 9.2. As can be read from Fig. 9.4, the expected lo-
cation of a galaxy at z ~ 3 in a color-color diagram with
this set of filters is nearly independent of the type and
star-formation history of the galaxy. Hence, sources in
the relevant region of the color-color diagram are very
good candidates for being galaxies at z ~ 3. The redshift
needs to be verified spectroscopically, but the crucial
point is that the color selection of candidates yields
a very high success rate per observed spectrum, and thus
spectroscopic observing time at the telescope is spent
very efficiently in confirming the redshift of distant gal-
axies. With the commissioning of the Keck telescope
(and later also of other telescopes of the 10-m class),
spectroscopy of galaxies with B < 25 became possi-
ble (see Fig. 9.5). Employing this method, more than
1000 galaxies with 2.5 < z < 3.5 have been detected
and spectroscopically verified to date.
9. The Universe at High Redshift
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Fig. 9.3. Top panel: a U-band drop-out galaxy. It is clearly de-
tected in the two redder filters, but vanishes almost completely
in the U-filter. Bottom panel: in a single CCD frame, a large
number of candidate Lyman-break galaxies are found. They
arc marked with circles here; their density is about 1 per square
From the spectra shown in Fig. 9.5, it also becomes
apparent that not all galaxies which fulfill the selection
criteria also show a Lya emission line, which provides
one possible explanation for the lack of success in earlier
searches for high-redshift galaxies using narrow-band
filters. The spectra of the high-redshift galaxies which
were found by this method are very similar to those of
starburst galaxies at low redshift. Obviously, the galax-
ies selected in this way feature active star formation.
Due to the chosen selection criterion, such sources are,
of course, preferentially selected, since star formation
produces a blue spectrum at (rest-frame) wavelengths
above 1216 A; in addition the luminosity of galaxies in
Fig. 9.4. Evolutionary tracks of galaxies in the (G — X) -
(U n — G) color-color diagram, for different types of galaxies,
as obtained from population synthesis models. All evolution-
al \ tracks start at z — 0, and the symbols along the curves
mark intervals of Az = 0.1. The colors of the various gal-
axy types are very different at lower redshift, but for z > 2.7,
the evolutionary tracks for the different types nearly coin-
cide - a consequence of the Lya absorption in the intergalactic
medium. Hence, a color selection of galaxies in the region be-
tween the dotted and dashed curves should select galaxies
with z > 3. Indeed, this selection of candidates has proven to
be very successful; more than 1000 galaxies with z ~ 3 have
been spectroscopically verified
the UV range strongly depends on the star-formation
The Correlation Function of LBGs. For a large vari-
ety of objects, and over a broad range of separations, the
correlation function of objects can be described by the
power law (7.19), with a slope of typically y ~ 1.7.
However, the amplitude of this correlation function
varies between different classes of objects. For exam-
9.1 Galaxies at High Redshift
C23 0000-263 z=3.19<
_±^
Fig. 9.5. Spectra of two galaxies ai - - 3.
detected by means of the U-drop-out tech-
nique. Below each spectrum, the spectrum
of a nearby starburst galaxy (NGC 4214)
- shifted to the corresponding redshift - is
plotted; it becomes apparent that the spectra
of galaxies at .: ~ 3 are very similar to those
of present-day galaxies. One of the two
U-drop-out galaxies features a strong Lya
emission line, the other shows absorption at
the respective wavelength
4000 5000 6000 7000
Wavelength (A)
- A U i ii |..J -
5000 5500 6000 6500 7000
Wavelength (A)
pie, we saw in Sect. 8.2.4 that the power spectrum of
galaxy clusters is larger by about a factor 7 than that of
galaxies (see Fig. 8.12); the same ratio holds of course
for the corresponding correlation functions. As we ar-
gued there, the strength of the correlation depends on
the mass of objects; in the simple picture of biasing
shown in Fig. 8.3, the correlation of objects is larger the
rarer they are. High-mass peaks exceeding the density
threshold needed for gravitational collapse have a lower
mean density than low-mass peaks, so they are there-
fore expected to be more biased (see Sect. 8.1.3) and
thus more strongly correlated.
In fact, these qualitative arguments can be sub-
stantiated with numerical simulations, as well as with
quantitative analytical estimates. From the LSS simu-
lations described in Sect. 7.5.3, one can identify dark
matter halos (employing, e.g., the overdensity crite-
rion which follows from the spherical collapse model
in Sect. 7.5.1) and compare their correlation function
with that of the overall dark matter distribution. From
such analyses it is concluded that massive halos are
more strongly clustered than the dark matter itself,
whereas low-mass halos are less correlated. The di-
vision between these two cases occurs roughly at the
mass scale M H ,(z) - see (7.34) - which describes the
non-linear mass scale at a given epoch. Therefore, mea-
suring the correlation function of objects and comparing
it with the correlation of dark matter at the correspond-
9. The Universe at High Redshift
ing redshift, the characteristic mass of the halos in which
these objects reside can be determined, as well as their
bias.
The Halo Mass of LBGs. If we consider the spatial dis-
tribution of LBGs, we find a large correlation amplitude.
The (comoving) correlation length of LBGs at redshifts
1.5 < z < 3.5isro ~4.2/j _1 Mpc, i.e. , not very different
from the correlation length of L* -galaxies in the present
Universe. Since the bias factor of present-day galaxies
is about unity, implying that they are clustered in a sim-
ilar way to the dark matter distribution, this result then
implies that the bias of LBGs at high redshift must
be considerably larger than unity. This conclusion is
based on the fact that the dark matter correlation at high
redshifts was smaller than today by the factor £>+(z).
Thus we conclude that LBGs are rare objects and thus
correspond to high-mass dark matter halos. Comparing
the observed correlation length r ( ) with numerical sim-
ulations, the characteristic halo mass of LBGs can be
determined, yielding ~ 3 x 10 U M Q at redshifts z ~ 3,
and ~ 1O 12 M at z ~ 2. Furthermore, the correlation
length is observed to increase with the luminosity of the
LBG, indicating that more luminous galaxies are hosted
by more massive halos, which are more strongly biased
than less massive ones. If these results are combined
with the observed correlation functions of galaxies in
the local Universe and at z ~ 1 , and with the help of
numerical simulations, then this indicates that a typical
high-redshift LBG will evolve into an elliptical galaxy
by today.
Proto-Clusters. Furthermore, the clustering of LBGs
shows that the large-scale galaxy distribution was al-
ready in place at high redshifts. In some fields the
observed overdensity in angular position and galaxy
redshift is so large that one presumably observes
galaxies which will later assemble into a galaxy clus-
ter - hence, we observe some kind of proto-cluster. We
have already shown such a proto-cluster in Fig. 6.47.
Galaxies in such a proto-cluster environment seem to
have about twice the stellar mass of those LBGs outside
such structures, and the age of their stellar population
appears older by a factor of two. This result indicates
that the stellar evolution of galaxies in dense environ-
ments proceeds faster than in low-density regions, in
accordance with expectations from structure formation.
It also reveals a dependence of galaxy properties on the
environment, which we have seen before manifested
in the morphology-density relation (see Sect. 6.2.9).
Proto-clusters of galaxies have also been detected at
higher redshifts up to z ~ 6, using narrow-band imaging
searches for Lyman-a emission galaxies.
Whereas the clustering of LBGs is well described
by the power law (7.19) over a large range of scales,
the correlation function exhibits a significant deviation
from this power law on very small scales: the angular
correlation function exceeds the power law at A6 < 1" ,
corresponding to physical length-scales of ~ 200 kpc.
It thus seems that this scale marks a transition in the
distribution of galaxies. To get an idea of the physical
nature of this transition, we note that this length-scale
is about the virial radius of a dark matter halo with
M~ 3 x 10 n M o , i.e., the mass of halos which host the
LBGs. On scales below this virial radius, the correla-
tion function thus no longer describes the correlation
between two distinct dark matter halos. An interpreta-
tion of this fact is provided in terms of merging: when
two galaxies and their dark matter halos merge, the re-
sulting dark matter halo hosts both galaxies, with the
more massive one close to the center and the other one
as "satellite galaxy". The correlation function on scales
below the virial radius thus indicates the clustering of
galaxies within the same halo, whereas on larger scales,
where it follows the power-law behavior, it indicates the
correlation between different halos.
Winds of Star-Forming Galaxies. The inferred high
star-formation rates of LBGs implies an accordingly
high rate of supernova explosions. These release part of
their energy in the form of kinetic energy to the inter-
stellar medium in these galaxies. This process will have
two consequences. First, the ISM in these galaxies will
be heated locally, which slows down (or prevents) fur-
ther star formation in these regions. This thus provides
a feedback effect for star formation which prevents all
the gas in a galaxy from turning into stars on a very
short time-scale, and is essential for understanding the
formation and evolution of galaxies, as we shall see in
Sect. 9.6. Second, if the amount of energy transferred
from the SNe to the ISM is large enough, a galactic wind
may be launched which drives part of the ISM out of the
galaxy into its halo. Evidence for such galactic winds
has been found in nearby galaxies, for example from
9.1 Galaxies at High Redshift
neutral hydrogen observations of edge-on spirals which
show an extended gas distribution outside the disk. Fur-
thermore, the X-ray corona of spirals (see Fig. 3.18) is
most likely linked to a galactic wind in these systems.
Indeed, there is now clear evidence for the presence
of massive winds from LBGs. The spectra of LBGs of-
ten show strong absorption lines, e.g., of Civ, which
are blueshifted relative to the velocity of the emission
lines in the galaxy. An example of this effect can be
seen in the spectra of Fig. 9.5, where in the upper panel
the emission line of Civ is accompanied by an absorp-
tion to the short-wavelength side of the emission line.
Such absorption can be produced by a wind moving
out from the star-forming regions of the galaxy, so that
its redshift is smaller than that of the emission regions.
Characteristic velocities are ~ 200 km/s. In one case
where the spectral investigation has been performed in
most detail (the LBG cB58; see Fig. 9.13), the outflow
velocity is ~ 255 km/s, and the outflowing mass rate
exceeds the star-formation rate. Whereas these obser-
vations clearly show the presence of outflowing gas,
it remains undetermined whether this is a fairly local
phenomenon, restricted to the star-formation sites, or
whether it affects the ISM of the whole galaxy.
Connection to QSO Absorption Lines. A slightly
more indirect argument for the presence of strong
winds from LBGs comes from correlating the absorp-
tion lines in background QSO spectra with the position
of LBGs. These studies have shown that whenever
the sightline of a QSO passes within ~ 40 kpc of an
LBG, very strong Civ absorption lines (with column
density exceeding 10 14 cm~ 2 ) are produced, and that
the corresponding absorbing material spans a velocity
range of Av > 250 km/s; for about half of the cases,
strong Civ absorption is produced for impact parame-
ters within 80 kpc. This frequency of occurrence implies
that about 1/3 of all Civ metal absorption lines with
N > 10 14 cm -2 in QSO spectra are due to gas within
~ 80 kpc from those LBGs which are bright enough to
be included in current surveys. It is plausible that the
remaining 2/3 are due to fainter LBGs.
The association of Civ absorption line systems with
LBGs by itself does not prove the existence of winds in
such galaxies; in fact, the absorbing material may be gas
orbiting in the halo in which the corresponding LBG is
embedded. In this case, no outflow phenomenon would
be implied. However, in that case one might wonder
where the large amount of metals implied by the QSO
absorption lines is coming from. They could have been
produced by an earlier epoch of star formation, but in
that case the enriched material must have been expelled
from its production site in order to be located in the
outer part of z ~ 3 halos. It appears more likely that
the production of metals in QSO absorption systems
is directly related to the ongoing star formation in the
LBGs. We shall see in Sect. 9.2.5 that clear evidence for
superwinds has been discovered in one massive star-
forming galaxy at z ~ 3.
Finally, we mention another piece of evidence for
the presence of superwinds in star-forming galaxies.
There are indications that the density of absorption lines
in the Lya forest is reduced when the sightline to the
QSO passes near a foreground LBG. This may well be
explained by a wind driven out from the LBG, pushing
neutral gas away and thus leaving a gap in the Lya forest.
The characteristic size of the corresponding "bubbles"
is ~ 0.5 Mpc for luminous LBGs.
Lyman-Break Galaxies at Low Redshifts. One might
ask whether galaxies similar to the LBGs at z ~ 3 ex-
ist in the current Universe. Until recently this question
was difficult to investigate, since it requires imaging
of lower redshift galaxies at ultraviolet wavelengths.
With the launch of GALEX an appropriate observatory
became available with which to observe galaxies with
rest-frame UV luminosities similar to those of LBGs.
UV-selected galaxies show a strong inverse correlation
between the stellar mass and the surface brightness in
the UV. Lower-mass galaxies are more compact than
those of higher stellar mass. On the basis of this cor-
relation we can consider the population of large and
compact UV-selected galaxies separately. The larger
ones show a star- formation rate of a few M Q /yr; at
this rate, their stellar mass content can be built up on
a time-scale comparable to the Hubble time, i.e., the
age of the Universe. These galaxies are typically late-
type spiral galaxies, and they show a metallicity similar
to our Galaxy. In contrast, the compact galaxies have
a lower stellar mass and about the same star- formation
rate, which allows them to generate their stellar popu-
lation much faster, in about 1 Gyr. Their metallicity is
smaller by about a factor of 2. These properties of these
compact UV-selected galaxies are quite similar to those
9. The Universe at High Redshift
of the LBGs seen at higher redshifts, and hence, they
may be closely related to the LBG population.
Lyman-Break Galaxies at High Redshift. By varia-
tion of the filter set, drop-outs can also be discovered
at larger wavelengths, thus at accordingly higher red-
shifts. The object selection at higher z implies an
increasingly dominant role of the Lya forest whose den-
sity is a strongly increasing function of redshift (see
Sect. 8.5.2). This method has been routinely applied up
to z ~ 4.5, yielding so-called B-drop-outs. Galaxies at
considerably higher redshifts are difficult to access from
the ground with this method. One reason for this is
that galaxies become increasingly faint with redshift,
rendering observations substantially more problematic.
Furthermore, one needs to use increasingly redder fil-
ter sets. At such large wavelengths the night sky gets
significantly brighter, which further hampers the de-
tection of very faint objects. For detecting a galaxy at
redshift, say, z — 5.5 with this method, the Lya line,
now at A s» 7900 A, is located right in the I-band, so
that for an efficient application of the drop-out tech-
nique only the I- and z-band filters or NIR-filters are
viable, and with those filters the brightness of the night
sky is very problematic (see Fig. 9.6 for an example of
a drop-out galaxy at very high redshift). Furthermore,
candidate very high-redshift galaxies detected as drop-
outs are very difficult to verify spectroscopically due to
their very low flux. In spite of this, we will see later that
the drop-out method has achieved spectacular results
even at redshifts considerably higher than z ~ 4, where
the HST played a central role. But the new generation
of 10-m class telescopes, equipped with instruments
sensitive in the appropriate wavelength regimes, can
also reveal a population of high-redshift drop-out candi-
dates. In particular, the Subaru telescope, which carries
a wide-field camera, has produced a deep field survey
with several broad-band filters and two narrow-band
filters situated at wavelengths of 8840 A and 9840A,
respectively, which is ideally suited to selecting z ~ 6
LBGs. A survey conducted with Subaru has detected
about 12 LBG candidates at this redshift. Calculating
the spatial number density of these objects indicates
that luminous star-forming galaxies were rarer by an
order of magnitude at z ~ 6 than at z ~ 3.
9.1.2 Photometric Redshift
The Lyman-break technique is a special case of
a method for estimating the redshift of galaxies (and
QSOs) by multicolor photometry. This technique can
be employed due to the spectral break at A = 912 A
and A. = 1216 A, respectively. Spectra of galaxies also
show other characteristic features. As was discussed
in detail in Sect. 3.9, the broad-band energy distribu-
tion is basically a superposition of stellar radiation.
A stellar population of age > 10 8 yr features a 4000- A
• 8185/105
•
•
•
•
•
t .
■
•
•
R
•
•
V
•
B
•
Fig. 9.6. A galaxy at z = 5.74, which is
visible in the narrow-band filter (upper left
panel) and in the I- and z-band (located
between the two horizontal dashes), but
which does not show any flux in the three
filters at shorter wavelength
break because, due to a sudden change in the opacity
at this wavelength, the spectra of most stars show such
a break at about 4000 A (see Fig. 3.47). Hence, the ra-
diation from a stellar population at A. < 4000 A is less
intense than at 1 > 4000 A; this is the case particularly
for early-type galaxies (see Fig. 3.50).
If we assume that the star-formation histories of gal-
axies are not too diversified, galaxies will not be located
at an arbitrary location in a multidimensional color di-
agram; rather, they should be concentrated in certain
regions. In this context the 4000-A break and the Lya-
break play a central role, as is illustrated in Fig. 9.7.
Once these characteristic domains in color space where
(most of) the galaxies are situated are identified, the
redshift of galaxies can be estimated solely from their
observed colors, since they are functions of the redshift.
The corresponding estimate is called the photometric
red shift.
More precisely, a number of standard spectra of
galaxies (so-called templates) are used, which are ei-
ther selected from observed galaxies or computed by
population synthesis models. Each of these template
spectra can then be redshifted in wavelength, from
which a K-correction (see Sect. 5.6.1) results. For each
template spectrum and any redshift, the expected galaxy
colors are determined by integrating the spectral energy
distribution, multiplied by the transmission functions of
the applied filters, over wavelength (see Eq. A. 25). This
set of colors can then be compared with the observed
colors of galaxies, and the set best resembling the ob-
servation is taken as an estimate for not only the redshift
but also the galaxy type.
The advantage of this method is that multicolor pho-
tometry is much less time-consuming than spectroscopy
of individual galaxies. In addition, this method can be
extended to much fainter magnitudes than are achiev-
able for spectroscopic redshifts. The disadvantage of
the method becomes obvious when an insufficient num-
ber of colors are available, since then the photometric
redshift estimates can yield a completely wrong z. One
example for the occurrence of extremely wrong red-
shift estimates is provided by a break in the spectral
energy distribution. Depending of whether this break
is identified as the Lyman-break or the 4000-A break,
the resulting redshift estimates will be very different.
To break the corresponding degeneracy, a sufficiently
large number of filters must be available to probe the
Observed Wavelength (A) (z=1 .0)
Observed Wavelength (A) (z=3.2)
Fig. 9.7. The bottom panel illustrates again the principle of
the drop-out method, for a galaxy at z ~ 3.2. Whereas the
Lyman-a forest absorbs part of the spectral flux between (rest-
frame wavelength) 912 A and 1216 A, the flux below 912 A
vanishes almost completely. By using different combinations
of filters (top panel), an efficient selection of galaxies at other
redshifts is also possible. The example shows a galaxy at z — 1
where the 4000-A break is utilized, which occurs in stellar
populations after several 10 7 yr (see Fig. 3.47) and which is
considered to be one of the most important features for the
method of photometric redshift
spectral energy distribution over a wide range in wave-
lengths. As a general rule, the more photometric bands
that are available and the smaller the uncertainties in the
measured magnitudes, the more accurate the estimated
redshift. Normally, data from four or five photometric
bands are required to obtain useful redshift estimates.
In particular, the reliability of the photometric redshift
benefits from data over a large wavelength range, so
that a combination of several optical and NIR filters is
desirable.
The successful application of this method also de-
pends on the type of the galaxies. As we have seen
in Sect. 6.6, early-type galaxies form a relatively well-
defined color-magnitude sequence at any redshift, due
9. The Universe at High Redshift
to their old stellar populations (manifested in clusters of
galaxies in form of the red cluster sequence), so that the
redshift of this type of galaxy can be estimated very ac-
curately from multicolor information. However, this is
only the case if the 4000-A break is located in between
two of the applied filters. For z > 1 this is no longer the
case in the optical range of the spectrum. Other types
of galaxies show larger variations in their spectral en-
ergy distribution, depending, e.g., on the star-formation
history.
Photometric redshifts are particularly useful for sta-
tistical purposes, for instance in situations in which the
exact redshift of each individual galaxy in a sample is
of little relevance. However, by using a sufficient num-
ber of filters a redshift accuracy of Az ~ 0.03(1 + z) is
achievable, as demonstrated in Fig. 9.8 by a compari-
son of photometric redshifts with redshifts determined
spectroscopically for galaxies in the field of the
HDF-North.
9.1.3 Hubble Deep Field(s)
The HDF-N. In 1995, an unprecedented observing pro-
gram was conducted with the HST. A deep image in four
filters (U300, B450, V606, and \%\a) was observed with
the Wide Field/Planetary Camera 2 (WFPC2) on-board
HST, covering a field of ~ 5.3 arcmin 2 , with a total
exposure time of about 10 days. This resulted in the
deepest sky image of that time, displayed in Fig. 9.9.
The observed field was carefully selected such that it
did not contain any bright sources. Furthermore, the po-
sition of the field was chosen such that the HST was
able to continually point into this direction, a criterion
excluding all but two relatively small regions on the
sky, due to the low HST orbit around the Earth. An-
other special feature of this program was that the data
became public immediately after reduction, less than
a month after the final exposures had been taken. As-
tronomers worldwide immediately had the opportunity
HDF-N
8z =0.06(1 +z)
Fig. 9.8. Photometric redshift versus the spectroscopic red-
shift for galaxies in the HDF-North. Photometric data in four
optical and two NIR bands have been used here. We see how
accurate photometric reckhifts can be - their quality depends
on the photometric accuracy in the individual niters, the num-
ber of filters used, the redshift and the type of the galaxy, and
also on details of the applied analysis method
Fig. 9.9. The Hubble Deep Field (North), at its time by far the
deepest image of the sky. In December 1995, the HST was
pointed to this field for about 10 days, and observations were
conducted in four different filters. The raw and reduced data
were made public!} available worldwide as early as Ian. 15,
1996. In this image, which spans about 5 square arcminutes,
about 3000 galaxies are \ isible, extending over a wide range
in redshift
to scientifically exploit these data and to compare them
with data at other frequency ranges or to perform their
own follow-up observations. Such a rapid and wide re-
lease was uncommon at that time, but is now seen more
frequently. Rarely has a single data set inspired and mo-
tivated a large community of astronomers as much as
the Hubble Deep Field (North) - HDF(N) - did.
Follow-up observations of the HDF(N) - have been
made in nearly all accessible wavelength ranges, so
that it is the best-observed region of the extragalac-
tic sky. The field contains ~ 3000 galaxies, six X-ray
sources, 16 radio sources, and fewer than 20 stars. For
more than 150 galaxies in this field, redshifts have been
determined spectroscopically, and about 30 have been
found at z > 2. Never before could galaxy counts be
conducted to magnitudes as faint as it became possible
in the HDF-N (see Fig. 9.10); several hundred galaxies
per square arcminute could be photometrically analyzed
in this field.
Detailed spectroscopic follow-up observations were
conducted by several groups, through which the HDF
became, among other things, a calibration field for pho-
tometric redshifts (see, for instance, Fig. 9.8). Most
galaxies in the HDF are far too weak to be analyzed
spectroscopically, so that one often has to rely on
photometric redshifts.
HDF-S and the Hubble Ultra-Deep Field. Later, in
1998, a second HDF was observed, this time in the
southern sky. In contrast to the HDF-N, which had been
chosen to be as empty as possible, the HDF-S contains
a QSO. Its absorption line spectrum can be compared
with the galaxies found in the HDF-S, by which one
hopes to obtain information on the relation between
QSO absorption lines and galaxies. In addition to the
WFPC2 camera, the HDF-S was simultaneously ob-
served with the cameras STIS (51" x 51" field-of-view,
where the CLEAR "filter" was used, which has a very
broad spectral sensitivity; in total, STIS is considerably
more sensitive than WFPC2) and NICMOS (a NIR cam-
era with a maximum field-of-view of 51" x 51") which
had both been installed in the meantime. Nevertheless,
the impact of the HDF-S was smaller than that of the
HDF-N; one reason for this may be that the requirement
of the presence of a QSO, combined with the need for
a field in the continuous viewing zone of HST, led to
a field close to several very bright Galactic stars. This
AB Magnitude
Fig. 9.10. Galaxy counts from the HDF and other surveys.
Solid symbols are from the HDF, open symbols from various
ound ' ' ob 1 1 ilion I Is ui cpi nl ] li lio i
from models in which the specli a I :ni y di tribution of the
galaxies does not evolve - the counts lie significantly above
these so-called non-evolution models: clearly, the galaxy pop-
ulation must be evolving. Note that the counts in the different
color filters are shifted by a factor 10 each, simply for display
purposes
circumstance makes photometric observations from the
ground very difficult, e.g., due to stray light.
In 2002, an additional camera was installed on-board
HST. The Advanced Camera for Surveys (ACS) has,
with its side length of 3. '4, a field-of-view about twice
as large as WFPC2, and with half the pixel size (0"05)
it better matches the diffraction-limited angular reso-
lution of HST. Therefore, ACS is a substantially more
powerful camera than WFPC2 and is, in particular, best
suited for surveys. With the Hubble Ultra-Deep Field
(HUDF), the currently, and presumably for quite some
years to come, deepest image of the sky was observed
and published in 2004 (see Fig. 9.11). The HUDF is, in
all filters, deeper by about one magnitude than the HDF.
The depth of the ACS images in combination with the
9. The Universe at High Redshift
Hubble Ultra Deep Field
Fig. 9.11. The Hubble Ultra-Deep Field,
a field of ~ 3.'4 x 3.' 4 observed by the ACS
camera. The limiting magnitude up to which
sources are detected in this image is about
one magnitude fainter than in the HDF.
More than 10 000 galaxies are visible in the
image, many of them at redshifts z > 5
NASA, ESA, S. Beckwith (STScI) and The HUDF Team STScl-PRCD4-07a
relatively red filters that are available provides us with
an opportunity to identify drop-out candidates at red-
shift z ~ 6; several such candidates have already been
verified spectroscopically.
One of the immediate results from the HDF was the
finding that the morphology of faint galaxies is quite
different from those in the nearby Universe. Locally,
most luminous galaxies fit into the morphological Hub-
ble sequence of galaxies. This ceases to be the case
for high-redshift galaxies. In fact, galaxies at z ~ 2 are
much more compact than local luminous galaxies, they
show irregular light distributions and do not resemble
any of the Hubble sequence morphologies. By redshifts
z ~ 1, the Hubble sequence seems to have been partly
established.
Further Deep-Field Projects with HST: GOODS,
GEMS, COSMOS. The great scientific harvest from
the deep HST images, particularly in combination
with data from other telescopes and the readiness to
make such data available to the scientific community
for multifrequency analyses, provided the motivation
for additional HST surveys. The GOODS (Great Ob-
servatories Origins Deep Surveys) project is a joint
observational campaign of several observatories, cen-
tering on two fields of ~ 16' x 10' size each that have
been observed by the ACS camera at several epochs be-
tween 2003 and 2005. One of these two regions contains
the HDF-N, the other a field that became known as the
Chandra Deep Field South (CDF-S). The Chandra satel-
lite observed both GOODS fields with a total exposure
time of ~ 1 x 10 6 s and ~ 2 x 10 6 s, respectively. Also,
the Spitzer observatory took long exposures of these two
fields. In addition, several ground-based observatories
are involved in this survey, for instance by contributing
an ultra-deep wide field image (~ 30' x 30') centered
on the CDF-S. The data themselves and the data prod-
ucts (like object catalogs, color information, etc.) are
all publicly available and have already led to a large
number of scientific results. Even larger surveys using
9.1 Galaxies at High Redshift
STRAIGHT ARC
(not multiply imaged)
m'
. ■ 1 ■ - » • -WM
■"©■ ©<j-
> .*.,
« ^k» n
the HST (GEMS, a field of 30' x 30' centered on the
CDF-S, and the 2 deg 2 COSMOS survey) will further
improve the statistics of the results obtained from the
HUDF and GOODS.
The multiwavelength approach by GOODS yields an
unprecedented view of the high-redshift Universe. Al-
though these studies and scientific analyses are ongoing
(at the time of writing), quite a large number of very
high-redshift (z > 5) galaxies have already now been
discovered and studied: a sample of more than 500 I-
band drop-outs has been obtained from deep ACS/HST
images. Lyman-break galaxies at z ~ 6 seem to have
stellar populations with masses and lifetimes compara-
ble to those at z ~ 3. This implies that at a time when the
Universe was 1 Gyr old, a stellar population with mass
~ 3 x 10 10 M o and age of a few hundred million years
(as indicated by the observed 4000-Angstrom break)
was already in place. This, together with the apparently
high metallicity of these sources, is thus another indica-
tion of how quickly the early Universe has evolved. The
z ~ 6 galaxies are very compact, with half-light radii of
~ 1 kpc, and thus differ substantially from the galaxy
population known in the lower-redshift Universe.
9.1.4 Natural Telescopes
Galaxies at high redshift are faint and therefore diffi-
cult to observe spectroscopically. For this reason, the
brightest galaxies are preferentially selected (for de-
tailed examination), i.e., basically those which are the
most luminous at a particular z - resulting in undesired,
but hardly avoidable selection effects. For example,
those Lyman-break galaxies at z ~ 3 for which the red-
HST/WFPC2/F814\
Fig. 9.12. A particularly interesting and efficient way tc
galaxies at high redshift is provided by the strong lensing effect in
clusters of galaxies. Since a gravitational lens can magnify the light
of background galaxies (by magnification of the solid angle), one
can expect to detect apparently brighter galaxies at high redshift
in the background of clusters. Here, an HST image of the cluster
Abell 2390 is shown, in which several lens systems are visible. On
the left, the central region of the cluster is shown. Three systems
with a strong lens effect in this cluster are presented in the blow-ups
at top. In the center, the so-called "straight arc'* is visible which has
a redshift of about 0.91 . On the right and left, two multiply imaged
systems are displayed, the images indicated by letters; the two
sources associated with these images have redshifts of z = 4.04
and 4.05, respectively
iMffi?
"MWlyJ
445
Fig. 9.13. The image on the left was taken by the Hubble Space
Telescope. It shows the cluster of galaxies MS 15 12+36, which
has a redshift of z — 0.37. To the right, and slightly above the
central cluster galaxy, an extended and apparently very blue
object is seen, marked by an arrow. This source is not phys-
ically associated with the cluster but is a background galaxy
at a redshift of z = 2.72. With this HST image it was proved
that this galaxj is strongly lensedby the cluster and, by means
of this, magnified by a factor of ~ 30. Due to the magnilica-
shift is verified spectroscopically are among the most
luminous of their kind. The sensitivity of our telescopes
is insufficient in most cases to spectroscopically analyze
a rather more typical galaxy at z ~ 3.
The magnification by gravitational lenses can sub-
stantially alter the apparent magnitude of sources;
gravitational lenses can then act as natural (and inex-
pensive!) telescopes. Examples are the arcs in clusters
of galaxies: many of them have a very high redshift, are
magnified by a factor > 5, and hence are brighter by
about ~ 1 .5 mag than they would be without the lens
effect (see Fig. 9.12). It should be mentioned that a fac-
tor of 5 in magnification corresponds to a factor 25 in
the exposure time required for spectroscopy. '
An extreme example of this effect is represented
by the galaxy cB58 at z = 2.72, which is displayed in
Fig. 9.13. It was discovered in the background of a gal-
1 This factor of 25 make the difference belwecn ; in observation that is
feasible ami one thai is nol. Whereas the proposal for a spectroscopic
observation of 3 hours exposure time at an 8-m telescope may be
successful, a similar proposal of 7 s hours would be hope lessK doomed
to failure.
445 450 455 Waveleng
Wavelength (nm)
tion, this Lyman-break galaxy is the brightest normal galaxy
at redshift z ~ 3, a fact that can be profitably used for a de-
tailed spectroscopic analysis. On the right, a small section
from a high-resolution VLT spectrum of this galaxy is shown.
The Lya transition of the galaxy is located at X = 4530 A,
visible as a broad absorption line. Absorption lines at shorter
wavelengths originate from the Lya-forest along the line-of-
sighl ( indicated by short vertical lines) or by metal lines from
the galaxy itself (indicated by arrows)
axy cluster and is magnified by a factor ~ 30. Hence,
it appears brighter by more than three magnitudes than
a typical Lyman-break galaxy. For this reason, the most
detailed spectra of all galaxies at z ~ 3 have been taken
of this particular source.
One can argue that there is a high probability
that the flux of the apparently most luminous sources
from a particular source population is magnified by
lensing. The apparently most luminous IRAS galaxy,
F10214+47, is magnified by a factor ~ 50 by the
lens effect of a foreground galaxy (where the exact
value of the magnification depends on the wavelength,
since the intrinsic structure and size of the source is
wavelength-dependent - hence the magnification is dif-
ferential). Other examples are the QSOs B 1422+231
and APM 08279+5255, which are among the brightest
quasars despite their high redshifts. In both cases, mul-
tiple images of the QSOs were discovered, verifying
the action of the lens effect. Their magnification, and
therefore their brightness, renders these sources pre-
ferred objects for QSO absorption line spectroscopy
(see Fig. 5.40). The Lyman-break galaxy cB58 men-
9.2 New Types of Galaxies
Fig. 9.14. A section of the galaxy cluster Abell2218
(z = 0.175), observed with the HST in four different filters.
Thr; region was selected because the magnification by the
g] a\ itational lens effect for sources at high redshift is expected
to be very large here. This fact has been established by a de-
tailed mass model of this cluster which could be constructed
from the geometrical constraints provided by the numerous
arcs and multiple images (Fig. 6.33). The red lines denote
the critical curves of this lens for source redshifts of z = 5,
6.5, and 7. A double image of an extended source is clearly
visible in the NIR image (on the right); this double image
was not detected at shorter wavelengths - the expected po-
sition is marked by two ellipses in the two images on the
left. The direction of the local shear, i.e., of the expected im-
age distortion, is plotted in the second image from the right;
the observed elongation of the two images a and b is com-
patible with the shear field from the lens model. Together
with the photometry of these two images, a redshift between
z — 6.8 and z = 7 is derived for the source of this double
image
tioned previously is another example. One important
result of such investigations of high-redshift sources
should be mentioned here. These galaxies and QSOs
have a high metal abundance, from which we conclude
that star formation must have already set in during a very
early phase of the Universe. We will later return to this
point.
The magnification effect is also utilized deliberately,
by searching for highly redshifted sources in fields
around clusters of galaxies. For a massive cluster, one
knows that distant sources located behind the cluster
center are substantially magnified. It is therefore not
surprising that some of the most distant galaxies known
have been detected in systematic searches for drop-
out galaxies near the centers of massive clusters. One
example of this is shown in Fig. 9.14, where a galaxy
at z ~ 7 is doubly imaged by the cluster Abell2218
(see Fig. 6.33), and by means of this it is magnified by
a factor ~ 25.
shifts. We have argued that LBGs are galaxies with
active star formation. Moreover, the UV radiation from
their newly-born hot stars must be able to escape from
the galaxies. From observations in the local Universe we
know, however, that a large fraction of star formation
is hidden from our direct view, since the star-formation
region is enveloped by dust. The latter is heated by ab-
sorbing the UV radiation, and re-emits this energy in the
form of thermal radiation in the FIR domain of the spec-
trum. At high redshifts such galaxies would certainly not
be detected by the Lyman-break method.
Instrumental developments opened up new wave-
length regimes which yield access to other types of
galaxies. Two of these will be described in more detail
here: EROs (Extremely Red Objects) and submillimeter
(sub-mm) sources, the latter often being called SCUBA
galaxies because they were first observed in large num-
bers by the SCUBA camera. But before we discuss these
objects we will first investigate starburst galaxies in the
relatively local Universe.
9.2 New Types of Galaxies
The Lyman-break galaxies discussed above are not the
only galaxies that are expected to exist at high red-
9.2.1 Starburst Galaxies
One class of galaxies, the so-called starburst galaxies,
is characterized by a strongly enhanced star-formation
rate, compared to normal galaxies. Whereas our Milky
9. The Universe at High Redshift
Way is forming stars with a rate of ~ 3M Q /yr, the star-
formation rate in starburst galaxies can be larger by
a factor of more than a hundred. Dust heated by hot stars
radiates in the FIR, rendering starbursts very strong FIR
emitters. Many of them were discovered by the IRAS
satellite ("IRAS galaxies"); they are also called ULIRGs
(ultra-luminous infrared galaxies).
The reason for this strongly enhanced star forma-
tion is presumably the interaction with other galaxies
or the result of merger processes, an impressive exam-
ple of which is the merging galaxy pair known as the
'Antennae" (see Fig. 9.15). In this system, stars and star
clusters are currently being produced in very large num-
bers. The images show a large number of star clusters
with a characteristic mass of 10 5 M o , some of which are
spatially resolved by HST. Furthermore, particularly lu-
minous individual stars (supergiants) are also observed.
The ages of the stars and star clusters span a wide range
and depend on the position within the galaxies. For in-
stance, the age of the predominant population is about
5-10 Myr, with a tendency for the youngest stars to be
located in the vicinity of strong dust absorption. How-
ever, stellar populations with an age of 100 and 500 Myr,
respectively, have also been discovered; the latter pre-
sumably originates from the time of the first encounter
of these two galaxies, which then led to the ejection
of the tidal tails. This seems to be a common phe-
nomenon; for example, in the starburst galaxy Arp220
(see Fig. 1.12) one also finds star clusters of a young
population with age < 10 7 yr, as well as older ones with
age ~ 3 x 10 8 yr. It thus seems that during the merging
process several massive bursts of star-cluster formation
are triggered.
It was shown by the ISO satellite that the most
active regions of star formation are not visible on op-
tical images, since they are completely enshrouded by
Fig. 9.15. The Antenna galaxies. On the left, the "true" op-
tical colors are shown, whereas in the right hand image (he
reddish color shows Ho- emission. This pair of merging gal-
a.\ic; (also see Fig. 1.13 and Fig. 3.4 for other examples of
merging gaia.\ic>.) is forming an enormous number of young
stars. Both the UV emission (bluish in the left image) and
the Hff radiation (reddish in the right image) are considered
indicators of star formation. The individual knots of bright
emission are not single stars but star clusters with typically
10 5 M Q ; however, it is also possible to resolve individual stars
(red and blue supergiants) in these galaxies
9.2 New Types of Galaxies
dust. A map at 15 |xm shows the hot dust heated by
young stars (see Fig. 9.16), where this IR emission is
clearly anticorrelated with the optical radiation. Ob-
viously, a complete picture of star formation in such
galaxies can only be obtained from a combination of
optical and IR images.
Combining deep optical and NIR photometry with
MIR imaging from the Spitzer telescope, star-forming
galaxies at high redshifts can be detected even if they
contain an appreciable amount of dust (and thus may
fail to satisfy the LBG selection criteria). These studies
find that the comoving number density of ULIRGs with
Lir > 10 12 L© at z ~ 2 is about three orders of magni-
tude larger than the local ULIRG density. These results
seem to imply that the high-mass tail of the local gal-
axy population with M > 10 n M Q was largely in place
at redshift z ~ 1.5 and evolves passively from there on.
We shall come back to this aspect below.
Observations with the Chandra satellite have shown
that starburst galaxies contain a rich population of very
luminous compact X-ray sources (Ultra-luminous Corn-
Fig. 9.16. The Antenna galaxies: superposed on the optical
HST image are contours of infrared emission at 15 u,m, mea-
sured by ISO. The strongest IR emission originates in optically
dark regions. A large fraction of the star formation in this gal
axy pair (and in other galaxies?) is not visible on optical
image:, because it is hidden by dust absorption
pact X-ray Sources, or ULXs; see Fig. 9.17). Similar
sources, though with lower luminosity, are also de-
tected in the Milky Way, where these are binary systems
with one component being a compact star (white dwarf,
neutron star, or black hole). The X-ray emission is
caused by accretion of matter (which we discussed in
Sect. 5.3.2) from the companion star onto the compact
component.
Some of the ULXs in starbursts are so luminous,
however, that the required mass of the compact star by
far exceeds 1M if theEddington luminosity is assumed
as an upper limit for the luminosity (see Eq. 5.23).
Hence, one concludes that either the emission of these
sources is highly anisotropic, hence beamed towards us,
or that the sources are black holes with masses of up to
~ 200M o . In the latter case, we may just be witness-
ing the formation of supermassive black holes in these
starbursts.
This latter interpretation is also supported by the fact
that the ULXs are concentrated towards the center of
the galaxies - hence, these BHs may spiral into the
galaxy's center by dynamical friction, and there merge
to a SMBH. This is one of the possible scenarios for the
formation of SMBHs in the cores of galaxies, a subject
to which we will return in Sect. 9.6.3.
9.2.2 Extremely Red Objects (EROs)
As mentioned several times previously, the population
of galaxies detected in a survey depends on the selec-
tion criteria. Thus, using the Lyman-break method, it
is mainly those galaxies at high redshift which feature
active star formation and therefore have a blue spec-
tral distribution at wavelengths longwards of Lya that
are discovered. The development of NIR detectors en-
abled the search for galaxies at longer wavelengths. Of
particular interest here are surveys of galaxies in the
K-band, the longest wavelength window that is reason-
ably accessible from the ground (with the exception of
the radio domain).
The NIR waveband is of particular interest because
the luminosity of galaxies at these wavelengths is not
dominated by young stars. As we have seen in Fig. 3.48,
the luminosity in the K-band depends only weakly on
the age of the stellar population, so that it provides
a reliable measure of the total stellar mass of a galaxy.
neutron stars \ black holes >1
Msol B.H.
|
%^^y NGC46Sr*
■ ■- M81 (disk) \\
\
Fig. 9.17. Ultra-luminous Compact X-ray Sources (ULXs) in
-.uu buixt galaxies. Upper left: the discrete X-ray sources in the
Antenna galaxies; the size of the image is 4' x 4'. Lower left:
optical (image) and (inlaid) Chandra image of the starbursl
galaxy NGC 253. Four of the ULXs are located within one
kiloparsec from the center of the galaxy. The X-ray image is
2.'2 x 2.'2. Upper right: 5' x 5' Chandra image of the starbursl
37 37.5 38 38.5 39 39.5 40 40.5 41
Log(l_x)(0.1-10.0keV)
gala\\ M82: the diffuse uidialion tied') is emilled b\ gas at
T ~ 10 6 K which is heated by the starburst and flows out from
the central region of the galaxy. It is supposed that M82 had
a collision with its companion M81 (see Fig. 6.7) within the
last 10 8 yr, by which the starburst has been triggered. Lower
right: the luminosity function of the ULXs in some starburst
galaxies
Characteristics of EROs. Examining galaxies with
a low K-band flux, one finds either galaxies with low
stellar mass at low redshifts, or galaxies at high redshift
with high optical (due to redshift) luminosity. But since
the luminosity function of galaxies is relatively flat for
L < L*, one expects the latter to dominate the surveys,
due to the larger volume at higher z. In fact, K-band sur-
veys detect galaxies with a broad redshift distribution.
9.2 New Types of Galaxies
bb.jb J iiyy .^Hgi dLL B ra a.
h l
1 "r ;
*
♦
* **•*.*'
S3 I :
v**^
^^My!'
*
• »?.«•<**
fv,,,T,
redshift
Fig. 9.18. Redshift distribution of galaxies with AT S
20, a
measured in ihe K20 survey. 'The shaded histogram represents
galaxies for which the redshift was determined solely by pho-
tometric methods. The bin at z < contains those 9 galaxies
for which it has not been possible to determine z. The peak at
z ~ 0.7 is produced by two clusters of galaxies in the fields of
the K20 survey
In Fig. 9.18 the z -distribution of galaxies in the K20
survey is shown. In this survey, objects with K s < 20
have been selected in two fields with a combined area of
52 arcmin 2 , where K s is a filter at a wavelength slightly
shorter than the classic K-band filter. After excluding
stars and type 1 AGNs, 489 galaxies were found, 480 of
which have had their redshifts determined. The median
redshift in this survey is z « 0.8.
Considering galaxies in a (R— K) vs. K color-
magnitude diagram (Fig. 9.19), one can identify
a population of particularly red galaxies, thus those
with a large R — K. These objects have been named Ex-
tremely Red Objects (EROs); about 10% of the galaxies
in K-selected surveys at faint magnitudes are EROs, typ-
ically defined by R— K > 5. Spectroscopic analysis of
these galaxies poses a big challenge because an object
with K — 20 and R - K > 5 necessarily has R > 25,
i.e., it is extremely faint in the optical domain of the
spectrum. With the advent of 10-m class telescopes,
spectroscopy of these objects has become possible in
recent years.
K
Fig. 9.19. Color-magnitude diagram, i.e., R — K as a function
of K, for sources in ten fields around clusters of galaxies. We
see that for faint magnitudes (roughly K > 19), a population
of sources with a very red color (about R — K > 5.3) turns up.
These objects are called EROs
The Nature of EROs: Passive Ellipticals Versus
Dusty Starbursts. From these spectroscopic results,
it was found that the class of EROs contains rather dif-
ferent kinds of sources. To understand this point we will
first consider the possible explanations for a galaxy with
such a red spectral distribution. As a first option, the ob-
ject may be an old elliptical galaxy with the 4000-A
break being redshifted to the red side of the R-band
filter, i.e., typically an elliptical galaxy at z > 1.0. For
these galaxies to be sufficiently red to satisfy the selec-
tion criterion for EROs, they need to already contain
an old stellar population by this redshift, which implies
a very high redshift for the star formation in these ob-
jects; it is estimated from population synthesis models
that their formation redshift must be Zf orm > 2.5. A sec-
ond possible explanation for large R — K is reddening
by dust. Such EROs may be galaxies with active star
formation where the optical light is strongly attenuated
by dust extinction. If these galaxies are located at a red-
shift of z ~ 1, the measured R-band flux corresponds to
a rest-frame emission in the UV region of the spectrum
where extinction is very efficient.
9. The Universe at High Redshift
Spectroscopic analysis reveals that both types of
EROs are roughly equally abundant. Hence, about half
of the EROs are elliptical galaxies that already have, at
z ~ 1, a luminosity similar to that of today's ellipticals,
and are at that epoch already dominated by an old stel-
lar population. The other half are galaxies with active
star formation which do not show a 4000- A break but
which feature the emission line of [Oil] at X = 3727 A,
a clear sign of star formation. Further analysis of EROs
by means of very deep radio observations confirms the
large fraction of galaxies with high star- formation rates.
Utilizing the close relation of radio emissivity and FIR
luminosity, we find a considerable fraction of EROs to
beULIRGsatz- 1.
at high elevations. In the submillimeter (sub-mm) range,
the long wavelength domain of thermal dust radiation
can be observed, which is illustrated in Fig. 9.20.
Since about 1998 sub-mm astronomy has experi-
enced an enormous boom, with two instruments having
been put into operation: the Submillimeter Common
User Bolometer Array (SCUBA), operating at 450 |xm
and 850 |xm, with a field-of-view of 5 arcmin 2 , and the
Max-Planck Millimeter Bolometer (MAMBO), oper-
ating at 1300 |xm. Both are bolometer arrays which
initially had 37 bolometers each, but which since then
have been upgraded to a considerably larger number
of bolometers. Figure 9.21 shows a 20' x 17' MAMBO
image of a field in the region of the COSMOS survey.
Spatial Correlations. EROs are very strongly cor-
related in space. The interpretation of this strong
correlation may be different for the passive ellipticals
and for those with active star formation. In the for-
mer case the correlation is compatible with a picture in
which these EROs are contained in clusters of galaxies
or in overdense regions that will collapse to a cluster in
the future. The correlation of the EROs featuring active
star formation can probably not be explained by clus-
ter membership, but the origin of the correlation may
be the same as for the correlation of the LBGs.
The number density of passive EROs, thus of old
ellipticals, is surprisingly large compared with expecta-
tions from the model of hierarchical structure formation
that we will discuss in Sect. 9.6.
9.2.3 Submillimeter Sources:
A View Through Thick Dust
FIR emission from hot dust is one of the best indica-
tors of star formation. However, observations in this
waveband are only possible from space, such as was
done with the IRAS and ISO satellites. Dust emission
has its maximum at about 100 |i,m, which is not ob-
servable from the ground. At longer wavelengths there
are spectral windows where observations through the
Earth's atmosphere are possible, for instance at 450 \im
and 850 |xm in the submillimeter waveband. However,
the observing conditions at these wavelengths are ex-
tremely dependent on the amount of water vapor in the
atmosphere, so that the observing sites must by dry and
The Negative K-Correction of Submillimeter
Sources. The emission of dust at these wavelenghs is
described by a Rayleigh-Jeans spectrum, modified by
an emissivity function that depends on the dust prop-
erties (chemical composition, distribution of dust grain
sizes); typically, one finds
S v oc v 2+ti with 6 ~ 1 . . . 2 .
E °r-
Guiderdo-i *: al. Hffi'O:
Blair, et al. (1999c)
jj. # x
QArp 220(1)
"l-.lrl L..1 ,l
;;r/v...-'t:iii+fK>s?
^6 f
;SMMJ02399-0134
#
[~ /.i'{,V-.i+.i7>i,n
Uovft- sal :L)
x APM CW273+5255 (L)
\r
O i
\-
* A Jo
■ BH
202-0725 [H) D '
^ ~ ^ JT
GFI
4C
i : -041 1 i
1 17(H)
\ I
- ' " v*^->>
ISC
435+635 (H)
u
10 '
10 2
u'°
10' 4
Rest frequency / Hz
Fig. 9.20. Spectral energy distribution of some dusty gal
axics with known redshift z (symbols), together with two
model spectra (curves). l ; ourl\ pcsofgalax) are distinguished:
(I) IRAS galaxies at low z; (S) luminous sub-mm galaxies;
(L) distant sources that arc magnified In the gra\ national lens
ii i iii' i nultipl mil >\ (H) AGNs. Only a few sources
among the lens systems (presumably due to differential mag
imii i ii ii ind di ' i . i in i 'i iii i id m hi u mi h 1
spectra
9.2 New Types of Galaxies
10
-10
Fig. 9.21. The image shows a field of 20' x 17' in the region of
the COSMOS survey, observed by the 1 17-channel MAMBO
instrument at the IRAM 30-m telescope on Pico Veleta. Coded
in color is the signal-to-noise ratio of the map, where the noise
level is about 0.9 mjy per 11" beam. About a dozen sources
with S/N > 4 are visible
This steep spectrum for frequencies below the peak
of the thermal dust emission at X ~ 100 \im implies
a very strong negative K-correction (see Sect. 5.6.1)
for wavelengths in the sub-mm domain: at a fixed ob-
served wavelength, the rest-frame wavelength becomes
increasingly smaller for sources at higher redshift, and
there the emissivity is larger. As Fig. 9.22 demonstrates,
this spectral behavior causes the effect that the flux in
the sub-mm range does not necessarily decrease with
redshift. For z < 1, the l/D 2 -dependence of the flux
dominates, so that up to z ~ 1 sources at fixed lumi-
nosity get fainter with increasing z. However, between
z ~ 1 and z ~ Zfl at the sub-mm flux as a function of
redshift remains nearly constant or even increases with
z, where zn- d i depends on the dust temperature T&; for
T d ~ 40 K and X ~ 850 |xm one finds z flat - 8. We there-
fore have the quite amazing situation that sources appear
brighter when they are moved to larger distances. This
is caused by the very negative K-correction which more
than compensates for the l/D 2 -decrease of the flux.
Only for z > Zn- dt does the flux begin to rapidly decrease
with redshift, since then, due to redshift, the correspond-
5x10 12 L o ;n =1; O A =0
0.1
10
850^m
175/jm
£=1.0, 1.5, 2.0 as
thickness increases
5x10 le l^;n =1; Q,
Redshift
Fig. 9.22. Predicted flux from dusty galaxies as a function
of redshift. The bolometric luminosity of these galaxies is
kept constant. The solid red and the blue dashed curves show
the flux at X = 850 p,m and X = 175 pm, respectively. On the
right, the index fi of the dust emissivity is varied, and the
temperature of the dust 7d = 38 K is kept fixed. On the left,
Redshift
/i = 1.5 is fixed and the temperature is varied. It is remarkable
how flat these curves are over a very wide range in redsliifi.
in particular at 850 pm; this is due to the very strong nega
tive K-correction which derives from the spectral behavior of
thermal dust emission, shown in Fig. 9.20
9. The Universe at High Redshift
ing rest-frame frequency is shifted to the far side of the
maximum of the dust spectrum (see Fig. 9.20). Hence,
a sample of galaxies that is flux-limited in the sub-mm
domain should have a very broad z-distribution. The
dust temperature is about T& ~ 20 K for low-redshift
spirals, and T A ~ 40 K is a typical value for galax-
ies at higher redshift featuring active star formation.
The higher T&, the smaller the sub-mm flux at fixed
bolometric luminosity.
Counts of sub-mm sources at high Galactic latitudes
have yielded a far higher number density than was pre-
dicted by galaxy evolution models. For the density of
sources as a function of limiting flux S, at wavelength
A = 850 |xm, one obtains
N(> S) ~ 7.9 x 10 J i
V 1 mJy /
dcg "-
(9.1!
The Identification of SCUBA Sources. At first, the
optical identification of these sources turned out to be
extremely difficult: due to the relatively low angular
resolution of SCUBA and MAMBO the positions of
sources could only be determined with an accuracy of
~ 15". A large number of faint galaxies can be iden-
tified on deep optical images within an error circle of
this radius. Furthermore, Fig. 9.22 suggests that these
sources have a relatively high redshift, thus they should
be very faint in the optical. An additional problem is
reddening and extinction by the same dust that is the
source of the sub-mm emission.
The identification of SCUBA sources was finally ac-
complished by means of their radio emission, since
about half of the sources selected at sub-mm wave-
lengths can be identified in very deep radio observations
at 1.4 GHz. Since the radio sky is far less crowded
than the optical one, and since the VLA achieves an
angular resolution of ~ 1" at k — 20 cm, the optical
identification of the corresponding radio source be-
comes relatively easy. One example of this identification
process is shown in Fig. 9.23. With the accurate radio
position of a sub-mm source, the optical identification
can then be performed. In most cases, they are very faint
optical sources indeed, so that spectroscopic analysis
is difficult and very time-consuming. Another method
for estimating the redshift results from the spectral en-
ergy distribution shown in Fig. 9.20. Since this spectrum
seems to be nearly universal, i.e., not varying much
among different sources, some kind of photometric red-
shift can be estimated from the ratio of the fluxes at
1 .4 GHz and 850 |xm, yielding quite accurate values in
many cases.
Until 2004, redshifts had been measured for about
100 sub-mm sources with a median of roughly z ~ 2.5.
In some of these sources an AGN component, which
heats the dust, was identified, but in general newly born
stars seem to be the prime source of the energetic pho-
Fig. 9.23. The sub-mm galaxy SMM J09429+4658. The three
images on the right have a side length of 30" each, centered
on the center of the error box of the 850 pm observation. The
smaller image on left is the difference of two HST images
in red and infrared filters, showing the dust disk in the spiral
galaxy HI . The second image iron! the left dispki\ s an R -band
image, superposed with the contours of the SCUBA 850 pm
a. The second image from the right is an I-band image,
superposed with the contours of radio emission at 1.4 GHz,
and the right-most panel shows a K-band image. The radio
contours show emission from the galaxy HI (z = 0.33), but
also weaker emission right at the center of the sub-mm map.
In the K-band, a NIR source (H5) is found exactly at this
position. It remains unclear which of these two sources is the
sub-mm source, but the ratio of sub-mm to 1 .4-GHz <
would be atypical if H I is identified with the sub-mr
9.2 New Types of Galaxies
tons which heat the dust. The optical morphology and
the number density of the sub-mm sources suggest that
we are witnessing the formation of elliptical galaxies in
these sub-mm sources.
Additional support for this idea is provided by the
fact that the sub-mm galaxies are typically brighter and
redder than (rest-frame) UV-selected galaxies at red-
shifts z ~ 2.5. This indicates that the stellar masses in
sub-mm galaxies are higher than those of LBGs.
A joint investigation of z ~ 2 sub-mm galaxies at
X-ray, optical and MIR wavelengths yields that these
sources are not only forming stars at a high rate, but that
they also already contain a substantial stellar popula-
tion with M ~ 10 11 Af , roughly an order of magnitude
more massive than LBGs at similar redshifts. The large
AGN fraction among sub-mm galaxies indicates that the
growth of the stellar population is accompanied by ac-
cretion and thus the growth of supermassive black holes
in these objects. Nevertheless, the relatively faint X-ray
emission from these galaxies suggests that either their
SMBHs have a mass well below the local relation be-
tween M, and stellar properties of (spheroidal) galaxies,
or that they accrete at well below the Eddington rate.
Furthermore, the typical ratio of X-ray to sub-mm lumi-
nosity of these sources is about one order of magnitude
smaller than in typical AGNs, which seems to imply
that the total luminosity of these sources is dominated
by the star-formation activity, rather than by accretion
power. This conclusion is supported by the fact that the
optical counterparts of sub-mm sources show strong
signs of merging and interactions, together with their
larger size compared to optically-selected galaxies at
the same redshifts. This latter point shows that the emis-
sion comes from an extended region, as expected from
star formation in mergers, rather than AGN activity.
9.2.4 Damped Lyman-Alpha Systems
In our discussion of QSO absorption lines in Sect. 5.6.3,
we mentioned that the Lya lines are broadly classed
into three categories: the Lya forest, Lyman-limit
systems, and damped Lya systems, which are sepa-
rated by a column density of N m ~ 10 17 cm -2 and
Nm ~ 2 x 10 20 cm -2 , respectively. The origin of the
Lya forest, as discussed in some detail in Sect. 8.5,
is diffuse highly ionized gas with fairly small den-
sity contrast. In comparison, the large column density
of damped Lya systems (DLAs) strongly suggests that
hydrogen is mostly neutral in these systems. The rea-
son for this is self-shielding: for column densities of
Nm >2x 10 20 cm~ 2 the background of ionizing pho-
tons is unable to penetrate deep into the corresponding
hydrogen "cloud", so that only its surface is highly
ionized. Interestingly enough, this column density is
about the same as that observed in 21 -cm hydro-
gen emission at the optical radius of nearby spiral
galaxies.
DLAs can be observed at all redshifts z < 5. For
z > 5 the Lya forest becomes so dense that these
damped absorption lines are very difficult to identify.
For z < 1 .6 the Lya transition cannot be observed from
the ground; since the apertures of optical/UV telescopes
in space are considerably smaller than those on the
ground, observing low-redshift DLAs is substantially
more complicated than that of higher z.
The Neutral Hydrogen Mass Contained in DLAs.
The column density distribution of Lya forest lines is
a power law, given by (8. 1 8). The relatively flat slope of
P ~ 1.6 indicates that most of the neutral hydrogen is
contained in systems of high column density. This can
be seen as follows: the total column density of neutral
hydrogen above some minimum column density Af min is
T dN
J dN m
: J dN m Hffl
and is, for /J < 2, dominated by the highest column den-
sity systems. In fact, unless the distribution of column
densities steepens for very high Nm, the integral di-
verges. From the extended statistics now available for
DLAs, it is known that dN/dN m attains a break at col-
umn densities above Nm > 10 21 cm -2 , rendering the
above integral finite. Nevertheless, this consideration
implies that most of the neutral hydrogen in the Universe
visible in QSO absorption lines is contained in DLAs.
From the observed distribution of DLAs as a function
of column density and redshift, the density parameter
Qm in neutral hydrogen as a function of redshift can be
9. The Universe at High Redshift
inferred. Apparently, Q m ~ 10~ 3 over the whole red-
shift interval < z < 5, with perhaps a small redshift
dependence. Compared to the current density of stars,
this neutral hydrogen density is smaller only by a factor
~ 3. Therefore, the hydrogen contained in DLAs is an
important reservoir for star formation, and DLAs may
represent condensations of gas that turn into "normal"
galaxies once star-formation sets in. Since DLAs have
low metallicities, typically 1/10 of the Solar abundance,
it is quite plausible that they have not yet experienced
much star formation.
The Nature of DLAs. This interpretation is supported
by the kinematical properties of DLAs. Whereas the
fact that the Lya line is damped implies that its ob-
served shape is essentially independent of the Doppler
velocity of the gas, velocity information can never-
theless be obtained from metal lines. Every DLA is
associated with metal absorption line systems, cover-
ing low- and high-ionization species (such as Sill and
Civ, respectively) which can be observed by choosing
the appropriate wavelength coverage of the spectrum.
The profiles of these metal lines are usually split up into
several components. Interpreted as ionized "clouds",
the velocity range Av thus obtained can be used as an
indicator of the characteristic velocities of the DLA.
The values of Av cover a wide range, with a median of
~ 90 km/s for the low-ionization lines and ~ 190 km/s
for the high-ionization transitions. The observed distri-
bution is largely compatible with the interpretation that
DLAs are rotating disks with a characteristic rotational
velocity of v c ~ 200 km/s, once random orientations
and impact parameters of the line-of-sight to the QSO
are taken into i
Search for Emission from DLAs. If this interpretation
is correct, then we might expect that the DLAs can also
be observed as galaxies in emission. This, however, is
exceedingly difficult for the high-redshift DLAs. Noting
that they are discovered as absorption lines in the spec-
trum of QSOs, we face the difficulty of imaging a high-
redshift galaxy very close to the line-of-sight to a bright
QSO (to quote characteristic numbers, the typical
QSO used for absorption-line spectroscopy has B ~ 18,
whereas an L„ -galaxy at z ~ 3 has B ~ 24.5). Due to the
size of the point-spread function this is nearly hopeless
from the ground. But even with the resolution of HST, it
is a difficult undertaking. Another possibility is to look
for the Lya emission line at the absorption redshift,
located right in the wavelength range where the DLA
fully blocks the QSO light. However, as we discussed for
LBGs above, not all galaxies show Lya in emission, and
it is not too surprising that these searches have largely
failed. To data, only three DLA have been detected in
emission, with two of them seen only through the Lya
emission line at the trough of the damped absorption
line, but with no observable continuum radiation. This
latter fact indicates that the blue light from DLAs is con-
siderably fainter than that from a typical LBG at z ~ 3,
consistent with the interpretation that DLAs are not
strong star-forming objects. One of these three DLAs,
however, is observed to be considerably brighter and
seems to share some characteristics of LBGs, including
a high star-formation rate. In addition, two DLAs have
been detected by [Om] emission lines. Overall, then,
the nature of high-redshift DLAs is still unclear, due to
the small number of direct identifications.
For DLAs at low redshifts the observational situa-
tion is different, in that a fair fraction of them have
counterparts seen in emission. Whereas the interpreta-
tion of the data is still not unambiguous, it seems that
the low-redshift population of DLAs may be composed
of normal galaxies.
The spatial abundance of DLAs is largely unknown.
The observed frequency of DLAs in QSO spectra is
the product of the spatial abundance and the absorption
cross-section of the absorbers. This product can be com-
pared with the corresponding quantity of local galaxies:
the detailed mapping of nearby galaxies in the 21 -cm
line shows that their abundance and gaseous cross-
section are compatible with the frequency of DLAs
for z < 1.5, and falls short by a factor ~ 2 for the
higher-redshifts DLAs.
9.2.5 Lyman-Alpha Blobs
The search for high-redshift galaxies with narrow-band
imaging, where the filter is centered on the redshifted
Lya emission line, has revealed a class of objects which
are termed "Lyman-a blobs". These are luminous and
very extended sources of Lya emission; their charac-
teristic flux in the Lya line is ~ 10 44 erg/s, and their
typical size is ~ 100 kpc. Some of these sources show
9.3 Background Radiation at Smaller Wavelengths
no detectable continuum emission in any broad-band
optical filter.
The nature of these high-redshifts objects is currently
unknown. Suggested explanations are wide-ranging,
including a hidden QSO, strong star formation and as-
sociated superwinds, as well as "cold accretion", where
gas is accreted onto a dark matter halo and hydrogen is
collisionally excited in the gas of temperature ~ 10 4 K,
yielding the observed Lya emission. It even seems
plausible that the Lyman-a blobs encompass a range
of different phenomena, and that all three modes of
powering the line emission indeed occur.
Two of these Lya blobs were discovered by narrow-
band imaging of the aforementioned proto-cluster of
LBGs at z = 3.09. Both of them are sub-mm sources and
therefore star-forming objects; the more powerful one
has a sub-mm flux suggesting a star-formation rate of
~ 1000M o /yr. Spatially resolved spectroscopy extend-
ing over the full ~ 100 kpc size of one of the Lya blobs
shows that across the whole region there is an absorption
line centered on the Lya emission line. The optical depth
of the absorption line suggests an Hi column density of
~ 10 19 cm -2 , and its centroid is blueshifted relative to
the underlying emission line by ~ 250 km/s. The spa-
tial extent of the blueshifted absorption shows that the
outflowing material is a global phenomenon in this ob-
ject - a true superwind, most likely driven by energetic
star formation and subsequent supernova explosions in
these objects.
9.3 Background Radiation
at Smaller Wavelengths
The cosmic microwave background (CMB) is a n
of the early hot phase of the Universe, namely thermal
radiation from the time before recombination. As we
extensively discussed in Sect. 8.6, the CMB contains
a great deal of information about our Universe. There-
fore, one might ask whether background radiation also
exists in other wavebands, which then might be of sim-
ilar value for cosmology. The neutrino background that
should be present as a relic from the early epochs of
the Universe, in the form of a thermal distribution of all
three neutrino families with T ss 1.9 K (see Sect. 4.4.2),
is likely to remain undiscovered for quite some time
due to the very small cross-s
n of these low-energy
Indeed, apparently isotropic radiation has been found
in wavelength domains other than the microwave regime
(Fig. 9.24). Following the terminology of the CMB,
these are called background radiation as well. However,
the name should not imply that it is a background radi-
ation of cosmological origin, in the same sense as the
CMB. From the thermal cosmic history (see Sect. 4.4),
no optical or X-ray radiation is expected from the early
phases of the Universe. Hence, for a long time it was
unknown what the origin of these different background
radiations may be.
At first, the early X-ray satellites discovered a back-
ground in the X-ray regime (cosmic X-ray background,
CXB). Later, the COBE satellite detected an apparently
isotropic radiation component in the FIR, the cosmic
infrared background (CIB).
In the present context, we simply denote the flux in
a specific frequency domain, averaged over sky posi-
tion at high Galactic latitudes, as background radiation.
Thus, when talking about an optical background here,
10 10 10 5 K J 10° 10- 5
Fig. 9.24. Spectrum of cosmic background radiation, plotted
as vl v versus wavelength. Besides the CMB, background ra
diation exists in the radio domain (cosmic radio background.
CRB), in the infrared (CIB), in the optical/UV (CUVOB), in
the X-ray (CXB), and at gamma-ray energies (CGB). With the
exception of the CMB, probably all of these backgrounds can
be understood as a superposition of the emission from discrete
sources. Furthermore, this figure shows that the energj den
sity in the CMB exceeds that of other radiation components,
as was assumed when we considered the radiation density in
the Universe in Chap. 4
9. The Universe at High Redshift
we refer to the sum of the radiation of all galaxies and
AGNs per solid angle. The interpretation of such a back-
ground radiation depends on the sensitivity and the
angular resolution of the telescopes used. Imagine, for
instance, observing the sky with an optical camera that
has an angular resolution of only one arcminute. A rel-
atively isotropic radiation would then be visible at most
positions in the sky, featuring only some very bright or
very large sources. On improving the angular resolution,
more and more individual sources would become visi-
ble - culminating in the observations of the Ultra-Deep
Fields - and the background could then be identified
as the sum of the emission of individual sources. In
analogy to this thought-experiment, one may wonder
whether the CXB or the CIB can likewise be understood
as a superposition of radiation from discrete sources.
9.3.1 The IR Background
Observations of background radiation in the infrared are
very difficult to accomplish. First, it is problematic to
measure absolute fluxes due to the thermal emission of
the detector. In addition, the emission by interplanetary
dust (and by the interstellar medium in our Milky Way)
is much more intense than the infrared flux from extra-
galactic sources. For these reasons, the absolute level of
the infrared background has been determined only with
relatively large uncertainties, as displayed in Fig. 9.25.
The ISO satellite was able to resolve about 10% of
the CIB at k = 175 \im into discrete sources. Also in
the sub-mm range (at about 850 \im) almost all of the
CIB seems to originate from discrete sources which
consist mainly of dust-rich star-formation regions (see
Sect. 9.2.3, where the source population in the sub-mm
domain was discussed).
In any case, no indication has yet been found that the
origin of the CIB is different from the emission by a pop-
ulation of discrete sources, in particular of high-redshift
starburst galaxies. Further resolving the background ra-
diation into discrete sources will become possible by
future FIR satellites such as, for instance, Herschel.
9.3.2 The X-Ray Background
In the 1970s, the first X-ray satellites discovered not
only a number of extragalactic X-ray sources (such as
AGNs and clusters of galaxies), but also an apparently
isotropic radiation component, the CXB. Its spectrum
is a very hard (i.e., flat) power law, cut off at an energy
above ~ 40 keV, which can roughly be described by
I v oc E~ l
exp
(9.3)
with £0 ~ 40 keV. Initially, the origin of this radiation
was unknown, since its spectral shape was different
from the spectra of sources that were known at that time.
For example, it was not possible to obtain this spectrum
by a superposition of the spectra of know AGNs.
ROSAT, with its substantially improved angular reso-
lution compared to earlier satellites (such as the Einstein
observatory), conducted source counts at much lower
fluxes, based on some very deep images. From this, it
was shown that at least 80% of the CXB in the energy
range between 0.5 keV and 2keV is emitted by dis-
crete sources, of which the majority are AGNs. Hence
it is natural to assume that the total CXB at these low
X-ray energies originates from discrete sources, and
observations by XMM-Newton seem to confirm this.
However, the X-ray spectrum of normal AGNs is
different from (9.3), namely it is considerably steeper
(about S v oc v~° 7 ). Therefore, if these AGNs contribute
the major part of the CXB at low energies, the CXB
at higher energies cannot possibly be produced by
the same AGNs. Subtracting the spectral energy of
Fig. 9.25. Measurement of, and limits to, the CIB. Squares
denote lower limits derived from the integration of observed
source counts, diamonds are upper limits from flux n
ments, and other symbols show absolute flux n
The shaded yellow range indicates the current observational
limits to the CIB
9.3 Background Radiation at Smaller Wavelengths
the AGNs found by ROSAT from the CXB spectrum
(9.3), one obtains an even harder spectrum, resembling
very closely that of thermal bremsstrahlung. Therefore,
it was supposed for a long time that the CXB is, at
higher energies, produced by a hot intergalactic gas at
temperatures of k B T ~ 30 keV.
This model was excluded, however, by the precise
measurement of the thermal spectrum of the CMB by
COBE, showing that the CMB has a perfect blackbody
spectrum. If a postulated hot intergalactic gas were able
to produce the CXB, it would cause significant devia-
tions of the CMB from the Planck spectrum, namely by
the inverse Compton effect (the same effect that causes
the SZ effect in clusters of galaxies - see Sect. 6.3.4).
Thus, the COBE results clearly ruled out this possibility.
By now, the nature of the CXB at higher energies has
also essentially been determined (see Fig. 9.26), mainly
through very deep observations with the Chandra satel-
lite. An example of a very deep observation, the Chandra
Deep Field South, is shown in Fig. 9.27. From source
counts performed in such fields, about 75% of the CXB
in the energy range of 2keV < E < lOkeV could be
resolved into discrete sources. Again, most of these
sources are AGNs, but typically with a significantly
harder (i.e., flatter) spectrum than the AGNs that are
producing the low-energy CXB. Such a flat X-ray spec-
trum can be produced by photoelectric absorption of an
intrinsically steep power-law spectrum, where photons
closer to the ionization energy are more efficiently ab-
sorbed than those at higher energy. According to the
_!_'_3AX__
_ASCA_2_
S erg s _1 cnr 2 [2-1 OkeV]
Fig. 9.26. In the left panel, the total intensity of discrete
sources with an individual flux > 5 in the energy range
2keV < E < lOkeV is plotted (thick curve), together with
Ihc uncertainly range (between the two thin curves). Most
of the data are from a 3 x 10 5 s exposure of the Chandra
Deep Field. The dashed lines show different measurements
of the CXB flux in this energy range; depending on which
of these values is the correct one, between 60% and 90%
of the CXB in the Chandra Deep Field at this energy is
resolved into discrete sources. In the right panel, the hard-
ness ratio HR - specifying the ratio of photons in the energy
range 2 keV < E < 10 keV to those in 0.5 keV < E < 2 keV,
HR = (5 >2 keV - S<2 kev)/(5> 2 keV + S<2 kev) - is plotted as
a function of redshift, for 84 sources in the Chandra Deep
1
- SSCODOSDOOO
-
- "-x
-
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iogN " =23 o\ ° o
- o \
° o \
-
,logN H =22CD ° N ^
-^ O
o * J
: 0^-°.^
XRB(r=i.4) ~ e ~- :
-0.5
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8 °
- 1
©O Q3DXED0HD OO
:
Field with measured redshifls. This plot indicates that the HR
decreases with redshift; this trend is expected if the X-ray
spectrum of the AGNs is affected by intrinsic absorption. The
dashed curves show the expected value of HR for a source with
an intrinsic povs er law spectrum I v oc v -0 - 7 , which is observed
tin ough an abst >rbing layer with a hydrogen column density of
Nn, by which these curves are labeled. Since low-energy pho-
tons are more strong!} absorbed by the photoelectric effect
than high-energy ones, the absorption causes the spectrum to
become harder, thus flatter, at relatively low X-ray energies.
Tins implies an increase of the HR (also see the bottom panel
of Fig. 6. 16). This died is smaller for higher redshift sources.
since the photon energy at emission is then larget b\ a iactot
of(l+z)
the z > 5.7 QSOs in Fig. 9.28, and since an appreciable
fraction of homogeneously distributed neutral gas in the
intergalactic medium can be excluded for z < 5, from
the tight upper bounds on the strength of the Gunn-
Peterson effect (Sect. 8.5.1) the Universe must have
been reionized between the recombination epoch and
the redshift z ~ 6.5 of the most distant known sources.
From the WMAP results (see Sect. 8.7. 1) one concludes
that reionization must have taken place at very high
redshift, z — 15.
Fig. 9.27. The Chandra Deep Field South, a deep X-ray image
of a 16' x 16' field with an exposure time of 10 6 s - one of
the deepest X-ray images ever obtained. Most of the sources
\ isible in Ihis field are AGNs, but galaxies, groups, and clusters
are also detected. The photon energy is color-coded, from
lower to higher energies in red, yellow, and blue. One of the
sources in this field is a very distant QSO of Type 2. A radial
variation of the PSF in the field is visible by the increasing
size of individual sources towards the edges
classification scheme of AGNs discussed in Sect. 5.5,
these are Type 2 AGNs, thus Seyfert 2 galaxies and
QSOs with strong intrinsic self-absorption. We should
recall that Type 2 QSOs have only been detected by
Chandra - hence, it is no coincidence that the same
satellite has also been able to resolve the high-energy
CXB.
9.4 Reionization of the Universe
After recombination at z ~ 1100, the intergalactic gas
became neutral, with a residual ionization of only
~ 10~ 4 . Had the Universe remained neutral we would
not be able to receive any photons that were emitted
bluewards of the Lya line of a source, because the
absorption cross-section for Lya photons is too large
(see Eq. 8.16). Since such photons are observed from
QSOs, as can be seen for instance in the spectra of
Fig. 9.28. Spectra of five QSOs at redshifts z > 5.7, discovered
in multicolor data from the Sloan Digital Sky Survey. The
positions of the most important emission lines are marked.
Pai (ieularh remarkable is the complete lack of flux blueu ards
of the Lya emission line in some of the QSOs. indicating
a strong Gunn-Peterson effect. However, this absorption is
not complete in all QSOs, which points at strong variations in
the density of neutral hydrogen in the intergalactic medium
at these high redshifts. Either the hydrogen density varies
strongly for different lines of sight, or the degree of io
is very inhomogeneous
9.4 ReionizationoftheUniv
This raises the question of how this n
curred, in particular which process was responsible for
it. The latter question is easy to answer - reionization
must have happened by photoionization. Collisional
ionization can be ruled out because for it to be effi-
cient the IGM would need to be very hot, a scenario
which can be excluded due to the perfect Planck spec-
trum of the CMB - the argument here is the same as
above, where we excluded the idea of a hot IGM as the
source of the CXB. Hence, the next question is where
the energetic photons that caused the photoionization of
the IGM come from.
Two kinds of sources may account for them - hot
stars or AGNs. Currently, it is not unambiguously clear
which of these is the predominant source of energetic
photons causing reionization since our current under-
standing of the formation of supermassive black holes
is still insufficient. However, it is currently thought that
the main source of photoionization photons is the first
generation of hot stars.
9.4.1 The First Stars
Following on from the above arguments, understanding
reionization is thus directly linked to studying the first
generation of stars. In the present Universe star forma-
tion occurs in galaxies; thus, one needs to examine when
the first galaxies could have formed. From the theory of
structure formation, the mass spectrum of dark matter
halos at a given redshift can be computed by means of,
e.g., the Press-Schechter model (see Sect. 7.5.2). Two
conditions need to be fulfilled for stars to form in these
halos. First, gas needs to be able to fall into the dark
halos. Since the gas has a finite temperature, pressure
forces may impede the infall into the potential well.
Second, this gas also needs to be able to cool, condens-
ing into clouds in which stars can then be formed. We
will now examine these two conditions.
The Jeans Mass. By means of a simple argument, we
can estimate under which conditions pressure forces are
unable to prevent the infall of gas into a potential well.
To do this, we consider a slightly overdense spherical
region of radius R whose density is only a little larger
than the mean cosmic matter density 75. If this sphere
is homogeneously filled with baryons, the gravitational
binding energy of the gas is about
GMM b
|£gravl — ,
where M and Mb denote the total mass and the baryonic
mass of the sphere, respectively. The thermal energy of
the gas can be computed from the kinetic energy per
particle, multiplied by the number of particles in the
is the speed of sound in the gas, which is about the av-
erage speed of the gas particles, and [im p denotes the
average particle mass in the gas. For the gas to be bound
in the gravitational field, its gravitational binding energy
needs to be larger than its thermal energy, | E gmv \ > E t h,
which yields the condition GM > c^R. Since we have
assumed an only slightly overdense region, the relation
M ~ p R 3, between mass and radius of the sphere ap-
plies. From the two latter equations, the radius can be
eliminated, yielding the condition
/c>\" 2 1
Thus, as a result of our simple argument we find that
the mass of the halo needs to exceed a certain threshold
for gas to be able to fall in. A more accurate treatment
yields the condition
M>M > m — (i) vr (9 - 5)
In the final step we defined the Jeans mass Mj, which
describes the minimum mass of a halo required for the
gravitational infall of gas. The Jeans mass depends on
the temperature of the gas, expressed through the sound
speed c s , and on the mean cosmic matter density 75. The
latter can easily be expressed as a function of redshift,
75(z) = 75 (l+z) 3 .
The baryon temperature has a more complicated de-
pendence on redshift. For sufficiently high redshifts, the
small fraction of free electrons that remain after recom-
bination - the gas has a degree of ionization of ~ 1 0~ 4 -
9. The Universe at High Redshift
provide a thermal coupling of the baryons to the cosmic
background radiation, by means of Compton scattering.
This is the case for redshifts z > z t > where
V 0.022/ '
hence, T b (z) « T(z) = 7b (1 + z) for z > z t . For smaller
redshifts, the density of photons gets too small to main-
tain this coupling, and baryons start to adiabatically cool
down by the expansion, so that for z<Zi we obtain
approximately 7b oc pj oc (1+z) 2 .
From this temperature dependence, the Jeans mass
can then be calculated as a function of redshift. For
Zt ^ z ^ 1000, M] is independent of z because c s oc
T 1/2 cx (l+z) 1/2 andpoc (1+z) 3 , and its value is
whereas for i
(l + z) 2 K,
V o.i5 y "'
t we obtain, with 7b — 1.7 x 10~ 2
«— >m
/Q b h 2 \
10.022/
Cooling of the Gas. The Jeans criterion is a necessary
condition for the formation of proto-galaxies. In order
to form stars, the gas in the halos needs to be able to
cool further. Here, we are dealing with the particular sit-
uation of the first galaxies, whose gas is metal-free, so
metal lines cannot contribute to the cooling. This means
that cooling can only happen via hydrogen and helium.
Since the first excited state of hydrogen has a high en-
ergy (that of the Lya transition, thus E ~ 10.2 eV), this
cooling is efficient only above T > 2 x 10 4 K. However,
the halos which form at high redshift have low mass,
so that their virial temperature is considerably below
this energy scale. Therefore, atomic hydrogen is a very
inefficient coolant for these first halos, insufficient to
initiate the formation of stars. Furthermore, helium is
of no help in this context, since its excitation tempera-
ture is even higher than that of hydrogen. The problem
resulting from these arguments has become even worse
given the WMAP discovery of a higher i
redshift than previously estimated.
Only in recent years has it been discovered that
molecular hydrogen represents an extremely important
component in cooling processes. Despite its very small
transition probability, Ilj dominates the cooling rate of
primordial gas at temperatures below T ~ 10 4 K - see
Fig. 9.29 - where the precise value of this temperature
depends on the abundance of H2.
By means of H2, the gas can cool in halos with a tem-
perature exceeding about T yn > 3000 K, corresponding
mass of M > 10 4 M o ; the exact values depend on
he redshift. In these halos, stars may then be able to
However, these stars will certainly be different
rom those known to us, because they do not con-
;ain any metals. Therefore, the opacity of the stellar
plasma is much lower. Such stars, which at the same
mass presumably have a much higher temperature and
luminosity (and thus a shorter lifetime), are called Pop-
ulation III stars. Due to their high temperature they are
much more efficient sources of ionizing photons than
stars with "normal" metallicity.
T[K]
Fig. 9.29. Cooling rate as a function of the temperature for
a gas consisting of atomic and molecular hydrogen (with
0.19S abundance) and of helium. The solid curve describes
the cooling by atomic gas, the dashed curve that by molcculai
hydrogen; thus, the latter is extremely important at tempera-
tures below ~ 10 4 K. At considerably low er tempera! tires the
gas cannot cool, hence no star formation will take place
9.4 ReionizationoftheUniv
9.4.2 The Reionization Process
Dissociation of Molecular Hydrogen. The energetic
photons from these Population III stars are now capable
of ionizing hydrogen in their vicinity. More important
still is another effect: the binding energy of H2 is only
1 1 .26 eV. Since the Universe is transparent for photons
with energies below 13.6 eV, photons with 11.26 eV <
E Y < 13.6 eV can propagate very long distances and
dissociate molecular hydrogen. This means that as soon
as the first stars have formed in a region of the Universe,
molecular hydrogen in their vicinities will be destroyed
and further star formation will then be prevented. 2
Metal Enrichment of the Intergalactic Medium.
Soon after Population III stars have formed, they will
explode as supernovae. Through this process, the met-
als produced by them are ejected into the intergalactic
medium, by which the initial metal enrichment of the
IGM occurs. The kinetic energy transferred by SNe to
the gas within the halo can exceed its binding energy,
so that the baryons of the halo can be blown away and
further star formation is prevented. Whether this effect
may indeed lead to gas-free halos, or whether the re-
leased energy can instead be radiated away, depends
on the geometry of the star- formation regions. In any
case, it can be assumed that in those halos where the
first generation of stars was born, further star forma-
tion was considerably suppressed, particularly since all
molecular hydrogen was destroyed.
We can assume that the metals produced in these first
SN explosions are, at least partially, ejected from the
halos into the intergalactic medium, thus enriching the
latter. The existence of metal formation in the very early
Universe is concluded from the fact that even sources at
very high redshift (like QSOs at z ~ 6) have a metallicity
of about one tenth the Solar value. Furthermore, the Lya
forest also contains gas with non-vanishing metallicity.
Since the Lya forest is produced by the intergalactic
medium, this therefore must have been enriched.
The Final Step to Reionization. For gas to cool in ha-
los without molecular hydrogen, their virial temperature
needs to exceed about 10 4 K (see Fig. 9.29). Halos of
ie needs less than 1 % of the
this mass form with appreciable abundance at redshifts
of z ~ 10, as follows, e.g., from the Press-Schechter
model (see Sect. 7.5.2). In these halos, efficient star for-
mation can then take place; the first proto-galaxies will
form. These will then ionize the surrounding IGM in
the form of Hll regions, as sketched in Fig. 9.30. The
corresponding Hll regions will expand because increas-
ingly more photons are produced. If the halo density is
sufficiently high, these Hll regions will start to overlap.
Once this occurs, the IGM is ionized, and reionization
is completed.
We therefore conclude that reionization is a two-
stage process. In a first phase, Population III stars form
through cooling of gas by molecular hydrogen, which
is then destroyed by these very stars. Only in a later
epoch and in more massive halos is cooling provided by
atomic hydrogen, then leading to reionization.
A Luminous J-band Drop-Out? The aforementioned
fact that, even at redshifts as large as z ~ 6, massive
galaxies with a fairly old stellar population were already
in place shows that there was an epoch of intense star
formation at even earlier times. This clearly suggests
that these galaxies must have played an important role
in the reionization process of the Universe. In fact, in
the HUDF an object was found that appears to be a J-
band drop-out, with no radiation seen at wavelengths
shorter than the J-band. Spectroscopy of this source
revealed the presence of a strong 4000-A break, not only
giving further support to its high redshift of z ~ 6.5, but
also indicating that the source contains a post-starburst
stellar population. The bolometric luminosity of this
source is estimated to be ~ 10 12 L Q , and using a Salpeter
initial mass function (see Eq. 3.67), a stellar mass of
~6x 10 n M Q is obtained. The strong 4000- Angstrom
break indicates that the spectral energy distribution is
dominated by A0 stars of masses < 3M Q . This provides
a clear indication of an old age of the population of
> 300 Myr, implying that the stars must have formed at
redshifts z > 9, but possibly at even higher redshift.
We can estimate the comoving volume that this gal-
axy was able to reionize, based on its high luminosity.
With all uncertainties entering such an estimate (such as
the escape fraction of ionizing photons from the galaxy),
it is concluded that this galaxy could ionize a volume
of ~ 10 5 Mpc 3 . This needs to be compared to the high-
redshift volume within which such a source would have
9. The Universe at High Redshift
Neutral Hydrogen •
•
•
z~30
* First stars form
* H 2 dissociates
— -2
i-4
• •
•
z~15
* Stars form in more
massive halos
-6
• W HII
z~10
* Hll-regions overlap
* UV intensity rises
-2
• T^-IO'K
# T VJ >10 4 K
-3
Fig. 9.30. Left: a sketch of the geometry of r
shown: initially, relatively low-mass halos collapse, a first
generation of stars ionizes and heats the gas in these halos.
By heating, the temperature increases so strongly (to about
T ~ 10 4 K) that gas can escape from the potential wells; these
halos may never again form stars efficiently. Only when more
massive halos have collapsed will continuous star formation
set in. Ionizing photons from this first generation of hot stars
produce HII regions around their halos, which is the onset of
~ ; regions in which hydrogen is ionized will
grow until they start to overlap; at that time, the flux of ionizing
photons will strongly increase. Right: the average spectrum of
photons at the beginning of the reionization epoch is shown;
here, it has been assumed that the flux from the radiation
source follows a power law (dashed curve). Photons with an
energy higher than that of the Lya transition are strongly sup-
pressed because they are efficiently absorbed. The spectrum
near the Lyman limit shows features which are produced by
the combination of breaks corresponding to the various Lyman
lines, and the redshifting of the photons
been detected in the HUDF. The result is that these two
volumes are quite comparable. Hence, it seems that this
galaxy was capable of reionizing "its" volume of the
Universe. Again we should warn that this is a prelimi-
nary conclusion, based on a single object; nevertheless,
it indicates that we might be seeing direct evidence for
early reionization, in accordance with the results from
WMAP.
Prospects for Observing Reionization Directly. We
note that only a small fraction of the baryons needs
to burn in hot stars to ionize all hydrogen, as we can
easily estimate: by fusing four H-nuclei (protons) to
He, an energy of about 7 MeV per nucleon is released.
However, only 13.6 eV per hydrogen atom is required
for ionization.
Furthermore, we point out again that the very dense
Lya forest seen towards QSOs at high redshift is
no unambiguous sign for approaching the redshift of
reionization, because a very small fraction of neu-
tral atoms (about 1%) is already sufficient to produce
a large optical depth for Lya photons. Direct observa-
tion of reionization will probably be quite difficult; an
illustration of this is sketched in Fig. 9.31.
We have confined our discussion to the ionization
of hydrogen, and disregarded helium. The ionization
energy of helium is higher than that of hydrogen, so
that its ionization will be completed later. From the
statistical analysis of the Lya forest and from the anal-
ysis of helium absorption lines in high-redshift QSOs,
a reionization redshift of z ~ 3 .2 for helium is obtained.
With the upcoming Next Generation Space Telescope
(which has more recently been named the James Webb
9.5 The Cosmic Star-Formation History
Fig. 9.31. Sketch of a potential observa-
tion of reionization: light from a very
distant QSO propagates through a par-
tialis ionized Universe; at locations where
it passes through HII regions, radiation
will get through - flux will be visible at
the corresponding wavelengths. When the
Hit regions start to overlap, the normal
Lya forest will be produced
Space Telescope, JWST), one hopes to discover the first
light sources in the Universe; this space telescope, with
a diameter of 6.5 m, will be optimized for operation at
wavelengths between 1 and 5 (im.
9.5 The Cosmic Star-Formation History
The scenario for reionization as described above should,
at least for the main part, be close to reality, with its
details still being subject to intense discussion. In partic-
ular, the star- formation rate of the Universe as a function
of redshift can be computed only by making relatively
strong model assumptions, because too many physi-
cal processes are involved; we will elaborate on this
n the next section. However, observations of galax-
at very high redshifts have also been accomplished
recent years, through which it has become possible
to empirically trace the star-formation rate up to large
ivdshifts.
9.5.1 Indicators of Star Formation
We define the star-formation rate (SFR) as the mass of
the stars that are formed per year, typically given in
units of M Q /yr. For our Milky Way, we find a SFR of
~ 3M Q /yr. Since the signatures for star formation are
obtained only from massive stars, their formation rate
needs to be extrapolated to lower masses to obtain the
full SFR, by assuming an IMF (initial mass function;
see Sect. 3.9.4). Typically, a Salpeter-IMF is chosen be-
tween 0.1 M Q < M < 100M Q . We will start by listing
the most important indicators of star formation:
• Emission in the far infrared (FIR). This is radiation
emitted by warm dust which is heated by hot young
stars. For the relation of FIR luminosity to the SFR,
observation yields the approximate relation
SFRfir ^ Lfir
M /yr ~ 5.8 x 10 9 L o '
» Radio emission by galaxies. A very tight correlation
exists between the radio luminosity of galaxies and
their luminosity in the FIR, over many orders of mag-
nitude of the corresponding luminosities. Since L FT r
is a good indicator of the star-formation rate, this
should apply for radiation in the radio as well (where
we need to disregard the radio emission from a poten-
tial AGN component). The radio emission of normal
galaxies originates mainly from supernova remnants
(SNRs). Since SNRs appear shortly after the be-
ginning of star formation, caused by core-collapse
supernovae at the end of the life of massive stars in
a stellar population, radiation from SNRs is a nearly
instantaneous indicator of the SFR. Once again from
observations, one obtains
SFRi.4GHz ^ L 1.4GHz
M /yr 8.4 x 10 27 ergs" 1 Hz" 1 '
• Ha emission. This line emission comes mainly from
the Hll regions that form around young hot stars. As
an estimate of the SFR, one uses
SFR Hg _ L Ha
M /yr 1.3 x 10 41 ergs _1 '
• UV radiation. This is only emitted by hot young stars,
thus indicating the SFR in the most recent past, with
SFR UV ^uv
M /yr ~ 7.2 x 10 27 ergs" 1 Hz" 1 '
9. The Universe at High Redshift
Applied to individual galaxies, each of these es-
timates is quite uncertain, which can be seen by
comparing the resulting estimates from the various
methods (see Fig. 9.32). For instance, Ha and UV pho-
tons are readily absorbed by dust in the interstellar
medium of the galaxy or in the star-formation regions
themselves. Therefore, the relations above should be
corrected for this self-absorption, which is possible
when the redding can be obtained from multicolor data.
It is also expected that the larger the dust absorption,
the stronger the FIR luminosity will be, causing devi-
ations from the linear relation SFR FTR oc SFRuv- After
the appropriate corrections, the values for the SFR de-
rived from the various indicators are quite similar, but
still have a relatively large scatter.
There are also a number of other indicators of star
formation. The fine-structure line of singly ionized car-
bon at X = 157.7 (im is of particular importance as it is
one of the brightest emission lines in galaxies, which
can account for a fraction of a percent of their total lu-
minosity. The emission is produced in regions which are
subject to UV radiation from hot stars, and thus associ-
ated with star-formation activity. Due to its wavelength,
this line is difficult to observe and has, until recently,
been detected only in star-forming regions in our Gal-
axy and in other local galaxies. However, recently this
§
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Fig, 9,32, Correlations of the slar formation rales in a sample
of galaxies, as derived from observation in different wave-
bands. In all four diagrams, the dashed line marks ihc identity
relation SFR, = SFR2; as is clearly seen, using the Ha lumi-
nosity and UV radiation as star-formation indicators seems
SFR 1.4GHz ( M V r1 )
to underestimate the SFR. Since radiation may be absorbed
by dust at these wavelengths, and also since the amount of
warm dust probably depends on the SFR itself, this effect
can be corrected for, as shown by the solid curves in the four
9.5 The Cosmic Star-Formation History
line was detected from the most distant QSO known, at
z = 6.42, where it is redshifted into the submillimeter
part of the spectrum. This not only suggests that the host
galaxy of the QSO undergoes an intense burst of star
formation, but also that the material in this host galaxy
is already significantly enriched with metals.
9.5.2 Redshift Dependence of the Star Formation:
The Madau Diagram
The density of star formation, Psfr, is defined as the
mass of newly formed stars per year per unit (comov-
ing) volume, typically measured in M Q yr _1 Mpc~ 3 .
Therefore, psfr as a function of redshift specifies how
many stars have formed at any time. By means of the
star-formation density we can examine the question, for
i 1 1 sta nee, of whether the formation of stars began only at
relatively low redshifts, or whether the conditions in the
early Universe were such that stars formed efficiently
even at very early times.
Investigations of the SFR in galaxies, by means of
the above indicators, and source counts of such star-
forming galaxies, allow us to determine psfr- The plot
of these results (Fig. 9.33) is called a "Madau diagram".
In about 1996, Piero Madau and his colleagues accom-
plished, for the first time, a determination of the SFR at
-U.b
I , I
I , | ■
It '-±-
-1.0
-1.5
UV
Ha
-2.0
□
IR
Radio
-?5
high redshifts from Lyman-break galaxies, where the in-
trinsic extinction was neglected in these first estimates.
Correcting for this extinction (for which the progress
in submillimeter astronomy has been extremely impor-
tant, as we saw in Sect. 9.2.3), a nearly constant psfr is
found for z > 1 , together with a decline by about a factor
of 10 from z ~ 1 to the present time. These results have
more recently been confirmed by investigations with
the Spitzer satellite, observing a large sample of galax-
ies at FIR wavelengths. Whereas the star- formation rate
density at low redshifts is dominated by galaxies which
are not very prominent at FIR wavelength, this changes
drastically for redshifts z > 0.7, above which most of
the star-formation activity is hidden from the optical
view by dust. From this we conclude that most stars in
our neighborhood were already formed at high redshift:
star formation at earlier epochs was considerably more
active than it is today.
Although the redshift-integrated star-formation rate
and the mass density of stars determined from galaxy
surveys, as displayed in Fig. 9.34, slightly deviate from
each other, the degree of agreement is quite satisfac-
tory if one recalls the assumptions that are involved
in the determination of the two quantities: besides (he
uncertainties discussed above in the determination of
the star-formation rate, we need to mention in particu-
lar the shape of the IMF of the newly formed stars for
the determination of the stellar mass density. In fact,
Fig. 9.34 shows that we have observed the formation of
essentially the complete current stellar density.
Fig. 9.33. The comoving star-formation density psfr as
a function of redshift, where the different symbols denote
different indicators used for the determination of the star-
fonnalion rate. This plot, known as the "Madau diagram".
shows the history of star formation in the Universe. Clearly
visible is the decline for z < 1; towards highci redshifts.
/>' seems to remain nearh constant. Tire curve is an empirical
fit to the data
Fig. 9.34. Redshift evolution of the
as measured from vari
specifies the integrated
density in stars,
galaxy surveys. The solid curve
■formation density from Fig. 9.33
9. The Universe at High Redshift
A simple argument supports the idea that a signifi-
cant fraction of cosmic star formation occurs in sources
which are hidden from our view. From Fig. 9.25 we
conclude that the energy density of the FIR background
radiation is of the same order as that of the optical/UV
background. The latter originates from star-forming re-
gions, from which energetic optical/UV photons can
escape and which can be observed as starbursts. In con-
trast to that, the CIB comes from the dusty regions in
sub-mm sources, heated by hot and thus newly formed
stars. The comparable energy density in these two ra-
diation fields then indicates that both modes of star
formation - with and without strong dust obscuration -
are about equally abundant. In fact, more than half of
cosmic star formation seems to hide in such dust regions.
The relative proportion of star formation in dusty re-
gions seems to be a strong function of redshift. Whereas
in the local Universe very luminous infrared galaxies are
rare, their abundance increases rapidly with redshift, at
least out to z ~ 1 . The total comoving infrared luminos-
ity density evolves as oc (1 + z) 4 out to z ~ 1, whereas
the ultraviolet luminosity density increases more slowly,
oc(l+z) 25 . Together, this then indicates that dusty
and hidden star formation was even more important
at higher redshifts than it is today, and it dominates the
star-formation activity beyond z ~ 0.7.
Whereas most of the star formation in the local
Universe occurs in spiral and irregular galaxies at
a modest rate (so-called quiescent star formation), the
star-formation activity at higher redshifts was domi-
nated by bursts of star formation, as evidenced in the
sub-mm galaxies and in LBGs. At a redshift z ~ 1, the
latter has apparently ceased to dominate, yielding the
strong decline of the star-formation rate density from
then until today. This behavior may be expected if bursts
of star formation are associated with the merging of gal-
axies; the merger rate declines strongly with time in
models of the Universe dominated by a cosmological
constant. This transition may also be responsible for the
onset of the Hubble sequence of galaxy morphologies
after z~ 1.
The derivation of the star-formation rate as a func-
tion of redshift is largely drawn from galaxy surveys
which are based on color selection, such as LBGs, EROs
and sub-mm galaxies. The possibility cannot be ex-
cluded that additional populations of galaxies which are
luminous but do not satisfy any of these photometric se-
lection criteria are present at high redshift. Such galaxies
can be searched for by spectroscopic surveys, extend-
ing to very faint magnitude limits. This opportunity now
arises as several of the 10-m class telescopes are now
equipped with high multiplex spectrographs which can
thus take spectra of many objects at the same time. One
of them is VIMOS at the VLT, another is DEIMOS on
Keck. With both instruments, extensive spectroscopic
surveys are being carried out on flux-limited samples
of galaxies. Among the first results of these surveys is
the finding that there are indeed more bright galaxies
at redshift z ~ 3 than previously found, by about a fac-
tor of 2, leading to a corresponding correction of the
star- formation rate at high redshifts. In a color-color
diagram, these galaxies are preferentially located just
outside the selection box for LBGs (see Fig. 9.4). Given
that this selection box was chosen such as to yield a high
reliability of the selected candidates, it is not very sur-
prising that a non-negligible fraction of galaxies lying
outside, but near to it are galaxies at high redshift with
similar properties.
9.6 Galaxy Formation
and Evolution
The extensive results from observations of galaxies at
high redshift which were presented earlier might sug-
gest that the formation and evolution of galaxies is quite
well understood today. We are able to examine galaxies
at redshifts up to z ~ 6 and therefore observe galaxies at
nearly all epochs of cosmic evolution. This seems to im-
ply that we can study the evolution of galaxies directly.
However, this is true only to a certain degree. Although
we have now found a large number of galaxies at nearly
every redshift, the relation between galaxies at different
redshifts is not easily understood. We cannot suppose
that galaxies seen at different redshifts represent vari-
ous subsequent stages of evolution of the same kind of
galaxy. The main reason for this difficulty is that differ-
ent selection criteria need to be applied to find galaxies
at different redshifts.
We shall explain this point with an example. Ac-
tively star-forming galaxies with z > 2.5 are efficiently
detected by applying the Lyman-break criterion, but
only those which do not experience much reddening by
9.6 Galaxy Formation and Evolution
dust. Actively star-forming galaxies at z ~ 1 are discov-
ered as EROs if they are sufficiently reddened by dust.
The relation between these two galaxy populations de-
pends, of course, on how large the fraction of galaxies is
whose star-formation regions are enshrouded by dense
dust. To determine this fraction, one would need to find
Ly man-break galaxies at z ~ 1, or EROs at z ~ 3. Both
observations are virtually impossible today, however.
For the former this is because the Lyman break is then
located in the UV domain of the spectrum and we have
no sufficiently sensitive UV observatory available. For
the latter it is because the rest wavelength correspond-
ing to the observed R-band is so small that virtually
no optical radiation from such objects would be visi-
ble, rendering spectroscopy of these objects impossible.
In addition to this, there is the problem that galaxies
with 1.3 < z < 2.5 are difficult to discover because, for
objects at those redshifts, hardly any spectroscopic indi-
cators are visible in the optical range of the spectrum -
both the 4000- A break and the A = 3727 A line of [Oil]
are redshifted into the NIR, as are the Balmer lines of
hydrogen, and the Lyman lines of hydrogen are located
in the UV part of the spectrum. For these reasons, this
range in redshift is also called the "redshift desert". 3
Thus, it is difficult to trace the individual galaxy pop-
ulations as they evolve into each other at the different
redshifts. Do the LBGs at z ~ 3 possibly represent an
curlv stage of today's ellipticals (and the passive EROs
at z ~ 1), or are they an early stage of spiral galaxies?
The difficulties just mentioned are the reasons why
our understanding of the evolution of the galaxy popu-
lation is only possible within the framework of models,
with the help of which the different observational facts
are being interpreted. We will discuss some aspects of
such models in this section.
Another challenge for galaxy evolution models are
the observed scaling relations of galaxy properties. We
expect that a successful theory of galaxy evolution can
predict the Tully-Fisher relation for spiral galaxies, the
fundamental plane for ellipticals, as well as the tight
correlation between galaxy properties and the central
black hole mass. This latter point also implies that the
evolution of AGNs and galaxies must be considered in
'Spectroscopy in [he .NIR is possible in principle, but the high level
of night sky brightness and. in particular, the large number ol atmo
spheric transition lines renders spectroscopic observations in the .NIK
much more time con aiming than optical spectroscopy.
parallel, since the growth of black hole mass with time
is expected to occur via accretion, i.e., during phases
of activity in the corresponding galaxies. The hierarchi-
cal model of structure formation implies that high-mass
galaxies form by the merging of smaller ones; if the
aforementioned scaling relations apply at high redshifts
(and there are indications for this to be true, although
with redshift-dependent pre-factors that reflect the evo-
lution of the stellar population in galaxies), then the
merging process must preserve the scaling laws, at least
on average.
9.6.1 Expectations from Structure Formation
Cosmological V-body simulations predict the evolution
of the dark matter distribution as a function of redshift,
in particular the formation of halos and their merger pro-
cesses. At the beginning of the evolution, gas follows
the dark matter. However, as soon as the gas becomes
dense enough, physical effects like heating by dissi-
pation, friction, and cooling start to play a prominent
role. Since dark matter is not susceptible to these pro-
cesses, the behavior and the spatial distribution of the
two components begins to differ.
In a CDM model, halos of lower mass form first; only
later can more massive halos form. This "bottom-up"
scenario of structure formation follows from the shape
of the power spectrum of density fluctuations, which
itself is defined by the nature of dark matter - namely
cold dark matter. The formation of halos of increasingly
higher mass then happens by the merging of lower-mass
halos. Such merging processes are directly observable;
the Antennae (see Fig. 9.15) are only one very promi-
nent example. Merging should be particularly frequent
in regions where the galaxy density is high, in clusters
of galaxies for instance. As shown in Fig. 6.45 for one
cluster, a large number of such merging processes are
detected in galaxy clusters at high redshift.
If the gas in a halo can cool efficiently, stars may
form, as we previously discussed in the context of reion-
ization. Since cooling is a two-body process, i.e., the
cooling rate per volume element is on p 2 , only dense
gas can cool efficiently. One expects that the gas, hav-
ing a finite amount of angular momentum like the dark
matter halo itself, will initially accumulate in a disk, as
a consequence of its own dissipation. The gas in the disk
9. The Universe at High Redshift
then reaches densities at which efficient star formation
can set in. In this way, the formation of disk galaxies,
thus of spirals, can be understood qualitatively.
However, the formation of disk galaxies by dissipa-
tional collapse of gas inside relaxed dark matter halos is
not without problems. The hierarchical nature of struc-
ture growth implies that due to subsequent merging
events, the disks can be significantly perturbed or even
destroyed. Furthermore, the disks can lose angular mo-
mentum in the course of galaxy collisions. It is likely
that an understanding of the formation of disk galaxies
requires additional ingredients; for example, disks may
form as a result of gas-rich mergers, where the resulting
angular momentum of the baryons is sufficient to form
a rotating and flat structure through dissipation.
9.6.2 Formation of Elliptical Galaxies
Properties of Ellipticals. Whereas the formation of
disk galaxies can be explained qualitatively in a rela-
tively straightforward way, the question of the formation
of ellipticals is considerably more difficult to answer.
Stars in ellipticals feature a very high velocity disper-
sion, indicating that the gas out of which they have
formed cannot have kinematically cooled down before-
hand into a disk by dissipation. On the other hand, it
is hard to comprehend how star formation may pro-
ceed without gas compression induced by dissipation
and cooling.
In Sect. 3.4.3 we saw that the properties of ellipticals
are very well described by the fundamental plane. It is
also found that the evolution of the fundamental plane
with redshift can almost completely be explained by
passive evolution of the stellar population in ellipticals.
In the same way, we stated in Sect. 6.6 that the ellipticals
in a cluster follow a very well-defined color-magnitude
relation (the red cluster sequence), which suggests that
the stellar populations of ellipticals at a given redshift
all have a similar age. By comparing the colors of stellar
populations in ellipticals with models of population syn-
thesis, an old age for the stars in ellipticals is obtained,
as shown in Fig. 3.49.
Monolithic Collapse. A simple model is capable of co-
herently describing these observational facts, namely
monolithic collapse. According to this description,
the gas in a halo is nearly instantaneously trans-
formed into stars. In this process, most of the gas
is consumed, so that no further generations of stars
can form later. For all ellipticals with the same red-
shift to have nearly identical colors, this formation
must have taken place at relatively high redshift,
z > 2, so that the ellipticals are all of essentially
the same age. This scenario thus requires the for-
mation of stars to happen quickly enough, before
the gas can accumulate in a disk. The process of
star formation remains unexplained in this picture,
however.
Minor Mergers. We rather expect, according to the
model of hierarchical structure formation, that massive
galaxies form by the mergers of smaller entities. Con-
sider what may happen in the merging of two halos or
two galaxies, respectively. Obviously, the outcome of
a merger depends on several parameters, like the relative
velocity, the impact parameter, the angular momenta,
and particularly the mass ratio of the two merging ha-
los. If a smaller galaxy merges with a massive one,
the properties of the dominating galaxy are expected
to change only marginally: the dark halo gains slightly
more mass from the companion, the stars of which are
simply added to the stellar population of the massive
galaxy. Such a "minor merger" is currently taking place
in the Milky Way, where the Sagittarius dwarf galaxy
is being torn apart by the tidal field of the Galaxy, and
its stars are being incorporated into the Milky Way as
an additional population. This population has, by itself,
a relatively small velocity dispersion, forming a cold
stream of stars that can also be identified as such by
its kinematic properties. The large-scale structure of
the Galaxy is nearly unaffected by a minor merger like
this.
Major Mergers and Morphological Transformations
of Galaxies. The situation is different in a merger pro-
cess where both partners have a comparable mass. In
such "major mergers" the galaxies will change com-
pletely. The disks will be destroyed, i.e., the disk
population attains a high velocity dispersion and can
transform into a spheroidal component. Furthermore,
the gas orbits are perturbed, which may trigger mas-
sive starbursts like, e.g., in the Antenna galaxies. By
means of this perturbation of gas orbits, the SMBH in
9.6 Galaxy Formation and Evolution
Fig. 9.35. The galaxy Centaurus A. The optical image i
played in grayscales, the contours show the radio <
and in red, an infrared image is presented, taken by the ISO
satellite. The ISO map indicates the distribution of dust, which
is apparently that of a barred spiral. It seems thai this elliptical
galaxy features a spiral that is stabilized by the gravitational
field of the elliptical. Presumably, this galaxy was formed in
a merger process; this may also be the reason for the AGN
activity
the centers of the galaxies can be fed, initiating AGN
activity, as is presumably seen in the galaxy Centau-
rus A shown in Fig. 9.35. Due to the violence of the
interaction, part of the matter is ejected from the gal-
axies. These stars and the respective gas are observable
as tidal tails in optical images or by the 21 -cm emis-
sion of neutral hydrogen. From these arguments, which
are also confirmed by numerical simulations, one ex-
pects that in a "major merger" an elliptical galaxy may
form. In the violent interaction, the gas is either ejected,
or heated so strongly that any further star formation is
suppressed.
This scenario for the formation of ellipticals is ex-
pected from models of structure formation. Thus far
it has been quite successful. For instance, it provides
a straightforward explanation for the Butcher-Oemler
effect (see Sect. 6.6), which states that clusters of gal-
axies at higher redshift contain a larger fraction of blue
galaxies. Because of the particularly frequent mergers
in clusters, due to the high galaxy density, such blue
galaxies are transformed more and more into early-
type galaxies. However, galaxies in clusters may also
lose their gas in their motion through the hot inter-
galactic medium, by which the gas is ripped out due
to the so-called ram pressure. The fact that the frac-
tion of ellipticals in a cluster remains rather constant
as a function of redshift, whereas the abundance of
SO galaxies increases with decreasing z, indicates the
importance of the latter process as an explanation of
the Butcher-Oemler effect. In this case, the gas of the
disk is stripped, no further star formation takes place,
and the spiral galaxy is changed into a disk galaxy
without any current star formation - hence, into a gal-
axy that features the basic properties of SO galaxies
(see Fig. 9.36). On the other hand, we have seen in
Sect. 3.2.5 that many ellipticals show signs of complex
evolution which can be interpreted as the consequence
of such mergers. Therefore, it is quite possible that the
formation of ellipticals in galaxy groups happens by vi-
olent merger processes, and that these then contribute
to the cluster populations by the merging of groups into
clusters.
This model also has its problems, though. One of
these is that merger processes of galaxies are also ob-
served to occur at lower redshifts. Ellipticals formed
in these mergers would be relatively young, which is
hardly compatible with the above finding of a consis-
tently old age of ellipticals. However, ellipticals are
predominantly located in galaxy clusters whose mem-
bers are already galaxies with a low gas content. In the
merging process of such galaxies, the outcome will be
an elliptical, but no starburst will be induced by merging
because of the lack of gas - such mergers are sometimes
called "dry mergers". In this context, we need to men-
tion that the phrase "age of ellipticals" refers to the age
of their stellar populations - the stars in the ellipticals
are old, but not necessarily the galaxies themselves.
The importance of dry mergers was recognized more
recently for a number of reasons. First, wide-field imag-
ing with HST, using mosaics of single fields, have shown
a large number of pairs of spheroidal galaxies at z < 0.7
which show signs of interactions. A dramatic example
of this is also seen in Fig. 6.45, where several gravi-
tationally bound pairs of early-type galaxies are seen
in the outskirts of a cluster at z — 0.83. These pairs
will merge on a time-scale of < 1 Gyr. Second, numer-
Fig. 9.36. An HST image of NGC 4650A, a polar-ring galaxj .
Spectroscopy shows that the inner disk-like part of the galaxy
rotates around its minor axis. This part of the galaxy is sur-
rounded by a ring of stars and gas which is intersected by the
polar axis of the disk and which rotates as well. Hence, the
inner disk and the polar ring have angular momentum vectors
that are pretty much perpendicular to each other; such a con-
figuration cannot form from the "collapse" of the baryons in
a dark matter halo. Instead, the most probable explanation for
the formation of such special galaxies is a huge collision of
l\\o galaxies in ihc past. Originall} the disk ma) ha\e been [he
disk of the more massive of the two collision partners, w hereas
the less massive galaxy has been torn apart and its material
has been forced into a polar orbit around the more massive
galaxy. New stars have then formed in the disk, visible here
in the bluish knots of bright emission. Since the polar ring is
deep inside the halo of the other galaxy, the halo mass distri-
buiion can be mapped out to large radii using the kinematic;
of the ring
ical simulations indicate that gas-free mergers preserve
the fundamental plane, in the sense that the merging
of two ellipticals that live on the fundamental plane
will lead to merger remnant that lies on there as well.
Third, dry mergers may provide the explanation for the
structural differences between high-luminosity and low-
luminosity ellipticals. The former ones tend to have
boxy isophotes and very little rotation, whereas the lat-
ter tend to have more disky isophotes and substantial
rotational support (see Fig. 3.9). In this model, the disky
ellipticals form from mergers of gas-rich disk galaxies;
the merger remnants will then contain a population of
young stars that were formed in the process of merging,
and depending on the relative orientation of the angu-
lar momentum vectors of the two galaxies, the merged
galaxy may have a substantial contribution of rotational
support. In contrast, no new stars are formed in the
merger of two early-type galaxies, preserving the age
of the stellar population. Since early-type galaxies are
on average more massive that late-type ones, the rem-
nants of dry mergers are expected to be more massive
on average than those of gas-rich mergers. 4
Redshift Evolution of Ellipticals. The luminosity func-
tion of luminous red galaxies up to z ~ 1 is not much
different from their local luminosity function, whereas
that of blue galaxies shows a very strong evolution. We
deduce from this that the formation of ellipticals is al-
ready concluded at a very early time. As also shown
by the Madau diagram (Fig. 9.33), most of the star
formation takes place at high redshifts, whereas the
local Universe is rather quiet in comparison. In a Uni-
verse with low density, the evolution of the growth
factor D + (z) over time (see Fig. 7.3) also implies that
cosmic evolution will slow down considerably after
z ~ 1, hence the majority of mergers happen at higher
redshifts. In fact, the fraction of irregular and peculiar
galaxies increases with decreasing magnitude, hence
with increasing mean redshift.
Indeed, the evolution of massive ellipticals seems
to have occurred even earlier than z ~ 1. Until recently,
the radio galaxy presented in Fig. 6.44 was the most dis-
tant known elliptical galaxy, with a redshift of z = 1 .55.
In the K20 survey (see Sect. 9.2.2), four EROs were
recently discovered with redshifts 1.6 < z < 1.9. The
spectroscopic verification of these objects is extremely
challenging because they are very faint at optical wave-
lengths (R < 24), and furthermore, at these redshifts
no distinct spectral signatures are visible in the optical.
The redshifts of these four EROs were determined by
a correlation of their spectra with the spectrum of the
aforementioned galaxy LBDS 53W091 (Fig. 6.44). The
spectra of these galaxies can, in the framework of pop-
ulation synthesis (Sect. 3.9), be explained by a stellar
population with an age of about one to two billion years,
and they are similar to the spectra of EROs at z ~ 1 , with
the exact age of the population depending on the as-
sumed metallicity. HST images of these objects strongly
indicate that their morphology also identifies them as
early-type galaxies. The stellar mass in these galaxies,
4 The fact that spectacular images of merging galaxies show niainK
gas rich mergers (Mich as in Fig. 9.15 or Fig. 1.13) can be attributed
to selection effects. On the one hand, gas-rich mergers lead to mas-
sive star formation. \ iclding an increased luminosity of (he systems,
whereas dr> mergers basically preserve the luminosity. On the other
hand, gas rich mergers can he recognized as such lor a longer period
of time than dry ones, owing to the clearly visible tidal tails traced b\
luminous newly formed stars.
9.6 Galaxy Formation and Evolution
derived from the NIR magnitude, is M* > 10 n M Q , so
they are comparable to elliptical galaxies in the local
Universe. Although this is only a small sample of ob-
jects, it is possible to estimate from them the density of
massive early-type galaxies at these high redshifts. This
yields a density which is comparable to that of massive
star-forming galaxies at similar redshift. This means that
at z ~ 2, not only were a large fraction of the stars which
are visible today formed (Fig. 9.34), but also that a com-
parable fraction of the stars were already present at that
time in the form of an old stellar population. The cosmic
mass density of stars in these early-type galaxies is about
10% of the current density in systems with stellar masses
above ~ 10 n M Q . The early appearance of massive el-
lipticals at such high redshifts, with number densities as
observed, are difficult to explain by hierarchical models
of galaxy evolution, which we will discuss next.
9.6.3 Semi-Analytic Models
One can try to understand the above qualitative argu-
ments in greater detail and quantitatively. Note that
this is not possible by means of a cosmological hy-
drodynamic simulation: the physical processes that
determine the formation of stars in galaxies occur on
very small length-scales, whereas the evolution of struc-
tures, which defines, e.g., the merger rate, happens on
cosmological scales. Hence it is impossible to treat both
scales together in a single simulation. Furthermore, the
physical laws determining the behavior of gas (hydro-
dynamical processes such as shock fronts and friction;
radiation processes) are too complicated to be mod-
eled in a detailed simulation, except in those which
are confined to a single galaxy. In addition, many of
the gas processes are not understood sufficiently well
to compute their effects from basic physical laws. Star
formation is just one example of this, although arguably
the most important one.
To make progress, we can parametrize the functional
behavior of those processes which we are unable to de-
scribe with a quantitative physical model. To give one
example, the star-formation rate in a galactic disk is ex-
pected (and observed) to depend on the local surface
mass density 27 g of gas in the disk. Therefore, the star-
formation rate is parametrized in the form M„ — A Eg ,
and the parameters A and /J adjusted by comparison
of the model predictions with observations. Such semi-
analytic models of galaxy formation and evolution have
in recent years contributed substantially to our under-
standing and interpretation of observations. We will
discuss some of the properties of these models in the
following.
Merger Trees. In the CDM model, massive halos are
formed by the merging of halos of lower mass. An ex-
tension to the Press-Schechter theory (see Sect. 7.5.2)
allows us to compute the statistical properties of these
merger processes of halos. By means of these, it is then
possible to generate a statistical ensemble of merger
histories for any halo of mass M today. Each individual
halo is then represented by a merger tree (see Fig. 9.37).
Alternatively, such merger trees can also be extracted
from numerical simulations of structure formation, by
following the mass assemble history of individual ha-
los. The statistical properties of halos of mass M at
redshift z are then obtained by analyzing the ensemble
of merger trees. Each individual merger tree specifies
Fig. 9.37. A typical merger tree, as expected in a hierarchical
CDM model of structure formation. The time axis runs from
top to bottom. A massive halo at the present time to has formed
in mergers of numerous halos of lower mass, as indicated in
the figure. One defines ihe lime of halo formation as the lime t\
at which one of the sub-halos had reached half the mass of the
9. The Universe at High Redshift
the merger processes that have led to the formation of
a particular halo.
Cooling Processes and Star Formation. In a halo
which does not undergo any merger process at a given
time, gas can cool, where the cooling rate is determined
by the chemical composition and the density of the
gas. Besides atomic radiation, bremsstrahlung (free-
free radiation) is also relevant for cooling, in particular
at higher temperatures. If the density is sufficiently
high and cooling is efficient, gas can be transformed
into stars. Star formation is parametrized by a factor
of proportionality between the star-formation rate and
the rate at which gas cools. The newly formed stars are
associated with a "disk component".
Feedback. Shortly after the formation of stars, the more
massive of them will explode in the form of supernovae.
This will re-heat the gas, since the radiation from the
SN explosions and, in particular, the kinetic energy of
the expanding shell, transfers energy to the gas. By this
heating process, the amount of gas that can efficiently
cool is reduced; this reduction increases with the star-
formation rate. This leads to a self-regulation of star
formation, which prevents all the gas in a halo from
being transformed into stars. This kind of self-regulation
by the feedback from supernovae (and, to some extent,
also by the winds from the most massive stars) is also
the reason why the star formation in our Milky Way
is moderate, instead of all the gas in the disk being
involved in the formation of stars.
Suppression of Low-Mass Galaxies. Besides heating
by the feedback process described above, the gas in
a halo can also be heated by intergalactic UV radiation
which is produced by AGNs and starbursts, and which
is responsible for maintaining the high ionization level
in the intergalactic medium (see Sect. 8.5). This radia-
tion has two effects on the gas: first, the gas is heated by
photoionization due to the energetic photons, and sec-
ond, the degree of ionization in the halo gas is increased.
Both effects act in the same direction, by impeding an
efficient cooling of the gas and hence the formation of
stars. For halos of larger mass, intergalactic radiation
is of fairly little importance because the corresponding
heating rate is substantially smaller than that occurring
by the dissipation of the gas. For low-mass halos, how-
ever, this effect is important. In this case, gas heating
can be strong enough for the generated pressure to pre-
vent the infall of gas into the gravitational potential of
the dark halo. For this reason, one expects that halos of
lower mass have a lower baryon fraction than that of
the cosmic mixture, f b — Q b /Q m . The actual value of
the baryon fraction depends on the details of the merger
history of a halo. Quantitative studies yield an average
baryon mass of
— P
M b = r- , (9.8)
[l + (2 1 / 3 -l)M c /M]
where Mq ~ 1O 9 M is a characteristic mass, defined
such that for a halo with mass Mq, M b /M = /b/2. For
halos of mass smaller than Mq, the baryon fraction is
suppressed, whereas for halo masses ^ Mq, the baryon
fraction corresponds to the cosmic average.
The low baryon fraction in low-mass halos is also
expected because of the different shape of the halo mass
spectrum in the Press-Schechter model, compared to the
shape of the luminosity function of galaxies. The former
is roughly oc M~ 2 for masses below M*, whereas the
galaxy luminosity function behaves like ocL~ l . This
different functional form is obviously not compatible
with a constant mass-to-light ratio.
Another process for the suppression of baryons in
low-mass halos is feedback; the transfer of kinetic en-
ergy from SN explosions to the gas can eject part of
the gas from the potential well of the halo, and this
effect becomes more efficient the smaller the bind-
ing energy of the gas is, i.e., the less massive the
halo is. The suppression of the formation of low-mass
galaxies by the effects mentioned here is a possible
explanation for the apparent problem of CDM sub-
structure in halos of galaxies discussed in Sect. 7.5.5.
In this model, CDM sub-halos would be present, but
they would be unable to have experienced an effi-
cient star-formation history - hence, they would be
dark.
Whereas the abundance of dark matter sub-halos in
galaxies no longer presents a problem for CDM models
of structure formation, the spatial distribution of satellite
galaxies around the Milky Way requires more explana-
tion. As we mentioned in Sect. 6.1.1, the 11 satellites
of the Galaxy seem to form a planar distribution. Such
a distribution would be extremely unlikely if the satellite
population was drawn from a near-isotropic probability
9.6 Galaxy Formation and Evolution
distribution. Therefore, the planar satellite distribution
has been considered as a further potential problem for
CDM-like models. However, using semi-analytic mod-
els of galaxy formation, combined with simulations of
the large-scale structure, a different picture emerges.
Since galaxies preferentially form in filaments of the
large-scale structure, the accretion of smaller mass ha-
los onto a high-mass halo occurs predominantly in the
direction of the filament. The most massive sub-halos
therefore tend to form a planar distribution, not unlike
the one seen in the Milky Way's satellite distribution.
The anisotropy of the distribution of massive satellites
may also serve to explain the Holmberg effect.
Major Mergers. In the framework of semi-analytic
models, a spheroidal stellar population may form in
a "major merger", which may be defined in terms of the
mass ratio of the merging halos (e.g., larger than 1:3)
- the disk populations of the two merging galaxies are
dynamically heated to commonly form an elliptical gal-
axy. The gas in the two components is heated by shocks
to the virial temperature of the resulting halo, which
suppresses future star formation.
Minor Mergers. If the masses of the two components in
a merger are very different, the gas of the smaller com-
ponent will basically be accreted onto the more massive
halo, where it can cool again and form new stars. By
this process, a new disk population may form. In this
model, a spiral galaxy is created by forming a bulge
in a "major merger" at earlier times, with the disk of
stars and gas being formed later in minor mergers and
by the accretion of gas. Hence the bulge of a spiral
is, in this picture, nothing but a small elliptical galaxy,
which is also suggested by the very similar characteris-
tics of bulges and ellipticals, including the fact that both
types of object seem to follow the same relation between
black hole mass and velocity dispersion, as explained
in Sect. 3.5.3.
The merging process of the two components does
not occur instantaneously, but since the smaller galaxy
will have, in general, a finite orbital angular momentum,
it will first enter into an orbit around the more massive
component. One example of this is the Sagittarius dwarf
galaxy, but also the Magellanic Clouds will, in a distant
future, merge with the Milky Way in this way. By dy-
namical friction, the satellite galaxy then loses its orbital
Fig. 9.38. On the left, the distribution of dark matter resulting
from an JV-body simulation is shown. The dark matter halos
identified in this mass distribution were then modeled as the
location of galaxy formation - the formation of halos and their
merger history can be followed explicitly in the simulations.
Semi-analytic models describe the processes which are most
important for the gas and the formation of stars in halos, from
which a model for the distribution of galaxies results. In the
panel on the right, the resulting distribution of model galax-
ies is represented by colored dots, where the color indicates
the spectral energy distribution of the respective galaxy: blue
indicates galaxies with active star formation, red arc galaxies
which are presently not forming any new stars. The latter are
particularl) abundant in clusters of galaxies - in agreemenl
with observations
9. The Universe at High Redshift
energy, and the tidal component of the gravitational field
removes stars, gas, and dark matter away from it; the
Magellanic Stream (see Fig. 6.6) is presumably the re-
sult of such a process. Only after several orbits - the
number of which depends on the initial conditions and
on the mass ratio - will the satellite galaxy finally merge
with the larger one.
Results from Semi- Analytic Models. The free parame-
ters in semi-analytic models - such as the star-formation
efficiency or the fraction of energy from SNe that
is transferred into the gas - are fixed by comparison
with some key observational results. For example, one
requires that the models reproduce the correct normal-
ization of the Tully-Fisher relation and that the number
counts of galaxies match those observed. Although
these models are too simplistic to trace the processes
of galaxy evolution in detail, they are highly successful
in describing the basic aspects of the galaxy population,
and they are continually being refined. For instance, this
model predicts that galaxies in clusters basically con-
sist of old stellar populations, because here the merger
processes were already concluded quite early in cos-
mic history. Therefore, at later times gas was no longer
available for the formation of stars. Figure 9.38 shows
the outcome of such a model in which the merger his-
tory of the individual halos has been taken straight from
the numerical N-body simulation, hence the spatial lo-
cations of the individual galaxies are also described by
these simulations.
By comparison of the results from such semi-analytic
models with the observed properties of galaxies and
their spatial distribution, the models can be increas-
ingly refined. In this way, we obtain more realistic
descriptions of those processes which are included in
the models in a parametrized form. This comparison is
of central importance for achieving further progress in
our understanding of the complex processes that are
occurring in galaxy evolution, which can neither be
studied in detail by observation, nor be described by
more fundamental simulations.
As a result of such models, the correlation func-
tion of galaxies as it is obtained from the Millennium
simulation (see Sect. 7.5.3) is presented in Fig. 9.39,
in comparison to the correlation function observed
in the 2dFGRS. The agreement between the model
and the observations is quite impressive; both show
a nearly perfect power law. In particular, the correla-
tion function of galaxies distinctly deviates from the
correlation function of dark matter on small scales,
implying a scale-dependent bias factor. The question
arises as to which processes in the evolution of galax-
ies may produce such a perfect power law: why does
the bias factor behave just such that £ g attains this sim-
ple shape. The answer is found by analyzing luminous
and less luminous galaxies separately, or galaxies with
and without active star formation - for each of these
subpopulations of galaxies, § g is not a power law. For
this reason, the simple shape of the correlation func-
tion shown in Fig. 9.39 is probably a mere coincidence
("cosmic conspiracy").
Another result from such models is presented in
Fig. 9.40, also from the Millennium simulation (see
Sect. 7.5.3). Here, one of the most massive dark mat-
ter halos in the simulation box at redshift z — 6.2 is
shown, together with the mass distribution in this spatial
region at redshift z — 0. In both cases, besides the dis-
tribution of dark matter, the galaxy distribution is also
Fig. 9.39. Correlation function of galaxies at z = (filled
circles connected by the solid curve), computed from the Mil-
lcnnium simulation in coinbinalion with semi anahlie models
of galaxy evolution. This is compared to the observed galaxy
correlation function as derived from the 2dFGRS (diamonds
with error bars). The dashed curve shows the convlalion
function of matter
Fig. 9.40. In the top panels, one of the most massive halos
at z — 6.2 from the Millennium simulation (see Fig. 7.12) is
shown, whereas in the bottom panels, the corresponding distri-
bution in this spatial region at z = is shown. Thus, this eai I\
massive halo is now located in the center of a very massive
galaxy cluster. In the panels on the left, the mass distribution is
displayed, wall die corresponding distribution of galaxies as
determined from a semi analytic model, which is shown in the
right hand panels. Galaxies at z = 6.2 are all blue since their
stellar population must be young, whereas at z = 0, most gal-
axies contain an old stellar population, here indicated by the
red color. Each of the panels shows the projected distribution
in a cube with a comoving side length of I0h~ [ Mpc
9. The Universe at High Redshift
displayed, obtained from semi-analytic models. Mas-
sive halos which have formed early in cosmic history
are currently found predominantly in the centers of very
massive galaxy clusters. Assuming that the luminous
QSOs at z ~ 6 are harbored in the most massive ha-
los of this epoch, we can deduce that these may today
be identified as the central galaxies in clusters. This
may provide an explanation as to why so many cen-
tral, dominating cluster galaxies show AGN activity,
though with a smaller luminosity due to small accretion
rates.
Abundance and Evolution of Supermassive Black
Holes. Within the framework of these models, predic-
tions are also made about the statistical evolution of
SMBHs in the cores of galaxies. When two galaxies
merge, their SMBHs will also coalesce after some time,
where an accurate estimate for this time-scale is diffi-
cult to obtain. Owing to the high initial orbital angular
momentum, the two SMBHs are, at the beginning of
a merger, on an orbit with rather large mutual separa-
tion. By dynamical friction (see Sect. 6.2.6), caused by
the matter distribution in the newly formed galaxy, the
pair of SMBHs will lose orbital energy after the merger
of the galaxies, and the two black holes will approach
each other. Since this process takes a relatively long
time, and since a massive galaxy will, besides a few
major mergers, undergo numerous minor mergers, it is
conceivable that many of the black holes that were orig-
inally the nuclei of low-mass satellite galaxies are today
still on orbits at relatively large distances from the center
of galaxies.
In this model, phases in the evolution of galaxies ex-
ist in which two SMBHs are located close to the center.
Indeed, there are a number of indications that such gal-
axies are actually observed. For instance, galaxies with
two active nuclei have been found. Also a class of radio
sources exists with an X-shaped morphology (instead
of the usual bipolar radio structure), which can be in-
terpreted as pairs of active SMBHs. Another signature
of a binary system of SMBHs would be a periodicity
in the emission, reflecting the orbital period. In some
AGNs, periodic variations in the brightness have in fact
been detected, the blazar OJ 287 being the best known
example, with a period of 1 1.86 years.
When two SMBHs merge, the initially wide orbit
shrinks, in the final stages due to the emission of grav-
itational waves. This will cause the orbits to become
more circular, as well as a decrease of the separation of
the two black holes. According to the theory of black
holes, there is a closest separation at which an orbit still
is stable. Once the separation has shrunk to that size, the
merging occurs, accompanied by a burst of gravitational
wave emission. If the two SMBHs have the same mass,
each of them will emit the same amount of gravitational
wave energy, but in opposite directions, so that the net
amount of momentum carried away by the gravitational
waves is zero. However, if the masses are not equal, this
cancellation no longer occurs, and the waves carry away
a net linear momentum. According to momentum con-
servation, this will yield a recoil to the merged SMBH,
and it will therefore move out of the galactic nucleus.
Depending on the recoil velocity, it may return to the
center in a few dynamical time-scales. However, if the
recoil velocity is larger than the escape velocity from
the galaxy, it may actually escape from the gravitational
potential and become an intergalactic black hole. The
importance of this effect is not quantitatively known,
since the amplitude of the recoil velocity as determined
from theoretical models is uncertain. It is zero for equal
masses, and very small if one of the black hole masses
is much smaller than the other. The recoil velocity at-
tains a maximum value when the mass ratio of the two
SMBHs is about 1/3.
The mass increase of SMBHs in the course of cosmic
history then has two different origins, first the merg-
ing with other low-mass SMBHs as a consequence of
merger events, and second the accretion of gas that
leads to the activity of SMBHs. Hierarchical models
of galaxy evolution with central SMBHs are able to
both reproduce, under certain assumptions, the corre-
lation (see Sect. 3.5.3) between the SMBH mass and
the properties of the spheroidal stellar component, and
to successfully model the integrated AGN luminos-
ity and the redshift-dependent luminosity function of
AGNs.
In the course of the merger of two SMBHs, an intense
emission of gravitational waves will occur in the final
phase. The space project LISA, which is planned to be
launched sometime after 2013, is capable of observing
the emission of gravitational waves from such merger
processes, essentially throughout the visible Universe.
Hence, it will become possible to directly trace the
merger history of galaxies.
9.6 Galaxy Formation and Evolution
9.6.4 Cosmic Downsizing
■ formation to increasingly lower-mass
The hierarchical model of structure formation predicts
that smaller-mass objects are formed first, with more
massive systems forming later in the cosmic evolution.
As discussed before, there is ample evidence for this
to be the case; e.g., galaxies are in place early in the
cosmic history, whereas clusters are abundant only at
redshifts z < 1. However, looking more closely into the
issue, apparent contradictions are discovered. For ex-
ample, the most massive galaxies in the local Universe,
the massive ellipticals, contain the oldest population of
stars, although their formation should have occurred
later than those of less massive galaxies. In turn, most
of the star formation in the local Universe seems to
be associated with low- or intermediate-mass galaxies,
whereas the most massive ones are passively evolving.
Now turning to high redshift: for z ~ 3, the bulk of star
formation seems to occur in the LBGs, which, accord-
ing to their clustering properties (see Sect. 9.1.1), are
associated with high-mass halos. The study of passively
evolving EROs indicates that massive old galaxies were
in place as early as z ~ 2, hence they must have formed
very early in the cosmic history. The phenomenon that
massive galaxies form their stars in the high-redshift
Universe, whereas most of the current star formation
occurs in galaxies of lower mass, has been termed
"downsizing".
This downsizing can be studied in more detail us-
ing redshift surveys of galaxies. The observed line
width of the galaxies yields a measure of the char-
acteristic velocity and thus the mass of the galaxies
(and their halos). Such studies have been carried out
in the local Universe, showing that local galaxies
have a bimodal distribution in color (see Sect. 3.7.2),
which in turn is related to a bimodal distribution in
the specific star- formation rate. Extending such stud-
ies to higher redshifts, by spectroscopic surveys at
fainter magnitudes, we can study whether this bi-
modal distribution changes over time. In fact, such
studies reveal that the characteristic mass separat-
ing the star-forming galaxies from the passive ones
evolves with redshift, such that this dividing mass
increases with z. For example, this characteristic
mass decreased by a factor of ~ 5 between z = 1 .4
and z = 0.4. Hence, the mass scale above which most
galaxies are passively evolving decreases over time,
Studies of the fundamental plane for field ellipticals
at higher redshift also point to a similar conclusion.
Whereas the massive ellipticals at z ~ 0.7 lie on the
fundamental plane of local galaxies when passive evo-
lution of their stellar population is taken into account,
lower-mass ellipticals at these redshifts have a smaller
mass-to-light ratio, indicating a younger stellar popula-
tion. Also here, the more massive galaxies seem to be
older than less massive ones.
Another problem with which models of galaxy for-
mation are faced is the absence of very massive galaxies
today. The luminosity function of galaxies is described
reasonably well by a Schechter luminosity function, i.e.,
there is a luminosity scale L* above which the number
density of galaxies decreases exponentially. Assuming
plausible mass-to-light ratios, this limiting luminosity
translates into a halo mass which is considerably lower
than the mass scale M* (z = 0) above which the abun-
dance of dark matter halos is exponentially cut-off. In
fact, the shape of the mass spectrum of dark matter halos
is quite different from that of the stellar mass (or lumi-
nosity) spectrum of galaxies. Why, then, is there some
kind of maximum luminosity (or stellar mass) for galax-
ies? It has been suggested that the value of L„ is related
to the ability of gas in a dark matter halo to cool; if the
mass is too high, the corresponding virial temperature
of the gas is large and the gas density low, so that the
cooling times are too large to make gas cooling, and thus
star formation, efficient. With a relatively high cosmic
baryon density of Q\, — 0.045, however, this argument
fails to provide a valid quantitative explanation.
The clue to the solution of these problems may come
from the absence of cooling flows in galaxy clusters.
As we saw in Sect. 6.3.3, the gas density in the inner
regions of clusters is large enough for the gas to cool
in much less than a Hubble time. However, in spite of
this fact, the gas seems to be unable to cool, for oth-
erwise the cool gas would be observable by means of
intense line radiation. This situation resembles that of
the massive galaxies: if they were already in place at
high redshifts, why has additional gas in their halos
(visible, e.g., through its X-ray emission, and expected
in structure formation models to accrete onto the host
halo) not cooled and formed stars? The solution for
this problem in galaxy clusters was the hypothesis that
9. The Universe at High Redshift
AGN activity in their central galaxy puts out enough
energy to heat the gas and prevent it from cooling
to low temperatures. Direct and indirect evidence for
the validity of this hypothesis exists, as explained in
Sect. 6.3.3.
A similar mechanism may occur in galaxies as well.
We have learned that galaxies host a supermassive black
hole whose mass scales with the velocity dispersion of
the spheroidal stellar component and thus, in elliptical
galaxies, with the mass of the dark matter halo. If gas
accretes onto these halos, it may cool and, on the one
hand, form stars ; on the other hand, this process will lead
to accretion of gas onto the central black hole and make
it active again. This activity can then heat the gas and
thus prevent further star formation. If the time needed
to cool the gas and form stars is shorter than the free-fall
time to the center of the galaxy, stars can form before
the AGN activity is switched on. In the opposite case,
star formation is prevented. A quantitative analysis of
these two time-scales shows that they are about equal
for a halo of mass ~ 2 x 10 ll h~ l M Q , about the right
mass-scale for explaining the cut-off luminosity L* in
the Schechter function.
Accounting for the AGN feedback explicitly in semi-
analytic models of galaxy evolution provides a good
match to the observed galaxy luminosity function in
the current Universe. Furthermore, such models closely
reproduce the observed evolution of the stellar mass
function, which provides a framework for understand-
ing the "downsizing" phenomenon. In fact, since the
mass of the central black hole was accumulated by ac-
cretion, and since the total energy output that can be
generated in the course of growing a black hole to a mass
of ~ 1O 8 M is very large, it should not be too surpris-
ing that this nuclear activity has a profound impact on
the galaxy hosting the SMBH. The fact that the hosts of
luminous QSOs show no signs of strong star formation
may be another indication that the AGN luminosity pre-
vents efficient star formation in its local environment.
9.7 Gamma-Ray Bursts
Discovery and Phenomenology. In 1968, surveillance
satellites for the monitoring of nuclear test ban treaties
discovered y -flashes similar to those that are observed
in nuclear explosions. However, these satellites found
that the flashes were not directed from Earth but from
the opposite direction - hence, these y-flashes must
be a phenomenon of cosmic origin. Since the satellite
missions were classified, the results were not published
until 1973. The sources were named gamma-ray hursts
(GRB).
The flashes are of very different duration, from a few
milliseconds up to ~ 100 s, and they differ strongly in
their respective light curves (see Fig. 9.41). They are ob-
served in an energy range from ~ 100 keV up to several
MeV, sometimes to even higher energies.
The nature of GRBs had been completely unclear
initially, because the positional accuracy of the bursts
as determined by the satellites was far too large to allow
an identification of any corresponding optical source.
The angular resolution of these y-detectors was many
degrees (for some, a 2n solid angle). A more precise
position was determined from the time of arrival of the
bursts at the location of several satellites, but the error
box was still too large to search for counterparts of the
source in other spectral ranges.
Early Models. The model favored for a long time in-
cluded accretion phenomena on neutron stars in our
Galaxy. If their distance was D ~ 100 pc, the corre-
sponding luminosity would be about L ~ 10 38 erg/s,
thus about the Eddington luminosity of a neutron star.
Furthermore, indications of absorption lines in GRBs
at about 40keV and 80keV were found, which were
interpreted as cyclotron absorption corresponding to
a magnetic field of ~ 10 12 Gauss - again, a characteris-
tic value for the magnetic field of neutron stars. Hence,
most researchers before the early 1990s thought that
GRBs occur in our immediate Galactic neighborhood.
The Extragalactic Origin of GRBs. A fundamental
breakthrough was then achieved with the BATSE exper-
iment on-board the Compton Gamma Ray Obsen atory,
which detected GRBs at a rate of about one per day
over a period of nine years. The statistics of these
GRBs shows that GRBs are isotropically distributed
on the sky (see Fig. 9.42), and that the flux distribution
N(> F) clearly deviates, at low fluxes, from the F~ l - 5 -
law. These two results meant an end to those models
that had linked GRBs to neutron stars in our Milky Way,
which becomes clear from the following argument.
9.7 Gamma-Ray Bursts
5 gamma-ray ticularly noted. All these light curses appear to be very
s should be par- dissimilar
Neutron stars are concentrated towards the disk of the case, the distribution might possibly be isotropic, but
Galaxy, hence the distribution of GRBs should feature the flux distribution would necessarily have to follow
a clear anisotropy - except for the case that the typical the law N(> F) on F~ 3/2 , as expected for a homoge-
distance of the sources is very small (< 100 pc), much neous distribution of sources, which was discussed in
smaller than the scale-height of the disk. In the latter Sect. 4.1.2. Because this is clearly not the case, a dif-
9. The Universe at High Redshift
2704 BATSE Gamma- Ray Bursts
Fluence, 50-300 keV (ergs cm 2 )
Fig. 9.42. Distribution of gamma-ray bursts on the sphere
as observed by BATSE, an instrument on-board the CGRO-
satellite, during the about nine year mission; in total, 2704
GRBs are displayed. The color of the symbols represents the
observed strength (fluence, or energy per unit area) of the
bursts. One can see that the distribution on the sky is isotropic
to a high degree
ferent distribution of sources is required, hence also
a different kind of source.
The only way to obtain an isotropic distribution for
sources which are typically more distant than the disk
scale-height is to assume sources at distances consid-
erably larger than the distance to the Virgo Cluster,
hence D ^> 20 Mpc; otherwise, one would observe an
overdensity in this direction. In addition, the deviation
from the N(> F) <x F~ 3/2 -law means that we observe
sources up to the edge of the distribution (or, more pre-
cisely, that the curvature of spacetime, or the cosmic
evolution of the source population, induces deviations
from the Newtonian counts), so that the typical distance
of GRBs should correspond to a relatively high redshift.
This implies that the total energy in a burst has to be
£~ 10 51 to 10 54 erg. This energy corresponds to the
rest mass Mc 2 of a star. The major part of this energy
is emitted within ~ 1 s, so that GRBs are, during this
short time-span, more luminous than all other y-sources
in the Universe put together.
Identification and Afterglows. In February 1997, the
first identification of a GRB in another wavelength band
was accomplished by the X-ray satellite Beppo-SAX.
Within a few hours of the burst, Beppo-SAX observed
the field within the GRB error box and discovered
a transient source, by which the positional accuracj was
increased to a few arcminutes. In optical observations of
this field, a transient source was then detected as well,
very accurately defining the position of this GRB. The
optical source was identified with a faint galaxy. Op-
tical spectroscopy of the source revealed the presence
of absorption features at redshift z — 0.835; hence, this
GRB must have a redshift equal or larger than this.
For the first time, the extragalactic nature of GRBs was
established. In fast progression, other GRBs could be
identified with a transient optical source, and some of
them show transient radiation also at other wavelengths,
from the radio band up to X-rays. The lower-energy ra-
diation of a GRB after the actual burst in gamma-rays
is called an afterglow.
GRBs can be broadly classified into short- and
long-duration bursts, with a division at a duration of
^burst ~ 2 s. The spectral index of the short-duration
bursts is considerably harder at y-ray energies that
that of long-duration bursts. Until 2005, only after-
glows from long-duration bursts had been discovered.
Long-duration bursts occur in galaxies at high redshift,
typically z ~ 1 or higher, with the highest-redshift burst
identified to date having a redshift of z = 6.3. In one
case, an optical burst was discovered about 30 sec-
onds after the GRB, with the fantastic brightness of
V ~ 9 mag, at a redshift of z — 1.6. For a short period
of time, this source was apparently more luminous than
any quasar in the Universe. Thus, during or shortly af-
ter the burst at high energies, GRBs are also very bright
in the optical.
Fireball Model. Whereas the distance of the sources,
and therefore also their luminosity, was then known,
the question of the nature of GRBs still remained unan-
swered. One model of GRBs quite accurately describes
their emission characteristics, including the afterglow.
In this fireball model, the radiation is released in the rela-
tivistic outflow of electron-positron pairs with a Lorentz
factor of y > 100. However, different hypotheses exist
as to how this fireball is produced, hence what the phys-
ical origin of a GRB might be. One of these states that
a GRB is caused by the merger of two neutron stars, or
a neutron star and a black hole. In this case, the emission
will probably be highly anisotropic, so that estimates of
the luminosity from the observed flux, based on the
assumption of isotropic emission, may not be correct.
9.7 Gamma-Ray Bursts
Hypernovae. At the present time, the sources of long-
duration GRBs have pretty much been identified. With
the accurate location obtained from observations of their
afterglows, it was found that they occur in star-forming
galaxies, not in passive early-type galaxies. This find-
ing is similar to that of core-collapse supernovae which
are also found only in galaxies with star-formation ac-
tivity. It was therefore speculated that the origin of
GRBs is closely linked to star formation. The discov-
ery of a coincidence of several GRBs with supernova
explosions suggests that long-duration GRBs are ex-
traordinarily energetic explosions of stars, so-called
hypernovae. Even if the emission is highly anisotropic,
as expected from the fireball model, the corresponding
energy released by the hypernovae is very large.
For the purpose of identifying GRBs, the SWIFT
satellite was launched in November 2004. This satel-
lite is equipped with three instruments: a wide-field
gamma-ray telescope to discover the GRBs, an X-ray
telescope, and a UV-optical telescope. Within a few
seconds of the discovery of a GRB, the satellite targets
the location of the burst, so that it can be observed by
the latter two telescopes, obtaining an accurate posi-
tion. This information is then immediately transmitted
to the ground, where other telescopes can follow the af-
terglow emission. SWIFT is expected to discover about
100 GRBs per year and to obtain significantly improved
statistics of their afterglow light curves and redshifts.
Already in its first year of operation, a large number
of GRBs were found by SWIFT, including the one at
z = 6.3.
Counterparts of Short-Duration GRBs. SWIFT has
allowed the identification of four short-duration GRBs
in 2005. In contrast to the long-duration bursts, some
of these seem to be associated with elliptical galaxies;
this essentially precludes any association with super-
nova explosions. In fact, for one of these short burst,
very sensitive limits on the optical brightness explicitly
rules out any contribution from a supernova explosion.
Furthermore, the host galaxies of short bursts are at sub-
stantially lower redshift, z ~ 0.2. Given that both kinds
of GRBs have about the same observed flux (or energy),
this implies that short-duration bursts are less energetic
than long-duration ones, by approximately two orders
of magnitude. All of these facts clearly indicate that
short- and long-duration GRBs are due to different pop-
ulations of sources. The lower energies of short bursts
and their occurrence in early-type galaxies with old stel-
lar populations are consistent with them being due to the
merging of compact objects, either two neutron stars, or
a neutron star and a black hole.
io. Outlook
In the final chapter of this book, we will dare to give
an outlook for the fields of extragalactic astronomy and
cosmology for the next few years from the perspective
of 2006.
Progress in (extragalactic) astronomy is achieved
through information obtained from increasingly im-
proving instruments and by refining our theoretical
understanding of astrophysical processes, which in turn
is driven by observational results. It is easy to fore-
see that the evolution of instrumental capabilities will
continue rapidly in the near future, enabling us to per-
form better and more detailed studies of cosmic sources.
A few examples illustrating this statement will be given
here. The size of optical wide-field cameras had reached
a value of ~ (20 000) 2 pixels by 2003 with the install-
ment of Megacam at the CFHT. This multi-chip camera
allows the mapping of one square degree of the sky
Fig. 10.1. Wide-field cameras, attached to telescopes on sites
with excellent atmospheric conditions, can obtain detailed
images of a large number of objects simultaneously. This
is illustrated here with the CFH12K camera at the CFHT.
Numbers in each panel, which show subsequent enlargements,
denote the number of pixels displayed, where the pixel size is
0'/2
a single exposure and, with a pixel size of 0"2, it
well matched to the excellent seeing conditions typ-
ically met on Mauna Kea (see Fig. 10.1). Additional
nstruments with similar characteristics have been re-
cently finished or are about to be commissioned. One
of them is OmegaCAM, a square-degree camera at the
newly-built VLT Survey Telescope on Paranal. Further-
more, the development of NIR detectors is rapid, and
soon wide-field cameras in the NIR regime will be con-
siderably larger than current ones. For instance, in 2007
the new 4-meter telescope VISTA will go into opera-
tion on Paranal, which will be equipped initially with
a single instrument, a wide-field NIR camera. The com-
bination of deep and wide optical and NIR images will
no doubt lead to great strides in astronomy. For exam-
ple, in the field of galaxy surveys, accurate photometric
redshifts will become available. The same holds true
for weak gravitational lensing or the search for very
rare objects, for which surveying large regions of the
sky is obviously necessary.
Within only a decade, the total collecting area of
large optical telescopes has increased by a large factor,
as is illustrated in Fig. 10.2. At the present time, about
10 telescopes of the 10-meter class are in operation, the
first of which, Keck I, was put into operation in 1993. In
addition, the development of adaptive optics will allow
us to obtain diffraction-limited angular resolution from
ground-based observations (see Fig. 10.3).
In another step to improve angular resolution, optical
and NIR interferometry will increasingly be employed.
For example, the two Keck telescopes (Fig. 1.28) are
mounted such that they can be used for interferometry.
The four unit telescopes of the VLT can be combined,
either with each other or with additional (auxiliary)
smaller telescopes, to act as an interferometer (see
Fig. 1.31). The auxiliary telescopes can be placed at
different locations, thus yielding different baselines
and thereby increasing the coverage in angular reso-
lution. Finally, the Large Binocular Telescope (LBT),
which consists of two 8.4-meter telescopes mounted
on the same platform, was developed and constructed
for the specific purpose of optical and NIR interfer-
ometry and had first light in October 2005. Once in
operation, expected to occur by the end of 2006, this
Peter Schneider,
Outlook
In: Peter Schnei
er.Extrag
alactic Astr
onom>
and Co
mho
ogy. pp
407^11
4 (2006)
DOI: 10.1007/1
614371_1
© Spring
-|--\er
ig Belli
iHe
delhei-L
2006
Fig. 10.2. The collecting area of large op-
tical telescopes is displayed. Those in the
Northern hemisphere are shown on the left,
whereas southern telescopes are shown on
the right. The joint collecting area of these
telescopes has been increased by a large fac-
tor over the past decade: only the telescopes
shown in the upper row plus the 5-meter
Palomar telescope and the 6-meter SAO
were in operation before 1993. If, in addi-
tion, the parallel development of detectors is
considered, it is easy to understand why ob-
servational astronomy is making such rapid
progress
Fig. 10.3. This figure illustrates the evolu-
tion of angular resolution as a function of
time. The upper dotted curve describes the
angular resolution that would be achieved
in the case of diffraction-limited imaging,
which depends, at fixed wavelength, only
on the aperture of the telescope. Some his-
torically important telescopes arc indicated.
The lower curve shows the angular res-
olution actually achieved. This is mainly
limited by atmospheric turbulence, i.e.,
seeing, and thus is largely independent
of the size of the telescope. Instead, it
mainly depends on the quality of the at-
mospheric conditions at the observatories.
For instance, we can clearly recognize
how the opening of the observatories on
Mount Palomar, and later on Mauna Kea,
La Silla and Paranal have lead to break-
throughs in resolution. A further large step
was achieved with HST, which is unaf-
fected by atmospheric turbulence and is
therefore diffraction limited. Adaptive op-
tics and interferometry will characterize the
next essential improvements
telescope will start a new era in high-resolution optical
astronomy.
The Hubble Space Telescope has turned out to be
the most successful astronomical observatory of all
time (although it certainly was also the most expen-
sive one). This success can be explained predominantly
by its angular resolution, which is enormously supe-
rior compared to ground-based telescopes, and by the
significantly reduced night-sky brightness, in partic-
ular at longer wavelengths. The importance of HST
for extragalactic astronomy is not least based on the
characteristics of galaxies at high redshifts. Before the
launch of HST, it was not known that such objects are
small and therefore have, at a given flux, a high sur-
face brightness. This demonstrates the advantage of the
high resolution that is achieved with HST. Several ser-
vice missions to the observatory led to the installment
of new and more powerful instruments which have con-
tinuously improved the capacity of HST. At present,
the future of HST is very uncertain. After the fatal dis-
Fig. 10.4. Artist's impression of the 6.5-meter James Webb
Space Telescope. Like the Keck telescopes, the mirror is seg-
mented and protected against Solar radiation by a giant heat
shield. Keeping the mirror and the instruments permanently
in the shadow will permit a passive cooling at a temperature
of ~ 35 K. This will be ideal for conducting observations at
NIR wavelengths, with unprecedented sensitivity
aster of the Space Shuttle Challenger, NASA initially
canceled the next planned servicing mission; this mis-
sion is vital for HST since its gyroscopes need to be
replaced. In addition, this servicing mission was sched-
uled to bring two new powerful instruments on-board,
further increasing the scientific capabilities of HST. At
present (2006) it is unclear whether this servicing mis-
sion will be launched, thereby prolonging the lifetime
of HST for several more years and bridging the time
until JWST will be launched (see below).
Fortunately, the successor of HST is already at an
intensive stage of planning and is currently scheduled
to be launched in 2013. This Next Generation Space
Telescope (which was named James Webb Space Tele-
scope - JWST) will have a mirror of 6.5-meters diameter
and therefore will be substantially more sensitive than
HST. Furthermore, JWST will be optimized for obser-
vations in the NIR (1 to 5 |im) and thus be able, in par-
ticular, to observe sources at high redshifts whose stellar
light is redshifted into the NIR regime of the spectrum.
The Spitzer Space Telescope already operates at NIR
and MIR wavelength. Despite the fact that Spitzer car-
ries only a 60 cm mirror, it is far more sensitive and effi-
cient in this wavelength regime than previous satellites.
We hope that JWST will be able to observe the first
galaxies and the first AGN, i.e., those sources responsi-
ble for reionizing the Universe. Besides a NIR camera,
JWST will carry the first multi-object spectrograph in
space, which is optimized for spectroscopic studies of
high-redshift galaxy samples and whose sensitivity will
exceed that of all previous instruments by a huge factor.
Furthermore, JWST will carry a MIR instrument which
is being developed for imaging and spectroscopy in the
wavelength range 5 |im < A. < 28 [im.
A new kind of observatory is planned for X-ray as-
tronomy where the focal length will be so large as to
require two spacecraft. One of them will carry the mir-
ror system, whereas the other will host the instruments.
Operating such a telescope will require that the separa-
tion between the telescope and the focal plane be kept
constant with very high precision. This poses a techno-
logical challenge for formation flight; formation flights
also need to be mastered for future IR interferometers
in space. The Next Generation X-ray Telescope will be
capable of observing galaxy clusters to the highest red-
shifts and to extend the studies of AGNs to much lower
luminosities than is currently possible. In particular we
hope to study gas physics in the close vicinity of the
event horizon of black holes.
Far-infrared astronomy will receive its next boost
in 2008, when the Herschel satellite will be launched
by ESA. Its 3.5 meter mirror will provide a far bet-
ter sensitivity in this wavelength regime than previous
FIR telescopes. Herschel will be launched together with
the Planck satellite, which will yield a far more de-
tailed image of the microwave sky than even WMAP.
While mainly targeted at measurements of the CMB
anisotropy, with better angular resolution and far better
wavelength coverage than WMAP, Planck will not only
be a very important mission for cosmology; its sky sur-
vey at many frequencies will also benefit many other
fields of astronomy. The discovery of galaxy clusters
by means of the Sunyaev-Zeldovich effect should be
mentioned as just one example.
There will also be revolutionary developments in
radio astronomy. New mm and sub-mm telescopes,
such as the recently commissioned APEX, will pro-
vide much more detailed maps of the dust emission
from star-forming regions than before. APEX will con-
duct a Sunyaev-Zeldovich survey for galaxy clusters
and therefore follow a new strategy for selecting clus-
ters. In a way, this provides a connection to the future
Planck mission. In particular, we expect a large number
of clusters at high redshift which are of special value
when using clusters as cosmological probes. Towards
the end of this decade, ALMA (Atacama Large Mil-
limeter Array, Fig. 10.5), a 64 antenna interferometer
operating at mm and sub-mm wavelengths, \\ ill start to
make its first observations. Its enormously increased an-
gular resolution and sensitivity will allow us to study,
among other issues, the dust emission and molecules
of very high redshift galaxies and QSOs. Furthermore,
future telescopes constructed in the Antarctic would
provide further opportunities for infrared and sub-mm
astronomy owing to the extremely dry atmosphere.
At even longer wavelengths, a technological revo-
lution will take place. Currently being developed are
concepts for radio telescopes whose radio lobes will
be digitally generated on computers. Such digital radio
interferometers not only allow a much improved sensi-
tivity and angular resolution, but they also enable us to
observe many different sources in vastly different sky
regions simultaneously. LOFAR will be the prototype
of such an instrument and will operate at frequencies
below about 200 MHz. In the more distant future, the
Square Kilometer Array (SKA) will be a much larger
observatory - its name is derived from its effective
collecting area. SKA will provide a giant boost to as-
tronomy; for the first time ever, the achievable number
density of sources on the radio sky will be comparable
to or even larger than that in the optical. The limits of
such instruments are no longer bound by the properties
of the individual antennas, but rather by the capacity
of the computers which analyze the data. To exploit
the full capacity of these digital radio interferometers,
a giant evolution in the hardware and software of such
supercomputers will be required.
Fig. 10.5. Artist's impression of the Ataca-
ma Large Millimeter Array (ALMA) which
is currently being built on the Llano de
Chajnantor in Chile, a plateau at 5000 me-
ters altitude (this is also the site of APEX).
The 64 antennas will have a diameter of
12 meters each. They will be operated in
an interferometric mode, and they will start
a totally new era in (sub-)mm astronomy,
owing to the large collecting area and the
excellent atmospheric conditions at this site
New windows to the Universe will be opened. The
first gravitational wave antennas are already in place,
and their next generation will probably be able to
discover the signals from relatively nearby supernova
explosions. With LISA, mergers of supermassive black
holes will become detectable throughout the visible
Universe, as we mentioned before. Giant neutrino detec-
tors will open the field of neutrino astronomy and will be
able, for example, to observe processes in the innermost
parts of AGNs. Observatories for cosmic rays are being
built. The Pierre- Auger observatory in Argentina is one
such example that has been in operation since 2004: in
particular, it will study the highest-energy cosmic rays.
Parallel to these developments in telescopes and
instruments, theory is progressing steadily. The contin-
uously increasing capacity of computers available for
numerical simulations is only one aspect, albeit an im-
portant one. New approaches for modeling, triggered
by new observational results, are of equal impor-
tance. The close connection between theory, modeling,
and observations will become increasingly important
since the complexity of data requires an advanced
level of modeling and simulations for their quantitative
interpretation.
The huge amount of data obtained with current and
future instruments is useful not only for the observers
taking the data, but also for others in the astronomi-
cal community. Realizing this fact, many observatories
have set up archives from which data can be retrieved.
Space observatories pioneered such data archives, and
a great deal of science results from the use of archival
data. Examples here are the use of the HST deep fields
by a large number of researchers, or the analysis of
serendipitous sources in X-ray images which led to the
EMSS (see Sect. 6.3.5). Together with the fact that an
understanding of astronomical sources usually requires
data taken over a broad range of frequencies, there is
a strong motivation for the creation of virtual observato-
ries: infrastructures which connect archives containing
astronomical data from a large variety of instruments
and which can be accessed electronically. In order for
such virtual observatories to be most useful, the data
structures and interfaces of the various archives need to
become mutually compatible. Intensive activities in cre-
ating such virtual observatories are ongoing; they will
doubtlessly play in increasingly important role in the
future.
One of the major challenges for the next few
years will certainly be the investigation of the very
distant Universe, studying the evolution of cosmic ob-
jects and structures at very high redshift up to the
epoch of reionization. To relate the resulting insights
of the distant Universe to those obtained more lo-
cally and thus to obtain a consistent view about our
cosmos, major theoretical investigations will be re-
quired as well as extensive observations across the
whole redshift range, using the broadest wavelength
range possible. Furthermore, the new astrometry satel-
lite GAIA will offer us the unique opportunity to
study cosmology in our Milky Way. With GAIA,
the aforementioned stellar streams, which were cre-
ated in the past by the tidal disruption of satellite
galaxies during their merging with the Milky Way,
can be verified. New insights gained with GAIA will
certainly also improve our understanding of other
galaxies.
The second major challenge for the near future
is the fundamental physics on which our cosmolog-
ical model is based. From observations of galaxies
and galaxy clusters, and also from our determinations
of the cosmological parameters, we have verified the
presence of dark matter. Since there seem to be no
plausible astrophysical explanations for its nature, dark
matter most likely consists of new kinds of elemen-
tary particles. Two different strategies to find these
particles are currently being followed. First, experi-
ments aim at directly detecting these particles, which
should also be present in the immediate vicinity of the
Earth. These experiments are located in deep under-
ground laboratories, thus shielded from cosmic rays.
Several such experiments, which are an enormous tech-
nical challenge due to the sensitivity they are required
to achieve, are currently running. They will obtain
increasingly tighter constraints on the properties of
WIMPS with respect to their mass and interaction
cross-section. Such constraint will, however, depend
on the mass model of the dark matter in our Galaxy.
As a second approach, the Large Hadron Collider at
CERN will start operating in 2007 and should estab-
lish a new energy range for elementary particle physics.
In particular, we hope to find evidence for or against
the validity of the supersymmetric model for parti-
cle physics, as an extension of the current standard
model. Indeed, we might expect the detection of the
lightest supersymmetric particle, the neutralino, which
would be an excellent candidate for the dark matter
particle.
Whereas at least plausible ideas exist about the nature
of dark matter which can be experimentally tested in the
coming years, the presence of a non- vanishing density
of dark energy, as evidenced from cosmology, presents
an even larger mystery for fundamental physics. Though
from quantum physics we might expect a vacuum en-
ergy density to exist, its estimated energy density is
tremendously larger than the cosmic dark energy den-
sity. The interpretation that dark energy is a quantum
mechanical vacuum energy therefore seems highly im-
plausible. As astrophysical cosmologists, we could take
the view that vacuum energy is nothing more than
a cosmological constant, as originally introduced by
Einstein; this would then be an additional fundamental
constant in the laws of nature. From a physical point of
view, it would be much more satisfactory if the nature
of dark energy could be derived from the laws of fun-
damental physics. The huge discrepancy between the
density of dark energy and the simple estimate of the
vacuum energy density clearly indicates that we are cur-
rently far from a physical understanding of dark energy.
To achieve this understanding, we might well assume
that a new theory must be developed which unifies quan-
tum physics and gravity - in a manner similar to the way
other 'fundamental' interactions (like electromagnetism
and the weak force) have been unified within the stan-
dard model of particle physics. Deriving such a theory
of quantum gravity turns out to be enormously prob-
lematic despite intensive research over several decades.
However, the density of dark energy is so incredibly
small that its effects can only be recognized on the
largest length-scales, implying the necessity of further
astronomical and cosmological experiments. Only as-
tronomical techniques are able to probe the properties
of dark energy empirically.
To investigate the nature of dark energy, two different
approaches are currently seen as the most promising:
studying the Hubble diagram of type la supernovae,
and cosmic shear. To increase the sensitivity of both
methods substantially, a satellite mission is currently
being planned which will allow a precision application
of these methods by conducting wide-field multi-
color photometry from space. This will yield accurate
lightcurves of SNIa, as well as accurate shape mea-
surements of very faint galaxies which are needed for
cosmic shear studies. Furthermore, there are several
planned ground-based projects to build telescopes, or in-
struments for existing telescopes, which predominantly
aim at applying these two cosmological probes. One
of them is the Large Synoptic Survey Telescope, an
8-meter telescope with a 7 square degree field camera.
Although inflation is currently part of the standard
model of cosmology, the physical processes occurring
during the inflationary phase have not been under-
stood up to now. The fact that different field-theoretical
models of inflation yield very similar cosmological
consequences is an asset for cosmologists: from their
point-of-view, the details of inflation are not imme-
diately relevant, as long as a phase of exponential
expansion occurred. But the same fact indicates the
size of the problem faced in studying the process of
inflation, since different physical models yield rather
similar outcomes with regard to cosmological observ-
ables. Perhaps the most promising probe of inflation is
the polarization of the cosmic microwave background.
since it allows us to study whether, and with what
amplitude, gravitational waves were generated during
inflation. Predictions of the ratio of gravity wave energy
to that of density fluctuations are different in different
physical models of inflation. After the Planck satellite
has been put in orbit, a mission which is able to mea-
sure the CMB polarization with sufficient accuracy to
test inflation will probably be considered.
Another cosmological observation poses an addi-
tional challenge to fundamental physics. We observe
baryonic matter in our Universe, but we see no signs of
appreciable amounts of antimatter. If certain regions in
the Universe consisted of antimatter, there would be ob-
servable radiation from matter-antimatter annihilation
at the interface between the different regions. The ques-
tion therefore arises, what processes caused an excess
of matter over antimatter in the early Universe? We can
easily quantify this asymmetry - at very early times, the
abundance of protons, antiprotons and photons were all
quite similar, but after proton-antiproton annihilation at
T ~ 1 GeV, a fraction of ~ 10~ 10 - the current baryon-
to-photo ratio - is left over. This slight asymmetry of the
abundance of protons and neutrons over their antipar-
ticles in the early Universe, often called baryogenesis,
has not been explained in the framework of the standard
model of particle physics. Furthermore, we would like to
understand why the densities of baryons and dark mat-
ter are essentially the same, differing by a mere factor
of- 6.
The aforementioned issues are arguably the best ex-
amples of the increasingly tight connection between
cosmology and fundamental physics. Progress in either
field can only be achieved by the close collaboration be-
tween theoretical and experimental particle physics and
astronomy.
Finally, and perhaps too late in the opinion of some
readers, we should note again that this book has as-
sumed throughout that the physical laws, as we know
them today, can be used to interpret cosmic phenomena.
We have no real proof that this assumption is correct,
but the successes of this approach justify this assump-
tion in hindsight. If this assumption had been grossly
violated, there would be no reason why the values of the
cosmological parameters, estimated with vastly differ-
ent methods and thus employing very different physical
processes, mutually agree. The price we pay for the ac-
ceptance of the standard model of cosmology, which
results from this approach, is high though: the stan-
dard model implies that we accept the existence and
even dominance of dark matter and dark energy in the
Universe.
Not every cosmologist is willing to pay this price.
For instance, M. Milgrom introduced the hypothesis
that the flat rotation curves of spiral galaxies are not due
to the existence of dark matter. Instead, they could arise
from the possibility that the Newtonian law of gravity
ceases to be valid on scales of lOkpc - on such large
scales, and the correspondingly small accelerations, the
law of gravity has not been tested. Milgrom's Modified
Newtonian Dynamics (MOND) is therefore a logically
possible alternative to the postulate of dark matter on
scales of galaxies. Indeed, MOND offers an explanation
for the Tully-Fisher relation of spiral galaxies.
There are, however, several reasons why only a few
astrophysicists follow this approach. MOND has an
additional free parameter which is fixed by matching
the observed rotation curves of spiral galaxies with the
model, without invoking dark matter. Once this param-
eter is fixed, MOND cannot explain the dynamics of
galaxies in clusters without needing additional matter -
dark matter. Thus, the theory has just enough freedom
to fix a problem on one length- (or mass-)scale, but ap-
parently fails on different scales. We can circumvent the
problem again by postulating warm dark matter, which
would be able to fall into the potential wells of clus-
ters, but not into the shallower ones of galaxies, thereby
replacing one kind of dark matter (CDM) with another.
In fact, the consequences of accepting MOND would
be far reaching: if the law of gravity deviates from the
Newtonian law, the validity of General Relativity would
be questioned, since it contains the Newtonian force law
as a limiting case of weak gravitational fields. General
Relativity, however, forms the basis of our world mod-
els. Rejecting it as the correct description of gravity,
we would lose the physical basis of our cosmological
model - and thus the impressive quantitath e agreement
of results from vastly different observations that we
described in Chap. 8. The acceptance of MOND there-
fore demands an even higher price than the existence of
dark matter, but it is an interesting challenge to falsify
MOND empirically.
This example shows that the modification of one as-
pect of our standard model has the consequence that
the whole model is threatened: due to the large internal
consistency of the standard model, modifying one as-
pect has a serious impact on all others. This does not
mean that there cannot be other cosmological models
which can provide as consistent an explanation of the
relevant observational facts as our standard model does.
However, an alternative explanation of a single aspect
cannot be considered in isolation, but must be seen in its
relation to the others. Of course, this poses a true chal-
lenge to the promoters of alternative models: whereas
the overwhelming majority of cosmologists are work-
ing hard to verify and to refine the standard model and to
construct the full picture of cosmic evolution, the group
of researchers working on alternative models is small 1
and thus hardly able to put together a convincing and
consistent model of cosmology. This fact finds its jus-
tification in the successes of the standard model, and in
the agreement of observations with the predictions of
this model.
We have, however, just uncovered an important so-
ciological aspect of the scientific enterprise: there is
1 1 lowever. there has been a fairly recent increase in research activity
on MOM!. This was triggered mainly iiy the tact that after many years
of research, a theory called TcVcS llbi feasor Vector-Scalar field) was
invented, containing General Relath ity. MOM) and Newton's law in
the respective limits - though at the cost ot introducing three new
arbitral") functions.
a tendency to 'jump on the bandwagon'. This results in
the vast majority of research going into one (even if the
most promising) direction - and this includes scientific
staff, research grants, observing time etc. The conse-
quence is that new and unconventional ideas have a hard
time getting heard. Hopefully (and in the view of this au-
thor, very likely), the bandwagon is heading in the right
direction. There are historical examples to the contrary,
though - we now know that Rome is not at the center
of the cosmos, nor the Earth, nor the Sun, nor the Milky
Way, despite long epochs when the vast majority of
scientists were convinced of the veracity of these ideas.
Appendix
A. The Electromagnetic Radiation Field
In this appendix, we will briefly review the most impor-
tant properties of a radiation field. We thereby assume
that the reader has encountered these quantities already
in a different context.
A.l Parameters of the Radiation Field
The electromagnetic radiation field is described by the
specific intensity /„, which is defined as follows. Con-
sider a surface element of area dA. The radiation energy
which passes through this area per time interval df from
within a solid angle element d&> around a direction de-
scribed by the unit vector n, with frequency in the range
between v and v + d v, is
dE = I v dA cos(9dfdwdv
(A.l)
I d&> /„ cos 9 .
(A.2)
The flux that we receive from a cosmic source is defined
in exactly the same way, except that cosmic sources
usually subtend a very small solid angle on the sky.
In calculating the flux we receive from them, we may
therefore drop the factor cosO in (A.2); in this con-
text, the specific flux is also denoted as S v . However,
in this Appendix (and only here!), the notation S v will
be reserved for another quantity. The flux is measured
in units of erg cm -2 s" 1 Hz -1 . If the radiation field is
isotropic, F v vanishes. In this case, the same amount
of radiation passes through the surface element in both
directions.
The mean specific intensity J v is defined as the
average of I v over all angles,
4jt J
where 9 describes the angle between the direction n of
the light and the normal vector of the surface element.
Then, d A cos 9 is the area projected in the direction of
the Mailing light. The specific intensity depends on the
considered position (and, in time-dependent radiation
fields, on time), the direction n, and the frequency v.
With the definition (A.l), the dimension of I v is energy
per unit area, time, solid angle, and frequency, and it is
typically measured in units of erg cm~ 2 s~ ' ster~ l Hz~ l .
The specific intensity of a cosmic source describes its
surface brightness.
The specific net flux F v passing through an area el-
ement is obtained by integrating the specific intensity
over all solid angles,
so that, for an isotropic radiation field, /„ = J v . The
specific energy density u v is related to J v according to
u v =^J v (A.4)
where u v is the energy of the radiation field per vol-
ume element and frequency interval, thus measured in
erg cm -3 Hz -1 . The total energy density of the radia-
tion is obtained by integrating u v over frequency. In the
same way, the intensity of the radiation is obtained by
integrating the specific intensity /„ over v.
A.2 Radiative Transfer
The specific intensity of radiation in the direction of
propagation between source and observer is constant,
as long as no emission or absorption processes are oc-
curring. If s measures the length along a line-of- sight,
the above statement can be formulated as
d.v
(A.5)
An immediate consequence of this equation is that the
surface brightness of a source is independent of its
distance. The observed flux of a source depends on
its distance, because the solid angle, under which the
source is observed, decreases with the square of the
distance, F v oc D~ 2 (see Eq. A.2). However, for light
propagating through a medium, emission and absorp-
tion (or scattering of light) occurring along the path
over which the light travels may change the specific in-
tensity. These effects are described by the equation of
radiative transfer
d/ v
v Iv -
(A.6)
The first term describes the absorption of radiation
and states that the radiation absorbed within a length
interval As is proportional to the incident radiation.
Peter Schneider. The lileetromattnctie Radiation Field.
DOI:10.1007/11614371_A t !
A. The Electromagnetic Radiation Field
The factor of proportionality is the absorption coef-
ficient k v , which has the unit of cm -1 . The emission
coefficient j v describes the energy that is added to the
radiation field by emission processes, having a unit
of erg cm -3 s _1 Hz -1 ster -1 ; hence, it is the radiation
energy emitted per volume element, time interval, fre-
quency interval, and solid angle. Both, k v and j v depend
on the nature and state (such as temperature, chemi-
cal composition) of the medium through which light
propagates.
The absorption and emission coefficients both ac-
count for true absorption and emission processes, as
well as the scattering of radiation. Indeed, the scatter-
ing of a photon can be considered as an absorption that
is immediately followed by an emission of a photon.
The optical depth x v along a line-of-sight is defined
as the integral over the absorption coefficient,
= J ds'K v ( S '),
where sq denotes a reference point on the sightline from
which the optical depth is measured. Dividing (A. 6)
by k v and using the relation dx v — k v ds in order to
introduce the optical depth as a new variable along the
light ray. the equation of radiative transfer can be written
as
field by emission, accounted for by the r'-integral. Only
a fraction exp (x' v — r v ) of this additional energy emitted
at r' reaches the point r, the rest is absorbed.
In the context of (A. 10), we call this a formal solution
for the equation of radiative transport. The reason for
this is based on the fact that both the absorption and
the emission coefficient depend on the physical state of
the matter through which radiation propagates, and in
many situations this state depends on the radiation field
itself. For instance, k v and j v depend on the temperature
of the matter, which in turn depends, by heating and
cooling processes, on the radiation field to which it is
exposed. Hence, one needs to solve a coupled system
of equations in general: on the one hand the equation of
radiative transport, and on the other hand the equation
of state for matter. In many situations, very complex
problems arise from this, but we will not consider them
further in the context of this book.
A.3 Blackbody Radiation
For matter in thermal equilibrium, the source func-
tion S v is solely a function of the matter temperature,
where the source function
S v =
(A.9)
S v =
B V {T), or j v = B v {T) K
(A. 11)
is defined as the ratio of the emission and absorption
coefficients. In this form, the equation of radiative trans-
port can be formally solved; as can easily be tested by
substitution, the solution is
/„(T V ) = / V (0)exp(-T„)
+ £ dx' v exp (r' v - x v ) S v (r' v ) ■ (A. 10)
o
This equation has a simple interpretation. If I v (0) is the
incident intensity, it will have decreased by absorption
to a value I v (0) exp (— t v ) after an optical depth of r v .
On the other hand, energy is added to the radiation
independent of the composition of the medium (Kirch-
hoff's law). We will now consider radiation propagating
through matter in thermal equilibrium at constant tem-
perature T. Since in this case S v = B V (T) is constant,
the solution (A. 10) can be written in the form
7 v (Tv) = /v(0)exp(-r v )
+ B V (T) I dr' v exp«-T v )
= /„(0) exp(-T y ) + B y (r) [l-exp(-T„)] .
(A. 12)
From this it follows that /„ = B V (T) is valid for suffi-
ciently large optical depth r v . The radiation propagating
through matter which is in thermal equilibrium is de-
scribed by the function B V (T) if the optical depth is
sufficiently large, independent of the composition of
the matter. A specific case of this situation can be il-
lustrated by imagining the radiation field inside a box
whose opaque walls are kept at a constant tempera-
ture T. Due to the opaqueness of the walls, their optical
depth is infinite, hence the radiation field within the box
is given by /„ = 5„(r). This is also valid if the volume is
filled with matter, as long as the latter is in thermal equi-
librium at temperature T. For these reasons, this kind
of radiation field is also called blackbody radiation.
The function B V (T) was first obtained in 1900 by
Max Planck, and in his honor, it was named the Planck
function; it reads
2h P v 3 1
- eW -r (A - 13)
where h P — 6.625 x 10~ 27 erg s is the Planck constant
and &b = 1.38 x 10~ 16 ergK -1 is the Boltzmann con-
stant. The shape of the spectrum can be derived from
statistical physics. Blackbody radiation is defined by
/„ = B V (T), and thermal radiation by S v = B V (T). For
large optical depths, thermal radiation converges to
blackbody radiation.
The Planck function has its maximum at
B V (T) =
a 2.82,
(A. 14)
i.e., the frequency of the maximum is proportional to
the temperature. This property is called Wien 's law. This
law can also be written in more convenient units,
The Planck function can also be formulated de-
pending on wavelength X — c/v, such that B^(T) dX =
B v (T)dv,
Bx(T) =
(A. 16)
2h P c 2 /X 5
~ exp (hpc/Xlc B T) - 1 '
Two limiting cases of the Planck function are of par-
ticular interest. For low frequencies, h P v <§C k s T, one
can apply the expansion of the exponential function for
small arguments in (A. 13). The leading-order term in
this expansion then yields
B V (T) ss B™(T) =
which is called the Rayleigh 1 an approximation of Hoe
Planck function. We point out that the Rayleigh-Jeans
equation does not contain the Planck constant, and this
law had been known even before Planck derived his
log v [Hz]
Fig.A.l. The Planck function (A. 13) for different tempera-
tures T. The plot shows B V (T) as a function of frequency v,
where high frequencies are plotted towards the left (thus large
\\ :i\ elengths towards the right). The exponentially decreasing
Wien part of the spectrum is visible on the left, the Rayleigh-
Jeans part on the right. The shape of the spectrum in the
Rayleigh-Jeans part is independent of the temperature, which
is determining the amplitude however
exact equation. In the other limiting case of very high
frequencies, h P v ^> k B T, the exponential factor in the
denominator in (A. 13) becomes very much larger than
unity, so that we obtain
(A. 17)
2/ipi
called the Wien approximation of the Planck function.
The energy density of blackbody radiation depends
only on the temperature, of course, and is calculated by
A. The Electromagnetic Radiation Field
integration over the Planck function,
4jt f An ,
u=— dvB v (T) = — B(T)=aT 4 , (A.19)
c J c
where we defined the frequency-integrated Planck
function
B(T) = / dv B V {T) = — T 4 , (A.20)
J 4tt
and where the constant a has the value
8jt 5 £ 4 ,, , ,
a= -4= 7.56 x 10~ 15 ergcm" 3 Kr 4 . (A.21)
I5c 3 hp
The flux which is emitted by the surface of a blackbody
per unit area is given by
F = I dv F v = it I dv B V (T) = ttB{T) = o- SB r 4 ,
where the Stefan-Boltzmann constant ctsb has a value
of
a precise definition. Since no historical astronomical
observations have been conducted in other wavelength
ranges, because these are not accessible to the unaided
eye, only optical astronomy has to bear the historical
burden of the magnitude system.
A.4.1 Apparent Magnitude
We start with a relative system of flux measurements
by considering two sources with fluxes S\ and Si. The
apparent magnitudes of the two sources, m\ and ni2,
then behave according to
rs,\ 5, = .4( mi - M2 ) _
s 2
(A.24)
-2.5 log
This means that the brighter source has a smaller ap-
parent magnitude than the fainter one: the larger the
apparent magnitude, the fainter the source. 1 The fac-
tor of 2.5 in this definition is chosen so as to yield the
best agreement of the magnitude system with the visu-
ally determined magnitudes. A difference of \Am\ = 1
in this system corresponds to a flux ratio of ~ 2.51, and
a flux ratio of a factor 10 or 100 corresponds to 2.5 or 5
magnitudes, respectively.
A.4 The Magnitude Scale
Optical astronomy was being conducted well before
methods of quantitative measurements became avail-
able. The brightness of stars had been cataloged more
than 2000 years ago, and their observation goes back
as far as the ancient world. Stars were classified into
magnitudes, assigning a magnitude of 1 to the brightest
stars and higher magnitudes to the fainter ones. Since
the apparent magnitude as perceived by the human eye
scales roughly logarithmically with the radiation flux
(which is also the case for our hearing), the magni-
tude scale represents a logarithmic flux scale. To link
these visually determined magnitudes in historical cat-
alogs to a quantitative measure, the magnitude system
has been retained in optical astronomy, although with
A.4.2 Filters and Colors
Since optical observations are performed using a com-
bination of a filter and a detector system, and since the
flux ratios depend, in general, on the choice of the filter
(because the spectral energy distribution of the sources
may be different), apparent magnitudes are defined for
each of these filters. The most common filters are shown
in Fig. A.2 and listed in Table A. 1 , together with their
characteristic wavelengths and the widths of their trans-
mission curves. The apparent magnitude for a filter X is
defined as m x , frequently written as X. Hence, for the
B-band filter, m B = B.
Next, we need to specify how the magnitudes mea-
sured in different filters are related to each other, in
order to define the color indices of sources. For this
! is confusing. particular!) to someone just
becoming familiar u ith astronomy, and it frequently causes confusion
and errors, as well as problems in
: to get alone « ith that.
A.4 The Magnitude Scale
Fig. A.2. Transmission curves of various
filter-detector systems. From top to bot-
tom: the filters of the NICMOS camera and
the WFPC2 on-board HST, the Washing-
ton filter system, the filters of the EMMI
instrument at ESO's NTT, the filters of
the WFI at the ESO/MPG 2.2-m telescope
and those of the SOFI instrument at the
NTT, and the Johnson-Cousins filters. In
the bottom diagram, the spectra of three
stars with different effective temperatures
are displayed
Table A.l. For some of the best-established filter systems - Johnson, Stromgren, and the filters of the Sloan Digital Sky
Surveys - the central (more precisely, the effective) wavelengths and the widths of the filters are listed
SDSS u' g- 1 i'
1? S
354
57
139
138
762
152
913
95
A. The Electromagnetic Radiation Field
purpose, a particular class of stars is used, main-
sequence stars of spectral type AO, of which the star
Vega is an archetype. For such a star, by definition,
U — B —V — R — I — ..., i.e., every color index for
such a star is defined to be zero.
For a more precise definition, let T x (v) be the
transmission curve of the filter-detector system. T x {v)
specifies which fraction of the incoming photons with
frequency v are registered by the detector. The apparent
magnitude of a source with spectral flux S v is then
( fdvT x (v)SS
V fdvT x (v) ,
(A.25)
where the constant needs to be determined from
reference stars.
Another commonly used definition of magnitudes
is the AB system. In contrast to the Vega mag-
nitudes, no stellar spectral energy distribution is
used as a reference here, but instead one with
a constant flux at all frequencies, S™ f = S^ B =
2.89 x 10~ 21 ergs -1 cm -2 Hz -1 . This value has been
chosen such that AO stars like Vega have the same mag-
nitude in the original Johnson V-band as they have in
the AB system, rriy B — m v . With (A.25), one obtains
for the conversion between the two systems
fdvT x (v)S™
f dv t x (v) s; ega
= : m A B^Vega ■
(A.26)
For the filters at the ESO Wide-Field Imager, which
are designed to resemble the Johnson set of filters, the
following prescriptions are then to be applied: Uab —
[/vega + 0.80; Sab = ^Vega - 0.1 1; Vab = VvegaJ Rab =
tf Vega + 0. 19; / AB = / V ega + 0.59.
A.4.3 Absolute Magnitude
The apparent magnitude of a source does not in itself tell
us anything about its luminosity, since for the determi-
nation of the latter we also need to know its distance D
in addition to the radiative flux. Let L v be the specific
luminosity of a source, i.e., the energy emitted per unit
time and per unit frequency interval, then the flux is
given by (note that from here on we switch back to the
notation where S denotes the f
by F earlier in this appendix)
S v =
4-itD 2 '
., which was denoted
(A.27)
where we implicitly assumed that the source emits
isotropically. Having the apparent magnitude as a mea-
sure for S v (at the frequency v defined by the filter
which is applied), it is desirable to have a similar mea-
sure for L v , specifying the physical properties of the
source itself. For this purpose, the absolute magnitude
is introduced, denoted as M x , where X refers to the
filter under consideration. By definition, M x is equal
to the apparent magnitude of a source if it were to
be located at a distance of 10 pc from us. The abso-
lute magnitude of a source is thus independent of its
distance, in contrast to the apparent magnitude. With
(A.27) we find for the relation of apparent to absolute
magnitude
- Ah
1 pc/
-5 = a*.
(A.28)
where we have defined the distance modulus p in the
final step. Hence, the latter is a logarithmic measure of
the distance of a source: p = for D = 10 pc, p — 10
for D = 1 kpc, and p = 25 for D = 1 Mpc. The dif-
ference between apparent and absolute magnitude is
independent of the filter choice, and it equals the dis-
tance modulus if no extinction is present. In general, this
difference is modified by the filter-dependent e:
coefficient - see Sect. 2.2.4.
A.4.4 Bolometric Parameters
The total luminosity L of a source is the integral
of the specific luminosity L v over all frequencies.
Accordingly, the total flux S of a source is the frequency-
integrated specific flux S v . The apparent bolometric
magnitude m\, \ is defined as a logarithmic measure of
the total flux,
-2.5 log S +
(A.29)
where here the constant is also determined from ref-
erence stars. Accordingly, the absolute bolometric
magnitude is defined by means of the distance mod-
ulus, as in (A.28). The absolute bolometric magnitude
A.4 The Magnitude Scale
depends on the bolometric luminosity L of a source via
M bo i = -2.5 log L + const. (A.30)
The constant can be fixed, e.g., by using the parame-
ters of the Sun: its apparent bolometric magnitude is
m obol = —26.83, and the distance of one Astronomical
Unit corresponds to a distance modulus of /x = —3 1 .47.
With these values, the absolute bolometric magnitude
of the Sun becomes
M obol = m 0bol -/Lt = 4.74, (A.31)
so that (A.30) can be written as
M bol = 4.74 -2.5 log (■£-} , (A.32)
and the luminosity of the Sun is then
L e = 3.85 xlO 33 erg s" 1 . (A.33)
The direct relation between bolometric magnitude and
luminosity of a source can hardly be exploited in prac-
tice, because the apparent bolometric magnitude (or the
flux 5) of a source cannot be observed in most cases. For
observations of a source from the ground, only a lim-
ited window of frequencies is accessible. Nevertheless,
in these cases one also likes to quantify the total lumi-
nosity of a source. For sources for which the spectrum
is assumed to be known, like for many stars, the flux
from observations at optical wavelengths can be extrap-
olated to larger and smaller wavelengths, and so m bo i
can be estimated. For galaxies or AGNs, which have
a much broader spectral distribution and which show
much more variation between the different objects, this
is not feasible. In these cases, the flux of a source in
a particular frequency range is compared to the flux
the Sun would have at the same distance and in the
same spectral range. If M x is the absolute magnitude of
a source measured in the filter X, the X-band luminosity
of this source is defined as
= 10" 1
Lqx ■
(A.34)
Thus, when speaking of, say, the "blue luminosity of
a galaxy", this is to be understood as defined in (A.34).
B. Properties of Stars
In this appendix, we will summarize the most important
properties of stars as they are required for understanding
the contents of this book. Of course, this brief overview
cannot replace the study of other textbooks in which the
physics of stars is covered in much more detail.
B.l The Parameters of Stars
To a good approximation, stars are gas spheres, in the
cores of which light atomic nuclei are transformed into
heavier ones (mainly hydrogen into helium) by ther-
monuclear processes, thereby producing energy. The
external appearance of a star is predominantly character-
ized by its radius R and its characteristic temperature T.
The properties of a star depend mainly on its mass M.
In a first approximation, the spectral energy distri-
bution of the emission from a star can be described
by a blackbody spectrum. This means that the specific
intensity I v is given by a Planck spectrum (A. 13) in
this approximation. The luminosity L of a star is the
energy radiated per unit time. If the spectrum of star
was described by a Planck spectrum, the luminosity
would depend on the temperature and on the radius
according to
L = 4jvR 2 ctsb T 4 ,
(B.I)
where (A. 22) was applied. However, the spectra of stars
deviate from that of a blackbody (see Fig. 3.47). One
defines the effective temperature T e{{ of a star as the
temperature a blackbody of the same radius would need
to have to emit the same luminosity as the star, thus
(B.2)
ct sb r e 4 ff = -
4ttR 2
The luminosities of stars cover a huge range; the weak-
est are a factor ~ 10 4 times less luminous than the Sun,
whereas the brightest emit ~ 10 5 times as much en-
ergy per unit time as the Sun. This big difference in
luminosity is caused either by a variation in radius or
by different temperatures. We know from the colors of
stars that they have different temperatures: there are
blue stars which are considerably hotter than the Sun,
and red stars that are very much cooler. The temper-
ature of a star can be estimated from its color. From
the flux ratio at two different wavelengths or, equiva-
lently, from the color index X— Y = m x — m Y in two
filters X and Y, the temperature T c is determined such
that a blackbody at T c would have the same color in-
dex. T c is called the color temperature of a star. If the
spectrum of a star was a Planck spectrum, then the
equality T c — T e g would hold, but in general these two
temperatures differ.
B.2 Spectral Class, Luminosity Class,
and the Hertzsprung-Russell
Diagram
The spectra of stars can be classified according to the
atomic (and, in cool stars, also molecular) spectral lines
that are present. Based on the line strengths and their
ratios, the Harvard sequence of stellar spectra was intro-
duced. These spectral classes follow a sequence that is
denoted by the letters O, B, A, F, G, K, M; besides these,
some other spectral classes exist that will not be men-
tioned here. The sequence corresponds to a sequence
of color temperature of stars: O stars are particularly
hot, around 50 000 K, M stars very much cooler with
Tc ~ 3500 K. For a finer classification, each spectral
class is supplemented by a number between and 9. An
Al star has a spectrum very similar to that of an A0 star,
whereas an A5 star has as many features in common
with an A0 star as with an F0 star.
Plotting the spectral type versus the absolute magni-
tude for those stars for which the distance and hence
the absolute magnitude can be determined, a strik-
ing distribution of stars becomes apparent in such
a Hcrlzsprung- Russell diagram (HRD). Instead of the
spectral class, one may also plot the color index of the
stars, typically B — V or V — I. The resulting color-
inagiiitiide diagram (CMD) is essentially equivalent to
an HRD, but is based solely on photometric data. A dif-
ferent but very similar diagram plots the luminosity
versus the effective temperature.
In Fig. B.l, a color-magnitude diagram is plotted,
compiled from data observed by the HIPPARCOS satel-
lite. Instead of filling the two-dimensional parameter
space rather uniformly, characteristic regions exist in
Peter Schneider. Properties of Stars.
In: Peter Schneider. Extragalactic Astronomy and Cosmology, pp. 425^429 (2006)
DOI: 10.1007/1 1614371_B © Springer- Verlag Berlin Heidelberg 2006
B. Properties of Stars
Since stars exist which have, for the same spectral
type and hence the same color temperature (and roughly
the same effective temperature), very different lumi-
nosities, we can deduce immediately that these stars
have different radii, as can be read from (B.2). There-
fore, stars on the red giant branch, with their much
higher luminosities compared to main-sequence stars
of the same spectral class, have a very much larger ra-
dius than the corresponding main-sequence stars. This
size effect is also observed spectroscopically: the grav-
itational acceleration on the surface of a star (surface
gravity) is
Fig. B.l. Color-magnitude diagram for 41453 indixidual
stars, whose parallaxes were determined by the HIPPARCOS
satellite with an accuracy of better than 20%. Since the stars
shown here are subject to unavoidable strong selection effects
favoring nearby and luminous stars, the relative number den-
sity of stars is not representative of their true abundance. In
particular, the lower main sequence is much more densely
populated than is visible in this diagram
such color-magnitude diagrams in which nearly all stars
are located. Most stars can be found in a thin band called
the main sequence. It extends from early spectral types
(O, B) with high luminosities ("top left") down to late
spectral types (K, M) with low luminosities ("bottom
right"). Branching off from this main sequence towards
the "top right" is the domain of red giants, and below the
main sequence, at early spectral types and very much
lower luminosities than on the main sequence itself, we
have the domain of white dwarfs. The fact that most
stars are arranged along a one-dimensional sequence -
the main sequence - is probably one of the most impor-
tant discoveries in astronomy, because it tells us that the
properties of stars are determined basically by a single
parameter: their mass.
GM
(B.3)
We know from models of stellar atmospheres that the
width of spectral lines depends on the gravitational ac-
celeration on the star's surface: the lower the surface
gravity, the narrower the stellar absorption lines. Hence,
a relation exists between the line width and the stellar
radius. Since the radius of a star - for a fixed spectral
type or effective temperature - specifies the luminos-
ity, this luminosity can be derived from the width of the
lines. In order to calibrate this relation, stars of known
distance are required.
Based on the width of spectral lines, stars are clas-
sified into luminosity classes: stars of luminosity class I
are called supergiants, those of luminosity class III are
giants, main-sequence stars are denoted as dwarfs and
belong to luminosity class V; in addition, the classifi-
cation can be further broken down into bright giants
(II), subgiants (IV), and subdwarfs (VI). Any star in the
Hertzsprung-Russell diagram can be assigned a lumi-
nosity class and a spectral class (Fig. B.2). The Sun is
a G2 star of luminosity class V.
If the distance of a star, and thus its luminosity, is
known, and if in addition its surface gravity can be
derived from the line width, we obtain the stellar mass
from these parameters. By doing so, it turns out that for
main-sequence stars the luminosity is a steep function
of the stellar mass, approximately described by
l ( m y- 5
Therefore, a main-sequence star of M = 10M o
~ 3000 times more luminous than our Sun.
B.3 Structure and Evolution of Stars
Fig.B.2. Schematic color-magnitude diagram in which the
spectral types and luminosity classes are indicated
B.3 Structure and Evolution of Stars
To a very good approximation, stars are spherically sym-
metric. Therefore, the structure of a star is described by
the radial profile of the parameters of its stellar plasma.
These are density, pressure, temperature, and chemical
composition of the matter. During almost the full life-
time of a star, the plasma is in hydrostatic equilibrium,
so that pressure forces and gravitational forces are of
equal magnitude and directed in opposite directions, so
as to balance each other.
The density and temperature are sufficiently high in
the center of a star that thermonuclear reactions are ig-
nited. In main-sequence stars, hydrogen is fused into
helium, thus four protons are combined into one 4 He
nucleus. For every helium nucleus that is produced this
way, 26.73 MeV of energy are released. Part of this
energy is emitted in the form of neutrinos which can
escape unobstructed from the star due to their very
low cross-section. 1 The energy production rate is ap-
proximately proportional to T 4 for temperatures below
about 15 x 10 6 K, at which the reaction follows the
so-called pp-chain. At higher temperatures, another re-
action chain starts to contribute, the so-called CNO
cycle, with an energy production rate which is much
more strongly dependent on temperature - roughly
proportional to T 20 .
The energy generated in the interior of a star is trans-
ported outwards, where it is then released in the form
of electromagnetic radiation. This energy transport may
take place in two different ways: first, by radiation trans-
port, and second, it can be transported by macroscopic
flows of the stellar plasma. This second mechanism of
energy transport is called convection; here, hot elements
of the gas rise upwards, driven by buoyancy, and at the
same time cool ones sink downwards. The process is
similar to that observed in heating water on a stove.
Which of the two processes is responsible for the en-
ergy transport depends on the temperature profile inside
the star. The intervals in a star's radius in which energy
transport takes place via convection are called convec-
tion zones. Since in convection zones stellar material is
subject to mixing, the chemical composition is homoge-
neous there. In particular, chemical elements produced
by nuclear fusion are transported through the star by
convection.
Stars begin their lives with a homogeneous chemi-
cal composition, resulting from the composition of the
molecular cloud out of which they are formed. If their
mass exceeds about O.O8M , the temperature and pres-
sure in their core are sufficient to ignite the fusion of
hydrogen into helium. Gas spheres with a mass below
~ O.O8M will not satisfy these conditions, hence these
objects - they are called brown dwarfs - are not stars in
The detection of neutrinos from the Sun in terrestrial detectors was
the linal proof for the energy production mechanism being nuclear
fusion. llowcAcr. the measured rate of election neutrinos from the
.mi ' ml i ill I ,i | i d h nn '' i m. d I I hi ii
neutrino problem kept pin sieists and astrophysicists busy i or decades.
1 1 ii i nil i hi ii i I
in this ease could electron neutrinos transform into another son ol
neutrino along the \va\ from the Sun to us. Recently, these neutrino
oscillations were confirmed: neutrinos ha\e a \er_\ small but linitc resl
mass. For their research in the lieldol Solar neutrinos. Ra\ mond t)a\ is
nd M n i in ill half of the Nobel Pri
in Plnsics ill 2002. I he oilier hall was aw aided lo Rieardo Giaeconi
for his pioneering work in the field of X-ray astronomy.
B. Properties of Stars
logf '
M bol
^^l£> 120 Mm
g^T^^i> 8b M B
' -T^l -i -T~
->
->
//m-
_m
" 25 Mg^^^^
'/Ma-
"'>'/'////
15 M ®viill^
' ////// /,.
==3|p-
12 M®^^^
9M(3 ^^
^ ^^
^irm'r
7M >§|^
5M \
z<jMff/-
4M
3M ^ -\
-
2,5M ®^^^~X
M
2M ^^^^\
Jf
1JM@ ^^N
X = 0.70
Z = 0.02
1 .5 M. H > ^^
1.3M S V^
1.15 M V
V
imA
-
04 07 09B0 B1 B2 B3
0.85 M,A
B5 B8 A0A3 A7 F1 F8G2K
W
Fig. B.3. Theoretical temperature-luminos-
ity diagram of stars. The solid curve is the
zero age main sequence (ZAMS), on which
stars ignite the burning of hydrogen in their
cores. The evolutionary tracks of these stars
are indicated by the various lines which are
labeled with the stellar mass. The hat. heel
areas mark phases in which the evolution
proceeds only slowly, so that many stars are-
observed to be in these areas
4.7 4.6 4.5 4.4 4.3 4.2 4.1 4.0 3.9 3.8 3.7 3.6
log Tie
a proper sense. 2 At the onset of nuclear fusion, the star
is located on the zero-age main sequence (ZAMS) in the
HRD (see Fig.B.3). The energy production by fusion
of hydrogen into helium alters the chemical composi-
tion in the stellar interior; the abundance of hydrogen
II the mass of a frown dwarf exceeds 0.0 1 3M .. the central density
and temperature are high enough to enable the fusion of deuterium
(heavj hydrogen) into helium. However, the abundance of deu-
terium is smaller by several orders of magnitude than that of normal
hydrogen, rendering the fuel reservoir of a brown dwarf very small.
decreases by the same rate as the abundance of helium
increases. As a consequence, the duration of this phase
of central hydrogen burning is limited. As a rough es-
timate, the conditions in a star will change noticeably
when about 10% of its hydrogen is used up. Based on
this ct itcrion, the lifetime of a star on the main sequence
can now be estimated. The total energy produced in this
phase can be written as
£ MS = 0.1 xMc 2 x 0.007,
(B.5)
B.3 Structure and Evolution of Stars
where Mc 2 is the rest-mass energy of the star, of which
a fraction of 0. 1 is fused into helium, which is supposed
to occur with an efficiency of 0.007. Phrased differently,
in the fusion of four protons into one helium nucleus, an
energy of ~ 0.007 x 4m p c 2 is generated, with m p denot-
ing the proton mass. In particular, (B.5) states that the
total energy produced during this main-sequence phase
is proportional to the mass of the star. In addition, we
know from (B.4) that the luminosity is a steep function
of the stellar mass. The lifetime of a star on the main se-
quence can then be estimated by equating the available
energy E M s with the product of luminosity and lifetime.
This yields
Ems _ _ . _ 9 M/M Q
fas = —r- % 8 x 10 T u y f
< L0 y
/ M N
yr.
(B.6)
Using this argument, we observe that stars of higher
mass conclude their lives on the main sequence much
faster than stars of lower mass. The Sun will remain on
the main sequence for about eight to ten billion years,
with about half of this time being over already. In com-
parison, very luminous stars, like O and B stars, will
have a lifetime on the main sequence of only a few mil-
lion years before they have exhausted their hydrogen
fuel.
In the course of their evolution on the main sequence,
stars move away only slightly from the ZAMS in the
HRD, towards somewhat higher luminosities and lower
effective temperatures. In addition, the massive stars in
particular can lose part of their initial mass by stellar
winds. The evolution after the main-sequence phase de-
pends on the stellar mass. Stars of very low mass, M <
0.7M o , have a lifetime on the main sequence which
is longer than the age of the Universe, therefore they
cannot have moved away from the main sequence yet.
For massive stars, M > 2.5M , central hydrogen
burning is first followed by a relatively brief phase in
which the fusion of hydrogen into helium takes place
in a shell outside the center of the star. During this
phase, the star quickly moves to the "right" in the
HRD, towards lower temperatures, and thereby expands
strongly. After this phase, the density and temperature
in the center rise so much as to ignite the fusion of
helium into carbon. A central helium-burning zone will
then establish itself, in addition to the source in the shell
where hydrogen is burned. As soon as the helium in the
core has been exhausted, a second shell source will form
fusing helium. In this stage, the star will become a red
giant or supergiant, ejecting part of its mass into the
ISM in the form of stellar winds. Its subsequent evolu-
tionary path depends on this mass loss. A star with an
initial mass M < 8M Q will evolve into a white dwarf,
which will be discussed further below.
For stars with initial mass M < 2.5 M Q , the helium
burning in the core occurs explosively, in a so-called he-
lium flash. A large fraction of the stellar mass is ejected
in the course of this flash, after which a new stable equi-
librium configuration is established, with a helium shell
source burning beside the hydrogen-burning shell. Ex-
panding its radius, the star will evolve into a red giant or
supergiant and move along the asymptotic giant branch
(AGB) in the HRD.
The configuration in the helium shell source is unsta-
ble, so that its burning will occur in the form of pulses.
After some time, this will lead to the ejection of the
outer envelope which then becomes visible as a plan-
etary nebula. The remaining central star moves to the
left in the HRD, i.e., its temperature rises considerably
(to more than 10 5 K). Finally, its radius gets smaller
by several orders of magnitude, so that the the stars
move downwards in the HRD, thereby slightly reduc-
ing its temperature: a white dwarf is born, with a mass
of about 0.6M Q and a radius roughly corresponding to
that of the Earth.
If the initial mass of the star is > 8 M Q , the tempera-
ture and density at its center become so large that carbon
can also be fused. Subsequent stellar evolution towards
a core-collapse supernova is described in Sect. 2.3.2.
The individual phases of stellar evolution have very
different time-scales. As a consequence, stars pass
through certain regions in the HRD very quickly, and for
this reason stars at those evolutionary stages are never or
only rarely found in the HRD. By contrast, long-lasting
evolutionary stages like the main sequence or the red
giant branch exist, with those regions in an observed
HRD being populated by numerous stars.
C. Units and Constants
In this book, we consistently used, besides astro-
nomical units, the Gaussian cgs system of units,
with lengths measured in cm, masses in g, and en-
ergies in erg. This is the commonly used system
of units in astronomy. In these units, the speed of
light is c — 2.998 x 10 10 cms -1 , the masses of pro-
tons, neutrons, and electrons are m p — 1.673 x 10~ 24 g,
m n = 1.675 x 1CT 24 g, and m e = 9.109 x 1CT 28 g, re-
spectively.
Frequently used units of length in astronomy
include the Astronomical Unit, thus the average
separation between the Earth and the Sun, where
1 AU= 1.496 x 10 13 cm, and the parsec (see Sect. 2.2.1
for the definition), 1 pc = 3.086 x 10 18 cm. A year
has 1 yr = 3.156 x 10 7 s. In addition, masses are typi-
cally specified in Solar masses, 1M© = 1.989 x 10 33 g,
and the bolometric luminosity of the Sun is
L = 3.846 x lO^ergs" 1 .
In cgs units, the value of the elementary charge
is e = 4.803x 10~ 10 cm 3 / 2 g 1/2 s" 1 , and the unit of
the magnetic field strength is one Gauss, where
1 G = 1 g 1/2 cm" 1 / 2 s" 1 = 1 erg 1 / 2 cm" 3 / 2 . One of the
very convenient properties of cgs units is that the en-
ergy density of the magnetic field in these units is given
by p B — B 2 /(8jz) - the reader may check that the units
of this equation is consistent.
X-ray astronomers measure energies in electron
Volts, where 1 eV = 1.602 x 10 12 erg. Temperatures can
also be measured in units of energy, because k B T has
the dimension of energy. They are related according
to 1 eV = 1.161 x 10 4 &b K. Since we always use the
Boltzmann constant k B in combination with a tem-
perature, its actual value is never needed. The same
holds for Newton's constant of gravity which is al-
ways used in combination with a mass. Here one has
GM e c- 2 = 1.495 x 10 5 cm.
The frequency of a photon is linked to its energy ac-
cording to h P v — E, and we have the relation 1 eV hp 1 =
2.418 x 10 14 s" 1 = 2.418 x 10 14 Hz. Accordingly, we
can write the wavelength X — c/v — h P c/E in the form
leV
1.2400 x 10" 4 cm= 12 400 A.
Peter Schneider. Units and Constants
DOI: 10.1007/11614371_C © Springer-
D. Recommended Literature
In the following, we will give some recommendations
for further study of the literature on astrophysics. For
readers who have been in touch with astronomy only
occasionally until now, the general textbooks may be
of particular interest. The choice of literature presented
here is a very subjective one which represents the pref-
erences of the author, and of course it represents only
a small selection of the many astronomy texts available.
D.2 More Specific Literature
More specific monographs and textbooks exist for the
individual topics covered in this book, some of which
shall be suggested below. Again, this is just a brief
selection. The technical level varies substantially among
these books and, in general, exceeds that of the present
D.l General Textbooks
There exist a large selection of general textbooks in
astronomy which present an overview of the field at
a non-technical level. A classic one and an excellent
presentation of astronomy is
• F. Shu: The Physical Universe: An Introduction
to Astronomy, University Science Books, Sausalito,
1982.
Turning to more technical books, at about the level of
the present text, my favorite is
• B.W. Carroll & DA. Ostlie: An Introduction to
Modern Astrophysics, Addison Wesley, Reading,
1996;
its ~ 1400 pages cover the whole range of astronomy.
The text
• M.L. Kutner: Astronomy: A Physical Perspective,
Cambridge University Press, Cambridge, 2003
also covers the whole field of astronomy. A text with
a particular focus on stellar and Galactic astronomy is
• A. Unsold & B. Baschek: The New Cosmos,
Springer- Verlag, Berlin, 2002.
The recently published book
• M.H. Jones & R.J. A. Lambourne: An Introduction
to Galaxies and Cosmology, Cambridge University
Press, Cambridge, 2003
covers the topics described in this book and is also
highly recommended; it is less technical than the present
text.
Astrophysical Processes
• M. Harwit: Astrophysical Concepts, Springer, New-
York, 1988,
• G.B. Rybicki & A. P. Lightman: Radiative Processes
in Astrophysics, John Wiley & Sons, New York,
1979,
• F. Shu: The Physics of Astrophysics I: Radiation,
University Science Books, Mill Valley, 1991,
• F. Shu: The Physics of Astrophysics II: Gas
Dynamics, University Science Books, Mill Valley,
1991,
• S.N. Shore: The Tapestry of Modern Astrophysics,
Wiley- VCH, Berlin, 2002,
• D.E. Osterbrock: Astrophysics of Gaseous Nebu-
lae and Active Galactic Nuclei, University Science
Books, Mill Valley, 1989.
Furthermore, there is a three-volume set of books,
• T. Padmanabhan: Theoretical Astrophysics: I. As-
trophysical Processes. II. Stars and Stellar Systems.
III. Galaxies and Cosmology, Cambridge University
Press, Cambridge, 2000.
Galaxies and Gravitational Lenses
• L.S. Sparke & J.S. Gallagher: Galaxies in the Uni-
verse: An Introduction, Cambridge University Press,
Cambridge, 2000,
• J. Binney & M. Merrifield: Galactic Astronomy,
Princeton University Press, Princeton, 1998,
• J. Binney & S. Tremaine: Galactic Dynamics,
Princeton University Press, Princeton, 1987,
Peter Schneider. Recommended Literature.
DOI: 10.1007/11614371J.) B ,!
D. Recommended Literature
• R.C. Kennicutt, Jr., F. Schweizer & J.E. Barnes: Gal-
axies: Interactions and Induced Star Formation,
Saas-Fee Advanced Course 26, Springer- Verlag,
Berlin, 1998,
• B.E.J. Pagel: Nucleosynthesis and Chemical Evo-
lution of Galaxies, Cambridge University Press,
Cambridge, 1997,
• F. Combes, P. Boisse, A. Mazure & A. Blanchard
Galaxies and Cosmology, Springer- Verlag, 2001,
• P. Schneider, J. Ehlers & E.E. Falco: Gravitational
Lenses, Springer- Verlag, New York, 1992,
• P. Schneider, C.S. Kochanek & J. Wambsganss:
Gravitational Lensing: Strong, Weak & Micro, Saas-
Fee Advanced Course 33, G. Meylan, P. Jetzer & P.
North (Eds.), Springer- Verlag, Berlin, 2006.
Active Galaxies
• B.M. Peterson: An Introduction to Active Galac-
tic Nuclei, Cambridge University Press, Cambridge,
1997,
• R.D. Blandford, H. Netzer & L. Woltjer: Active
Galactic Nuclei, Saas-Fee Advanced Course 20,
Springer- Verlag, 1990,
• J. Krolik: Active Galactic Nuclei, Princeton
University Press, Princeton, 1999,
• J. Frank, A. King & D. Raine: Accretion Power in As-
trophysics, Cambridge University Press, Cambridge,
2002.
Cosmology
• M.S. Longair: Galaxy Formation, Springer- Verlag,
Berlin, 1998,
• J. A. Peacock: Cosmological Physics, Cambridge
University Press, Cambridge, 1999,
• T. Padmanabhan: Structure Formation in the Uni-
verse, Cambridge University Press, Cambridge,
1993,
• E.W. Kolb and M.S. Turner: The Early Universe,
Addison Wesley, 1990,
• S. Dodelson: Modern Cosmology, Academic Press,
San Diego, 2003,
• P.J.E. Peebles: Principles of Physical Cosmology,
Princeton University Press, Princeton, 1993,
< G. Borner: The Early Universe, Springer- Verlag,
Berlin, 2003,
< A.R. Liddle and D.H. Lyth: Cosmological Inflation
and Large-Scale Structure, Cambridge University
Press, Cambridge, 2000.
D.3 Review Articles, Current Literature,
and Journals
Besides textbooks and monographs, review articles on
specific topics are particularly useful for getting ex-
tended information about a special field. A number of
journals and series exist in which excellent review ar-
ticles are published. Among these are Annual Reviews
of Astronomy and Astrophysics (ARA&A) and Astron-
omy & Astrophysics Reviews (A&AR), both publishing
astronomical articles only. In Physics Reports (Phys.
Rep.) and Reviews of Modern Physics (RMP), astro-
nomical review articles are also frequently found. Such
articles are also published in the lecture notes of interna-
tional summer/winter schools and in the proceedings of
conferences; of particular note are the Lecture Notes of
the Saas-Fee Advanced Courses. A very useful archive
containing review articles on the topics covered in this
book is the Knowledgebase for Extragalactic Astron-
omy and Cosmology, which can be found at
http://nedwww.ipac.caltech.edu/level5.
Original astronomical research articles are published in
the relevant scientific journals; most of the figures pre-
sented in this book are taken from these journals. The
most important of them are Astronomy & Astrophysics
(A&A), The Astronomical Journal (AJ), The Astro-
physical Journal (ApJ), Monthly Notices of the Royal
Astronomical Society (MNRAS), and Publications of
the Astronomical Society of the Pacific (PASP). Besides
these, a number of smaller, regional, or more special-
ized journals exist, such as Astronomische Nachrichten
(AN), Acta Astronomica (AcA), or Publications of the
Astronomical Society of Japan (PASJ). Some astronom-
ical articles are also published in the journals Nature
and Science. The Physical Review D and Physical Re-
view Letters contain an increasing number of papers on
astrophysical cosmology.
D.3 Review Articles, Current Literature, and Journals
The Astrophysical Data System (ADS) of NASA
which can be accessed via the Internet at, e.g.,
http://cdsads.u-strasbg.fr/abstract_service.html
http://adsabs.harvard.edu/abstract_service.html
provides the best access to these (and many more)
journals. Besides tools to search for authors and key-
words, ADS offers also direct access to older articles
that have been scanned. The access to more recent
articles is restricted to IP addresses that are asso-
ciated with a subscription for the respective jour-
nals.
An electronic archive for preprints of articles is freely
accessible at
http://arxiv.org/archive/astro-ph.
This archive has existed since 1992, with an increas-
ing number of articles being stored at this location. In
particular, in the fields of extragalactic astronomy and
cosmology, more than 90% of the articles that are pub-
lished in the major journals can be found in this archive;
a large number of review articles are also available here.
astro-ph has become the primary source of information
for asl
E. Acronyms Used
In this appendix, we compile some of the acronyms that
CFRS
Canada-France Redshift Survey (Sect.
are used, and references to the sections in which these
8.1.2)
acronyms have been introduces or explained.
CGRO
Compton Gamma Ray Observatory
(Sect. 1.3.5)
2dF(GRS)
Two-Degree Field Galaxy Redshift
CHVC
Compact High- Velocity Cloud (Sect.
Survey (Sect. 8.1.2)
6.1.3)
ACBAR
Arcminute Cosmology Bolometer Ar-
CIB
Cosmic Infrared Background (Sect.
ray Receiver (Sect. 8.6.5)
9.3.1)
AGO
Abell, Corwin & Olowin (catalog of
CLASS
Cosmic Lens All-Sky Survey (Sect.
clusters of galaxies, Sect. 6.2.1)
3.8.3)
ACS
Advanced Camera for Surveys (HST
CMB
Cosmic Microwave Background (Sect.
instrument)
8.6)
AGB
Asymptotic Giant Branch (Sect. 3.9.2)
CMD
Color-Magnitude Diagram
AGN
Active Galactic Nucleus (Chap. 5)
(Appendix B)
ALMA
Atacama Large Millimeter Array
COBE
Cosmic Background Explorer (Sect.
(Chap. 10)
8.6.4)
APEX
Atacama Pathfinder Experiment
CTIO
Cerro Tololo Inter-American Observa-
(Chap. 10)
lorv
AU
Astronomical Unit
DASI
Degree Angular Scale Interferometer
BAL
Broad Absorption Line (-Quasar, Sect.
(Sect. 8.6.4)
5.6.3)
EdS
Einstein-de Sitter (Sect. 4.3.4)
BATSE
Burst And Transient Source Experi-
EMSS
Extended Medium Sensitivity Survey
ment (CGRO instrument, Sect. 9.7)
(Sect. 6.3.5)
BBB
Big Blue Bump (Sect. 5.4.1)
EPIC
European Photon Imaging Camera
BBN
Big Bang Nucleosynthesis (Sect.
(XMM-Newton instrument)
4.4.4)
EROS
Experience pour la Recherche d'Objets
BCD
Blue Compact Dwarf (Sect. 3.2.1)
Sombres (microlenses collaboration,
BH
Black Hole (Sect. 5.3.5)
Sect. 2.5)
BLR
Broad-Line Region (Sect. 5.4.2)
ESA
European Space Agency
BLRG
Broad-Line Radio Galaxy (Sect. 5.2.3)
ESO
European Southern Observatory (Sect.
BOOMERANG Balloon Observations Of Millimetric
1.3.3)
Extragalactic Radiation and Geo-
FFT
Fast Fourier Transform (Sect. 7.5.3)
physics (Sect. 8.6.4)
FIR
Far Infrared
CBI
Cosmic Background Imager (Sect.
FJ
Faber- Jackson (Sect. 3.4.2)
8.6.5)
FOC
Faint Object Camera (HST instrument)
CCD
Charge Coupled Device
FORS
Focal Reducer / Low Dispersion Spec-
CDM
Cold Dark Matter (Sect. 7.4.1)
trograph (VLT instrument)
CERN
Conseil European pour la Recherche
FOS
Faint Object Spectrograph (HST instru-
Nucleaire
ment)
CfA
Harvard-Smithsonian Center for As-
FP
Fundamental Plane (Sect. 3.4.3)
trophysics
FR(I/II)
Faranoff-Riley Type (Sect. 5.1.2)
CFHT
Canada-France-Hawaii Telescope
FWHM
Full Width Half Maximum
(Sect. 1.3.3)
GC
Galactic Center (Sect. 2.3, 2.6)
Peter Schneider,
Acronyms Used.
In: Peter Schnek
er, Extragalactic Astronomy and Cosmology, pp. 437^39 (2006)
DOI: 10.1007/1
614371_E © Springer- Verlag Berlin Heidelberg 2006
E. Acronyms Used
GRB Gamma-Ray Burst (Sects. 1.3.5, 9.7)
GUT Grand Unified Theory (Sect. 4.5.3)
Gyr Gigayear = 10 9 years
HB Horizontal Branch
HCG Hickson Compact Group (catalog of
galaxy groups, Sect. 6.2.8)
HDF(N/S) Hubble Deep Field (North/South)
(Sects. 1.3.3,9.1.3)
HDM Hot Dark Matter (Sect. 7.4. 1)
HEAO High-Energy Astrophysical Observa-
tory (Sect. 1.3.5)
HRD Hertzsprung-Russell Diagram (Ap-
pendix B)
HRI High-Resolution Imager (ROSAT in-
strument)
HST Hubble Space Telescope (Sect. 1.3.3)
HVC High- Velocity Cloud (Sect. 2.3.6)
IAU International Astronomical Union
ICM Intracluster Medium (Chap. 6)
IGM Intergalactic Medium (Sect. 8.5.2)
IMF Initial-Mass Function (Sect. 3.9.1)
IoA Institute of Astronomy (Cambridge)
IR Infrared (Sect. 1.3.2)
IRAS Infrared Astronomical Observatory
(Sect. 1.3.2)
ISM Interstellar Medium
ISO Infrared Space Observatory (Sect.
1.3.2)
IUE International Ultraviolet Explorer
JCMT James Clerk Maxwell Telescope (Sect.
1.3.1)
JVAS Jodrell Bank-VLA Astrometric Sur-
vey (Sect. 3.8.3)
JWST James Webb Space Telescope (Chap.
10)
KAO Kuiper Airborne Observatory (Sect.
1.3.2)
LBG Lyman-Break Galaxies (Sect. 9.1.1)
LCRS Las Campanas Redshift Survey (Sect.
8.1.2)
LHC Large Hadron Collider
LISA Laser Interferometer Space Antenna
(Chap. 10)
LMC Large Magellanic Cloud
LOFAR Low-Frequency Array (Chap. 10)
LSB galaxy Low Surface Brightness galaxy (Sect.
7.5.4)
LSR
Local Standard of Rest (Sect. 4.2.1)
LSS
Large-Scale Structure (Chap. 8)
MACHO
Massive Compact Halo Object (and
collaboration of the same name,
Sect. 2.5)
MAMBO
Max-Planck Millimeter Bolometer
(Sect. 9.3.2)
MAXIMA
Millimeter Anisotropy Experiment
Imaging Array (Sect. 8.6.4)
MDM
Mixed Dark Matter (Sect. 7.4.2)
MIR
Mid-Infrared
MLCS
Multicolor Light Curve Shape (Sect.
8.3.1)
MMT
Multi-Mirror Telescope
MS
Main Sequence
MW
Milky Way
NAOJ
National Astronomical Observatory of
Japan
NFW
Navarro, Frenk & White (-profile, Sect.
7.5.4)
NGC
New General Catalog (Chap. 3)
NGP
North Galactic Pole (Sect. 2.1)
NICMOS
Near Infrared Camera and Multi-
Object Spectrometer (HST instrument)
NIR
Near-Infrared
NLR
Narrow-Line Region (Sect. 5.4.3)
NLRG
Narrow-Line Radio Galaxy (Sect.
5.2.3)
NOAO
National Optical Astronomy Observa-
tory
NRAO
National Radio Astronomy Observa-
tory
NTT
New Technology Telescope
OGLE
Optical Gravitational Lensing Ex-
periment (microlenses collaboration,
Sect. 2.5)
ovv
Optically Violently Variable (Sect.
5.2.4)
PL
Period-Luminosity (Sect. 2.2.7)
PLANET
Probing Lensing Anomalies Network
(microlenses collaboration, Sect. 2.5)
PN
Planetary Nebula
POSS
Palomar Observatory Sky Survey
PSF
Point-Spread Function
PSPC
Position-Sensitive Proportional
Counter (ROSAT instrument)
QSO
Quasi-Stellar Object (Sect. 5.2.1)
E. Acronyms Used
RASS ROSAT All Sky Survey (Sect. 6.3.5)
RCS Red Cluster Sequence (Sect. 6.6)
REFLEX ROSAT-ESO Flux-Limited X-Ray sur-
RGB Red Giant Branch (Sect. 3.9.2)
ROSAT Roentgen Satellite (Sect. 1.3.5)
SAO Smithsonian Astrophysical Observatory
SCUBA Submillimeter Common-User Bolome-
ter Array (Sect. 1.3.1)
SDSS Sloan Digital Sky Survey (Sect. 8.1.2)
SFR Star-Formation Rate (Sect. 9.5. 1)
SGP South Galactic Pole (Sect. 2.1)
SIS Singular Isothermal Sphere (Sect. 3.8.2)
SKA Square Kilometer Array (Chap. 10)
SN(e) Supernova(e) (Sect. 2.3.2)
SNR Supernova Remnant
SMC Small Magellanic Cloud
SMBH Supermassive Black Hole (Sect. 5.3)
STIS Space Telescope Imaging Spectrograph
(HST instrument)
STScI Space Telescope Science Institute (Sect.
1.3.3)
SZ Sunyaev-Zeldovich (effect, Sect. 6.3.4)
TF Tully-Fisher (Sect. 3.4)
UDF
Ultra-Deep Field (Sect. 9.1.3)
ULIRG
Ultra-Luminous Infrared Galaxy (Sect.
9.2.1)
ULX
Ultra-Luminous Compact X-ray Source
(Sect. 9.2.1)
UV
Ultraviolet
VLA
Very Large Array (Sect. 1.3.1)
VLBI
Very Long Baseline Interferometer (Sect.
1.3.1)
VLT
Very Large Telescope (Sect. 1.3.3)
VST
VLT Survey Telescope (Sect. 6.2.5)
WD
White Dwarf (Sect. 2.3.2)
WIMP
Weakly Interacting Massive Particle
(Sect. 4.4.2)
WFI
Wide Field Imager (camera at the
ESO/MPG 2.2-m telescope, La Silla,
Sect. 6.5.2)
WFPC2
Wide Field and Planetary Camera 2 (HST
instrument)
WMAP
Wilkinson Microwave Anisotropy Probe
(Sect. 8.6.5)
XMM
X-ray Multi-Mirror Mission (Sect. 1.3.5)
XRB
X-Ray Background (Sect. 9.3.2)
ZAMS
Zero Age Main Sequence (Sect. 3.9.2)
F. Figure Credits
Chapter i
1.1 Credit: ESO
1.2 Credit: NASA, The NICMOS Group (STScI,
ESA) and The NICMOS Science Team (Univ. of
Arizona)
1.5 Source: http://adc.gsfc.nasa.gov/mw/
mmw_product.html#viewgraph
Credit: NASA's Goddard Space Flight Center
Radio Continuum (408 MHz): Data from ground-
based radio telescopes (Jodrell Bank Mark I
and Mark IA, Bonn 100-meter, and Parkes 64-
meter). Credit: Image courtesy of the NASA
GSFC Astrophysics Data Facility (ADF). Refer-
ence: Haslam, C. G. T, Salter, C. J., Stoffel, H.,
& Wilson, W. E. 1982, Astron. Astrophys. Suppl.
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Ser., 62, 365 Hartmann, Dap, & Burton, W. B.,
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Radio Continuum (2.4-2.7 GHz): Data from
the Bonn 100-meter, and Parkes 64-meter
radio telescopes, contact/credit: Roy Dun-
can, ccroy@yowie.cc.uq.edu.au References: Dun-
can, A. R., Stewart, R. T, Haynes, R. F, &
Jones, K. L. 1995, Mon. Not. Roy. Astr. Soc, 277,
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vey.html http://www.atnf.csiro.au/database/
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Molecular Hydrogen: Data from the Columbia/
GISS 1.2 m telescope in New York City, and a twin
telescope on Cerro Tololo in Chile contact/credit:
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ences: Dame, T. M., Hartmann, Dap, & Thad-
deus, P. 2001, Astrophysical Journal, 547, 792.
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composite survey) from ADC archives: ftp://adc.
gsfc.nasa.gov/pub/adc/archives/catalogs/8/8039/
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Satellite (IRAS) Credit: Image courtesy of the
NASA GSFC Astrophysics Data Facility (ADF).
Reference: Wheelock, S. L., et al. 1994, IRAS Sky
Survey Atlas Explanatory Supplement, JPL Publi-
cation 94-1 1 (Pasadena: JPL). Online data access:
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leased IRAS data products: http://space.gsfc.
nasa.gov/astro/iras/iras_home.html
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line Information: http://sd-www.jhuapl.edu/MSX/
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sawitz, D., & Weiland, J. 1995, COBE Diffuse
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the ADF http://space.gsfc.nasa.gov/astro/cobe/
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1.6 Credit: S. Hughes & S. Maddox - Isaac Newton
Group of Telescopes
1.7 Credit: M. Altmann, Observatory Bonn University
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Credit: John Bahcall (Institute for Advanced
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Schneider (University of Arizona) and Nick Scov-
ille (California Institute of Technology), and
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Credit: STScI and The Hubble Heritage Project
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Credit: NOAO/AURA/NSF
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Credit: Optical: La Palma/B. McNamara/X-Ray:
NASA/CXC/SAO
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index.html
Credit: STScI und das Hubble Heritage Project
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COBE_Home/DMR_Images.html
Credit: COBE/DRM Team, NASA
o.edu/imagegallery/php/
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Credit: NRAO/AUI
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hires/ao004.jpg Courtesy of the NAIC - Arecibo
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1.20 left: Source: http://www.mpifr-bonn.mpg.de/
public/images/lOOm.html
Credit: Max Planck Institute for Radio Astronomy
1.20 right: Source: http://www.nrao.edu/imagegallery/
php/level3 .php?id=4 1 2
Credit: NRAO/AUO
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Credit: NRAO/AUO
1.22 Source: http://www.mpifr-bonn.mpg.de/staff/
bertoldi/mambo/intro.html
Credit: Max Planck Institute for Radio Astronomy
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Credit: Joint Astronomy Centre
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about/earlyhisi.shlml
Credit: Courtesy
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1 20396_index_ 1 _m.html
Credit: ESA / www.esa.int
1.25 Source: R. Wainscoat
Credit: R. Wainscoat, Hawaii University
1.26 Credit: ESO
1.27 Source: http://hubblesite.org/newscenter/
newsdesk/archive/releases/1996/01/image/d.
Credit: R. Williams (STScI), the Hubble Deep
Field Team and NASA
1.28 Source: R. Wainscoat.
Credit: R. Wainscoat, Hawaii University
1.29 Credit: ESO
1.30 Credit: ESO
1.31 Credit: ESO
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PIFICONS/rosat-transparent.gif
Credit: www.xray.mpe.mpg.de /
www.mpe.mpg.de
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resources/illustrations/craftRight.html
Credit: NASA/CXC/SAO
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science-e/www/area/index.cfm?fareaid=23.
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epo/gallery/cgro/
Credit: NASA
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Credit: ESA / www.esa.int
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milchstrasse.html
Credit: Stephan Messner, Observatory Brenner-
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sandage.html
Credit: Allan Sandage, The Observatories of the
Carnegie Institution of Washington
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Credit: G.A. Tammann, Astronomical Institute,
Basel University, Switzerland
2.10 Credit: Unsold Baschek, Der Neue Kosmos,
Springer- Verlag, Berlin Heidelberg New York
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2.11 Credit: Schlegel, D.J., Finkbeiner, D.P. & Davis,
M., ApJ 1999, 500, 525
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Credit: Reprinted, with permission, from the
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426, 427.
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the National Science Foundation operated under
cooperative agreement with Associated Universi-
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Naval Research Laboratory is supported by the
U.S. Office of Naval Research
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Image Credit: NASA, ESA, and The Hubble
Heritage Team (STScI/AURA) Acknowledgment:
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tory, and the National Astronomical Observatory
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3.48 Credit: Chariot, Lecture Notes in Physics Vol. 470,
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Credit: Steve Maddox Nottingham Astronomy
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Will Sutherland http://www-astro.physics.ox.ac.
uk/index.html
George Efstathiou Director Institute of Astron-
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with follow-up by Gavin Dalton, Astrophysics
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Credit: Walter Jaffe/Leiden Observatory, Holland
Ford/JHU/STScI, and NASA
5.31 Credit: Urry & Padovani 1995, PASP, 107, 803
5.32 Source: http://www.spacetelescope.org/images/
html/opo9943e.html.
Credit: NASA/ESA and Ann Feild (Space Tele-
scope Science Institute)
5.33 left: Source: http://hubblesite.org/newscenter/
newsdesk/archive/releases/2003/03/image/c.
Credit for WFPC2 image: NASA and J. Bahcall
(IAS)
5.33 right: Source: http://wwwll.msfc.nasa.gov/
news/news/photos/2000/photos00-308.htm.
Credit: NASA/CXC/H. Marshall et al.
5.34 Source: http://hubblesite.org/newscenter/
new sdesk/archive/releases/ 1 999/43/
Credit: NASA, National Radio Astronomy Ob-
servatory/National Science Foundation, and John
Biretta (STScI/JHU)
5.35 left: Source: http://heasarc.gsfc.nasa.gov/docs/
objects/heapow/archive/active_galaxies/
pksl 127_chandra.html.
Credit: X-ray: NASA/CXC/A. Siemiginowska
(CfA) & J. Bechtold (U. Arizona);
Radio: Siemiginowska et al. (VLA)
5.35 right: Source: http://chandra.harvard.edu/photo/
2001/0157blue/ NASA/SAO/ R. Kraft et al.
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5.38 Credit: Chaffee et al. 1988, ApJ 335, 584
5.39 Credit: Rauch 1998, ARA&A 36, 267
5.40 Credit: Sargent, Wallace L. W.; Steidel, Charles
C; Boksenberg, A. 1989, ApJS 69, 703
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lines: Probing the universe: Proceedings of the
QSO Absorbtion Line Meeting, Baltimore, MD,
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Heidelberg
Chapter 6
6.1 Source: http://pupgg.princeton.edu/ groth/
Credit: Seldner, Siebers, Groth and Peebles, 1977,
A. J., 82, 249
6.2 Credit: Sharp, N.A. 1986, PASP 98, 740
6.3 left: Source: http://subarutelescope.org/
Pressrelease/1999/01/index.html#hcg40
Credit: Subaru Telescope, National Astronomical
Observatory of Japan (NAOJ)
6.3 left: Source: http://www.eso.org/outreach/
press-rel/pr-2002/phot- 18-02.html.
Credit: ESO
6.4 Credit: Eva Grebel, Astronomical Institute,
University of Basel, Switzerland
6.5 Source: http://www.noao.edu/image_gallery/
html/im0562.html
Credit: NOAO/AURA/NSF
6.6 Credit: Brans et al. 2005, A&A 432, 45
6.7 Source: http://www.robgendlerastropics.com/
M8182.html.
Credit: Robert Gendler
6.10 left: Source: http://www.astro.uni-bonn.de/
~maltmann/actintgal.html.
Credit: M. Altmann
6.10 right: Source: http://hubblesite.org/newscenter/
newsdesk/archive/releases/2002/22/image/
Image.
Credit: NASA, J. English (U. Manitoba), S. Huns-
berger, S. Zonak, J. Charlton, S. Gallagher (PSU),
and L. Frattare (STScI) Science Credit: NASA,
C. Palma, S. Zonak, S. Hunsberger, J. Charl-
ton, S. Gallagher, P. Durrell (The Pennsylvania
State University) and J. English (University of
Manitoba)
6.11 Credit: Goto et al. 2003, astro-ph/03 12043
6.12 Reused with permission from I. K. Baldry, M.
L. Balogh, R. Bower, K. Glazebrook, and R.
C. Nichol, in Color bimodality: Implications for
galaxy evolution, Rolan d E. Allen (ed), Confer-
ence Proceeding 743, 106 (2004). Copyright 2004,
American Institute of Physics
6.13 left: Source: http://heasarc.gsfc.nasa.gov/docs/
rosat/gallery/clus_coma.html
Credit: S. L. Snowden USRA, NASA/GSFC
6.13 right:
Credit: Briel et al. 2001, A&A 365, L60
6.14 Source: http://wave.xray.mpe.mpg.de/rosat/
calendar/ 1 997/may.
Credit: Max-Planck-Institute for extraterrestrial
Physics, Garching
6.15 Source: http://hubblesite.org/newscenter/
newsdesk/archive/releases/ 1 998/26/
Credit: Megan Donahue (STSCI) / Ground.
Credit: Isabella Gioia (Univ. of Hawaii), and
NASA
6.16 Source: http://www.astro.uni-bonn.de/ reiprich/
act/gcs/
Credit: Thomas Reiprich
6.17 Source: http://www.astro.uni-bonn.de/ reiprich/
act/gcs/
Credit: Thomas Reiprich
6.18 Source: http://chandra.harvard.edu/photo/2002/
0146/
Credit: NASA/IoA/ J. Sanders & A. Fabian
6.19 Credit: Peterson et al. 2003, astro-ph/03 10008
6.20 Source: http://wave.xray.mpe.mpg.de/rosat/
calendar/ 1994/sep Copyright: Max-Planck-
Institute for extraterrestrial Physics, Garching
6.21 Source: http://chandra.harvard.edu/photo/2001/
hcg62/
Credit: NASA/CfA/ J. Vrtilek et al.
6.22 Credit: Markevitch et al. 2002, ApJ 567, L2
6.23 Credit: Carlstrom et al. 2002, ARA&A 40, 643
6.24 Credit: Grego et al. 2001, ApJ 552, 2
6.25 Credit: Stanford et al. 2001, ApJ 552, 504
6.26 Source: http://www.xray.mpe.mpg.de/rosat/
survey/sxrb/12/ass.html Image
Credit: M.J. Freyberg, R. Egger (1999), "ROSAT
PSPC All-Sky Survey maps completed", in Pro-
ceedings of the Symposium "Highlights in X-ray
Astronomy in honour of Joachim Triimper's 65th
birthday", eds. B. Aschenbach & M.J. Freyberg,
MPE Report 272, p. 278-281
6.27 Credit: Finoguenov et al. 2001, A&A 368, 749
6.28 Source: http://www.astro.uni-bonn.de/ reiprich/
Credit: Reiprich & Bohringer 2002, ApJ 567, 716
6.29 Credit: Lin et al. 2004, ApJ 610, 745
6.30 Source: http://serweb.oamp.fr/kneib/hstarcs/
hst_a370.html.
Credit: Jean-Paul Kneib
6.31 Source: http://www.eso.org/outreach/press-rel/pr-
1998/pr-19-98.html
Credit: ESO
6.32 Credit: Fort, B. & Mellier, Y. 1994, A&AR 5, 239
6.33 top: Source: http://hubblesite.org/newscenter/
newsdesk/archive/releases/ 1 995/ 1 4/image/a
Credit: W. Couch (University of New South
Wales), R. Ellis (Cambridge University), and
NASA
6.33 left: Source: http://hubblesite.org/newscenter/
newsdesk/archive/releases/ 1 996/ 1 0/image/a
Credit: W.N. Colley and E. Turner (Prince-
ton University), J.A. Tyson (Bell Labs, Lucent
Technologies) and NASA
6.34 Source: http://hubblesite.org/newscenter/
newsdesk/archive/releases/2003/0 1 /image/b.
Credit: NASA, N. Benitez (JHU), T Broadhurst
(Racah Institute of Physics/The Hebrew Uni-
versity), H. Ford (JHU), M. Clampin (STScI),
G. Hartig (STScI), G. Illingworth (UCO/Lick
Observatory), the ACS Science Team and ESA
6.35 Credit: C. Seitz, LMU Miinchen
6.35 Credit: Optical Image: HST/NASA, Colley et al.
6.36 Shear Field and Mass Reconstruction, C. Seitz,
LMU Miinchen
6.37 Credit: Squires et al. 1996, ApJ 461, 572
6.38 bottom left:
Credit: Luppino & Kaiser 1997, ApJ 475, 20
6.38 right:
Credit: Hoekstra et al. 2000, ApJ 532, 88
6.39 bottom:
Credit: Trager et al. 1997, ApJ 485
6.40 bottom: Source: http://www.noao.edu/outreach/
press/prOl/prOl 1 l.html
Credit: Lucent Technologies' Bell
Labs/NOAO/AURA/NSF
6.41 Credit: Gioia et al. 2001, ApJ 553
6.42 Credit: Margoniner et al. 2001, ApJ 548, L143
6.43 Credit: Gladders & Yee 2000, AJ 120, 2148
6.44 Credit: Spinrad et al. 1997, ApJ 484, 581
6.45 Source: http://www.astro.cz/apod/ap990722.html
Credit: P. van Dokkum, M. Franx (U. Groningen /
U. Leiden), ESA, NASA
6.46 Credit: 327; Mullis et al. ApJ 623, L85, 2005
6.47 Source: http://www.eso.org/outreach/press-rel/
pr-2002/pr-07-02.html
Credit: George Miley, Observatory Leiden Uni-
versity, The Netherlands
Chapter 7
7.1 Source: http://www.mso.anu.edu.au/2dFGRS/
Credit: The 2dF Galaxy Redshift Survey team,
http://magnum.anu.edu.au/ TDFgg/
7.2 Source: http://cfa-www.harvard.edu/ huchra/zcat/
Credit: John Huchra
7.4 Credit: Tucker et al. 1997, MNRAS 285, L5
7.7 Credit: Eke et al. 1996, MNRAS 282, 263
7.8 Credit: Bahcall & Fan 1998, ApJ 504, 1
7.9 Credit: Springel et al. 2005, astro-ph/0504097
7.10 Source: http://www.mpa-garching.mpg.de/
galform/virgo/int_sims/index.shtml.
Credit: "The simulations in this paper were
carried out by the Virgo Supercomputing Con-
sortium using computers based at Computing
Centre of the Max-Planck Society in Garching
and at the Edinburgh Parallel Computing Centre.
The data are publicly available at www.mpa-
garching.mpg.de/galform/virgo/
int_sims"
7.11 Source: http://www.mpa-garching.mpg.de/
galform/virgo/hubble/index.shtml.
Credit: The simulations in this paper were carried
out by the Virgo Supercomputing Consortium us-
ing computers based at the Computing Centre of
the Max-Planck Society in Garching and at the Ed-
inburgh parallel Computing Centre. The data are
publicly available at http://www.mpa-garching.
mpg.de/galform/virgo/hubble
7.12 Credit: Springel et al. 2005, astro-ph/0504097
7.13 Credit: Navarro, Frenk & White 1997, ApJ 490,
493
7.14 Credit: Navarro, Frenk & White 1997, ApJ 490,
493
7.15 Credit: Navarro, Frenk & White 1997, ApJ 490,
493
7.16 Credit: Lin et al. 2004, ApJ 610, 745
7.17 Credit: Moore et al. 1999, ApJ 524, L19
7.18 Credit: Moore et al. 1999, ApJ 524, L19
7.19 Credit: Fassnacht et al. 1999, AJ 117, 658
7.20 left:
Credit: Koopmans et al. 2002, MNRAS 334
7.20 right: Source: http://cfa-www.harvard.edu/castles/
Individual/MG20 1 6.html.
Credit: http://cfa-www.
harvard.edu/castles/ (C.S. Kochanek, E.E. Falco,
C. Impey, J. Lehar, B. McLeod, H.-W. Rix)
Chapter 8
8.1 Source: http://cfa-www.harvard.edu/~huchra/zcal
Credit: John Huchra
8.2 Credit: Lin et al. 1 996, ApJ 47 1 , 6 1 7
8.3 Credit: Peacock 2003, astro-ph/0309240
8.4 Credit: Peacock & Dodds 1994, MNRAS 267,
1020
8.5 left:
Credit: Peacock 2003, astro-ph/0309240
8.5 right:
Credit: Peacock 2001, astro-ph/0105450
8.6 top:
Credit: Hamilton 1997, astro-ph/9708102
8.7 Credit: Hawkins et al. 2003, MNRAS 346, 78
8.8 Credit: Conolly et al. 2002, ApJ 579, 42
8.9 Credit: Dekel 1 994, ARA&A 32,371
8.10 Source: http://www.eso.org/outreach/press-rel/
pr-1999/phot-46-99.html.
Credit: ESO
8.12 Credit: Schuecker et al. 2001, A&A 368, 86
8.13 Credit: Filippenko & Riess 2000, astro-ph/
0008057
8.14 Source: http://www-supernova.lbl.gov/public/
figures/stretch_hamuy. gif .
Credit: S. Perlmutter
8.15 Credit: Riess et al. 2004, ApJ 607, 665
8.16 Credit: Riess et al. 2004, ApJ 607, 665
8.17 Credit: Riess et al. 2004, ApJ 607, 665
8.18 Source: http://www.cfht.hawaii.edu/News/
Lensing/ Image
Credit: Canada-France-Hawaii Telescope Corpo-
ration
8.19 Source: http://www2.iap.fr/LaboEtActivites/
ThemesRecherche/Lentilles/arcs/
cosmicshearstatus.html.
Credit: Yannick Mellier, Institut d'Astrophysique
de Paris
8.20 Credit: van Waerbeke et al. 2001, A&A 374, 757
8.21 Credit: Miralda-Escude et al. 1996, ApJ 471, 582
8.22 Credit: Dave 2001, astro-ph/0 105085
8.23 Credit: Weinberg et al. 1998, astro-ph/9810142
8.24 Credit: Hu & Dodelson 2002, ARA&A 40, 171
8.25 Credit: Hu & Dodelson 2002, ARA&A 40, 171
8.26 Credit: Bennett et al. 2003, ApJS 148, 97
8.27 Credit: de Bernardis et al. 2000, astro-ph/ 0004404
8.28 Credit: Netterneld et al. 2002, ApJ 571, 604
8.29 Credit: Wang et al. 2002, Phys. Rev. D 68, 123001
8.30 Credit: Bennett et al. 2003, ApJS 148, 1
8.31 Credit: Bennett et al. 2003, ApJS 148, 1
8.32 Credit: Spergel et al. 2003, ApJS 148, 175
8.33 Source: http://space.mit.edu/home/tegmark/
sdsspower.html.
Credit: M. Tegmark
8.34 Credit: Contaldi et al. 2003, Phys. Rev. Lett. 90,
1303
8.35 Source: http://supernova.lbl.gov/ adapted from
Knop et al. 2003, ApJ 598, 102
Chapter 9
9.1 Credit: Fan et al., 2003, AJ 125, 1649
9.2 Credit: Steidel et al., 1995, AJ 1 10, 2519
9.3 Source: http://www.astro.caltech.edu/~ccs/
ugr.html.
Credit: C. Steidel, Caltech, USA
9.4 Credit: Steidel et al., 1995, AJ 1 10, 2519
9.5 Credit: Steidel et al., 1996, AJ 462, L17
9.6 Credit: Hu et al., 1999, ApJ Letters, 522, L9
9.7 Credit: Adelberger 1999, astro-ph/9912153
9.8 Credit: Benitez 2000, ApJ 536,571
9.9 Credit: R. Williams (STScI), the Hubble Deep
Field Team and NASA
9.10 Source: Ferguson et al. 2000, ARA&A38, 667
Credit: Reprinted, with permission, from the An-
nual Review of Astronomy & Astrophysics, by
Annual Reviews www.annualreviews.org
9.11 Source: Space Telescope Science Institute, NASA
Credit: S. Beckwith & the HUDF Working Group
(STScI), HST, ESA, NASA
9.12 Source: http://serweb.oamp.fr/kneib/hstarcs/
hst_a2390.html
Credit: Jean-Paul Kneib, Laboratoire d' Astro-
physique de Marseille
9.13 left: Copyright Stella Seitz, LMU Munchen
9.13 right:
Credit: ESO
9.14 Credit: Kneib et al., 2004, ApJ 607, 697
9.15 Credit: Whitmore et al. 1999, AJ 116, 1551
9.16 Source: http://www.casca.ca/lrp/ch5/en/
chap520.html.
The image taken from: "Canadian Astronomy and
Astrophysics in the 21st century", "The Origins
of Structures in the Universe" Courtesy Christine
Wilson, McMaster University
9.17 left: Source: http://chandra.harvard.edu/photo/
2001/0120true/index.html.
Credit: NASA/SAO/ G. Fabbiano et al.
9.17 bottom left: Source: http://chandra.harvard.edu/
photo/200 1/00 1 2/index.html
Credit: X-ray: NASA/SAO/CXC, Optical: ESO
9.17 top right: Source: http://chandra.harvard.edu/
photo/200 l/0094true/
Credit: NASA/SAO/ G. Fabbiano et al.
9.17 bottom right: Source: http://chandra.harvard.edu/
photo/200 1/0 120true/lum_functions.jpg.
Credit: SAO/CXC/ A. Zezas
9.18 Credit: Cimatti et al., 2002, A&A 391, LI
9.19 Credit: Smith et al. 2002, astro-ph/0201236
9.20 Source: Blain et al. 1999, astro-ph/9908111
9.21 Credit: F. Bartoldi, MPIfR
9.22 Source: Blain et al. 1999, astro-ph/9908111
9.23 Source: Blain et al. 1999, astro-ph/9908111
9.24 Credit: Hauser & Dwek, ARA&A 2001 39, 249
9.25 Credit: Hauser & Dwek, astro-ph/0105539
9.26 Source: Tozzi et al. 2001, ApJ 562, 42
Credit:
9.27 Source: http://chandra.harvard.edu/photo/2001/
cdfs/Cdfs_scale.jpg.
Credit: NASA/JHU/AUI/
R. Giacconi et al.
9.28 Credit: Fan et al., 2004, AJ 128, 515
9.29 Credit: Barkana & Loeb 2000, astro-ph/00 10468
9.30 Credit: Barkana & Loeb 2000, astro-ph/00 10468
9.31 adapted from: Barkana & Loeb 2000, astro-
ph/00 10468
9.32 Credit: Hopkins et al. 2001, astro-ph/0103253
9.33 Credit: Bell 2004, astro-ph/0408023
9.34 Credit: Bell 2004, astro-ph/0408023
9.35 Credit: Mirabel et al. 1998, astro-ph/9810419
9.36 Source: http://antwrp.gsfc.nasa.gov/apod/
ap990510.html.
Credit: J. Gallagher (UW-M) et al. & the Hubble
Heritage Team (AURA/ STScI/ NASA)
9.37 Credit: Lacey & Cole 1993, Mon. Not. R. Astron.
Soc. 262, 627-649
9.38 Source: http://www.mpa-garching.mpg.de/
galf orm/gif/index . shtml .
Credit: G. Kauffmann, VIRGO Kollaboration,
MPA Garching
9.39 Credit: Springel et al., 2005, astro-ph/0504097
9.40 Credit: Springel et al., 2005, astro-ph/0504097
9.41 Source: http://heasarc.gsfc.nasa.gov/docs/objects/
grbs/grb_profiles.html
Credit: J.T. Bonnell, GLAST Science Support
Center, NASA Goddard Space Flight Center,
Greenbelt, Maryland, USA
9.42 Source: http://www.batse.msfc.nasa.gov/batse/
grb/skymap/images/fig2_2704.pdf
Credit: Michael S. Briggs, NASA
Chapter io
10.1 Source: http://www.cfht.hawaii.edu/Instruments/
Imaging/CFH 1 2K/images/NGC34 86-CFH 1 2K-
CFHT-1999.jpg.
Credit: Dr. Jean-Charles Cuillandre, Canada-
France-Hawaii Telescope Corporation, Hawaii,
USA
10.2 Credit: ESO
10.3 Credit: ESO
10.4 Source: http://jwstsite.stsci.edu/gallery/
telescope, shtml
Credit: Courtesy of Northrop Grumman Space
Technology
10.5 Source: http://www.eso.org/outreach/press-rel/
pr-2003/pr-04-03.html.
Credit: ESO
Appendix A
A.l Source: T. Kaempf & M. Altmann, Observatory
of Bonn University
A.2 from Girardi et al. 2002, A&A 195, 391
Appendix B
B.l Source: ESA Web Page of Hipparcos-Projekts
B.2 Source: http://de.wikipedia.org
B.3 from: Maeder & Meynet 1989, A&A 155, 210
Subject Index
AB magnitudes 422
Abell catalog see clusters of galaxies
Abell radius 229
absorption coefficient 418
absorption lines in quasar spectra
219-222,331,361,365
- classification 219-221
- Lyman-a forest see Lyman-!* forest
- metal systems 219,221,361
accelerated expansion of the Universe
154, 327
, 186-188,249,371
ndisk 186-188,191,195
acoustic peaks 339
active galactic nuclei 89, 175-222
active galaxies 10, 89, 175-215
- absorption lines see absorption
- anisotropic emission 207
- big blue bump (BBB) 195, 202
- binary QSOs 204
- BL Lac objects 185,211
-black hole 185-194
-blazars 185,196,210
- broad absorption lines (BAL) 219,
221
- Type 2 QSO 208, 382
- unified models 183, 207-215
- variability 178, 184-186, 198, 202,
211
- X-ray emission 207,214,380
adaptive optics 80
Advanced Camera for Surveys (ACS)
26, 365
age of the Universe 17, 154, 271,
274, 353
age-metallicity relation 55
ALMA 410
Andromeda galaxy (M31) 14, 87
Anglo- Australian Telescope (AAT)
25
angular correlations of galaxies
318-319
angular-diameter distance 157, 158
anthropic principle 173
APEX 410
Arecibo telescope 19
ASCA 31
astronomical unit 37
asymmetric drift 58
asymptotic giant branch (AGB) 429
lines 177, 181-182,
196
- broad-line region (BLR) 207
- classification 182-185, 207
- energy generation 109
- host galaxy 183, 185, 202-204, 207
- in clusters of galaxies 25 1
- luminosity function 182,216-219
- narrow line region (NLR) 20 1
- OVV (optically violently variable)
184-185,211
- QSO (quasi-stellar object) 183
- quasars 10, 178-183
- radio emission 178-181, 207
- radio galaxies 183-184
- Seyfert galaxies 11, 177, 183, 191
Baade's window 54, 78
background radiation 379-382
- infrared background (CIB) 380,
390
e background see cosmic
e background
- of ionizing photons 333
bar 55,89
baryogenesis 412
baryonic oscillations 337
- at low redshifts 344
baryons 4, 161, 163, 165, 334, 352
beaming 210-212
Beppo-SAX 32, 404
biasing 313,314,324,359
Big Bang 3,15,152,153,157
BL Lac objects see active galaxies
black hole demography 206
black holes 109,191
- binary systems and merging
400-402
-inAGNs 81,185-194,383
- in galaxies 3, 8, 109-1 13, 191, 202,
218,371,400-402
- in the Galactic center 6, 80-85,
194, 205, 218
- scaling with galaxy properties 3,
111-113,206,207,397
- Schwarzschild radius 109, 186, 193
blackbody radiation 418^120
- energy density 419
bolometric magnitude see magnitude
BOOMERANG 342, 348
bosons 160
bottom-up structure formation 293
boxiness in elliptical galaxies 96
b re m s s t rahlung 242-244
brightness of the night sky 236, 362
brown dwarfs 427
bulge 54, 92, 98, 397
-of the Milky Way 54-55
Butcher-Oemler effect 270, 393
Canada-France Redshil'l Surve>
(CFRS) 312
Canada-France-Hawaii Telescope
(CFHT) 25,312
Center for Astrophysics (CfA) Survey
311
Cepheids 44
- as distance indicators ■ ! !. 63.
115-116
- period-luminosity relation 115
Chandra satellite 31, 104, 202, 214,
248, 381
Chandrasekhar mass 49, 328
chemical evolution 49, 138-140
clusters of galaxies 12, 223-273,
321-324
- Abell catalog 228-230
- Abell radius 229
- baryon content 323
-beta-model 261
- catalogs 228-230, 255-256, 322
-classification 231
- color-magnitude diagram 271
- Coma cluster 12, 228, 242
- cooling flows 248-252, 263, 401
- core radius 232, 233
- dark matter 223, 234, 248, 323
- distance class 230
- evolution effects 270-273
- galaxy distribution 231-233
- galaxy luminosity function 230
- galaxy population 393
- HIFLUGCS catalog 255, 256
- hydrogen clouds 237
- intergalactic stars 236-237
- intraeluster medium 223. 236.
242-256
- luminosity function 256, 270
- mass determination 13, 233-234,
246, 248, 257, 262, 266, 322
- mass-luminosity relation 258
- mass-temperature relation 256
- mass-to-light ratio 234, 266
- mass-velocity dispersion relation
257
- morphology 23 1
- near-infrared luminosity 259
- normalization of the power spectrum
291,322,324
- number density 322, 330
- projection effects 229,255
- richness class 230
- scaling relations 256-260, 322
- Virgo Cluster 12, 228
- X-ray radiation 13
- Zwicky catalog 229
COBE 22,168,336,341,346
cold dark matter (CDM) 286
- substructure 396
collisionless gas 94
color excess 41
color filter 420-422
color index 41,420
color temperature 425
color-magnitude diagram 39, 373,
Coma cluster of galaxies 12, 1 1 7, 228
comoving coordinates 146, 280
comoving observers 146, 155
Compton Gamma Ray Observatory
(CGRO) 32,402
Compton scattering 193
- inverse 168, 193, 214, 252, 381
n index of the NFW profile
- density fluctuations 145, 277-335
- expansion equation 147, 149-155
- expansion rate 146, 155
- homogeneous world models
141-174
- Newtonian cosmology 146-148
: formation 16, 17, 279,
299
convection 427
convergence point 38
cooling fronts 25 1
cooling of gas 384, 391, 396
- and star formation 384, 396
- the role of molecular hydrogen 384
Copernicus satellite 30
correlation function 283-284, 288,
316,338,360,398
- related to biasing 358
correlation length 283
cosmic microwave background 3,16,
142, 252, 253, 336, 379
-dipole 114,316,320
- discovery 168, 336
- fluctuations 277, 278, 309,
336-349
- foreground emission 342
- measuring the anisotropy 341-349
-origin 168
- polarization 307, 348
- primary anisotropics 336-337
- redshift evolution 156
- secondary anisotropies 336-338,
341
-spectrum 156,381,383
-temperature 156
cosmic rays 51-54,180
- acceleration 53
- energy density 54
cosmic shear see gravitational lenses
cosmic variance 346-347
cosmological constant 4, 15, 148,
149, 327, 344, 349
cosmological parameters,
determination 291,309-355
cosmological principle 145
cosmology 14, 17, 141-174, 309,
277-354
- components of the Universe
149-151
icalar 151
391
COSMOS survey 367
D n -a relation 108
dark energy 4, 149, 327, 328, 353,
412
dark matter 3, 63, 64, 102, 165-166,
353,411-412
- cold and hot dark matter 286-287,
351,391
- in clusters of galaxies 1 3 , 223 , 248
- in galaxies 8, 100, 101
- in the Universe 17, 165, 281, 336
dark matter halos 101, 290-293, 391
- number density 291-293, 322, 383
- substructure 302-306
- universal mass profile 298-302
de Vaucouleurs law 55, 90, 98
deceleration parameter 153,154
declination 35
deflection of light see gravitational
density contrast 278, 289
density fluctuations in the Universe
277
-origin 307
density parameter 148, 151, 154, 171,
322, 330, 336, 344, 349, 351, 353
deuterium 163
- in QSO absorption lines 165
- primordial 165
diskincss in elliptical galaxies 96
distance determination 1 14, 253
- of extragalactic objects 104,
114-117
- within the Milky Way 36-44
distance ladder 114
distance modulus 40, 422
distances in cosmology 155,
157-159,216
Doppler broadening 196
Doppler effect 38
Doppler favoritism 211
Doppler shift 10
Doppler width 11,196
downsizing 401
drop-out technique see Lyman-break
technique
dust 41,208
- extinction and reddening 41,328,
373
- gray dust 329
- infrared emission 5 1, 90, 102, 196,
208, 369, 374
-warm dust 196,208,371,374
dwarf galaxies see galaxies
dynamical friction 235, 238, 397, 400
dynamical instability of A' body
systems 85
dynamical pressure 47
early-type gala:
Fddingto
Eddington efficiency 205, 206
Eddington luminosity 193-194, 205,
207,371,402
effective radius R e 55, 90, 107
effective temperature 425, 426
FiTclsbcrg radio iclescope 19
Einstein obscrvalon 3 I. 255
Einstein radius 6>e see gravitational
Einstein-de Sitter model
elliptical galaxies see galaxies
emission coefficient 418
energy density of a radiation field 417
equation of radiative transfer 40, 417
equivalent width 181,332
expansion rate see cosmology
Extended Medium Sensitivity Survey
(EMSS) 255
extinction 40, 328
- extinction and reddening 40, 328
- extinction coefficient 41,422
extremely red object (ERO) 37 1-374,
391,394
Fanaroff-Riley classification 178,
184
Faraday rotation 52, 2 1 2
feedback 360, 396
fermions 160
Fingers of God 316,317
fireball model see gamma-ray bursts
flatfield 236
flatness problem 172, 174, 307
flux 417
free-free radiation 242
Freeman law 99,105
free-steaming 286
Fricdmann equations 149, 289
Friedmann-Lemaitre model 15, 149
fundamental plane 107-108,319,
392
FUSE 30, 335
GAIA 38,411
Galactic center 6, 77-84
-black hole 80-85
- distance 45, 56
- flares 82
Galactic coordinates 35-36
Galactic latitude 35
Galactic longitude 35
Galactic poles 35
galaxies 7-8, 87-140
- bimodal color distribution 119
- brightness profile 90-92, 98
- cD galaxies 90, 92, 230, 237, 249,
264
- chemical evolution 138-140
- color-color diagram 357
- dwarf galaxies 90, 224
-E+A galaxies 241
- early-type galaxies 8
- elliptical galaxies 88, 90-98
blue compact dwarfs (BCD's) 90
classification 88, 90
composition 92-93
counter-rotating disks 96
dark matter 101
dwarf ellipticals (dE's) 90
dwarf spheroidals (dSph's) 90
dynamics 93-95
formation 392-395
indicators for complex evolution
95-98, 393
mass-to-light ratio 96
stellar orbits 94
- halos 100
- Hubble sequence 88
- IRAS galaxies 368, 370
- irregular galaxies 88, 224
- late-type galaxies 8
- low surface brightness galaxies
(LSBs) 99, 301
- luminosity function 117-119
- Lyman-break galaxies 367, 391
- morphology-density relation 360
- post-starburst galaxies 241
- SO galaxies 88, 393
- satellite galaxies 101, 224, 304, 360
- scaling relations 104-109, 1 16, 319
- SCUBA galaxies 374-377
- spheroidal component 111, 113
- spiral galaxies 88, 98-104
bars 88
bulge 98, 102, 397
corona 103
dark matter 100
early-type spirals 98
normal and barred 88
rotation curve 8 , 1 00- 1 02, 1 04
spiral structure 103
stellar halo 100
stellar populations 102
thick disk 100
- starburst galaxies 1 1, 89, 203, 358,
369-371
- substructure 302-306, 396
- ULIRG (ultra-luminous infrared
galaxy) 24, 90, 208, 370
galaxy evolution 17, 390-402
galaxy groups 223, 228, 237-238
- compact groups 237, 238
- diffuse optical light 238
GALEX 31,361
gamma-ray bursts
- afterglows 404
- fireball model 404
- hypernovae 405
- short- and long-duration bursts 404
G-dwarf problem 140
globular clusters 55, 102
GOODS project 366
gravitational instability 278-282
gravitational lenses 65, 121-131
- clusters of galaxies as lenses
260-269, 368
- cosmic shear 329-330
- critical surface mass density 122,
128
- Einstein radius 124, 126, 261
- Einstein ring 67, 125, 126, 128
- galaxies as lenses 121-131
- Hubble constant 130,254
- luminous arcs 260-264, 356
- magnification 67-69, 123, 367-369
- mass determination 126, 128, 130
- microlensing effect 64-77
- multiple images 65, 66, 123,
125-129, 261, 368
- point-mass lenses 123
- search for clusters of galaxies 269
- shear 265
- substructure 304-306
- weak lensing effect 102, 264-269,
329-330, 338
gravitational redshift 337
gra\ italional waves 307, 400
Great Attractor 320
Great Debate 87
Great Wall 277,311
Green Bank Telescope 20
groups of galaxies 14
growth factor D + 292, 322
Gunn-Peterson test 382
! larrison-Zeldovich fluctuation
spectrum 285, 307, 339, 344
HEAO-1 31
heliocentric velocity 38
helium abundance 142, 164-165
Herschel satellite 380,410
Hertzsprung-Russell diagram (HRD)
133
llickson compact groups 237
hierarchical structure formation 293,
303, 374, 391-392
high- velocity cloud (HVC) 56, 227
horizon length 171, 339
horizon problem 171, 173, 307
hot dark matter (HDM) 286
Hubble classification of galaxies
88-89, 366, 390
Hubble constant H 9, 1 14, 1 16, 1 17,
154, 344, 352
- scaled Hubble constant h 10
Hubble Deep Field(s) 26, 364-367
Hubble diagram 9, 158, 325, 326, 328
Hubble Key Project 116
Hubble law 8, 9, 1 14, 146, 156, 325
Hubble radius 145
Hubble sequence see galaxies
Hubble Space Telescope (HST) 25,
26, 110,212,261,362,409
Hubble time 144
hypemovae 405
hypervelocity stars 84
inflation 173-174, 412, 307-412
1 mass function (IMF) 1 32, 387
Integral satellite 33
:egrated Sachs-Wolfe effect (ISW)
338-340, 349
fractions of galaxies 218
interferometry 21
itergalactic medium 331-332,334,
357, 382, 385
mization parameter 205
IRAS 23,311,321,342,370,374
•gular galaxies see galaxies
ISO 24, 370, 374, 380
isochrones 133
isophote 88
isothermal sphere 123, 232-233, 247,
261
□
375
HII re
:il
HIPPARCOS 37, 38, 426
Holmberg effect 225, 397
horizon 170-171,287
IUE 30
I nn Clerl i , 11 Telesi op
(JCMT) 22
lames Webb Space Telescope (JWST)
409
Jeans mass 383-384
jets 178, 190, 191, 196, 211-215, 251
Keck telescope 2, 25, 27, 357, 407
King models 233, 247
Kirchhoff's law 418
Kormendy relation 92
large-scale structure of the Universe
307, 309
- galaxy distribution 309-32 1 , 344
- numerical simulations 293-297
- power spectrum 284-289,291,
307,313-318,330,335,338
Las Campanas Redshift Survey
(LCRS) 311
last-scattering surface 168
late-type galaxies 88
light cone 142
light pollution 18
Limber equation 318
line transitions: allowed, forbidden,
semi-forbidden 201
linearly extrapolated density
fluctuation field 281
linearly extrapolated power spectrum
286
LISA 400,411
Local Group 14, 223-228
- galaxy content 224-225
- mass estimate 225-227
local standard of rest (LSR) 57
LOFAR 410
Lorentz factor 190
luminosity
- bolometric 423
- in a filter band 423
luminosity classes 425^126
luminosity distance 157, 158, 325,
327
luminosity function 117
- of galaxies 1 17-1 19, 230, 322,
372, 394, 396
-of quasars 216-219,333
luminous arcs see gravitational lenses
Lyman-a blobs 378
Lyman-a forest 219,220,351,357,
385,386
- as a tool for cosmology 335-336
- damped Lya systems 220, 332
- Lyman-limit systems 220, 332
Lyman-break galaxies see galaxies
Lyman break method 362,390
MACHOs 64, 70-74
Madau diagram 389-390, 394
Magellanic Clouds 14, 70, 224
-distance 115
Magellanic Stream 56, 225, 398
magnification see gravitational lenses
magnitude 420-423
- absolute magnitude 422
- apparent magnitude 420-422
- bolometric magnitude 422-423
main sequence 40, 426, 428
Malmquist bias 1 1 8
MAMBO 22
maser 78
mass segregation 236
mass spectrum of dark matter halos
291-293, 297
mass-to-light ratio 50, 135, 227, 238,
322
- in clusters of galaxies 322-323
-of galaxies 96,100,106,108
MAXIMA 342
merger tree 395
merging of galaxies 140, 203, 370,
390-393
- dry mergers 240, 393
- major merger 392, 397
- minor merger 392, 397
MERLIN 21
metallicity 44, 47, 50, 133, 138
metals 44
microlensing see gravitational lenses
Milky Way 35-85
- annihilation radiation 54, 77
- bulge 54
- center see Galactic center
- chemical composition 47
- dark halo 5, 70, 73
-disk 46-51
- distribution of dust 5 1
- gamma radiation 54
-gas 46,50-51,222
- halo 55-56
- hypervelocity stars 84
- kinematics 57-64
- magnetic field 51-53
- rotation curve 5,59-64
- structure 4, 44-56
- thick disk 46, 50
- thin disk 46, 50
Millennium simulation .296. 398
mixed dark matter (MDM) 288
Modified New Ionian D\ namies
(MOND) 413
mm in« cluster parallax 38-39
narrow-band photometry 356, 379
Near Infrared Camera and Multi-Object
Spectrograph (NICMOS) 26
neutrinos 48, 162
- masses 166, 288, 352
- radiation component of the Universe
162
- Solar neutrinos 427
neutron stars 48, 74
New General Catalog (NGC) 87
New Technology Telescope (NTT) 26
non-linear mass-scale 292
El
Olbers' paradox 142
Oort constants 61
optical depth 41,418
optically violently variables (OVV)
pair production and annihilation
161-163
Palomar Observatory Sky Survey
(POSS) 228
parsec 4, 37
passive evolution of a stellar populatioi
137
peculiar motion 114, 116
peculiar velocity 57, 280, 306-307,
316-321
period-luminosity relation 43, 44,
perturbation theory 279
photometric redshift 362-365
Pico Veleta telescope 22
Planck function 156,419
Planck satellite 349, 410
planetary nebulae 49, 429
- as distance indicators 1 16
polarization 51
Population III stars 384, 385
population synthesis 132-137, 392
power spectrum, normalization 291,
313,330,336,353
Press-Schechter model 297, 383, 395
primordial nucleosynthesis (BBN)
15, 163-166, 309
- baryon density in the Universe 323
proper motion 38, 80
proto-cluster 360, 374
proximity effect 332
pulsating stars 43
QSOs s
quasars
e galaxies
ive galaxies
radial velocity 38
radiative transfer equation 4 1 7-4 1 8
radio galaxies see active galaxies
random field 282-283
- Gaussian random field 285
- realization 282
Rayleigh-Jeans approximation 374,
119
162
115
recombination 16, 166-168, 337, 382
red cluster sequence (RCS) 27 1 , 274,
364, 392
red giants 426, 429
redshift 10, 142
- cosmological redshift 155-157,
188
redshift desert 391
redshift space 316
redshift surveys of galaxies 309-321
reionization 168, 331, 332, 337, 348,
351, 382-387
relaxation time-scale 235
ie\ erberation mapping 198, 204
right ascension 35
ROSAT 31, 104, 255, 258, 322, 380
ROS AT All Sky Survey (RASS) 3 1 ,
255, 324
rotation measure 52
rotational flattening 93
RR Lyrae stars 44, 56
Sachs- Wolfe effect 337, 339
Sagittarius dwarf galaxy 6, 227, 392
Saha equation 167
satellite galaxies see galaxies
scale factor see Universe
scale-height of the Galactic disk 46,
47
scale-length of the Galactic disk 47
S chechter lumino sity function 118,
230, 323
SCUBA 22
secondary distance indicators
116-117,319
seeing 19, 25, 37
self-shielding 377
service mode observing 29
Seyfert galaxies see active galaxies
SgrA* see Galactic center
shape parameter r 288, 314, 319,
324, 336
shells and ripples 96
shock fronts 53, 180, 214, 251
Silk damping 337, 339
singular isothermal sphere 232, 261
Sloan Digital Sky Survey (SDSS)
217,239,313,319,351
SOFI camera 26
softening length 294
sound horizon 337, 339, 340
source counts in an Euclidean universe
144
source function 418
specific energy density of a radiation
field 417
specific intensity 156, 417
spectral classes 425-426
spherical collapse model 289-290
spiral arms 98, 103
- as density waves 103
spiral galaxies see galaxies
Spitzer Space Telescope 24
Square Kilometer Array (SKA) 410
standard candles 49, 116, 324-325
star formation 51, 374, 383,
387-389, 392
- and color of galaxies 136, 274,
363, 390
- cosmic history 387-390
- quiescent star formation 390
- star-formation rate 89, 133, 203,
333,361,387
sUirbursl galaxies see galaxies
Stefan-Boltzmann law 420
stellar evolution 392
stellar populations 46, 47
Subaru telescope 27, 362
Sunyaev-Zeldovich effect 338,341
- distance determination 253
- Hubble constant 254
superclusters 277, 322
superluminal motion 188-191, 207,
210
supernovae 47, 429
- as distance indicators 49, 116,
324-329
- at high redshift 325-326
- classification 48
- core-collapse supernovae 48
- evolutionary effects 328
- metal enrichment of the ISM
47-49, 385
-SN1987A 48,115
- supernovae Type la 49
surface brightness fluctuations 116
surface gravity 426
SWIFT 405
synchrotron radiation 52,179-181,
196,212,214
synchrotron self-absorption 181
synchrotron self-Compton radiation
214
tangent point method 61-63
tangential velocity 38
thermal radiation 419
Thomson scattering 193, 337, 341
three-body dynamical system 85
tidal tails 370, 393
time dilation 328
transfer function 285-288, 314
trigonometric parallax 37-38
Tully-Fisher relation 3 1 9
- baryonic 106
Two-Degree Field Survey 217, 277,
314,316
Two-Micron All Sky Survey (2MASS)
259, 321
two-photon decay 167
ultra-luminous compact X-ray sources
(ULXs) 371
ultra luminous infrared galaxies
(ULIRGs) .see galaxies
Universe
- age 144, 153, 154
- critical density 148, 171, 298
-density 15,17,351,353
- density fluctuations 17, 277
- density parameter 155,165,171,
322, 323, 330
- Einstein-de Sitter model 17, 155,
159,281,292
- expansion 8, 146, 147, 161, 279
- scale factor 146, 152
- standard model 3, 169-173, 302,
309, 346
- thermal history 1 60- 1 69
vacuum energy see dark energ)
velocity dispersion 46
- in clusters of galaxies 232
-in galaxies 93,113,123
Very Large Array (VLA) 21,212,
376
Very Large Telescope (VLT) 28
Very Long Baseline Array (VLBA)
21
Very Long Baseline Interferometry
(VLB I) 21
violent relaxation 235, 290
Virgo Cluster of galaxies 12, 1 14,
116, 117,228
virial mass 256
virial radius 256, 298
virial theorem 13,187,233
wedge diagram 310,316
virial velocity 298
white dwarfs 49, 73, 328, 426, 429
virtual observatory 411
Wide-Field and Planetary Camera
VISTA 407
(WFPC2) 26
VLT Survey Telescope (VST) 407
wide-field cameras 265, 407
voids 12,277,311
width of a spectral line 181
Voigt profile 221
Wien approximation 419
Wien's law 419
P71
WIMPs 161,165-166,411
yj
WMAP 22, 168, 345-352, 382
warm-hot intergalactic medium 334
wave number 284
□
weak lensing effect see gravitational
lenses
X-factor 51
XMM-Newton 31,202,248
X-ray background 168
X ray binaries 188
Zeeman effect 52
zero-age main sequence (ZAMS) 428
Zone of Avoidance 36, 223, 320
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