IAAORDIAL
COSAAOLOGY
Giovanni Montani
Marco Valerie Battisti
Riccardo Benini
Giovamti Imoonente
y x .
^
World Scientific
PRIMORDIAL
COSMOLOGY
This page is intentionally left blank
PRIMORDIAL
COSMOLOGY
Giovanni Montani
ENEA - C.R. Frascati, ICRANet and Dipartimento di Fisica,
Universita di Roma "Sapienza", Italy
Marco Valerio Battisti
Dipartimento di Fisica, Universita di Roma "Sapienza", Italy
and
Centre de Physique Theorique, Luminy, Marseille, France
Riccardo Benini
ICRA and Dipartimento di Fisica,
Universita di Roma "Sapienza", Italy
Giovanni Imponente
Queen Mary, University of London, UK
f World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING ■ SHANGHAI ■ HONG KONG • TAIPEI • CHENNAI
4-
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Primordial cosmology / by Giovanni Montani ... [et al.].
Include, bibliographical reference, and index.
ISBN-13: 978-981-4271-00-4 (hardcover : alk. paper)
ISBN-10: 981-4271-00-4 (hardcover : alk. paper)
1. Big bang theory. 2. Singularities (Mathematics) 3. Cosmoloay-Mathematical models.
I. Montani, Giovanni.
QB991.B54P752011
523.1--dc22
2010027346
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Ltbran .
Copyright © 201 1 by World Scientific Publishing Co. Pte. Ltd.
Ml riyjtts reserved. This book, or parts thereof, may not be 1 1 /> ornt oi In any in
electronic or me: , i ■ < tuition stora c mill rctt
system now kit
For photocopying of material in this volume, plea
Clearance Center, Inc., 222 Rosewood Drive, Danve:
nholocop) is not required from lite publisher.
Printed in Singapore.
4-
The authors dedicate this Book to all the people who, like
them, regarded and will regard the understanding of the Early
Universe as the greatest challenge of their lives.
This page is intentionally left blank
Preface
Understanding the Origin and Evolution of the Universe is certainly one
of the most ambitious and fascinating attempts of the human intelligence.
This intellectual and scientific adventure is not only the major approach
to give answers to fundamental questions concerning our existence but also
this activity provides our intelligence with the special role to be the only
(known) tool the Universe has to investigate itself, acquiring awareness on
the Reality essence.
Despite this ambitious path could discourage from addressing any per-
spective, the scientific investigation of the Universe has reached surprising
and impressive achievements and offers many reliable answers about our
origin.
This volume summarizes the most important results of the scientific
cosmology, describing the observational knowledge about the Universe evo-
lution and how it allowed the derivation of a theoretical paradigm, able of
predictions beyond detected phenomena.
Our analysis is based on a rigorous mathematical characterization of the
cosmological topics, finding in the Einstein theory of General Relativity the
privileged descriptive physical tool. For the wide and coherent cosmological
scenario, this Book is built up as a reference for both students interested to
an introductory path, and also for specialists who desire to deepen selected
topics.
This Book faces the analysis of many aspects of Modern Cosmology,
starting from the presentation of well-grounded assessments on the observed
Universe and their theoretical interpretation, up to the discussion of very
speculative topics concerning the nature of the Cosmological singularity,
which are timely for the scientific debate.
The content can be successfully approached by any reader having a
viii Primordial Cosmology
certain familiarity with the concepts and the formalisms at the ground of
General Relativity. The presentation is purposely self-consistent when char-
acterizing some canonical topics with a pedagogical perspective, as well as
when serving an advanced profile for the subjects requiring a wider back-
ground knowledge.
Some peculiar approaches to modern cosmology are treated in their
general aspects and included only to provide the reader with a broad vi-
sion of the contemporary lines of research, although averting from the core
perspective of the Book, and referring to the specific literature for details.
We widely illustrate a series of modern cosmological issues, re-enforcing
the idea that the highly symmetric nature of our Universe, as observed at
very large scales, is not a primordial notion but results from an evolution-
ary process of very general initial conditions.
The first part of the Book (Chapters 1 and 2) is devoted to a historical
picture of the notion of Universe across the centuries and then addresses the
fundamental formalism of General Relativity and differential geometry for
the modern approaches to the Early Universe providing the basic notions
for the non-specialist reader. The so-called Physical Cosmology (Chapters
3-6) is faced in details, introducing the structure and the evolution of the
isotropic Universe, according to the Standard Cosmological Model. Par-
ticular attention is dedicated to the phenomenology of the Universe with
respect to the i.niplicat ions for the interpretation of the features implied by
the theoretical prescriptions of different models. This aim is also pursued by
analyzing the inflationary paradigm as well as a detailed treatment of the
density inhomogeneities faced by the perturbation theory to the isotropic
Universe and by the paradigm of the quasi-isotropic solution.
The part of the Book entitled Matht matical Cosmology (Chapters 7-9)
gives a wide discussion of the general features of the Universe near the
singularity, when the isotropy and homogeneity assumptions are removed.
We start with a geometrical characterization of the homogeneous three-
dimensional spaces, as arranged in the Bianchi classification, whose dy-
namics is treated in the framework of General Relativity, also by means of
a Hamiltonian formulation. In this respect, our presentation is focused on
the study of the chaotic dynamics of the Bianchi type VIII and IX models
near the singularity. Here, the discussion clearly separates well-established
results from timely reformulations of the problem or aspects yet open to
scientific investigation. Finally, we endow this part by a deep analysis
of the generic inhomogeneous solution near the cosmological singularity,
Preface ix
derived in analogy to the chaotic homogeneous model and implementing
the parametric role played by the space coordinates. The nature of the
space-time foam, characterizing the generic Universe asymptotically to the
singular point, is outlined toward a proper statistical picture. This study
of cosmologies more general than the isotropic Universe is also extended to
multidimensional issues, in view of the interest raised through the recent
literature.
The last part of the Book is focused on Quantum Cosmology (Chapters
10-12) and it touches very timely questions in applying Quantum Gravity
proposals to the Universe origin and its primordial evolution. We discuss
different approaches to the quantization of the gravitational field, concen-
trating the investigation on the canonical method, both in the original
Wheeler-DeWitt and in the more promising Loop Quantum Gravity ap-
proaches. However, different, points of view, like the path integral quan-
tization and the generalized Heisenberg non-commutative algebras proce-
dures are also taken into account and implemented on specific models. Our
quantum description of the Universe is very general and addresses all the
most relevant features, especially in view of the interpretative shortcomings
of the different approaches. The recent and outstanding success of Loop
Quantum Cosmology in determining the existence of a Big Bounce at the
Universe birth is eventually traced with care. Indeed, the quantum nature
of the cosmological singularity (its removal or survival) is a central theme
and many different issues are contrasted in view of their motivating hy-
potheses. A part of such Section treats the interpretation of a semiclassical
Universe and its tendency to isotropization.
The material presented in the Chapters on quantum cosmology clarifies
the motivation for an intense study of the mathematical cosmology, because
the general character of the cosmological models in the regime asymptotic
to the initial singularity makes them as the most appropriate Cor the im-
plementation of a quantum theory to the Universe birth. The idea that
the classical Universe comes out of the Planck epoch via a semiclassical
limit, say when its volume expectation value is large enough, implies that
we must have a clear understanding of such general dynamics already at a
classical level. The possibility to link the anisotropic and inhomogeneous
cosmologies to the isotropic Universe, underlying the Standard Cosmologi-
cal model, can be recognized in the inflation scenario. In fact, the vacuum
energy responsible for the de Sitter phase of the inflating Universe is a
strong isotropic term, able to stretch the inhomogeneities at scales much
x Primordial Cosmology
larger than the physical horizon and to suppress anisotropic features.
This scenario prescribes that the Universe is born from a singularity-
free generic inhomogeneous model; when its volume is probabilistically large
enough, it transits to a quasi-classical dynamics, and then it is reconciled
to the Standard Cosmological Model by the inflationary process. Such a
point of view constitutes the leading perspective suggested and partially
demonstrated by this Book.
We would like to express our gratitude to Dr Massimiliano Lattanzi
for his precious contribution to the part of this Book devoted to Physical
Cosmology, especially in view of the effort made to link the theoretical
framework to the present knowledge of the observed Universe.
Dr Nakia Carlevaro and Dr Francesco Cianfrani are thanked for their
help in writing Sec. 3.6 on the Lemaitre-Tolmann-Bondi model. F.C. is
also thanked for his comments and suggestions on the part of the Book
concerning Quantum Cosmology.
We would like to acknowledge Dr Simone Speziale for his valuable sug-
gestions on that part in which we review the main features of the Loop
Quanti lm Gravity theory.
Finally, a special thought of G.M. is devoted to the memory of Prof.
Kensuke Yoshida, who appreciably encouraged him to write the review
article 1 , from which the project of this Book arose.
Giovanni Montani
Marco Valerio Battisti
Riccardo Benini
Giovanni Imponente
1 Montani, G., Battisti, M.V., Benini, R. and Imponente, G.(2008). Classical and Quan-
tum Features of the Mixmaster Singularity, Int. J. Mod. Phys. A23, pp. 2353 - 2503,
doi:10.1142/S0217751X08040275.
k = 8ttG
Einstein constant
h = c=i
natural units
+,-,--)
metric signal iu'c
ftj
metric tensor
/la/3
spatial metric tensor
ds 2
space-time line element
dl 2
spatial line element
M
space-time manifold
E
space-like surface
5
actiou
£
Lagrangian density
H
Hamiltonian density
ff
Hubble parameter
P
energy density
P
pressure
P
particle momentum
*,j,fe,...
space-time indices
a, /3,7, . . .
spatial indices
I,J,K,...
internal 4-d indices
a,b,c, . . .
internal 3-d indices
This page is intentionally left blank
Contents
Preface
List of Figures
Historical and Basic Notions
Historical Picture 3
1.1 The Concept of Universe Through the Centuries 3
1.1.1 The ancient cultures 3
1.1.2 Ancient Greek and the Mediterranean 5
1.1.3 The Hellenistic era 9
1.1.4 The philosophical point of view in the Old Age . . 11
1.1.5 The Middle Age dogmas 11
1.1.6 The Renaissance revolution 15
1.1.7 The Scientific Revolution 16
1.1.8 The Enlightenment Era 23
1.2 The XIX Century Knowledge 25
1.2.1 Geometrical formalisms 25
1.2.2 Difficulties for the birth of a real cosmology: 01-
bers' paradox 27
1.2.3 Luigi Bianchi and the developments of differential
geometry 28
1.2.4 Einstein vision of space-time 28
1.3 Birth of Scientific Cosmology 30
1.3.1 Einstein proposal of a static Universe 31
Primordial Cosmology
1.3.2 Galaxies and their expansion: The Hubble's dis-
covery 32
1.4 The Genesis of the Hot Big Bang Model 34
1.4.1 Recent developments 37
1.4.2 Discovery of the accelcral ion 40
1.4.3 Generic nature of the cosmological singularity:
The Cambridge and the Laudau School 41
1.4.4 The inflationary paradigm 43
1.4.5 The idea of non-singular cosmology: The cyclic
Universe and the Big Bounce 44
1.5 Guidelines to the Literature 46
Fundamental Tools 51
2.1 Einstein Equations 52
2.2 Matter Fields 54
2.2.1 Perfect fluid 54
2.2.2 Scalar field 56
2.2.3 Electromagnetic field 58
2.2.4 Yang-Mills fields and 6-sector 60
2.3 Hamiltonian Formulation of the Dynamics 63
2.3.1 Canonical General Relativity 64
2.3.2 Hamilton-Jacobi equations for gravitational field . 70
2.3.3 The ADM reduction of the dynamics 71
2.4 Synchronous Reference System 73
2.5 Tetradic Formalism 74
2.6 Gauge-like Formulation of GR 77
2.6.1 Lagrangian formulation 77
2.6.2 Hamiltonian formulation 80
2.6.3 On the gauge group of GR 84
2.7 Singularity Theorems 85
2.7.1 Definition of a space-time singularity 85
2.7.2 Fluid kinematics 86
2.7.3 The Ravchaudhuri equal ion . 88
2.7.4 Singularity Theorems 90
2.8 Guidelines to the Literature 92
Physical Cosmology 93
3. The Structure and Dynamics of the Isotropic Universe 95
3.1 The RW Geometry 96
3..1..1 DeiinJtiou of isolropy 97
3.1.2 Kinematics of the isotropic Universe 98
3.1.3 The particle motion 99
3.1.4 The Hubble law 100
3.1.5 The Hubble length and the cosmological horizon . 103
3.1.6 Kinetic theory and thermodynamics in the expand-
ing Universe: The hot Big Bang 106
3.2 The FRW Cosmology 112
3.2.1 Field equations for the isotropic Universe 112
3.2.2 Asymptotic solution toward the Big Bang 114
3.2.3 The de Sitter Solution 118
3.2.4 Hamiltonian dynamics of the isotropic Universe . 119
3.3 Dissipative Cosmologies 122
3.3.1 Bulk viscosity 122
3.3.2 Matter creation in the expanding Universe .... 124
3.4 Inhomogeneous Fluctuations in the Universe 127
3.4.1 The meaning of cosmological perturbations .... 127
3.4.2 The Jeans length in a static uniform fluid 131
3.4.3 The Jeans length in an expanding Universe .... 132
3.5 General Relativistic Perturbation Theory 136
3.5.1 Perturbed Einstein equations 137
3.5.2 Scalar-vector-tensor decomposition and Fourier
expansion 140
3.5.3 Perturbed conservation equations 143
3.5.4 Gauge modes 144
3.5.5 Evolution of scalar modes 147
3.5.6 Adiabatic and isocurvature perturbations 149
3.5.7 Imperfect fluids 151
3.5.8 Kinetic theory 154
3.6 The Lemaitre-Tolmann-Bondi Spherical Solution 159
3.7 Guidelines to the Literature 162
4. Features of the Observed Universe 165
4.1 Current Status: The Concordance Model 166
Primordial Cosmology
The Large-Scale Structure 168
4.2.1 Deviations from homogeneity 169
4.2.2 Dark matter 170
4.2.3 The power spectrum of density fluctuations .... 172
The Acceleration of the Universe 175
The Cosmic Microwave Background 176
4.4.1 Sources of anisotropy 179
4.4.2 The power spectrum of CMB anisotropies .... 180
4.4.3 Acoustic oscillations 183
4.4.4 Effect of the cosmological parameters 187
Guidelines to the Literature 191
5. The Theory of Inflation
The Shortcomings of the Standard Cosmology 194
5.1.1 The horizon and flatness paradoxes 195
5.1.2 The entropy problem and the unwanted relics para-
dox 197
The Inflationary Paradigm 200
5.2.1 Spontaneous symmetry breaking and the Higgs
phenomenon 200
Presence of a Self-interacting Scalar Field 205
5.3.1 Coupling of the scalar field with the thermal bath 211
Inflationary Dynamics 213
5.4.1 Slow-rolling phase 214
5.4.2 The reheating phase 215
Solution to the Shortcomings of the Standard Cosmology 221
5.5.1 Solution to the horizon and flatness paradoxes . . 221
5.5.2 Solution to the entropy problem and to the un-
wanted relics paradox 224
General Features 225
5.6.1 Slow-rolling phase 225
5.6.2 Reheating phase 226
5.6.3 The Coleman- Weinberg model 227
5.6.4 Genesis of the seeds for structure formation . . . 228
Possible Explanations for the Present Acceleration of the
Universe 233
5.7.1 Dark energy 234
5.7.2 Modified gravity theory 236
Guidelines to the Literature 239
6. Inhomogeneous Quasi-isotropic Cosmologies 241
6.1 Quasi-isotropic Solution 242
6.2 The Presence of Ultrarelativistic Matter 242
6.3 The Role of a Massless Scalar Field 246
6.4 The Role of an Electromagnetic Field 252
6.5 Quasi-isotropic Inflation 257
6.5.1 Geometry, matter and scalar field equations . . . 258
6.5.2 Inflationary dynamics 259
6.5.3 Physical considerations 262
6.6 Quasi-isotropic Viscous Solution 264
6.6.1 Form of the energy density 265
6.6.2 Comments on the adopted paradigm 266
6.6.3 Solutions of the 00-Einstein and hydrodynamical
equations 270
6.6.4 The velocity and the three-metric 272
6.7 Guidelines to the Literature 274
Mathematical Cosmology 277
7. Homogeneous Universes 279
7.1 Homogeneous Cosmological Models 280
7.1.1 Definition of homogeneity 280
7.1.2 Application to Cosmology 283
7.1.3 Bianchi Classification and Line Element 286
7.2 Kasner Solution 288
7.2.1 The role of matter 291
7.3 The Dynamics of the Bianchi Models 292
7.3.1 Bianchi type II: The concept of Kasner epoch . . 294
7.3.2 Bianchi type VII: The concept of Kasner era . . . 297
7.4 Bianchi Types VIII and IX Models 298
7.4.1 The oscillatory regime 298
7.4.2 Stochastic properties and the Gaussian distribution 301
7.4.3 Small oscillations 304
7.5 Dynamical Systems Approach 308
7.5.1 Equations for orthogonal Bianchi class A models . 310
7.5.2 The Bianchi I model and the Kasner circle .... 312
7.5.3 The Bianchi II model in vacuum 315
xviii Primordial Cosmology
7.5.4 The Bianchi IX model and the Mixmaster attrac-
ted theorem 317
7.6 Multidimensional Homogeneous Universes 320
7.6.1 On the non-diagonal cases 323
7.7 Guidelines to the Literature 325
8. Hamiltonian Formulation of the Mixmaster 327
8.1 Hamiltonian Formulation of the Dynamics 328
8.2 The Mixmaster Model in the Misner Variables 331
8.2.1 Metric reparametrization 332
8.2.2 Kasner solution 333
8.2.3 Lagrangian formulation 334
8.2.4 Reduced ADM Hamiltonian 334
8.2.5 Mixmaster dynamics 335
8.3 Misner-Chitre Like Variables 339
8.3.1 The Jacobi metric and the billiard representation 341
8.3.2 Some remarks on the billiard representation . . . 344
8.4 The Invariant Liouville Measure 344
8.5 Invariant Lyapunov Exponent 346
8.6 Chaos Covariance 348
8.6.1 Shortcomings of Lyapunov exponents 349
8.6.2 On the occurrence of fractal basin 351
8.7 Cosmological Chaos as a Dimensional and Matter Depen-
dent Phenomenon 353
8.7.1 The role of a scalar field 353
8.7.2 The role of a vector field 355
(S.8 Isol ropizat iou Mechanism 356
8.9 Guidelines to the Literature 359
9. The Generic Cosmological Solution Near the Singularity 361
9.1 Inhomogeneous Perturbations of Bianchi IX 362
9.2 Formulation of the Generic Cosmological Problem .... 366
9.2.1 The Generalized Kasner solution 367
9.2.2 Inhomogeneous BKL solution 368
9.2.3 Rotation of the Kasner axes 372
9.3 The Fragmentation Process 373
9.3.1 Physical meaning of the BKL conjecture 375
9.4 The Generic Cosmological Solution in Misner Variables . 375
Hamiltonian Formulation in a General Framework .... 378
9.5.1 Local dynamics 380
9.5.2 Dynamics of inhomogeneities 381
The Generic Cosmological Problem in the Iwasawa
Variables 383
9.6.. I Asymptotic freezing of the Iwasawa variables . . . 385
9.6.2 Cosmological billiards 386
Multidimensional Oscillatory Regime 388
9.7.1 Dilatons, p-forms and Kac-Moody algebras .... 390
Properties of the BKL Map 391
9.8.1 Parametrization in a generic number of dimensions 392
9.8.2 Ordering properties 394
9.8.3 Properties of the BKL map in the v space .... 397
Guidelines to the Literature 398
Quantum Cosmology 401
10. Standard Quantum Cosmology 403
10.1 Quantum Geometrodynamics 405
10.1.1 The Wheeler-DeWitt Theory 405
10.1.2 Relation with the Path Integral Quantization . . . 410
10.2 The Problem of Time 412
10.2.1 Time before quantization 413
10.2.2 Time after quantization 418
10.2.3 Timeless physics 419
10.3 What is Quantum Cosmology? 421
10.3.1 Minisuperspace models 422
10.3.2 Interpretation of the theory 424
10.3.3 Quantum singularity avoidance 425
10.4 Path Integral in the Minisuperspace 427
10.5 Scalar Field as Relational Time 430
10.6 Interpretation of the Wave Function of the Universe . . . 433
10.6.1 The semiclassical approximation 434
10.6.2 An example: A quantum mechanism for the
isotropization of the Universe 437
10.7 Boundary Conditions 441
10.7.1 No-boundary proposal 443
10.7.2 Tunneling proposal 445
Primordial Cosmology
10.7.3 Comparison between the two approaches 446
10.8 Quantization of the FRW Model Filled with a Scalar Field 447
10.9 The Poincare Half Plane 451
10.10 Quantum Dynamics of the Taub Universe 453
10.10.1 Classical framework 453
10.10.2 Quantum framework 454
10.11 Quantization of the Mixmastcr in the Misner Picture . . . 457
10.12 The Quantum Mixmaster in the Poincare Half Plane . . . 459
10.12.1 Continuity equation and the Liouville theorem . . 460
10.12.2 Schrodinger dynamics 462
10.12.3 Eigenfunctions and the vacuum state 464
10.12.4 Properties of the spectrum 466
10.13 Guidelines to the Literature 469
Generalized Approaches to Quantum Mechanics 473
11.1 The Algebraic Approach 474
11.1.1 Basic elements 475
11.1.2 GNS construction and Fell theorem 478
11.2 Polymer Quantum Mechanics 480
11.2.1 From Schrodinger to polymer representation ... 481
11.2.2 Kinematics 483
11.2.3 Dynamics 485
11.3 On the Existence of a Fundamental Scale 487
11.4 String Theory and Generalized Uncertainty Principle . . . 488
11.5 Heisenberg Algebras in Non-Commutative Snyder Space-
Time 491
11.6 Quantum Mechanics in the GUP Framework 494
11.7 Guidelines to the Literature 498
Modern Quantum Cosmology 499
12.1 Loop Quantum Gravity 500
12.1.1 Kinematics 502
12.1.2 Implementation of the constraints 506
12.1.3 Quantum constraints algebra 510
12.2 Loop Quantum Cosmology 511
12.2.1 Kinematics 511
12.2.2 Quantum dynamics and Big Bounce 516
12.2.3 Effective classical dynamics 519
12.3 Mixmaster Universe in LQC 521
12.3.1 Loop quantum Bianchi IX 522
12.3.2 Effective dynamics 526
12.4 Triangulated Loop Quantum Cosmology 528
12.4.1 The triangulated model 528
12.4.2 Isotropic sector: FRW 531
12.4.3 Anisotropic sector: Bianchi IX 532
12.4.4 Full dipole model 533
12.4.5 Quantization of the model 535
12.5 Snyder-Deformed Quantum Cosmology 536
12.6 GUP and Polymer Quantum Cosmology: The Taub
Universe 540
12.6.1 Deformed classical dynamics 541
12.6.2 Deformed quantum dynamics 542
12.7 Mixmaster Universe in the GUP Approach 546
12.8 Guidelines to the Literature 550
Bibliography 553
Index 585
This page is intentionally left blank
List of Figures
2.1 Construction of the lapse function and of the shift vector ... 65
4.1 Luminosity distance vs. rodshifl 175
4.2 CMB frequency spectrum 176
4.3 Microwave sky 177
4.4 Acoustic oscillations 183
4.5 CMB anisotropy spectrum from WMAP7 189
5.1 Higgs potential 201
5.2 Old inflation potential 207
5.3 Slow rolling potential 208
5.4 Temperature-dependent Higgs potential 210
5.5 Behavior of the energy density during reheating 217
7.1 Kasner indices in terms of the parameter 1/u 287
7.2 The Kasner Circle 310
7.3 Dynamics of fi and S in the Bianchi I model with matter . . 312
7.4 Bianchi II transitions in £ plane 313
7.5 Bianchi IX evolution 314
7.6 Region where the 5-d BKL mechanism breaks down 320
8.1 Equipotential lines of the Bianchi type VIII model 332
8.2 Equipotential lines of the Bianchi type IX model 333
8.3 Reduced configurational space Tq(£,6) 338
8.4 Fractal structure in Mixmaster dynamics 348
9.1 Domain of validity for the 4-d Kasner parametrization 391
9.2 3-d representation of pi of the 4-d Kasner model 392
xxiv Primordial Cosmology
10.1 Ground state of an isotropizing Universe 437
10.2 The Hartle-Hawking instanton 440
10.3 Triangular domain Uq(u,v) in the Poincare plane 448
10.4 Dynamics of the Taub Universe in the (r, x)-plane 450
10.5 WDW wave packet for the Taub Universe 452
10.6 r„ (a) for three different values of the parameter k n 455
10.7 Approximate domain for the quantum dynamics of Bianchi IX 460
10.8 Wave function of the ground-state of the Mixmaster 462
11.1 Region allowed by the generalized uncertainty principle .... 486
12.1 Effective LQC vs classical dynamics for the FRW Universe . . 517
12.2 GUP wave- function for the Taub Universe 540
12.3 Polymer wave packet |* (x, r) | of the Taub Universe 541
12.4 Velocity of the wall of Bianchi IX in the GUP framework ... 545
PART 1
Historical and Basic Notions
In these Chapters, we present the fundamental concepts relevant for the
further developments of the topics.
Chapter 1 is devoted to provide a historical picture concerning the notion
of Universe through the centuries.
Chapter 2 gives a pedagogical review of the fundamental formalisms re-
quired for the self-consistency of the presentation.
This page is intentionally left blank
Chapter 1
Historical Picture
This Chapter is devoted to draw the historical path of Cosmology from the
first written evidences in the ancient cultures up to contemporary science
and serves as an introduction to the topics of modern Cosmology which
constitute the main core of the Book.
Although synthesized, we will provide a vision of how the concept of
Cosmology as a realm outside the daily experience evolved from a religious
or philosophical task toward a well-grounded scientific subject of investiga-
tion and discussion. The history of this evolution has been slow since the
experimental hints pushing to find explanation to natural phenomena have
been very limited up to the last century, leaving Cosmology to the area
of Astronomy at most, and only in regions of the world where the social
environment supported such kind of science. Finally, since the object of
investigation cannot be reproduced in a laboratory or allows any "second
try" for any test, the theoretical approach has often been influenced by
personal beliefs, traditions and not-scientific related issues.
To give a peculiar introduction to Theoretical Cosmology, this Chapter
stresses how the relationship of the human beings evolved with respect
to the celestial phenomena, enlarging their (and our) view of the space
surrounding the planet where we live and following how they pushed further
and further the borders of the Universe around them.
1.1 The Concept of Universe Through the Centuries
1.1.1 The ancient cultures
The perception of the nature as vital environment had a variegate evolution
and aim, either with the passing of the centuries or in the different geo-
4 Primordial Cosmology
graphical areas. As a first instance, the modern science finds its two main
roots in the Greek-Roman basin and in the Oriental world. Nevertheless,
when the latter flourished, showing a written tradition which dated back to
around 1300-1200 B.C. with the religious and philosophical setting of Hin-
duism with the first doctrines in the Rig Veda sacred books and thereafter
in the Upanishad around the IX- VIII century B.C., the former was still in a
embryonic stage. The principal effort of the understanding pursued by the
philosophers in the Eastern area was devoted to deal with existential prob-
lems, considering the sense of knowledge towards ones' salvation and free-
dom. The role of the Universe was a paradigm, a reference for the human
behavior. On the other side, around the Mediterranean sea, other cultures
showed attention to the natural phenomena starting from the Chaldee who,
even before 2000 B.C., addressed much efforts towards the comprehension
of celestial phenomena, taking records of eclipses, constellations shapes, as-
tral conjunctions, and considering a primordial Zodiac. In a nearby area,
the Egyptians, strong of the emerging mathematical techniques, developed
an accurate record of stars and constellations arising in particular from the
precise orientation evident in the pyramid foundations, as well as also in
Mesopotamia, under the kingdom of Hammurabi of Babylon. Such cul-
tures received a strong influence from the Oriental connections, were the
privilege of a sacerdotal caste and relied on myths regarding the origin
of the Universe, plenty of fantastic representations with a strong religious
and traditionalistic role. Much differently, on the Greek side, grown in the
Hellenistic age, the basic demand was toward a break with the preceding
beliefs, in a rational investigation of the natural phenomena. Although it
is natural that the Greek science was aware of other traditions, like for ex-
ample a catalogue of the possible astronomical phenomena, Egyptian and
Mesopotamians used them to study the cosmic order and its influence on
the daily life of their kingdoms (essentially as horoscopes, observation of
time). The Greeks tended to a more theoretical approach, with the pur-
pose of understanding the causes of the astral movements, the origin of
the eclipses, passing iron: a description to a tentative rational explanation,
getting toward the fundamental abstraction provided by the concept of a
formula and the law, i.e. the inclusion of the infinite possible cases of re-
alizing a certain situation. The socio-economical environment of the Greek
cities, where the republican society is much more dynamic than the absolute
power kingdoms in other areas, will reflect in the vision of the Universe as
an evolutionary process, rationally quesl ionable, whose basis is not under
the control of a religious group but can be debated by the free thought of
Historical Picture 5
the philosophers.
The shape and the borders of the physical world changed with the size
of the observable Universe, thus limited to the astronomical observable
phenomena and leaving all the remaining to the metaphysical one where
gods and myths had a crucial role, often communicating with the sensible
experience. The concept of Universe was intended to include both of them
and the principal effort was to point out its origin.
A common aspect of all cultures when trying to explain or describe
the whole Universe is that it must be emerging from a limited and compact
description, which can be a basic element (water, earth, fire) or made (from
a breathe, or by a god) by a clearly defined subject.
The analysis of the human activities and the comprehension of the
existing world from the natural events in relation to the social and eco-
nomic strategies is pursued by a unique figure provided by the philosopher.
Physics will be considered as a branch of philosophy up to the end of the
XVII century and on many topics both of them remain deeply related to
Religion.
1.1.2 Ancient Greek and the Mediterranean
Aristotle mentioned that the poet Hesiod (around 700 B.C.) was the first to
search for a unique principle for all the things, <CFirst existed chaos, then
the earth . . . and finally the love among the gods»: although the origin of
the Universe finds an answer from the myth, it opened the way to several
explanations devoted to find a fundamental one, unrelated to the activities
of the common life but essentially metaphysical, beyond the nature. The ac-
tual Universe is regarded as an ordered state and the Cosmology is devoted
to find the unity which guarantees its origin and its equilibrium.
During the VI century B.C. in the Ionian region, flourished a strong soci-
ety based in several important centers such as Miletus, Ephesus, Colophons,
Samos, with a class of merchants willing to expand in the Mediterranean
area, going from Black Sea to Egypt, to the Caucasus, Sicily, Spain and
France with a mentality open to overcome the limits of the magic beliefs and
devoted to a more accurate care to the rational observations of the natural
phenomena. In this environment a group of thinkers called Pre-Socratic
(also known as Pre-Sophists) is formed, whose main topic of investigation
is the cosmological problem: the Ionians pose at the basis of everything a
unique and eternal reality, provided by the arche (i.e. a principle) for the
matter from which everything comes out. The primordial force is provided
6 Primordial Cosmology
of an intrinsic force allowing all movements.
This cultural current has been initiated by Thales (around 585 B.C.)
who seeks in the water the basic element from which everything comes out.
In the same years Anaximander didn't look at a specific material element
but to the infinity, the infinite amount of matter, the origin of all what
is observed. This principle is never ending and undying, it is outside the
world and comprises: he developed an idea for the process of the matter
generation through successive separations, breaking the uniqueness of the
primordial infinity and providing a law governing the nature. His idea of
the Earth is in the form of a cylinder and the human beings come from
inside fishes where some of them developed and finally they were thrown
outside to live by themselves.
In the years 530-490 B.C. Pythagoras and his School were active in
Magna Grecia (across southern Italy) and funded the modern mathemat-
ics, introducing basic concepts, the first rigorous demonstrations providing
abstractions from many empirical situations. Every geometrical figure is
a deployment of points and the numbers measure the order of everything.
The true nature of the world comes as a measurable order of basic elements
and the opposition which is manifest in the real world can be comprehended
in a natural cosmic harmony from which everj I lung proceeds.
A deep crisis of the Pythagoreans philosophers sprung with the discov-
ery of the infinity in mathematics and in numerics. Their astronomical
knowledge was rather evolved, as based on the sphericity of the Earth and
of the celestial bodies. Such shape, as maximum expression of harmony,
was rephrased also in the model for the sky: the Earth moving around
a central Fire, together with all other celestial bodies. All celestial bodies
were classified as the sky of the fixed stars, the five planets (Saturn, Jupiter,
Mars, Mercury, Venus), the Sun, Moon, Earth and anti-Earth (hypothetical
planet in order to reach the sacred number of 10). Only few decades later
Aristarchus from Samos (III century B.C.) proposed the Sun as the center
of the celestial spheres, thus clearly anticipating the Copernican heliocen-
As we see from what described, the concept itself of Universe was limited
to what was the knowledge of existence beyond the Earth: in this view we
have the two extremes of the local astronomy on one side and on the other
a more effective ontological concept about the origin of everything, with the
concept of far (fixed) stars in between. Although in the subsequent evolu-
tion the astronomical knowledge deviated toward an Earth-centric system
following a peculiar line of debate, this frame remained almost unchanged
Historical Picture 7
up to the first astronomical evidences of different orbiting systems and ce-
lestial bodies beyond our galaxy.
The concept of eternal evolution, within the harmony of the opposites,
dominates the thought of Heraclitus from Ephesus (VI-V cent. B.C.),
whose idea of becoming identifies alternating eras of destruction-production,
assessing a strong difference with the philosophies from the East.
At this stage, the concept of Universe and that of Cosmos are still deeply
different from the idea in the readers' mind: Cosmos, in ancient Greek "or-
nament" is considered as a decoration of the celestial sphere and its physical
reality is deeply mixed and indistinct from its ontological counterpart.
One of the first proposals regarding the origin of matter derived from
the physical experience lies in the idea of Anaxagoras (mid V century B.C.).
He argued about the absence of either a minimal or a maximal size, from
the tiniest pieces of matter to the whole existing Universe, where the divine
Nous, intelligence, orders the original seeds properly mixing them as they
appear in the world. From the original chaos of such seeds a swirling
movement produced the Earth and the stars result from the lightening of
the particles coming from it and even the Sun.
The concept of a minimum dimension for the particles constituting the
matter has been proposed by Democritus (ca. 460 B.C. ca. 370 B.C.), with
the atoms (i.e. non divisible pari icles) which are unchangeable, eternal and
indestructible. They chaotically move in every direction, whirling in infinite
ways and assembling in an infinite number of ways which are born and die
perpetually. The impulse to a mechanistic attitude in the investigation
of the nature lies in the search for the causes of the events, looking for
quantitative properties and definition of objective properties.
The influence on the evolution of thought of Socrates first and then
of his disciple Plato in the environment of the Academia in Athens relied
mainly on the investigation of ethics and other problems focused on man,
his interior and relations, recovering in the cosmological picture the pri-
mordial link between celestial bodies and the divinities.
A new effort toward the investigation about the system of the nature
and science was by the Plato disciple Aristotle (about 384-322 B.C.). A
systematic classification of all branches of the knowledge provided a clear
statement of the role of Physics, well separated from Metaphysics, Theol-
ogy and other branches of philosophy. The fundamental topic of his inves-
tigation is the being in movement with all its qualities and properties: he
classified three species of motion, from the center of the world upward, from
8 Primordial Cosmology
the high downward, referred to generation and corruption of compounded
substances, perishable and mutating. The third species is the circular one,
which has no contraries, thus the substances moving with this peculiar mo-
tion are necessarily unchangeable, un-generable, incorruptible. Aristotle
regards the ether as compounding celestial bodies and being the only ele-
ment in circular movement, different from all other elements. This opinion
regarding a material for the celestial objects distinct from the remainder
of the Universe and therefore not subject to birth, death and alteration, as
indeed for the matter encountered by experience, will last for a long time
in the western culture, finally revised and abandoned in the XV century by
Nicolaus Cusanus (Nicholas of Kues). For Aristotle, the physical Universe,
comprising the sky made of ether and the sub-lunar world made by the four
elements (water, air, earth, fire), is perfect, unique, finite and eternal. The
basic elements are displaced in a natural order: at the center of the world
the earth is the heaviest element; around it there are the spheres referring
to the decreasing order of weight, water, air and fire. The fire constitutes
the outer zone of the sub-lunar sphere and around it we have the first ethe-
real (or celestial) sphere, the one of the Moon. In his view, the Universe
is a priori perfect and thus finite. In fact, infinity would be related to
an uncompleted and unfinished property, lacking a part and which could
eventually be added of something. Moreover, for him a real thing cannot
be infinite as anything in the real life has a direction and a well defined
position in the space: since no physical reality can be infinite, the sphere
of the fixed stars marks the limits of the Universe beyond which there is
no space. This is the maximum volume and no line can cross its diameter:
other worlds cannot exist besides our and since the space cannot be empty,
even vacuum cannot exist, either infra-cosmic (between common objects)
either extra-cosmic, as one allocating the Universe itself. In this scheme, if
it is meaningful to ask where is an object, this is not true for the Universe:
it is indeed the container of all what exists. This is a revolutionary view
that strongly adheres with the modern speculations. The concept of time,
strongly related to that of space, is analyzed as the property of the becom-
ing and the changing of the common things. From this point of view, the
world as a whole is perfect and finiteness is eternal, without an origin and
without an end. Eternity is seen as different from the infinite duration of
time, it is the atemporal existence of the immutable, thus the world has
never been generated and will never be destroyed, comprising all its alter-
ations. Aristotle does not formulate a cosmogony since time is eternal and
comprises all single local events: the world is eternal as well and has not
Historical Picture 9
an origin.
The ideas of Aristotle had an enormous impact on the following thinkers
on several topics related to science, such as biology and logics, for several
centuries: although in the Middle Age the Arab philosophers were more
influential, they considered him as the principal expression of the human
reason, he will shape and feed the philosophy up to the XVI century all
over Europe in the university studies. A deep criticism to his ideas started
with the birth of the modern science, when his astronomical and physical
theories were found unable to describe the world under the evidence of the
discoveries performed in such years.
1.1.3 The Hellenistic era
The IV century B.C., in particular after the death of Alexander the Great
in 323 B.C., saw his immense empire divided in three big reigns as Macedo-
nia, Egypt and Asia characterized by a universalistic cultural progression,
with the birth of several new places for the social and cultural life. In this
fertile environment of social changes, there was a great attention to the
particular sciences, separated from the speculative philosophy of the past.
Alexandria of Egypt became the paradigmatic city giving an enormous im-
pulse to increase and feed the cultural activities, with the 700,000 volumes
in its Bibliotheca, its Museum with a center of studies and research, an
astronomical observatory, a zoo, a botanical garden and the first anatomy
tables. It hosted scientists and teachers paid by the government which
could devote their time to free research. Several other cities followed the
example of Alexandria, whose magnificence will continue to be renowned
even after its destruction in 641 A.C.
The peculiar environment of a society split between a diffuse wealthy
governance and intellectual class and, on the other side, the abundance of
slavery lead to a clear separation between science and technique, thus polar-
izing the investigations of philosophers toward very speculative attitudes.
For example, Zeno of Citium (336-35 B.C.-264-63 B.C.), initiator of the
Stoic school was very keen for the role of science, whose basic concept relies
on an immutable, rational, perfect and necessary order, coinciding also with
a religious point of view. The whole world life performs a cycle, even the
stars turn around up to the same initial position: everything started with
a conflagration and the destruction of all existing beings. At that point,
another identical cycle restarts.
10 Primordial Cosmology
An important step ahead, with the purpose of containing the religious
influence on the cosmic vision, was performed by Epicurus (341 B.C. -270
B.C.). He aimed at leaving out any role of the gods in the nature design,
envisaged an infinite number of worlds, each one with a birth and a death
and each of them constituted by a finite number of atoms moving in an
infinite vacuum.
These cosmological speculations introduced a great bloom for all sci-
ences in the Hellenistic era, lasting approximately from 300 B.C. to 145
B.C.: in this year the Museum was destroyed during the civil war, the in-
tellectual elite had to abandon Alexandria starting a period of decadence,
marked by a first fire in the Bibliotheca in 48 B.C. The only exceptions were
offered during the II century A.D. by Claudius Ptolemaeus for astronomy
and Galen for medicine.
The Alexandrine era (after III century B.C.), had three great mathe-
maticians, Euclid, Archimedes and Apollonius, whose efforts in organizing
the knowledge and rigorous approach to calculus and geometry provided
the basis also of the important astronomic speculations which remained
valid and undiscussed up to the XX century.
Although from Plato and Aristotle the geocentric prevailed on the helio-
centric system, Heraclides Ponticus (387 B.C. -312 B.C.), disciple of Plato
and friend of Aristotle, strongly supported a hybrid system: it was geocen-
tric for the Sun and all planets except Venus and Mercury which, in order to
explain their anomalies, turned around the Sun in uniform circular motion.
A few years later, Aristarchus of Samos (310 B.C. -230 B.C.) extended
such system on three fundamental hypotheses: the absolute motionless of
the fixed stars sphere, the perfect stillness of the Sun at its center and the
annual movement of the Earth on a circle centered on the Sun. He admitted
that the fixed stars sphere had a radius enormously larger than that of the
terrestrial orbit. Such theory, based on different sphere dimensions for the
motion of inner and outer planets, was shortly thereafter opposed as it
was contrary to the initial appearance of observations. The efforts of the
following scientists were devoted to support the religious tradition.
Hipparchus (190 B.C. -120 B.C.) made a first stellar catalogue, counting
approximately 800-900 stars. Although he introduced new hypotheses to
support the geocentrism, he admitted that the Earth was not exactly at
the center of the celestial sphere and that the motion of planets was on
epicycles, i.e. the combination of two circumferences moving one inside the
other. This theory was eventually fit by Ptolemaeus (II century B.C.), on
the basis of the summary of all existing observations of planetary motions.
Historical Picture 11
In this period, for the first time, the notion of the number of celestial
bodies, their nature and the radius of the spheres where they are supposed
to move are the point of interest of the scientists and are critically consid-
ered. We will see how similar progressions in the astronomical observations
will bring, several centuries after, to modern cosmology.
1.1.4 The philosophical point of view in the Old Age
The first centuries (I- VII A.C.) are marked by the growth and consolidation
of the new religion which diffuses around the Mediterranean with a strong
impact on the evolution of the philosophy, with the efforts of Christianity
and its evolution in the contemporary society: the concepts of Universe,
space and time are rescaled to the human dimensions, the purpose of the
research is to give a foundation to the new ideas and to defend, from the
apologetics and through the Patristic, from the persecutions to the achieve-
ments got in the common knowledge. The purpose of the research is based
on the dualism God/man, and thus reason, faith and soul. The specula-
tions of the philosophers regarding the nature and its origin are formulated
starting from the Biblical story: as in the work of Augustine of Hippo,
saint, (354-430 A.C), God, being the basis of all existing things, created
the Universe and everything else. The speculations are led by ethical and
religious reasoning, without pursuing a specific attention to the physical as-
pects of the narration, giving rise to an eternal and perfect Universe with all
characteristics set according to some preliminary fundamental statements,
simply regarding the role and attributes of God.
In the same framework, although the physical aspects of the Universe
creation are not the focus of the research, a central point of investigation is
the role of time: even admitting the action of God for creating the whole
world, before this event time did not exist and thus the concept of eternity
takes form, opposed to the fugacity of the temporal evolution over the
human time-scales.
1.1.5 The Middle Age dogmas
A change of attitude shapes the research of the philosophers in the Middle
Age: the problem of the Scholasticism in the Christian Europe is to bring
the man to comprehend the revealed truth: the religious tradition is the rule
of the research and since everything has been revealed through the sacred
Biblical books, philosophy has to support the common work towards the
12 Primordial Cosmology
divine revelation, excluding the need of formulating a new system or new
explanations to explain some phenomena. This attitude perfectly reflects in
the rigid hierarchies of the Medieval society. Starting from the VIII century,
many economical and cultural exchanges fade up to starting again in the
XI century, opening the way to a criticism of a rigid cosmic order. The idea
of a supernatural relation to the human power and to the initiative of the
single person takes place in the following years, up to the end of the XIII
century.
In this environment, it is worth mentioning the role of Johannes Scotus
Eriugena (c. 810-877) who for the first time denied that the sky was made of
incorruptible and not-generable ether (according to Aristotle) and proposed
an astronomical system with the Earth still in the center, but all planets
orbiting around the sun.
For decades, most of the efforts remain devoted to address theological
questions, especially regarding the role of the man in the created world,
whose origin is granted as created by God at the beginning of time.
In the same period and across XI and XII centuries the Greek philos-
ophy and science had been inherited also by the Arab world, where many
thinkers gave a great impulse in astronomy and mathematics earlier than
the western Schools. In particular, they had a rational approach to sev-
eral basic problems which were considered as reasoning paradigms by the
European philosophers, conscious of the limits and drawbacks provided by
the heavy influence of the tradition. The translation and diffusion of the
Greek works, served as basis for the development of two clear philosophical
approaches represented by two prominent thinkers, Avicenna and Averroes,
who were the most prominent exponents of the so-called Neoplatonic and
Aristotelian evolutions.
Avicenna (Abu Ali Sina Balkhi or Ibn Sina), (around 980-1037), medical
doctor and philosopher, exposed the principle of the necessity of the being:
everything happens necessarily and could not happen in a different way.
The role of God is to shape the natural events and is the first origin of
all physical processes. The astrological predict ions arc thus fully justified
since the action of God is directly on the asters and from them it expands
to the other levels of the nature: if the human beings could have a perfect
knowledge of the stellar (and planetary) evolution, they could know the
events on the Earth without mistake. The predictions fail due to this
imperfect knowledge of the details of the movements of the celestial bodies.
Although at this stage there is no clear distinction between stars and
planets, the philosophers point to the sky as the intermediary place with
Historical Picture 13
the divinity, recognizing its role either for the religious speculations, or for
the human events.
The arab Spanish-born philosopher Averroes (Ibn-Rashid) , (1126—
1198), devoted much of his efforts to discuss the contemporary philoso-
phers, particularly hostile to Avicenna, starting from the same principle of
the necessity of all that exists. He considered the Aristotelian doctrine as
the scientific and demonstrative counterpart of the Muslim religion, which
on the other hand is seen as the simplified version suitable for uneducated
people. Nevertheless, since the world itself is necessary because it is cre-
ated by God, it is eternal as well and cannot have originated within the
time. The order of the world is also necessary, and the human being has
not capability or freedom of action. Although such last concept could seem
incompatible with the free research or aim for new explanations of the nat-
ural events, it remains at the basis of Renaissance confidence to discover a
necessary order in all manifestations of the nature.
The Arab and Judaic thinkers, mainly active in Spain and Egypt,
strongly influenced the inheritance over the thinkers in central Europe,
since the doctrines of Plato and Aristotle were transmitted through their
studies. This had the effect of splitting the philosophers between those
who opposed Aristotelianism in favor of Platonism and those who desired
to merge several aspects of them.
In particular, we note the role of Robert Grosseteste (1175-1253) active
in Oxford and bishop, for his speculations on natural philosophy. He stated
that the study of the nature must be based on mathematics, and reduced
the whole Physics to a theory of the light, which is the primary form of
the bodies: since the light diffuses in all directions, it is equivalent to the
corporeity itself, similar to the extension of the matter in the three space
dimensions.
Thomas Aquinas (1225-1274) inserts himself in the Scholastic debate
regarding theology and philosophy, essentially for the integration of rational
thinking and faith, finding a solution in the subordination of the reason to
faith, proving itself as an instrument for the theological truth. His approach
to the demonstration of the existence of God finds the first proof as a cos-
mological one, relating the existence of any movement in the Universe to
the existence of God: given that every movement of any object has been ac-
tivated by that of another one, this provides a potentially infinite chain, i.e.
the first principles of action, commonly attributed to God. Similarly, the
movement of the celestial spheres also comes from Him. For what regards
the creation, the only conclusion Thomas offered was the impossibility of
14 Primordial Cosmology
demonstrating either the beginning of time or the world eternity.
Towards the end of the XIII century a restored interest in the philoso-
phy of nature arose across Europe, dividing theological questions from an
autonomous effort of the reason for the problems of the physical world. In
every cultural field, following the new Aristotelian spirit, the experimental
research proposes new methods and new questions.
The most important representative of this experimental approach of
the XIII century is Roger Bacon (c. 1210-1292). Although his results in
physics, in particular optics, astronomy and mathematics did not find out-
standing originality, his great contribution relies on declaring the sources
of any knowledge: reason and experience. Nevertheless considering natural
(from external experience) and supernatural (from interiority experiences)
truths and acting much as an alchemist looking for marvelous discoveries,
he can be considered as the precursor of the modern science. In fact, he
gave maximum value to the experimental approach to research, giving to
mathematics the fundamental guiding role for it, in order to give certainty
to the finding of other sciences.
The role or philosophy is to clarify the limits of the human science
domain for John Duns Scotus (c. 1266-1308), who assessed how science
and faith refer to different levels of truth. The human mind aims at com-
prehending the rational aspects of nature adopting a theoretical approach
based on the freedom of reasoning, while the faith is based on a more practi-
cal level, regarding I lie human behavior and possibility of actions, needless
of doubt. Following Aristotle and the Arab philosophers, he declared the
ideal of a necessary science, fully relying on principles based on evidence
and on rational demonstrations. Statin; 1 , that all attributes regarding God
are matter of faith and cannot be demonstrated, Duns opened the way to
the rise of the Renaissance, proposing an approach of division between re-
ligion and science which was, on the contrary, the main issue of the early
Scholasticism.
The rise of a new class of merchants and bankers, opposed to a static
theological society infrastructure, manifested in a new interest toward na-
ture and science. In a changing society divided among the secular role of
the Catholic church fight in;; against the Emperor, William of Ockham (c.
1290- c. 1348) lived contributing to establishing a strong basis to a rad-
ical empiricism as the foundation for the philosophical investigation. For
the first time, he exposed the limits of the theological proofs regarding
the existence of God, on the basis of a total heterogeneity of science and
ii ih i hi 1 1 I Miii in; light on the lack of necessity in every step of the
Historical Picture 15
proofs. In his criticism of the concept of cause and effects, which can be
related among themselves only on the basis of the experience, the natu-
ral events evolve according to necessary laws, independently of the divine
action. Metaphysics loses its power to explain everything, giving the re-
searcher the role to describe how phenomena happen, avoiding to enquire
about their essence or purpose. In his anti-metaphysics theology, the Uni-
verse has been created by God without any pre-existing logical rule, and
therefore God could have given the world also a different set of rules, at his
freedom: philosophers can only accept the world as it is, without trying to
find a metaphysics explanation.
Since the nature is the domain of the human knowledge, the experience
loses its magical character, as in the past, to be accessible to all human
beings: Ockham, for the first time, rules out the belief of the different nature
attributed to celestial bodies and sub-lunar ones. On the basis of a principle
of economy (the so-called razor, i.e. to avoid all unnecessary concepts to
describe some phenomena), he asserted how all bodies are made of the same
kind of matter. He also admitted the possibility of a multi-world Universe,
where all local characteristics (up, down, center, etc.) would appropriately
have their meaning, in that local Universe. The whole Universe can be
infinite and eternal, as any celestial revolution can indefinitely repeat and
the creation itself can be excluded as being highly improbable, opening new
ways to the philosophical investigation.
1.1.6 The Renaissance revolution
The XV and XVI centuries are characterized by a radical change toward the
modern age, in an enlargement of the view of the world, coherently with
new geographical discoveries, new inventions and new efforts by a urban
society, powered by banks and social transformations.
The Humanism, preparing the way to the Renaissance, explicits the new
culture which breaks the previous perspective of the human being with re-
spect to the life and the world, eventually opening the way to different
interpretations of the nature apart from the religion dogmas, though recon-
sidering the efforts made by the Greek philosophers. The culture is now
organized by the local communities as Academiae, where the use of the latin
language is considered as a means of intellectual and international way to
circulate the ideas, also with the help of print. The naturalism at the basis
of the Renaissance marks the active role of the humans as a full part of the
nature and their attitude and interest in its study, though frequently the
16 Primordial Cosmology
nature will be approached by a magic perspective.
Nicolaus Chrypffs (disarms, 1401-1464), was the first prominent
philosopher stating the role of mathematics to evaluate the proportion of
knowledge with respect to ignorance. He was the first to refuse the idea that
the celestial part of the world possesses an absolute perfection and thus be
un-generable and incorruptible. The world (considered as the whole Uni-
verse) has no center nor circumference but comprises all the space. Since
there is no center, the center is everywhere, the Earth is not at the center of
the world and moves of a rather perfect circularity. The Sun is another star,
made of more pure elements and all movements of the nature are arranged
so as to guarantee the highest possible order, maximally approaching the
circular shape.
Giovanni Pico della Mirandola (1463-1494) states how the Cabbala can
help to penetrate the divine mysteries and the astrology can be used to
understand the mathematical rules of the Universe, though the human ac-
tions cannot be influenced by the celestial bodies since they are free and
full of dignity.
The main idea of Renaissance is deeply characterized by a re-birth of
the human beings within the nature, fully introduced in the world and thus
giving large space to investigation of the natural world, in an early stage
through magic and dually with The philosophy of the nature.
1.1.7 The Scientific Revolution
The Scientific revolution marking 1 be following centuries can be chronolog-
ically set between the publication of Copernicus De revolutionibus orbium
coelestium (On the Revolutions of the Celestial Spheres, 1543) and New-
ton's Principia (1687). In this lapse of time some paradigms express the
new approach to nature, in particular:
• the nature has an objective order, i.e. its character has nothing to
do with a spiritual dimension of investigation, and thus with the
human purposes and needs. The Universe has no human attributes
nor qualities.
• The nature has a causal order, in the sense that there is a constant
relation between one (or more) facts and the only type of cause
is the efficient one. The Science does not investigate the purposes
for which events happen, but is devoted to study the causes that
produce them.
Historical Picture 17
> The Nature is a set of causal relations and, finally,
> the facts are governed by laws, and the Nature is the compound of
the laws which govern all phenomena and make them predictable.
Correspondingly, the Science is an experimental knowledge based on
the experience, it is mathematical, as it can quantify and express itself by
formulas, it is accessible to everyone and its only purpose is the objective
knowledge of the world and its rules. Such knowledge permits to eventually
modify the world according to the human purposes, thus providing strong
links between scientists and technicians.
The scientific revolution marked a deep philosophical change in the ap-
proach to the vision of the Universe. In fact, the Greek-medieval cosmol-
ogy, arising from Aristotle and Ptolemaeus, conceived the world as unique,
closed, finite, made as concentric spheres, geocentric and divided into two
qualitatively distinct parts. Such vision was based on a unique Universe, as
the only existing one, with all possible matter aggregated in a single place.
It was closed, as a limited sphere from the level of the fixed stars in the sky,
and outside it there was nothing, except the realm of God. It was finite,
since the concept of infinity could not be considered as reality. Finally, the
Universe was represented as two different cosmic zones, one perfect and
one imperfect. The former was the super-lunar world of the skies made of
ether, a divine element incorruptible and perennial, characterized only by
a circular movement. The sub-lunar world, on the other hand, was made
by the four elements (air, water, earth, fire) having rectilinear motion and
marked by generation and corruption. Such framework was coherent with
the metaphysics and religious justification, linked to the creationist doctrine
and the human role stated by the sacred books.
Nicolaus Copernicus (1473-1543) provided the first strong criticism
to the geocentric Ptolemaic system in his De Revolutionibus orbium
coelestium. As a theorist of celestial mechanics, he considered such sys-
tem as too complex and reconsidered alternative theories in the books of
the Greek philosophers, thus recovering the heliocentric idea which, in his
view, offered a deep simplification in the mathematical evaluation of the
celestial bodies movements. Although in his scheme the Solar system re-
covered what it is known today, with the Sun at the center and the planets
orbiting around it, his vision of the Universe was limited to the sphere of the
fixed stars, leaving a unique and closed spherical Universe with a perfect
circular uniform motion.
18 Primordial Cosmology
Although several thinkers opposed the ideas of Copernicus, mainly on
the religious and philosophical point of view, his ideas were extended by
Tycho Brahe (1546-1601) who proposed a Solar system with the planets
orbiting around the Sun and all together orbiting around the Earth in the
center, overcoming any conflict with the Bible.
Johannes Kepler (1571-1630) started his investigations on astronomy
exalting the harmony of a Solar system centered on the Sun as the im-
age of God with the planets at the edges of a regular polyhedron. The
mathematical difficulties in relating such assumptions with the astronom-
ical observations led him to dismiss such Pythagorean approach to the
symmetry of cosmos. In the following, he considered only physical forces
instead of intelligent motions, and attributed to the world and to the mat-
ter necessarily a geometrical order. His major achievement springs from
Tycho's observations stating the laws for the planets' motion over ellipses,
thus confirming the role of mathematical proportions for the description
of the natural objectivity. So far, the Universe is still limited to the solar
system and to the fixed stars with their immensity and incommensurability.
Although Copernicus left the discussion about the possible infinity of the
Universe to philosophers, Kepler's Universe is still finite.
Although not an astronomer nor a mathematician, a new vision of the
Universe was proposed by Giordano Bruno (1548-1600) who forced the Lu-
cretius thought about ancient atomism toward the intuition of the infinity
of the Universe based on the Copernican discoveries. His hypothesis was
based on the similarity to the solar system for all stars in the sky, thus with
a Universe composed of a multiplicity of systems similar to ours, with an
unlimited number of Suns in different places of the space, though stating
that they have never been observed. The revolution was initiated and ready
to influence the following philosophers and astronomers.
The Universe has no borders and is open in all directions, while the
fixed stars are dispersed in the unlimited space. Coherently, other parts of
the sky are made of the same kind of matter, with no hierarchy about the
quality of the component substances, in a unique and homogeneous space.
Such view deeply influenced the perception of the thinkers, either
philosophers or poets, about the role of the human beings relative to the
position of the Earth in the Universe, inducing a debate regarding the dif-
ficult coexistence of the new cosmology with the Bible sentences about the
movement of the planets or the position of the Sun. Thus, a great impulse
on the view of the world for the contemporary thinkers was given by a vi-
sionary proposal, opening the way to the following evolution (in particular
Historical Picture 19
by Galilei) which would have shaped the following years, although its af-
firmation was based more on the fascination evoked by the new ideas than
on the physical observations which were still far to come.
Although the Brunian proposal was included in the arguments for the
magnification of the religious orthodoxy, with an infinite Universe testify-
ing the infinity of the Being who created it, the Church continued to be
reluctant to accept the Copernican view of the world, canceling the con-
demnation for the Copernican writers only in 1757, allowing the printing
of books teaching the Earth motion in 1822 and taking off from the list of
forbidden books the De Revolutionibus only in 1835.
We see how the new theories prepared the way to the modern view of
the Universe, although yet without experimental ground, and changed the
way the celestial space was perceived, leaving to the Earth (and the human
beings on it) a marginal role in the whole context of infinite planets, stars
and worlds.
Starting from mathematical studies, Galileo Galilei (1564-1642) was the
first to use the spyglass (1609) for his discoveries in Astronomy enthusias-
tically announced in his Sidereus Nii.nc.iii.s (1610), promptly supported by
Kepler. In contrast with the hierarchies of the Church, he received an am-
munition (1616) which did not stop him from publishing // Saggiatore about
the problems of cometary motions. He also continued his work on the Dia-
log over the two maximum systems of the world (discussing the comparison
between the Ptolemaic and the Copernican cosmological systems) which
led him under trial (1632) and forced him to abjure, publicly dismissing his
results in 1633 and to retire in Arcetri (near Florence) to write his last book
about dynamics, finally published in Holland. His point of view about the
relation between the religious truth and the scientific truth was explained
in the Copernican letters where he detailed how both truths derive from
God and any contradiction among them must be apparent, since the Bible
and the science approach different aspects of the knowledge. Thus, for what
regards the investigation of the Nature, the Bible interpretation must adapt
to the scientific evidence.
His si uclies about Mechanics and the motion of bodies were the sclent ilk-
counterpart to the Brunian intuit ions about the cosmos. The principle of
inertia was valid for the terrestrial as well as for the astronomical dynamics,
though leaving the idea of a finite Cosmos, explaining I he indefinite motion
of all planets. The idea of circular orbits for the planetary motions was
retained in his picture outlining a reminiscence of a theological influence.
20 Primordial Cosmology
The formulation of a unique science of motion, with the negation of a
different nature between circular and rectilinear motions (so far considered
typical of the lunar and sublunar worlds, respectively), brought to refuse
the existence of a different structure between the sky and the Earth. So
far, a possible difference was based on the possible motions on them.
Galilei was the first to explore, thanks to his spyglass, the Lunar surface,
finding it similar to the terrestrial one, with valleys, craters, mountains and
shadows. The discovery of the four Jupiter satellites, orbiting together with
the parent planet around the Sun, suggested that also the Earth, together
with the Moon, was orbiting around the Sun and could be experimentally
verified by the observations done with the new telescope. The discovery
of the Venus phases induced the idea that all the planets receive their
light from the Sun orbiting around it. Finally, and most relevantly for
our discussion, Galilei used the telescope to discover that after the fixed
stars, visible by naked eye, a huge number of other stars existed, never seen
before but observable with the new instrument. The galaxy itself was only
an aggregation of uncountable stars scattered in groups, similarly to the
nebulae which were composed of other stars.
The analysis of the structure of the Universe, al least the part surround-
ing the Earth, was ready to be boosted on an observal ional basis, drastically
changing the view of cosmology and the evolution of science.
In the Dialogue, in 1632, Galilei confuted all the current theses against
the Earth motion, introduced the principle of inertia, explained the Earth
rotation and liually exposed his theory for the sea tides.
The convergence of the technical expertise in assembling the telescope
with its scientific usage provided the formal scheme which imprinted the
modern science, from phenomena observation to mathematical measure-
ments of data, hypotheses, ycrijicat ion and finally lormulat ion of a physical
law. Galilei was persuaded about the mathematical structure of the Uni-
verse, with the necessity of a geomet ideal description for deciphering its real
structure.
The impulse to the physical investigations were now related to a new
crucial possibility to apply the observations also to Astronomy which, on
a mathematical basis, was open to explore a natural world which was no
longer split between the terrestrial and the one of the celestial spheres, but
characterized by general laws where all parts were correlated by a causal
relat ion.
Historical Picture 21
Rene Descartes (1596-1650), mathematician, physicist and philosopher,
formalized the scientific method founding the way to proceed in the research
about the concept of doubt, an approach that would have marked the fol-
lowing centuries. His view of the Universe was based on a Euclidean space,
arising from the identification of the concept of matter with its own exten-
sion. Thus, the infinity of the space implied the infinity of the matter, its
infinite divisibility and continuity, without holes nor vacuum. He refused
the concept of forces acting at distance among different bodies, leaving the
Universe to the principle of inertia and of conservation of momentum, in a
fully deterministic framework scaling up to include the first impulse from
God in the evolution of the Universe. This idea led him to describe the
motion of the celestial bodies as immersed in the ether filling the entire
Universe, with all movements due to a scries of vortices including all pla-
nets, in order to satisfy a mechanistic description for an infinite continuum.
The unique engine of the big machine constituting the world is the original
momentum, distributed in different ways among the bodies through their
collisions.
A new vision of the cosmic order was proposed by Baruch de Spinoza
(1632-77), who considered an identification of the Nature as the order go-
verning all substances and their movements. Thus, it assumes a role as
God-Nature, geometrically ordering the whole Universe with its laws. His
criticism of the former philosophers, including Galilei, addressed the fina-
lism, viewing it as a prejudice preventing a correct interpretation of the
world scheme.
With Newton, the scientific Revolution initiated by Copernicus and
Galilei gets its final form, either on the methodological approach, either
for its contents, thus outlining the Universe picture which is familiar to the
modern view and which, after Einstein, is called "classical Physics".
Isaac Newton (1642-1727), mathematician and physicist, worked in op-
tics, inventing the reflection telescope, and starting from the results of
Huygens on pendulum oscillations and on the experimental techniques de-
veloped during the second half of XVII century, investigated several aspects
of physics and gravitation, summarized in his Philosophiae Naturalis Prin-
cipia Mathematica, published in 1687 with the support of the astronomer
Halley.
In this work, he recognized the identity of the motion of the planets
with the fall of heavy bodies on Earth, in the formulation of the universal
gravitation law, finally explaining (he planets' motion around the Sun and
of the satellites around their corresponding planets.
22 Primordial Cosmology
The infinitesimal calculus was the missing concept to unify Physics and
Mathematics, and its use allowed Newton to correct the Kepler laws of mo-
tion, taking into consideration the attraction exerted by the Sun together
with the attraction among the planets. This way he could explain a per-
turbative mean for the planets motion, so that, for example, the Earth is
not moving along a perfect ellipsis but along one perturbed by the action of
other planets around. On a broader point of view, he stated the similarity
of the motion of bodies on Earth with that of planets in the sky, a picture
in which still missed the initial principle of motion. Newton thus admitted
as the first cause the creational action of the divinity, providing the initial
momentum.
The dynamics received a definite form, once introduced the concept of
mass for the generalization of the concept of force, consequently extending
the laws of mechanics to the entire Universe.
A focal point for mechanics is the idea of an absolute motion, with
reference to empty space, funded on absolute space and time, mathemati-
cally fluent uniformly, without relation to anything external, related to an
absolute space, always similar to itself and stationary.
His formulation of mechanics and dynamics led to exclude other forces,
apart from gravity, acting on the movement of celestial bodies. Moreover,
the formulation of scientific induction, prescribing the extension of a law
verified for a limited number of cases to all possible cases, opened to the de-
scription of all parts of the Universe with the same law. His method stated
also that the propositions got by induction from phenomena must be con-
sidered exactly or approximately as true until other phenomena eventually
confirm or show any exception.
The scientific method with the support of calculus provided a new ba-
sis to the description of the celestial phenomena. The Universe resulting
from the Newton's investigations appears as a real physical environment
whose phenomena are governed everywhere by the same laws, which can be
formulated in a precise mathematical language. More than modifying the
notion of Cosmology, Newton imprinted with his new method the possibil-
ii (or tlj i in i i lo addn I hi ci< >bser vat ions in a self-consistent
theoretical framework. However, the uudei'staudinj", of the laws governing
the gravitational interactions at a local level allowed to extrapolate their
validity everywhere in the Universe and therefore to approach the analysis
of its structure on a new perspective.
Historical Picture 23
The strong scientific imprint to the interpretation of natural phenom-
ena did not prevent Newton from mixing some Hermeticism ideas (from the
Hellenistic Egyptian tradition) about the relations of attraction and repul-
sion between particles, gained from a strong interest for alchemy. Thus, his
view of an invisible force acting on large distances was seen as the attempt
to introduce some occult component in the natural picture.
His relation with religion was twofold: his interest in the Bible was
exerted trying to extract any information regarding nature or referable
to some scientific measurement, though he viewed God as the clockmaker
of the Universe. The complexity of the planetary motions could not be
simply ascribed to natural phenomena but should have been designed by
an intelligent being.
Nevertheless, this line of thinking would have found several crucial dif-
ficulties which prevented the birth of a modern notion of Cosmology before
the formulation of General Relativity by Albert Einstein.
A new vision of the cosmos was under way: Galilei and Newton had
opened the broadest possibility of development to astronomy.
The needs of navigation lead to address the new notions more accurately,
thus inducing the foundation of new observatories. The first was established
in Paris by Louis XIV and granted to the Italian astronomer Giandomenico
Cassini. A few years later, in 1675 the Greenwich observatory was built
by Charles II, directed by Edmond Halley (1656-1742). His studies were
devoted to the motion of the comets, demonstrating how they belong to
the solar system and move on eccentric orbits. In 1675 the Danish Olaf
Romer, working at the Paris observatory, noted that the eclipses of the
Jupiter satellites on certain periods of the year happened earlier and on
other periods later than those predicted through the computed tables. Thus
he ascribed such phenomenon to the different distance from the Earth of
such satellites, finally providing a measurement of the speed of light as
308,000 km/s, indeed very accurate, a result which was vainly searched by
Galilei.
1.1.8 The Enlightenment Era
During the XVIII century, a new and specific attitude to relate to the
rationality sprung out as a philosophical counterpart of the Scientific Revo-
lution. The Enlightenment was characterized by a new role of the reason in
the society with an emerging role of the bourgeoisie, together with a novel
24 Primordial Cosmology
perception of the science and its influence on the real life of the people (in-
cluding trade, economics and technical evolution), rising in the hierarchy
of the activities related to knowledge.
The role of the experience in the philosophical investigation mitigates
the rationalistic and idealistic impulse to the theoretical exploitation of the
physical laws underlying the observable phenomena.
In the 1700, during the philosophical celebration of science and of its
methods, an important development involved mathematics and astronomy.
Leonhard Euler (1707-1783) born in Swiss and active in Russia at the
court of Catherine I the Great in St Petersburg, showed his trust in the
mathematics devoting all his efforts in developing the infinitesimal calculus
towards the application to several phenomena.
In the same years, Joseph Louis Lagrange (1736-1816) (born Giuseppe
Lodovico Lagrangia, in Turin, Italy), first in Berlin and finally in Paris, was
appreciated by Napoleon who named him to the Legion of Honour for his
results in applied calculus. His work Mecanique Analytique (Berlin, 1788)
summarized all topics of classical mechanics treated so far, since the epoch
of Newton.
On the basis of the great impulse to mathematics, also the celestial me-
chanics established by Newton developed and flourished. The most promi-
nent astronomer of this period was Friedrich Wilhelm Herschel (1738-1822),
who discovered Uranus, enlarging the borders of the Solar system, limited
so far to Saturn since the ancient times. After this, he discovered the Sun's
mil hi iii i i n i 11 planets with it and proved the rotation of the
Saturn ring, measured also its period. His work comprised the catalogue
of a large number of nebulae and finally his studies about the Milky Way
lead to view it as a quantity of stars disc-shaped with a diameter equal to
five times its width.
In the same years, Giuseppe Piazzi (1746-1826) discovered Ceres, the
first planetesimal between Jupiter and Mars, whose orbit would have been
calculated several months later by the German mathematician C.F. Gauss.
The perception of the Cosmos was surpassing the limit related to the
observational capacities which were increasing year after year, providing a
novel consciousness about the perspectives on new ways to cover.
The experiments carried out by Henry Cavendish (1731-1810), opened
the path to measuring the weight of the Earth and of the celestial bo-
dies. Through the use of a torsion pendulum he computed the value of the
gravitational constant g characterizing the Newton's law.
Finally, the role of the mathematician Gaspard Monge (1746-1818) pro-
Historical Picture 25
vided a way to the description of phenomena thanks to the invention of the
descriptive geometry, allowing to treat on a bidimensional surface (like a
sheet of paper) tridimensional displacements of bodies.
Other scientific fields were improved under the impulse of the adop-
tion of calculus and its extensions outside mathematics, as for thermology,
electrology, chemistry and biological sciences. The effort to explain and or-
ganize the current knowledge in several topics lead to classify and organize
the observed phenomena in catalogues and systematic classifications of the
natural world, with applications to animals, plants, basic constituents of
matter, celestial events.
A huge work of collection in this line was pursued by Georges-Louis
Leclerc, Comte de Buffon (1707-1788) with his Histoire naturelle, in 44
volumes, including several volumes devoted to quadrupeds, birds and mi-
nerals and finally to the theory of the Earth and the general characters
of the plants, of the animals and of humans. He explained the origin of
the Earth from the impact of a comet with the Sun and this idea inspired
several thinkers thereafter.
The most important thinker influenced by such ideas was Immanuel
Kant (1724-1804) in his work General History of Nature and Theory of the
Heavens (Allgemeine Naturgeschichte una 1 Theorie des Himmels) (1755) de-
scribing the formation of the whole cosmic system from a primordial nebula,
in accordance with the Newtonian physics. Moreover, he investigated the
role of mathematics for the description of physical phenomena, overcoming
the explanation provided by Galilei, who based his epistemology on the
existence of God, but attributing to the nature of space and time an in-
trinsic geometrical and arithmetical configuration: if the concept of space
itself is Euclidean, the theorems of Euclid's geometry apply to the whole
phenomenological world.
Analogous theories about the formation of the solar system were also
proposed, in the same years, by Johann Heinrich Lambert (1728-1777) and
Pierre-Simon Laplace (1749-1827).
1.2 The XIX Century Knowledge
1.2.1 Geometrical formalisms
The new science arising from the scientific revolution expressed its potential
during the 1800s. Despite the strong link between science and philosophy
in the previous era, in the XIX century the latter tends to reduce the strong
26 Primordial Cosmology
link with the experimental foundation leaving the science fragmenting in
several specific topics, often without communication between them. Around
the mid of the century, thanks to the progress in the mathematical abstrac-
tion and modeling, the mechanistic ideas permeate again the different fields
of Physics, posing the basis for the following unification approaches.
In particular, Laplace expressed the view of the current state of the
Universe as the effect of the previous state and the cause of the following
one (1812, Thcoric an ah/ 1 1 que des probabilites). In his Mechanique celeste,
published in five volumes from 1799 up to 1825, he addressed the stability
of the Solar system, with the purpose to match the astronomical data under
the point of view of the laws of motion. He also pursued the idea intro-
duced by Kant about the origin of the Solar system based on the nebular
hypothesis giving rise to a planetary system and discussing the possibility
that gravitational forces could not act instantaneously, although without
success.
The mechanism thus formulated gave rise to a vision of the Universe
where reversibility would always be possible, thus preventing the idea of an
evolution or degradation of the nature at large scales.
The independent path taken by the evolution of mathematics revealed
the basis of a new formal approach to scientific problems, leading to a
level of abstraction which would bring to a process of unification for the
phenomena based on more generic formal structures.
For the first time, the mathematicians discussed non-Euclidean geome-
tries, without the necessity of a strong link with the real world. In par-
ticular, the analysis of the fifth postulate of Euclidean geometry was re-
considered, attributed to Proclus Lycaeus (412-485): given a straight line,
it is possible to draw one and only one line parallel to it and passing on
a point external to the first line. Independently, Karl Friederich Gauss
(1777-1855), Nicolaj Ivanovic Lobacevskij (1793-1856) and Janos Bolyai
(1802-1860), founded the new hyperbolic geometry. Analogously, by the
end of the XIX century, Bernard Riemann formulated the elliptic geometry
which would have been fundamental for the modern gravitational physics
introducing the General Relativity Theory and a new Cosmology.
From an evolutionary point of view, the history of the Universe was con-
sidered from a new perspective once the analysis of thermodynamics and
entropy were applied to the Universe as a whole: time was seen as asymmet-
ric and the new ideas lead to consider that if the Universe can be regarded
as an isolated system, it must evolve toward a progressive thermal death.
Scientists started to apply the consequences of mathematical abstraction
Historical Picture 27
to the physical world thus posing new challenges to metaphysics.
The abstraction pursued by the research in mathematics brought to
investigate the geometry as a new field, assuming the character of an a
priori synthetic science. Bernhard Riemann (1826-1866), student of Gauss,
lectured about a new multi-dimensional geometry Uber die Hypothesen
welche der Geometric zu Grunde liegen (On the hypotheses which lie at the
foundation of geometry), later published in 1868, describing manifolds with
any dimension and any type of curvature, constant or variable.
A different approach to geometry, strongly based on an axiomatic per-
spective, independently on any hypothesis about the physical space, was
investigated by David Hilbert (1862-1943) in his work Grundlagen der Ge-
ometric (1899, Foundations of Geometry), developing non-Euclidean ge-
ometries by using purely formal methods.
1.2.2 Difficulties for the birth of a real cosmology: Olbers'
paradox
In 1826, the German astronomer Heinrich Wilhelm Olbers (1758-1840)
assessed the paradox regarding the consequences of an infinite Universe
over the night sky: if the Universe has an infinite extension (as proposed
by Newton to prevent a collapse), contains an infinite number of stars and
exists from an infinite time, the sky should not be dark at night, since every
point of the sky should be covered by the emission of the light by some star,
though far and distant. Such paradox had already been stated by Kepler in
1610 but all possible explanations were destined to fail or to induce other
paradoxes.
The solutions proposed over the years spanned different points of view:
Otto von Guericke (1602-1686) had previously proposed that the dark-
ness is caused by an endless void between the stars. Olbers thought that
light would be absorbed by clouds of dust in the interstellar medium while
William Thomson (1824-1907) (known as Lord Kelvin) a few years later
introduced the idea that stars would have started their life a finite amount
of time ago, thus introducing also a limit on the size of the observable
Universe.
Notwithstanding such proposals, a definitive answer was awaiting from
the observations made by Hubble in the Twenties of the XX century.
28 Primordial Cosmology
1.2.3 Luigi Bianchi and the developments of differential ge-
ometry
Under the influence of Riemann and Sophus Lie (1842-1899), across the
change of century the Italian School of mathematics provided several efforts
in the field of differential geometry, algebra and topology. In 1898, Luigi
Bianchi (1865-1928) derived the classification which brings his name of the
isometries classifying the Riemannian tridimensional spaces into nine non-
equivalent (Lie) groups which is at the basis of the extension to Cosmology
performed six decades later by the Landau School. Friend and colleague
of Bianchi, Gregorio Ricci-Curbastro (1823-1925) promoted a group focus-
ing on tensor calculus, involving also Tullio Levi-Civita (1873-1941), thus
opening the way to the formalism of differential calculus with coordinates,
later becoming the language for General Relativity.
1.2.4 Einstein vision of space-time
Addressing the new discoveries about the electromagnetism phenomena and
about the nature of the light speed, Albert Einstein (1879-1955) published
in 1905 a short memory Uber einen die Erzeugung und Verwandlung des
Lichtes betreffenden heuristischen Gesichtspunkt (On the electrodynamics
of bodies in motion) which, formalized in 1908 by the mathematician Her-
mann Minkowski (1864-1909), provided a geometrical interpretation of the
basic postulates of Special Relativity. Einstein worked on extending the
relativity principle to accelerated systems, exposing in 1916 the new theory
of gravitation, known as General Relativity.
The ideas of Einstein constituted a conceptual revolution since the no-
tions of space and time acquired an intrinsic relative character, being in-
fluenced by the matter field living in the background environment. This
striking contrast with the Newton picture of spacetime (however contained
in General Relativity in the proper non-relativistic limit) opened new per-
spectives to reconsider the mechanisms governing the Universe genesis and
evolution.
The great impact of General Relativity on the concepts of matter and
spacetime was mainly due to the synthesis of the geometrical description of
the spacetime manifolds by the tensorial formulation of the laws of nature.
On a physical ground, this correspondence is able to conjugate the General
Relativity Principle (i.e. all the physical laws stand in the same form in all
the reference systems) with the geometrodynamics (i.e. the gravitational in-
Historical Picture 29
teraction implies the spacetime evolution coupled to the energy-momentum
source describing the physical entities).
Einstein started with the idea of generalizing the Special Relativity the-
ory to non-inertial systems, but his deep understanding of the Equivalence
principle allowed the full development of the gravitational theory in a space-
time geometrodynamics.
The formulation of Special Relativity started by the experimental evi-
dence of a constant, frame independent, value of the speed of light. Hence,
Einstein recognized that a limiting velocity of the signals implied the rela-
tive nature of two events simultaneity.
Despite what is often believed, General Relativity also arose from an
experimental evidence, i.e. the proportionality of the inertial and the grav-
itational masses through a universal factor (conventionally set to unity).
Einstein deeply considered the unnatural physical scenario coming out
from the equivalence of these very different concepts. Indeed the gravita-
tional mass is nothing more than the charge of the gravitational field, in
principle fully uncorrelated from the dynamical concept of inertial mass.
Thus Einstein formulated the Equivalence Principle as the local physical
correspondence between a non-inertial system and an inertial one, endowed
with a suitable gravitational field: both the interactions (non-inertial forces
and gravity) have the same property to exert a force independent of the
inertial (i.e. gravitational) mass of the test body. The Equivalence Princi-
ple not only links the non-inertial to the gravitational force in a common
physical scenario, but suggests the idea that these force fields have an en-
vironment character, well dressed by their geometrical origin.
Despite the ideas of the new theory were well defined in the Einstein
mind, the walk to the proper mathematical formulation of General Rela-
tivity was somewhat long and also required the important contribution of
Marcel Grossmann (1878-1936), who supported Einstein with his rigorous
mathematical hints on the formal description of the physical phenomena.
At the end of this conceptual and formal path, Einstein summarized the
non-inertial and the gravitational forces into the metric tensor describing
the spacetime manifold, with the fundamental distinction that the inertial
forces arise from changes of reference system, while the gravitational field
is associated to a spacetime curvature.
Wo emphasize bow Einstein derived his revolutionary theories from very
well-known facts (constant speed of light and mass equivalence), but he was
able to cast these notions toward a new physics providing the "correct an-
swers to the good questions" . However, we cannot forget how Einstein was
30 Primordial Cosmology
influenced by the spirit of his time, especially in the concept of a relativis-
ts world. Indeed two important facts must support the clever intuition
of a physicist: the existence of a philosophical background favorable to
the development of his ideas and the possibility to adopt well-grounded
mathematical formalisms. Einstein could make account on both factors.
1.3 Birth of Scientific Cosmology
Although Cosmology, in a strict sense, is one of the most ancient disciplines
of speculation, only in the XX century it has acquired a proper scientific
character, on the basis of the new conceptual instruments provided by the
Theory of Relativity and from Particle Physics, together with the recent
observational means allowed by new telescopes and from the introduction
of radioastronomy.
The Newtonian notion of absolute space, together with the law of the
static gravitational field were not able to give rise to a modern notion of
Universe. In particular, Olbers' paradox constituted a very serious no-go
argument to the construction of a coherent picture for the observed sky in
the framework of a stable gravitational configuration.
General Relativity, in agreement to the idea of a dynamical space, de-
formed by the matter and energy distributions contained in it, offered a
deeply new scenario to describe the origin of the Universe, born from a pri-
mordial phenomenon and emerging as an expanding space. In this context,
Olbers' paradox is easily solved because an observer can receive light from
the distance traveled by a photon from the Universe birth, which in this
framework is finite. Furthermore, these photons, coming from far galaxies,
are redshifted by the Universe expansion and they are observed with lower
energy than they were emitted. In this respect, first the discovery of the
galaxies (i.e. understanding that the observed nebulae were outside the
Milky Way), and later the Hubble demonstration of their recession, were
milestones in the definition of a modern view of the cosmological paradigm.
Einstein himself, without the notion of expanding galaxies, had difficul-
ties in accepting that his theory privileged non-stationary Universes with
respect to the static one initially proposed.
Thus, we can say that the geometrical framework of the gravitational
field in the Einstein picture was naturally able to read the Book of the Ori-
gin, in view of the link between the gravitational interaction and anything
else present in the Universe; for this reason in the Einstein formulation
Historical Picture
General Relativity is often called environment interaction.
1.3.1 Einstein proposal of a static Universe
From a theoretical point of view, the birth of modern Cosmology can be
traced back to 1917 (on the wave of GR, started in 1914), when Einstein
proposed a mathematical Cosmology based on his theory of General Rela-
tivity. As a first attempt to build a descripl ion of the Universe based on the
equations of General Relativity, his model was based on three assumptions.
The first was that, on the largest scales, the Universe is spatially homo-
geneous and isotropic, i.e. that no preferred observers exist. One reason for
this assumption was certainly philosophical in nature, since in some sense
it embodies the Copernican Principle, i.e. the Earth does not occupy a
special place in the Universe. This assumption is called the cosmological
principle, a term coined by Edward Milne (1896-1950). Another reason for
introducing the cosmological principle, and for its success before proving as
a good approximation, was that it simplified the mathematical treatment
of Einstein field equations. In fact, at the time of its introduction, the cos-
mological principle did not properly describe the observed Universe, until
then limited to our galaxy. It was already well known that stars in the
Milky Way are not homogeneously distributed. The cosmological principle
was more than a theoretical prejudice or a simplifying working hypothesis,
until the observations showed that there were other galaxies beyond the
Milky way and that they were indeed homogeneously distributed.
The second assumption was that the Universe has a closed spatial ge-
ometry and thus a constant positive curvature, ensuring a finite volume,
although without boundaries, like the surface of a sphere.
Finally, the third assumption was that the Universe is static, i.e. it
does not change with time. This can also be considered either a theoret-
ical/philosophical prejudice or a simplifying assumption. In particular, it
avoided the embarrassment of dealing with a "creation" event. When taken
together, the cosmological principle, expressing the space invariance of the
Universe, and the static Universe assumption, expressing time invariance,
are sometimes called the "perfect cosmological principle" .
As Einstein himself realized, a shortcoming of his static model is that
the equations of General Relativity do not admit any solution compatible
with these three assumptions. In order to obtain the static scheme, Ein-
stein modified the field equations introducing a cosmological constant term,
which can be interpreted as a gravitationally repulsive term acting at large
32 Primordial Cosmology
distances. Later, when the redshift of the galaxies was observed, thus prov-
ing that the Universe is not static but is indeed expanding, he regretted
the introduction of the cosmological constant as the "greatest blunder" of
his life 1 .
In 1917, the astronomer Willem de Sitter (1872-1934) found another
solution to Einstein field equations (with a cosmological constant) that
satisfies the cosmological principle and describes an expanding empty Uni-
verse. A notable feature of the de Sitter solution is that, even if expanding,
it is however stationary since it admits a time-independent representation.
In the following years, other relativistic cosmological models were devel-
oped. In particular, both the Russian mathematician Aleksandr A. Fried-
mann (1888-1925) and the Belgian astronomer Georges Lemaitre (1894-
1966) independently discovered, in 1922 and 1927 respectively, the solu-
tions to the Einstein equations that describe a Universe filled with matter.
They assumed the validity of the cosmological principle but dropped the
assumption of time-independence, considering both positively and nega-
tively curved spaces. The Friedmann-Lemaitre models predict that a pair
of objects move with a relative velocity proportional to their distance, thus
anticipating the discovery of the Hubble law. Friedmann and Lemaitre
also determined the physical conditions for an open Universe, indefinitely
expanding, and on the other hand for a closed Universe, destined to a
contraction, depending on the amount of matter contained in it.
1.3.2 Galaxies and their expansion: The Hubble's discovery
A turning point along the path that led to the birth of modern Cosmology
was the realization that the spiral nebulae (from the latin word for "cloud" ,
at that time used to denote any astronomical object with a diffuse appear-
ance, as opposed to a star) visible in the sky are indeed galaxies, similar
to the Milky Way, and discovering a whole new level in the hierarchical
structure of the Universe.
The controversy about the galactic or extragalactic nature of the spiral
nebulae was settled down in the 1920s. In 1922, the Estonian astronomer
Ernest Opik (1893-1985) estimated the distance of the Andromeda Nebula
M31 placing it well outside the Milky Way 2 . The American astronomer
1 Ironically, recent observations suggest that the Universe is currently undergoing a
phase of accelerated expansion that could be driven by a cosmological constant-like
component.
2 The obtained value, D ~ 450 kpc is actually quite close to the modern determination
of 770 kpc.
Historical Picture 33
Vesto Slipher (1875-1969) showed that the recession velocities of the spiral
nebulae, as estimated by the Doppler shift of their spectral lines, were far
higher than those of the other known astronomical objects. The definitive
evidence for the extragalactic nature of the nebulae was found in 1923-24 by
Edwin Hubble (1889-1953), who resolved the stars inside the Andromeda
and the Triangulum (M33) nebulae. In particular, since some of these stars
were variable Cepheids, he could measure the distance of the two nebulae
(the luminosity of Cepheid stars is correlated to the period of variation of
the luminosity itself, making them standard candles for the determination
of distances). Hubble found that both M31 and M33 lie at a distance of
approximately 300 kpc from the Earth, again placing them far beyond the
borders of our galaxy. In 1926, using the method of the number count, he
also quantitatively showed a homogeneous distribution of galaxies in space,
without any observable boundary. This discovery was the experimental
verification of Milne's cosmological principle discussed above, and together
with the realization that spiral nebulas are "island Universes" of their own,
it can be considered the 20th century version of the Copernican revolution,
showing that the Earth occupies no special place at all in the Universe.
However, the contribution to observational cosmology for which Hubble
is best known is the discovery of the expansion of the Universe. Building
on the studies of Slipher (who in the 1910s already observed the puzzling
fact that the spectra of most galaxies are shifted towards the red), and
aided by fellow astronomer Milton Humason (1892-1972), Hubble was able
to measure the spectral shifts and the distances of a sample of roughly
50 galaxies. The relationship between redshift and distance was first pub-
lished in a short paper by Hubble alone in 1929 and then in a longer paper
authored by Hubble and Humason in 1931. The conclusion was that the
redshift of a galaxy is directly proportional to its distance. By interpret-
ing the redshift as a Doppler shift, this relationship, that bears the name of
Hubble law, can be restated as a proportionality between a galaxy's distance
and its velocity, thus providing evidence for the expansion of the Universe.
Many theorists made several attempts to interpret the expansion on the
ground of Einstein's theory of General Relativity. In fact, an expanding so-
lution of Einstein equations, compatible with the homogeneity and isotropy
requirements of the cosmological principle, had already been found by de
Sitter in 1917. However, there were two shortcomings associated to the
interpretation of Bubble's findings in terms of the de Sitter solution. The
first refers to the requirement of the expansion pattern to be the same as
seen from any galaxy, and thus in order to fulfill the cosmological principle,
34 Primordial Cosmology
one has to set particular initial conditions. The second shortcoming is that
the de Sitter solution describes an empty Universe, and it is at variance
with the observation that galaxies are spread throughout the space. At
that time, the only widely known matter-filled cosmological model was the
Einstein static model, that of course does not predict any expansion. The
solution to the conundrum was in the matter-filled, expanding Universe so-
lution to the General Relativity field equations already found by Friedmann
in 1922. His solution was independently rediscovered by Lemaitre in 1927,
and the latter's work was later brought to more general attention in the
early 30s by Eddington and de Sitter. The final step was the demonstration
of Robertson (1935) and Walker (1936) that the line element adopted by
Friedmann is the more general line element in a spatially homogeneous and
isotropic spacetime.
1.4 The Genesis of the Hot Big Bang Model
Hubble's discovery led to dismiss the static model of the Universe, while
introducing the idea of an evolving Universe like that described by the
Friedmann-Lemaitre models. It was soon realized that the extrapolation
of the cosmological evolution backwards in time implies that matter and
radiation, nowadays sparsely scattered through space, were concentrated
in a remote period of time. In fact, a feature of the Friedmann models
is the existence of an instant of time in the past when the dimension of
the Universe extrapolates to zero and the matter and radiation densities
correspondingly diverge. Friedmann himself calculated this time for our
Universe to be some ten billions years in the past, although it is not clear
how much physical significance he attributed to the initial singularity. The
first to actually put forward the idea that the Universe expanded from a
very dense state was, in 1931, Lemaitre, who used to call the initial state the
"primeval atom". Remarkably, he also speculated on the possibility that
the early hot and dense phase should leave some relic radiation, that he
described as "the vanished brilliance of the origin of the worlds" . However,
the Lemaitre theory was mainly developed during the 1940s by Russian-
born American physicist George Gamow (born Georgy Antonovich Gamov,
1904-1968), who had briefly been a student of Friedmann in St. Petersburg.
Gamow advocated the theory that the chemical elements present in the
Universe were synthesized in a very early phase of the Universe, when it
was dense and hot enough for thermonuclear reactions to take place. This
Historical Picture 35
was opposed to the theory according to which the chemical elements are
produced in stars. We now know that both theories are right, since in the
early Universe only the lightest elements (mainly hydrogen and helium) are
produced, while the heavier elements are produced in stars. Gamow and
his student Ralph Alpher (1921-2007) first exposed this theory in 1948 in
the so-called "a/37" paper, signed by Alpher, Bethe 3 and Gamow. In that
paper, the authors argue that "various nuclear species must have originated
[...] as a consequence of a continuous building-up process arrested by a
rapid expansion and cooling of the primordial matter". This model was
later given the name of hot Big Bang model, although this was originally
intended as a pejorative monicker, coined by those who opposed the theory.
It is also a sort of misnomer, since the term "Big Bang" seems to point to
a single event localized in space that, in the past, triggered the expansion.
This popular undo standing of the model is wrong. The first reason is
that the Big Bang did not happen at a specific point, in space, as it should
be obvious from the cosmological principle. Secondly, and maybe more
importantly, the term "Big Bang" does not refer to the initial singularity
that is present in the Friedmann models. Although usually cosmologists
indulge in the habit of calling "Big Bang" the singularity, properly speaking
the term should preferably be used to describe the very hot and dense
primeval phase of the expansion, regardless of the presence of singularity.
The difference is more than semantic since, as we shall see below, we have
compelling evidence for the physical reality of the hot phase, while the same
certainly cannot lie said for t be singularity. In fact, the hot Big Bang model
would still be true even if the singularity did not occur.
Later in 1948 Alpher, joined by Robert Herman (1914-1997), and
Gamow independently realized that their theory implied that, at the time
of the synthesis of the chemical elements, the Universe was filled by a black-
body radiation with an associated thermal energy of the order of 1 MeV.
This radiation was actually providing the main contribution to the energy
density of the Universe at that stage of evolution. The present-day tem-
perature of this radiation field can be estimated and it turns out to be
around a few Kelvins, so that the maximum of its spectrum should be in
the microwave range. This blackbody with T ~ few K is what today we
call the Cosmic Microwave Background (( 1MB). The presence of the CMB
was a definite prediction of the hot Big Bang model, and thus it could be
3 Hans Bethe (1906-2005) did not actually contributed to the paper; his name was
added humorously by Gamow in order to create a joke with the first three letters of the
Greek alphabet.
36 Primordial Cosmology
used to test its validity. Unfortunately, for some time the Gamow-Alpher
picture was put aside since it became clear that the elements' primordial
build up could not proceed past 4 He, making the stellar synthesis picture
more appealing.
In the same years, the Big Bang theory was opposed to the so-
called steady-state theory, formulated by Hermann Bondi (1919-2005) and
Thomas Gold (1920-2004) and further developed by Fred Hoyle (1915-
2001). Although nowadays referred to as an ■■alternative cosmology", at
the time the steady-state theory was much more credited, among scien-
tists, than its hot Big Bang competitor. The steady-state model was based
on the perfect cosmological principle, i.e. on the notion that the Universe
should be the same at every point in space and at every instant of time. In
order to reconcile this assumption with the, by the time accepted, cosmo-
logical expansion, the theory postulated the continuous creation of matter,
in order to balance for the dilution caused by the expansion and maintain
the same average density. A steady-state Universe has no beginning nor
end in time. One point in favor of the steady-state model was the fact that
the age of some astronomical object seemed to be larger than the age of
the Universe as estimated in the framework of the hot Big Bang model.
Today, we know that this was due to the severe overestimate of the Hubble
constant (to which the age of the Universe is inversely proportional) .
The debate, as it should happen in natural sciences, was settled by
observations. In the early '60s, many scientists (re-)realized that the obser-
vation of a microwave blackbody radiation would have provided compelling
evidence for the Big Bang. In particular, both the group of Robert Dicke in
Princeton and that of Yakov Zel'dovich in Moscow independently arrived at
this conclusion. In 1964, Dicke and collaborators (among them Jim Peebles
and David Wilkinson) were actually planning to search for a microwave ra-
diation of cosmological origin, building a dedicated radiometer. However,
they were unaware that the CM IJ radiatiou had actually already been ob-
served by a radiometer at the Bell Telephone Laboratories in Holmdel, just
50 kilometers away from Princeton.
The radiometer had originally been used for the first experiments on
satellite communications, and in 1963 Arno Penzias and Robert Woodrow
Wilson started to prepare it for radio astronomy observations.
After removing known sources of noise (like for example radio broad-
casting), they were left with an apparently inexplicable residual, isotropic
noise, corresponding to a 3.5 K excess antenna temperature.
Alter learning of the ongoing efforts of the Princeton group, they started
Historical Picture 37
to guess the possible cosmological implications of their findings. They con-
tacted Dicke and a joint meeting with the Princeton group was organized,
where the conclusion was reached that the origin of the 3.5 K signal was
extraterrestrial. The two groups decided to publish their results indepen-
dently but at the same time. Thus, in the same volume of the Astrophysical
Journal two papers appeared, one authored by Penzias and Wilson, the
other by Dicke, Peebles, Roll and Dickinson. In the first, conservatively
titled "A Measurement of Excess Antenna Temperature at 4080 Megacy-
cles per Second", the existence of the isotropic signal was reported. In the
second, the signal was interpreted as the relic radiation from the hot Big
Bang. The discovery of the CMB dealt the final blow in the steady-state
model and led to the definitive acceptance of the hot Big Bang theory.
Later, in 1978, Penzias and Wilson received the Nobel Prize for their
discovery.
1.4.1 Recent developments
1.4.1.1 Observed isotropy
After the observation of the CMB, much effort was devoted to the pre-
cise determination of its frequency spectrum, in order to confirm the black
body shape. This was at least in part due to the claims by the supporters
of the steady-state theory that an isotropic microwave background can be
generated by the scattered, redshifted light of very distant galaxies. This
background would not have a thermal spectrum, however. In the same
years, during the 70s, theoretical cosmologists started to realize that small
inhomogeneities, of order of one part in 10 4 — 10 5 , should have been present
in the primordial plasma, that eventually grew (through gravitational in-
stability) and formed the galactic structures that we observe today. These
small inhomogeneities should leave an imprint in the CMB radiation, pro-
ducing anisotropics approximately at the same level. Then, during the
80s the observational efforts converged on the measurements of the an-
gular fluctuations in the CMB, other than to the ultimate determination
of its frequency spectrum. Both these goals were reached by the COsmic
Background Explorer (COBE), a NASA satellite launched in 1988. COBE
carried two instruments, the Far-InfraRed Absolute Spectrometer (FIRAS)
and the Differential Microwave Radiometer (DMR). FIRAS provided the
definitive measurement of the frequency spectrum of the CMB, showing
that it is a nearly perfect black body with a temperature T = 2.723 K (it
38 Primordial Cosmology
is actually the most accurate blackbody that is observed in nature), while
DMR observed for the first time the large scale CMB anisotropies, detect-
ing temperature fluctuations of some tens of microkelvins. Two of COBE's
principal investigators, John Mather and George Smoot, were awarded the
Nobel Prize in 2006 for their work on the experiment.
Following the observations of COBE, in the 90s many ground-based and
balloon-borne experiments were designed to measure the CMB anisotropies
at smaller angular scales with respect to those that were accessible by
COBE. The determination of the exact anisotropy pattern could discrimina-
te between different theories for the origin of the primeval seeds from which
cosmic structures originated. In particular, the two most credited theories,
cosmic strings and inflation, predicted different anisotropy patterns. In
the cosmic strings scenario, the power spectrum of the anisotropies should
appear as nearly featureless, while in the inflationary scenario, it should ex-
hibit a characteristic alternation of peaks and dips, caused by the presence
of coherent acoustic waves in the early Universe. The existence of at least
one peak was hinted by several experiments during the 90s, and finally in
2000 the BOOMERanG experiment, led by Andrew E. Lange and Paolo
de Bernardis, detected the presence of multiple peaks and provided a pre-
cise determination of the position of the first acoustic peak, at an angular
scale of roughly one degree. The BOOMERanG results implied that the
spatial geometry of the Universe is nearly flat and confirmed inflation as
the leading theory for the origin of primordial fluctuations. BOOMERanG
also provided, for the first time, fairly tight constraints on the value of the
cosmological parameters, thus marking the birth of precision cosmology.
In the last decade, the CMB anisotropy spectrum has been measured
with increased precision and down to smaller angular scales. A new ob-
servational target is the pattern of the fluctuations in the polarization of
the CMB photons, that itself encodes many information on the Universe,
like those related to the formation of the first stars or to the presence of
a relic background of gravitational waves. In 2001, NASA launched an-
other CMB space mission, called Wilkinson Microwave Anisotropy Probe
(WMAP) in tribute to David Wilkinson. WMAP has obtained the most
precise measurement of the CMB temperature fluctuations to date and pro-
vides the tightest constraints on the values of the cosmological parameters.
In 2009, the ESA mission Planck was launched, and is expected to provide
the ultimate measurement of the CMB temperature fluctuations.
Historical Picture 39
1.4.1.2 Dark matter
Another striking feature of the Standard CosLuoloiaca] Model is the pres-
ence in the Universe of a large amount of non-baryonic matter, accounting
for roughly 25% of the total matter-energy content of the Universe. The
existence of this "dark matter" was suggested by the Swiss astronomer Fritz
Zwicky (1898-1974) to explain the motion of galaxies inside the Coma clus-
ter. In particular, he found that the mass of the cluster, estimated on the
basis of galactic motions, was 400 times larger than the visible mass inside
the cluster. This was known as the "missing mass" problem, and Zwicky -
although not taken seriously at that time - suggested that it was due to the
presence of a matter component not interacting with light. The existence
of dark matter remained a hypothesis for 40 years, until the '70s, when
the American astronomers Vera Rubin and W. Kent Ford Jr. used a new
spectrograph to measure the rotation curves of galaxies, i.e. the orbital
velocities of stars inside galaxies as a function of their distance from the
center, with unprecedented accuracy. They found that the mass required
to explain the observed curves was roughly 10 times larger than the visible
mass of the galaxy, and that this mass extended far beyond the visible edge
of the galaxy. This is considered the first strong observational hint for the
existence of dark matter. From the 70s until today, a bulk of evidence has
been gathered confirming its presence (for example, the CMB anisotropy
spectrum, or the galaxy power spectrum) although a direct detection is
still missing. The best evidence to date is provided by the Bullet cluster
observed by the Chandra X-ray Observatory. The Bullet cluster consists of
two merging clusters passing one through the other. The hot (collisional)
plasma inside the clusters can be clearly seen in the X-rays to be stuck in the
middle of the two colliding objects due to its collisional nature. However,
the total mass distribution in the cluster can be estimated by studying the
gravitational lensing of background objects, and it shows that the centers-
of-mass of the two clusters are separated. This indicates that most of the
mass in the clusters is in the form of a dark, collisionless component. The
common scientific view is that dark matter is made by Weakly Interact-
ing Massive Particles (WIMPs). Although there is no WIMP candidate in
the framework of the Standard Model (SM) of particle physics, neverthe-
less they are predicted by many extensions of the Standard Model itself.
The most popular candidate is the neutralino, appearing in supersymmet-
ric extensions of the SM. Some proposals have also been made trying to
explain the missing mass problem not through the presence of dark matter,
40 Primordial Cosmology
but instead through modifications to the theory of gravity. However, these
theories have to face the problem that the presence of dark matter can be
inferred by observations made at many different scales, from galactic, to
cluster, to the largest cosmological scales (probed by the CMB).
1.4.2 Discovery of the acceleration
Another important development in contemporary Cosmology took place in
the late '90s, when a very puzzling fact emerged from the observations of
distant type la Supernovae (SNIa). SNIa are standard candles, i.e. object
of known intrinsic luminosity, so that they can be used to build a Hub-
ble diagram. Since they are very luminous, they can be observed up to
very large distances and used to probe the expansion history deeper in
the past. In 1998 two groups, the Supernova Cosmology Project and the
High- z Supernova Search, measured the supernovae Hubble diagram and
independently reported evidence that distant SNIa are less luminous than
they would be in a decelerating Universe, implying that the Universe is
now accelerating. This fact, albeit strange, can be easily accommodated in
the framework of the Standard Cosmological Model. In fact, even if neither
matter nor radiation can give rise to an accelerated expansion, a component
with negative pressure could. A natural candidate is the cosmological con-
stant that Einstein introduced to obtain a static Universe and that he later
regretted as the "greatest blunder" of his life. In fact, when interpreted
in the framework of a Friedmann Universe with matter and a cosmological
constant, the SNIa data provide compelling evidence for the presence of
the latter. Unfortunately, this raises some very problematic issues from
the point of view of quantum field theory. The observational evidence for
the acceleration has grown in the last years and is now well established.
However, its theoretical interpretation is still unclear and can probably be
regarded as one of the biggest open problems in cosmology nowadays. One
possibility is that, as stated above, the expansion is caused by a component
with negative pressure (like a dynamical scalar field), generically dubbed
"dark energy" . Another is that the theory of General Relativity fails at the
cosmological scales and has to be replaced by a more general theory of the
gravitational interaction. The third possibility is that the observed accel-
eration is just an artefact due to the effect of small-scale inhomogeneities
on the propagation of photons.
Historical Picture 41
1.4.3 Generic nature of the cosmological singularity: The
Cambridge and the Landau School
The problem regarding the existence and nature of the singularity was
widely studied during the second half of the XX century, providing many
interesting features about the possible beginning of the Universe and still
leaving many intriguing topics unanswered. The existence of a singularity
came out after the works by Roger Penrose and Stephen Hawking, regarding
the analysis of geodesies in different conditions: the impossibility, in some
cases, of an indefinite continuation suggested the presence of a singularity
in the general solution of the Einstein equations (see Sec. 2.7).
The Landau School was created by L. D. Landau (1908-1968) at the
Science Academy in Moscow. This group of scientists gave an important
contribution to the development of Relativistic Cosmology in the '60s and
'70s of the last century. Apart from the work of Landau on superfluids
(he got the Nobel prize in 1962 for this study) and a few other issues, the
Landau School was surprisingly in the excellent capability to extract signif-
icance from the implementation of a theory for the synthesis of new physics.
The studies in Cosmology we are going to refer to are an excellent exam-
ple of the technical power that this team of scientists had in implementing
General Relativity.
Within the Landau school, at the beginning of the '60s, was pursued a
detailed and deep analysis of the general solutions of the Einstein equations
when evolving toward the singularity, either considering the instability of
density perturbations, either Rddressiug I be generality of the properties of
the solutions themselves.
The two most important results obtained in the investigations of the
early Universe can be recognized in the Lifshitz analysis of the gravitational
stability of the isotropic case and in the discovery by Belinskii, Lifshitz
and Khalatnikov (BKL) of the chaotic behavior of the generic cosmological
solution near the initial singularity.
The Lifshitz results demonstrated the stability of the Friedmann-
Robcri sou- Walker (1'RW) Universe when the volume expands, ensuring
how this highly symmetric solution can represent the present Universe.
The BKL analj sis clarified the existence of a past time-like singularity
as a general feature of the Einstein equations under cosmological hypothe-
ses. This result has to be considered in comparison and contrast with the
general theorems due to the Hawking School. Such rigorous framework, de-
scribed in Sec. 2.7, has a powerful nature, but being of topological nature
42 Primordial Cosmology
(i.e. concerning the behavior of world lines), it is not able to characterize
the physical properties of the singularity as a space-time pathology. The
investigations of the Landau School allowed one to determine a piecewise
analytical representation of the generic cosmological solution near the ini-
tial singularity, so properly characterized as a real space-time feature for a
very wide class of models (the word generic can be qualitatively understood
as absence of any specific symmetry). The nature of the BKL achievements
and the modern developments in this line of research are discussed in great
detail in the third part of the book, namely Mathematical Cosmology.
Indeed, an important step in the development of a general point of
view on the origin of a non-symmetric Universe is constituted by the work
of Lifshitz and Khalatnikov in 1963. In this study, they derived the so-
called generalized Kasner solution, extending the intrinsic anisotropic exact
solution of the Einstein equations for the Bianchi I model (provided by
Kasner in 1921) to the inhomogeneous sector and asymptotically to the
singularity.
This is the building block of the generic cosmological solution, being its
analytical segment, iterated to the cosmological singularity in a sequence
of infinite alternation of equivalent regimes (known as Kasner epochs). De-
spite this generalized Kasner solution is derived by imposing a condition
which limits its generality, Lifshitz and Khalatnikov did not realize the
underlying scenario at that stage of understanding and claimed that the
generic solution had to be asymptol i< ally Kasucr-likc.
Only at the end of the '60s. when Bclinskii and Khalatnikov investi-
gated deeper the behavior of the homogeneous model, became clear that
the instability of a Kasner epoch could result in the transition to a new one
with different values for the metric parameters. From this new intuition
arose first the BKL study of the Bianchi type VIII and IX model and then
the extension of this prototype to the generic case.
The very surprising feature was the discovery that the iteration of equiv-
alent regimes were associated to the appearance of chaotic properties of the
time evolution. Much later studies, mainly due to A. A. Kirillov and G.
Montani outlined how the stoehast icily of the time evolution induces a
chaotic morphology of the spatial slices too. This phenomenon associated
to the inhomogeneous BKL solution makes the spacetime near the singu-
larity like a foam of statistical nature.
After the work of Charles Misner in 1969, the BKL dynamics was called
as Mixmaster Universe (with particular reference to the Bianchi IX cos-
mological model), in view of its Hamiltonian representation as a particle
Historical Picture 43
randomizing in a closed potential. Over the last three decades the interest
for the Mixmaster evolution of the early Universe remained high, both for
the attention devoted to such behavior by the dynamical system theory,
and because its generality suggests that it trace very well the dynamical
conditions under which the Universe was borne, especially in view of a
quantum gravity scenario.
Indeed, the Mixmaster Universe properties are still intensely studied
providing a deep insight about the Universe and offering a test field for
different methods of analysis coming from several fields to a generic prob-
lem. Quite often, an evolutionary property derived for a specific topic is
extended and tested (sometimes in a speculative way) also in the Mixmaster
model.
In particular, a wide interest emerged on the chaos features shown by
the Mixmaster Universe, giving rise to several tests from different points
of view, such as dynamical systems" approaches, discrete mathematics and
numerical calculations.
The possibility to reconcile the exotic nature of the generic cosmological
solution with the regular homogeneous and isotropic model of the hot Big
Bang is today recognized in the inflationary scenario, formulated at the
end of the '70s and the beginning of the '80s of the last century. This
paradigm, discussed in details in Chap. 5 was introduced by Guth and
Linde to overcome the shortcomings of the Standard Cosmological Model.
1.4.4 The inflationary paradigm
After the hot Big Bang scenario became the standard model of cosmol-
ogy, the theoretical effort was mainly aimed at two different goals. The
first, more phenomenological, was to develop the theoretical tools that are
necessary to extract meaningful, observable predictions. The second, more
philosophical, was to understand some puzzling facts about the standard
model which offered a coherent picture, confirmed by the presence of the
CMB, of the history of the Universe from the time of nucleosynthesis until
today. The supporting evidence in favor of the hot Big Bang model has
grown through the years and to date there are no observations that are
at variance with it. However, some paradoxes arise that cannot be solved
in the framework of the standard model. Simply put, these paradoxes are
mainly related to the fact that, in order to evolve into its present state, the
Universe should have started from very peculiar initial conditions in the
early phases. This was first noted by Zel'dovich in the early '70s. In the
44 Primordial Cosmology
following years, it was realized that an early de Sitter phase of exponential
expansion (nowadays called an inflationary era) would solve these prob-
lems. In 1980, Alexei Starobinsky and Alan Guth independently proposed
two mechanisms to generate the exponential expansion. 4 In Starobinsky's
model, this is caused by quantum corrections to gravity that become im-
portant at very high energy. In Guth's model, inflation is caused by the
fact that the early Universe is trapped in a metastable, false vacuum state;
the expansion is driven by the vacuum energy associated to the false vac-
uum. Unfortunately, Guth himself realized that this model suffers from
the "graceful exit" problem, namely the fact that the phase transition be-
tween the false and true vacua never takes place and inflation never ends.
The graceful exit problem was solved independently shortly after by An-
drei Linde on one hand and Andreas Albrecht and Paul Steinhardt on the
other. In their variant (referred to as "new inflation" or "slow-roll infla-
tion" ) of the original model, inflation is driven by the energy density of a
scalar field slowly rolling down its potential. Other than solving the stan-
dard model paradoxes, inflation also offers a mechanism for the generation
of the primordial density fluctuations. To date, inflation is a successful
paradigm that has received its confirmation mainly by the measurements
of the CMB anisotropy spectrum. On the other hand, from the theoretical
point of view, there are still many open questions: for example, we still
do not know what the scalar liold allowedly responsible for the inflationary
expansion (the inflaton) is. The ambition is that the inflationary scenario
will some day be embedded in a more general theory, like supersymmetry
or string theory. We conclude by stressing that the inflationary expansion
can be produced by many differenl mechanisms so that inflation is more
correctly referred to as a paradigm or a scenario; within this scenario, many
different models exist, and can be discriminated through the observations.
1.4.5 The idea of non-singular cosmology: The cyclic Uni-
verse and the Big Bounce
We conclude the historical picture of the development of a modern cosmol-
ogy by discussing the inflationary scenario and the observation of an ac-
celerating Universe (collocated mainly 30 and ten years ago, respectively)
4 However, it should be noted that Starobinsky was probably mostly motivated by the
goal of avoiding the initial singularity. Thus, in his work there is no reference to the
standard model paradoxes nor to the exponential expansion as a possible solution of the
paradoxes themselves.
Historical Picture 45
because they represent the most significant theoretical and observational
progresses achieved by cosmologists.
Indeed the great efforts made in the recent years to improve and com-
plete our knowledge of the Early Universe gave rise to very promising new
points of view on the nature of the singularity (see for instance the pre-Big
Bang scenarios predicted by String cosmology) , although still at the center
of the contemporary debate about their prediction capability.
However, let us characterize some recent developments in Loop Quan-
tum Cosmology , like the possible existence of a Big Bounce for the Planck-
ian evolution of the isotropic Universe. This attention is not motivated by
the conviction that these studies are completely settled down and predic-
tive, but in view of their peculiar features, presented in Chap. 12.
Despite some theoretical shortcomings, like the problem of entropy, the
idea of a cyclic (closed) Universe, oscillating between a Big Bounce and a
turning point, seemed to Einstein and other theoreticians a very pleasant
alternative to the Big Bang singularity. In this respect, the results obtained
by Ashtekar and collaborators are a very encouraging issue in favor of this
cyclical idea. The isotropic Big Bounce has been derived implementing the
ideas and formalism of Loop Quantum Gravity (mainly due to Ashtekar,
Smolin and Rovelli). This canonical approach to the quantization of the
gravitational field has the great merit of starting from a continuous descrip-
tion of the spacetime manifold, nonetheless recovering the discrete struc-
ture of the space, in terms of discrete spectra of the geometrical operators,
like areas and volumes. The kinematical sector of Loop Quantum Gravity
resembles a non-Abelian gauge theory and it allows the extension to the
gravitational field of the so-called Wilson loops approach for strongly cou-
pled Yang-Mills theories. However, the dynamical implementation of the
super-Hamiltonian quantum constraint contains a certain level of ambigu-
ity, e.g. the non-unitary equivalence of theories corresponding to different
values of the Immirzi parameter, entering the canonical variables definition.
The application of Loop Quantum Gravity to the minisuperspace of a
homogeneous cosmological model, expectedly implies a non-singular behav-
of the quantum Universe, as a direct consequence of the cut-off scale
mposed on the Universe volume, by the tnuiiuial (taken Planekian) value of
ts operator spectrum. Indeed, the Friedmann-Robertson- Walker geometry
acquires in Loop Quantum Cosmology a non-singular behavior as described
in terms of a free massless scalar field (the kinetic term of the inflaton field)
playing the role of a relational time. The semiclassical picture of this non-
singular Universe can be restated in the form of a maximal critical energy
46 Primordial Cosmology
density for the asymptotic approach to the initial instant.
As we already stressed in Sec. 1.4, this cut-off on the maximal available
temperature of the primordial Universe does not affect the theory of the
Hot Big Bounce, because its scale is much greater than the physical regions
of interest for the Standard Cosmological Model predictions like inflation,
baryogenesis and nucleosynthesis. In this new scenario the idea of a cyclical
Universe takes new vigor and is substantiated by a precise quantum and
semiclassical scenario.
Although the Big Bounce theory is a promising perspective and deserv-
ing many attempts to extend its applicability to more general cosmological
models (up to the generic quantum Universe), nevertheless its derivation
is affected by some open issues. In fact, the restriction of the Loop Quan-
tum Gravity theory to the minisuperspace has the non-trivial implication to
replace the non-Abelian SU{2) by an Abelian U(l) symmetry, unable to en-
sure the discreteness of the volume spectrum. The possibility to recover the
Big Bounce from the minisuperspace dynamics relies on the introduction
by hands of the space discreteness as a natural, but not direct, consequence
of the full theory equipment. These shortcomings of imposing the symme-
tries of the isotropic model before quantizing its dynamics prevent the Big
Bounce to be self-consistently derived, but do not seem able to affect the
impact of this issue on the modern idea of a primordial Universe.
1.5 Guidelines to the Literature
The history of Cosmology is based on a variety of sources which cannot
be compactly summarized for what concerns the most ancient documents.
Since the approach to the Cosmos was borne as the vision of the philoso-
phers along the centuries on the basis of the limited physical evidences avail-
able, some reference can be searched consulting textbooks on history of phi-
losophy, specialized for the different historical periods, such as Guthrie [213]
for the Greek era and Grant for the Middle Ages [204] or covering up to
the 19th century [205].
The works of most authors, although reprinted from time to time
up to the XX century, are nowadays directly accessible by sev-
eral online services, such as books.google.com which is a generic
source of original documentation or by other online libraries, such as
http://digital.lib.lehigh.edu/planets/ for some works by Coperni-
cus and Brahe.
Historical Picture 47
A summary of the fundamental works by Galileo Galilei writings is given
by [1, 183-185, 192] and by Isaac Newton by [362,363].
To read the revolutionary ideas of Riemann on a new foundation of
geometry, among several translations, one can refer to [390,391], while the
pioneering study on differential geometry obtaining the classification of the
non-equivalent Lie groups is [86].
The most relevant papers by Einstein during the second decade of the
XX century can be found as [160-166].
In the 70s, Penrose and Hawking introduced the theorems on the
singularity [231,232] in the fertile environment of mathematical cosmol-
ogy on the path from the first results by Kasner [266] towards the
main results of the Landau School, which can be found in the papers
[58, 65, 66, 273, 274, 312, 313, 315, 345].
An overall vision over the last century is provided by the book of Longair
[325] while the physical Cosmology progress is summarized as follows.
The work of Starobinsky on the possibility of avoiding the initial sin-
gularity by taking into account higher-order curvature corrections to the
Einstein-Hilbert action can be found in [426]. The idea of an inflationary
Universe as a solution to the standard model paradoxes was proposed by
Guth in his 1981 paper [212]. The "slow-roll" inflation was invented shortly
after by Linde [320] and Albrecht & Steinhardt [2].
A very detailed account of the birth and development of physical cos-
mology, until the discovery of the CMB, can be found in the first part of
the book by Peebles [378].
The first cosmological solution to the equations of General Relativity
(with the addition of a cosmological term), describing a static homogeneous
and isotropic Universe was found by Einstein in 1917 [165]. The term
"Cosmological Principle" that denotes Einstein's assumption of isotropy
and homogeneity was coined nearly 20 years later, in 1935. by Milne in his
book [343]. Also in 1917, de Sitter derived the solution for an expanding,
empty Universe [143]. The solution dosoj:ibu)!>, an expanding Universe filled
with matter and radiation was derived by Fricdma 1 1 n first in 1922 for a space
with positive curvature [176] and then in 1924 for a space with negative
curvature [177]. Lemaitre independently re-derived these solutions and was
the first to make the very important connection between the expansion of a
matter-filled Universe and the observed redshift of the galaxies [305] . The
original work is written in French and was later translated into English
[307]. The translation came after papers by Eddington [157] and de Sitter
[144] drew attention on Lemaitre's work that had gone unnoticed at that
48 Primordial Cosmology
time. A few years later, the line element used by Friedmann and Lemaitre
was shown independently by Robertson [395] and Walker [458] to be the
most general element compatible with the requests of spatial isotropy and
homogeneity. This important result is actually geometrical in nature and
does not rely at all on the theory of General Relativity.
The spectroscopic observations performed by Slipher during the 1910s
showed that the spectra of most spiral nebulae are redshifted towards the
red, indicating that they are receding from us [420]. Slipher's results also
implied that the recession velocities of the nebulae were much bigger than
those of other astronomical objects, hinting to their extragalactic nature.
In 1922 Opik made an estimation of the distance of the Andromeda nebula
that placed it outside the Milky Way [367]. This result was confirmed by
Hubble's observations of the Andromeda and Triangulum nebulae reported
in 1926 [252,254]. In the same year Hubble also showed that galaxies
are distributed homogeneously and without a visible edge [253]. In 1929,
Hubble reported the first evidence for a proportionality between the redshift
of a galaxy and its distance [250]. Stronger evidence was given in a 1931
paper by Hubble and Humason, where what we know today as Hubble law
was formulated for the first time [251].
The notion that the Universe started from a very dense and hot state was
first put forward by Lemaitre with his theory of the "primeval atom" [306] .
In 1946, Gamow proposed the idea that the chemical elements have been
synthesized in the early Universe [187]. This idea was developed in the
a/?7 paper [6]. The implication of the presence of a thermal radiation
associated with the hot phase was realized independently by Alpher [5]
and Gamow [188, 189]. Gamow gave an order-of-magnitude estimate of the
present temperature of the radiation; his calculations were refined by Alpher
and Herman [7] . The competitor theory of the steady state Universe was
proposed by Bondi and Gold [101] and by Hoyle [242], independently. The
fact that the presence of a cosmic blaokbodv radial ion with T ~ 1 — 10 K was
potentially detectable and that it would have constituted a confirmation of
the hot Big Bang theory was pointed out again nearly 20 years later, in 1964,
by Doroshkevich & Novikov [156]. Shortly after, the CMB was observed by
Penzias & Wilson [381] and correctly identified as the relic of the hot Big
Bang by Dicke and collaborators [152].
The results of COBE-FIRAS (http://lambda.gsfc.nasa.gov/
product/cobe/f iras_overview. cfm) on the CMB frequency spectrum
and those of COBE-DMR (http://lambda.gsfc.nasa.gov/product/
cobe/dmr_overview. cfm) on the CMB angular anisotropics can be found
Historical Picture 49
in [173, 337, 338] and [423, 469], respectively. The observations of
BOOMERanG (http://cmb.phys.cwru.edu/boomerang/) and their cos-
mological interpretation were discussed in [139,140,303,339,361]. The latest
results of the WMAP satellite (map.gsfc.nasa.gov), after several years of
observations, can be found in [72,198,264,291,304,460]. For more informa-
tion on the Planck mission, we refer the reader to Planck's homepage.
http : //www . sciops . esa . int/index . php?pro j ect=PLANCK.
The original paper (in German) by Zwicky on the motion of galaxies
inside the Coma cluster can be found in Ref. [471]. The results reported
there can also be found in a later paper (in English) [472]. The work of
Rubin & Ford on the rotation curves of galaxies can be found in [403,404].
The observations of the Bullet cluster and their dark matter interpretation
have been reported in [122].
The first evidence for the acceleration of the Universe from the
SNIa observations was reported in 1998 by the High- z Supernova Search
(www.cfa.harvard.edu/supernova/HighZ.html) [392] and the Supernova
Cosmology Project (www.supernova.lbl.gov) [386] teams. For a recent
review on both the observational and theoretical status, see [178].
This page is intentionally left blank
Chapter 2
Fundamental Tools
This Chapter is devoted to introduce of some basic aspects of General
Relativity (GR) and of its formalism, completed by some selected topics
which are relevant for later analyses presented in this Volume. The study
of this Chapter endows the reader with some fundamental tools, necessary
for understanding some technical and conceptual passages in the discussion
of cosmological issues.
We start with a very schematic review of the ideas and of the formalism
at the ground of GR and then we provide a rather detailed discussion of
the type of matter fields, emphasizing the features of impact in the study
of primordial Cosmology.
The Hamiltonian formulation for the dynamics of the gravitational field
is faced outlining the constrained structure of GR in the phase space. This
formulation is at the ground of the canonical quantum gravity in the metric
approach, presented in Chap. 10.
We devote Sec. 2.4 to the description of the synchronous reference be-
cause of the particularly simple form that the cosmological problem assumes
in this special coordinate frame. Then, the tetradic formalism is illustrated
to characterize the existence of a local Lorentz gauge symmetry, motivating
a first-order formulation of the Einstein-Hilbert action. Such revised frame-
work for GR is addressed starting from the Hoist gravitational Lagrangian,
introducing the Ashtekar-Barbero-Immirzi variables in the Hamiltonian pic-
ture. The paradigm of Loop Quantum Gravity, which will be discussed in
detail in Chap. 12, is based on such analysis.
Finally, we provide a schematic discussion of the singularity theorems,
developed on a topological setting, to fix the conditions under which a
singular space-time point appears. This study has to be regarded as com-
plementary to the behavior of the Universe, asymptotically to the initial
52 Primordial Cosmology
singularity, with reference to the work of the Landau School and presented
in Part 3 (entirely devoted to mathematical cosmology).
2.1 Einstein Equations
The main issue of the Einstein theory of gravity is the dynamical char-
acter of the space-time metric, described within a fully covariant scheme.
i ti< 1 ioiii linn n i 'ii i 1 m m fold M, endowed with space-time coor-
dinates x % and a metric tensor gij(x k ), its line element reads as
ds 2 = gijdx l dx 3 . (2.1)
This quantity fixes the Lorentzian notion of distances. The motion of a free
test particle on M corresponds to the solution of the geodesic equation
where u 1 = dx l /ds is its four- velocity, defined as the vector tangent to the
curve x l (s), and L* 7 = g" n Tji m are the Christoffel symbols given by
r,* m = r ljm = ± (d j9lm + d l9jm - d m9lj ) . (2.3)
The geodesic character of a curve requires to deal with a parallel transported
tangent vector u l . However, for a Lorentzian manifold this curve is an
extremal for the distance functional, i.e. it is provided by the variational
principle
'/*-«/*^??- - < 2 - 4 '
If a test particle has zero rest mass, its motion is given by ds = and there-
fore an affine parameter must be introduced to describe the corresponding
trajectory. The equivalence principle is here recognized as the possibility to
have vanishing ( 'Unstolfd symbols at a given point of M. (or along a whole
geodesic curve). The space-time curvature is ensured by a non- vanishing
Riemann tensor
R l jki = d k r 3l - d^ + r#r{„ fc - r? k r ml , (2.5)
with the physical meaning of tidal forces acting on two free-falling observers
and whose effect is expressed by the geodesic deviation equation
M 'Vz( U fc V fcS 4 ) = W 3 i m uiu l s m , (2.6)
Fundamental Tools 53
s t being the vector connecting two nearby geodesies. The Riemann tensor
obeys the algebraic cyclic relations
Rijkl + Riljk + Riklj = , (2.7)
and the first order differential equations
VmRijkl + ^iRijmk + VkR l3 l m = . (2.8)
The constraint (2.8) is called the Bianchi identity and it is identically satis-
fied as soon as one expresses the Riemann tensor in terms of the metric g^ .
On the other hand it can be considered as a real equation for the Riemann
tensor when the covariant derivatives are only expressed in terms of the
metric.
Contracting the Bianchi identity with g ' g' J , we get the equation
VjG{=Q, (2.9)
where the Einstein tensor Gij is defined as
Gij = Rij ~ \R9H , (2-10)
in terms of the Ricci tensor R^ = R l uj and of the scalar of curvature
R = g ij Rij.
In order to get the Einstein equations in the presence of a matter field
described by a Lagrangian density C m , we must fix a proper action for the
gravitational field, i.e. for the metric tensor g^ of the manifold M. A
gravity-matter action, satisfying the fundamental requirements of a covari-
ant geometrical physical theory of the space-time, takes the Einstein-Hilbert
matter form
S = S g + S m = -— [ d 4 x v ^g~(R-2n£ m ) , (2.11)
2k Jm
where g is the determinant of the metric tensor g,,j and k is the Einstein
constant. The variation of the action (2.11) with respect to g lJ leads to the
field equations in the presence of matter as
Gij = R^ - ]-Rg tl = nTij. (2.12)
Here T^ denotes the energy-momentum tensor of a generic matter field and
reads as
2 ( 6{V=g£ m ) d S(V=gC m ) \
As a consequence of the Bianchi identity (2.9), we find the conservation
law VjT 4 J = 0, which describes the motion of matter and arises from the
54 Primordial Cosmology
Einstein equations. The gravitational equations thus imply the equations
of motion for the matter itself. The whole content of the Einstein theory
can be summarized as follows.
The space-time is defined as a four- dimensional manifold M on which
a Lorentzian metric tensor g t j is assigned. The space-time (metric) is a
dynamical entity which evolves in tandem with matter according to the
Einstein equations (2.12). The. physical lams are background independent,
i.e. they must retain the same tensor form for any assigned reference frame.
Such statement is known as the Principle of General Relativity.
Finally, by comparing the static weak field limit of the Einstein equa-
tions (2.12) with the Poisson equation of the Newton theory of gravity, we
get the form of the Einstein constant in terms of the Newton constant G as
k = 8ttG. We emphasize that the whole analysis discussed so far regards the
Einstein geometrodynamics formulation of gravity. In the gravity-matter
action, the cosmological constant term is considered as vanishing, although
it would be allowed by the paradigm of GR. This choice is based on the
idea that such term should come out from the matter contribution itself,
expectably on a quantum level.
2.2 Matter Fields
In GR, continuous (macroscopic) matter fields are described by the energy-
momentum tensor X^ introduced above in Eq. (2.13). Here, we will mainly
focus our attention on tensor fields. In particular, we will discuss the rel-
evant cases of the perfect fluid, the scalar, the electromagnetic and the
Yang-Mills fields.
2.2.1 Perfect fluid
The energy-momentum tensor of a perfect fluid is given by
^^(P + p^-Pg^, (2.14)
where ui is a unit time-like vector field representing the four-velocity of
the fluid. The scalar functions p and P denote the energy density and the
pressure, respectively, as measured by an observer in a locally inertial frame
co-moving with the fluid. These two quantities can be related to each other
by an equation of state of the form P = P{p). Since no term describing
heat conduction or viscosity is introduced here, the fluid is considered as
Fundamental Tools 55
perfect. For the isothermal early Universe, an appropriate equation of state
can be cast as
P=( 7 -l)p, (2.15)
where 7 is the polytropic index.
The equations of motion for a perfect fluid on a curved background can-
not be in general derived from a Lagrangian formulation. They can be con-
structed by the conservation law of the corresponding energy-momentum
tensor (2.14) expressed as
V fe T PF t = V fe [{P + p) u lU k - PS*} = , (2.16)
which can be restated as
u t V k [(p + P) u k ] +(p + P) u k V kUl = d t P . (2.17)
Multiplying Eq. (2.17) by u % and making use of the relation u l \7 k Ui = (a
direct consequence of the normalization u l Ui = 1), one obtains the scalar
equation
V fc [{p + P) u k ] =u l d l P. (2.18)
Substituting this relation into Eq. (2.17), we arrive at the equations of
motion for the perfect fluid flow
U " V ^ = (^p) & P ~ ^ k dkP) ■ (2.19)
In the particular case when P = 0, we deal with a dust, whose elements fol-
low geodesic trajectories (this is also true if P = const.). It is worth noting
that, from Eq. (2.19), the pressure effects prevent the geodesic motion of a
perfect fluid and then the co-moving frame cannot also be a synchronous
one, because it would be a geodesic reference (hoc Sec. 2.4).
We stress that for a homogeneous isotropic space, i.e. where the pressure
is time dependent only, the right-hand side of Eq. (2.19) vanishes when it, =
(1,0, 0, 0). In this case, the co-moving system should also be a synchronous
reference and in the isotropic case the energy-momentum tensor in the co-
moving frame would read as
T PF = diag(p,-P,-P,-P). (2.20)
We can conclude that the only models admitting a co-moving synchronous
reference are the homogeneous spaces.
56 Primordial Cosmology
2.2.2 Scalar field
The Lagrangian density for the linear, relativistic, scalar field theory reads
£ =i(d fe <^>-m 2 <£ 2 ) , (2-21)
and on a curved space-time the dynamics can be implemented by the min-
imal substitution rule rjij — > gij and di — > V,, i.e. we deal with the Klein-
Gordon equation
g ij V i (d j <p)+m 2 4> = 0. (2.22)
The corresponding energy-momentum tensor is then expressed as
T^ = drfdjcf) - ]- gij (d k cf>d k <j) - m 2 <f> 2 ) . (2.23)
Let us now look for a Lagrangian formulation of the perfect fluid dy-
namics, based on a scalar degree of freedom, able to reproduce the features
of the desired energy-momentum tensor. We consider a massless scalar field
(j) whose dynamics is governed by the Lagrangian density
C*=\ (9 ik di4>dk^) C , (2.24)
C being a free parameter. Using the definition of the energy-momentum
tensor (2.13), from Eq. (2.24), we get
T?j = C (g kl d k <f>d l ^) < ~ 1 d^djcp - C^ . (2.25)
Comparing this expression with the perfect fluid energy-momentum tensor
(2.14), we can identify the fundamental quantities p, P and Ui as follows
P=\ {<?%<&<!>)* ,
«, 5 , di<t> ■ (2.26c)
One can immediately recognize that this scheme allows to reproduce a
perfect fluid with an equation of state of the form P = p/{2Q — 1). Therefore
the parameter £ is related to the polytropic index 7 by the relation
C = W^)- (2 ' 27)
Fundamental Tools 57
It is worth noting that, according to the identifications (2.26), the variation
of the action C^ (2.24) for the scalar field provides the equations V J T^ = 0.
The particular case £ = 1 corresponds to a massless Klein- Gordon field
and it is associated to the equation of state P = p, where the sound speed
v s = y^dP/dp equals the speed of light. Since this case has a clear physical
interpretation in terms of a fundamentally free field, we can generalize it
by considering a self-interacting scalar field <fi, described by the action
C^= l -d k ^d k 4>-V^). (2.28)
Applying the same analysis as above to this self-interacting case, we gain
the new identifications
P=\{g jl d j cl>d l <j>)+V{4>), (2.29a)
P = ^ l d 3 4>d^-V(4>), (2.29b)
u l= - p£ . (2.29c)
One can realize how the self-interacting scalar field has the characteris-
tic feature of being associated to different equations of state in different
dynamical regimes. This follow.-, iron! the relations
where the notation KT stands for a kinetic term-like expression. When the
potential term V(</>) is negligible with respect to the kinetic one KT we
recover the equation of state P ~ p. On the other hand, in the opposite
regime (V(<j>) 3> KT) we deal with the condition P ~ —p. We will see in
Chap. 5 the relevance of this property of the self-interacting scalar field
when its dynamics is implemented at the cosmological level.
Let us observe that, if we take V(<j>) = ^m 2 (f> 2 , we recover the case of the
free Klein-Gordon field. In this sense, a generic potential term describes a
self-interaction of the field 0, as naturally arises in a quantum perturbation
theory. In fact, when we can treat the non-quadratic terms as small correc-
tions, we can define asymptotic free states at t — >• — oo and study scattering
processes among the scalar particles, which generate the out-going free
modes at t — > oo. When the theory is fully non-perturbative, the particle
interpretation is not feasible and we have to speak of self-interacting scalar
58 Primordial Cosmology
modes. Indeed, around a local minimum, say at (p = </> m i n , the potential
term admits the expansion
V{4>) * V(cf> min ) + £ ( ^ ) {4> - mi „) 2 . (2.31)
■-(—) (
Without any loss of generality, redefining m ; n = and V(4) m i n ) = 0,
we get to the dominant order a Klein-Gordon field, under the identifica-
tion in 2 = ((i 2 V/(i</> 2 )0 = min . This is the classical picture underlying the
quantum notion of particle mass as the effect of small fluctuations around
a vacuum state (classically the lowest minimum) of a given field.
However we emphasize that, when the gravitational interaction is in-
cluded in this paradigm, the redefinition of the minimum value of the po-
tential describing a zero energy density is no longer allowed. In fact, the
background metric tensor is sensitive to such vacuum energy density, unless
the whole quantum dynamics of the field can be regarded as a test one over
that background. The relevance of this consideration will be discussed in
Chap. 5, where we will deal with the inflationary scenario and with the
transition of a scalar field from a false to the true vacuum state.
Finally we remark that, as it takes place for an electromagnetic field,
also the boson scalar dynamics has a classical character under suitable
conditions. In fact, when we deal with a free scalar lickl having extremely
high occupation numbers characterizing its states, we can properly address
its evolution as a classical one, retaining the quantum effects just as small
perturbations to the background field. This is at the ground of our classical
consideration on the self-interacting scalar field and we will see in the study
of the inflation paradigm (see Chap. 5), how this quasi-classical picture is
relevant on a cosmological level.
2.2.3 Electromagnetic field
The electromagnetic Maxwell field is described by the Lagrangian density 1
(Mo = £o 1 = AlT )
C EM = --^F ij F i \ (2.32)
and the electromagnetic energy-momentum tensor reads as
T T = ^ (FikF k j + l - 9ii F kl F H \ . (2.33)
x Let us remember that the universal relation c 2 /io£o = 1 among the speed of light c, the
electric permittivity of vacuum eo and the magnetic permeability of vacuum jio holds.
Fundamental Tools 59
Here F = dA denotes the curvature 2-form associated to the connection
1-form A = Aidx 1 , i.e.
Fu = S7 l A j - VjAi = diA, - djAi . (2.34)
We note that the trace of the energy-momentum tensor defined as in
Eq. (2.33) is identically vanishing, i.e. g tj T^ M = 0. From the minimal
substitution rule, the Maxwell equations in a curved space-time become
\7 t F kl = -4ir.J k , (2.35a)
V {i F jk] = d {i F jk] = , (2.35b)
where J k denotes the current density four-vector of electric charge and
the square brackets around the indices are the compact notation for anti-
symmetrization. The continuity equation (namely, the conservation of the
electric charge) V& J k = follows from the antisymmetry of the Faraday
tensor Fij.
We will express F^ in terms of the electric and magnetic fields. Such
decomposition allows a fluid description for the Maxwell field. Given an
observer moving with a four- velocity m (such that u l m = 1), the quantities
Ei = F ]t u\ B ^ = \ tiJklF ik u l (2.36)
are the electric and the magnetic fields, respectively, measured by the ob-
server (eijki denotes the totally antisymmetric pseudo-tensor on curved
space-time). Note that EiU 1 = Biu 1 = 0, so that Ei and B{ are space-
like vector fields. The electromagnetic tensor can be decomposed as
T ™=G-t)' < 2 - 37 »
Here
W = ^-(EiE* + BiB 1 ) (2.38)
is the energy density of the field, Si is \]ic elect romaguclic Poynting vector
S t = e t3k i&B k u l , (2.39)
while
a af3 = -L (E a E + B a B p - 4TrW5 al3 ) , (2.40)
is the Maxwell stress tensor. Equation (2.37) provides a fluid description of
the electromagnetic ik'ld aud mauifests its intrinsic anisotropic nature. In
particular, the Maxwell field corresponds to an imperfect fluid with energy
density W, anisotropic stresses given by a a p and an energy- flux vector
represented by 5».
Finally, we give the world line for a charged particle moving in the
electromagnetic field
u l Viu l = ^-F I3 Uj , (2.41)
m
where q and m denote the charge and the mass of the particle, respectively.
60 Primordial Cosmology
2.2.4 Yang-Mills fields and ©-sector
We now introduce the concept of non-Abelian gauge fields in view of the
later comparison in Sec. 2.6 between the first-order formulation of gravity
with such non-linear theories.
Let us consider a set of fields t A = t A (x'). having a generic nature
(scalars, spinors, etc., encoded in the generic set of internal indices A)
whose Lagrangian density, in Minkowski space-time, has the form
£+ = \n ij di^ ■ d^ A - V(\ $ |) , (2.42)
where i\) ' denotes the hermitian conjugated of tp , while • is the product
on the internal space. It is immediate to recover the invariance of this
Lagrangian density under the internal unitary SU(N) transformations
(V A )' = Uip A = exp{i\<d a T a )i> A , a=l,...,N 2 -l, (2.43)
where U G SU(N), 6 a are constant parameters and A is a coupling con-
stant. The generators T a of the symmetry group SU(N) are Hermitian
matrices satisfying the su(N) Lie algebra
[T a , T b ] =iC abc T c , (2.44)
in which C abc are called structure constants. Since the transformation
(2.43) describes a global symmetry of the theory, the G a are independent
of the coordinates x l . In the limit of an infinitesimal transformation, i.e.
6 a -> 5<d a < 1, we get
{ip A )' = [l + i\5G a T a ]ip A , (V At )' = V At [1 - iXSQ a T a ] . (2.45)
The existence of such internal symmetries is an observed feature of the
Lagrangians associated to elementary particle physics. The SU{2) group
describing the isospin was historically discovered from the independence of
the nuclear interaction with respect to the electric charge, i.e. by guess-
ing that protons and neutrons were different states associated to the same
particles. The generators of the isospin symmetry, today recognized as a
fundamental one in the Standard Model of elementary particles, are given
by T a = cr a /2, where a a are the Pauli matrices. The structure constants of
this group are given by C abc = e abc , e abc denoting the totally antisymmetric
tensor on the internal indices.
Let us now promote the parameters a (and thus also the infinitesimal
ones (56 a ) to space-time functions, i.e. a = €) a (x' t ). The Lagrangian
density of the theory (2.42) is no longer invariant under this local gauge
transformation because the term
^ r] ij (d l ^T a d 5Q a 4! A - d i 6G a ip A ^T a d j i; A ) (2.46)
Fundamental Tools 61
does not cancel out. Local invariance is restored only by introducing a set
of vector fields A? (x : > ) and redefining the Lagrangian density as
£«, = \^V^ ■ V^ A - V(\ $ A |) . (2.47)
Here T>i denotes the covariant (gauge) derivative which explicitly acts as
V^ A = dnp A + i\A°;T a tp A , V^ = d^ - i\Ay A ^T a . (2.48)
The invariance of the Lagrangian under the local transformation of tp
and xj) \ called a gauge transformation, is ensured by the corresponding
variation of the gauge vector field Af , as
(A?)' = A1 - d t 5Q a + i\C a bc SQ b A c l . (2.49)
This transformation rule corresponds to the electromagnetic gauge pre-
scription plus an additional term containing the structure constants. This
reflects the non-Abelian character of the Yang-Mills fields here introduced
(i.e. Cabc 7^ 0). The EM case is associated to the U{1) symmetry group,
for which C abc = (i.e. it is an Abelian field).
The picture traced above needs to be completed by specifying the dy-
namical properties of the gauge vector fields. In analogy to the electromag-
netic case, we can define a gauge tensor by means of repeated applications
of the covariant derivative. For instance, we get
[Vi , Vj] = i\T a (d l A a j - djA* - \C a hc A\A c ^) = i\T a F^ . (2.50)
The antisymmetric tensor F^, which takes values in the SU(N) group, is
known as the field strength and transforms according to
(Fij)' = UF ij U~ 1 . (2.51)
An important difference with the electromagnetic case is the quadratic na-
ture of the gauge tensor in the Yang-Mills potential vector fields. Further-
more, it can be checked that such gauge tensor is not invariant under the
transformation (2.49) and thus it is not a physical observable, differently
from the linear case of a Faraday tensor, which is gauge invariant.
A natural choice for the Yang-Mills Lagrangian density is the quadratic
gauge- invariant term 2 5 a bF^F hlJ = —2Ti\FijF l ' J ]. The complete La-
grangian density takes the form
^+ym = W 3V ^ ■ V ^ A ~ V (\ ^ I) " Wb F tjF biJ ■ (2-52)
because TrpFijF^lJ- 1 } = Tr^i™]
62 Primordial Cosmology
We see how a local symmetry of the matter dynamics implies the presence of
gauge vector fields (bosons) carrying the interaction related to that specific
symmetry. In fact, the Lagrangian density (2.52) contains the free evolu-
tion of the matter and boson fields, but also their reciprocal interaction,
emerging from the covariant gauge derivative.
In the Hamiltonian formulation of a Yang-Mills field, the components
Aq behave as Lagrangian multipliers, whose variation yields the Gauss con-
straints
G a = d a E« + iC abc A b a E ac = , (2.53)
E% denoting the canonically conjugate momenta to the variables A a a .
The analysis above is referred to a Minkowski space-time, but it can
almost straightforwardly be extended to a curvilinear coordinate system
or to a real curved space-time. In fact, the metric r]ij can be replaced
by a tensor g t j , while the covariant structure of the theory is restored by
means of the covariant derivative. We remark that the antisymmetry of
F?j implies the cancellation of the Christoffel symbol from its expression
which, therefore, retains again the form (2.50).
We conclude with a brief discussion about the structure of the vacuum
in non-Abelian gauge theories. Let us introduce the dual field strength
tensor *Fij defined as
*Fij = \^ U Fm (2.54)
and satisfying (lie Biauchi identity
We can construct the topological (-barge (}. that is
Qoc J d 4 xTr[-kF i:j F ij } (2.56)
a topological invariant closely related to the physical vacuum of a Yang-
Mills theory. Q is invariant with respect to any local variation 5Ai whether
the equations of motion are satisfied or not. In fact,
SQ ex - / d i xTr[*F ij (V i 6A j - V j SA 1 )} = f d i xTr[*F ij V i 6A j ]
' r r (2-57)
= / cPxd'Tr^FijdA 3 ] - / d 4 xTr[{V l * F^SA 3 ] =0.
The first term in Eq. (2.57) is zero because there are not surface contribu-
tions while the second one vanishes because of the Bianchi identity (2.55). A
Fundamental Tools 63
quantity which satisfies this property is called a topological invariant. The
topological invariant charge Q is closely related to the physical vacuum of
a Yang-Mills theory.
In gauge theories the vacuum is usually defined by the conditions
Fij = 0. However, it turns out that there are infinite topologically dis-
tinct vacua in the SU(N) gauge theories (assuming that all vector poten-
tials decrease faster than l/\x\ at large distances). Furthermore, distinct
vacua are inequivalent because of the Gauss constraint (2.53). This result is
mainly due to the (topological) equivalence of SU{2) with the three-sphere
S 3 . The gauge mapping S 3 — > SU(2) is characterized by an integer W,
known as winding number. The different states \W) are related to each
other by a unitary transformation corresponding to the generators of the
Gauss constraint. Since it is unitary, its eigenvalues are given by exp(i0),
€ [0, 2tt). It is possible to show that the physical vacuum of non-Abelian
gauge theories is given by
|0)= JT exp(*We)|W>- ( 2 - 58 )
The charge (2.56) is related to the winding number W which in turn can
be obtained by a spatial integration of the 3-form A A A A A. Requiring
that gauge potentials tend to a pure gauge at large distances, and fixing
the gauge by Aq = 0, it is possible to show that
Q = W\ 00 , (2.59)
where oo stands for spatial infinity. The topological charge (2.56) can thus
be interpreted as the winding number of the pure gauge configuration to
which A a tends.
Summarizing, it is possible to add to the Yang-Mills Lagrangian density
a term Tr[*FijF tJ ]. This new term does not influence the classical equa-
tions of motion and does not contribute to the energy-momentum tensor.
However, it modifies the quantum dynamics which depends on the action.
For example, the vacuum-to-vacuum transition (9| exp(-iWi)|0) in QCD
is sensitive to the topological charge Q. These different physical scenarios
are known as the 0-sectors of QCD.
2.3 Hamiltonian Formulation of the Dynamics
The aim of this Section is to analyze the Hamiltonian formulation of GR
in the metric formalism. This program, that culminates in the formulation
64 Primordial Cosmology
given by Arnowitt, Deser and Misner in 1962, allows to identify some pe-
culiar features of the Einstein theory. The corresponding Hamilton- Jacobi
theory and the so-called ADM reduction of the dynamics will also be dis-
cussed.
2.3.1 Canonical General Relativity
The generally covariant system par excellence is the gravitational field in
GR, being an invariant under arbitrary changes of the space-time coor-
dinates (four-dimensional diffeomorphisms). The canonical formulation of
GR assumes a global hyperbolic topology for M (the physical space-time) ,
allowing the splitting
M = RxE, (2.60)
£ (the three-space) being a compact three-dimensional manifold. As follows
from standard theorems, all physical space-times possess such a topology.
This way, M can be foliated by a one-parameter family of embeddings
X t : £ — > M, t € K, of £ in M. As a consequence, the mapping X :
KxE^M, defined by (x, t) -> X u is a diffeomorphism of R x £ to M . A
useful parametrization of the embedding is given by the deformation vector
field
Y\X) = dX% ^ t] = N{X)n\X) + N\X) , (2.61)
where A*(X) = N a d a X i . As soon as the vector field Y l is everywhere
time-like, it can be interpreted as the "flow of time" throughout the space-
time. In Eq. (2.61), ri 1 is the unit vector field normal to £4, i.e. the relations
gijnW = 1 (2.62a)
giXdcXJ = , (2.62b)
hold. The quantities N and N a are known as the lapse function and the
shift vector, respectively. The space-time metric gij induces a spatial metric,
i.e. a three-dimensional Riemannian metric tensor h a s on each £ t , by
h aP = -g^ d a X i d p X j . (2.63)
The space-time line-element adapted to this foliation thus reads as
ds 2 = N 2 dt 2 - h af3 (dx a + N a dt){dx li + N li dt) . (2.64)
This formalism is known as the ADM procedure. The geometrical meaning
of N and N a is the following: the lapse function N specifies the proper
Fundamental Tools 65
time separation between the hypersurfaces -X*(£) and X t +dt(^) measured
in the direction n l normal to the first hypersurface. On the other hand,
the shift vector N a measures the displacement of the point X t+ dt(x a ) from
the intersection of the hypersurface Xt+dti^) with the normal geodesic
drawn from the point X t (x a ) (see Fig. 2.1). In order to have a future
directed foliation of the space-time, the lapse function TV must be positive
everywhere in the domain of definition.
Figure 2.1 Geometric interpretation of the lapse function and of the shift vector: ri l is
the unit vector field normal to St. From this emerges the link between the lapse function
N and time diffeomorphisms, and between N a and spatial diffeomorpliisms.
In the canonical analysis of GR, the Riemannian metric h a p on E t plays
the role of the fundamental configuration variable. The rate of change of
h a p with respect to the time label t is related to the extrinsic curvature of
the hypersurface E t by the relation
A'„
~\-nh a0
(2.65)
where L a denotes the Lie derivative along the vector field a. In the c
a splitting as in Eq. (2.60), Eq. (2.65) explicitly reads as
K a0 (x,t) =
^K,
- (U-v/iWJ
- W a Np - VpN,^ .
66 Primordial Cosmology
Let us pull-back the Einstein Lagrangian density by the adopted foliation
X : M x £ — > M and express the result X* : M — > M x S in terms
of the extrinsic curvature K a p, the three- metric h a p, N and iV Q . This
procedure yields the so-called Gauss-Codazzi relation, which relates the
four-dimensional Ricci scalar 4 R to three-dimensional one 3 R; it explicitly
stands as
X* (V^5 4 ^) = NVh({K^f - K a pK al3 - 3 r)
+2 j- (VhK2) + dp(K«NP - h ali d a N^j ,
where y/^g = N\fh, h = det h a p.
We are able to re-cast the original Hilbert action into a 3 + 1 form simply
by dropping the total differential expressed by the last two terms on the
r.h.s. of Eq. (2.67) as
S g (h,N,N a )= J C 3+1 dtd 3 x
ilxE
= -— [ NVh({K^f - K afi K ati - 3 r) dt d 3 x . (2.68)
By performing a Legendre transformation of the Lagrangian density £3+1
appearing in Eq. (2.68), we obtain the corresponding Hamiltonian density.
Let us note that the action (2.68) does not depend on the time derivatives
of N and iV Q ; therefore, using the definition (2.66) and the fact that 3 R
does not contain time derivatives, we obtain that the conjugate momenta
are given by
IP*(x t t) = *%ti- = ^ (h°*iq - K*) , (2.69a)
n(x,t) = -^- = 0, (2.69b)
SN(x,t)
IL a (x, t) = 5Cz+1 = . (2.69c)
K ' 6N a (x,t)
From Eqs. (2.69), it follows that not all conjugate momenta are indepen-
dent, i.e. one cannot solve for all velocities as functions of coordinates
and momenta: one can express ft a g in terms of /?, a . : j, N, N a and H a P , but
the same is not possible for TV and N a . In other words, we deal with the
so-called primary constraints
C{x 7 t) =U(x,t) =0, C a {x,t) =U a (x,t) =0, (2.70)
Fundamental Tools 67
where "primary" emphasizes that the equations of motion have not been
used to obtain these relations.
According to the theory of constrained Hamiltonian systems, let us in-
troduce the new fields A(x, /■) and A" (.?;./) as the Lagrange multipliers for
the primary constraints, making the Legendre transformation invertible.
The corresponding action is thus given by
S g = dt d 3 x\h a pU al3 + NU + N a U a
- (AC + \ a C a + N a H a + NH) J .
H = g a0l6 U a ^W 5 ~^r 3 R, (2.72a)
H a = -2/i Q7 V /3 lF /3 , (2.72b)
(2.72c)
Equation (2.72a) defines the super-Hamillonian. Eq. (2.72b) the super-
momentum, while Eq. (2.72c) defines the super-metric Q a p-y8 on the space of
the three-metrics. Varying action (2.71) with respect to the two conjugate
momenta II and II Q we obtain
N{x,t) = X(x,t), N a (x,t) = X a {x,t), (2.73)
ensuring that the trajectories of the lapse function and of the shift vector
in the phase space are completely arbitrary.
The classical canonical algebra of the system can be expressed in terms
of the standard Poisson brackets as
{h a p(x,t),h 7 s(x',t)} = (2.74a)
{U a/3 (x, t), IT"V, £)} = (2.74b)
{h lS (x, t),Ii ap {x\ t)} = S^6 3 (x - x 1 ) . (2.74c)
From Eq. (2.71), we can define the Hamiltonian of the system as follows
\ a C a + N a H a + NH)
(2.75)
= |C(A) + (7(A) + H(N) + H(N)j .
The variations of the action (2.71) with respect to the Lagrange multipliers
A and A" reproduce the primary constraints (2.70). The consistency of
Hs //
68 Primordial Cosmology
the dynamics is ensured preserving C and C a during the evolution of the
system, i.e. by requiring
C(x,t) = {C(x,t),U} = 0, C a (x,t) = {C a (x,t),H} = 0. (2.76)
However, the Poisson brackets in Eq. (2.76) do not vanish but are equal
to T-i{x,t) and 'H 0! (x,t), respectively, and therefore the consistency of the
motion leads to the secondary constraints by means of the equations of
uiol ion
H(x,t) = 0, H a {x,t) = 0. (2.77)
Let us observe that the Hamiltonian of the Einstein theory is constrained as
H«0, being weakly zero, i.e. vanishing on the constraint surface (defined
as the surface where the constraints hold): this is not surprising since we
are dealing with a generally covariant system.
A problem that in general can arise is that the constraint surface could
not be preserved under the motion generated by the constraints themselves,
but this is not the case for the Einstein theory: the Poisson algebra of the
super-momentum H a and of the super-Hamiltonian H, computed using the
relations (2.74), is in fact closed. In other words, the set of constraints is a
first class set, i.e. the Poisson brackets of the Hamiltonian H with any of
the constraints weakly vanish. This can be gained from the relations
{H, H(f)} = H(L$f) - H(l f N) , (2.78)
{H, H(f)} = H(Ljff) + H(N(N, f, h)) , (2.79)
/ being a smooth test function, and N a (NJ,h) = h a ^(Ndpf - fdpN).
These equations are equivalent to the Dirac algebra
{H(f),H(f>)}=H(L f f>), (2.80a)
{H(f),H(f)} = H(L f f), (2.80b)
{H (/), H(f')} = H(N(f, /', h)) , (2.80c)
which underlines the canonical formulation of any field theory based on a
diffeomorphism (Diff(AI)) invariant action, like the Einstein theory.
Three remarks on this algebra are in order.
(i) Because of Eq. (2.80a), H(f) generates a sub-algebra which can
be identified with the Lie algebra diff(E) of the spatial diffeomor-
phism group Diff(E) of the Cauchy surface E. This way the super-
momentum constraint H a = is also called the spatial diffeomor-
phism constraint.
Fundamental Tools 69
(ii) Equation (2.80b) states that the super-Hamiltonian constraint
(which is also called the scalar constraint) H = is not Diff(£)-
invariant. This constraint, or more precisely its Hamiltonian flow,
generates a gauge motion which can be identified with the evolu-
tion generated by vector fields orthogonal to the spatial surfaces
s t .
(iii) The relation (2.80c) implies that the Dirac algebra (2.80) is not
a Lie algebra in the strict sense. Although the right-hand side of
this equation is proportional to the diffeomorphism constraint, the
coefficients are not constants but have a highly non-trivial phase-
space dependence through the metric tensor h a p(x, t). This feature
is not a problem in the classical framework, but is one of the key
difficulties in constructing a quantum theory of the gravitational
field in the canonical framework (see Sees. 10.1 and 12.1).
Let us make some considerations on the formulation described above.
The Hamiltonian of the theory in Eq. (2.75) is not a standard Hamilto-
nian but a linear combination of constraints. From Eqs. (2.78) and (2.79),
it is possible to show that rather than generating time translations, the
Hamiltonian generates space-time diffeomorphisms, whose parameters are
the completely arbitrary functions N and N a , and the corresponding mo-
tions on the phase-space have to be regarded as gauge transformations.
An observable is defined as a function on the constraint surface that is
gauge invariant; more precisely, in a system with first class constraints an
observable can be described as a phase-space function that has weakly van-
ishing Poisson brackets with the constraints. In our case, O is an observable
if and only if
{O,H(A,A Q ,A Q ,A)}«0, (2.81)
for generic A, A", N a and N. By this definition, one treats on the same
footing the ordinary gauge invariant quantities and the constants of motion
with respect to the evolution along the foliation associated to N and N a .
The basic variables of the theory, h a p and H al3 , are not observables as
they are not gauge invariant. In particular, no observables for GR are
known, except for the particular situations characterized by asymptotically
flat boundary conditions. Let us remark that the equations of motion
*-<*"- jnlib' ^-''--raj' (2 - 82)
together with the eight constraints (2.70) and (2.77) are completely equiv-
alent to the Einstein equations in vacuum given by Rij = 0.
70 Primordial Cosmology
Let us finally remark that the canonical framework is manifestly gener-
ally covariant since it is faithfully represented in terms of the Dirac algebra
(2.80) . One never uses, in the Hamiltonain construction, the notion of a
background metric, and the space-time diffeomorphisms invariance is not
violated at any point. Although a splitting between space and time is
performed, all the splittings are simultaneously considered (this feature is
reflected by the presence of constraints) and the diffeomorphisms invariance
is preserved. Such invariance should not be confused with the Poincare one.
The Poincare invariance is not a gauge symmetry of GR and refers only to
a special solution (the flat solution) of the vacuum Einstein equations. The
gauge group of the theory is Diff(A^), which is background independent
since it needs a differential manifold M rather than a metric one (M,gij).
Only when the space-time manifold is equipped with asymptotically flat
boundary conditions, the Poincare group (and its generators) can be prop-
erly defined.
2.3.2 Hamilton- J acobi equations for gravitational field
The formulation of the Hamilton- Jacobi theory for a covariant system is
simpler than the conventional non-relativistic version. In fact, in such case
the Hamilton-Jacobi equations are expressed as
«(«..|;) =0, (2-83)
where % and S(q a ) denote the Hamiltonian and the Hamilton function,
respectively. For GR, the Hamilton-Jacobi equal ions arising from the super-
Hamiltonian and super- momentum read as
- 1^- 3 R = (2.84a)
Hf a S* = -2/i Q7 V^-p^ = 0. (2.84b)
These four equations, together with the primary constraints (2.70), com-
pletely define the classical dynamics of the theory.
Let us point out how, through a change of variable, we can define an
internal time-like coordinate. Writing h a p = T 4 ^ 3 u a p, with r = h l / A and
&etu a f) = 1, from the scalar Hamilton-Jacobi Eq. (2.84a), the following
relation stands
3 (5S\ 2 2 5S SS 2/ ~
16 V or J t z ' 5u a fj du^s
Fundamental Tools 71
where the potential term V = V(u a p, Vr, Vm q ^) comes out from the spatial
Ricci scalar and V refers to spatial gradients only. As we can see from
Eq. (2.85), t has the correct signature for an internal time-like variable
candidate; when dealing with cosmological settings, this variable turns out
to be a power of the isotropic volume of the Universe (see Chap. 3).
2.3.3 The ADM reduction of the dynamics
The ADM reduction of the dynamics relies on the possibility of identifying
a temporal parameter as a functional of the geometric canonical variables.
Let us enumerate the degrees of freedom of the gravitational field. There
are 20 phase-space functions, given in the 3 + 1 formalism by the set (A, II),
(N a ,U a ) and (h a p, IP' 3 ), subjected to eight first-class constraints (II =
0,II Q = 0,H = 0,7i a = 0). Since each first-class constraint eliminates
two phase-space variables, we remain with four of them, corresponding to
the two physical degrees of freedom of the gravitational field, i.e. to the
two independent polarizations of a gravitational wave in the weak field
limit. Apart from A and N a (and their vanishing momenta II and II Q ), we
deal with 12 x oo 3 variables (h a p(x, t),H a ^(x, t)). We can remove 4 x oo 3
variables by means of the secondary constraints (2.77). The remaining
4 x oo 3 non-physical degrees of freedom, in analogy with the Yang-Mills
theory, must be eliminated by imposing some sort of gauge on the lapse
function and on the shift vector.
This procedure can be implemented in three steps.
(i) Perform a canonical transformation
(^.r^^P^f,^), (2.86)
where A = 1,2,3,4 and r = 1,2. Here \ defines a particular
choice of the space and time coordinates, 3 Pa are the corresponding
canonically conjugate momenta and the four phase-space variables
(4> r , 7r r ) represent the physical degrees of freedom of the system. We
emphasize that these "physical" fields are however not Dirac ob-
servables, in the sense defined above. Consequently, the symplectic
structure of the theory is determined by
{x A (x,t),P B (x',t)} = 6p 3 (x-x'), (2.87a)
{<j> r (x,t),ir s (x',t)} = 6 r s 6 3 (x-x') (2.87b)
3 These fields can be interpreted as defining an embedding of S in A4 via some para-
metric equations.
Primordial Cosmology
while all other Poisson brackets do vanish,
(ii) Express the super-momentum and super-Hamiltonian in terms of
the new fields, and then write the Lagrangian density as
£' 3+1 (N,N a ,x A ,P A ,<f> r ,ir r ) = P A dtX A + v r d t <f - NH' - N a U' a .
(2.88)
(iii) Remove 4 x oo 3 variables arising from the constraints H = and
H a = by solving the equations
P A (x,t) + h A (x,t; X ,4>,ir)=0 (2.89)
with respect to P A and by inserting them back in Eq. (2.88). After
removing the remaining 4 x oo 3 non-dynamical variables we obtain
the so-called reduced Lagrangian density
£red = TTrdt^ - h A d tX A , (2.90)
where the lapse function and the shift vector do not play any role,
but only specify the form of the functions d t X A ■ Once the con-
straints are solved, the evolution of \ A is n °t related anymore to
the parametric time t. Thus, in Eq. (2.90) we can choose the con-
ditions x \ x -, t) = A'/H-*')- oblainiu"'," reduced Ramiltonian, i.e.
H rcd = J d tX
%X A h A (xu4>,ir)d s x. (2.91)
From Eq. (2.91) one can derive the equations of motion as
d t <f> r = {4> r ,U led }^, (2.92a)
d t n s = {TT s ,H, cd } < p, 7r (2.92b)
where the notation {. . .}^ j7T refers to the Poisson brackets evaluated
in the reduced phase space with coordinates given by the physical
fields <jf and tt s only.
This is an operative prescription for solving the constraints on a classical
level, pulling out all the gauges, and obtaining a canonical description for
the physical degrees of freedom only. It is worth noting that such procedure
violates the geometrical structure of GR, since it removes part of the metric
tensor. Even if this is not a problem at a classical level, it poses several
questions when implemented at the quantum one (especially in the reduced
phase-space quantization).
Fundamental Tools
2.4 Synchronous Reference System
In this Section we will focus our attention on one of the most interesting
reference systems, i.e. the synchronous one. This reference is defined by
the following choices for the metric tensor g t j
9oo = 1 , (2.93a)
g 0a = 0, (2.93b)
and thus, in the canonical framework, the conditions N = 1 and N a =
in Eq. (2.64) have to be taken into account. The condition in Eq. (2.93a)
is allowed by the freedom to rescale the variable t with the transformation
^Jgoodt, in order to reduce 500 to unity and setting the time coordinate x° =
t as the proper time at each point of space. The condition (2.93b) is possible
because of the non- vanishing h = det(/i a/ g) and allows the synchronization
of clocks at different points of space. The corresponding line element is
then provided by the expression
ds 2 = dt 2 - h a p (x, t) dx a dx li (2.94)
and in such reference system the time-like curves along the ^-direction result
to be geodesies of the space time. Indeed the four- vector u l = dx l /ds, which
is tangent to the t- lines, has components u° = 1, u a = and automatically
satisfies the geodesic equation
^ + r>V = r* = o. (2.95)
The choice of such reference is always possible and is not unique. Let us
consider a generic iuUuUoshnal displacement, i.e.
t' = t + Z(x,t), x a ' = x a +i a {x,t). (2.96)
It is easy to show that, if
«9 t £ = 0^' =£ + £(£") (2.97a)
d a £ = => x' a =x a + d^ I h al3 dt , (2.97b)
then the new four-metric tensor g'-- = g, r) — 2V(,£ ; ). satisfies Eq. (2.94) (the
round brackets around indices imply a symmetric linear combination).
In this reference frame, the Einstein equations in mixed components
read as
Rl = ^-KJ - K 5 a K a s = K ( T ° - ]-T\ (2.98a)
R° a = \7 a K s s - V 7 i^2 = kT° (2.98b)
74 Primordial Cosmology
The extrinsic curvature- A' (v£ j (2.66) explicitly reads as K a p = —d t h a p/2,
while 3 Rp is the three-dimensional Ricci tensor obtained from the metric
h a p and stands, in terms of the spatial Christoffel symbols F?g, as
3 R aP = d^lp - d a t s ps + r^f £ A - f £ t f £„ (2.99)
rI/3 = \h l5 {d a hsp + d p h aS - d 5 h a p) . (2.100)
In this analysis the spatial metric is used to raise and lower indices within
the spatial sections. From Eq. (2.98a) it is straightforward to derive, even
in the isotropic case, the Landau-Raychaudhuri theorem, stating that the
metric determinant h must monotonically vanish in a finite instant of time.
However, wc want, to stress that the singularity in this reference system is
not physical and can be removed by a coordinate transformation.
2.5 Tetradic Formalism
The tetradic formalism consists in replacing the metric tensor <?y with four
linearly independent covariant vector fields e\ = ej(x k ). Let (M,gij)be the
space-time four-dimensional manifold and e a one-to-one correspondence
on it, i.e. e : M — > TM. More precisely, e maps tensor fields on M
to tensor fields on the Minkowski tangent space TM. The four linearly
independent fields e\ (tetrads or vierbein 4 ) are an orthonormal basis for
the local Minkowski space-time and satisfy the only condition
e\e i j = r II j, (2.101)
rjij being a symmetric, constant matrix with signature (+,—,—,—). Let
us introduce the reciprocal (dual) vectors e l1 , such that eje l j = 5j. By
definition of e\ and by Eq. (2.101), the condition ejeJj = 5f is also verified.
This way, Lorentzian indices are lowered and raised by the matrix rju, and
the vector fields ej are related to the metric tensor g^ by the relation
9lJ {x k ) = mje\(x k )e](x k ) . (2.102)
From this perspective, the gravitational field is a 1-form e 1 = e\dx % with
values i.ii I he Minkowski space-time. From a physical point of view, the
tetrad e\ describes the departure of a space-time manifold from being flat.
The projections of a vector field A 1 along the four e\ are denoted as
"vierbein components" and read as A] = e\Ai and A 1 = ejA 1 = r] IJ Aj. In
4 Capital latin letters denote Lorentzian indices.
Fundamental Tools 75
particular, for the partial differential operator we have di = e)di. The gen-
eralization to a tensor of any number of covariant or contravariant indices
is straightforward.
Using the tetradic fields ef we can rewrite the Lagrangian of the Einstein
theory in a more elegant and compact form. Firstly, we notice that the
tetradic fields ef define a connection u> IJ = —to JI . This is a 1-form, known
as spin connection, with values in the Lie algebra of the Lorentz group
50(3, 1) and uniquely determined by the II Cartan structure equation. In
the torsion-free case, this equation reads as
Tl = %4 + u^ = 0, =► u = w(e) , (2.103)
where the (Lorentz algebra valued) 2-form T 1 = 7~j dx l dx^ denotes the
torsion field. Equation (2.103) admits the following solution
u{ J = e /j V,e/ . (2.104)
Let us introduce the curvature of the spin connection R ij (lu), a Lorentz
valued 2-form defined by the I Cartan structure equation, i.e.
R IJ tj M = d {i uj]{ + J {lK ojf , (2.105)
where the anti-symmetrization regards only the spatial indices and R IJ (w)
is related to the Riemann curvature tensor by
R i j ki(9) = eiejjR IJ ki(")- (2-106)
The action (2.11) of GR, in the absence of matter fields, can be recast
in the form
S g (e) = -^ [ ee)e j J R IJ ij (e)d i x
^K J M
(2.107)
where e = \J—g denotes the determinant of ef. When a theory depends
only on the metric gij or on tetrads ef, as in this case, we deal with the
so-called second-order formalism.
As is well known, GR also admits a first-order formulation (d la Palat ini)
in which the tetrads ef and the spin connection u>f J are considered as
independent variables. The Einstciii-Hilbcri action then reads as
S P (e,u) = ~ I ee\e j J R IJ lJ (u)d 4 x. (2.108)
The variation of Eq. (2.108) with respect to the connection ojf J gives the
II Cartan structure equation (2.103), and thus the second-order formalism
is recovered. It is worth noting that in the presence of matter, however,
the two formalisms are not equivalent if the Lagrangian of the matter fields
sr 29, 2010 11:22
76 Primordial Cosmology
contains connections (for instance, fermion fields). Notably, the action
(2.108) is invariant under space-time diffeomorphisms of M as well as local
50(3, 1) (Lorentz) transformations.
Variation of the action (2.108) with respect to the gravitational field e 7
leads to the Einstein equations in vacuum
ijf-iiJef = 0, (2.109)
where the Ricci tensor Rj is defined as R( = R IJ ij e j J , while R = Rfe)
denotes the Ricci scalar. In the presence of matter, the Einstein equations
in the tetradic formalism read as
R{ - -Re{ = kT^, (2.110)
T/ = Tije I: > being the tetradic projection of the energy- momentum tensor
(2.13). From a geometric point of view, we are dealing with a Lorentz
vector bundle over the space-time manifold where the spin connection uj( j
is the connection on the bundle.
The analysis of the tetradic formalism is completed by introducing the
Ricci coefficients ^ijk = —Jjik and their linear combinations Xijk =
-Xikj, i.e.
-rijK = V k e Ii J I e k K, (2-Hla)
Xijk = lux - Iikj ■ (2.111b)
The Riemann and the Ricci tensors can be expressed in terms of Jijk and
of Xijk as
RlJKL = d L "f U K ~ d K lUL + 1IJM {l M KL ~ J M LKJ , ,
M M l/.llzaj
+ lIMKl JL ~ lIMLl JK i
Ru = -\(d K X u K + d K Xj! K + djX K Ki+
+d I X K KJ + X KL jX KLI + X KL jX KLI + (2.112b)
- -^j KL ^ikl + X K kl Xjj L + X K kl Xjj L J .
Finally, the relation between the spin connection bj\ J and the Ricci coeffi-
cients is given by
Fundamental Tools 77
2.6 Gauge-like Formulation of GR
This Section is devoted to the analysis of the more recent formulation of the
Einstein theory, due to Ashtekar (and generalized by Barbero and Immirzi).
Such formulation reveals a structural identity between GR and Yang-Mills
theories and finds the most important application to quantum gravity, since
it opens the possibility of using the Wilson loops technique to quantize the
gauge fields also in the case of gravity. As a matter of fact, this new formal-
ism leads to the Loop Quantum Gravity theory, which can be considered as
the most advanced implementation of the canonical approach to quantum
gravity and will be discussed in Sec. 12.1.
Both the Lagrangian and the Hamiltonian formulations are restated in
details through this Section also paying attention to recent debates on this
approach.
2.6.1 Lagrangian formulation
As we have seen in Sec. 2.2.4, in Yang-Mills theories it is possible to add a
topological term to the action which does not change the classical equations
of motion because its integrand can be expressed as a total derivative of
a 3-form and this property holds also in the gravitational case. Let us
consider the integral
5 TT (e, W ) = -L f d^xee^j *R IJ tj (u;)
1 ,. UM (2.114)
— __ / ^4„,i ?' IJ r>KL I, n
where sijkl is the Levi-Civita tensor on the tangent space and * is the
Hodge dual operator defined in Eq. (2.54). Stt is a topological term, but
in a sense weaker than the Yang-Mills case, as it identically vanishes on
the histories (trajectories) where the II Cartan structure equation (2.103)
holds. In fact, its integrand is equal to zero because of the Bianchi cyclic
identity (2.7) Ri[jki] = and then
eeU^e^KLR^^) = ef ef e* kl R ijKL (oj) = e^ kl R ijkl {e) . (2.115)
In the last equality of Eq. (2.115) the spin connection (2.104) is taken into
account, and therefore this is only true when the II Cartan structure equa-
tion (2.103) holds, and therefore the second-order formalism is restored.
This way, Stt (2.114) can be added to the Palatini action (2.108) without
78 Primordial Cosmology
affecting the equations of motion, obtaining the Hoist action
S K (e,ui) = S P + S T T = -^J dtxeeietj (r ij t] -- * R IJ ^ .
(2.116)
The coupling constant 7 7^ is called the Immirzi parameter and does not
affect the classical theory. Indeed, the Yang-Mills and the Hoist gravita-
tional theories present, in some respects, the same features. In both cases
it is possible to add a term that does not change the equations of motion
but induces a canonical transformation in the classical phase-space that
cannot be unitarily implemented at a quantum level (see Chap. 12). The
quantum theory built on the Hoist action (2.116), i.e. the Loop Quantum
Gravity theory, has inequivalcnt 7-seetors resembling the inequivalent 6-
sectors in the Yang-Mills one (see Sec. 2.2.4). The Immirzi parameter can
be considered, in this respect, as the analogous of the 6-angle in QCD.
Before analyzing the canonical formulation of this theory, let us discuss
the relevant cases 7 = ±i leading to the original formulation of GR proposed
by Ashtekar in 1986. A generic tensor T IJ is called self-dual (respectively
anti-self-dual) if it satisfies
T IJ = =R * T IJ = T -e IJ KL T KL . (2.117)
When the Immirzi parameter is fixed to 7 = — i, we are naturally led to
consider as basic connections the complex quantities
A\ J {uj) = uj\ j -i*u{ J , (2.118)
instead of the spin connections uj( j . The new variables A{ j {lo), called
self-dual spin connection, are the Ashtekar connections. If F IJ ij(A) is the
curvature 2-form (2.50) of the self-dual spin connection A\ J , i.e.
F IJ tJ (A) = d {i A]{ + A\ iK Af , (2.119)
it is not difficult to show that F IJ ij(A) is related to the curvature R IJ i:j (uj)
of the spin connection uj\ j by
F IJ ij(A) = R IJ l3 - i * R IJ i:i . (2.120)
The curvature of the Ashtekar connections, i.e. the Yang-Mills field strength
of A\ J , is just the self-dual part of the spin-connection curvature. This
quantity is exactly the term in brackets in Eq. (2.116) in the 7 = — i case.
Considering the change of coordinates from (ej.ujf J ) to (ej,Aj J ), the
action in Eq. (2.116) rewrites as
S H (e,A) = -^- [ d A xee\eijF IJ l0 {A). (2.121)
*K J M
Fundamental Tools
We can consider Sn(e,A) as the starting point for the Ashtekar gravita-
tional theory. The equations of motion which follow are given by
e ijkl ejjF IJ k i = 0, (2.122a)
f 6 iK 6 JL + l - e KLIJ \ e ijkl V k (e iiej j) = , (2.122b)
where T>i denotes the covariant derivative (2.48) defined by the connec-
tion in Eq. (2.118). Let us stress once again that this formalism is com-
pletely equivalent to the Einstein formulation of GR. Indeed, if the couple
(e{(x k ), Aj J (x k )) satisfies the equations of motion (2.122), then the metric
tensor (2.102) is a solution of the vacuum Einstein equations. The inverse
statement is also true.
By means of Eqs. (2.122), the Ashtekar connection A\ J (x k ) results in
the self-dual part of the spin connection defined in Eq. (2.104), i.e.
A\ J {e) = e Ij V ie j - % - e IJ KL e Kj Vief . (2.123)
The geometric interpretation of this framework is the following: the com-
plex Lorentz group (and also its algebra) splits into two complex 50(3; C)
groups, the self and anti-self dual ones. More precisely, there exists an iso-
morphism between the direct sum of these two reduced algebras and the
original complex Lorentz algebra, i.e.
80(1,3; C) = so(3; C) © so(3; C). (2.124)
The connection on the SO(l, 3; C) vector bundle over the space-time man-
ifold splits into two independent components, the self dual and the anti-
self-dual. These are independent since the self-dual part of the curvature is
the curvature of the self-dual connection, i.e. Eq. (2.120) holds. It is worth
stressing that the difference between the self-dual curvature and the real
one is nothing but the topological term Stt (2.114). Since the complexifi-
cation of the Lorentz algebra decomposes as in relation (2.124), not all the
components of the Ashtekar connection (2.118) are independent. In order
to deal with real rather than complex GR, one has to impose the reality
condition
A\ J + (A^)l J = 2w\ J . (2.125)
Although the original Ashtekar connection has a clear geometric interpre-
tation (see below in Sec. 2.6.3), it takes values in the Lie algebra of a
non-compact group (namely the Lorentz group).
80 Primordial Cosmology
The most relevant result in this field has been obtained adopting real,
rather than complex, connections, the so-called Barbero-Immirzi connec-
tions
A{ j {lo)=lo i 1 j --*u( J (2.126)
7
defined for real values of 7. As we said, the 7 parameter induces a canonical
transformation of the form (2.126) which, in general, is a vector space
isomorphism on the Lorentz group. In the particular cases of 7 = ±i, this
map is a Lie algebra homomorphism.
2.6.2 Hamiltonian formulation
As in the metric case analyzed before, the starting point of the Hamiltonian
analysis of the Hoist theory is the 3 + 1 splitting of the space-time manifold
M. As usual, we assume that the space-time is globally hyperbolic and
that, from the Geroch theorem, can be foliated as M = M x £. Here it is
convenient to carry out a partial gauge fixing. Let the internal vector field
n 1 orthogonal to the spatial Cauchy surfaces £ be defined by wJni = 1 and
n 1 'e a j = 0. The time component of the tetrad field ef can be written as
e^ = Nn I + N a e I a . (2.127)
The gauge which is normally adopted is the time-gauge, i.e. the tetrad is
chosen such that
n 1 = (1,0,0,0). (2.128)
This choice implies that the spatial components of the tetrad e" (denoted
with small latin letters) span the space tangent to the Cauchy surfaces and
that e° = 0. The splitting reduces the tetrad fields e\ and their inverse e J j
to the following form
I _(NN«el\ ( AT 1 0\
Because of the adopted time-gauge, the boost sector of the (local) Lorentz
group is frozen out and the Lorentz invariance reduces to a local 50(3) ~
SU(2) invariance. The resulting implications will be discussed below.
Because of the splitting, the spin connection bj\ J defines two so(3)-
valued 1-forms on the spatial surfaces (whose metric h a p in Eq. (2.64) is
given by h a p = 5 a be%e b p) which read as
T^ = e ai e Q/ (*w/ J )nj, (2.130a)
K a a = e ai e aI wl J nj. (2.130b)
Fundamental Tools 81
These quantities have a natural geometric interpretation. T^ denotes a
so(3)-connection on E and furthermore, if the spin connection is a solution
of Eq. (2.103), T° satisfies the II Cartan structure equation induced on the
spatial surface: in this case, T° is said to be compatible with e a a . On the
other hand, K% = e al3 K a p stands for the extrinsic curvature 1-form. It is
the Lie derivative of h a p with respect to the normal vector to the spatial
slice and thus can be written as
K a a = {L R h aP )5 ab e li b . (2.131)
The next step is to consider the linear combination of the 1-forms in Eqs.
(2.130) as
Al=T°+>yK°, (2.132)
which is again a connection on E taking values in the Lie algebra so(3)
(namely, su(2)).
Let us now take into account the densitized triads E£ related to the
three-metric h a p by
K = \ £ aM e abc e b p e c 1 = ee a a =^\h\e a a . (2.133)
This is a vector density of weight 1 on E which takes values in the dual
of so(3). The densitized triads (2.133) carry information about the spatial
geometry (encoded in the three- metric), while the connections (2.132) de-
scribe the spatial curvature through the spin connections and the extrinsic
curvature. The most peculiar feature of this framework is that the two
quantities in Eqs. (2.132) and (2.133) span the phase space of GR. In fact,
they are canoiiicalh conjugate fields whose Poisson '.Markets read as
{A a a {x, t),E^(x', t)} = kS^6^S 3 (x - x 1 ) . (2.134)
The phase-space exactly resembles the one of a Yang-Mills theory, with
SU{2) as gauge group. Following the conventions of gauge theory, we can
call E" as the gravitational electric field since it is the momentum canoni-
cally conjugate to the connection A^, which is the configuration field of the
theory.
Given all this apparatus, it is now possible to rewrite the Hoist action
(2.116) in the appropriate canonical form, although we will skip the explicit
computation which is straightforward but rather tedious (the interested
reader is referred to the literature). The result is given by the 3 + 1 action
S H = - I dt f d 3 x [A%E% - (A^G a + N a H a + NTi)] , (2.135)
82 Primordial Cosmology
where A%, N a and N are the Lagrange multipliers. Let us note that Aq
results to behave as a multiplier according to gauge theories (see Sec. 2.2.4).
The term in brackets denotes the Hamiltonian (density) of GR which, as
in the metric case, is a linear combination of constraints. The diffeomor-
phisms (H a = 0) and scalar (H = 0) constraints rewrite, in the connection
formalism, as
■H a = E f iF^-(l+ 1 2 )K^G a , (2.136a)
n = ^f "^ (^^ - 2(1 + t 2 )^,)
+(1+72 K71K (2 - 136b)
respectively, and
F^ = 2d [a Af 3] + ~ie a bc A b a A c (2.137)
denotes the components of the curvature 2-form associated to the connec-
tion A^. In this formalism, a new constraint G a = arises with respect to
the metric approach and explicitly reads as
G a = V a E% = d a El - je ab c A b a E* = 0, (2.138)
and it is the analogous of the Gauss constraint (2.53) of Yang-Mills theories
which gets rid of the SU(2) degrees of freedom. If we ignore the two con-
straints (2.136) we deal with a SU{2) gauge theory. However, differently
from the Yang-Mills case, the Hamiltonian of GR is a linear combination
of constraints. In such new reformulation, the Einstein theory can be re-
garded, as a background independent SU{2) gauge theory.
In order to illustrate the meaning of the two constraints (2.136) and their
consequences, the terms proportional to G a generating internal rotations,
will be removed. As a matter of fact, since G a generates a sub-algebra of
the constraint algebra, the system described by Eqs. (2.136) without the
G a term, defines the same constraint surface in the phase space. Thus, it
is completely equivalent to work with the set of constraints
n a = El F^p = , (2.139a)
H = -A=Ktf U ab c F^ - 2(1 + 7 2 )* [a^]) = , (2.139b)
2y/\tl\
together with Eq. (2.138). As in the metric case, Eq. (2.139a) generates
spatial diffcomorpliisins along the vector field N a on the Cauchy surface
E, while Eq. (2.139b) generates the time evolution off S. These constraints
Fundamental Tools 83
exactly reflect the gauge freedom of the physical theory, in particular the
internal automorphism of the SU(2) gauge bundle and the diffeomorphism
invariance of the space-time, so constraining the system on a restricted
region of the phase space. Furthermore, a direct calculation of the Poisson
algebra between the constraints shows that Eqs. (2.139) and (2.138) are of
first class. Because we are dealing with a canonical transformation, such
constraint algebra coincides with the Dirac one (2.80) on the sub-manifold
G a = of the phase space.
Two remarks are in order.
(i) The Haniiltouia.n (scalar) constraint Eq. (2.139b) has an important
peculiarity: considering the 7 = ±i case, it reads as
after a rescaling of the factor 1/2 y|ft|. In other words, it becomes
polynomial and a huge simplifical ion occurs as soon as the original
Aslitekar variables (defined on the slicing, surface E)
A a a =Y a a ±iK a a , EP = ee%, (2.141)
are taken into account. Most of the initial excitement over the
Ashtekar discovery was exactly due to such feature. The price one
has to pay using these variables is that they are complex valued.
When the 7 parameter is real, A a a and E^ are both real valued and
can be directly interpreted as the canonical pair for the phase space
of a SU(2) gauge theory. On the other hand, when 7 is complex,
some reality conditions have to be imposed (see Eq. (2.125)) and
A a a + (A*)% = 2T a a , El + (E^f h = (2.142)
and guarantee that there is no doubling of the number of degrees
of freedom. This way, only SU(2) gauge transformations are al-
lowed but not general 5 SL(2,C) transformations. However, the
reality conditions (2.142) are non-polynomial and thus difficult to
implement in the quantum theory. Of course, these two reality
conditions are trivially satisfied when 7 is real, i.e. when we deal
with the real Barbero-Immirzi connection.
5 The complexification of the Lorentz group can be identified with its universal cover
SL(2,C).
Primordial Cosmology
(ii) All complex values of the Immirzi parameter 7 lead to Hamiltonian
formulations completely equivalent to the ADM formulation. In
fact, the framework described above can also be obtained from the
metric one by the use of a canonical transformation. Such construc-
tion consists of two steps: an extension of the ADM phase-space
passing through the tetradic formalism and a canonical transfor-
mation on such extended phase space. The parameter 7 enters in
this second step as a rescaling of the conjugate variables K^ and
E^. This way. as far as „S7 r (2) invariant observables are concerned
(i.e. considering the symplectic reduction with respect to the Gauss
constraint (2.138)), both the ADM and the gauge formulations are
completely equivalent to each other.
2.6.3 On the gauge group of GR
Let us now focus our attention on the (internal) gauge group of GR. Because
of the nature of the Lorentz and Poincare groups, it is generally argued that
the (local) gauge group of GR must be a non-compact one. The puzzle is
that, as we have seen, the Barbero-Immirzi Hamiltonian formulation uses a
real and compact gauge group, i.e. the SU{2) one. In order to investigate
this issue, the geometrical meaning of the connection defined in Eq. (2.132)
has to be analyzed.
We want to discuss the conceptual differences between the real and the
complex valued connections. Both are sw(2)-valued connections and the re-
lation with the metric variables has the same form in both cases. Nonethe-
less, an important difference occurs: only the Ashtekar connection (2.141)
is the (anti)self-dual piece of the pull-back to £ of the four-dimensional
spin connection uj( j and then has a covariant interpretation. In all other
cases (7 real), this is not true. The manifestly covariant origin of the
phase-space spanned by the Barbero-Immirzi connection is lost due to the
(partial) gauge fixing of the Lorentz group provided by the time gauge. Un-
less 7 = ±i, the Hoist action (2.116) leads to constraints of second class, i.e.
constraints which are not generators of gauge transformations. These con-
straints are solved by imposing the time-gauge which eliminates the boost
component of the Lorentz group leading to the 50(3) (or SU(2)) sub-group.
However, this reduction does not pose any difficulty. This criticism is only
of aesthetic nature since we are not interested in non-gauge-invariant ob-
jects; there will be no lack of it at quantum level. A space-time geometry
is the analogous of a trajectory in particle mechanics and trajectories do
Fundamental Tools
not play any essential role in quantum mechanics.
2.7 Singularity Theorems
In this Section we investigate the space-time singularities through the cel-
ebrated theorems given by Hawking and Penrose at the end of the '60s.
We will show that singularities are true, generic features of the Einstein
theory of gravity and how they arise under certain, quite general, assump-
tions. These theorems are of fundamental importance since they state that
GR has a limited range of validity out of which quantum gravity effects
could be required. In this respect, the initial cosmological singularity at
the beginning of our Universe is expected to be tamed by quantum proper-
ties, similarly to the instability problem of a classical hydrogen atom which
is solved by the existence of a finite energy ground-state of the electron.
The prediction of space-time singularities in GR implies the necessity to
work out a quantum theory of gravity able to solve such unphysical predic-
tions. As we will see in Sec. 12.2, Loop Quantum Cosmology faces exactly
this problem replacing the Big Bang of the Universe by a non-singular Big
Bounce.
After the definition of a space-time singularity, we will present some ba-
sic techniques and finally we will discuss the singularity theorems; we enter
the main aspects only without giving rigorous proofs, for which we refer
the reader to the original works. In this Section, we adopt the signature
(— , +, +, +) for coherence with the standard literature on this subject.
2.7.1 Definition of a space-time singularity
Let us clarify the meaning of singularity of a space-time. In analogy with
field theory, we can represent such a singularity as the ■■place" of the space-
time where the curvature diverges, or where some similar pathological be-
havior of the geometric invariants takes place. The characterization as
place, however, poses several problems: since in GR the space-time consists
of a manifold M and a metric g t j defined everywhere onX, a singularity
(as the Big Bang singularity of the isotropic cosmological solution or the
r = singularity in the Schwarzschild space-time) cannot be considered as
a part of the manifold itself. We can speak of a physical event only when a
manifold and a metric structure are defined around it. A priori, it is pos-
sible to add points to the manifold in order to describe the singularity as a
86 Primordial Cosmology
real place (as the boundary of the manifold), but apart from very peculiar
cases, no general notion or definition of a singular boundary exists. An-
other problem is that singularities in gravity are not always accompanied
by unbounded curvature as in the best known cases. Several examples of
singularities without diverging curvature can be given. In fact, as we will
see, this feature is not the basic mechanism behind singularity theorems.
The best way to clarify what a singularity means is the geodesic incom-
pleteness, i.e. the existence of geodesies which are inextensible at least in
one direction and thus have only a finite range for the affine parameter.
We can then define a singular space-time as the one possessing at least one
incomplete (time-like or null) geodesic curve.
2.7.2 Fluid kinematics
Now we will provide the reader with some basic notions of fluid kinematics.
We initially define the notion of congruence.
Definition 2.1 (Congruence). Let O be an open set of a space-time
manifold M. A congruence in O is defined as a family of curves such
that only one curve of this family passes through each point p G O.
There exist diiforoul lands of congruences, resembling the properties of the
tangent vector field £ l associated to the family of curves. In particular,
time-like, null, or space-like congruences are generated by nowhere van-
ishing time- like, null, or space- like vector fields £', respectively; geodesic
congruences are generated by vector fields which have 1 vanishing covariant
derivative £ J 'Vj£* = 0. Now we will be interested in time-like congruences,
while in the next subsection we will be dealing with geodesic congruences.
The treatment of the remaining congruences is conceptually similar and not
discussed here.
Given a unit time-like vector £* (i.e. f£j = —1), we can construct a
tensor hij that projects the other tensors onto their orthogonal components
hi = ga + tit, . (2.143)
Let us analyze the covariant derivative of £ 8 . From the relation
(eVjVZi + ViC) £' = (2.144)
it follows that the term in parentheses is orthogonal to £', i.e.
£*V fc && + Vj-& = tffijVkb = % + Uij . (2.145)
Fundamental Tools 87
In the last equality, we have introduced the so-called expansion tensor Oij ,
corresponding to the symmetric part, and the rorl/cilt/ levsor Wy, corre-
sponding to the antisymmetric one. We can also define the acceleration
vector £i = tfVjZi. If we further decompose Oij in a term proportional to
its trace (the expansion scalar) plus a trace-less part (the shear tensor
Ojj) its
Oij = -dhij + on , (2.146)
we arrive at the so-called kinematical decomposition, that explicitly reads
Vi& = fhj + o-ij + ojij - tej . (2.147)
It is worth noting that all the tensors defined above are orthogonal to the
vector field £* because they are constructed from the projection tensor hij .
Because of the physical meaning of 0, an average length scale L{t) can
be defined along the fluid flow lines by the equation L/L = 0/3, so that the
volume SV of any small fluid element evolves like L 3 along any flow line
and it is easy to obtain the equation
• 1 o l
(2.148)
Finally, the magnitude of o~ij and Wy are given respectively by
a 1 = -o-ijO- ij > , lu 2 = -uj l:j uj ij > . (2.149)
These quantities vanish if and only if the corresponding tensors vanish, i.e.
a 2 = O o-ij = and lu 2 = O w y = O %<9/]6 = 0. The last
equivalence implies that the vorticity w, 3 vanishes if and only if the flow
vector field £* is orthogonal to a family of hypersurfaces of the space-time.
The evolution equations for 0, o~ij and u>ij follow from the geodesic equation.
If we define B i3 = V^j, then we have that (y = £ k V k y)
i k v k B tJ = Bij = i k v k v t i-j = i k {v l v k i + R kij l b)
= v 4 (e fc v fc o) - (v^ fe )(v fc o) + Rk^ 'e fc 6
= Viij - B z k B k3 + R kij 'e fc 6 • (2.150)
In the simpler case of vanishing acceleration vector £*, we have that taking
the trace, the symmetric trace-free part, and the antisymmetric one of
Eq. (2.150), we obtain
±0 2 -2(a 2 -oj 2 )-R ki ee,
Primordial Cosmology
-an - {uiia\ + a a a\) - R lmjn CC
"« = - T «« - M™. « im <r-.) • (2.151c)
Equation (2.151a), known as the Raychaudhuri equation, is of funda-
mental importance in proving the singularity theorems and it will be ana-
lyzed in the following subsection.
2.7.3 The Raychaudhuri equation
Let us specialize the treatment discussed in the previous subsection to the
case of a geodesic congruence, and let us focus our attention on the right-
hand side of Eq. (2.151a): using Einstein equations, the last term can be
written as
r^H 1 = k (th - \t 9 ^ ee = k (r^ee + \t\ . (2.152)
Let us assume a physical criterion in order to prevent the stresses of matter
from becoming so large to make the right-hand side of Eq. (2.152) negative,
T l3 Ce>-\T. (2.153)
This condition is known as the strong energy condition and it is commonly
expected that every reasonable kind of matter should satisfy such condition.
From the Raychaudhuri equation (2.151a) one can see that, if the congru-
ence is non-rotating (w, 3 = 0) and the strong energy condition holds, 9
always decreases along the geodesies. More precisely, we get
<9+V<0, (2.154)
whose integral implies
9- 1 (t)>9o 1 + \t, (2.155)
where 9 is the initial value of 9. For negative values of 9 n (i.e. the congru-
ence is initially converging), 9 will diverge after a proper time not larger
than r < 3/|#o|- In other words, the geodesies must intersect before such
instant and form a caustic (a focal point). Of course, a singularity of 9 is
Fundamental Tools 89
nothing but a singularity in the congruence and not a space-time one, since
the smooth manifold is well-defined on caustics.
To get insight into the strong energy conditions, consider the simple
case of the perfect fluid as in Eq. (2.20). Thus Eq. (2.153) reads as
p + 3P>0, p + P>0, (2.156)
and it is satisfied for p > and for a negative pressure component smaller
than p in magnitude.
To translate the occurrence of caustics into space-time singularities, we
need to introduce some notions of differential geometry and topology. Let
7 be a geodesic with tangent v% defined on a manifold M. We call rf a
solution of the geodesic deviation equation 6 (2.6). and this constitutes a
Jacobi field on 7. If rf is non- vanishing along 7, but rf{p) = rf(q) =
(p, q G 7), then the two points p, q are said to be conjugate. It is possible
to show that a point q e 7 lying in the future of p e 7 is conjugate to p if
and only if the expansion of all the time-like geodesies congruence passing
through p approaches — oc at q. A point is then conjugate if and only if it
is a caustic of such congruence. A necessary hypothesis in this statement
is that the space-time manifold (A4,gij) satisfy ./i' ,,('(' > 0, for all the
time- like £'. Moreover, a necessary and sufficient condition for a time- like
curve 7, connecting p, q € M, to locally maximize the proper time between
p and q, is that 7 is a geodesic without any point conjugate to p between p
An analogous analysis can be made for time-like geodesic and a smooth
space-like hypersurface £. In particular, let 6 be the expansion of the
geodesic congruence orthogonal to £. Then, for 6 < and within a proper
time r < 3/|0|, there will be a point p conjugate to £ along the geodesic
orthogonal to £ (for a space-time (M,g;j) satisfying RijC& > 0). As
above, a time-like curve that locally maximizes the proper time between p
and £ has to be a geodesic orthogonal to £ without conjugate point to £.
The last step toward the singularity theorems is to prove the existence
of maximum length curves in globally hyperbolic space-times. We recall
that this is the case because they possess Cauchy surfaces in accordance
with the determinism of classical physics. Without entering the details, in
such a case a curve 7 for which r attains its maximum value exists, and a
necessary condition is that - bo a geodesic withoul conjugate points.
6 In this case, a minus sign appears on the right-hand side of (2.6) due to the different
90 Primordial Cosmology
2.7 .4 Singularity Theorems
Theorem 2.1. Let a space-time manifold (AA., gij) be globally hyperbolic
satisfying the condition
RijCC 3 > (2.157)
for all the time-like vectors t; 1 . Suppose that the expansion 9 of a Cauchy
surface everywhere satisfies < C < 0, for a constant C '.
Then, no past-directed time-like curves A from E can have a length
greater than 3/|C|.
Proof. If there is a past-directed time- like curve, then a maximum length
curve would also exist; this curve should be a geodesic, thus contradicting
the property that no conjugate point exists between E and p € A. Therefore
such curve cannot exist. In particular, all past-directed time-like geodesies
are incomplete. □
This theorem is valid in a cosmological context and expresses that, if the
Universe is expanding everywhere at a certain instant of time, then it must
have begun with a singular state at a finite time in the past. It is also
possible to show that the previous theorem remains valid also relaxing the
hypothesis that the Universe is globally hyperbolic. The price to be paid
is that E has to be assumed as a compact manifold (dealing with a closed
Universe) and, especially, that only one incomplete geodesic is predicted.
We will now discuss the most general theorem, which completely elimi-
nates the assumptions of a Universe expanding everywhere and the global
hyperbolicity of the space-time manifold (Ad, gij). On the other hand, we
lose any information about the nature of the incomplete geodesic. Such a
theorem, in fact, implies the existence of only one incomplete geodesic, i.e.
it does not distinguish between a time-like and a null geodesic.
Theorem 2.2. A space-time (Ad, gij) is singular under the following three
hypotheses:
(i) the condition R i jV l v : > > holds for all time-like or null vectors v %
(ii) no closed time-like curve exists
(Hi) at least one of the following properties holds:
a) (Ad, gtj) is a closed Universe
b) (M,gij) possesses a trapped surface 7
7 A trapped surface is a compact smooth space-like manifold, such that the expansion 9
of either outgoing eitl ling future directed null geodesies is everywhere negative.
Fundamental Tools 91
c) there exists a point p G M such that the expansion 9 of the fu-
ture or past directed null geodesies emanating from p becomes
negative along each geodesic in this congruence.
This theorem states that our Universe, as classically described, must be
singular. In fact, conditions (i)-(ii) hold and 9, for the past-directed null
geodesies emanating from us at the present time, becomes negative before
the decoupling time, i.e. the time up to when the Universe is well described
by the Friedmann- Robertson- Walker model.
The occurrence of a space-time singularity undoubtedly represents a
breakdown of the classical theory of gravity. The removal of such singu-
larities is a prerequisite for any fundamental theory, as expected in the
quantum formulation of the gravitational field. The singularity theorems
are very powerful instruments, although do not provide any information
about the nature of the predicted singularity. Unfortunately, we do not
have a general classification of singularities, i.e. many different types ex-
ist and the unbounded curvature is not the basic mechanism behind such
theorems.
Summarizing, we have shown that a space-time singularity in GR can
be defined following two criteria. The first one is the causal geodesic in-
completeness (global criterion) and the second one is the divergence of the
scalars built up from the Riemann tensor (local criterion). Although the
latter is useful to characterize a singularity, it is unsatisfactory since a space-
time can be singular without any pathological character of these scalars.
Not all singularities have large curvature but. most, importantly, diverging
curvature is not the assumption of the singularity theorems. Singularity
theorems, which demonstrate the geodesic incompleteness, are based on
general properties of differential geometry and topology, in addition with
positive curvature. The Einstein theory enters only in replacing positive
curvature with positive energy conditions affecting the Raychaudhuri equa-
tion. It is worth noting that no general mechanism able to demonstrate a
non-singular behavior for a space-time is available. Although violating the
energy conditions is an immediate way to avoid the singularity theorems,
this is not an enough general criterion.
92 Primordial Cosmology
2.8 Guidelines to the Literature
The Einstein theory of gravity described in Sec. 2.1 can be found in many
classical textbooks. We recommend Landau & Lifshitz [301] and Misner,
Thorne & Wheeler [347] for a introductory exposition while Hawking &
Ellis [228], Wald [456] or Weinberg [462] for a rigorous analysis.
The inclusion of macroscopic matter fields in GR as in Sec. 2.2 is dis-
cussed in the above textbooks. For what concerns the energy-momentum
tensor in GR, see the review [428], while regarding the Yang-Mills fields
(Sec. 2.2.4) we suggest the textbooks of Pokorski [388], Weinberg [463],
and the reviews [289,368].
The canonical formulation of GR, as well as the reduction of the dy-
namics, developed in Sec. 2.3 was formulated by Arnowitt, Deser & Misner
in [17-19] (for reviews see [197,262,263,438,456]). The general theory of
constrained systems can be found in the book of Dirac [153] and Henneux
& Teitelboim [237]. The Hamilton- Jacobi formalism for GR (Sec. 2.3.2) is
discussed in [193,241].
An exposition on the synchronous reference frame (Sec. 2.4) can be
found in Landau & Lifshitz [301].
The tctradic i'ommli sm presented in Sec. 2.5 is analyzed in the standard
textbooks [301,456] and in that by Chandrasekhar [116].
The reformulation of GR in terms of self-dual connection variables, pre-
sented in Sec. 2.6, has been proposed by Ashtekar in [21,22] (for reviews
see [396]). The real Ashtekar variables have been introduced in [37] and
generalized in [255,256]. The action for this new formulation of GR has
been proposed by Hoist in [240] and generalized in the presence of fermions
in [340,385]. There are several reviews and books on such topics. In par-
ticular, [28] starts from the Lag] in in foi mulation, while in [384,438] the
starting point is the Hamiltouiau framework. The gauge group of GR, dis-
cussed in Sec. 2.6.3, has been addressed in [3,407,408] and later clarified
in various works (see for example [4, 121,437]). For a comparison between
the geometrodynamics and the connection formalism see [298] .
A complete discussion on the space-time singularity theorems (Sec. 2.7)
is given in the textbooks by Hawking & Ellis [228] and Wald [456] while for
a recent review see [415]. A presentation of fluid kinematics can be found
for example in [125].
PART 2
Physical Cosmology
In these Chapters, written in collaboration with Dr Massimiliano Lattanzi
(Dipartimento di Fisica, Universita di Roma "Sapienza", Italy), we give
a wide description of the Universe evolution as provided by the Standard
Cosmological Model, including the inflationary paradigm and the dynamics
of small inhomogeneities.
Chapter 3 is dedicated to the analysis of the isotropic Universe evolu-
tion, tracing the kinematical and dynamical features of the Friedmann-
Robertson- Walker cosmology.
Chapter 4 provides a complete picture of the most relevant observational
facts at the ground of the present knowledge of our Universe. Significant
links between observations and theoretical predictions are marked.
Chapter 5 concerns the illustration of the inflationary scenario in its most
general (model independent) form. Starting from the shortcomings of the
Standard Cosmolojuca] Model, we describe the hypotheses and the predic-
tions characterizing the inflation.
Chapter 6 discuss the so-called quasi-isotropic solution, which represents a
natural inhomogeneous extension of the Friedmann- Robertson- Walker cos-
mology. In this model the inhomogeneities remain dynamically weak and
we study their evolution in presence of the main cosmological sources.
This page is intentionally left blank
Chapter 3
The Structure and Dynamics of the
Isotropic Universe
In this Chapter we will present the main features of the Standard Cosmo-
logical Model (SCM). The SCM is built upon the geometrical framework
of the homogeneous and isotropic Robertson- Walker (RW) geometry and
is able to explain the phenomenology that emerges by the direct observa-
tion of the Universe. Our discussion allows the reader to get a synthetic but
complete view of the most relevant kinematical and dynamical properties of
the expanding Universe, together with a description of the thermal history
associated to the evolution of the primordial thermal bath.
We start by analyzing the kinematical properties of the RW Universe
and showing how the main signature of an expanding Universe can be de-
rived without the need of implementing the Einstein dynamics. In partic-
ular, we describe the motion of free particles on an expanding background,
inferring the redshift of light, the Hubble law and the recession of galaxies
and, eventually, we will discuss the two fundamental causal scales governing
the propagation of signals and of physical interactions across the Universe.
Finally, we will introduce the Boltzmann equation on the expanding Uni-
verse, outlining how the macroscopic properties of the cosmological fluid
can be properly recovered from the microphysics of the elementary species
constituting the thermal bath.
The next step in our analysis of the SCM is the implementation of the
Einstein equations in correspondence to the highly symmetric RW geom-
etry, i.e. the study of the Friedmann dynamics describing the behavior
of the isotropic Universe FRW. Let us remark how we make reference to
the cosmological model as FRW and to the underlying geometry as RW.
By investigating the structure of the equations and via the derivation of
asymptotic solutions, we will characterize the nature of the Hot Big Bang,
with particular attention on the radiation-dominated era. Then, we will
96 Primordial Cosmology
devote some space to the discussion of the de Sitter solution, describing
the evolution of the Universe when it is dominated by a constant energy
density term (this regime will have a crucial role in the study of the infla-
tionary paradigm, addressed in Chap. 5). In view of the implementation of
a quantum cosmology framework, as it will in Chap. 10 and in Chap. 12,
we reformulate the Friedmann dynamics in the Hamiltonian formalism, by
stressing the emergence of a super-Hamiltonian constraint in place of the
original Friedmann equation.
The exact homogeneous dynamics is completed by discussing two rele-
vant examples of dissipative isotropic cosmologies, i.e. the evolution of the
cosmological fluid when the bulk viscosity effect or the possibility of matter
creation cannot be neglected.
The study of the RW Universe is eventually enriched by a rather de-
tailed description of the role played by the inhomogeneous perturbations
as seeds for the later structure formation. We will introduce the concept
of Jeans scale in the cases of a stationary and then of an expanding back-
ground. Then we pursue the fully relativistic perturbation theory, coupling
the perturbed dynamical system to the inhomogeneous component of the
Boltzmann equation.
The Chapter ends with a brief description of the inhomogeneous, spheri-
cally symmetric Tolmann-Bondi cosmology. We outline its main dynamical
features in the synchronous reference and provide a Lagrangian picture of
its geometrodynamics. The relevance of this class of Universes relies on
their ability to describe local inhomogeneous structures, properly matched
at the large scale with the RW metric.
3.1 The RW Geometry
In this section we will analyze the properties of the homogeneous and
isotropic Universe, whose geometry is properly described by the RW line
element. Our aim is to extract cosmological information from the struc-
ture of the line element describing the space-time, without imposing the
corresponding Einstein dynamics. In particular, we are interested in char-
acterizing the particle motion on an expanding background, in order to
fix its kinematic properties and to provide a physical insight on some phe-
nomenological issues of the observed Universe. Among the possible kine-
matic effects, we focus our attention on the motion of nearby galaxies (i.e.
the Hubble law), on the non-stationary dynamics of elementary particles
The Structure and Dynamics of the Isotropic Universe 97
(i.e. the redshift of the wavelengths), and on the causal structure charac-
terizing the propagation of physical signals (i.e. the Hubble length and the
cosmological horizon).
3.1.1 Definition of isotropy
Qualitatively, isotropy refers to the absence of preferred directions in space.
Before formulating in a precise way this notion, we have to stress that, at
each point, at most one observer can see the Universe as isotropic. In fact,
given a matter field filling the Universe, any observer in relative motion
with the matter will measure anisotropies in the expansion of matter. Let
us give a precise definition of isotropy.
Definition 3.1. A space-time is spatially isotropic at each point if there
exists a congruence of time-like curves (namely observers), with tangents
denoted by u l , such that: for any point p and for any two unit spatial
tangent vectors w\ and wl atp, there exists ait isomc.l.ry of g,j which leaves
p and u l at p fixed but rotates w\ in w\ .
In an isotropic Universe it is thus impossible to construct a (geomet-
rically) preferred tangent vector orthogonal to u % . In a homogeneous and
isotropic space-time (for the definition of homogeneity see Sec. 7.1) the
spatial surfaces of homogeneity must be orthogonal to the tangents u % to
the world lines of the isotropic observers.
We can define the isotropy group I p of a point p as the set of all isome-
tries leaving p fixed. Suppose now that at p we choose coordinates such
that gij(p) = r]ij. The group I p then leaves the Minkowski metric invariant,
i.e. the isotropy group must be a subgroup of the (homogeneous) Lorentz
group 5*0(3, 1). The dimension m of I p is thus m < 6 = dimS'0(3,l).
I p is a subgroup of the (full) symmetry group of the manifold (see Sec.
7.1). The Friedmann-RW (FRW) cosmological models are characterized by
a three-dimensional isotropy subgroup.
It is worth noting that a space isotropic around any point is necessar-
ily also homogeneous. The contrary is not true, i.e. homogeneous and
anisotropic spaces do exist. When referring to the isotropic Universe, it is
always also homogeneous.
98 Primordial Cosmology
3.1.2 Kinematics of the isotropic Universe
In agreement with the Cosmological Principle, stating that each observer
looks at the same Universe (i.e. both privileged space points and preferred
space directions are forbidden), we base the description of the Universe
kinematics on a non-stationary homogeneous and isotropic three-geometry.
The hypothesis of isotropy imposes that the three spatial directions
evolve with the same time law. while the space-time must be characterized
by vanishing g oi (if non-zero, this component would fix a preferred direction
because it transforms as a three-vector under spatial coordinate transfor-
mations). Thus, in any synchronous reference, we deal with the RW line
element
ds 2 = dt 2 -a 2 (t)dl 2 xw . (3.1)
The cosmic scale factor a(t) is the only degree of freedom available to the
dynamical problem and dl 2 ^ denotes the spatial line element of a three-
space with constant zero, positive or negative three-curvature, i.e.
9 2 + sin 2 6dtf) , (3.2)
•*»KW - "a/3 — — - 1 _ Kr2 ' ' V 1
with r, 6 and <f> being the usual spherical coordinates, while K denotes the
spatial curvature. When K ^ 0, it is always possible to set \K\ = 1, by
means of the redefinitions a — > a/\fK = a curv and r — > f = Kr. Unless
differently specified, we will refer to the cosmic scale factor as the curvature
radius of the Universe a curv , that for K ^ is a measurable quantity. In
the case K = 0, the curvature radius is infinite, so that the normalization
of the scale factor is completely arbitrary and has no physical meaning (i.e.
observable quantities like the redshift depend only on the ratio of the scale
factor measured at different times).
The three spatial line elements can be respectively interpreted as a
hyper-plane (K = 0), a hyper-sphere (K = 1), and a hyper-saddle
(K = —1), although the line element does not fix the global topological
properties of the three-space. Different choices for the topology are pos-
sible: for instance, either the hyper-plane (that is an open space) or the
closed torus are characterized by K = 0.
The RW geometries are often described in terms of an angle-like coor-
dinate x, defined as r = \ f° r K = 0, r = sinx for K = 1, or r = sinhx
for K = — 1 respectively, and defining the co- moving time coordinate r\ by
The Structure and Dynamics of the Isotropic Universe 99
means of dt = a(r])dn. In this picture, the line element rewrites as
ds 2 = a 2 {r,) (d V 2 - dl 2 RW ) ,
dlliw = d X 2 + a 2 K ( X ) (d6 2 + sin 2 8d<j) 2 ) ,
{sinhx 0<x<oo for if = -1 (3.3)
X < x < oo for if =
sinx < x < * for K = +1 .
This way, we deal with a conformal expression of the space-time metric and
the (6 — x) light-cone for if = is 7r/4 wide.
The time variation of the cosmic scale factor provides the evolution of
the Universe (expansion or contraction) and when a(to) = 0, also the metric
determinant is zero. However, at this level, such an instant to cannot yet
be recognized as a real space-time singularity (i.e. the determinant is not
an invariant quantity).
Because of the homogeneity hypothesis, the Ricci scalar is independent
of the spatial coordinates, and in terms of the non-zero components of the
Ricci tensor it is given by
(a a 2 K\
i?oo = -3- (3.5a)
R a0 = -(^+2^ + 2^\h™. (3.5b)
Since the cosmological implementation of this model has to take place
in the presence of a matter source describing the present or the primor-
dial Universe, the above quantities are never globally vanishing and their
behavior can reveal the presence of a physical singularity.
3.1.3 The particle motion
In this Section, we derive an important feature concerning the behavior of
the momentum of a particle moving in a RW space-time.
The trajectory of a test particle, moving across the Universe, is de-
scribed by the geodesic Eq. (2.2) associated to the RW metric (3.1). For
our purposes, we can limit our attention to the zero component only
^ + V = 0, (3.6)
100 Primordial Cosmology
where u 2 = a 2 h^u a u^ is the square of the modulus of the spatial velocity.
In a synchronous reference frame, the normalization condition for the four-
velocity can be stated as (m ) 2 = u 2 + 1, and hence we get u a du° = udu.
Thus, remembering that u°ds = dt, Eq. (3.6) rewrites as
This expression admits the relevant solution u oc 1/a. If we denote as
mo the rest mass of the particle, the modulus of its three-momentum is
p = m u oc 1/a. The momentum of a particle is a time-dependent quantity
and, for the particular case of an expanding Universe, it is redshifted by
the underlying dynamics. We have to stress that the above derivation does
not rely on the notion of non-vanishing element of proper time. Indeed,
since the differential ds does not appear in Eq. (3.7), the result does not
depend on the particular choice of the affine parameter. Thus this result
holds even in the case of a zero mass particle, like a photon. Indeed, for a
massless particle we get the relation
£=p=^oc±, (3.8)
£ denoting the energy and A the corresponding wavelength. If we consider
a photon emitted at a given time t e in the past and observed today at to,
the ratio of the coik S|Miini n 1 n th i ] i lie expression
In the case of an expanding Universe a(to) > a(t c ) and the observed
wavelength is larger than the wavelength at the emission, i.e. it is shifted
towards the red. The quantity z represents the amount of this redshift
and it is measurable, playing a role equivalent to the scale factor whose
variability follows the Universe kinematics.
The physical distance between a pair of co-moving observers scales with
the cosmic scale factor, exactly like the wavelength of photons. Thus any
intrinsic or co-moving length I becomes a measurable quantity of the ex-
panding Universe only if redefined as / p h ys = a(t)l. This non-stationary
feature reflects an intrinsic property of any system living over an expand-
ing geometry.
3.1.4 The Hubble law
We will now derive how the RW kinematics can explain the recession of
galaxies. Indeed, the Universe expansion accounts for the galaxy recession
The Structure and Dynamics of the Isotropic Universe 101
via the geodesic motion on the isotropic and homogeneous background.
Apart from small proper motions (random physical velocities) and local
gravitational interactions (which are able to form bounded systems), the
galaxy flow (the Hubble flow) can be properly described as the motion
of pressure-less particles (a dust system) that are freely falling on the ex-
panding geometry. However, the specific form of the Hubble law can be
reproduced only in the limit of galaxies close to our own (by convention
placed at r = and t = t ), i.e. for z -C 1.
Before entering into the details of the proof, let us stress that the galaxy
expansion, being a pure geometrical effect, does not provide any physical
motion. In other words, the co-moving spatial coordinates of any single
galaxy remain fixed. Moreover, when a gravitational system (for instance a
group of galaxies) has enough binding energy, it detaches from the Hubble
flow and forms a non-expanding substructure. This is exactly the reason
why within any galaxy, the space-time is essentially flat and inertial frames
are allowed. When a density perturbation, corresponding to the galactic
scale, reaches a certain critical value, it decouples from the cosmological
fluid and starts an independent (-volution as a bound system, retaining a
geodesic motion only as a whole.
Let us observe how looking far in the Universe (say, at a generic coor-
dinate r), we are lookiii;; backward in time (say, at a generic instant t), i.e.
when the scale factor of the Universe was a(t) < a(to) = ao- Assuming that
t is sufficiently close to to, the Taylor expansion
a(t) = ao + o| t=to (i -to) + ■■■ , (3-10)
leads to
— = T ^— = l-ifo(to -*) + -.., (3-11)
a 1 + z
where the definition of redshift Eq. (3.9) was used. In Eq. (3.11), the Hubble
parameter H(t) = - remains defined together with its present value, the
Hubble constant H = H(t = t ). The Hubble parameter measures the
(logarithmic) expansion rate of the Universe at a given time.
To obtain the Hubble law, we need to express (t — t), i.e. the time for
the light to go from the source to us, in terms of the same distance. Since
for a photon ds 2 = 0, we can write
if K =
r if if = 1 (3.12)
Jt a(t) J
^~~ —
1 sinhr if K = -I
102 Primordial Cosmology
which, in the limit r«l, this is simply equal to r. The spatial curvature
can thus be neglected when dealing with small values of (t — to), i.e. with
distances much smaller than the curvature radius of the Universe. At this
level, the space can be assumed to be flat. Inserting Eq. (3.10) on the
left-hand side of Eq. (3.12) wo get, to first order,
t - t = d+ ... , (3.13)
where d = a^r is the present distance to the source. The physical content
of this equality is straightforward: when an object is close enough (so that
we can neglect the spatial curvature along with the space expansion), the
time for the light signal to reach us is proportional to the distance from the
source. This allows us to rewrite Eq. (3.11) in the form (we use 1/(1 + z) ~
(1 - z) for z < 1)
z = H d + .... (3.14)
Interpreting the geometrical redshift of the photons emitted by a galaxy
as the Doppler effect due to a physical velocity v, we get the well-known
expression for the Hubble law
v = H d. (3.15)
Equation (3.15) can be generalized to account for higher order terms in the
scale factor expansion. However, this involves some subtleties related to
the notion of distance in cosmology, and to how distances are measured.
We have seen that the proper distance at the time t between us (r = 0)
and an object at coordinate position r is d(t) = a(t)r. Unfortunately, the
proper distance is not what is directly measured through observations. For
example, distances are often measured recurring to standard candles, i.e.
objects with known intrinsic luminosity (for example, supernovae la). If
the luminosity L is known and the flux F at the Earth is measured, one
can introduce a luminosity distance d\, as
This expression again is a definition of d\, . Its operative meaning is directly
related to an observable quantity (the flux) and thus it is an observable
quantity itself. The reason for the definition is that in a Minkowski space
dh coincides with d. However, in general this is not the case, once the effects
of the spatial curvature and of the expansion are taken into account. It can
be shown, using the conservation of the energy momentum tensor, that the
The Structure and Dynamics of the Isotropic Universe 103
luminosity distance d\, and the proper one d are related by (assuming for
simplicity K = 0)
d L = d{l + z). (3.17)
This formula can also be interpreted as follows: L/^im'^r 2 is the flux that
would be measured in the absence of expansion; the flux is reduced by a
factor 1+z for the redshifl of i he energ} of t he single photon (see Sec. 3.1.3),
and by another factor of 1 + 2 for the I hue dilat ion c licet between the source
and the observer.
Repeating the steps above, one finds that
H d L = z+^(l-q )z 2 + ..., (3.18)
where the deceleration parameter of the Universe qo is defined as
go "-^L,- (3 - i9)
We stress that the higher order terms in Eq. (3.18) depend on the distance
indicator written on the left-hand side of the equation.
Equation (3.18) is particularly useful when dealing with standard can-
dles since, in that case, d\, is the directly observable quantity. On the other
hand, if one deals with standard rulers, i.e. objects of known linear size,
the natural distance indicator is the angular diameter distance d\, defined
as the linear size of the object divided by the angle it subtends in the sky.
The Hubble law for c?a has a form similar to Eq. (3.18) but with a different
second (and higher) order term. For the dominant term, in the limit of
small z, all the forms of the Hubble law reduce to H n d = z. This traces the
fact that for z -C 1, all the distance indicators like d^ and c?a coincide with
the proper distance d (like it should be in a static Euclidean geometry).
We finally note that the exact knowledge of a(t) is required to derive the
exact relation between d^ (or d^) and z, which in turn requires to solve the
Einsetin equations.
The capability of the RW kinematics to reproduce the observed Hubble
law stands as a significant confirmation of the isotropy and homogeneity of
the observed Universe.
3.1.5 The Hubble length and the cosmological horizon
In the continuation of the Book, the description of the physical scenario
living on an isotropic, non-stationary background will often require to fix
104 Primordial Cosmology
the characteristic time and length scales of the system under study, in our
case the whole Universe. The fundamental cosmological time scale is given
by the inverse of the expansion rate, i.e. by the Hubble time H^ 1 = a/a.
The Hubble time roughly gives the time in which the scale factor doubles.
If the scale factors scales like a(t) oc t a , it follows that H^ 1 oc t. In
fact, in a Friedmann Universe (see Sec. 3.2) the Hubble time gives, apart
from numerical factors of order unity, the age of Universe. This relation
can however be deeply altered in more general cosmological models (for
example in the inflationary scenario, see Chap. 5).
The length associated to the Hubble time is the Hubble length Ln(t)
Ln(t) = Hit)- 1 , (3.20)
also called the Hubble radius. This is roughly the distance that a photon
can travel in an expansion time around t and is the scale to which the
characteristic length of any physical phenomenon has to be compared in
order to understand if it can coherently operate on cosmological scales. For
example, the comparison of the characteristic mean free path I of a given
particle species with the Hubble length (3.20) determines whether such
a species participates in the thermodynamical equilibrium (l <C Lh) or is
decoupled from it (F^> Ln). In other words, for a given process to be able to
maintain the equilibrium, its rate must be much faster than the geometrical
rate of curvature change or, equivalently, its time scale r must be much
smaller than the Hubble time H~ x . In this respect, the Hubble length
represents a real horizon for the microphysics of the expanding Universe.
The Hubble length, being associated to the space-time curvature of the
Universe, is in itself a physical scale 1 and can be measured today by direct
and indirect observation on the large scales (see Sec. 4.3).
Another relevant length is the one characterizing the maximal causal
distance at which physical signals can propagate in an expanding Universe,
starting at a finite initial instant of time, say t = 0. Such a distance, called
the particle horizon (often simply "the horizon") corresponds to the path
traveled by a photon emitted at t = and is calculated from the condition
for the propagation of a wave front, i.e. ds 2 = 0. In terms of the RW line
element (3.1), this condition can be stated as
dt = a{t)dl RW =► hw= I ^77- (3.21)
Jo a (t )
To get from such co-moving length the physical and measurable horizon,
we have to rescale it by the cosmic scale factor, obtaining
/•* dt!
Jo W.
ii
c
da = a(t) / — - . (3.22)
The Structure and Dynamics of the Isotropic Universe 105
Objects separated by a distance larger than da have never been in causal
contact and they cannot have been affected each other. In particular, this
implies that spatial regions which are separated more than one cosmolog-
ical horizon cannot be in thermal equilibrium. Moreover, for a power-law
expression of the scale factor a(L) x t a , the physical horizon is a finite quan-
tity for a < 1. As we shall see in Sec. 3.2, this is the case for an expanding
Universe filled with ordinary matter and radiation.
Also the co-moving horizon dn/a is always increasing, being the integral
of a positive-defined quantity. Since, by definition, co-moving distances are,
constants, the ratio between the horizon and any given distance decreases
backwards in time. In other words, things that are in causal contact to-
day were not necessarily so in the past. When studying how a causally
connected region at the present time should have looked in the past, one
cannot think of it as a unique causal region but, most likely, of a collection
of a large number of independent causal patches. This is at the origin of
the horizon paradox that affects the SCM and that will be addressed in
Chap. 5.
In the case of the power-law example considered above, the Hubble
length and the physical horizon are comparable quantities, i.e. dn =
«Lh/(1 — ex). They coincide when a = 1/2, corresponding to the cos-
mological scenario of a radiation dominated Universe.
We remark again that the equivalence of these two spatial scales is not
a general feature, as shown by the counterexample of a de Sitter phase
of expansion. In fact, even if the numerical values of such two quantities
roughly coincide in a Friedmann Universe, their physical meaning is deeply
different. The Hubble length gives the distance traveled by a photon in one
Hubble time, while the particle horizon is the distance traveled by a photon
during the whole life of the Universe. Points that are distant more than one
Hubble length have not been in causal contact for the last Hubble time or
so, while points that are distant more than one particle horizon have never
been in causal contact. Thus, the Hubble length is a strictly local quantity,
depending on the expansion rate at the time t only, while the particle
horizon is an integral quantity that receives contributions from all the past
expansion history. The value of the horizon can be dramatically altered
by contributions coming from a non-standard (with respect to Friedmann
models) behavior of the scale factor at t ~ 0. We can anticipate that the
horizon paradox is resolved in the inflationary scenario through an early
phase of de Sitter expansion that makes the particle horizon many orders
of magnitude larger than its Friedmann value.
106 Primordial Cosmology
3.1.6 Kinetic theory and thermodynamics in the expanding
Universe: The hot Big Bang
The early Universe, as described in the hot Big Bang theory (firstly formu-
lated by Gamow in the '40s), is characterized by a thermal bath in which all
fundamental particle species are embedded and are maintained at equilib-
rium by interactions with other species. The cosmological expansion implies
that the thermodynamical parameters of the macroscopic cosmological fluid
depend only on time. Thus, we can assume that the Universe expansion
proceeds through equilibrium stages and that it is characterized by a global
temperature T{L), as far as non-ideal fluid effects due to out-of-equilibrium
features (for instance dissipative mechanisms) can be neglected. Indeed, the
main part of the thermal history of the Universe can be well represented
as equilibrium phases and the cosmological fluid is well-modeled by a per-
fect one, even if some steps of the early cosmology are associated to phase
transitions or species decays and decoupling, which require an appropriate
out-of-equilibrium treatment.
The kinetic theory of particles on the RW background is described by
the relativistic Boltzmann equation, with the usual Liouvillc operator is
generalized to curved space-time as
where / = f(x z ,p t ) denotes the distribution function on the (relativistic)
phase-space and the collision operator C[f] is the collision integral that
characterizes the change in the distribution function, in a unit of proper
time, due the particle interactions. Making use of the geodesic equation to
describe the particle acceleration 1 dp 1 /ds, Eq. (3.23) for a RW background
can be restated. The homogeneity of the space prevents any spatial de-
pendence in the distribution function, while isotropy requires that it can
depend on the three-momentum only through its magnitude or, equiva-
lently, on the energy E = p° , i.e. / = /(£, E). Thus, Eq. (3.23) for the
isotropic Universe can be rewritten as
4f-^!= c «- (3 - 24)
Equation (3.24) provides a complete microscopical description of the matter
filling the Universe, once a specific form of the collision integral is given.
1 Here we adopt the notion of a free-falling scalar particle, assuming that any other
effect (like, for example, the spin) averages to zero over the cosmological fluid.
The Structure and Dynamics of the Isotropic Universe 107
In order to switch to a macroscopic description, one has to integrate the
Boltzmann equation over the momentum space.
Let us analyze the case when the collision integral is negligible, which
is certainly appropriate when the mean free path of the particles is much
larger than the Hubble length. However, as far as we are dealing with
species in thermal equilibrium, the collision integral can also be neglected
because interacting particles admit the same distribution function and the
matrix elements, governing the microphysics, are in general invariant under
time reversal. 2
Dividing Eq. (3.24) by E and observing that the relativistic dispersion
relation implies dE/dp = p/E, we get
Recalling the definition of the particle number density
n=-^jd 3 pf(t,E), (3.26)
where g do! denotes the number of degrees of freedom of the particle, we
can derive a macroscopic law from the Boltzmann equation. By integrating
Eq. (3.25) over momentum space (d 3 p = Airp 2 dp because of the isotropy
assumption) and integrating the second term by parts (the distribution
function has to vanish for diverging p), we obtain an equation for the num-
ber density of the form
^ + 3-n = =* n(t)<x±. (3.27)
dt a or
Thus the Universe expands and the particle number density decays ac-
cording to the increase of the spatial volume (V ~ a 3 ). In other words,
the number of particles contained inside a coordinate domain is conserved
during the expansion.
Even if the result in Eq. (3.27) has been derived assuming a vanishing
collision term, it is indeed more general. In particular, it holds as long as
the collisions conserve the particle number, ensuring that, even if C[f] ^ 0,
one has J(C[f]/E) d 3 p = 0, i.e. the collision term is zero once integrated
over momentum space. For what concerns the evolution of the number
density, the presence of a collision iniogni] has to be taken into account in
those processes which do not preserve the number of interacting particles,
as particle decays or annihilations. For definiteness, let us consider a species
.. the- explicit form of the collision integral identically
108 Primordial Cosmology
X decaying to species Y, plus some other particle species Z whose evolution
we are not interested in. Let r d be the mean lifetime of species X. The
effect of the decay X — > Y + Z can be schematically described modeling the
collision term in Eq. (3.24) as C[f] = ±{E/r d )fx, where the (+) and (-)
sign holds for the Y's and X's respectively. The two Boltzmann equations
for the X's and Y's write as
(3.28a)
df 2L _a df x _ fx
dt a dp Td
dfy a dfy , fx ,„ O0 , s
-7- p-^— = + — . 3.28b
dt a dp r d
Repeating the same steps leading to Eq. (3.27), we get the following equa-
tions for the number densities
^ + 3 (H + H d ) n x = (3.29a)
^ + 3Hn Y - 3H d n x = (3.29b)
where we have defined H d = l/3r d and the number densities finally evolve
n x {t) oc —e^ Hdt (3.30a)
n y (i)«^(l-e- 3 ^*) . (3.30b)
Let us also note how the Boltzmann equations for the X's and Y's, ei-
ther in their unintegrated or integrated form, can be summed to obtain
a single equation for the total density n x +y = n x + ny perfectly iden-
tical to Eq. (3.27), i.e. with vanishing collision term. This implies that
n x+ y oc a~ 3 (as it can be directly verified from the solutions Eq. (3.30a)
and Eq. (3.30b)). This is not surprising, because in each decay process a
particle X is destroyed and a Y is created, so that there is no net change
in the total number of X and Y.
Similarly to number density, we can obtain a macroscopic relation in-
volving the energy density and the pressure of the cosmological fluid by
multiplying Eq. (3.25) by E. Again, we consider a vanishing collision term
although the final result still holds as long as the collisions conserve the
energy. Noting the conjugate character of the variables t and E, we get
The Structure and Dynamics of the Isotropic Universe 109
As before, we will apply the integral operator J d 3 p = J Q 4np 2 dp (the an-
gular integration gives An because of the isotropy condition) to this equa-
tion. Multiplying Eq. (3.31) by p 2 , after some manipulation we rewrite it
as (dE/dp = p/E)
Ft ^ Ef ^ ~ ~a [| ^ " 3p2Ef - P i f }=°- (3 - 32)
Bearing in mind the definitions of the energy density p and of the pres-
sure P as
P=-^jd 3 pEf(t,E), (3.33)
p =wrJ d3p &w> E) (3 - 34)
and integrating Eq. (3.32) over the momenta, we get the following macro-
scopic equation, called the continuity equation
^+3^(p + P) = 0. (3.35)
The thermodyuamical interpret at iou of Eq. (3.35) is straightforward in
terms of the first law of thermodynamics applied to the Universe. Since
the Universe, by definition, cannot exchange heat with an external source,
SQ = (i.e. the expansion is adiabatic) and the first law reads as dU =
—PdV. The internal energy U inside a co-moving volume satisfies U = pV,
so that the first law can be restated as V dp + (p + P)dV = 0. Since the
volume V oc a 3 , we have that dV = Wda/a and then
dp + 3{p + P)— = 0. (3.36)
The assumption to deal with an adiabatic expansion follows from the ab-
sence of external heat sources (by definition of Universe). On the other
hand, on a local level, the adiabatic character of the Universe follows from
the impossibility to exchange heat between spatial points being at the same
temperature (this in turn is due to the homogeneity of the Universe).
( 'on i pa ring S lie kinetic deiin.jiioiis (3.33) aud (3.34). an equation of state
P = P{p) can be defined in two limiting cases of cosmological interest.
(i) When the temperature of the Universe is much smaller than the
rest mass m of the particles, we deal with the limit E ~ m, p -C
m, so that we get the dust-like relations p ~ mn, P ~ 0. Such
approximation well describes the present Universe, where galaxies,
apart from proper motions and local interactions, resemble a free
falling dust fluid. In this case, Eq. (3.35) implies p oc 1/a 3 , as
expected by the behavior of the number density.
110 Primordial Cosmology
(ii) When the temperature of the Universe is much larger than the rest
mass of the particles we get the relations E ~ p and hence P ~ p/3.
Such ultrarelativistic equation of state properly describes the be-
havior of the very early Universe, when only highly energetic par-
ticles were present. In this case, Eq. (3.35) provides the expression
for p as p oc a~ 4 .
The two limits discussed here concern the matter dominated and radi-
ation dominated Universe, respectively.
In general, one can assign a generic equation of state of the form (2.15)
for the isothermal Universe. Another common way to write the equation of
state is P = wp, where w = 7 — 1 is called the equation of state parameter.
Henceforth, we require the polytropic index 7 to fulfil the condition 7 < 2,
to ensure a non-superluminar sound velocity v s = \J dP j dp = y/j — 1 for
the cosmological fluid. For such choice of the equation of state, the relation
between the energy density p and the cosmic scale factor a, as obtained
from Eq. (3.35), is p oc a~ 37 . The matter and the radiation-like behaviors
discussed above correspond to 7 = 1 (w = 0) and 7 = 4/3 (w = 1/3),
respectively. Equation (3.35) can also be derived by the conservation law
of the energy-momentum tensor (2.14) on the RW background, clarifying
the assumption that the cosmological fluid is properly represented by a
perfect one.
The energy density of the very early Universe scales as 1/a 4 and implies
that the limit a — > corresponds to a physical singularity of the RW space-
time, associated to the instant (say t = 0) when the Universe was born.
A peculiar case would correspond to P = — p (7 = 0), when the Universe
energy density is not diverging and, from (3.35), remains constant as p =
PA = const. From Eq. (2.14), such equation of state corresponds to a matter
source described by an energy-momentum tensor of the form
T lk = p A g t k , (3.37)
that properly mimics a cosmological constant term.
Let us now link the energy density of the Universe to its temperature
and hence fix the expression of the latter in terms of the cosmic scale factor.
For a species in kinetic equilibrium, the distribution function takes the form
dof x
2tt 3 \E- p\
where p denotes the chemical potential, while the signs (+) and (— ) pertain
to fermions or bosons, respectively. In the radiation (photon-like) approx-
The Structure and Dynamics of the Isotropic Universe 111
imation, i.e. K-qT 3> m and K-qT 3> p., the Universe energy density reads
Prad = ^g*(T)T\ (3.39)
T being the photon temperature. The quantity g*(T) is a measure of the
effective number of degrees of freedom contributing to the radiation energy
density and is given by
^(T)=E<° f (^) 4 + E^ F 0f (^) 4 , (3-40)
where the letters B and F indicates Bose-Einstein and Fermi-Dirac species,
respectively and the factor 7/8 arises from the difference between the
fermion and the boson statistics. At sufficiently high temperature, when all
the fundamental particles are in thermal equilibrium, the function g* (T) is
weakly depending on temperature and can be approximated by a constant
value gl ad (the same situation takes place even in later phases, far from the
energy thresholds that correspond to the annihilation and disappearance
of a given particle species). Remembering that for a radiation-dominated
Universe the energy density behaves as p oc 1/a 4 , we can infer the inverse
proportionality relation between the temperature and the scale factor, i.e.
T oc 1/a. The cosmological singularity emerging for a — > is associated to
a diverging temperature, as a consequence of the radiation nature of the
very early Universe. This consideration is at the ground of the concept that
the Universe was born in a hot Big Bang.
Let us now briefly discuss the behavior of entropy on a RW background.
The relation between entropy and the other thermodynamic quantities is
TdS = dU+ P dV. (3.41)
Introducing the entropy density s = S/V and recalling that U = pV , this
equation can be rewritten as
dp=(Ts-p- P)dV + Tds . (3.42)
Since p does not depend on volume but only on temperature, the coefficient
in front of dV must vanish and thus
.-*±£ (3.43)
so that the entropy inside a co-moving volume is S = a?{p + P)/T. Since
heat transfer between a co-moving region and its surroundings is not pos-
sible because of the assumption of homogeneity, it follows that the entropy
112 Primordial Cosmology
inside a co-moving volume is conserved and then that the entropy density
scales like s ex a -3 . This quantity is dominated by the contribution of
relativistic particles for which P = p/3, so that Eq. (3.43) gives
S = lz 9 * sT ^ (3 ' 44)
wlu'iT' //.,,, is doiiiK'd similarly to g*, namely
'E#(¥\ +Ej.
The conservation of S implies that g*, s ^'Vr ! = (oust. Since g* s is nearly
always constant (it varies when a given particle species disappears from the
thermal bath), Toe l/o.
We conclude by noting that g* and g* s coincide when all the species
have a common temperature T and this is nearly always the case during
the history of the Universe so that one can usually take <7* = g* s . The
only notable exception to this is given by cosmological neutrinos, that have
a present temperature T v ~ 1.9 K, different from the photon temperature
T 7 ~ 2.7 K.
3.2 The FRW Cosmology
3.2.1 Field equations for the isotropic Universe
Let us now analyze the form that the Einstein equations acquire when the
hypotheses of homogeneity and isotropy are retained, as for the line element
(3.1)-(3.2). In order to specify the Einstein equations (2.12) we take the
cosmological fluid as co-moving with the synchronous reference. This choice
is possible, even in the presence of pressure, because of the high symmetry
of the RW geometry, i.e. the spatial gradients of pressure are identically
zero and its time derivative terms cancel out of the fluid equations of motion
(see Sec. 2.2.1). Thus, u l = <5q and the components of T? take the diagonal
form presented in Eq. (2.20).
The Einstein equations are obtained using Eqs. (3.4), (3.5) and (2.20).
In particular, the 00 component of the field equations takes the form
^ 2 =(^) 2 = fp-f , (3-46)
which is usually called the Friedmann t quation. The aa components reduce
to three identical equations by virtue of the Universe isotropy, i.e.
K
-G)
The Structure and Dynamics of the Isotropic Universe 113
Substituting Eq. (3.46) into Eq. (3.47), we get the equation for the Universe
acceleration
~a = ~l {p + W) - (3 ' 48)
Equations (3.46) and (3.48) can be accompanied by the continuity equation
(3.35), that however is dependent on the other two. In fact, each of the three
equations (3.46), (3.48) and (3.35) can be obtained combining the other
two. In general, to describe the dynamics of the Universe it is convenient
to choose the Friedmann equation and the continuity equation. The former
provides a link between matter and geometry, while the latter closes the
dynamical problem, fixing the behavior of the energy density in terms of the
scale factor, once an equation of state for the cosmologica] (Juid is assigned.
We observe that Eq. (3.46) and Eq. (3.48), when calculated at the
present instant of time, provide simple expressions for the Hubble constant
Hq and for the deceleration parameter qo (3.19) in terms of the actual value
for the Universe radius of curvature, energy density and pressure. For in-
stance, for K = 0, we have
Ho = sJ^Po (3.49a)
q = \ (1 + 3w) , (3.49b)
where we have used the equation of state (2.15).
Let us infer some properties of the isotropic Universe dynamics, from a
qualitative analysis of this fixed set of equations. First of all, from (3.48)
it comes out that, as far as the Universe evolution is dominated by matter
described by an equation of state P > —p/3, the expansion has to decel-
erate. Thus, the evidences that the Universe is presently accelerating, as
discussed in Sec. 4.3, lead to a serious revision of our understanding about
the nature of the matter filling the present Universe or, alternatively, of the
notion of Friedmann dynamics.
Equation (3.46) establishes a relation between the square of the Hubble
function, the energy density and the spatial curvature. While the former
two are positive by definition, the latter is fixed by the curvature sign. For
K = 0,-1, there is no time where H, i.e. d, vanishes and no turning
point in the Universe expansion arises. The radiation dominated Universe
emerges from the hot Big Bang, then passes through an equilibrium era and
finally ends its life in a decelerating matter dominated phase: no re-collapse
of the space can take place. When K = +1, the Hubble function vanishes in
114 Primordial Cosmology
correspondence to a given instant t tp , such that o tp = a(t tp ) = \/{3/npt p ),
p t p = p{ttp)- Both in a radiation and matter dominated Universe, the
second time-derivative of the scale factor, evaluated at t = t tp , is negative
in view of Eq. (3.48). Thus, the above value <2t p is a maximum for the
Universe expansion. The subsequent evolution of the Universe is described
by a recollapse phase to a singularity where a = 0. We can conclude that
the flat and negatively curved spaces are characterized by an indefinite
expansion from the Big Bang, ending in a deceleral ing rarefied Universe. On
the other hand, the closed R\Y geometry expands from a Big Bang, reaches
a maximum value and then recollapses to Big Crunch, opening fascinating,
although debated, perspectives for a cyclic Universe. Furthermore, let us
note that, in correspondence to a tp and p tp , it is possible to establish the
existence of a static Universe with such structural parameters. However,
this configuration, which would have been appropriate for the XIX Century
notion of cosmology, has a purely mathemal is al meaning, in view of its well-
known instability.
Finally, if we divide Eq. (3.46) by H 2 and define the quantities
(3.50)
(3.51)
_3H 2
Pcrit
it rewrites as
n —s4
(3.52)
Here p cr ; t denotes the Universe critical density, i.e. the density it would
have for K = 0. The quantity 57 is called the density parameter of the
Universe and it is larger, equal and smaller than unity, for the closed, flat
and negatively curved RW models, respectively.
The present value of the critical density is p" rit = 1.03 x 10~ 29 g/cm 3 ~
5.8 x 10~ 6 GeV/cm 3 , while Q,q is equal to unity within a few percent; its
sign has still to be determined.
3.2.2 Asymptotic solution toward the Big Bang
Once the relation p = — — (p = const.) is assigned, the Friedmann
Eq. (3.46) admits analytical solutions that reproduce the qualitative be-
haviors described above. Since we are mainly interested in the asymptotic
The Structure and Dynamics of the Isotropic Universe 115
behavior of the Universe towards the Big Bang (a — > 0), the spatial curva-
ture term is negligible with respect to the energy density, as long as 7 > 2/3
and therefore one can approximate the Friedmann equation as
The solution of this equation reads
a(t) = (t) ^ , t= -,J- , (3.54)
where i is an integration constant, fixed arbitrarily by the generic value
p. Often t is conveniently taken as the age of the Universe, so that today
the scale factor remains fixed to unity, i.e. a = 1. This solution shows
that the singularity appears in correspondence to any positive value of the
parameter 7 and is characterized by a divorcing energy density of the form
om-f-sk- (3 ' 55)
Let us stress how the energy density, being an observable quantity, is inde-
pendent of the (arbitrarily chosen) value of p.
We now introduce the Planck length lp = y^n/Sir = 0(lO~ 33 cm) and
the associated Planck time tp = lp = O (lCU 44 s). The Planck length is
the only combination of the three fundamental constants G, c and h with
the dimensions of a length, and represents the length scale where both
quantum physics and GR are relevant. Since we do not have yet a settled-
down theory of quantum gravity, the Planck length should be regarded as
the limit where our understanding of physics starts to be deeply speculative.
Equation (10.30) can be rewritten as
M-*^ -£?(*)'• (3 ' 56)
where we have further introduced the Planck energy density pp defined as
p P = i- = O(10 93 g/cm 3 ) = O (10 117 GeV/cm 3 ) . (3.57)
t p
The era between the Big Bang and the Planck time, called the Planck
era, is expected to correspond to a quantum evolution of the Universe
(see Chap. 10). In this temporal region, the predictivity of the Friedmann
equation is lost in favor of non-deterministic concepts, like the Universe
wave function. Nevertheless, the very small value of the Planck time (in
116 Primordial Cosmology
comparison to the Universe age) allows us to extrapolate backward the
classical dynamics, disregarding here the finite nature of this value.
From the solution (3.54), we get the Hubble length and the cosmological
horizon as
L n = H- l = - h = ^-t (3.58)
Jo a (t) 37-2
These two lengths are of the same order for a Friedmann Universe and the
Hubble time H^ 1 provides a good estimate for the age of the Universe. The
cosmological horizon would become diverging when ", < 2/3, i.e. the same
range of equation of state where the Universe would accelerate, according
to Eq. (3.48). We will discuss later the possible physical interest of such
peculiar mailer behavior.
The radiation dominated Universe When addressing the Universe
evolution near the singularity, the thermal energy overcomes the rest mass
energy of any species (apart from Planck mass particles) and that the con-
dition K-qT 3> m holds; in agreement with the discussion of Sec. 3.1.6, the
energy content of the Universe is dominated by ultrarclativistic particles,
with equation of state P = p/3, i.e. 7 = 4/3. The scale factor takes the
explicit form
■4
o(t) = y = , (3.60)
that provides a coincidence of the Hubble length with the cosmological
horizon, i.e.
L H = d H = 2t. (3.61)
This expression allows us to rewrite the energy density (3.56) in the simple
form
^ d= 327 v ^J ' (3 - 62)
Finally, the temperature of the Universe is related by Eq. (3.39) to the
radiation energy density and reads as
■i^w) i' (3 ' 63)
where T\\ m denotes the temperature above which all fundamental particle
species are present and in thermal equilibrium. For the Standard Model of
The Structure and Dynamics of the Isotropic Universe 117
elementary particles, such temperature is T^ ~ 300 GeV and the corre-
sponding value of g* is gf 1 = 106.75.
The radiation dominated Universe is characterized by the birth of the
Universe in the form of a hot Big Bang and the expansion from this initial
state decelerates with time. In fact the Universe has a diverging geomet-
rical velocity when it emerges from Big Bang (on a physical point of view,
having a Planck value), accordingly to the law a oc l/o, but it is drastically
suppressed by a deceleration proportional to the expanding volume, i.e.
a oc — 1/a 3 . Such diverging character of the geometrical velocity is required
to bring the volume of the Universe from U = 0ina = 0toa finite value,
in any arbitrarily instant close to t = 0. In view of further developements,
any physical length (for instance an inhomogeneity scale) evolves as the
scale factor, i.e. L p h ys oc a. Since the cosmological horizon is finite and it
behaves like da oc a 2 , close enough to the singularity each physical scale is
super-horizon sized, i.e.
lim J^- = o . (3.64)
On the contrary, when the Universe expands, increasingly large physical
scales enter the cosmological horizon, or equivalently the Hubble length.
In general, the ratio dn/^phys scales like a~^~ , so the behavior just de-
scribed takes place if 7 > 2/3 or, equivalently, w > —1/3. In general, if
in the Universe are present both a radiation and a non-relativistic matter
component, with equation-of-state parameters w = 1/3 and w = respec-
tively, at some point the Universe will be matter-dominated, even if it was
radiation-dominated at the beginning. In fact, the radiation density p ra d
scales like a -4 while the matter density p m scales like a -3 , so the former
decreases faster than the latter during the expansion.
The time t cq when p m = p ra ,i is called the time of "matter-radiation
equality" and separates two different regimes in the evolution of the Uni-
verse. The corresponding redshift z cq = z(t cq ) can be expressed in terms of
measurable quantities, namely the present densities p^ and p° ad of matter
and radiation. At the time of equivalence, by definition, p m (t cq ) = p r ad(^oq)
and. using (lie scaling of p m and p ra d with redshift, one gets
< (1 + ^ oq ) 3 = p° ad (1 + z cq ) 4 , (3.65)
118 Primordial Cosmology
where fi^ = p° m / >° rit and ft° ad = p° ad /p° rit denote the present density
parameters of matter and radiation, respectively. The observations (see
Sec. 4.4) show that fi^ ~ 0.25 and fi° ad ~ 8 x 1CT 5 (the latter value
includes the contribution from photons and three neutrino families), so
that z c „ ~ 3000.
3.2.3 The de Sitter Solution
A notable cosmological solution of Einstein equations is the de Sitter one,
describing an empty Universe with a cosmological constant. Here, we will
investigate the geometrical structure of the de Sitter space-time and its
cosmological implement at iou. Let us consider a live-dimensional Minkowski
space-time, with line element
ds 2 = ■q IJ dx I dx J = r) t:j dx l dx j - (dx 4 ) 2 , (3.67)
where the indices /, J run from to 4. A sphere of radius I in such space-
time admits the equation
mj x I x J = jjyxV - (x 4 ) 2 = I 2 . (3.68)
In order to calculate the metric induced on this sphere, we solve Eq. (3.68)
to obtain x A as
h^ijx^i - I 2 (3.69)
f -rl-r-i
(3.70)
y/VijX^J - I 2
Hence, the line element (3.67) can be restated into that of a Minkowski
To characterize on a cosmological level this line element, let us consider
the change of coordinates
where £" denotes three-dimensional spherical coordinates; :
of variables the line element (3.71) rewrites as
GW
The Structure and Dynamics of the Isotropic Universe 119
where <if2§ denotes cMr W in the closed case. We are dealing with a non-
stationary metric in a synchronous reference and, under the identification
a(t) = I cosh(t/l), such line element coincides with the RW one (3.1), having
the spatial geometry associated to K = 1, as in Eq. (3.2). This form of
the metric describes the de Sitter space-time associated to an isotropic
closed Universe. Indeed, this line element corresponds to a solution of the
Friedmann Eq. (3.46), associated to the presence of a cosmological constant
term A = 3/Z 2 , i.e.
(§)'-?-?■
The de Sitter model describes a singularity-free Universe, which collapses
towards a minimal volume for a = I (but this value can be arbitrarily
re-scaled) and then re-expands indefinitely. The disappearance of the sin-
gularity is a typical effect of the presence of a cosmological constant. In
fact, also in the case of a flat RW dynamics we get the peculiar behavior
associated to the scheme
a(t) = a e*/' . (3.76)
In this case the volume of the Universe vanishes only in the limit t — ¥ — oo
and no real singularity appears. Such flat de Sitter model will be crucial
in the study of the inflationary scenario, as we shall see in Chap. 5. An
important feature of this model is that it would correspond to an equation
of state P = —p, as discussed before in Sec. 3.1.6 and therefore it accelerates
as a ex a. Furthermore, this model has a constant Hubble length L H = L
which significantly differs from the cosmological horizon, which is indeed
diverging. Such discrepancy in the behavior of the two fundamental lengths
confers an important dynamical and physical role to a de Sitter phase of the
Universe evolution. In fact, during such phase, the physical lengths increase
with respect to the physical Hubble horizon, even becoming super-horizon
3.2.4 Hamiltonian dynamics of the isotropic Universe
We will now restate the dynamics of the FRW Universe in the framework
of the Hamiltonian formulation of gravity, developed in Sec. 2.3. This ap-
proach is relevant for the canonical quantization of this cosmological model,
120 Primordial Cosmology
as we shall discuss in Sec. 10.1, when dealing with the Wheeler-DeWitt
paradigm. The analysis below allows us to outline interesting features of
the equations of motion in a generic time gauge.
Even if we could recover the Hamiltonian formulation for the highly
symmetric case of the isotropic Universe by simply imposing the appropri-
ate restrictions to the general formulation of Sec. 2.3, nevertheless we will
address the problem starting from the Lagrangian approach. Indeed, the
homogeneity hypothesis allows to integrate out the spatial dependence of
the three-geometry from the action integral. In the case of a closed Universe
(K = +1), this spatial integral has value 2tt 2 and in the flat and negative
curvature cases (that correspond to an open space with infinite volume,
unless a non-trivial topology is imposed), we choose for convenience to in-
tegrate over a fiducial volume of the same value. This is possible because
such a choice does not affect the variational principle.
Let us consider the ADM line element for the homogeneous and isotropic
model
ds 2 = N(t) 2 dt 2 - a(t) 2 dli w , (3.77)
where the term dl 2 ^ is provided by Eq. (3.2) and we included the lapse
function N(t), but not the shift vector iV" because the former ensures the
time reparametrization of the dynamics while the latter is forbidden by the
isotropy condition (it behaves as a spatial vector and thus would single out a
preferred direction). Differently from the generic case discussed in Sec. 2.3,
here we derive the Hamiltonian equations starting from the gravitational
action for the isotropic case, i.e. we will not use the Gauss-Codazzi relation
(2.67). In the presence of an energy density p = p(a), the action takes the
form
Srw = J** dt [^ (aa 2 + aa 2 + KN 2 a) - 2n 2 Npa^ . (3.78)
The geometrical part of this action is obtained by direct substitution of the
RW metric (3.77) into the Einstein-Hilbert action, with the prescription
above for the value of the spatial integral.
The second derivatives with respect to time of the scale factor can be
eliminated using the relation a 2 'd = (a 2 a)' — 2ad 2 . Integrating out the total
derivative, Eq. (3.78) reads as
-i:
-i:
dtC RW (N, a, a)
The Structure and Dynamics of the Isotropic Universe 121
Let us perform a Legendre transformation, by denning the momentum p a
conjugate to the scale factor a as
d£*w I2n 2 . K N Pa
P ^ — = ~^N aa => a = -V2^~- (3 - 80)
Hence the Hamiltonian function -/VHrw = Pa& — £rw is obtained using
Eq. (3.80) and the action (3.79) then rewrites as
-s:
It { Pa a - NH R w)
dt\paa-N _ ^_ a + 2 ^V 3
(3.81)
24tt 2 a
The Hamilton equations explicitly road as
A __ w ^__^4 + *!*™_^«!ff2, ( 3.s2b)
<9a 247T 2 a 2 k da
while the variation of Eq. (3.81) with respect to N provides the following
ivlat ion
V l 1447T 4 487T 4
In the particular case N = 1, Eq. (3.83) coincides with the Friedmann
Eq. (3.46) by virtue of Eq. (3.80). Meanwhile, substituting Eq. (3.82a) into
Eq. (3.82b), and making use of the continuity equation (3.35) in the form
^l = -3a 2 P, (3.84)
the system (3.82) is equivalent to the space component of the Einstein Eqs.
(3.47).
The variations with respect to the lapse function N and to the conjugate
variables (a, p a ) provide three independent equations, yielding a complete
representation of the dynamics of the isotropic Universe. Equation (3.84),
which allows the identification of the pressure function in the equations
of motion, has been inferred for the consistence of the Hamilton and Ein-
stein systems, but in (he La;>,rau<',i.ai> formulation it must be thought of as
preliminarily solved to give the relation p = p(a).
In the Hamillouia.n formulation, the dynamics of the isotropic Universe
resembles that of a one-dimensional point particle, with generalized coordi-
nate a and momentum p a , whose dynamics is governed by a potential term
122 Primordial Cosmology
fixed via the Hamiltonian constraint (3.83) that also states the vanishing
nature of the particle energy. 3 The correspondence between the Einstein
equations and the equations for a particle motion is a general property
of the homogeneous cosmologies, as we shall see in Chap. 8, and offers
an interesting scenario for the canonical quantization of this cosmological
dynamics.
3.3 Dissipative Cosmologies
It is interesting to analyze two relevant examples of dissipative effects that
could be able to alter the standard dynamical features discussed in 3.1, i.e.
the presence of a bulk viscosity in the cosmological fluid and the possibility
of matter creation during the expansion of the Universe.
These two dissipative effects have different origin, but both provide the
same dynamical feature of dealing with a negative pressure term. Such
phenomenological issue can have deep implications on the evolution of the
early Universe, both with respect to the nature of the singularity, and to
the morphology of its causal structure. We will discuss in some details
the cosmological pictures emerging from the inclusion of these phenomena,
treated via suitable hydrodyuamica.] approaches that make possible their
description.
3.3.1 Bulk viscosity
When discussing viscosity effects on the Universe dynamics, we have to
distinguish the so-called shear viscosity from the bulk viscosity features. In
fact, the former emerges as a result of the reciprocal friction that different
parts of a system (for example, different layers of a fluid) exert each other.
On the other hand, the latter is a macroscopic measurement of the reaction
of a fluid to compression or rarefaction and can be related to the difficulty of
the system (in our case the Universe) to maintain the thermal equilibrium
as it expands.
From this simple description of the two viscosity terms, it appears nat-
ural that the shear viscosity coefficient must vanish for a perfectly homo-
geneous and isotropic cosmological fluid, for which friction among layers is
absent by definition (this effect however survives in the presence of inho-
3 We stress however that wi u< d< ding \ itli icon n inn L IJ.ami.1 i u framework, i.e.
"H. is not straightforwardly related to the energy.
The Structure and Dynamics of the Isotropic Universe 123
mogeneous perturbations, see Sec. 3.4). On the contrary, a bulk viscosity
term is still permitted by the requirements of isotropy and homogeneity. In
particular, the bulk viscosity coefficient expectedly receives significant con-
tributions from the primordial phases of the Universe, when the expansion
rate is very high and non-equilibrium features were possibly more relevant.
Since such region of evolution corresponds to a non-trivial kinetic theory
of the cosmological medium (in analogy with what discussed in Sec. 3.1.6),
following the Landau School we consider a fluid-dynamical scheme. This
approach is based on expressing the bulk viscosity coefficient as a function
of thermodynamical parameters, mainly the energy density of the Universe.
This way, we provide a phenomenological description of the off-equilibrium
behavior of the primordial Universe which allows a simple enough charac-
terization of the viscous cosmologies. Finally, we will not take into account
effects due to the finite value of the speed of light, which are indeed relevant
when a causal thermodynamics of the Universe is involved. These correc-
tions, ensured by introducing in the problem suitable relaxation times, are
important just near the Big Bang, but nevertheless the analysis presented
below provides a proper description of the qualitative features emerging in
a viscous isotropic cosmology.
The energy-momentum tensor of a fluid characterized by bulk viscosity
takes the form
T^ v = (p + P + U v )u iUj - (P + U v ) 9ij , (3.85)
where H v corresponds to a negative pressure-like contribution H v =
— CViu\ The bulk viscosity coefficient C can be expressed as proportional to
some power s of the energy density of the fluid, i.e. C = Co/A Co and s being
constant parameters of the phenomenological model under consideration.
Since in a co-moving frame the relation V^w* = 3H holds, the continuity
equation (3.35) for the viscous Universe writes as
p = -3(p + P-3( p s H)H. (3.86)
Limiting our attention to the case of a flat model, we can make use of
the Friedmann equation (3.46) to replace the Hubble parameter inside the
parentheses obtaining
p = -3 (p + P - Vlte(op s+1/2 ) H ■ (3.87)
By virtue of the equation of state (2.15), Eq. (3.87) eventually rewrites
p = -3p ( 7 - V3^CoP s ~ 1/2 ) H . (3.88)
124 Primordial Cosmology
From Eq. (3.88), the parameter s must obey the inequality s < 1/2 for
large values of the energy density. In fact, for s > 1/2, in the asymptotic
limit p — ¥ oo (as expected near the Big Bang), the bulk viscosity term
would dominate the continuity equation. Such situation would conflict
with the well-grounded idea that viscous effects are a phenomenological
outcome of small deviations from thermal equilibrium. To describe strong
modifications of the equilibrium, probably occurred in the early Universe,
the kinetic treatment would indeed be necessary.
On the other hand, for the case s < 1/2, the same asymptotic limit
would imply a negligible contribution towards the Big Bang singularity.
Thus, on a dynamical level, the most interesting situation corresponds to
the value s = 1/2, for which the continuity equation rewrites as
p = -3p (7 - V3^Co) H = -ZipH . (3.89)
In the present model, the main effect of the bulk viscosity on the dynamics
of the isotropic Universe consists of re-scaling the index 7 as 7 — > 7' =
7-V3^Co-
In general, the bulk viscosity is expected to represent only a perturba-
tion to the perfect fluid behavior and in turn Co to be small. This implies
that 7' ~ 7 = 4/3 and that no strong deviations from the standard radia-
tion dominated behavior are induced.
3.3.2 Matter creation in the expanding Universe
The rapid time variation of the scale factor a(t) during the early phases
of the Universe evolution implies that the cosmic gravitational field, like
any other rapidly changing field, is able to create particles, as a result of
a quantum effect induced on the microscopic matter fields. On a funda-
mental level, this rate of particle creation could be calculated by analyzing
the states of certain fields, living in the expanding Universe. However, the
passage from a microscopic description of the physics of a non-stationary
background to a phenomenological macroscopic characterization of the cos-
mological fluid is highly non-trivial. Thus, like in the case of bulk viscosity,
the problem of matter creation, induced by the Universe expansion, has to
be addressed on the basis of a phenomenological model.
The proper framework to treat the cosmological particle creation was
identified by Prigogine during the '60s, and involves treating the Universe
as an open thermodynamical system. We have seen in Sec. 3.1.6 how the
continuity equation (3.35) is equivalent to the first principle of thermody-
The Structure and Dynamics of the Isotropic Universe 125
namics applied to a spatially isothermal and isoentropic co-moving volume.
In order to take into account the possibility of particle creation, we have
to restate such thermodynamical principle including a non-zero chemical
potential p
dU = 5Q- PdV + pdN , (3.90)
where N denotes the particle number. Expressing U = pV and using the
second law of thermodynamics 4 SQ = TdS = T(adN + Nda) (where S
denotes the entropy inside the co-moving volume V and a = S/N is the
relative entropy per particle), the above equation rewrites as
b P)^ + (Ta + p)^f. (3.91)
Observing that the chemical potential is defined as the Gibbs free-energy
G = U + PV — TS per unit particle, i.e. G = pN, we have that
(p + P)V= {Ta + p)N (3.92)
so that Eq. (3.91) rewrites as
*-f*-(, + *)(i-££)£. <->
We are now led to replace the standard request for an isoentropic Universe
with the weaker condition that only the entropy per particle be conserved,
i.e. da = d(S/N) = 0. The entropy and the particle number in a co-moving
volume are then linked by a direct proportionality (S oc N). Since in the
Universe, treated as an open thermodynamical system, the number of par-
ticles changes along the evolution, the total entropy is no longer a conserved
quantity and varies accordingly to the processes of matter creation. Deal-
ing with a conserved entropy per particle reduces Eq. (3.93) to the simpler
In other words, the effect of matter creation is described by an additional
negative pressure term II mc = — (p + P)dlnN/dhiV. The analysis of the
dynamical implications associated to such negative pressure requires the
4 We remind the reader that the Second Law of Thermodynamics states that the equality
5Q = TdS holds for reversible processes only. However, since U, V, S and N are
functions of state, and thus independent of the particular process, Eq. (3.91) and the
following ones hold for any transformation. On the other hand, in the presence of
s like particle creation, entropy is not conserved (dS ^ 0) but the
is still adiabatic (SQ = 0).
126 Primordial Cosmology
specification of a phenomenological expression for dlnN/dlnV. In this
respect, let us rewrite Eq. (3.94) in the form of a continuity equation as
This equation generalizes Eq. (3.35) in the presence of matter creation and
offers the proper tool to get the modified relation between the energy den-
sity and the cosmic scale factor p = p(a), necessary to solve the Friedmann
equation (which here, as in the bulk viscosity case, retains the usual form
(3.46)).
To fix the form of the particle creation rate, we focus our attention on
the case of a flat RW model (K = 0) and observe that, since the particles are
created by the time variation of the cosmological field, a suitable expression
for the ansatz we are searching for reads as
ldlnTV (H\ 2b (p\ b
3di^r s UJ = bJ ' (3 - 96)
where b is a free parameter of the theory and the two constants H and
p are related by H 2 = up/3. Substituting this ansatz in Eq. (3.95), and
eliminating the synchronous time in favor of the dimensionless variable
x = 3 In a, we get the final form of the revised continuity equation
dp
dx
(p + P) 1- m . (3.97)
Considering a generic equation of state of the form (2.15), Eq. (3.97) takes
the integrable form
and its solution, restated as p(a), explicitly reads as
p(a) = P - , (3.99)
[1 + Aa 3 *]*
where A is a constant. The dynamical implications of such relation can be
qualitatively inferred without solving the Friedmann equation. The most
relevant modification arises in the finite constant value p, taken as the en-
ergy density for a — > 0, in correspondence to any choice of the index 7.
In the present scenario with matter creation, the Universe was born by a
singularity-free solution, characterized by a de Sitter phase emerging from
t — > —00, where the scale factor would asymptotically vanish. However,
The Structure and Dynamics of the Isotropic Universe 127
for a sufficiently large value of a, i.e. when Aa 3f>7 3> 1, the energy density
regains the standard form p oc a~ 37 and the features of an isoentropic Uni-
verse with a fixed value of N are recovered. We see how the present picture
has the merit to reconcile a non-singular de Sitter-like Universe in the ear-
liest cosmological phases with a standard picture of the later evolution. Of
course, one can expect that in order to completely reproduce the Standard
Cosmological Model at later times, a certain fine-tuning of the parameters
would be required.
We conclude this section by stressing how the analysis of the dissipative
cosmologies developed so far was based on the study of the continuity equa-
tion modified to account for an additional negative pressure term. Indeed,
in Sec. 3.1.6, we saw how this equation macroscopically accounts for the
microscopic structure of the Boltzmann equation and therefore our phe-
nomenological approaches are nothing more than an effective theory of a
really complex microphysics, that describes only the general features of the
investigated phenomena.
Hence, replacing the old thermostatic pressure with the restated pres-
sure term, the Einstein equations for the isotropic Universe preserve the
same structure studied in Sec. 3.2.1. Such similarity of the isotropic dy-
namics in the dissipative and non-dissipative cases is at the ground of the
qualitative dynamical considerations outlined in the two subsections above.
3.4 Inhomogeneous Fluctuations in the Universe
3.4.1 The meaning of cosmological perturbations
The idea of a perfectly homogeneous and isotropic Universe is mainly a
mathematical notion, hardly reconciled with the morphology of the present
Universe, unless (initially small) deviations from homogeneity are allowed
on different physical scales. Indeed, as we shall see in Chap. 4, at galactic
scale the Universe dumpiness appears as a deep modification of the RW
geometry. At this level, we speak of non-linear nature of the density fluc-
tuations, because the ratio of the matter fluctuations Sp to the average
background density p is much larger than unity, i.e. Sp/p 3> 1. In this
situation, the dynamics of the Universe cannot be recovered from a lin-
ear perturbation theory of the RW metric and non-linear features of the
Einstein equations have to be involved. On sufficiently large scales, greater
than some hundred of Megaparsecs, the linearity is recovered and the notion
128 Primordial Cosmology
of a homogeneous and isotropic Universe becomes solid. Even if the fluctua-
tions on a given scale are very large, we expect that they were smaller in the
past, so that in the early Universe the fluctuations were still in the linear
regime. Thus, when Sp/p <C 1, like it is today at the scale of superclus-
ters of galaxies or like it was at the time of recombination (the very small
temperature fluctuations of the CMB trace correspondingly small density
contrasts) at all scales of cosmological interest, we have a natural approach
to describe the evolution of inhomogeneous perturbations, either if they are
scalar fluctuations in the energy density, pressure and velocity distributions,
as well as if they take the form of vector or tensor disturbances, describing
inhomogeneous rotational velocity fields (vortices) or gravitational waves,
respectively.
Before facing the proper approaches to perturbation dynamics, we pro-
vide a physical insight on why not only inhomogeneities are necessary to
explain the observed Universe, but they arc also an unavoidable feature of
the primordial Universe. In fact, even if we start with a very smooth and
isotropic Universe at very early times, a certain degree of fluctuations of
the thermodynamical parameters on different Hubble volumes is necessarily
implied. The independent evolution of such fine-tuned micro-causal regions
has to magnify the relative fluctuations, since the gravitational instability
amplifies the density contrast on a given scale, enforcing the dumpiness of
the cosmological fluid. At the end, a certain significant perturbation spec-
trum has to be generated at later stages of evolution even if we start with
a very fine-tuned homogeneous Universe.
To shed light on the impossibility to have 1 a perfectly uniform Universe,
we stress that it cannot remain in thermal equilibrium arbitrarily close to
the initial singularity. Indeed, the mean free path of a given particle species
l s is provided by the relation
l s ~ , (3.100)
n s and a s being the number density of the species and the cross-section
of the interactions that are maintaining the equilibrium. Since n s oc T 3
and typically a s oc (E s )~ 2 ~ 1/T 2 (here (E s ) denotes the mean thermal
energy of the particles), then the mean free path scales like l s oc a. Since
I Ue Bubble leugtb decreases faster toward the singularity, a primordial in-
stant has to exist when the mean free path is, on average, greater than the
microphysical scale and, in practice, it diverges. In such scenario, the Uni-
verse cannot be regarded as in a real thermal equilibrium and the request of
smoothness does not appear well-grounded. A numerical estimate provides
The Structure and Dynamics of the Isotropic Universe 129
the constraint T > O(10 16 GeV) to allow a similar situation. Since this re-
gion of the Universe evolution is expected to be in a pre-inflationary phase,
the modern idea on the origin of primordial fluctuation origin escapes, in
part, this argument, as we shall see in Chap. 5. On the other hand, it well
stresses the necessity of a certain degree of inhomogeneity even when the
primordial Universe evolution is addressed on a purely classical level (for a
discussion of quantum gravity implications in this respect, see Chap. 10).
On the basis of the considerations above, we see why the request for a
fine-tuned uniform Universe (say immediately after the Planck era) can
constitute a physical puzzle, indeed at the ground of the horizon paradox
that we will discuss in Chap. 5.
In order to face the problem of cosmological perturbations within the
framework of GR, we have to write the metric tensor in the form
9ik = 9ik + lik (3.101)
where (jik denotes the RW background term, while 7^ represents a small
perturbation (7^ -C (jik) which describes the ripples of the space-time asso-
ciated to the inhomogeneous features. Starting from such metric tensor, one
constructs the first order linearized Einstein tensor SGf, properly simplified
by the choice of a suitable gauge, such as the synchronous gauge
7,0 = 0. (3.102)
On the same level, we can perturb the cosmological fluid by a density fluc-
tuation 5p, a pressure fluctuation SP and a four- velocity disturbance 5u l ,
never co-moving with the background flow. From these quantities, together
with -fik , we can build up the first-order linearized energy- momentum tensor
5Tf and hence fixing the perturbation dynamics by the linearized equations
bG\ = k<5T*. Of course, this sot of equations is coupled to the background
dynamics and, in the case the matter behavior is provided by a kinetic the-
ory, we have to couple also the Boltzmann equation, itself separated into
background and first-order components.
As far as the scale of perturbations is much smaller than the Hubble hori-
zon and the fluid velocity fields are non-relativistic, we can neglect all GR
effects and then we deal with a very simplified system of equations, describ-
ing the gravitational instability of the Universe. In such non-relativistic
scheme, we can neglect the back-reaction of the matter perturbations on
the full tensorial structure of the gravitational field and we limit our atten-
tion to the Newton potential <& only (i.e. 700)- The expansion can be taken
into account as a background effect.
130 Primordial Cosmology
The system of non-relativistic equations consists of the continuity equa-
tion 5
d t p + V- {pv) = 0, (3.103)
(v being the velocity field) which ensures the mass conservation of the fluid.
The second Newton law of mechanics takes the form of the Euler equation
<9 t v + («-V)v + -VP + V$ = 0, (3.104)
P
while the relation between the matter distribution and the Newton potential
is provided by the Poisson equation
V 2 $=|p, (3.105)
where V is the Laplace operator.
In Sec. 3.4.3, we will take into account the Universe expansion in the
perturbation dynamics by retaining the same scheme outlined above. This
analysis, in spite of its non-relativistic character, has a real cosmological
predictivity and it is at the ground of a significant physical insight.
In order to familiarize with the Jeans approach to the gravitational
stability of a uniform static fluid, as discussed in the next subsection, we
propose a qualital ive analysis of the mechanism by which a density contrast
evolves. Let us consider an exceeding matter fluctuation 5p > over the
background level p. The self-gravity of this matter blob tends to induce a
collapse toward the formation of a structure, but such a force is contrasted
by the pressure gradients, stretching and Ualleuuij>, the denser region. The
fate of the matter density, collapse or disappearance, depends on the net
resultant of such two forces and, if they are near equilibrium, the fluctuation
will expectably oscillate.
We can estimate the condition for a collapse in the case of a spherical
inhomogeneity. In fact, the disturbances in the fluid density can lead to the
collapse only if their propagation velocity, coinciding with the sound speed
r. s . = (dP /dp) 1 ' 2 , is smaller than the limiting value
-71. <"»>
I being the radius of the spherical matter blob, say its linear size. Observing
that the mass of the system can be written as M ~ (p + Sp)l 3 ~ pi 3 , the
5 In accordance with the standard literature on the topic, we adopt the vectorial nota-
tion. Operation are intended as in the standard Euclidean case.
The Structure and Dynamics of the Isotropic Universe 131
collapse condition can take place only for matter condensations having a
linear size (i.e. a matter content) sufficiently high, namely
<~»VI- (3 - im)
This length scale is called the Jeans length and represents a natural sep-
aration between the perturbations which are obliged to undergo acoustic
oscillations and those sufficiently large to be able to collapse because of
their self-gravitation and to become the seeds for structure formation. Be-
low we will provide a more rigorous mathematical derivation of this simple
10, by stressing its relevance in a cosmologira.l implement at ion.
3.4.2 The Jeans length in a static uniform fluid
Let us now study the linear dynamics of a perturbation (5p, 5v) around a
configuration of the fluid characterized by a uniform matter density p =
const., zero velocity v = and constant sound velocity v s = const., so
that P = const, and SP = v 2 Sp. Under these hypotheses, the continuity
Eq. (3.103) to first order in perturbation terms takes the form
d t 5p + pV-v = 0. (3.108)
In the same way, the Euler equation (3.104) acquires the linearized form
2
d t 5v + ^V5p + \75$ = 0, (3.109)
P
where #$ denotes the gravitational potenl ial associated to the perturbation
and is related to 5p via the Poisson equation as
V 2 d$=-8p. (3.110)
Taking the divergence of the vector in Eq. (3.109), we get the scalar equation
d t { V • 5v) + -4 V 2 5p + V 2 <5$ = . (3.111)
Using Eq. (3.108) to express the divergence of 6v and Eq. (3.110) to remove
the Laplacian of (5$, we arrive at a linear second order equation in dp only
d 2 t 5p - v 2 s V 2 Sp - ^pSp = . (3.112)
The solution of this equation can be expanded in a Fourier integral so that
we can study the behavior of a generic mode
132 Primordial Cosmology
where A and w denote the wavelength and the frequency of a plane wave,
respectively, while ft denotes the unit vector of its propagation direction.
Substituting Eq. (3.113) into Eq. (3.112), we get the key relation
Vl
A 2 8tt 2
^\TI-^- (3-114)
As far as the wavelength of the perturbation is sufficiently small, the fre-
quency is real and the corresponding modi 1 oscillates; on the other hand,
when the Jeans condil ion
/8tt 2
A>Aj = u s W — (3.115)
holds (this value of Aj closely resembles the previous estimate (3.107)),
the frequency becomes an imaginary number u> = ±ia. In this region of
wavelengths, exponentially growing modes appear for u = ia (as well as
absorbed ones for lo = —ia) and the corresponding matter perturbation
can collapse, according to
Pin exp I
5 p = S Pin exp<—n-r + at\. (3.116)
In view of the hypotheses considered here, this analytical behavior is
predictive if 8p/p <C 1 only. The relevance of this analysis relies on the
information that matter perturbations with a linear dimension much larger
than the Jeans length are gravitationally unstable and proceed towards a
collapse in stable structures. Despite this result has been obtained for a
static uniform medium, the Jeans length can be extrapolated to the case
of an expanding Universe, preserving its present meaning: this is the main
task of the following section.
3.4.3 The Jeans length in an expanding Universe
In order to extend the previous Jeans analysis to an expanding background,
we have to search for an exact solution of the three equations (3.103),
(3.104) and (3.105), under the hypothesis of a matter dominated Universe
P « p, 7 ~ 1.
For 7 = 1, the unperturbed energy density p scales as 1/a 3 and hence
d t p = p = —3Hp, and then the continuity Eq. (3.103) provides
\7-v = 3H => v = Hf. (3.117)
The Structure and Dynamics of the Isotropic Universe 133
The background velocity of the cosmological fluid corresponds to the ge-
ometrical velocity of expansion. On the same level, Eq. (3.105) gives the
relation
d§ = -of. (3.118)
6
The current choice for the background quantities automatically solves
Eq. (3.104), as far as Eq. (3.48) is taken into account for the case P ~ 0. Fi-
nally, the background dynamics is completed by Eq. (3.46), which provides
the scale factor evolution.
Linearizing Eq. (3.103) with respect to perturbations, we get
d t Sp + 3HSp + Hr-VSp + pV ■ Sv = , (3.119)
while the Euler Eq. (3.104) up to first order reads as
d t 5v + HSv + H(r-\7)5v = -^Wp- V<5$. (3.120)
P
Actually, the sound velocity is no longer a constant as in the previous Sub-
section. Once assigned a polytropic relation for the perturbation behavior,
like P ex / o 4 / 3+e (e > 0), we get the time evolution of the sound velocity as
r 2 x p l /- ! + c . Finally, we get the perturbed Poisson equation
V 2 <5$ = ^Sp => V<5$ = (^Spj f. (3.121)
Taking the curl of Eq. (3.120) and observing that the right-hand side van-
ishes, the following relation follows
d t (V A Sv^j + 2HV ASv + H(f- V) (vAffi)=0. (3.122)
This equation involves only the rotational component of the velocity per-
turbation and, as we shall see below, it carries a clear physical information.
Analogously, we can take the divergence of equation (3.120) obtaining
d t (v-Sv) +2H\7-5v + H(r-V)(v ■ Sv) =-^\7 2 Sp- -dp, (3.123)
where we also made use of Eq. (3.121). If we now define the fractional
density contrast as S = Sp/ ' p, then Eq. (3.119) takes the simpler form
dtS + Hf- V8 + V -8v = 0. (3.124)
It is now convenient to expand the spatial dependence of the quantities
S and v in plane waves of the expanding Universe. More precisely, let us
134 Primordial Cosmology
analyze the behavior of each single mode of the Fourier transform of the
spatial dependence of these objects, i.e.
5 = 5 k (t)exp{ik phys n-f} (3.125)
5v = v k (t) exp {ik phys n ■ r} . (3.126)
Here fc p h ys = 27r/A p hys denotes the wave number associated to the phys-
ical wavelength A p hy S which is proportional to the scale factor since
d(X~ h )/dt = —H\~ h . In view of the expansion above as in (3.125),
Eqs. (3.122), (3.123) and (3.124) rewrite respectively as
v l l + Hv l l = (3.127a)
vl + ^p) h (3.127b)
4 = -»Wfc> (3.127c)
where the superscripts || and _L denote the velocity components, parallel
(vl = (n ■ i>k)n) and transverse (v^ = ilk — cj!) to the plane wave direction
n, respectively.
Equation (3.127a) states that the rotational modes of the perturbations
decay because of Universe expansion, i.e. vl oc 1/a. The compressional
modes, described by the remaining two equations, have a non-trivial dy-
namics and wo can shod light on it by deriving Eq. (3.127c) with respect
to t, getting
5 k = -ik phys (vi - Hv£) . (3.128)
Using Eq. (3.127b) and taking into account again Eq. (3.127c), we obtain
the final fundamental equation for the density contrast
5 k + 2H5 k + (« s 2 fcp\ ys - |p) 8 k = , (3.129)
which reproduces the Jeans dispersion relation (3.115) for a = const.
In order to obtain some information about the perturbation fate, we
have to specify Eq. (3.129) in correspondence to a given cosmological model.
Since the region of Universe evolution during which the Jeans mass is the
relevant scale for the matter dynamics is the one characterized by a negli-
gible spatial curvature, then we can deal with the K = model, without
significant loss of generality. For such model with P ~ 0, the scale fac-
tor behaves as a a t 2 ' 3 and therefore the energy density takes the time
dependence
*=d?' (3 - 130)
s of the Isotropic Universe
while the sound velocity v s acquires the time evolution v s ~ t 1//3 £ . Hence
Eq. (3.129) explicitly rewrites as
*♦(«£-£>
3t -' 1,2-2, ->,2/-- ' ( 3 - 131 )
where the constant C takes account for the amplitude of the first term
in parentheses of Eq. (3.129). As time goes by, this term decreases more
rapidly than np/2 and becomes, sooner or later, negligible. Such situation
corresponds just to the time dependent Jeans condition
A phys »^ — . (3.132)
It is immediate to recognize that, under such conditions Eq. (3.131)
admits the solution 5k ~ a(t) ~ t 2 ^ 3 , which increases with time. A more
careful analysis, based on the behavior of the Bessel functions entering the
solution of Eq. (3.131) would refine the inequality (3.132) by a multiplying
factor ^3/2.
Let us now briefly discuss the solution of the equation for the density
contrast during the radiation era. We are interested in the behavior of the
density contrast of matter components (for example baryons), while the
main contribution to total density of the Universe is given by the photon
component. In this case, Eq. (3.129) takes the form (assuming that the
radiation component is unperturbed)
<5t / 4:TT 2 V 2 k \
4 + y+ -^--PJ4 = 0, (3.133)
where in this case p and 5k denote the background density of matter and
the corresponding density fluctuation, respectively.
Let us consider the limit of very large wavelength of the perturbation.
i.e. perturbations well above the Jeans length, so that the first term in
brackets can be neglected. Moreover, the density of photons p 1 during the
radiation-dominated era can bo calculated from the Friedmann equation
and is equal to
P7 = i ^. (3.134)
The matter density p can be written as ep 7 with e <C 1, and then the
second term in brackets is of order e5k/t 2 . This is much smaller than the
first derivative term, which is of order 5k /t 2 , so that it can also be neglected.
The equation can thus be approximated as
4 + -^ = 0, (3.135)
Primordial Cosmology
and admits the solution
S k = S k (t in )\l + Alog(-pj}. (3.136)
Thus the general solution is the superposition of a constant plus a logarith-
mic term. This means that the perturbations can grow at most logarithmi-
cally, i.e. much slower than they would do in a matter-dominated Universe.
This result, known as Meszaros effect, is at the basis of the fact that the
onset of structure formation has to wait until the Universe becomes matter
dominated. 6 The physical reason for this result can be identified with the
faster expansion during the radiation-dominated era and then to its more
relevant damping effect.
The analysis here addressed shows that the Jeans length retains the
same physical meaning and essentially the same structure even in the ex-
panding matter-dominated Universe. Furthermore, we got the important
information about the capability of the gravitational instability to mag-
nify density contrasts during the matter-dominated era, according to the
law 5 ~ a, and on the reduced (logarithmic with time) growth of fluctua-
tions during the radiation-dominated era. Such behavior explains how the
small perturbations of the early matter-dominated Universe can increase
to reach the non-linear regime and therefore they are the seeds from which
the structures in the presently observed Universe have been formed.
3.5 General Relativistic Perturbation Theory
In this Section we will deal with the full, general relativistic treatment
of the evolution of small perturbations over a RW background. As we
shall see, this involves writing the metric as the RV\ metric plus a (small)
perturbation. Then we compute the perturbed Einstein tensor SGij, the
perturbed energy-momentum tensor STij, and finally obtain the perturbed
Eiusleiu equations ;>,ovcnung the evolution of the metric, as well as of the
matter-energy. In the following we will neglect the spatial curvature, i.e.
we will suppose to deal with perturbations at scales much smaller than the
curvature radius of the Universe. This is not a strong limitation, since we
know that \CIk\ ^ 10 -2 , so that the present curvature radius is at least a
6 Of course, modes that enter the horizon much earlier than the time of matter-radiation
equality can grow appreciably during the radiation-dominated era, and this has to be
taken into account in detailed calculations.
The Structure and Dynamics of the Isotropic Universe 137
hundred times larger than the Hubble length, and the flat approximation
is appropriate even for modes that are well above the horizon (i.e. pertur-
bations with wavelength much bigger than La). The approximation was
even better in the past, when the curvature was less important.
3.5.1 Perturbed Einstein equations
We take the unperturbed metric to be the flat RW metric as in Eq. (3.2)
where we adopt Cartesian coordinates for the spatial part of the metric,
i.e. h™ = 5 a f). The only non- vanishing Christoffel symbols are 7
f ° ap = aa5 a p (3.137a)
f^ = ±8<$ (3.137b)
while the non-vanishing components of the unperturbed Ricci tensor are
R 00 = -3-, (3.138a)
R a p = (2a 2 + aa)5 aP . (3.138b)
Let us consider a small perturbation 7^- to the RW metric g~ij as in
Eq. (3.101), and adopt the gauge (3.102) to fix some of the components of
jij. Equation (3.102) states that the perturbed metric is also synchronous,
i.e. g o = goo = 1 an d goa = goa = and for this reason such gauge is
called synchronous gauge. It was the one used in the original paper in 1946
by Lifshitz on cosmological perturbations, and it is still very popular today,
mainly because of the numerical stability of the equations written in this
gauge. A drawback of this choice is that the condition (3.102) does not fully
exhaust the gauge freedom: this gives rise to spurious, unphysical gauge
modes among the solutions to the equations that need to be recognized and
eliminated.
The perturbation to the inverse of the spatial metric 7 a/3 is related to
7a/3 by 7 Q/3 = — a _4 7 Q/ 3, as it can be noted by writing g a/3 = g a/3 +7"^ and
imposing that g a ^g^ = o~a-
As anticipated above, our goal is to write the perturbed Einstein equa-
tions that we will analyze in the more convenient form
8Rij = k ISTij - 1<S (ft 3 T) . (3.139)
7 Also in this Section, overbar is adopted to denote an unperturbed quantity.
sr 29, 2010 11:22
138 Primordial Cosmology
The perturbations to the Christoffel symbols are given, to first order in the
small quantities j a/ 3, by
ST° 00 = (3.140a)
ST° a0 = (3.140b)
5T% = (3.140c)
^l f 3 = -\i afi (3.140d)
«To8 = - ^2 (7^-2^/3) (3.140e)
ST ^ = ~Ja 2 ~ (9m7q ^ + ^ 7mq " dal ^ ] (3 - W0f)
The perturbations to the Ricci tensor can be expressed in terms of the 5T l - k
5R tl = 5R k ikj = d e (STij) - dj (ST e ie ) + (5T e me ) T™
+ T e mi (ST%) - (6T e mj ) T™ - T l mj {8T™ t ) . (3.141)
Using Eqs. (3.140) above, we get
SRoo = ^ ka " 2^7aa + 2 (^ - ^\ 7c J (3.142a)
6Ro a = \d t [^ {derm - d pla A (3.142b)
11a a 2
SR a0 = --7a/3 + 2" (7a/3 - 7 W ^) + ^ (- 2 7a/3 + 7 W <W)
+ ^ (^7^ + 3A7a/» " ^7^ - S M ^7 aM ) (3.142c)
Here, and for the rest of this Chapter, we adopt the convention that re-
peated indices are always summed, even if they are both covariant or con-
travariant. For example, we have that 7 MM = 7 M „<^ = Eu = n7w Of
course, this is not the trace ^ = "f a p9 a ^ of 7 Q ^, although the two are
related by 7^ = --f^/a 2 .
Let us turn our attention to the right-hand side of the Einstein equa-
tions. Defining the source tensor Sij
SijEETij-lg^T, (3.143)
The Structure and Dynamics of the Isotropic Universe 139
the perturbation SSij to the source tensor is given by
SSa = 5T tJ - )p l3 f - \g l3 bT . (3.144)
Neglecting for the moment dissipative processes like viscosity or heat con-
duction, we consider for the energy-momentum tensor T t j the perfect fluid
form (2.14), that we rewrite here for convenience
T ij = (p + P)u i u j -Pg ij . (3.145)
The perturbation to the energy-momentum tensor can be described in terms
of the perturbed density p = p + Sp, pressure P = P + 6P and four- velocity
Ui = Hi + Sui of the cosmological fluid. In particular, one has
ST i:j = (dp + 8P)uiUj + {p + P){uiSuj + u 3 5 Ul ) - P ll3 - SPg t3 . (3.146)
We recall that the zeroth-order four-velocity is given by Ui = (1, 0, 0, 0) and
moreover, the identity UiU 1 = 1 (that is true at all orders) can be perturbed
to give
= 5{u iU l ) = 5{g l3 u lUj ) = S(g 00 u u ) = 25u . (3.147)
This yields for the perturbations to the energy-momentum tensor
5T 00 = Sp, (3.148a)
6T aa = (p + P)Su a , (3.148b)
5T aP = -Pj a p + a 2 5PS a(s . (3.148c)
The trace of the energy-momentum tensor is given by T = p — 3P so that
the associated perturbation is given by
8T = Sp-3SP. (3.149)
By putting together Eqs. (3.142), (3.148) and (3.149), we finally get the
perturbed Einstein equations
(00) j aa -2-% a + 2[^--) laa = K a 2 {5p + 36P), (3.150a)
(a/3) 7 Q/3 - - (7 a/3 - 7„m<W) + 2 ^(27a/3 - 7 W <M
- ^ {d a d filaa + d^d^ la p - d a dfftf - d a d platl ) (3.150c)
= K [(p-P)h af3 -a 2 (Sp-SP)5 af3 }.
140 Primordial Cosmology
These equations can be further simplified by introducing the rescaled metric
perturbation j a p = jap/a 2 , so that Eqs. (3.150) rewrite as
(00) % a + 2-% a = K {5p + 35P), (3.151a)
(0o) d a ^ -d % = 2 K (p + P)6u a , (3.151b)
{a/3) %p + - (3%p + 7mm<W) ~ ^(dadpy^ + d,,d^ a p
- d a d^%p - 0„0/»7a„) = k(SP - S P )5 a p. (3.151c)
In deriving Eq. (3.151c), we have used the two background equations (3.46)
and (3.47) to express the unperturbed density p and pressure P in terms
of the scale factor a(t) and of its time derivatives.
3.5.2 Scalar-vector-tensor decomposition and Fourier ex-
pansion
It is convenient to decompose the perturbation ~„ ; into its scalar, vector
and tensor components. This corresponds to split j a p into parts that be-
have differently under spatial rotations with the advantage that the scalar,
vector and tensor components are decoupled and thus evolve one indepen-
dently from the other. The spatial metric perturbation (as well as any other
symmetric tensor) can be decomposed as
%p = |<W + lip + "iU + 7<tf, (3-152)
where 7 = j aa is the Euclidean trace of 7 Q/ g. The three components 7%,
7^3 and 7^ are respectively called the longitudinal, solenoidal and trans-
verse parts of 7a/3, and are traceless by construction. In addition, they
satisfy the conditions
eafrdfid^ = (3.153a)
dadplif} = (3.153b)
a Q 7^ = 0. (3.153c)
In other words, the first condition states that the divergence of the longitu-
dinal part 7]^ is longitudinal itself (i.e. curl- free); the second means that
the divergence of the solenoidal part 7^3 is transverse (i.e. divergence- free);
the third that the transverse part 7^ is, as the name suggests, transverse. 8
8 In fact, another possible nomenclature is to call y^„ doubly longitudinal - | , singly
longitudinal and 7J3 doubly transverse.
The Structure and Dynamics of the Isotropic Universe 141
From the conditions above, it follows that 7JL can be expressed in terms
of a scalar field p as
lip = (d a df, - \5 a pd,d v \ n, (3.154)
while 7^ can be expressed in terms of a transverse vector field V a
xfe p = d a V p + dpV a ; d a V a = . (3.155)
The components 7 and 7% (or equivalently p) represent the scalar part
of ja/3, la/3 ( or equivalently V a ) represents its vector part, and finally 7^,
that cannot be obtained from the gradient of a scalar or a vector, represents
its tensor part.
A similar decomposition can be done for the velocity perturbation 6u a
which can be divided into a parallel part Sua and a transverse, divergence-
less part Su^ . The parallel can be expressed as the divergence of a scalar
field Su, so that we have
Su a = d a (Su) + 5u^ (3.156)
d a 5ui = 0. (3.157)
After the metric perturbation and the velocity field have been decom-
posed in this way, the Einstein equations decompose as (V 2 = (),,(),,)
• Scalar modes:
^ + 2-j = k(S P + 3SP) (3.158a)
d a (7 - V 2 /i) =3n(p + P) d a 5u (3.158b)
7 - V 2 /i + 3- (27 - V 2 /i) - 4 y2 (7 " V 2 /i) = 3k(SP - 5p)
(3.158c)
d a dJii + 3%+^V 2 n-^)=0. (3.158d)
The scalar modes are compressional modes, involving the pertur-
bations to the density, pressure and to the irrotational part of the
velocity field of the fluid. They are thus the most interesting, since
they are related to the growth of density fluctuations and then to
the linear phase of structure formation. We note that we have four
equations for the five unknowns 7, fi, Sp 7 5P and Su. In order to
close the system we have, if possible, to specify the equations of
state P = P(p). For example, for particles with very low thermal
142 Primordial Cosmology
velocities (like cold dark matter) we can simply put P = (and
fa = as well), while for ultrarelativistic particles like photons or
light neutrinos we can use P = p/3. In general, however, a proper
kinetic treatment should be performed to close the system, cou-
pling the Einstein equations to the Boltzmann equation describing
the evolution of the perturbations to the energy-momentum tensor,
as we shall see at the end of this section.
• Vector modes:
\7 2 V a = -2k(p + P)5u^, (3.159a)
d a (v p + 3^Vfij=0. (3.159b)
The vector modes represent the vorticity components of the fluid.
As we will show below, the conservation of the energy-momentum
tensor implies that for a perfect fluid the quantity (p + P)5u^ de-
cays as a -3 . For this reason vector modes are in general not very
relevant for the cosmological evolution.
• Tensor modes:
lip + ^llp ~ ^ y2 7J/3 = • (3.160)
It is easy to recognize that this is the wave equation with a damp-
ing term proportional to H. In fact, the tensor modes represent
gravitational waves propagating in the expanding Universe.
Einstein equations in Fourier space. The next step in simplifying the
perturbation equations is to Fourier transform the spatial dependence of all
the quantities involved, i.e. the metric and stress-energy perturbations. For
example, we consider the Fourier transform ///, .(/.) of //(.?. /.)
p k (t)= f t i(x,t)e i ^d 3 x (3.161)
and analogously for 7, 7II, 7 T , dp, 5P, Su and 6u^. With a slight abuse
of notation, we drop the subscript k and keep using the same symbol for a
given quantity and its Fourier transform. By doing this, the above equations
rewrite as
The Structure and Dynamics of the Isotropic Universe 143
• Scalar modes:
^ + 2-^ = k(S P + 3SP) (3.162a)
i / + k 2 fi = 3n(p + P)5u (3.162b)
k 2 V a = 2k (p + P) 5ui (3.163a)
V a +3-V a = Q. (3.163b)
Tensor modes:
72^=0.
3.5.3 Perturbed conservation equations
Now we will derive the conservation equations satisfied by 5p, SP and
5u a for a perfect fluid. The conservation of the energy momentum ten-
sor VjTj = gives, to first order in perturbations, the expression
<5(V;t;) = 8(diTj) - T%5t; + fiiSTf - sr^fi + sr^fj = o . (3.165)
The j = component of this equation gives the equation of energy conser-
vation
d t 6T$ + d a ST* + 3-5T ° - -8T2 = l(p + P), (3.166)
while the j = a component gives the equation of momentum conservation
d t ST° + dpST? - aaST* + 2-ST° = . (3.167)
For a perfect fluid, the conservation equations rewrite as
8p + 3-(8p + 8P)- ^f-d a 5u a = l(p+P), (3.168a)
d t [{p + P)Su a ] + 3-{p + P)Su a - d l SP = . (3.168b)
144 Primordial Cosmology
Recalling the decomposition of Su a into its longitudinal and transverse
components, we can split the conservation equations into scalar and vector
parts. The equation of energy conservation is already purely scalar and
simply rewrites as
Sp + 3^(5p + SP) - ^^\7 2 5u = l(p + P). (3.169)
The part of the momentum conservation equation that is proportional to a
longitudinal vector (and then to the derivative of a scalar) is
d t [(p + P)Su\ +3-(p + P)Su-6P = 0. (3.170)
Finally, the part of the momentum conservation equation that is propor-
tional to a transverse vector is
d t [(p + P)8ui] + 3-(p + P)5ut = . (3.171)
In particular, this last equal ion implies that the quantity (p + P)5u^ scales
like 1/a 3 .
The conservation equations are not independent of the field equations.
However, in the case of non-interacting fluids, they are satisfied separately
by each component. For example, if we consider a Universe filled by non-
relativistic matter and by radiation not interacting with each other, we can
write separate energy and momentum conservation equations for matter
and for radiation. In this case, the conservation equations really carry
additional information with respect to the field equations. Moreover, when
performing a numerical integration of the field equations, it is useful to use
the conservation equation to check the validity of the numerical solution.
3.5.4 Gauge modes
Here we will show how the synchronous condition 7^0 = does not ex-
haust all the gauge freedom in the Einstein equations. Let us consider a
coordinate transformation
x l -+ X H = x i + e i {x 3 ) (3.172)
where e is a small quantity. This induces a transformation in the metric
tensor given by
, , ., , , dx l dx m , . de l de e , _ ,
In this equation, x and x' that appear on the opposite sides correspond
to the same physical point, which has different coordinate labels in the
The Structure and Dynamics of the Isotropic Universe 145
two reference frames. However, we are interested in the change, after the
transformation, of the value gij evaluated at the same coordinate value
x, which will correspond in general to two different physical points. The
values of the metric tensor in the two points x and x' (both referring to the
transformed frame) are related by
g' ij (x')=g' ij (x) + ^ f ^e e (x). (3.174)
Then, putting together Eq. (3.173) and Eq. (3.174) we get
U-) = 9^)-m d ^-m d ^-%\ (3.175)
where gij is the unperturbed metric. In other words, this last formula is
taking into account that the metric tensor evaluated at a given coordinate
value is changing because the metric tensor at a given physical point is
changing according to Eq. (3.173), and also because the physical point
associated to the coordinate value has changed according to Eq. (3.174).
Now we can attribute all the change in g t j to a change in the perturba-
tion 7^, so to leave the unperturbed metric unchanged
A7«C«0 = iifr) ~ 7tf (*) = 9ij(x) - 9ij (x)
_ de e _ 8e e d 9ij t
= - 9u w- 9ti w-d? e - (3 - 176)
As usual, the spatial part of Cj can be decomposed into a parallel and a
transverse part as
e a = el + ei = d a E + e^ . (3.177)
Let us consider a gauge transformation
dt
W)
with e-^ arbitrary. Using Eq. (3.176), it is straightforward to check that
after this transformation the new metric is still synchronous, i.e. A700 =
Aj a o = 0. However, the spatial part of the metric perturbation changes by
the quantity
A-fap = 2aaeoS a i3 — d a ep — dpe a
= 2aae S al 3 - 2d a d E - (d a ej + dpe£)
= ( 2doe - -V 2 £ J 5 a p - 2 (d a dp - ^V 2 J E - (d a ej + dpe^) ,
eo{x,t) = F(xi) E(x,t) = -a 2 (t)F(x^JJ^ (3.178)
146 Primordial Cosmology
where the expression on the last line corresponds to the by now familiar
decomposition of A-f a p into a trace, a longitudinal and a solenoidal part in
that order (the transverse part is missing, however). It is now clear that the
scalar, vector and tensor perturbations transform according to (we switch
to the rescaled perturbation ~f a p = ^ a p/a 2 )
/\j = G-e --L\7 2 E, (3.180a)
A / u=-^, (3.180b)
-L
AV a = — f, (3.180c)
A7J0 = 0. (3.180d)
Tensor perturbations are unaffected by gauge transformations, though, on
the other hand, the scalar and vector perturbations do not, in general,
remain unchanged under a gauge transformation. In particular, under the
transformation defined by Eq. (3.178) the scalar perturbations transform
A7 = 6^ + 2V 2 j f|^, (3.181a)
Ap = 2F j -^-. (3.181b)
The components of the perturbed energy-momentum tensor transform in
the same way as j a0 (see Eq. (3.176)), i.e.
A(.T,) = -T^-f^-^, (3.182)
ensuring that the field equations remain unchanged after the gauge trans-
format ion.
Equation (3.182) implies that, under the transformation in Eq. (3.178),
the scalar energy-momentum perturbations Sp, SP and Su transform as
A{6p) = -Fp, A(SP) = -FP, A(Su) = -F. (3.183)
The fact that the field equations are invariant under a gauge transforma-
tion, but nevertheless not the metric and energy-momentum tensor pertur-
bations, implies that, if 7 or Sp are solutions, then 7 + A7 and Sp + A(Sp)
(with A7 and A(Sp) given by Eq. (3.181a) and by the first of Eq. (3.183), re-
spectively) are also solutions. Moreover, since the field equations are linear,
A7 and A(Sp) are solutions themselves. Of course, they cannot represent
any physical disturbance to the metric or to the density field, since they
The Structure and Dynamics of the Isotropic Universe 147
can be put to zero by a suitable coordinate transformation. However, the
gauge ambiguity can be removed if there is a component of the fluid, like
cold dark matter, whose particles have very small thermal velocities and
are thus essentially at rest in the co-moving frame. In this case, we know
that for such a fluid P, 5P and Su all vanish, so that the right, physical
gauge can be found as the one where P = 5P = Su = 0.
3.5.5 Evolution of scalar modes
Here we study the evolution of scalar perturbations, with particular regard
to their behavior outside the horizon. The evolution of scalar modes is
described by the four Eqs. (3.158) once an equation of state P = P(p)
has been specified. However, it is more convenient to trade two of the field
equations for the two conservation Eqs. (3.169) and (3.170); in particular,
considering Eq. (3.158a) the conservation equations allows to reduce by one
the number of unknowns, since the scalar perturbation p does not appear.
Considering adiabatic perturbations (that is appropriate for a single
fluid) we have that 5P = v^Sp, where v s is the sound speed of the fluid.
Equations (3.158a), (3.169) and (3.170) can be rewritten as
7 + 2- 7 -3 — (1 + 3v s 2 ) <5 = , (3.184a)
6 + 3-(v*-w)6-(l + w) (^jSu + | ) = , (3.184b)
Su-3-wSu 2—5 = 0. (3.184c)
These three equations form a closed system for the three unknowns 7, 6
and Su. The sound speed can be related to the equation of state parameter
by noting that
In most cases of interest, w is constant (for example, for non-relativistic
matter w = 0, while for ultrarelativistic matter w = 1/3), so that we
can safely assume that v 2 s = w and the equations reduce to (switching to
148 Primordial Cosmology
fc-space, so that V 2 — > — k 2 )
7 + 2-^-3— (l+3w)<5 = 0,
(3.186a)
8 + (l + w)(^6u-£) = 0,
(3.186b)
<5m - 3-w5u — <S = .
(3.186c)
H
The scalar velocity perturbation Su is more conveniently expressed in terms
of the quantity 0, defined through
diST l = (p + P)0 , (3.187)
so that = —\7 2 Su/a 2 (9 = k 2 5u/a 2 in fc-space) and the equations can be
recast as
7 + 2-^-3^(l + 3u;)<5 = 0, (3.188a)
S + (l + w)(e-^J=0, (3.188b)
a k 2 w
6+-(2-3w)0-- I T- -<5 = 0. (3.188c)
In order to study the behavior of the perturbations outside the horizon
(i.e., fcphys — > 0), we neglect the last term in Eq. (3.188c). Using the fact
that a/ a = H ex 1/t, the system of Eqs. (3.188) admits simple power-law
solutions of the form
7, 5 oc t a , (3.189)
floci"- 1 . (3.190)
Since the system is fourth-order, there will be four independent solutions
of this kind. In particular, during the matter-dominated (MD) era (w =
and H = 2/3i), the four solutions correspond to
' "' ! (MD) , (3.191)
so that a general solution for 6 is
5 = A + Bt- 1 + Ct- 1/3 + Dt 2/3 (MD). (3.192)
On the other hand, during the radiation-dominated (RD) era {w = 1/3 and
H = l/2t) the four solutions are
■{*-.H
(RD),
The Structure and Dynamics of the Isotropic Universe 149
so that a general solution for 5 is
S = A + Br 1 +Ct 1/2 + Dt (RD). (3.194)
In both matter- and radiation-dominated cases it can be shown that the
first two modes (those proportional to A and B) are unphysical, gauge
modes that can be put to zero by a suitable coordinate transformation (see
Sec. 3.5.4). On the contrary, the modes proportional to C and D represent
actual density perturbations. Thus, the fastest growing modes evolve like
t 2 ' 3 and t during the matter- and radiation-dominated eras, respectively.
In terms of the scale factor, since a oc t 2 / 3 (MD) and a oc t 1 / 2 (RD), the
density contrast is given by
(MD),
(RD).
(3.195)
This remarkable result also holds for imperfect fluids (see Sec. 3.5.7), be-
cause dissipative effects cannot operate on scales larger than the Hubble
horizon.
3.5.6 Adiabatic and isocurvature perturbations
In this section we briefly discuss how the density pert urbat ions in the early,
radiation-dominated Universe, are characterized. These primordial fluctu-
ations are the initial conditions from which the perturbations are evolved,
and can be defined in terms of their power spectrum (see Sec. 4.2.3). The
issue of how these primordial fluctuations are generated, and the prediction
for their power spectrum, will be dealt with in Sec. 5.6.4 in the framework
of the inflationary paradigm. Here we focus on the way in which the initial,
super-horizon perturbations can be decomposed according to their physical
properties.
There are two different kinds of primordial fluctuations, the so-called
adiabatic (or isoentropic) and isocurvature (or entropic) fluctuations. The
distinction between the two is that adiabatic perturbations are perturba-
tions in the total energy density of the system, while in the case of isocur-
vature perturbations the relative fluctuations between the different compo-
nents are arranged in order to compensate and leave the total energy density
unperturbed. It is clear that the latter can arise only in a system with two
or more distinct components. Then adiabatic perturbations represent fluc-
tuations in the intrinsic scalar curvature, 9 that is instead left unperturbed
9 For this reason, they are also called curvature perturbations.
150 Primordial Cosmology
in the case of isocurvature perturbations (hence the name 10 ). A generic
fluctuation can be written as a sum of these two kinds of perturbations.
For simplicity, let us consider the simple case where only a radiation
(w = 1/3) and a matter component (w = 0) are present. The adiabatic
condition in this case reads as
i<5m = ^ rad , (3.196)
where as above the density contrast 5{ of the ith component is defined as
Si = 5 pi J pi. Given that p m oc T 3 and p ra( j oc T 4 , while the number densities
"m.rad of both matter and radiation scale like T 3 , so that p m oc n m and
p ra d oc n rad , the adiabatic condition (3.196) implies
^m = ^»d = ^ j (3197)
where the last equality follows from the fact that the total entropy density
s is dominated by the radiation component and from Eq. (3.43). Then we
have
tf(— ) = — -^Ss = 0, » = m, rad. (3.198)
This explains the reason why Eq. (3.196) is called adiabatic condition: it
implies that the relative fluctuation between the number density of any
species and the entropy density vanishes, or, in other words, that the num-
ber of particles per comoving volume is left unperturbed. In general, in
the presence of N components of the system i\, 12, ■ ■ ■ , in, the adiabatic
condition is restated as
^ = ^ = ... = ^^ = ^, (3.199)
1 + Wil l + w l2 l+w lN 1 + w' V '
where 5 and w are defined in terms of the total density and pressure of the
fluid.
The deviation from the adiabatic condition (3.196) can be expressed by
defining the non-adiabatic fluctuation S as
S = S m - h rad (3.200)
so that in terms of the newly-defined quantity, the condition reads S = 0.
In order to have isocurvature perturbations, it is thus necessary that 5^0.
For an isocurvature fluctuation, 5p = and then
dp = Sp m + <5/9 ra d = S m p m + C> ra d/O ra d = => <5 ra d
duel nations were mainly referred to as isothermal fluctua-
really so, but tin lami 1 lien out of usage.
The Structure and Dynamics of the Isotropic Universe 151
SO that
S = 5 m (l + - A ^-\ ~<5 m ^0, (3.202)
where the last approximate equality holds during the radiation-dominated
era, when p rac i 3> p m - Once <5 ra d has been fixed, the most general mat-
ter fluctuation can be written as a combination of an adiabatic and an
isocurvature part
5 m = A + S (3.203)
where A = 3<5 rac i/4. The two quantities A and S, or, better, their primordial
power spectra Vj\.{k) and Vs(k) completely specify the initial conditions in
the early Universe from which the perturbations have evolved. This is
mathematically equivalent to specify the initial spectra for the matter and
radiation components V ra d(k) and V m (k).
For a system with N > 2 components, the entropic perturbation for
every pair of components (/, j) is defined as follows
5i . = -Jl- - -A- (i,j = l,...,N) (3.204)
and adiabatic perturbations are characterized by the vanishing of all the
<Sjj's. In general, there is one adiabatic perturbation mode and N — 1
isocurvature modes, corresponding to the original N degrees of freedom of
the system.
Loosely speaking, purely adiabatic fluctuations are present when the
different density perturbations all originate from the same, "fundamental"
fluctuation (so that, in some sense, there is only one degree of freedom in
the system); this is for example the case in single-field inflationary models,
where the density perturbation arise from primordial quant um fluctuations
in the scalar field il il i i >on i.lil I inflation On the contrary, as noted
above, isocurvature fluctuations need the presence of at least one more com-
ponent. This condition, albeit necessary, is not sufficient by itself for the
presence of isocurvature fluctuations: the absence of thermal equilibrium
between the extra degree of freedom and radiation is also required. Thus
isocurvature fluctuations can be generated in mull iple-field models of infla-
tion, or by some dark matter candidates like the axion, that were never in
thermal equilibrium with radiation.
3.5.7 Imperfect fluids
In the above derivation we assumed that the cosmological fluid can still
be described in terms of a perfect fluid. However, this is in general not
152 Primordial Cosmology
true, because dissipative effects like viscosity or heat conduction can in
principle be relevant, at least at the perturbation level. These effects can
be taken into account, by adding a term U t j , called anisotropic inertia, to
the energy-momentum tensor, i.e.
T}J = T^ F + Hy = (p + P)uiUj - Pg l3 + n 4i . (3.205)
This equation must be thought of as a definition for the anisotropic inertia
term, encoding the deviations from the perfect fluid behavior. Once the
extra term is introduced, an ambiguity arises in the definition of the density,
pressure and fluid velocity. This ambiguity is removed firstly by requiring
that TJq still gives the energy density of the fluid, i.e. Tqq = p. Secondly,
one requires that u a is the velocity of energy transport, so that pu a is the
energy current four-vector. The two conditions imply that only the spatial
components of IIjj are different from zero, i.e. IIoo = IIo Q = 0.
Let us define the three-dimensional tensor ir a p as 7r a ^ = — U a p/a 2 ; the
term "anisotropic inertia" refers to it a p also. The flat three-dimensional
metric is used to raise and lower the indices of 7r Q/ g, implying that II =
II* = 7r QQ . The anisotropic inertia can be decomposed into a scalar, a
vector and a tensor part as
TTa/3 = ^S a0 + (d a d fj - ^V 2 ) tt s + (d a nf + d p ^) + ^ (3.206)
d a n^ = , d a irl = , < Q = . (3.207)
The perturbation to the spatial part of the energy-momentum tensor is
5T a[j = -P lafi + a 2 6P5 aP - a 2 TT aP
= -Pl a ,3 + a 2 [ [5P - ^f + ^) 5 aP - d a d p n s
- {d a 4 + d^i) - tt^] , (3.208)
while the perturbation to the trace is
ST = Sp-35P + n aa . (3.209)
From the above expressions it follows that one of the two scalar degrees
of freedom in 7r Q/ g can be eliminated by including the term (V 2 7r s — 7r QQ )/3
into the pressure perturbation SP, i.e. by continuing to define a 2 5P as the
coefficient of 5 a0 in the sum {5T afj + Pjap). Of course this is not the only
e and Dynamics of the Isotropic Universe
way to eliminate the superfluous degrees of freedom; another popular choice
is to take 7r Q/ g to be traceless, TT aa = 0. In the following we will make the
first choice, and then assume that V 2 7r s — n aa = 0. Thus, the reader can
verify that under these hypotheses the perturbation equations are modified
as follows
• Scalar modes:
7 + 2-7 = k{S P + 35P - V 2 tt s ) ,
d a (7 - V 2 p) = 3k (p + P) d a 5u ,
7- - V 2 /i + 3^ (2J - V 2 /i) - ^V 2 (7 - V 2 p)
= 3k(SP-S P -\7 2 t: s ),
( a 1 7 7r s \
^ /i + 3-A + ^^vV - T^ + 6^—
\ a 3a^ 3a^ a^ /
= 2 K a Q ^^ s .
(3.210a)
(3.210b)
V 2 14 = -2k(p + P)c5w^,
d a (% + 3-V0) = 2 K d a iri
(3.211a)
(3.211b)
> Tensor modes:
7j, + 3^I /3 -^V 2 7j, = 2 K7 r^.
iveniencc, we also give the corresponding equations ii
• Scalar modes:
7 + 2-7 = k(6 P + 3SP + fcV),
7 + k 2 fi = 3k (p + P) <5m,
7 + fc 2 /i + 3- (27 + fc 2 /i) + —(7 + fc 2 p)
= 3k (SP - Sp + k 2 TT S )
a k 2 7
3a 2
(3.212)
Fourier space
(3.213a)
(3.213b)
(3.213c)
(3.213d)
Primordial Cosmology
k 2 V a = 2n(p + P)5u^ (3.214a)
V a + 3-14 = 2ktt^ (3.214b)
3.5.8 Kinetic theory
As we anticipated, the equations for the scalar perturbations contain more
unknowns than equations so that an equation of state P = P{p) has to
be assigned to close the system. This cannot always be adequate, as in
the case of mildly relativistic particles where it is not possible to assign a
simple equation of state. When one considers an imperfect fluid, the prob-
lem is also more evident., extending also to the vector and tensor modes.
One solution is of course to follow in this case a phenomenological ap-
proach, parametrizing in some way the dissipative effects encoded in the
anisotropic inertia tensor. For example, if one is concerned by viscosity ef-
fects, a shear viscosity coefficient can be introduced and expressed in terms
of other quantities (i.e. p and P). Moreover, sometimes the interactions
between different components of the fluid need to be taken into account,
like in the case of the cosmological baryon-photon fluid. As long as the
interactions between baryons and photons are very frequent, the two com-
ponents are tightly coupled and can be treated as a single fluid. However,
when the time-scale for the collisions is of order of the Hubble length, the
single-fluid approximation breaks down and dissipative effects have to be
taken into account.
The proper way to deal with this problem is to turn to a microscopi-
cal description of the energy-momentum tensor. The fundamental quantity
that describes, from a statistical point of view, the state of the fluid is
the distribution function in phase space f(x a , pp, t). The phase space is
described by three positions x a and their conjugate momenta pp. The dis-
tribution function is the density in phase space, i.e. given a six-dimensional
infinitesimal volume element dV = dx l dx 2 dx z dp\dp 2 dpz around the point
(x a , pp) at time t, containing dN particles, then
f(x a ,pp,t)dV = dN. (3.216)
The Structure and Dynamics of the Isotropic Universe 155
The time evolution of the distribution function is described by the Boltz-
mann equation (Sec. 3.1.6)
as as ox a as opp
The right-hand side represents the change in the distribution function due
to the effects of collisions in a unit of proper time (hence the subscript s).
For our purposes, it is more convenient to change the momentum variable
from the conjugate momentum p a to the proper momentum p' a measured
by a co-moving observer. The two are related by
Pa = ^^V" 3 = a (S a , - gf ) ^ = (S a , - g|) /, (3.218)
where we have also defined the rescaled momentum q a = ap' a . We note
that the indices of the proper and rescaled momenta are raised and lowered
using the flat metric, so that p' a = p' a and q a = q a . We can write q a as
q a = qn a n a , where q = y/q a q a , and the n a are unit vectors, i.e. n a n a = 1.
We can change the momentum variables from p % to (q,q a ) and replace 11
f(x a , p/3, t) by f(x a , q,np, £), so that the left-hand side of the Boltzmann
equation, i.e. the Liouville operator, rewrites as
d ., a .. dt df dx a df ^dqdf dn a df
— f(x a , q,n [3 , t) = — — + — — — - - + — — - + — — - — . 3.219
as as at as ox a as oq as on a
The term dq/ds can be computed from the geodesic equation as follows.
First of all, we note that
dq d — a dq a
ds ds ds
Multiplying both sides of Eq. (3.218) by (5 au +'f au /2a 2 ) q a can be expressed
in terms of p a and p a as
q a = (<W + %f)PP= (-« 2 <W + ?f ) /• (3-221)
Then it results (.hat
-_ (. WJrf + 1^ + f^ s + ( .^ + If) (^) .
(3.222)
11 Since x a and g a are not conjugate variables, d 3 xd 3 q is not the phase-space volume
element and fd 3 xd 3 q is not the particle number.
Primordial Cosmology
The geodesic equation gives
^f- = -rg. P y = -2-pV 4
ds J a
- ^ (fl^etf " 2^
m)pV
+ ^(2ft,7a.
- a a 7^)p M p"
, (3.223)
and then
£_(_w«* +£**,+ £%**)•
+ 2aap°p a - (d tla0 - 2% aP J p
V-^(2a, 7a ,-
dalnv)p tl p V ' ~
^7^PV
= -^VfltTa/? + V/7^ - \ip vlail
~ daj^p^p"
. (3.224)
Putting everything together, we
finally get
d 1 a 8 o ( * •
«7«A i Q
n P °% 8 ,
(3.225)
where we remember that 7 Q/ g =
= las/a 2 - For what concerns
dn a /ds. it
results that
dn a d / i
ds ds \
<7 / g ds q 2
dq
(3.226)
Since both dq/ds and dq a /ds ar
o (9(-, (> ,v), also rfn
a /ds is of order 0(7 a/3 )
at least. As we shall see, this
Let us write the distribution function f(x a , q^n^^ t) as the sum of an
unperturbed part / plus a small perturbation Sf = f^. The background
homogeneity implies that / cannot depend on the spatial position x a , while
the background isotropy implies that it can depend on the momentum only
through its magnitude q. Finally, recalling again that dq/ds = 0(j a g), the
zeroth-order Boltzmann equation for / rewrites as
f= d 7 f=P° d 4 = 0, (3-227)
ds ds at
so that / does not depend on time either (/ = /(<?)) and
f(x a , q, np, t) = f(q) + Sf(x a , q, np, t)
= f(q)[l + y(x a ,q,np,t)}.
We can assume the zeroth-order distribution function as given, in the co-
moving frame, by an equilibrium Bose-Einstein or Fermi-Dirac distribution
with temperature T and zero chemical potential
f=^Zs^ • ( 3 - 229 )
" 271-3
The Structure and Dynamics of the Isotropic Universe 157
This form of the distribution is appropriate for species that are at kinetic
equilibrium and with vanishing chemical potential. However, most of the
formulas in the following are independent of the particular choice of /.
The energy E that appears in Eq. (3.229) is the energy measured by a
co-moving observer so that it is related to the proper momentum p' a and to
q a by E 2 = p' a p' a +m 2 = (q/a) 2 + m 2 . Let us introduce the rescaled energy
variable e defined as the proper energy E times the scale factor a, so that
e = v/g 2 + a 2 m 2 . We note that Eq. (3.227) implies that, for a Fermi-Dirac
or Bose-Einstein distribution
Finally, we can rewrite the first-order Liouville operator (3.219) as
df n -\dV q a <9* 1 „ «• din /]
ds [ at ae ox a 2 H d In q J
considering that, to zeroth order, p = e/a and p a = —q a /a 2 . The term
(dn a /ds)(df/dn a ) in Eq. (3.219) does not appear because both (dn a /ds)
and (df/dn a ) are 0{j a p) and then their product is 0{{^ a[3 ) 2 ). Using
p°ds = dt, the Boltzmann equation reads as
9* 1 a d* , 1 a *~ din/ 1.
~m ~ ~ed^ + 2 n n ^dhTq- ~ 7 t[f] ( }
where, as usual, the subscript t in the collision term on the right-hand
side represents the fact that C t [f]dt gives the variation of the distribution
function due to the collisions in a small interval of time dt. The form
above makes explicit that the Boltzmann equation is coupled to the Einstein
equations by the presence of the metric perturbation ^a/3- In Fourier space,
the Boltzmann equation rewrites as
vj, + i±( ka n a )V + \n a n^ a p^l = -C t [f] (3.233)
ae 2 dmq j
where we continue to use \I/ to denote the Fourier transform of the pertur-
bation to the distribution function.
The next step to close the Einstein-Boltzmann system is to express the
components of the energy momentum tensor in terms of integrals of the
distribution function. In general, the energy-momentum tensor is related
to the distribution function by
T/ = -L / ^f(x a , pp, t)d 3 p (3.234)
158 Primordial Cosmology
where g = — a 6 (l — 7) is the determinant of the metric. For consistency,
we need to express the integrand in terms of q and n a . From Eq. (3.218),
the Jacobian matrix of the transformation from the p a to the q? , is equal
to dp a /dqP = (Sap — On s/-<i 2 )- s < > that the Jacobian is, to first order in 7,
equal to (1 — 7/2), and
d 3 p = (l " |) d 3 q = f 1 " |) ^dqd^l, (3.235)
where d 3 p/ ^f^g = a~ 3 d 3 q. Here dfl is the infinitesimal element of solid
angle around the direction n a . It results that
T °° = h I e/(<7) (1 + *)« 2d « d0 ' ( 3 - 236a )
T " = h i 5a ' 3 ~ 2 2 £ ) / qn0 ^ 1 + ^ q2dqdQ > (3.236b)
Tf = - -^ f Q n " n0 f(l + ^>)q 2 dqdVL , (3.236c)
where p° = p = e/a. In the second of these equations, the angular integral
of the unperturbed part is just J n npd£l that vanishes identically. We then
have
T° = Lf e f(q)(l + V)q 2 dqdn (3.237a)
T° = — / qn a f^q 2 dqdn (3.237b)
a A J
T^ = -\l q n * n P f(l + V) q 2 dqdn (3.237c)
which makes explicit that the convenience of replacing the conjugate mo-
mentum p a with the proper momentum q a is that the metric perturbations
disappear from the expressions of the components of Tj.
The mixed components of the energy-momentum tensor are related to
the energy density, pressure, velocity and anisotropic inertia of the fluid by
T ° = p = p + Sp (3.238a)
T° = (p + P)Su a (3.238b)
Tg Q = -P5% + n^ = -(P + 5P)5% + n^ . (3.238c)
Comparing with Eqs. (3.237), the background quantities p and P stand as
P = J / qWq 2 + a 2 m 2 f(q)dq
- \, f q* - ( 3 - 23 °)
P = — / q f(q)dq
3a 4 J y/qi + a 2 m 2
The Structure and Dynamics of the Isotropic Universe 159
where f dQ, = 4ir and J n a np = AirSap/'S. For the perturbations, it results
dp = -L J q 2 ^qZ+a 2 m 2 f{q)^dqdVL (3.240a)
(p + P)5u a = -1 J qn a f^q 2 dqdn (3.240b)
5P5?. - IK = -1- / — =U^^fVdqdn . (3.240c)
Once the collision term on the right-hand side of the Boltzmann equation
has been specified, the Einstein equations (3.213), (3.214) and (3.215) and
the Boltzmann equation 12 (3.233), together with the relations (3.240) form
a closed system for the coupled evolution of 7 a/ g and \I/ that can be solved
without any further assumption once initial conditions are given. In the
form presented here, this system takes the form of an integro-differential
system, since integrals of the distribution function / appear as sources of
the differential equations for 7 Q/ g.
3.6 The Lemaitre-Tolmann-Bondi Spherical Solution
The Lemaitre-Tolmann-Bondi (LTB) spherical solution can be thought of
as a generalization of the RW line element in which the requirement of
homogeneity is dropped, while that of isotropy is kept. It is worth noting
that the LTB model can be isotropic but not homogeneous (see Sec. 3.1.1)
because a preferred point is singled out, i.e. the space is isotropic only when
viewed from this particular point, but not from any other point. For this
reason, it is more correctly described as a spherically symmetric, inhomo-
geneous solution of the Einstein equations. In particular, it describes the
evolution of a zero-pressure spherical overdensity in the mass distribution,
and thus the resulting solution is different from the Schwarzschild one. In
the synchronous reference system (see Sec. 2.4), the spherically symmetric
line element can be written as
ds 2 = dt 2 - e 2a dr 2 - e 2/3 (d6 2 + sin 2 6 d<j) 2 ) , (3.241)
12 In general, for a fluid made of many uncoupled (or, more precisely, not perfectly
coupled) components, one should write a Boltzmann equation for every component.
These Boltzmann equations can be coupled, other than by the presence of the metric
perturbation, representing the effect of the gravitational field, by the collision terms,
representing the effects of impulsive interactions between the various components. For
example, the Boltzmann equations for baryons and photons are coupled by the collision
term for Thomson scattering. However, this does not change the sense of the disi
160 Primordial Cosmology
where a = a(r,t) and /3 = f3(r,t), while the identity y/^g = smOe a+213
follows.
Originally, this kind of solution was discussed under the assumption that
the perfect fluid energy-momentum tensor (2.14) is dominated by pressure-
less dust (P = 0) and by a cosmological constant term A. In this scheme,
the Einstein field equations rewrite as
kT^ = = -2/3' - 2/3/3' + 2d/3' = G? (3.242a)
kT; 1 = A = 2/3 + 3/3 2 + e" 2/3 - (/3') 2 e~ 2a = G\ (3.242b)
K T ° = up + A (3.242c)
=$ 2 + 2a$ + e~ 2fi - e- 2a [2f3" + 3(/3') 2 - 2a'/3'] = Cg ,
where the () and the ()' denote derivatives with respect to time and to
the radial coordinate r, respectively. Let us note that these are the only
independent equations because the following relation stands
G\ = G\ + \G\\'/2P . (3.243)
Since T° vanishes, Eq. (3.242a) rewrites as
/3'//3' = d t In /3' = a - /3 , (3.244)
which admits the solution /?' = f{r)e a ~ f3 giving e' 3 /?' = ^e' 3 = /(r)e a .
Let us now define the scale factor o(r, t) by using the parametrization
eP = ra(r,t), f( r ) = [l-r 2 K 2 ] 1 / 2 , (3.245)
where K = K{r). Using the expressions above, the LTB line element
(3.241) rewrites as follows
ds 2 = dt 2 - [K J 2 l dr 2 - (ar) 2 {d6 2 + sin 2 d<j> 2 ) . (3.246)
If a and K are independent of the radial coordinate r, Eq. (3.246) corre-
sponds to the RW line element (3.1) and it can be constructed if and only
if T{* vanishes. The function K 2 has been written as a square, to conform
with the standard notation for the isotropic models, but of course K 2 can
be negative, as in the open isotropic model.
The field equations (3.243) and (3.242b) rewrite now as
(up + A)[(ar) 3 ]' = 3[a 2 ar 3 + ar 3 K 2 }' , (3.247a)
2a a 2 K 2
A = — + — + — , (3.247b)
The Structure and Dynamics of the Isotropic Universe 161
respectively. The last expression, if multiplied by a 2 a, is a total time deriva-
tive, which can be integrated getting
a 2 a + aK 2 - -a 3 = F(r) . (3.248)
Let us now suppose that the cosmological constant term is small enough for
the function F(r) to be positive. We can also change the radial coordinates
according to
[ A \ ' [F(r)\ ' [F(r)\ '
(3.249)
(A being a constanl ) leaving the form of the line element (3.246) unchanged.
Furthermore, the right-hand side of Eq. (3.248), in the new coordinate
system, is constant. In this scheme, Eqs. (3.247a) and (3.248) rewrite
respectively as
«P( r '*) = 7^2' ( 3 - 25 °)
a 2 a + aK 2 --a 3 =A, (3.251)
where we have dropped the bars for the sake of simplicity.
Let us consider the Lagrangian formulation of LTB space-time. The
Lagrangian density can be obtained from the expression of the Ricci scalar
and by avoiding total derivatives it reads as
C LTB = — f (-r 3 Waa 2 + r 2 V'aa - V) dtdr , (3.252)
where
The momentum conjugated to a is given by
p a = —(-2Wr 3 aa + r 2 aV) , (3.254)
such that the Hamiltonian turns out to be
k p 2 n V 1 (V') 2 ra 2ir
162 Primordial Cosmology
3.7 Guidelines to the Literature
There are many textbooks on GR that deal with the kinematics and dy-
namics of the isotropic Universe, studied in Sees. 3.1 and 3.2, starting from
the classic one by Landau & Lifshitz [301] and by Weinberg [462]. The
more recent books by Kolb & Turner [290] and Peebles [378] are more di-
rectly devoted to physical cosmology, with the first one putting quite an
emphasis on topics at the interface between cosmology and particle physics.
Another book that covers a wide range of topics is that by Peacock [374].
The recent ones by Dodelson [155] and by Weinberg [464] include many
recent developments in the field.
A textbook on kinetic theory and the Boltzmann equation, introduced
in Sec. 3.1.6 is the book by Lifshitz & Pitaevskij [317]. The Boltzmann
equation and the topic of the kinetic theory and thermodynamics of the
expanding Universe are covered in the book by Bernstein [82] , as well as in
the above mentioned by Kolb & Turner, Dodelson, and Weinberg.
For what concerns the dissipative cosmologies studied in Sec. 3.3, we re-
fer the reader to Landau & Lifshitz Fluid Mechanics [300], and in particular
to Chap. XV. The issue of the effects of bulk viscosity on the cosmological
evolution has been studied in [41,42,62,67,111,331,371,461]. The phe-
nomenological description of the process of matter creation in an isotropic
expanding Universe was firstly formulated by Prigogine [351]. The influ-
ence of the matter creation term on the dynamics of the Universe has been
studied in [148,351].
There is a vast literature on the theory of cosmological perturbations,
discussed in Sec. 3.4. The Jeans mechanism, both in a static and in an
expanding Universe, is described in many textbooks, like Weinberg [462]
and Kolb & Turner [290]. The general relativistic treatment of small fluc-
tuations over a RW background was first formulated, in the synchronous
gauge, by Lifshitz [311]. This work was later reviewed by Lifshitz & Khalat-
nikov [312]. The conformal Newtonian gauge was introduced by Mukhanov
and collaborators in [358]. The gauge-dependent treatment (either in the
synchronous or Newtonian gauge, or in both) is summarized, among oth-
ers, in the books by Weinberg [462,464], Landau & Lifshitz [301], and
Dodelson [155], as well as in many reviews, like that of Bertschinger [84]
(see also [85]). The gauge invariant formulation, that we did not address
here, was formulated by Bardeen [38] and later reviewed by Kodama &
Sasaki [288]. The systematic treatment of the coupled Boltzmann and
Einstein equations, including all species of cosmological interest, has been
The Structure and Dynamics of the Isotropic Universe 163
given by Ma & Bertschinger [329] . A discussion of the Lemaitre-Tolmann-
Bondi solution can be found, among others, in the book by Peebles [378].
A comprehensive treatment of the evolution of cosmological perturbations,
including also evolution in the non-linear regime, is given by Padmanab-
han [370], which is devoted to cosmological structure formation.
This page is intentionally left blank
Chapter 4
Features of the Observed Universe
In this Chapter we review the present observational knowledge of the Uni-
verse, introducing the main cosmological observables and discussing how
these can be used to extract information on the main parameters describ-
ing the Universe.
We start by briefly reviewing the so-called ACDM (or "concordance")
cosmological model, that is able to account for all observations, although
yet not explaining the precise nature of the dark matter and dark energy
components that provide most of the energy content of the Universe.
The relevance of the distribution of large-scale structures in the Universe
is treated arguing how their existence is not at variance with the isotropy
and homogeneity requirements of the cosmological principle. We briefly
recall the main features of the mechanism of gravitational instability, and
show how the existence of galaxies already provides an indirect evidence
for the presence of a non-baryonic matter component. We introduce the
quantitative tools necessary to study the distribution of matter in the Uni-
verse, such as the two-point correlation function and its Fourier transform,
or the matter power spectrum.
In the following section, we will show how the Hubble diagram can
be used to infer the matter content of the Universe. The observations of
supernova la indicate that the Universe is accelerating and we will briefly
discuss the theoretical implications.
Finally, the last section is devoted to the study of the Cosmic Microwave
Background (CMB). We first discuss its black body frequency spectrum
that demonstrates that the early Universe was in thermal equilibrium, and
then its extreme isotropy and how the small anisotropies carry important
cosmological information. We briefly describe the mechanism by which
these anisotropies are produced and introduce the power spectrum of the
166 Primordial Cosmology
anisotropies, discussing how the acoustic oscillations present in the primeval
plasma left a distinct pattern in the spectrum, made of alternating peaks
and dips. We conclude by discussing the effect of the cosmological param-
eters on the spectrum, and giving their values obtained by the most recent
('MB observations.
4.1 Current Status: The Concordance Model
For many years, cosmology has been a data-starved science and, until a few
decades ago, the observational basis for the standard cosmological model,
although robust, consisted of just a handful of observations, basically given
by:
(i) the spectrum of distant galaxies is shifted towards the red;
(ii) the existence of an isotropic background of thermal radiation in
the microwave range;
(iii) the distribution of galaxies;
(iv) the measured abundances of the light elements.
In the last couple of decades, however, the observational data has grown
in quality and quantity. Observational cosmologists have been able to test
the above-mentioned pieces of evidence even further. The expansion his-
tory of the Universe has been probed up to redshifts of the order of 1.8.
Satellites like BOOMERanG and WMAP have measured the tiny angular
anisotropies in the temperature of the cosmic microwave background ra-
diation, disclosing a wealth of information about the Universe as it was
nearly 400,000 years after the Big Bang. Galaxy surveys, like the 2dF and
SDSS, have increased in volume, allowing to collect a sample of ~ 1 million
objects with measured spectra, thus mapping the distribution of matter in
the Universe with high precision. Finally, the abundances of light elements
have been measured with increasing precision.
In the last years, new ways have been designed for the ongoing study of
the Universe: for example the measure of abundance of neutral hydrogen
by radio telescope arrays looking at the characteristic 21 cm line emission,
to probe the "dark ages" in the history of the Universe; the polarization
of the cosmic microwave background, providing new informations to detect
gravitational waves produced in the early Universe: such background of
gravitational waves is also a possible target for detection by interferometers
like LISA, and for the so-called Pulsar Timing Arrays.
Features of the Observed Universe 167
The observational knowledge of the Universe has very much increased in
the last twenty years. Has our basic understanding of the cosmo increased
as well? The answer is twofold. On one hand, the observations point to a
very simple picture. Our Universe is very well described, at least at large
scales, by a flat Robertson- Walker geometry. Nowadays, its energy content
is given by some form of "dark" matter, making up roughly 20% of the total,
and by an equally unknown (and even more exotic) form of "dark" energy,
making up 75% of the total. Normal matter composes just the remaining
5%. The present Universe is very cold (2.7 K) but, since it is expand-
ing, it was much hotter in the past. The light elements (hydrogen, helium
and, to a lesser amount, lythium) present today were produced in this very
dense and hot phase, as hypothesized by Gamow, when the temperature
was around 10 MeV. The microwave radiation observed is the red-shifted
relic of this early phase, released when the free protons and electrons re-
combined to form neutral hydrogen atom, thus allowing the photons to
propagate freely. The structures observed today - galaxies, clusters, super-
clusters - have been grown by small "seeds" through the Jeans mechanism
of gravitational instability. According to the inflationary scenario, these
seeds have been produced from quantum fluctuations in the early Universe.
As we have noted above, this "concordance model" , as it is currently called,
can safely explain all the pieces of evidence brief!} discussed here. However,
the model is unsatisfactory in many ways. First of all, it is not yet known
what dark matter really is, although there is no shortage of well-motivated
particle physics candidates, starting from the supersymmetric neutralino.
There is hope that in the next decade or so the dark matter particle will
be detected either directly (by producing it in accelerators, or revealing it
in specifically-designed detection experiments) or indirectly (through the
observation of its dccav/auuibJlatiou products in an astrophysical or cos-
mological setting), thus shedding light on its nature. The situation is worse
for what concerns dark energy. In this case, it is fair to say that there is
not a strongly motivated candidate, at least from the theoretical point of
view, although many proposals have been made. From the observational
point of view, the simpler explanation of the currently available data in the
framework of an FRW model is still given by a cosmological constant-like
fluid. This problem seems to point out that either we miss something very
fundamental from the point of view of particle physics, or that the stan-
dard cosmological model is incomplete, or both. Many scientists have tried
alternative ways to explain the observations without invoking any dark en-
168 Primordial Cosmology
ergy component. 1 Since the main evidence for the presence of dark energy
comes from the dynamics of the Universe on the largest scales (in particular
from acceleration) , a natural approach is to assume that General Relativity
is modified on cosmological scales, as proposed by f(R) theories.
Another possibility is that the observed acceleration is an artifact, due
either to our location in an underdense region, or to the breaking of the
homogeneity assumption underlying the FRW model at small scales. Al-
though there is not yet a shared consensus if these approaches can provide a
satisfactory explanation to the acceleration, without at the same time spoil-
ing other observations, at the present time they still represent a possible
alternative to the dark energy models.
4.2 The Large-Scale Structure
The Standard Cosmology is based on the cosmological principle, namely
on the assumption that the Universe is homogeneous and isotropic. The
best evidence for this is the great degree of isotropy of the cosmic microwave
background radiation: the fractional dili.crc.ua' in 1 cnipcrature between two
directions in the sky is smaller than 10~ 4 . This is also a proof of homo-
geneity, because the temperature variations in the CMB track the density
fluctuations at the time of photon decoupling (see Sec. 4.4 below). This is
in fact an evidence that the Universe was very homogeneous and isotropic
at the time of last scattering, roughly 400,000 years after the big bang
(corresponding to a redshift z ~ 1100).
What about the present-day Universe? On a first look, it would seem
very far from homogeneity. If we look at the sky, we see stars that are
located into bound systems (the galaxies) separated by large, empty regions.
The density inside a galaxy is roughly 10 5 times larger than the average
density of the Universe, so that a galaxy cannot certainly be considered a
small fluctuation of the background density. From this point of view the
galaxies in some sense constitute the "elementary particles" of cosmology,
since they can be taken as free falling in the cosmological gravitational
field. Galaxies themselves tend to form groups called galaxy clusters (with
an average density 10 2 -10 3 times the background), which in turn can form
larger (not yet virialized) structures called superclusters. The density in
the superclusters is estimated to be of the order of the background density,
so that at those scales the density perturbations are presumably only in the
Home] hues even wil lioul dark matter, but this is a much harder I ask.
Features of the Observed Universe 169
mildly nonlinear regime.
However, even if the distribution of luminous matter in the Universe is
inhomogeneous, nevertheless these inhomogeneities become smaller as we
look at the Universe at the largest scales. More precisely, if we adopt a
coarse-grained description of the Universe, considering the density of lu-
minous matter averaged over a given fiducial volume, the fluctuations in
the coarse-grained density become smaller as the fiducial volume increases.
The existence of such a "homogeneity" scale has been somewhat debated
until recently, but now the scientific consensus is that the present Universe
is homogeneous on scales larger than ~ 100 Mpc.
The good approximation of the Universe homogeneity and isotropy
does not mean that the existence of structures is irrelevant for cosmol-
ogy. The origin and evolution of galaxies and larger structures is an issue
of paramount importance in modern cosmology, as it provides key informa-
tion on the evolution of the Universe. Here we will give some ideas about
galaxies formation so that the reader can understand the overall picture.
4.2.1 Deviations from homogeneity
If the Universe were perfectly homogeneous at all scales, it would not be
possible to form any kind of structure. Indeed, small deviations from ho-
mogeneity are needed as starting "seeds" from which structures are formed.
For the moment we will leave the existence of such primordial seeds as an
assumption, coming back later to how they formed. The mechanism of
growth for the initial inhomogeneities is the Jeans mechanism of gravita-
tional instability, that we have described in detail in Sec. 3.4. The idea is
that, if an overdense region is present in an otherwise homogeneous fluid,
it will correspond to a potential well of the gravitational field, attracting
other particles inside the well. This will increase the overdensity and fur-
ther deepen the well, and so on. However, this is just a part of the story; the
gravitational collapse is countered by the pressure forces inside the fluid,
that increase with the overdensity. The final fate of the initial small in-
homogeneity is decided by the (un)balance between gravity and pressure:
if gravity dominates, the inhomogeneity will grow, become non-linear and
eventually a structure will be formed; on the contrary, the amplitude of
the inhomogeneity will just oscillate (and eventually decay once "real-life"
dissipative effects are taken into account). These two behaviors are sepa-
rated by a critical length called the Jeans length Xj [given in Eq. (3.132)],
that is a function of the density and of the speed of sound of the fluid
170 Primordial Cosmology
(see Sec. 3.4). Perturbations with a linear size larger that Xj will collapse,
while those smaller than Xj will oscillate. Although the detailed behavior
is in general more complex, this simple picture captures the essence of the
mechanism of structure formation.
The full theory of cosmological perturbations has been developed in
Sec. 3.5, where we have written the equations describing the coupled linear
evolution of the perturbations in the metric and in the energy-momentum
tensors. Once the initial inhomogeneities are given, the only missing piece
of information to fully compute the linear evolution is the composition of
the Universe.
4.2.2 Dark matter
The evidence of galaxies existence is a strong hint to the fact that baryonic
matter is not the only kind of matter present in the Universe. In fact,
baryons were tightly coupled to photons via Thomson scattering until the
time of decoupling, when the CMB radiation was emitted (see Sec. 4.4).
Before that time, baryons and photons were behaving as a single fluid,
with a very large pressure given by the photon component. Such a large
radiation pressure was extremely effective in contrasting the gravitational
instability, making the Jeans length roughly equal to the size of the cosmo-
logical horizon. 2 This means that perturbations in the baryonic component
could not grow until decoupling, which occurred when the scale factor a of
the Universe was ~ 1CT 3 of its present value. However, the baryon density
contrast at the time of decoupling has to be of the same order of magnitude
as the temperature fluctuations observed in the CMB (the factor of 4 comes
from the proportionality to T 4 of the energy density):
** I „ 4 ^<10- 4 . (4.1)
Pb Idee T
Combining this with the fact that, as seen in Sec. 3.4.3, during the matter-
dominated era the perturbations grow linearly with a, we obtain that the
present density contrast should be ~ 10 _1 . This implies that the non- linear
evolution should not have started yet, and no structures would have formed
at all - actually, if it were so, they will not form before some other ten billion
years!
2 More quantitatively, this can be understood by noting that Eq. (3.132) with v s = \/3
(the speed of sound in an ultrarelativistic fluid, with equation of state P = p/3) and
p B =Pc = 3H 2 /k gives Xj -H- 1 .
Features of the Observed Universe 171
The solution to this apparent paradox is that some other kind of matter
exists that does not couple to photons (and hence it is "dark"). 3 The den-
sity fluctuations of such dark matter component are not hindered by the
pressure of photons and can start growing well before the time of decou-
pling, creating the potential wells where baryons will fall later, eventually
leading to the formation of galaxies.
Prom the point of view of theoretical particle physics, there is no short-
age of candidates for the role of dark matter. For the purpose of the present
section, it will suffice to say that nearly all candidates belong to one of two
broad classes: hot and cold dark matter. The distinction is based on the
damping length of the particles, namely on the characteristic length below
which dissipative effects become important and perturbations are erased.
In the case of collisionless dark matter, this damping effect is provided
by Landau damping, or free streaming. In other words, free streaming is
due to the fact that, in a collisionless fluid, the particles can stream from
overdense to underdense regions, in the process smoothing out the inhomo-
geneities. Since fast (hot) particles can cover larger distances, collisionless
damping is more important for dark matter candidates with a large ve-
locity dispersion. In more detail, one defines as hot dark matter (HDM)
those candidates with a damping length Ad of the order of the size of the
horizon at the time of matter-radiation equality Xeq- This is the case for
ultrarelativistic relics like neutrinos. On the other hand, cold dark matter
(CDM) candidates have Xjj <C Xeq- This is the case for non-relativistic
relics like the supersymmetric neutralino. The importance of the time of
matter-radiation equality is due to the fact that structure formation cannot
start earlier, since during the radiation dominated regime the fast cosmo-
logical expansion nearly freezes the growth of all fluctuations - including
those in the dark matter component. This is the Meszaros oifect described
at the end of Sec. 3.4.3.
The difference in the damping scale leads to different scenarios of struc-
ture formation between hot and cold dark matter-dominated Universes. In
the case of hot dark mat tor. all perl urbat ions below the (very large) damp-
ing length are erased, so that only the perturbations on the very largest
scales survive. This implies that the largest structures in the Universe (like
superclusters) are formed first, and smaller structures are formed later via
a fragmentation process. In particular, the first structures to form have a
3 The fact that it is not possible to form the present cosmological st
not the only reason to introduce dark mailer but we think it nicely illustrates that the
large-scale structure encodes fundamental information about the Universe.
172 Primordial Cosmology
mass of roughly 10 15 M Q , much larger than the typical mass of a galaxy
(~ 10 11 - 1O 12 M ). This is called a top-down process of structure for-
mation. On the other hand, for cold dark matter the damping length is
effectively zero so then 1 is no damping of small scale perturbations. Thus
small structures (on subgalactic scales, ~ 1O 6 M0) form first, and eventually
merge to form larger structures. This is called a bottom-up, or hierarchi-
cal, process of structure formation. The modern observations rule out the
HDM scenario for at least two reasons. The first one is the prediction of
more structures on large scales than actually seen. The second is that small
structures seem actually to be older than large structures. The currently
favored scenario is then with structures formed via a bottom-up process
driven by CDM. This does not however exclude that a small HDM fraction
could be present as a subdominant dark matter component.
4.2.3 The power spectrum of density fluctuations
The quantity describing the distribution of matter in the Universe is the
density contrast 6(x):
5(x)= P -^-l (4.2)
where p(x) is the density in x, and p is the average density of the Universe
(in general, we will use bars to denote background quantities). It is useful
to consider the Fourier transform 5k of 5(x)
6 k = S(x)e lk - X d 3 x. (4.3)
In the previous paragraph, we have given a very qualitative description
of how the quantity 5 evolves with time. In Sec. 3.4 we have given the exact
equations describing the full evolution of 6 in the linear regime 5<1.
The main question to answer is how to compare theory with observa-
tions: of course, a qualitative approach would be to compare the distri-
bution of galaxies in the real, observed sky with a map produced by a
numerical simulation. However, one does not compare the observed and
theoretical matter distributions per se, but rather their statistical proper-
ties. The basic quantity is then the two-point correlation function £(r)
£(f) ee (S(x)S(x+f)) = iy 5(x)S(x + r)d 3 x , (4.4)
Features of the Observed Universe 173
namely the autocorrelation function of the density field. Here the brackets
denote an average over some fiducial volume V. The two-point correlation
function evaluates how the density fluctuations in pairs of points separated
by f are correlated. We stress that £ does not depend on the absolute
position x, but on the points separation f, since by construction it is a
volume average. 4 Moreover, the isotropy of the universe implies that £
depends only on the modulus r of r, i.e. on the distance between the
points.
Taking the Fourier transform of the two-point correlation function, one
obtains the power spectrum P(k) as
P(k) = J f(r)e^ r d 3 r. (4.5)
It can be shown that the relation between the power spectrum and the
Fourier transform 5k of the density contrast
5 k 5l, = (2nfP(k)5l(k-k'), (4.6)
holds, where 5 Z D is the three-dimensional Dirac 5 function. It is then clear
that P(k) is related to the variance of the density field in fc-space:
The power spectrum at a given wave number A; is a measure of the
"dumpiness" at a scale A ~ 2n/k. In particular, let us compute the average
density inside a sphere of radius A centered around a given point in space.
This will smooth out any inhomogeneities at scales much smaller than A.
Then, let us move the center of the sphere and repeat the procedure for
every point in space. If the sample of values obtained in this way has a large
variance, then P(k ~ \/2ir) will be large as well, and vice versa. Repeating
the procedure for different sizes of the sphere will give the complete power
spectrum P(k).
The power spectrum is the key quantity when comparing the theory
with observations of the large scale distribution of galaxies. A fundamental
requirement of any successful cosmological model is then to predict a matter
power spectrum in good agreement with the observations. The fluctuations
4 Strictly speaking, the brackets in Eq. (4.4) should denote a
ever, the homogeneity of the Universe makes it reasonable to
average corresponds to a volume average.
Primordial Cosmology
(S(xf) = j^Jp(k)d 3 k = J^ , ^ldlnk, (4.8)
at a given scale are often expressed in terms of the more convenient dimen-
sionless quantity Af = k 3 P(k)/2n 2 , whose relevance is related to having
= D k 3 P(k)^
"(2tt;
so that A 2 (fc) can be regarded as the contribution to the real space variance
from a given logarithmic interval in k.
We remark that the two-point correlation function (or equivalently the
power spectrum) encodes all the information on the statistical properties of
the density field only if the fluctuations in the density field are Gaussian. A
non-Gaussian field is also defined by its higher order moments, starting from
the three-point correlation function (and so on), while for a Gaussian field
the higher-order moments are either vanishing (if odd) or can be expressed
in terms of the two-point correlation function (if even).
As a consequence of the central limit theorem, a field is Gaussian if
the phases of the different Fourier modes S k are uncorrelated and random.
Since the linear evolution does not change the phases, this amounts to the
requirement that the initial perturbations are Gaussian, as in the case of
fluctuations produced during inflation. However, in general, the Gaussian-
ity of the fluctuations is an assumption that has to be tested. This has been
done, finding that the initial fluctuations were indeed highly Gaussian.
Another property to consider when comparing theory with observations
is that the power spectrum P m defined above refers to the whole matter
distribution, i.e. including both baryons and dark matter. However, since
we cannot directly observe dark matter, what is measured is just the power
spectrum of the luminous matter, i.e. the galaxy power spectrum P ga i 7^
P,„. The minimal assumption to obtain P m from a measurement of P ga i is
Pgai = b 2 P m (4.9)
where b is a constant (i.e. scale-independent) bias parameter. Thus, the
matter and galaxy spectra coincide, apart from their overall normalization.
This assumption can be restated by saying that light faithfully traces mass.
If this is true, the only effect of the mismatch between the matter and galaxy
spectra is the introduction of an additional, effective parameter that has to
be taken into account when analyzing the experimental data. However, in
the past few years it has become increasingly clear that the assumption of
a scale-independent bias is unsatisfactory, and a dependence of the bias pa-
rameter on k has to be introduced to better model the relationship between
the dark and luminous matter distributions.
Features of the Observed Universe 175
4.3 The Acceleration of the Universe
The most puzzling fact about our Universe is probably its presently ac-
celerating expansion, whose main evidence comes from the observations of
distant type la supernovae (SNe la). SNe la are standard candles, i.e. they
are objects whose intrinsic luminosity is known, so that a measurement of
their flux allows a determination of their luminosity distance dh ■ By mea-
suring also the redshift of the SN, the distance-redshift relationship can be
reconstructed. Since it depends on the past expansion history, its determi-
nation allows to measure the energy budget of the Universe. In the small
redshift limit, the distance-redshift relationship is given by the Hubble law
z = H dL, as seen in Sec. 3.1.4. Going to larger redshift, some deviations
from the linear behavior can be observed. Another advantage of using SNe
la, apart from being standard candles, is their brightness (the typical lu-
minosity is of the order of that of a galaxy) so that they can be observed
up to high redshifts (currently, up to z ~ 1.8).
In the 1990s, two research groups, the Supernova Cosmology Project
and the High-z Supernova Search, independently reported evidence that
the Universe is accelerating. Both groups found that distant supernovae
are dimmer than they should if the Universe were decelerating. In the
framework of a FRW cosmology, their results pointed to S1a > 0. We know
from Sec. 3.2.1, and in particular from Eq. (3.49b) for the deceleration pa-
rameter, that, in the framework of FRW cosmology, the acceleration of the
Universe cannot be produced by a normal matter or radiation component,
i.e. a component with w > 0. In fact, acceleration requires w < —1/3 as it
can be seen from Eq. (3.48) or Eq. (3.49b).
Let us explain in more detail how the distance-redshift relationship can
constrain the matter-energy content of the Universe. For simplicity, we
restrict the discussion to the case of a fiat Universe (K = 0). We have seen
in Sec. 3.1.4 that the luminosity distance dj, of a source is related to the
coordinate distance r by Eq. (3.17), i.e. dr, = «or(l + z) = rf(l + z), where
d is the proper distance to the source today.
The proper distance d = a^r as a fund ion of redshift can be calculated
using the fact that for a photon ds 2 = dt 2 — a(t) 2 dr 2 = 0, so that
f r , , f to dt f ao da f z dz'
d = a dr =a — = a Q / -——-=/ — — , (4.10)
Jo Jt! a J ai a z H(a) J H{z')
where t\ and oi are the time and scale factor at the time the photon was
emitted, dt = da/a = da/aH and 1 + z = a /a. For a flat Universe
i by non-relativistic matter, radiation and by an exotic component
176 Primordial Cosmology
("dark energy") with a constant equation-of-state parameter Wd c (so that
its energy density pdc scales like (1 + z ) 3t ^ 1+WdB ' ) ), with respective density
parameters il m , il ra d and Side, the Friedmann equation Eq. (3.46) can be
put in the form
H(z) = H y/n m (l + zf + O rad (l + z) 4 + fi d0 (l + z)3(i+-de) ; (4.11
so that
d(z)
-Ho 1 j*
d L (z) = H^(l + z)J* j= = = J\ = === , (4.13)
Vn m (i + z'f + n rad (i + z'Y + n do (i + Z ')w+"*-) '
(4.12)
Using the fact that the radiation density is negligible at z <C z cq ~ 10 4 ,
and the flatness condition (implying Jldo = 1 — fi m ), we finally get for the
luminosity distance as a function of redshift
Vn m (i + ^') 3 + (i - fi m)(i + z') 3(1+Wde) ' '
which depends on the matter content of the Universe and on the dark
energy equation-of-state parameter u>d e - The cosmological constant case
can be recovered by putting w dc = -1 in Eq. (4.13). In Fig. 4.1, we
show the luminosity distance (multiplied by Hq) as a function of redshift
for different values of the matter content of the Universe and of the dark
energy equation-of-state parameter.
The relation between the luminosity distance and redshift is often more
conveniently restated as a relation between the distance modulus and red-
shift. The distance modulus \i is the difference between the apparent mag-
nitude m (defined as the logarithm of the ratio of the flux to a reference
flux) and the absolute magnitude M (defined as the logarithm of the ratio
of the luminosity to a reference luminosity), i.e.
^)=m-M = 51og 10 (d L /10pc). (4.14)
A plot of n(z) vs. z is called a Hubble diagram.
4.4 The Cosmic Microwave Background
The CMB is a unique cosmological observable providing a "snapshot" of the
Universe as it was nearly 400,000 years after the Big Bang, corresponding
to a redshift z ~ 1100, when the photons of the CMB last interacted with
matter; after that time, they have been freely streaming until the present
time. Thus, the CMB radiation carries a wealth of information about the
Features of the Observed Vnivei
ire 4.1 The effects of the matter content Qm and of the dark energy equal ion-oi-
e parameter w^ e on the luminosity distance d ^ as a function of rcdslu.lt .-..
physical conditions in the early Universe and a great deal of effort has
gone into measuring its properties since its sercndipit oils discover) by \ tiiu
Penzias and Robert Wilson in 1965. In fact, as wc have seen in Chap. 1,
the very existence of the CMB was enough to make the Hot Big Bang
model prevail over the Steady State Universe model. The first, fundamental
information that can be inferred from the observation of the CMB is that
the photons in the early Universe were in thermal equilibrium. This arises
from the blackbody spectrum of the CMB with a temperature of 2.725 K all
across the sky, providing the more perfect blackbody spectrum observed in
nature. Figure 4.2 reports the spectrum of the cosmic microwave radiation
measured by the Far InfraRed Absolute Spectrometer (FIRAS) instrument
on board the COBE satellite.
The CMB blackbody spectrum is a consequence of the fact that the fre-
quent Thomson scattering of photons over electrons maintained the thermal
equilibrium of the plasma. The scattering was very effective until the elec-
trons recombined with the free protons in the plasma to form neutral hydro-
gen atoms, when the temperature of the Universe was T ~ 3500 K ~ 0.3 eV,
corresponding to the value z ~ 1100; at that time, the photons scattered
over electrons for the last time. 5 For this reason, the spatial (hyper-) sur-
5 More precisely, since the Universe will later get reionized by the U V light of the first
stars, the photons have been able to scatter again before the present time.
Primordial Cosmology
CMB Frequency Spectrum
Figure 4.2 Intensity of the CMB radiation as a function of frequency, measured by
the FIRAS instrument, shown with 100<t errorbars. The solid curve is the theoretical
expectations for a blackbody at T = 2.725 K.
face at the time t(z = 1100) is called the last scattering surface. The reason
why the temperature at the time of last scattering is so much smaller than
the hydrogen ionization threshold Eh = 13.6 eV is the tiny value of the
baryon-to-photon ratio r\ = nbjn 1 ~ 10" 10 . Thus, even when the average
photon energy is well below 13.6 eV, there is still a sufficiently large number
of photons with energy E > Eh able to photoionize the hydrogen atoms
and prevent recombination. After recombination, the photons can travel
almost freely through the Universe, so that the present-day microwave sky
gives a faithful image of the last scattering surface, apart from the redshift
of the photon energy due to the cosmological expansion.
The second remarkable property of the CMB is its extreme isotropy.
The deviation AT(n) =T(h) —T from the average temperature at a given
direction n in the sky is everywhere < 200 /iK, corresponding to a fractional
deviation AT(n) / T < U)~ l . II us h igh degree of isotropy can be traced back
to the equally high homogeneity of the cosmological plasma at the time of
recombination. However, the CMB is not perfectly isotropic, implying that
Features of the Observed Universe
Figure 4.3 The microwave sky as seen by the WMAP experiment after
of observations. The image shows a temperature range of ±200/JKelvi
http://map.gsfc.nasa.gov/).
there were indeed small perturbations in the plasma. In some sense, one
should expect this, since, as explained in the previous section, galaxies
are thought to be formed through the growth of small primordial density
fluctuations. The anisotropies of the CMB are in fact related to the density
perturbations at the last scattering surface. Similarly to what happens for
the distribution of matter, a successful cosmologica] model should be able to
explain the angular distribution of CMB anisotropies. A further advantage
is given by the fact that the perturbations were still linear at the time of
recombination (as testified by AT/T <C 1), so that complications related to
the non-linear stages of evolution are absent. For all these reasons, a great
deal of observational effort has been invested to measure the anisotropy
pattern.
4.4.1 Sources of anisotropy
Let us briefly look at the mechaiuMii:-. ll).i/<m<>,b which density and veloc-
ity perturbations give rise to the temperature anistropies. It is customary
to distinguish between primary anistropies, already present at the time of
last scattering, and secondary anisotropies, created along the photon's path
from the last scattering surface to the observer. There are three sources
of primary anisotropies. The first ones are the density fluctuations them-
selves: where the plasma is denser, it is also hotter. The second source is
180 Primordial Cosmology
given by the velocity perturbations. A patch that is moving towards us will
appear hotter due to the Doppler shift, and vice versa a patch moving in
the opposite direction will seem colder. The third source is given by the
perturbations to the gravitational potential. Photons coming from poten-
tial wells will appear colder, since they lose more energy to climb out of
the well. This is known as Sachs- Wolfe (SW) effect. For what concerns the
secondary anisotropics, the most important is the integrated Sachs- Wolfe
(ISW) effect. The physical mechanism is exactly the same at the basis of
the SW effect, i.e. a difference in the gravitational potential, but this time
the difference arises from the time variation of the potential as the pho-
tons travel towards the observer (hence the term "integrated"). Since the
gravitational potentials are constant in a matter dominated Universe, this
effect is relevant either at early times (just after recombination), when the
radiation contribution to the total density is still important, or at late times
(close to the present day) when the contribution of dark energy becomes
relevant. The two effects are referred to as "early" and "late" ISW.
Another source of secondary anisotropies is reionization. Once the first
stars were formed, around redshift 10, the UV radiation emitted by stars
reionized the neutral hydrogen and helium in the Universe. We know from
the observations of the absorption spectra of distant quasars that the Uni-
verse was completely reionized at least from redshift z ~ 6. When the Uni-
verse is even partially reionized, free electrons are present and the CMB
photons are scattered again. Roughly speaking, this new "last" scattering
will mix up photons coming from different points of the last scattering sur-
face at z = 1100 and thus will tend to smear out anisotropies on scales
below the horizon at the time of reionization (i.e. an angular scale 9 ~ 5°
for a reionization occurring at z ~ 10). The temperature fluctuations below
this angular scale are suppressed by a factor e~ T (r is the optical depth
to the last scattering surface) which is the fraction of unscattered photons.
Finally, the Sunyaev-Zel'dovich effect generates secondary anisotropies due
to the scattering of the CMB photons over the free electrons present in the
hot intracluster medium.
4.4.2 The power spectrum of CMB anisotropies
As it was the case for the spatial fluctuations of the cosmological density
field, it is not possible to directly compare the exact pattern of temperature
fluctuations that we observe with the predictions of a certain theory. This is
because what we see in the sky is just one particular realization of the ran-
Features of the Observed Universe 181
dom process from which the temperature fluctuations originated. 6 Instead,
successful theories are required to predict the right statistical properties of
the temperature field. As usual, if the fluctuations are Gaussian, all the
statistical properties of the CMB are encoded in the power spectrum, re-
lated to the two-point correlation function. A difference with respect to
the power spectrum of galaxies is that the observed temperature field is
two-dimensional, depending only on the direction of observation n.
The first step is to expand the temperature field 6(n) = AT(h)/f in
spherical harmonics as
6(») = f;^a !m ^(n) I (4.15)
which is the analogue on the two-sphere of the Fourier transform in three-
dimensional space. The functions Y\ m form a complete basis over the two-
sphere and the set of a/ m encodes all the information present in the original
function. The spherical harmonics satisfy the ortonormality relation
J Y lm (h)Y} m ,{h)dSl = 5 w 5 mm , , (4.16)
where dfl is the infinitesimal element of solid angle spanned by n. This can
be used to invert Eq. (4.15) and write the ai m in terms of
= J Q{h)Y? m {h)dQ.. (4.17)
As noted above, we cannot make predictions directly for the a/ m [or equiva-
lent^ for 0(n)] but only for their statistical properties. The mean value of
the a; m 's vanishes because the average of fluctuations is zero by definition.
The temperature angular power spectrum C; is given by the variance of the
aim's as
{a lm a\, m ,)=8 w 5 mm ,Ci. (4.18)
If the temperature fluctuations (and thus the a; m 's) follow a Gaussian dis-
tribution, the C/'s completely define the temperature field, at least from a
statistical point of view. Let us stress one subtelty in the definition of the
C/'s, with important observational consequences. Since we have just one
Universe, we can observe just one temperature field and the corresponding
ai m 's. However, the definition of the Ci's involves an average operation —
in particular, the brackets in Eq. (4.18) denote an ensemble average, i.e.
6 This random process is related to the presence of quantum fluctuations in the early
Universe, as we shall see in more detail in Chap. 5.
182 Primordial Cosmology
an average over many independent realizations of the underlying random
process. If we were in a laboratory, we could repeat this process many times
under the same conditions observing the results at every realization. For
the CMB this is clearly impossible — so how can we perform the average?
As indicated by Eq. (4.18), the C/'s do not depend on to, i.e. for a given I
all the aim's have the same variance. This (as well as the constraints I = I'
and to = m') follows from the assumption of statistical isotropy, since the
Cj's cannot depend on the orientation of the coordinate system. For a given
/, the parameter to can assume 21 + 1 possible values, providing as many
samples drawn from the same distribution. We can construct the unbiased
estimator of Ci
C^^faLaim. (4.19)
The possibility to sample the distribution of the aim's only with a limited
number of values (equal to 2/+1 for a given I), implies an intrinsic limitation
to the measurement accuracy for a given Cj. This effect, known as cosmic
variance, is more important at low Vs where the number of a; m 's samples is
smaller. In particular, the minimum variance of a measured C; is given by
2Cf/(2l + 1), so that the relative uncertainty related to the cosi
(4.20)
where CV stands for "cosmic variance" . Another effect that introduces an
uncertainty in the measurement of the C;'s is the sample variance due to
the fact that an experiment does not observe the full sky, but covers a solid
angle A < Aty. Quantitatively, this increases the cosmic variance by a factor
of At:/ A. This is also relevant for full-sky experiments, since usually some
parts of the sky that are very contaminated by foreground emission, like
the Galactic plane, arc removed when analyzing CMB fluctuations. Both
cosmic and sample variance are present independently of the resolution and
sensitivity of the instrument. In particular, the cosmic variance represents
an intrisic limitation in the measure of the Cj's. For this reason, when
performing forecasts for the accuracy with which future CMB observations
will be able to constrain a given parameter, it is often customary to con-
sider as the most optimistic case that of an ideal, "cosmic variance-limited"
experiment, i.e. one where the only source of error is that in Eq. (4.20).
Features of the Observed Universe 183
4.4.3 Acoustic oscillations
We will now give a qualitative description of how the distinctive sequence of
oscillating peaks in the CMB angular spectrum is generated. This structure
is due to standing pressure waves in the plasma prior to recombination. At
that time, the photons were tightly coupled to baryons through Thomson
scattering and the pressure of the plasma was mainly given by that of the
photon component, i.e. P = p 7 /3. This corresponds to a sound speed v s of
the order of the speed of light, i.e. v s = \JdPfdp = l/\/l5, whose large value
prevented the growth of baryon density perturbations. In terms of the Jeans
mechanism (see Sec. 3.4), the Jeans length of the baryon-photon fluid is
comparable in size to the cosmological horizon. Then all the perturbations
inside the horizon at the time of recombination are oscillating, while those
outside are "frozen" to their initial values. If perturbations at all scales
have the same initial conditions as t — > 0, i.e. if all modes oscillate with the
same phase 0, some of them will be caught at a maximum or minimum of
the oscillation at recombination and will correspond to peaks in the angular
power spectrum (since the spectrum is proportional to the variance of the
temperature, both maxim;) and minima <uvc rise to a peak). On the other
hand, modes that are caught at the zero of the oscillation correspond to
dips in the spectrum. Since all waves have the same phase, the distance
between peaks and dips follows a harmonic pattern.
Let us give a more quantitative description of the physics of acoustic
oscillations. The equations for the evolution of temperature fluctuations in
the tight coupling limit were put in the form of an oscillator equation in
a classic work by Hu and Sugiyama in 1996. In a slightly simplified form,
neglecting for the moment the dynamical effects of baryons, the equation for
the temperature fluctuation Qk in Fourier space takes the forced oscillator
G k +v 2 k 2 Q k = J - , (4.21)
where dots denote derivatives with respect to the conformal time rj, k is the
wave number of the perturbation and the forcing term T takes into account
the effects of gravity. Neglecting the forcing term, the more general solution
to the homogeneous equation is simply Qk(v) = @fc cos(v s kr] + <p k ), where
the integration constants O^ and 4> k depend on the initial conditions. For
the moment, let us assume that 4> k does not depend on fc and that Gfc
is a simple featureless power law in k. As we shall see in Chap. 5, these
are the initial conditions predicted by inflation. We also take 4> k = 0, i.e.
&k(v = 0) = 0, thus neglecting the initial velocity perturbations. At a fixed
184 Primordial Cosmology
instant in time, and in particular at the time 77* of recombination (asterisks
denote quantities evaluated at recombination) the temperature distribution
as a function of k is given by
B k (v*) = e fe cos(fcs*), (4.22)
where s(r]) ~ v s n ~ T]/\/S is the distance a sound wave can travel in a time
interval 77, usually called the sound horizon (it is the acoustic equivalent of
the causal horizon discussed in Sec. 3.1.5).
The form of Qk(v*) shows that the acoustic oscillations generate a
cosine-like structure, superimposed on the featureless initial conditions 0fc.
The variance Q\ will exhibit a series of alternating peaks, starting with the
first peak at k = n/s* (corresponding to the mode with wavelength equal to
twice the sound horizon) and with subsequent peaks at integer multiples of
the first. At wavelengths much larger than the sound horizon, fcs* <C 1, the
perturbations will still be tracing their primordial values &k, as illustrated
in Fig. 4.4.
In the limit of constant gravitational potentials, the forcing term T is
equal to — A; 2 \&fc/3, where the curvature perturbation ^ k coincides with
the Newtonian gravitational potential at scales well below the horizon. In
this case, the oscillator equation can be put again in the homogeneous
form by defining an effective temperature Q' k = Qk + ^k- All the results
obtained until now still hold as long as they are stated in terms of the
effective temperature 6^, which is also the actual observed quantity. The
reason is that after recombination photons have to climb out their potential
wells to reach the observer, so that they lose energy proportionally to the
value of the gravitational potential. This is the Sachs- Wolfe effect at the
last scattering surface, briefly discussed above. In the case of time-varying
gravitational potentials (the case when the Universe is not perfectly matter-
dominated), the forcing term also includes terms proportional to the time
derivative of the potential which give rise to the integrated Sachs- Wolfe
effect also discussed above.
The effect of including baryons is twofold. First of all, they reduce the
sound speed to v s = 1/a/3(1 + R), where R ~ 3pb/4p 7 is the baryon-to-
photon momentum density ratio. This shifts all the peaks to larger k, i.e.
they are now at k = mry/3(l + R)/rj* instead of k = rnrVs/rj^. Secondly,
the baryons shift the zero of the acoustic oscillations to 0& = — (1 + R)'$ l k,
Features of the Observed Universe
Figure 4.4 Behavior of different fc-modes of temperature fluctuations as a function of
the conformal time r\. We show a mode that at recombination is caught at a minimum
of the oscillation (k = tt/s„, solid line), one that is caught in phase with the background
(k = 3?r/2s, , dashed line) and one that is caught at a maximum of oscillation (k = 2tt/s, ,
dotted line). These modes correspond to the first peak, first dip and second peak in the
anisotropy spectrum. We also show a mode with ks„ <C 1 (dot-dashed thin line), i.e.
with a wavelength much larger than the sound horizon, that had no time to evolve and
is still tracing the initial condition 0.
since they increase the inertia of the plasma. In the limit of constant R,
the effective temperature field at recombination is
e' fc (?7*) = e fc cos(fcs*) - R^ k , (4.23)
with s* ~ ?7*/^/3(l + R). This form breaks the symmetry between odd
peaks (corresponding to the maximum compression of the plasma) and
even peaks (corresponding to maximum rarefaction). In particular, odd
peaks are enhanced while even peaks are suppressed.
Another effect that should be taken into account is radiation damping,
better known as Silk damping, due to the fact that the baryon-photon fluid
is not a perfect fluid. In particular, shear viscosity and heat conduction
effects become important at scales below the mean free path of photons A 7 ,
in particular close to recombination when the tight coupling approximation
breaks down. Simply speaking, Silk damping is due to the fact that not-
so-tightly coupled photons can diffuse out of overdense regions and into
underdense regions and then cancel small-scale fluctuations in the radiation
186 Primordial Cosmology
density. The mean free path of photons is A 7 = l/(n e (TT) where n e is the
number density of electrons and ot is the Thomson cross section. In a
Hubble time -ff -1 , a photon will scatter on average n e <JT/H times, so that
the mean total distance traveled in that interval will be
(4.24)
We expect that perturbations below the damping scale Ad are canceled.
A careful numerical integration of the Boltzmann equation is required in
order to follow the evolution of Xd as the photons decouple from baryons and
A 7 — > oo, however the calculations show that inhomogeneities are damped
by a factor exp(— k 2 /k^), with the critical wave number kd of the order of
10/s*.
Finally, after recombination, photons are no longer coupled to the
baryons and can travel almost freely. This is when the secondary
anisotropies like the ISW effect and reionization, briefly discussed above,
come into play.
Let us briefly discuss how the 3-D field 0^ translates into the anisotropy
spectrum. In a flat Universe, a temperature spatial fluctuation with wave-
length A = 27r/fc will roughly correspond to angular fluctuations at the scale
9 ~ A/(?7o — V*) — V^o, where 770 is the conformal time today and thus
770 — V* i s the comoving distance between us and the last scattering surface.
When considering the multipole expansion of the temperature field on the
sphere [see Eq. (4.15)], a given angular scale corresponds (roughly) to a
multipole / ~ 1/8. Summarizing, inbotnogeuoities on a scale k are mapped
onto anisotropies at the multipole I ~ fer/o- In particular, the peaks will be
located at multipoles l n ~ mri]o/s.. t . Putting the numbers, 7 one gets that
the first peak should be at I ~ 180, in agreement with the observed position
of the first peak. In a non-flat Universe, a given angular scale k would not
be projected onto an angle 8 ~ (kr/o)^ 1 , but on a larger or smaller angle in
the case of a closed or open Universe, respectively. The position of the first
peak is, in fact, a powerful way to measure the curvature of the Universe.
The whole argument basically relies on the knowledge of the distance to the
last scattering surface and of the size of the acoustic horizon at decoupling,
so I ha I I ho angle under wliiob it gels project 01 1 provides informal ion on I he
spatial curvature. In jargon, the sound horizon at decoupling is a "standard
ruler" .
7 This can be done by taking s* = r/ t /\^3 and noting that during the matter dominated
era r) grows like (1 + z)~ 1/2 , so that 170/r?* ~ ^fzZ ~ 30.
Features of the Observed Universe 187
Of course, the general calculation of the exact temperature pattern 0^
will require to follow numerically the evolution of the perturbations, in-
cluding all the effects neglected so far (like varying gravitational potentials,
the dependence of the gravitational potentials themselves on the photon
density and thus on 6, etc.). Several codes have been developed to this
aim. The first was CMBFAST, written by U. Seljak and M. Zaldarriaga
(partially based on E. Bertschinger's COSMICS package). To date, the
most widely used is the CAMB code by A. Lewis and A. Challinor, itself
partially based on CMBFAST.
Let us discuss the issue of the initial conditions, and in particular the
assumption of coherent oscillations, i.e. that the phase <f>k is the same for
all fc-modes. The initial conditions have to be given by the theory that
explains how the primordial fluctuations have been generated, such as in-
flation which in fact predicts that </>*, is the same for all fc's. Although we
will discuss inflation in more detail in the next chapter, we can anticipate
that the reason is that all the perturbations are generated at the same
time independently of k. In models when this does not happen, for ex-
ample in topological defects models that were extensively studied during
the '90s as a possible alternative to inflation, the phases <j>k are uncorre-
cted. The resulting uncoherent oscillations lead to a washing out of the
acoustic peaks. In fact the observations of the peak structure of the CMB
anisotropy spectrum, first made by the BOOMERanG experiment in the
late '90s, ruled out topological detects as the main mechanism of generation
of the primordial fluctuations, in favor of the inflationary paradigm.
4.4.4 Effect of the cosmological parameters
The main features of the CMB anisotropy spectrum can be qualitatively
predicted through the arguments presented above, however the precise val-
ues of the Ci's depend on the cosmologii al parameters. This is the reason
why the CMB angular spectrum is such a powerful tool to measure the
cosuiologica] parameters. We will briefly discuss their effect.
The first step is to choose which parameters should be used to describe
our Universe. The simplest model able to explain the WMAP data is a
6-parameter, flat ACDM model with adiabatic initial conditions. This min-
imal model is often dubbed "vanilla" ACDM. The parameters of the vanilla
model are the physical 8 baryon density u>b = Qbh 2 (where h is the Hubble
8 The term physical density (denoted with w) is a jargon to indicate the density param-
eter ft = p/pc multiplied by h 1 . in order to eliminate the uncertainity related to the fact
188 Primordial Cosmology
constant 7Jn in units of 100 km s _1 Mpc -1 ), the physical cold dark matter
density uj c = Sl c /i 2 , the cosmological constant density £1a, the amplitude of
the primordial curvature perturbations A|j at the scale ko = 0.002 Mpc~ ,
the slope of the primordial spectrum n s , and the optical depth r to the
last scattering surface. Of these six parameters, three (wt>, <^c and Cl\)
describe the matter-energy content of the Universe, two (A^ and n s ) de-
scribe the initial conditions from which the perturbation evolution started,
and the remaining one (r) is related to the reionization of the Universe at
z ~ 10. The value of h is fixed requiring that the Universe should be flat,
i.e. Q, h + Q, c + tt A = (uj h + uj c )/h 2 + Q A = 1. A different combination of the
parameters could have been chosen, for example selecting il},. fl c and Hq
in place of ojb, ^c and SIa- In this case, ft\ would be the derived parameter
fixed by requiring flatness, i.e. Qa = 1 — ^b — ^c- In general, there is
no "correct" choice for the parameter set. However, one tries to choose
the parameters such that each of them has a unique, peculiar effect on the
anisotropy spectrum (even if that is not always possible). Thus, one can
disentangle the effect of each parameter, useful for pedagogical purposes,
other than when performing "real analyses" of the data. The one presented
here is a fairly common choice of the minimal parameter set. A good pa-
rameter choice for one particular observable, for example the CMB, could
not be as good for another one, for example the matter power spectrum.
For these reasons, the choice of the parameters used to describe the cosmo-
logical model is often a matter of compromise. The vanilla model is just
the simplest choice, so that it can be expanded by considering additional
parameters beyond the minimal set. Some examples include, but are not
limited to, the neutrino mass, 9 the effective number of relativistic species,
the running of the spectral index, tensor modes, the fraction of isocurvature
perturbations, the equation-of-state parameter for dark energy.
After this necessary caveats, let us examine the effect of the parameters
on the spectrum.
Dark matter density. The lower uj c , the longer the radiation-dominated
era lasts, 10 so that matter-radiation equality occurs closer to recombination.
Since the gravitational potentials decay during the radiation-dominated
that p c oc li 2 . Of course, there is n i m ll ihout the Q's.
9 The vanilla AC'DAJ indeed includes neutrino. I ml considers them massless. Even
though we know, from oscillation expej inieuts, that neutrinos do have a mass, this is a
reasonable approximation for the minimal cosmological model since the effects of a finite
10 Recall that 1 + z eq = w m /a) rad .
Features of the Observed Universe 189
era, this will lead to a smaller gravitational potential at recombination and
thus to a larger temperature fluctuation. 11 Another effect is the early
integrated SW that occurs immediately after recombination due to the
residual radiation. Both effects sum up to increase the amplitude of the
spectrum for smaller ui c .
Baryon density. Changing Wb also produces the effects described above
for uj c , since they only depend on the time of matter-radiation equality and
thus on uj m = cob + co c . However, Wb also has a very peculiar effect on the
spectrum since the presence of baryons is responsible for the alternating
structure of the peaks (odd peaks are higher than even peaks 12 ) and a
larger value of Wb makes this asymmetry stronger, enhancing odd peaks
and suppressing even peaks. This peculiar character makes this effect very
easy to be isolated, and in fact Wb is one of the parameters that are better
measured from the CMB. Another effect of an increased baryon density is a
reduced diffusion damping (the photon mean free path is smaller), so that
the small scale (high Z's) spectrum is larger.
Cosmological constant. The effect of a cosmological constant mainly
comes from the late integrated SW effect, because the potential is not
constant at late times, when the Universe is not perfectly matter dominated.
A large value of fl\ thus enhances the large-scale anisotropics. However,
since the spectrum is usually normalized at the large scales, the net effect
of increasing fl\ is to suppress the small-scale anisotropics.
However, all the parameters considered until now have a small effect
in the location of the peaks. In the case of co c and co^ this is due to their
effect on the sound horizon at recombination s* and on the conformal time
today 770, while the cosmological constant density fl\ has no effect on s*
but changes r/ .
Amplitude of the primordial curvature perturbations. The effect of
changing the amplitude of the primordial spectrum of fluctuations is to
modify the normalization of the CMB spectrum.
Spectral index. Changing the spectral index n affects the relative height
between the small and large scales. The Harrison-Zel'dovich spectrum n =
I generates a large-scale CMB spectrum that scales as [1(1+ 1)] _1 . This is in
fact the reason why the spectrum is usually plotted in terms of the quantity
II This somewhat counterintuitive result is due to the fact that, even if the smaller
potential corresponds to a less dense (i.e. colder) region, the photons have to climb a
less deep potential well and this will more than compensate for the smaller temperature.
In other words, 0j. becomes smaller but the observed temperature 0^ + \Pfc becomes
larger.
12 Before diffusion damping is taken into account.
Primordial Cosmology
Parameter
Mean WMAP
100tt b h 2
tt c h 2
okc+0-057
z,.zoa_Q 056
0.1109 ±0.0056
n A
0.734 ±0.029
n s
0.963 ±0.014
10 9 Al(k )
0.088 ±0.015
2.43 ±0.11
Table 4.1 Cosmological parameters
from WMAP7. Mean values of the
parameters of the minimal ( "vanilla" )
ACDM model from the analysis of
the 7-year WMAP data. The er-
rors show the 68% confidence region.
The primordial curvature fluctuation
A^j is normalized at the pivot point
k = 0.002Mpc _1 . Adapted from
Ref. [304].
Ci = 1(1 + l)Ci, so that at small Vs it will reach a plateau. In terms of Ci all
multipoles receive the same contribution from an initial Harrison-Zel'dovich
spectrum. If the spectrum is tilted (n/ 1), the contribution to Ci will scale
like l n ^ 1 . If the initial spectrum is blue (n > 1), the small scales will have
more (primordial) power with respect to the large scales; vice versa if the
spectrum is red (n < 1). Considering again that the spectrum is normalized
at small l's, we have that n > 1 (< 1) will increase (decrease) the overall
power at large Vs. Since it represents the slope of the spectrum, n can be
measured more and more precisely as smaller scales become accessible to
observations.
Optical Depth. A non-zero value of the optical depth r to the last scat-
tering surface represents the integrated effect of the scattering of photons
over free electrons after the Universe gets reionized at z ~ 10. As explained
above, this tends to cancel the anisotropies at scales below the horizon at
the time of reionization (roughly I > 10), with a suppression factor given
by e~ T .
The measurement of the CMB anisotropy spectrum is a powerful tool
to constrain the values of the cosmological parameters. In Tab. 4.1 we
show the values of the six parameters of the minimal ACDM determined
from the analysis of the 7-year WMAP data. In Fig. 4.5 we also show
the best-fit anisotropy spectrum along with the WMAP data. The picture
emerging from the CMB is that of a Universe with only 5% of baryons,
22% of cold dark matter and 73% made of a cosmological constant-like
component, consistently with that coming from other observations.
Features of the Observed Vnivei
Multipole Moment (I)
Figure 4.5 CMB anisotropy spectrum corresponding U
WMAP analysis. The points show the WMAP7 data w
4.5 Guidelines to the Literature
Many of the textbooks recommended in the previous Chapter also deal, to
some extent, with the phenomenology of the observed Universe. Since this
topic is strictly connected to the advances in the observational field, recent
books provide a picture of the current observational status. In general, we
refer the interested reader to Dodelson's book [155], that puts a focus on
the quantitative comparison between theory and observations, with some
bias towards the CMB and the matter distribution. It also provides an
excellent introduction to the basic techniques that are used in the analysis
of cosmological data.
The topic of large-scale structure discussed in Sec. 4.2 is the subject
of many textbooks, like the classic one by Peebles [377] or the more re-
cent one by Padmanabhan [370]. An introduction can also be found in the
already mentioned books by Kolb & Turner [290] and Peebles [378]. The
review [159] also covers in detail the topic of galaxj formation. The effect of
collision- less damping on the evolution of perturbations in a hot dark matter
component (massive neutrinos) was studied in [99,100]. We refer the reader
192 Primordial Cosmology
interested in the topic of dark matter to the review [83] . The galaxy power
spectrum has been measured in the last decade by the Two Degree Field
(2dF) galaxy survey (http://msowww.anu.edu.au/2dF\index{2dF}GRS/)
and by the Sloan Digital Sky Survey (SDSS) (www.sdss.org/). The cos-
mological implications of the observations of the 2dF and SDSS surveys are
discussed in [123,158,167,233,375,382,383,409,448] and [412,413,433,434],
respectively. The most recent determination of the homogeneity scale, using
the luminous red galaxy sample of the SDSS, can be found in [239].
The acceleration of the Universe, discussed in Sec. 4.3, including its
observational evidence and the possible interpretations, is the subject
of [178]. The papers reporting the first evidences for the acceleration
are [392] and [386]. See also the websites www.supernova.lbl.gov/ and
www. cf a. harvard. edu/supernova//HighZ. html More recent SNIa data
and their cosmological interpretation can be found in [296] .
An introduction to the physics of the CMB discussed in Sec. 4.4 can
be found in [246,249] and [295]. Here we did not address the topic of
the polarization of the CMB, for which we refer the interested reader
to [107,265,294]. The description of the recombination process was first
made by Peebles in [376] and the original paper by Silk on diffusion damp-
ing is of the same year [418]. The evolution of the perturbations in the
baryon-photon fluid was first computed in [380]. Dark matter was in-
troduced only later in the computations, see e.g. [453]. The physics of
acoustic oscillations was investigated in detail in a series of papers by
Hu & Sugiyama [247,248]. The computational framework for the inte-
gration of the coupled Einstein-Boltzmann equation has been established
by Ma & Bertschuiger [329]. L'be calculation of the CMB anisotropics
has been sped up after the introduction of the line-of-sight integration
approach by Seljak & Zaldarriaga [414] and of their computer program
CMBFast (lambda.gsfc.nasa.gov/toolbox/tb_cmbfast_ov.cfm). The
method has been refined by Lewis, Challinor & Lasenby [309] and im-
plemented in the code CAMB (camb. inf o/). The measurements of the
frequency spectrum of the CMB, shown in Fig. 4.2, were made by the FI-
RAS experiment on board the COBE satellite [173,337,338]. The most
recent measurements of the CM! ai o1 opj spectrum shown in Fig. 4.5,
along with their cosmological interpretation, made by the WMAP satel-
lite (map.gsfc.nasa.gov/) after seven years of observations, can be found
in [72,198,264,291,304,460].
Chapter 5
The Theory of Inflation
In this Chapter we will discuss the inflationary scenario, focusing on the
most general features of this paradigm for the evolution of the early Uni-
verse. The idea of an early phase of inflationary expansion was developed
between the end of the '70s and the beginning of the '80s, in order to
overcome some critical shortcomings of the Standard Cosmological Model.
Despite its standing success in solving basic paradoxes of the standard hot
Big Bang model, the inflationary scenario still has an ad hoc taste. This is
due to the need for a certain amount of fine-tuning of the model parameters
(especially concerning the flatness of the scalar field potential) and, on the
other hand, to the many alternative proposals for the detailed evolution of
the self-interacting scalar field at the ground of the whole idea.
However, inflation can be regarded as a cosmological I harry because its
basic framework is well-motivated at the level of fundamental physics and
its predictions, other than solving conceptual questions, are in agreement
with the present osbervational knowledge of the Universe.
In the following, we will concentrate on the most general features of
the inflationary scenario, which have been largely unaffected by the later
developments of the theory and are the most relevant for the primordial
history of the Universe, which is the main subject of this Book.
We start by discussing the basic shortcomings of the Standard Cos-
mology which require the introduction of a new paradigm. Then we will
provide a brief description of the ideas characterizing the theory of elemen-
tary particles which offer the physical motivation and the dynamical tools
to implement the key role of a phase transition during the evolution of the
Universe, making available a dominant vacuum energy. The real inflation-
ary evolution is implemented by describing the different dynamical regimes
in the evolution of the self-interacting scalar field. We show how the infla-
194 Primordial Cosmology
tion solves the main puzzles of the standard cosmology and the resulting
predictions. In particular, the mechanism by which this paradigm provides
a perturbation spectrum for the isotropic Universe is treated in some detail.
This Chapter ends with a brief discussion of the late acceleration of the Uni-
verse, a timely question related to the inflationary paradigm, which could
be explained by the presence of an exotic component called dark energy, or
by modifications to GR.
5.1 The Shortcomings of the Standard Cosmology
The SCM provides a successful representation of the Universe in terms of
the Robertson- Walker (RYV ) geometry underlying the large-scale evolution
of a homogeneous and isotropic thermal bath. As the Universe expands,
the temperature of the bath decreases and a series of departures from equi-
librium and phase transitions happen, like for example baryogenesis, nucle-
osynthesis and hydrogen recombination. In the latter phase of the evolution
of the Universe, corresponding to the matter-dominated regime, the Jeans
mechanism (see Sec. 3.4.3) explains the magnification of the primordial
density perturbations, eventually resulting in the formation of cosmological
structures once the perturbations reach the non-linear regime. The SCM is
expected to fail close to the Planck era, when the quantum gravity effects
have to be taken into account. Moreover, it has been argued that in the
present Universe the small-scale inhomogeneities could induce significant
deviations from the RW background even on very large scales, implying
that the homogeneity hypothesis at the basis of the SCM is not completely
correct. However, these are not failures of the SCM per se, but instead
just limits for its domain of applicability. On the other hand, the SCM
leads to paradoxical results when it is applied to the very early Universe
just after the Planck time. All such paradoxes, described in more detail in
the following, are somewhat related to the very particular initial conditions
needed to obtain the present day Universe.
In the following Chapters we will describe a rather different scenario
for the very early Universe with respect to the homogeneous and isotropic
framework. Inflation plays a crucial role in reconciling these very general
dynamical perspectives with the SCM phenomenology. However, we stress
that even if the main aim of this Book is to investigate a very general
(anisotropic and inhomogeneous) nature of the Big Bang, nevertheless the
request for an inflationary phase of the Universe arises from internal incon-
The Theory of Inflation 195
sistencies of the SCM, emerging as soon as even qualitative observational
evidences are critically analyzed.
This section is then devoted to the analysis and discussion of four funda-
mental shortcomings of the SCM, the so-called horizon and flatness para-
doxes and the entropy and unwanted relics problems.
5.1.1 The horizon and flatness paradoxes
The horizon paradox The evidence of a conceptual problem in the
understanding of the Friedmann-Robertson- Walker (J/TiW) Universe (es-
sentially characterized only by the radiation- and matter-dominated eras)
arose immediately after the discovery of the CMB and of its high degree
of isotropy (the temperature angular fluctuations are less than one part
in 10 4 , see Sec. 4.4). The observation of such isotropic thermal radiation
provided a compelling evidence in favor of the hot Big Bang theory, but it
was soon realized that the spatial uniformity of such black body was indeed
problematic.
To understand the paradox, one needs to relate the CMB isotropy to the
notion of causality. As explained in Sec. 3.1.5, in a Friedmann Universe the
size of the causal horizon coincides, apart from factors of order unity, with
the Hubble length Lh- Since H oc 1/t and during the matter-dominated
era a oc £ 2 / 3 , the Hubble length at the time of hydrogen recombination was
L H (t rc ) = H- C l = Hq X {1 + z rc )-V 2 = L°i(l + x rc )~ 3/2 , where z rc ~ 1100.
The corresponding physical distance today is d = (1 + z IC )Lu(t IC ) = (1 +
z rc r 1/2 £&.
To estimate how many independent causal regions are contained in
the CMB sphere, we observe that the latter has a surface of the order
47r 2 (ciJ _1 ) 2 , thus the number of the observed independent causal regions
^ (1 + z rc ) ~ 10 3 . (5.1)
VM*re)(H
In other words, when we look at the extremely uniform microwave sky, we
are actually looking at ~ 1000 independent causal regions at the time of
recombination, which never had the chance to be in thermal contact; in
spite of this, they all have the same temperature within one part in 10~ 4 .
The question at the basis of the paradox is: why have these regions such
a fine tuned temperature if they had never been in thermal contact among
themselves at the time when the CMB was emitted?
196 Primordial Cosmology
A possible answer is that the homogeneity was part of the initial con-
ditions. In fact, one could ask: Why is the CMB isotropy so strange if we
considered a RW geometry?
The point is that if we assign, at a given instant, the initial conditions
for the cosmological fluid, we unavoidably deal with uncertainties on the
fundamental matter fields. Despite their smallness, such uncertainties inde-
pendently evolve on disconnected causal regions. After a certain interval of
time, the matter fields eventually acquire a degree of inhomogeneity due to
the specific thermodynamical evolution of each horizon. Thus, to interpret
the CMB isotropy as a consequence of the initial conditions, we need an
estimate of the density contrast in a primordial stage, say at the Planck
era.
The fractional temperature fluctuation of the CMB is ST/T < 1CP 4
and traces the density fluctuations in the cosmological fluid at the time
of recombination t rc (see Sec. 4.4 for more details). Since the fluid was
in thermal equilibrium, one has p ex T 4 and 5 = Sp/p = A5T/T. Thus,
a reliable estimate of the degree of inhomogeneity of the Universe at the
recombination, for z ~ 10 3 , is given by the value 5 rc ~ 10~ 4 . As discussed
in Sec. 3.5.5, during the matter-dominated era the density perturbations
outside the horizon grow like a oc i 2 ' 3 and during the radiation-dominated
era as a 2 oc t, one can compute the density contrast <5p at the Planck era.
At the time of matter-radiation equality it was z ~ 10 4 [t cq ~ 0(lO 12 )s],
and therefore 5 cq ~ £>(10~ 5 ). Hence, at the Planck era [t P ~ 0(lO -44 s)j,
we get S P ~ 0(l(r 5 ) x t P /t cq ~ 0(l(r 61 )! Such a Planckian value of
the density contrast is too small to be physically acceptable as an initial
condition, especially in view of the quantum fluctuations characterizing
those primordial phases. The horizon paradox is therefore a real and deep
conceptual inconsistence of the SCM, unless an extreme fine tuning of the
initial conditions is accepted as an a priori prescription of the Nature.
The flatness paradox The present value of the spatial curvature of the
Universe is very small; as discussed in Chap. 4, the CMB observations
indicate the present value of the critical parameter Q = 1.01 ± 0.02, or,
in other words, \Q — 1| < 10~ 2 , thus the Universe is very close to being
flat, even if the sign of the curvature is still unknown. The flatness paradox
emerges from analyzing the structure of the relation (3.52), which is restated
here for convenience
H-1-5&. (5.2)
The Theory of Inflation 197
This formula, recalling that the matter and radiation energy densities be-
have as p m ~ 1/a 3 and p ra( j ~ 1/a 4 respectively, allows us to get the
behavior of the quantity (Q — 1) as a function of redshift. Today, the curva-
ture term in the Friedmann equation is negligible with respect to the matter
source and this was even more true in the past, since the curvature term
scales like a~ 2 while the matter and radiation terms scale as a~ 3 and a -4
respectively. The Friedmann Eq. (3.46) states the proportionality between
H 2 and p, so that we can write
Matter dominated Universe: (0. — 1) oc (1 + z)~ x (5.3)
Radiation dominated Universe: (fl — 1) ex (1 + z)~ 2 . (5.4)
If today we have |f2 — 1 1 ^S 10~ 2 , at the time of equivalence ;z oq ~ 10 4
we get |fi eq — 1| % 10~ 6 and finally we gain the surprising Planck value
(zp ~ 10 32 )
ftp-1 <0(1O -62 ). (5.5)
As for the density contrast in the previous subsection, we find again that
the initial condition compatible with the present flatness of the Universe
requires an extreme fine tuning of its initial value at the Planck time. From
a physical point of view, this result is equivalent to require that the Uni-
verse is appropriately described by the flat RW geometry, i.e. by K = 0.
Such situation is paradoxical since tiny, local density fluctuations (naturally
expected after a quantum regime of the Universe), would have affected dras-
tically the present structure of the cosmological space, but this is not the
case. Thus, as for the horizon paradox, we deal with a subtle conceptual
puzzle, which calls attention for its solution in a new cosmological frame-
5.1.2 The entropy problem and the unwanted relics paradox
The entropy problem The entropy problem can be stated noting that
the entropy of the observable Universe is enormous. We know from
Eq. (3.44) that the entropy density s ~ T 3 . The size of the observable
Universe, in a Friedmann Universe is roughly given by the Hubble length
Lh = H^ 1 and then the total entropy S inside the presently observable
Universe is
S = (T°L H ) 3 ~ 10 87 (5.6)
where we have used the present-day photon temperature T° ~ 2.73 K~
1(T 13 GeV and L H ~ 10 28 cm ~ 10 42 GeV -1 .
198 Primordial Cosmology
Such a large value of the entropy is especially puzzling if the expansion is
taken to conserve entropy. This would imply that the Universe has started
with an enormous entropy S ~ 10 87 , and this appears like a very particular
initial condition.
The unwanted relics paradox The unwanted relics paradox is related
to the fact that if we allow the early Universe to have an arbitrarily high
temperature at that time (or at least a temperature as high as the Planck
energy), we expect very heavy degrees of freedom (for example new particle
species predicted by unified gauge theories or by supersymmetry). Such
relics could survive until today with an actual abundance that would be in
gross contradiction with observations.
Let us briefly describe how to determine the present cosmological abun-
dance of a species that was in thermal equilibrium in the early Uni-
verse. Considering in the early Universe a very heavy (for example,
m x > 100 GeV) particle X, the X's were kept at equilibrium with the
other particles in the cosmological plasma (call them in general Y) by rapid
annihilations of the type XX -H> YY . This happens as long as the annihi-
lation rate r ann is fast enough with respect to the expansion rate, given by
the Hubble parameter H, thus r ann 3> H.
The annihilation rate r ann is given by nx(^ann'')i where {a ann v) is the
thermally averaged cross section multiplied by velocity. When the annihi-
lation rate drops below the expansion rate, i.e. r ann < H, the annihilations
are no longer effective in coupling the X's to the other species in the plasma,
and the number of X's per comoving volume is "frozen" at the value it had
at the time U. . such that r ann (if. .) — H{U, Q ). This process is called
freeze- out and after it, the number density nx is diluted by the Universe
expansion, i.e. nx(t > t{. .) = n x (t { . .)[a(t { . .) / a(t)} 3 .
In more detail, we can distinguish three phases:
• Initially, 1 r ann > H and T > m X - The first condition ensures
that the annihilations are very effective in maintaining the X's at
the equilbrium with the plasma. The second condition states that
the X's are ultrarelativistic, so that their number density nx — T 3 .
In particular, this implies that nx ~ ??~ (this is basically a conse-
quence of the equipartition theorem). Assuming for definitiveness
that (a ann v) = const. = cr 2 , we have that r ann oc T 3 . On the
other hand, during the radiation-dominated era, from the Fried-
The Theory of Inflation 199
mann equation one has that H = ^/k/7^/3 oc T 2 and the annihi-
lation rate decreases faster than the expansion rate.
• As the Universe expands and cools down, it enters a second phase,
corresponding to the regime when the annihilations are still ef-
fective, r ann 3> H, but the X's are non-relativistic, 3 T < nix-
The X's still have an equilibrium distribution, but the equi-
librium number density in the non-relativistic regime is nx —
(mxT) 3 / 2 exp(— mx/T), so that the number density exponentially
decreases with temperature. This is also a consequence of the fact
that when T < mx the average energy of the other particles is
not large enough to produce the X's, so that the annihilations
are not compensated by the inverse creation process, i.e. one has
XX -5- YY but not YY -> XX.
• Finally, when the temperature decreases enough, r ann < H and
the annihilations become ineffective, the X's are not destroyed and
their number per comoving volume is conserved.
Prom this picture one expects that the larger the annihilation cross
section the smaller the freeze-out abundance, because annihilations will
"switch off" later and the abundance will be more reduced with respect to
its high-temperature value. One expects that for fixed (a ann v), a smaller
mass will produce a larger relic abundance because the exponential decay
of the density will start at a correspondingly lower temperature T ~ mx ,
and thus closer to freeze-out, leaving less time for the annihilations to op-
erate. In fact, when the freeze-out process is worked out, the integral of
the relevant Boltzmann equation, leads to 4 :
N X oc 1 = -L- , (5.7)
m X {(Ta.nnV}f.o m X Cr
where the last equality holds for constant (cr ann v). The present number
density n x is also proportional to l/(mx<ro), so that the present energy
density p x = mxn x only depends on crn.
The "unwanted relics" paradox comes from the very small annihilation
cross section of very heavy particles, <j$ oc 1/mx, so that in the end the X's
3 We are implicitly assuming that the freeze-out happens when the particles are non-
relativistic, so that T ~ mx before r ann ~ H. This makes the X particle a cold relic.
The opposite case, when T ~ ni x is later than r ann ~ H (i.e. a freeze-out that occurs
when the particles are ultrarc'lativiHtk). is called that of a hot relic. In practice, a hot
relic skips the second phase so that its number density is frozen to the high temperature
4 There is actually also an additional, logarithmic dependence on i>i x that we neglect
200 Primordial Cosmology
density parameter is ttx = Px/Pc ^ 'nx- Very heavy, stable particles tend
to overdose the Universe, i.e. fix 3> 1, at variance with the observations
which indicate that f^tot — 1-
5.2 The Inflationary Paradigm
This Section is devoted to define the general framework of the inflationary
model. We start from the description of the spontaneous symmetry breaking
process at the ground of the inflationary phase transition and then we
develop the details of the scalar field dynamics.
5.2.1 Spontaneous symmetry breaking and the Higgs phe-
At low energies, the electromagnetic and weak interactions appear as sep-
arate physical phenomena, although the Standard Model of elementary
particles predicts a unified electroweak interaction which accounts for all
the observation in a common theoretical picture. However, the electroweak
model relies on a fundamental symmetry which is not directly observed
in Nature. The process which allows this transition from a more general
symmetry to a restricted one is known as spontaneous symmetry breaking
(SSB) and it corresponds to the fact that a quantum theory can be invariant
under a certain symmetry, while at the same time the associated vacuum
state is not invariant under such symmetry. The low energy limit, near
that vacuum state, is unable to reveal the global symmetry of the model,
that is then said to be spontaneously broken, because no additional terms,
violating this symmetry, are present in the Lagrangian of the model.
Furthermore, the observed gauge bosons Z and W± carrying the weak
interaction are massive particles, but they emerge from the massless bosons
of the electroweak model (only photon remain massless after the SSB pro-
cess is implemented on a linear combination of the fundamental four gauge
fields). Since a massive vector boson has three independent degrees of
freedom, instead of two like the corresponding massless field (for which
longitudinal states are forbidden), we have to answer the question:
where do the three degrees of freedom come from in the SSB of the elec-
troweak model?
An appropriate answer is provided by the so-called Higgs phenomenon,
according to which the scalar field responsible for the SSB transition pro-
The Theory of Inflation 201
vides also some degrees of freedom to give mass to the vector bosons; in
particular, we will see below how the single degree of freedom of a Goldstone
boson becomes available to induce longitudinal states of a vector field.
The SSB process and the Higgs phenomenon represent the fundamental
physics motivation for the inflationary paradigm, deriving a more detailed
discussion, despite not essential for the overall cosmological dynamics. For
the sake of simplicity, wc will treat these two concepts in the simplified
case of an Abelian U{1) symmetry, instead of the real non-Abelian scheme
of the electroweak model. Furthermore, we limit our presentation to a
semiclassical framework, avoiding additional quantum features. Despite
these simplifications, the SSB process and the Higgs phenomenon are traced
in the following analysis in all their formal elegance and power.
Let us consider the Lagrangian density of a complex scalar field <fi =
(</>!+ z0 2 )A/2
C 4 , = v t ii d i ^d j (t>-Vn{\<t>\), (5.8)
with the Higgs potential term
Vk = fH 2 + ^| 4 , (5.9)
a being a negative number (a = — /i 2 ) and A a positive one. The Lagrangian
density (5.8) is invariant under a rotation in the {<j)\, 02}-plane, i.e. for
<j>\ = </>i cosip + </>2 sin V' (5.10)
<t>2 = — <f>i sintp + 02 cos^ , (5-11)
where ip is a constant rotation angle (ip = const.).
The vacuum state of the field <fi has to correspond to the state of minimal
energy for the system. Considering that the energy density has the form
^ = |9 t 0| 2 + |V0| 2 + t/(|0|), (5.12)
the vacuum is obtained for a constant value of <j) which gives the lowest
local minimum of the potential. On a quantum level, the field fluctuates
around such a constant value, and in the semiclassical treatment one deals
with expectation values of the field around the vacuum, denoted as (<j>o)-
In correspondence to the quartic potential (5.9), the case a > yields
as vacuum state (<fio) = 0, which is invariant under the symmetry (5.11)
and no SSB takes place in the field dynamics. Instead, when a = —fi 2 , we
get an infinite array of degenerate vacuum states, namely those associated
to the circumference (<j>i) 2 + {4>2) 2 = (?*o) 2 = M 2 A-
202 Primordial Cosmology
In such situation, a SSB arises because the rotation (5.11) maps the
different vacuum states one onto another. Without loss of generality, if we
choose the vacuum state as (<fii) = (r ), ((fc) = (0), then it is manifestly
non-invariant under the fundamental symmetry of the model, while the
Lagrangian density is invariant under the same symmetry. In order to
outline two relevant implications of this SSB scenario, let us introduce a
new representation of the complex scalar field, having the form
4> = -=— = r exp(«0) . (5.13)
Using this parametrization of the field <fi, the Lagrangian density (5.8)
rewrites as
C^ = V ij (d.rd.r + r 2 di9dj9) + ^-r 2 - V . (5.14)
In such representation, the symmetry (5.11) reduces to the invariance of
the Lagrangian density (5.14) under the transformation 8 — > 6' = 8 + ip.
In order to analyze the features of the Lagrangian near the vacuum state
chosen above (that in the new variables reads as (r) = (r ) and (8) = (0)),
let us define the fields f = r— (r ) and 8 = 8, such that they have vanishing
i expectation values. In terms of these fields, adapted to the chosen
, the Lagrangian density (5.14) takes the form
C<t, =v 11 \d t rd 3 f +(f+ (rorfdiOdjO]
2 x ' (5-15)
From a careful analysis of the Lag):aii(>,'ian density, the two fundamental
statements follow:
- The model is no longer invariant under the symmetry r — > —r, be-
cause in the Lagrangian density (5.15) linear and cubic terms ap-
pear. Despite the full model is still invariant under this (discrete)
reflection symmetry, an observer living near the chosen vacuum
does not realize it: this is the real physical content of the SSB
phenomenon. Such discrete symmetry is equivalent, in the present
context, to the transformation 8 — ¥ 8 + ir, but we are interested in
its existence and in the spontaneous breaking as referred to field
r alone. In fact, we will show that the degree of freedom associ-
ated to 8 is absorbed by the longitudinal mode of a vector boson.
Therefore, the scalar field responsible for the inflationary scenario
has a potential invariant under the parity symmetry, see Fig. 5.1.
:r 29, 2010 11:22
The Theory of Inflation 203
However it is spontaneously broken near one of the two degenerate
minima, as outlined above.
- The scalar field 9 corresponds to a massless boson (coupled to the
field r): this feature induced by the SSB is known as the emergence
of a Goldstone boson.
Figure 5.1 The Higgs potential Vh(0) of the Lagrangian (5.15), also known ai
hat. The rotational invariance of Vn (</>) can be noticed, as well as the presence of infinite
equivalent minima.
Finally, to illustrate the Higgs mechanism, we promote the transformation
on 9 to a gauge symmetry, by requiring the angle ip to be a space-time
function, i.e. we deal with the change 9' = 9 + ip(x l ).
According to Sec. 2.2.4, the lost invariance of the Lagrangian density
under such local symmetry is restored by introducing an Abelian gauge
boson Ai, which correspondingly transforms as A/ = Ai — ditp and allows
to set the covariant gauge derivative T>i9 = d l 9 + A l . A Lagrangian density
204 Primordial Cosmology
invariant under the above gauge symmetry explicitly reads as
£ = v ij [difdjf + (f + (r )) 2 {diO + Ai) (djO + A,-)]
+ ^(r+(ro)f-j(r+(ro)f-^F l3 F^
(5.16)
where Fij = diAj — djAi is the gauge tensor associated to Ai, and g is the
associated coupling constant. Looking at the form of Eq. (5.16), we are
naturally led to define the new gauge boson Bi = Ai + did, which allows to
rewrite the electromagnetic tensor as F^ = diBj — djBi = Fij(Bi) (indeed
the definition of Bi corresponds to a gauge transformation for Ai), and
hence the Lagrangian density can be rewritten as
£ = r) ij (difdjf +(f+ (r )f BiBj)
+ Y {f+ (ro))2 " I (f + (ro))4 " -flFii^'&i) ■ ( 5 - 17 )
The 9 boson disappears from the theory, but its degree of freedom is in-
corporated within the massive boson B i7 which has an additional (longitu-
dinal) degree of freedom with respect to the original massless gauge boson
Ai . Such massive gauge boson is a typical feature of the Higgs phenomenon
associated to a SSB process. Summarizing, we started with a theory in-
variant under a given internal symmetry which is spontaneously broken by
the quartic Higgs potential and, near a vacuum state, such symmetry was
lost. However, if we upgrade that symmetry on a gauge level, the SSB pro-
cess is able to provide a mass for the corresponding boson, eliminating the
Goldstone boson from the theory. It is exactly by the non-Abelian version
of this Higgs mechanism that the Z and W± electroweak bosons acquire a
non-zero mass.
In the following subsection we will see how the SSB process provides a
physical framework to the inflationary paradigm, especially in view of the
transition from a single minimum of the Higgs potential to a configuration
with two degenerate minima, that happens in correspondence to the change
of the parameter a from a positive to a negative value. We will implement
the phase transition associated to the new dynamical regime through the
coupling of the Higgs field, i.e. the relic field f, with the thermal bath of
the primordial Universe.
The Theory of Inflation 205
5.3 Presence of a Self-interacting Scalar Field
The general idea at the ground of the inflationary paradigm is the presence
in the early Universe of a real self-interacting scalar field, whose potential
has a direct dependence on the temperature of the thermal bath (account-
ing, in a phenomenological way, for the relative interaction between these
two components), i.e. we deal with a Lagrangian density of the form
£ = Igf'diWrf ~ V{4>, T) . (5.18)
At sufficiently high temperatures (this concept will be more precise in the
next subsection) , the dynamics of the scalar field is dominated by its kinetic
term, while the potential term is expected to be negligible. According to
Sec. 2.2.2, the energy density and pressure of a scalar field </> = <f>(t) living
in an expanding isotropic Universe are given by the expressions
p^ = ^<j> 2 + V(0,T), (5.19)
P = ^ 2 -U(0,T), (5.20)
where the spatial gradients of the scalar field <j> were neglected, according
to the assumption of homogeneity. In fact, the gradient terms in the energy
density (5.19) and pressure (5.20) are of the form ~ (V0) 2 /a 2 , so that even
if they are initially present (at a perturbative level) they are redshifted away
by the expansion of the Universe. When specialized to the RW geometry
in a synchronous reference frame, the dynamics of the field is described by
the Euler-Lagrange equations obtained from the Lagrangian density in Eq.
(5.18)
4> + 3H<}>+— - = 0. (5.21)
d(f>
If the potential term is neglected, this equation gives </> oc 1/a 3 and
hence Eq. (5.19) yields p^ oc 1/a 6 . This result is compatible with the
analysis of Sec. 2.2.2, where it was shown how the potential- free scalar field
is isomorphic to a perfect fluid with equation of state P = p (i.e. 7 = 0, or
w = 1). For a — ¥ 0, the energy density of the field is much larger than the
radiation contribution, which scales as a~ 4 . Near the Big Bang (a — > 0),
the energy density p^ has a strong divergent behavior which is expected
to dominate the typical potential terms of the inflationary paradigm. A
qualitative constraint on the form of the potential close enough to the
206 Primordial Cosmology
singularity can be obtained by the Friedmann Eq. (3.46). Near the Big
Bang one can neglect the spatial curvature and therefore it reduces to
ff 2 = ^> 2 oc-L. (5.22)
From this equation, i.e. from a 2 /a 2 oc 1/a 6 , it follows that a oc i 1 / 3 .
Substituting this behavior of the scalar field back into the same equation
(5.21), we arrive at the explicit expression of <f>(t) as
^ 2 = ^^) = /|ln^o, (5.23)
where <fi is an integration constant. If the dependence of the potential term
on the temperature is assumed to be weak, as argued in the next subsection
(apart from the transition phase), in order to have a kinetic energy of the
field dominant close to the singularity (t — > 0) we get the condition
M« = .. (5.24)
The scalar field is treated as classical because of its scalar character and of
the high occupation numbers of its states, in agreement with its nature of
source for the geometrodynamics of the Universe. However, the regime in
which the kinetic term of this field dominates can overlap with the Planck
era, where the quantum features of both the field and the scale factor are
relevant (see Sec. 10.5).
As the Universe expands, the scalar field evolution is damped and,
sooner or later, the potential term will become important. In fact, mul-
tiplying Eq. (5.21) by <j> and recalling the form of the energy density (5.19),
one gets the decay law (valid also in the presence of a potential term)
P4, = -3H<j> 2 , (5.25)
which, for an expanding Universe (H > 0), implies the monotonic decrease
of the energy density associated to the scalar field. After an initial regime,
where the kinetic term of the scalar field dominates the Universe dynamics,
the damping due to the expansion eventually makes the field fall into a min-
imum of the potetial. When the scalar field is in a minimum, it behaves as a
perfect fluid with p<f, = const, and an equation of state parameter w = — 1.
When the energy density of the scalar field in the minimum is dominant,
it gives rise to a de Sitter phase of exponential expansion. This situation
is eventually going to happen because the density of any other component
(e.g. radiation) decreases with the expansion (an increasing energy density
The Theory of Inflation 207
would require w < — 1).
The idea behind the original model of inflation (now called old inflation),
proposed by Guth in 1980, is that, at some point in the past history of
the Universe, it was dominated by the energy density of a scalar field in
a minimum of its potential which started expanding exponentially. What
happens before this time is not really relevant for the inflationary paradigm,
because the effect of the exponential expansion is to get rid of the initial
conditions from which inflation itself started. However, the minimum where
the field is standing in is not the global minimum of the potential (the true
vacuum) but a local minimum (a false vacuum). At high temperatures, the
true vacuum is not accessible to the system (we will explain this in more
detail later) but it becomes accessible as the Universe expands and cools
down. When this happens, at some critical temperature T c , the field can
undergo a symmetry-breaking phase transition. At T < T c , the two minima
of the potential are separated by a barrier that the field has to overcome, for
example via tunneling, in order to complete the phase transition and evolve
toward the global minimum where p<p ~ 0, with a first-order transition.
Guth realized that the de Sitter phase of expansion allows one to solve
the shortcomings of the SCM, but the first-order character of the transition
was still problematic. A modification of Guth's original model, called new
inflation, was proposed by Linde and Albrecht & Steinhardt shortly there-
after. The basic idea behind new inflation is that the phase transition is a
second-order transition, i.e. it happens smoothly. This can be realized by
requiring that the potential is such that the do Sitter expansion does uoi
happen when the field is trapped in the false vacuum, but instead when
the field is slowly-rolling towards the true vacuum over a plateau (so that
(/>< V and w~ -1).
Inflation is not necessarily associated to symmetry breaking and to
phase transitions. For example, in the model of chaotic inflation proposed
by Linde, the scalar field potential has a single minimum, but the value of
the field is not homogeneous across the Universe. At the points where it is
displaced from the minimum, inflation occurs.
Now we will describe in more detail the old and new inflationary mod-
els. At some early time, the Universe density was dominated by a scalar
field in a minimum of its potential. The phase transition associated to
inflation consists of the appearance of a second local minimum in the po-
tential V(<f>, T). Initially, above the critical temperature T c , the energy
208 Primordial Cosmology
density associated to this second minimum is greater than that of the vac-
uum state. While the Universe expands and the temperature decreases as
T ex 1/a, the height of the second minimum decreases up to be degenerate
with the original vacuum state, at T = T c . Thus, if the potential is char-
acterized by a fundamental symmetry <p — > — (p, the two (correspondingly
symmetric) degenerate vacuum states realize a SSB scenario. Indeed, the
SSB process is not strictly necessary (see the considerations below), but it
has to be inferred because of the link between the inflaton (as the scalar
field responsible for the inflation is named) and the Higgs field, emerging
from a SSB framework. When the new minimum becomes lower than the
original, the scalar field, and hence the whole Universe, remains trapped
in the higher state, the false vacuum, in contrast to the real vacuum cor-
responding to the newly formed absolute local minimum. Since between
the two minima there is a barrier, the false vacuum becomes a metastable
state and the Universe can perform a phase transition from the false to the
true vacuum state. The process underlying this transition is, in general, a
quantum or thermal tunneling across the barrier, which has to take place
independently on each causal region of the Universe. The crucial dynamical
feature of the inflationary scenario is the de Sitter phase (see Sec. 3.2.3)
driven by the constant energy density (corresponding to the gap between
the two minima) that dominates the geometrodynamics. In old inflation,
the constant energy density manifests its effects during the phase in which
the Universe is trapped in the false vacuum, i.e. before the tunneling. This
point of view requires that the barrier is high enough to get a sufficiently
long de Sitter phase; a typical example is sketched in Fig. 5.2. However, in
this scheme, the different causal regions are associated to inflating bubbles,
which randomly undergo the transition from the false to the true vacuum.
When a bubble of true vacuum is created, the energy density corre-
sponding to the gap between the vacua is stored in the walls of the bubble
and is released when two bubbles collide. This provides the energy that
reheats the Universe (that has supercooled after the inflationary phase of
expansion) and allows the usual Friedmann evolution to start. This mecha-
nism poses the problem of whether the bubbles of the new phase can appear
fast enough to cover all the Universe, because the "voids" between the bub-
bles will be exponentially expanding, being still in the old, false vacuum
phase. The answer is negative: the bubbles never percolate, so that bubbles
of true vacuum continue to be created over a false vacuum background that
expands too fast, and inflation never ends. This is called the graceful exit
problem and is based on the bubble nucleation rate, i.e. the probability of
:r 29, 2010 11:22
World Scientific Book - 9in x 6in
PrimordialCosmology |
The Theory of Inflat:
V(0)
Figure 5.2 A typical potential able to induce inflation, as thought by Guth in its original
work. The potential exhibits two different minima, separated by a barrier that does not
allow the field ij> to evolve in the real vacuum through classical mechanisms.
undergoing the phase transition, which has to be small enough in order for
inflation to last enough to solve the SCM pardoxes. Moreover, the colli-
sions between bubbles tend to create topological defects (the transition is
strongly first order) so that a nucleation rate too large would also generate
too many topological defects.
The idea underlying the new inflation paradigm is instead that the de
Sitter phase takes place after the Universe has left the false vacuum. This
difference is crucial because the barrier between the two minima must no
longer be particularly high (as required for having a sufficiently long de
Sitter regime before the transition) and, moreover, the Universe undergoes
the tunneling effect when it still has a rather smooth dynamics. These two
combined effects, a lower barrier and a regular evolution of the scale factor,
allow to overcome the graceful exit problem and to avoid the appearance of
a large number of topological defects. Yet, it remains to be explained how
a constant cosmological term is induced in the dynamics to get the desired
de Sitter phase. A solution to this problem is offered by the idea that the
barrier between the two minima has a long plateau, on which the scalar
field classically performs a slow-rolling evolution. In such situation, the
:r 29, 2010 11:22
PrimordialCosmology |
i'ruiioi-iiiid Cosmology
kinetic energy of the scalar field is negligible with respect to the constant
potential energy on the plateau (see Fig. 5.3).
Figure 5.3 A possible potential able to induce slow-rolling. The slow-rolling c<
to the phase when the field "falls" from the higher minimum on the left into
n the right of the figure.
Thus, in the new inflation framework, the cosmological constant-like
energy is relevant during the phase transition of the Universe from the
false vacuum towards the real vacuum which is slowly approaching. Of
course, when the scalar field falls into the second potential well, the de
Sitter phase stops and the inflationary process ends with a different scenario
(see Sec. 5.4.2). It is possible to argue (see the next Subsection) that,
in agreement with the SSB paradigm, the new inflation model can take
place without the existence of a quantum tunneling through the barrier. In
fact, one could consider a simple transition in which a local minimum is
transformed into a really flat local maximum standing up on two degenerate
deep minima. In this case, the whole evolution remains essentially classical
and the slow-rolling of the field coincides with the field falling into one of
the two equivalent minima. The first phase of this falling is characterized
by an almost flat potential around its value on the two degenerate vacua
and its happening is ensured by the unstable nature of the maximum and
The Theory of Inflation 211
by the expectation that the transition from a minimum to a maximum
perturbs the scalar field, even if originally at rest at the minimum.
The price to pay for this new inflation paradigm is in a more stringent
fine tuning to be imposed on the form of the scalar field potential term at
the end of the phase transition.
5.3.1 Coupling of the scalar field with the thermal bath
We will now discuss how the phase transition associated to the SSB can
be triggered by the coupling of the scalar field with the underlying thermal
bath, whose temperature decreases as the Universe expands.
The presentation below has no aim to fix a specific model, but provides
a rather general paradigm, linking the true vacuum to the false vacuum
configuration throughout the embedding of a self-interacting scalar field
into an expanding background.
The main point is that the potential V fully determines the dynamics
of the field only in the zero-temperature limit. At finite temperature T^O
one should take into account the presence of a thermal bath of particles. In
this case the dynamics of the field is determined by the "finite-temperature
effective potential" Vt{4>), that is nothing else but the free energy density
F(<t>, T). Of course F(<j>, T = 0) = V{<j>). It can be shown that the effective
potential for scalar particles of mass m in the ultrarelativistic limit (T 3> m)
is
V T (cf>) = V{4>) - ^T 4 + ^T 2 [l + og)], (5.26)
where we recall that m 2 = m 2 (<f>) = d 2 V/d<f> 2 . The first additional
term is the free energy of a gas of spin massless bosons and does not
alter the dynamics since it does not depend on cf>. The effect of the
mass-dependent terms can instead be interpreted like as introducing a
temperature-dependent mass tot = y/d 2 Vr(<i>)/d(j) 2 . If we consider the
potential Vr for a Higgs field as in Eq. (5.9) so that m 2 = -y 2 + 3A(/> 2 , the
effective potential Vt in the high-temperature limit is
W)=-^ 2 + ^§V + ^ 4 = ^ 2 + ^ 4 , (5-27)
where we have omitted terms that do not depend on (f>, and thus do not
alter the field dynamics. The temperature-dependenl effective mass tot is
r-^^-
212 Primordial Cosmology
which is real at temperatures above the critical temperature T c = 2/i/\/X,
while it is imaginary below T c .
At sufficiently high temperatures, the Higgs field has an effective mass,
associated to the fluctuation around the minimum for </> = 0. However,
when the temperature of the Universe decreases enough, the SSB configu-
ration (with two degenerate minima) appears and the minimum is replaced
by a local maximum of the potential (see Fig. 5.4). Thus, the phase tran-
Figure 5.4 The potential (5.27) is depicted above for two different temperatures. Whe
T > 2fj,/y/\ (solid line) a unique true minimum in A exists and the field <j> ne s there
when T < 2fi/y/X (dashed line) the minimum in A becomes a local maximum, and tw
minima appear. This is a direct consequence of the cooling of the Universe during it
evolution.
sition associated to the SSB process can be realized by taking into account
the effects of the finite temperature and the presence of a thermal bath of
particles. The presence of a cosmological background of interacting parti-
cles at a given temperature alters the absolute zero of the energy density
and, when the Universe is very hot, it can hide the SSB configuration,
maintaining the false vacuum always stable.
The simple scenario depicted above opens a new point of view about
the nature of the inflationary paradigm. In the present framework, the de
Sitter phase is generated as the slow-rolling of the scalar field on the plateau
around the emerging maximum (requiring an appropriate fine tuning of the
The Theory of Inflation 213
parameters). Furthermore, no real barrier exists between false and true
vacuum, and the scalar field falls into one of the two degenerate local min-
ima in an essentially classical evolution, while the Universe remains smooth
during the whole process. Such point of view appears as an intriguing and
original interpretation of the SSB phase transition. The real inflationary
picture can follow more general and not exactly symmetric features and
therefore the scheme above must be thought of as the dominant component
of a mixed framework where, for instance, a tiny barrier can arise between
two non-perfectly equivalent vacua.
We can argue that, in a realistic scenario, the role played by the back-
ground temperature is relevant only in generating the SSB configuration,
while the evolution on the plateau is well approximated by a slow-rolling
on the following temperature-independent (and hence time-independent)
profile
VpiateauO) ^ PA~ -
where p\ and uj are constant quantities describing the gap of the energy
density between the local maximum and the minima, and describing the
departure from a pure de Sitter phase, respectively. We will use this form of
the potential to describe the slow-rolling phase and the dynamical paradigm
solving the paradoxes outlined in Sec. 5.1.
5.4 Inflationary Dynamics
In this Section we analyze the specific features of the scalar field dynamics
during the slow-roll, which give rise to the de Sitter phase of expansion. In-
tuitively, the evolution of the homogeneous scalar field (j>(t) over the plateau
of its potential can be understood as the behavior of a point-particle moving
on a horizontal potential profile. The scalar field coordinate will perform
a slow-roll, characterized by a small <j> and negligible o values. During this
phase, the Universe is dominated by the potential energy of the field (the
effective cosmological constant). We will discuss the implications of the
resulting exponential expansion of the Universe (see Sec. 3.2.3), allowing to
overcome the horizon and flatness paradoxes.
214 Primordial Cosmology
5.4.1 Slow-rolling phase
The conditions to be imposed on the system to get the desired de Sitter
regime can be summarized as follows.
(i) The effective cosmological constant energy density p\ ~ p^ dom-
inates the relativistic energy density of radiation p ra( j, as well as
any other contribution, i.e.
PA > Prad • (5.30)
This implies that inflation cannot start before the temperature
drops below a value T ~ pi .
(ii) The constant term of the potential of the scalar field dominates its
correction depending on <j> over the interval (fa, fa), corresponding
to the initial and final stages of the slow-roll region, i.e.
<pe {fa, fa) :p A »^ 4 , (5.31)
which can be easily satisfied requiring uj <C 1.
(iii) The acceleration of the 0-coordinate must be negligible in com-
parison to the velocity term associated to the damping due to the
Universe expansion, i.e.
4><^3Hfa (5.32)
Under these assumptions, the Friedmann Eq. (3.46) and the one for the
scalar field (5.21) take the form
(o) = i PA (5 ' 33a)
S-<j)-ufa = 0. (5.33b)
Such system can be solved with respect to the two unknowns a(t) and fat),
giving the explicit expressions
a(t) = a exp[iT t] , H* = J ^ = const. , (5.34)
««-fiFTT ,5 - 35)
ao and t* being two integration constants. It is immediate to check that all
the requirements on the dynamics are satisfied as long as t <S^ t* , so that
we impose if <C £*.
The Theory of Inflation 215
During the de Sitter phase, all the physical lengths (for instance the
particle wavelengths) are stretched to much larger distances due to the ex-
ponential behavior of the scale factor. Similarly, the energy density of the
relativistic species populating the early Universe is drastically decreased as
well as the corresponding temperature (T ex a -1 ex exp(—H*t)). The cru-
cial point here is the constant behavior in time of the microphysical horizon
Lh = a/a ~ (H*)^ 1 = const. As a consequence of the slow-rolling dynam-
ics, the Universe profile is deeply modified. In fact, the matter that before
the inflationary expansion was contained within a single Hubble radius is
redistributed after the de Sitter regime over a much larger region contain-
ing many Hubble lengths. These regions would be causally disconnected
according to the standard Friedmann evolution of the radiation dominated
Universe where dn — £h (see Sec. 3.1.5). Furthermore, the wavelengths of
particles undergo a strong redshift, resulting in an extreme cooling of the
cosmological fluid.
We emphasize that, despite the scalar field potential does not play any
dynamical role in Eq. (5.33b) during the de Sitter phase, we address the
evolution at an essentially classical level, instead of speaking of free self-
gravitating bosons. This can be done because we are assuming that the
energy density can be regarded as a classical object, hence ensuring the
classical nature of </>(£). As already emphasized, this is exactly the same
situation of an electromagnetic field, which appears as a classical entity in
view of the high photon density (i.e. we deal with a boson state or an
electromagnetic wave with an extremely high occupation number). Never-
theless, we will see in the following that both the free field character and
the quantum nature of the infiaton field contribute to the achievement of a
soli-consistent inflationary paradigm.
5.4.2 The reheating phase
When the slow-rolling of the scalar field ends, the evolution follows the
profile of the potential term close to the true vacuum and we can reliably
infer the fall of the scalar field into the well of the SSB configuration.
This regime corresponds to a very fast decay of the scalar field into the
local minimum describing the true vacuum configuration, as suggested by
the analogy of a massive point-particle moving over the potential profile.
After such fast transition, the evolution of the Universe is governed by the
damped oscillations which take place around the true vacuum. In fact, near
enough to the bottom of the potential well, the expression for V{(f>) admits
216 Primordial Cosmology
the usual quadratic representation
V(4>)^^ 2 (<f>-a) 2 , n =J(f^\ , (5.36)
where we have considered a Taylor expansion around the value <f> = a,
corresponding to the minimum configuration, assuming that in the true
vacuum V(4> = a) = 0. This is in agreement with the observations which
indicate that the vacuum energy density is at most of the order of the
present critical density, and thus very small with respect to any reasonable
energy scale for inflation (see Sec. 5.7 for further details). Here, /j,q denotes
the effective mass acquired by the scalar field during its small oscillations.
In this approximation, the scalai field <! > riamics is that of a free massive
boson living on an expanding background, i.e. Eq. (5.21) takes the form
4> a + 3H<j> a + vl<t>a = , (5.37)
where <f> a = <j> — a but, for the sake of convenience, in what follows we
will drop the subscript a, i.e. we shift the minimum to the origin of the <fr
axis. The physics underlying this equation is that of a super-cooled Bose-
Einstein condensate, whose constituents are very massive scalar bosons,
i.e. we assume that fi 3> Hf, where Hf = H(t = if) is an estimate of the
typical value of the Hubble function during this oscillatory period. The
field evolves very rapidly on a cosmological time scale and the existence of
the condensate is due to the very low temperature of the Universe after the
de Sitter phase.
Such system of very cold spin bosons is unstable, essentially because
the particles should have decay channels into particles with a lower mass,
which will turn out as relativistic components. In other words, the huge
effective mass that these bosons acquire as an effect of their small oscilla-
tions around the true vacuum is transformed by the decay processes into
energy of ultrarelativistic species, so that the Universe undergoes a strong
reheating phase.
The decay processes can be phenonionologiealh described by an average
characteristic time t^ = const., acting as a friction term in Eq. (5.37), which
has to be restated as
ij> + 3(H + H d )4> + n 2 4> = , (5.38)
where we set H& = l/(3rd). When (t — if) <C t<j, the decay is very slow on a
cosmological time scale and the damping of the field is due to the expansion.
On the contrary, when (t — if) > t^ the decay is fast on a cosmological time
The Theory of Inflation 217
scale and is responsible for the damping of the field. In the latter case, the
evolution of the scalar field around the minimum acquires the behavior of a
damped oscillator, due to the fundamental scalar particles instability, that
is to say
4> + 3H d ^> + nl(j> = . (5.39)
This equation admits the damped oscillating solution
4>{t) = A exp (- ^H d t) sin L 2 - ^Hj\ t + <J , (5.40)
where A and (fr are constant amplitude and phase, respectively. In the case
when the expansion damping is dominant, as well as in the general case
when both damping terms are relevant, the evolution of the field should be
derived by solving Eq. (5.38) coupled to the Friedmann equation.
In general, the solution always shows a damped oscillatory behavior,
although the damping is less severe than the exponential one found during
the decay phase. For example, if the scale factor evolves with time as
a oc £ 2 / 3 (as it happens during reheating), the corresponding solution for (j>
cj>(t) = - sin(/i t + 0o) • (5-41)
However, since /j,q is much larger than both H and H d , oscillations are
very rapid with respect to the damping limosoalo and thus the oscillatory
behavior can be integrated out by averaging over many periods. Therefore
we will generally write
<j>(t)=A(t)sm(»ot + (t>o) (5.42)
where the amplitude A(t) should be thought of as a slowly- varying function
of time, i.e. varying on a time scale much larger than (jl^ and can be taken
as constant over a single oscillation period, providing
( P4> ) ~ 00 2 + l -^ ~ \ {Ai^f ~ (4> 2 ) . (5.43)
Multiplying Eq. (5.38) by <j> and taking the average, we obtain the energy
loss of the scalar field as
fa) = -3(H + H d )( P4> ) , (5.44)
which admits the solution
W>^exp{-^1, (5.45)
218 Primordial Cosmology
where p^ is an integration constant. For simplicity, in the following we will
drop the brackets around p^, even if we are always dealing with a period
average. The decay process of these massive bosons results in an increase of
the relativistic species and the Universe is reheated. The law underlying the
increase of the relativistic component is dictated by energy conservation,
amended for the expansion damping, i.e.
Prad = -4#p rad + 3H dP4> . (5.46)
The evolution of p^ and p lad is obtained by solving the two coupled Eqs.
(5.44) and (5.46), where H is given by the Friedmann Eq. (3.46) as
H 2 = | ( P4 > + Prad) , (5.47)
taking into acccount that, after the de Sitter regime, the spatial curvature
is negligible (see Sec. 5.5.1).
Although a full solution to the coupled system has to be obtained nu-
merically, an analytical solution can however be found for if < t < t,j.
In this interval, neglecting the exponential term the energy density of the
scalar field is given by Eq. (5.45)
P^ P± (5.48)
so that the scalar field behaves as non-relativistic matter. In fact, during
the coherent oscillations, it can be seen that (<j> 2 ) = (2V(</>)) and thus P ~
holds. Immediately after the de Sitter phase, the total energy density is
provided by the massive bosons condensate (the radiation component has
been redshifted away so it is safe to assume that p ra d(tf) = 0) thus the
Universe behaves as matter-dominated. One can neglect p ra d on the right-
hand side of Eq. (5.47) and get the familiar result for the evolution of the
cosmic scale factor in a matter-dominated Universe a oc £ 2 / 3 and H = 2/3t.
Moreover, Eq. (5.47) also gives
P<t> = 3H 2 /k = -^ (5.49)
and the radiation energy density p ra d obeys the equation 5
Prad = -^Prad+4^§, (5.50)
that admits the solution with initial condition /9 ra d(if) = as
^.^[i-^)**]. (5.5!)
5 We need to keep the term oc H d in Eq. (5.46), even if we neglected it in Eq. (5.44),
because p^ 3> p ra( j. In other words, the condition H 3> H d does not imply Hp v&d 3>
Hdp<p', in fact the solution shows that Hp Illd < H d p^.
The Theory of Inflat:
This expression reaches a maximum for t ~ 1.8if and for t 3> U I
1/t. The radiation density has a sharp rise around t ~ if and then starts de-
creasing; however, it decreases more slowly than the matter density because
Prad oc t _1 ex a~ 3 / 2 while p^ ex a -3 , so that at some point the radiation
will end up dominating the Universe. This condition approximately real-
izes at the time t ~ t^ and the approximations used to derive this solution
break down. In fact, when the evolution enters the region t > t<i, the scalar
field energy density begins to exponentially decay and the radiation con-
tribution drastically rises, eventually dominating the Universe dynamics.
At such state reheating ends and the usual Friedmann expansion starts.
In Fig. 5.5 we show the exact solution, obtained by the numerical integra-
tion of Eqs. (5.44) and (5.46), compared with the approximated analytic
solution discussed so far.
Figure 5.5 Behaviour of the densities of the scalar field p^ (dashed lines) and radiation
Prad (solid lines) during reheating, assuming rj = lOOtf. The thick lines represent
the numerical solutions to Eqs. (5.44) and (5.46), while the thin lines represent the
approximated analytical solutions (5.48) and (5.51). The densities are normalized to the
value of p at t { .
Let us estimate the temperature at which the Universe is reheated. If
the decay time is much smaller than the Hubble time Hf 1 at the end
220 Primordial Cosmology
of the de Sitter phase, the decay of the scalar field is instantaneous (on a
cosmological time scale) and all the vacuum energy is instantly converted in
radiation. In this limiting case, there is no matter-dominated regime after
the exponential expansion. After reheating p ra( j ~ p^iU) ~ p\ and recalling
the relation between the energy density and temperature of radiation we
can estimate
where g* is estimated at the end of the reheating phase and the superscript
G stands for "good". If g* did not vary too much, according to this estimate
the temperature of the Universe at the end of the inflation scenario is of the
same order of magnitude it had when inflation started. The FRW standard
evolution is recovered although with a strongly stretched geometry. This
situation is not the most general possible one and for a discussion of the
so-called poor reheating, as opposed to the case of a good reheating discussed
below, in the next Section. In the general case with r > Hf 1 , the vacuum
energy is not immediately converted and is partially redshifted away by the
expansion during the coherent oscillations regime, so that the reheating
temperature is smaller than the value given by Eq. (5.52).
We will now discuss the evolution of the entropy during reheating. The
behavior of the entropy per comoving volume S during the coherent oscil-
lations regime can be inferred recalling that the entropy density s is related
to the radiation energy density by the relation s = 4p rac i/3T oc pj^ d (the
power-law behavior stands as long as g, t is nearly constant). Recalling that
during reheating p ra( j oc a -3 / 2 , the entropy S = sa 3 increases as a 15 / 8 .
However, most of the entropy production during reheating happens close
to if, when the radiation energy density sharply rises from zero to a finite
value.
If the decay process of the <f> bosons violates the symmetry parti-
cles/antiparticles, at the end of the reheating phase we get a net baryon
number density rig. Assuming that the decay of a single boson of mass
po produces £ baryons, we can fix the relation ns — ^n^, n 4> denoting the
boson number density. Since the o-partieles behave as a non-relativistic
species, we have
IT 2 4
P4> ~ "0Mo ~ Pa ~ —g* (T^) . (5.53)
Combining these relations and recalling the expression of the entropy den-
sity in terms of the radiation one, we get the ratio of the net baryon density
The Theory of Inflation 221
number to the entropy density as
~ = ^Tg. (5.54)
s 4^*0
This result allows to calculate the baryon asymmetry at the end of the re-
heating phase once £ and /zo are provided by the details of the SSB process.
5.5 Solution to the Shortcomings of the Standard Cosmol-
ogy
5.5.1 Solution to the horizon and flatness paradoxes
The feature of the de Sitter phase that allows to overcome the SCM para-
doxes is the constant character of the Hubble length in comparison to the
exponential behavior of physical scales. In particular, the value a{ of the
cosmic scale factor at the end of the de Sitter phase, is related to the value
o,; at the beginning of the slow-rolling phase by the relation
a f = aiexp[iT(if - U)] = a ; exp[£] , (5.55)
where £ = ln(af /ai) is called the e-folding of the inflationary process. The
exponential growth of the physical scales offers a proper framework to solve
the horizon paradox. In fact, it can be seen that the particle horizon at the
end of inflation is da ~ exp(£)/H* , i.e. it is exponentially larger than the
Hubble length at the same time, so that the numerical coincidence between
the two quantities does not hold in the presence of an inflationary phase
of expansion. We can thus explain the strong uniformity of the CMB in
the sky simply by requiring that all the material we are looking at was
initially contained within a single microphysical horizon before inflation
started. According to this point of view, the different Hubble volumes at
recombination were not in "local" causal contact (i.e. in the sense that they
were one outside the Hubble radius of the other, as explained in Sec. 3.1.5)
but nevertheless they were inside the respective particle horizons, meaning
that they had the possibility to interact sometime in the early Universe (we
recall that the particle horizon is an integral quantity, so it "has memory"
of the past history of the Universe, while the Hubble radius does not). In
other words, the scale corresponding to a Hubble length before inflation,
containing matter in thermal equilibrium, has been stretched to a very
large scale by the de Sitter phase of exponential expansion; in particular,
this scale is much larger than the Hubble length after inflation. In order to
see large inhomogeneities of the CMB, corresponding to the real causally
222 Primordial Cosmology
disconnected regions, we would have to wait a sufficiently long time (indeed,
a huge one even on a cosmological scale) , leaving time to the Hubble volume
to incorporate many microcausal horizons of the primordial Universe at £;.
Since the mismatch between the causal horizon (as estimated assuming
a Friedmann-like expansion) at the time of recombination and the Hubble
length today is not so severe, a modest amount of inflation (a small value
of 5) is required to solve the paradox in this form. However, we have seen
that to explain the observed homogeneity of the Universe one has to require
that this homogeneity was already present at the time when the Friedmann
expansion started. Assuming that inflation started at the Planck time tp
gives the most severe constraint on the amount of inflation necessary to
explain the observed homogeneity. In other words, one has to require that
the present Hubble scale .Ho -1 ~ O(10 28 cm) was, at the beginning of the
de Sitter phase, within the corresponding Hubble length. At the Planck
time the maximum causal distance was roughly given by the Planck length
Zp ~ 10~ 33 cm. After inflation, the corresponding scale d is inflated to a
value d(tf) = e £ lp. Assuming that the reheating is maximally efficient, so
that the Universe is reheated to a temperature corresponding to the Planck
energy Tp ~ 10 19 GeV, the scale factor of the Universe between the end
of inflation and today has increased by a factor 6 Tp/T ~ 10 32 . The scale
corresponding to a Planck length today is thus d(to) ~ I0 32 e £ lp ~ O.le^cm.
Requiring at least such length to be equal to the present Hubble radius
yields the inequality
0.1e £ cm > 10 28 cm (5.56)
that expresses the request that all the matter inside the present Hubble
radius was inside a single causal horizon at the time inflation began (as-
sumed at the Planck time). Equation (5.56) is equivalent to the following
condition on the number £ of e-folds
£>lnl0 29 ~67. (5.57)
If inflation started later than the Planck time, or if the temperature
after reheating was less than the temperature at the start of inflation, the
minimum number of e-folds required to solve the horizon paradox is re-
duced. In the following, when needed we will use £ = 60 as a typical value
for the number of e-folds. In conclusion we can claim that, once the request
on the number of c-foldiugs parameter has been satisfied, the inflationary
paradigm is a convincin; 1 , cxplanal ion for the deep conceptual problem un-
derlying the horizon paradox.
e will neglec
The Theory of Inflation 223
The solution of the flatness paradox is grounded to the dynamical be-
havior of the density parameter Q, described by Eq. (3.52) during the de
Sitter evolution. As already noted, the Hubble function H remains fixed
to its constant value H* while the scale factor of the Universe inflates. As
a result, the quantity Qk = & — 1 strikingly decreases, while the relation
between the initial and final values of Q — 1 is
ft f -l = (ft;- l)exp(-2£). (5.58)
Using the fact that VLk scales as a 2 during the radiation-dominated era
and like a during the matter dominated era, and assuming again that the
Friedmann phase of expansion starts at the Planck temperature, we have
that it has increased by a factor ~ 10 60 between the end of inflation and
the present time. This yields
|n„ - 1| = 10 60 |Oi - 1| e~ 2£ < 1(T 2 (5.59)
where the last inequality is the observational constraint on |f2o — 1| and the
minimum value of e-folds required to solve the flatness paradox is thus:
£ > In 10 31 + - |0 ; - 1| ~ 71 + - |0 ; - 1| . (5.60)
As it was for the horizon paradox, the minimum number of e-folds is smaller
if inflation takes place later than the Planck time or if reheating is not
maximally efficient. It is also interesting to note that, if the initial deviation
from flatness is not too large, the amount of inflation required to solve the
flatness and horizon paradoxes is approximately the same.
The solution to the flatness paradox relies on the fact that, starting
from a generic value of fli, not fine-tuned to a (unphysical) huge value, at
the end of the de Sitter phase flf is equal unity up to a very high degree of
approximation (see Eq. (5.58)). After tf, the critical parameter regains its
standard evolution and, until today, it has increased but did not yet have
the time to deviate from unity. Thus, in the framework of the inflationary
scenario, the observation of a critical parameter very close to unity does
not imply any fine-tuning on the initial conditions at the Planck era. On
the contrary, the observation of this feature is a good indication in favor of
the inflationary paradigm.
Summarizing, the de Sitter phase of the inflationary scenario has the ef-
fect to distribute thermalized matter on a large spatial scale and to stretch
the spatial geometry up to be indisl inguishable from the real case with zero
space curvature K = 0. The behavior of the microcausal horizon and of the
particle horizon are very different during the de Sitter regime a
224 Primordial Cosmology
the slow-rolling. In fact, while the former remains essentially constant, the
latter drastically increases, so that one can say that the inflaton dynamics
solves the horizon paradox because the cosmological horizon is exponen-
tially increased with respect to the value it would have in the- SCM.
5.5.2 Solution to the entropy problem and to the unwanted
relics paradox
Both the entropy problem and the unwanted relics paradox are resolved
by the decay of the scalar field during the reheating phase. The enormous
value of the entropy per comoving volume S ~ 10 87 that is measured today
is due to the heat produced during reheating. To illustrate this property,
let us consider a volume V\ at the onset of the inflationary expansion, when
the temperature of the Universe was T = T c , while the entropy contained
in Vi was
S; ~ T?Vi. (5.61)
When the de Sitter phase ends, just before the onset of the oscillations, the
entropy inside the volume is still equal to Si, because no entropy is produced
during the de Sitter phase. This implies that the linear size of the volume
has increased by a factor exp(5) and the temperature has decreased by the
same factor. After an efficient reheating, the Universe is brought back to a
temperature T r h — T c . Assuming for simplicity that the reheating happens
instantaneously, no further expansion occurs and so the initial volume has
increased to a value Vf = exp(3£)V;. The entropy inside the volume at the
beginning of the Friedmann phase is thus
S{ ~ T 3 h U f ~ T c 3 e 3£ Ui ~ e 3£ Si , (5.62)
showing how, after reheating, the entropy inside the volume is increased
by the huge value e 3£ . For the typical value £ = 60, this is e 180 ~ 10 78 .
The entropy inside a volume corresponding to the present Hubble radius
before inflation was at most ~ 10 9 , that is a far less impressive number
with respect to 10 87 . Putting the argument the other way around, if we
assume that the region with the size of the present Hubble radius had
initially an entropy of order unity, Si, then roughly 871n(10)/3 = 66 e-folds
of expansion are required to produce the observed entropy. We find again
that the requirements on the number of e-folds imposed by the horizon,
flatness and entropy paradoxes arc remarkably close to each other.
For what concerns the unwanted relics paradox, the solution lies in
the fact that the pre-inflationary abundance nx of any particle species X
The Theory of Inflation 225
is reduced by a factor exp(35) after the de Sitter phase. Of course this
also holds for photons, so that the U X to photon ratio" nx/n^ is actually
constant during the slow-roll. However, photons are produced copiously
during reheating (in fact, practically all the photons observed today were
produced at that time) while, if the reheating temperature is low enough,
the X's are not, so that their abundance with respect to photons is greatly
diluted. In particular, the final abundance of X's will be exp(— 35) times
their initial abundance. On the contrary, the abundance of photons just
after reheating is n 7 ~ T^ h ~ T;''. i.e. it is basically flic same as it was
before inflation. We stress again that it is fundamental, for this argument
to work, that the reheating temperature is low enough so that the X's stay
decoupled from the plasma and do not share the entropy transfer from the
scalar field. If this is not true, they would rapidly thermalize and we would
be back to the uncomfortable situation nx ~ n 7 . Another way to see the
solution to this paradox is to say that during reheating the specific entropy
per X particle S/Nx increases enormously. 7 The connection between these
two formulations is readily made by noting that nxjn 1 ~ n x /s = Nx/S
and that S increases by a factor e 3£ after reheating.
5.6 General Features
We now describe some general aspects of the inflationary scenario which do
not rely on the specific form of the potential, but only on the SSB profile
with a significant plateau.
5.6.1 Slow-rolling phase
Let us restate the equations of the coupled system (5.21) that describes
the evolution of the scale factor a(t) and of the scalar field </>(£) during the
slow-rolling phase, associated to the zero-temperature potential V(<p), i.e.
(a) H " = l V ^ ] (5 - 63a)
dV
3H<j> = --—. (5.63b)
7 The paradox itself can be restated in this form: the specific entropy per X particle is
expected to be of order unity, but it turns out to be much larger than that.
226 Primordial Cosmology
From the definition of the Hubble function, the e-folding £ remains defined
(o.()4)
with faj = (6(/i.f). Making use of Eqs. (5.63), the quantity £ rewrites as
aking use of Eqs. (5.63), the qua
r4>i H 2 r<h y
£ = -3y —d^=-nj —d<J>,
where the prime denotes the derivative with respect to fa Let v.
that in the interval (fan fa), the ratio V/V (i.e. H 2 /V) is nearly constant
and that V ~ V" (fa — 4>i). The integral above (5.65) can be evaluated as
£ = K ^i=Vr (5 - 66)
where the modulus accounts for the negativity of V" resulting from the
slow decreasing of the plateau from the maximum arising from the SSB
scenario. From the relation (5.66) we can see that an efficient de Sitter
phase, associated to a high value of the e-folding parameter (say £ ~ 60),
requires the constraint
\%\«H 2 - (5.67)
This requirement characterizes the slow-rolling phase and ensures that the
time evolution of the scalar field is very slow on a cosmological time scale.
5.6.2 Reheating phase
The subsequent stage of the scalar field dynamics is associated to a rapid
fall of the scalar field into the true vacuum well. This evolution takes place
on a scale smaller than the Hubble time and inequality (5.67) has to be
reversed towards the opposite condition
\w\ >>H2 - {5M)
As we have seen in Sec. 5.4.2, the second derivative of the potential term
fixes the square of the boson mass //g and therefore the condition (5.68)
states that the period of the coherent oscillations of the scalar field is much
smaller than the Hubble time. Indeed, the analysis in Sec. 5.4.2 of the re-
heating process is rather general since the main features do not depend on
the form of the zero-temperature potential V{fa). There we noted that if
The Theory of Inflation 227
the decay time of the boson species is much smaller than the Hubble time
at the end of inflation, a good reheating of the Universe is reached, in which
the entire energy density of these particles is transferred to the radiation
component. In fact, under the assumption H<± 3> H{, the mass density
of the Bose condensate is not significantly redshiftcd by the Universe ex-
pansion, before being transformed into reheating relativistic species. Since
the fall into the potential well is very rapid, the energy density of the su-
percooled bosons is approximately the vacuum one p\. If instead we are
in the case H^ <C H, i.e. t^ 3> £h, the Universe expansion has the net
effect of redshifting the value of the energy density which is going to reheat
the causally connected regions. Such redshift can be estimated in the time
interval from U (the beginning of the coherent oscillation phase) to tf + t^
(i.e. when the conversion of the condensate into relativistic particles affects
the evolution). We have the relations
(p ) t=tf ~ PA (5.69)
where we have taken into account that t^ 3> if. This value of the boson
mass density is significantly decreased as effect of the Universe expansion
and, when converted into the radiation component, we get the reheating
temperature as
(H 2 d 30p A \
(5.71)
In this case, after inflation the Universe is characterized by a much smaller
temperature than the case treated in Sec. 5.4.2 and we deal with a poor
reheating.
In order to deal with a phase of coherent oscillations performed by the
scalar field, the condition fi 3> H& must hold even in a general case.
5.6.3 The Coleman- Weinberg model
One of the first proposals for new inflation is the potential of the SU(5)
Coleman- Weinberg scenario. In this model, before the SSB process, the
inflaton field is a 24-dinionsiona] Higgs field, responsible for the decoupling
of the SU(5) interaction of a GUT into the Standard Model of elementary
particles SU(3) G ® SU(2) L <g> U{l)y. The relic scalar field <p is described
228 Primordial Cosmology
by the (one-loop zero-temperature) potential
y(0)=/3 + 2/? U 4 [ln( U 2 )-i] , (5.72)
where u = (p/T, (S ~ 2 x 10 15 GeV being the energy scale of the SSB
process) and f3 = B£ 4 /2, with B = 25q;gut/16 — 1CP 3 , «gut denoting
the coupling constant associated to the GUT interaction.
The dependence on the temperature of the full potential is characterized
by a small barrier, having a height of about 0(T 4 ) and a critical temper-
ature T c running between 10 14 GeV and 10 15 GeV. Nonetheless, the false
vacuum in 4> = remains metastable up to a temperature of about 10 9 GeV
and the phase transition is due to the one-loop radiative corrections.
Near <fi = 0, i.e. 4> <C E, the potential (5.72) admits a plateau in the
quartic form (5.29), with p\ = /3 and u is provided by the logarithmic
term, when calculated for a characteristic value (f>* of the plateau such that
u = 4Bln(0*/E) 2 -0.1.
The Coleman- Weinberg model is no longer a reliable candidate to pro-
vide the basis for inflation, because it is based on a SU(5) symmetry (almost
abandoned in GUT because it violates the limits on the proton lifetime)
and for its internal inconsistencies (unless the parameters of the model -
essentially u> - are fine-tuned) . However, this scheme remains an important
example, elucidating how the inflation proposal is strengthened from its
crossmatch with the fundamental particle physics background.
5.6.4 Genesis of the seeds for structure formation
Until now, our analysis was based on the idea that the scalar field is a func-
tion of time only, as required by the Universe homogeneity. This statement
is valid on a classical level, because the energy density of the self-interacting
bosons is, as a whole, the source for the spacetime geometry. Such situa-
tion does not hold when quantum fluctuations of the field are taken into
account. We have implicitly made reference to this inhomogeneous features
of the inflaton during the discussion of the quantum tunneling across the
barrier between the false and the true vacua (see Sec. 5.2). In fact, the mi-
crophysical causal structure requires independent evolution over causally
disconnected patches. The same concept is also applicable to the quantum
fluctuations of the inflaton during the de Sitter phase associated to the
slow-rolling regime. We will concentrate our attention on this stage of the
inflationary paradigm because the inhomogeneous scales generated during
The Theory of Inflation 229
the exponential growth of the scale factor are stretched to super-horizon
size, and can re-enter the Hubble scale at later times, consistently with the
requested spectrum of structure formation.
In the inflationary paradigm, the initial seeds for structure formation are
the quantum fluctuations of the scalar field. During the phase of inflation-
ary expansion, these fluctuations are stretched well above the microcausal
horizon and become classical curvature perturbations. When the Fried-
mann phase begins, the microcausal horizon starts growing faster than the
perturbation scale so that, after a certain time, the perturbation will re-
enter the horizon and will start to evolve as a density fluctuation. The
idea we are tracing is apparently surprising: galaxies originally arise from
quantum disturbances thai, evolved until the present time via the Universe
expansion and the gravitational instabilities] the possibility for a different
origin of the inhomogeneous fluctuations is forbidden because of the strong
cancellation of the initial conditions that inflation produces during the de
Sitter phase. For example, it could be thought that the origin of the pri-
mordial flucuations is in the radiation component present before inflation.
However, since the radiation energy density behaves as a -4 , it is strongly
depressed by the exponential expansion of the slow-rolling evolution. The
initial value of the radiation density would be suppressed by the huge factor
exp (-45) before the end of the de Sitter phase. The radiation component
of the Universe is suppressed so strongly that ils spectrum could never be
at the ground of the structure formation process. For a detailed discussion
of the inhomogeneity behavior during an exponential expansion of the Uni-
verse, see the study on the quasi-isotropic inflation of Sec. 6.5. It is exactly
this suppression of the energy density of all components of the cosmolog-
ical fluid, apart from the scalar field, that leads to search in the quantum
fluctuations of the scalar field itself the only reliable mechanism for the
generation of a spectrum of primordial in homogeneities.
In the following we will make use of the concepts introduced in Sec. 4.2.3
to characterize the fluctuations of a field. First of all, we split the field cf> in
the sum of an unperturbed, homogeneous part with a small fluctuation as
</>(x, t) = 4>{t) + 8<f>(x, t). (5.73)
By a Fourier transform of the spatial dependence in 5(f>, we can deal with
the perturbation in fc-space 5<j>k, where fc is the co- moving wave- number,
related to the physical wave-number by fc p i iyB = k/a(t). During the de Sitter
phase the slow-roll condition ensures that V" <C H 2 , so that the mass of
the field m 2 = V" can be neglected, i.e. one deals with a free massless and
230 Primordial Cosmology
minimally coupled boson field in a de Sitter space. The variance of the field
perturbations in fc-space is given by
W fe | 2 } = |S, ( 5 - 74 )
so that the power spectrum A^ is
A5=(|)\ (5.75)
In the last two equations, H is evaluated at the time when the mode k exits
the horizon. The perturbation of a mass 1 1 calai I eld obeys the evolution
equation in A:-space
S'<j> k + 3H5J> k + k 2 S(f> k = (5.76)
which shows how, when a mode is well outside the horizon (fc — > 0), the am-
plitude of the corresponding perturbation remains constant and the mode
re-enters the horizon with roughly the same amplitude it had when it left.
Since it can be shown that in the region of (co-moving) wavelengths
interesting for the structure formation (i.e. from 0.1 Mpc to 100 Mpc 8 ) the
Hubble constant during inflation remains nearly constant and we finally
obtain the suggestive feature of a scale independent fluctuation spectrum.
All the perturbations leave the physical horizon of the slow-rolling regime
with the same amplitude, almost independently of their size. When they
become super-horizon sized, the microphysics cannot affect any longer their
evolution and a process of freezing takes place with the net result of reducing
the quantum spectrum to a classical profile of curvature perturbations.
The perturbation of the energy density Sp k associated to the quantum
fluctuations in the field 5(f>k is
5 Pk = -^-5(j> k = V'5<t>k = -3H<p k 5(j> k , (5.77)
since (luring the de Sitter phase p^ ~ V and 3H(f> k + V' = 0.
After the end of inflation the Universe regains its standard Friedmann
like evolution and the microphysical horizon increases faster than the phys-
ical scales. The perturbations generated according to the mechanism de-
scribed above start to re-enter the Hubble horizon. To investigate the struc-
ture formation process, as predicted by the inflationary spectrum, we need
to calculate the form of such spectrum when the perturbations become sub-
horizon-sized again. The main difficulty in this task is the gauge-dependent
8 We recall that the value of the scale factor today is fixed, by convention, equal to 1,
so that the comoving scale- coincides wi1 h the present physical ones.
The Theory of Inflation 231
nature of the spectrum evolution; in particular, the density constrast 5p/p
is not gauge invariant. We can overcome this problem by using the gauge
invariant quantity ( introduced by Bardeen, which has the key property to
remain constant during the super-horizon evolution of the perturbations.
The point to be addressed is the link between the energy density fluctu-
ations and the value of ( in correspondence of the two horizon crossings.
When the mode is not too much outside the horizon, A < if -1 , so that the
perturbations of the metric can be neglected and ( is given by
An
(5.78)
Recalling that during the de Sitter regime the term in the denominator
of Eq. (5.78) is fixed by the scalar field energy density as p + P ~ <j> 2 , at
the first crossing (FC) we have
s*° = (e+p ( Y c = £ < *c (579)
V P J PA
where 5k = 5pk/p. On the other hand, at the second crossing (SC) of the
horizon, i.e. at the re-entrance into the Hubble radius, when the Universe
is either radiation- or matter-dominated, we have
4 sc = (^c) SC = (i + -x s<
Since C is time independent, C = C and thus Eqs. (5.79) and (5.80)
together yield
5 S k C = (1 + W)PA AFC „ PA^FC (5gl)
The density perturbation at horizon exit can be obtained combining Eqs.
(5.74) and (5.77) to get
FC H 2 <j>
h "-WV' (5 - 82)
where we omitted a numerical factor of order unity. The perturbation at
horizon re-entry is thus
and the power spectrum of density perturbations /S.\ = fc 3 |<5fc| 2 /27r 2 is fi-
nally
232 Primordial Cosmology
The perturbation spectrum predicted by inflation as an initial condition for
structure formation has a flat profile because it does not depend on k. This
is called the Harrison- Zeld'ovich (HZ) spectrum.
The fact that the Hubble parameter is actually varying (albeit slowly)
during the slow roll introduces a small scale dependence in the perturbation
spectrum so that the spectrum has a power-law form as
A 2 k = Ak"- 1 , (5.85)
where A is the amplitude, n is the spectral index, and we follow common
usage in making the HZ spectrum correspond to n = 1 . The spectral index
is given by
n-l = -6e + 2r/, (5.86)
where the slow-roll parameters < and // are defined as
1 v"
V=-~- (5.88)
As the name suggests, these parameters are related to the slow-roll since
the conditions for its occurrence are e <C 1 and \rj\ <C 1. The existence
of the slow-roll phase automatically implies that n — 1 -C 1, i.e. that the
spectrum is very close to the HZ one. A measurement of the spectral index
is a powerful tool to constraint the possible models of inflation, since its
value is related to the characteristics of the potential through the slow-roll
parameters. The value of the spectral index has actually been measured to
percent accuracy through the observations of the CMB anisotropy spectrum
made by the WMAP satellite (see Sec. 4.4) and it has been found to be less
than unity.
While the shape of the spectrum is quite a precise prediction of the the-
ory of inflation, the same cannot be said for its amplitude which has to be
fixed by requiring a satisfactory scheme of structure formation, with partic-
ular reference to the origin of galaxies. Perturbations at the galactic scale
fcgai ~ IMpc" 1 enter the horizon at z ~ 10 5 , in the radiation-dominated
era. As discussed in Sec. 3.4.3, the perturbations can grow only logarith-
mically in that regime, so for the moment we neglect the growth of the per-
turbation between horizon entry and matter-radiation equality (z oq ~ 10 4 ).
After equality, the perturbation grows as a (see again Sec. 3.4.3). A rea-
sonable condition is a density contrast of order unity, corresponding to the
The Theory of Inflation 233
beginning of the non-linear evolution, not latest that z ~ 10, which yields
l«?W
IC-H
-1(T 3 . (5.89)
Taking into account the logarithmic growth between the horizon re-entry
and the matter-radiation equality would alter the above estimate by a factor
m(W*sc) = 21n(a cq /asc)-5.
Let us conclude this Section by stressing how the inflationary paradigm
is able to provide a natural mechanism for the generation of density in-
homogeneities. The predicted spectrum has the striking feature to have
a nearly scale-independent form at the time when the perturbations re-
enter the horizon. This simple and convincing picture for the genesis of a
clumpy Universe must be regarded as one of the most appealing issues of
the inflation scenario, in addition to the solution of the SCM paradoxes.
5.7 Possible Explanations for the Present Acceleration of
the Uni
In this Section we will discuss some possible explanations for the present
observed acceleration of the Universe, introduced in Sec. 4.3, in order to
provide an overall picture of the contemporary ongoing lines of research.
In particular, we will first introduce the possibility that the acceleration
is caused by dark energy, namely an exotic component of the cosmological
fluid with an equation of state parameter w,ie < —1/3. We will focus on
quintessence models, in which the dark energy is a scalar field. Then, we
will describe modifications to GR, the so-called f{R) theories. A third
possibility, not treated here, is that the acceleration is just an artefact due
to the inhomogeneous structure of the Universe at small scales.
The ideas underlying the quintessence and the f(R) theory are some-
what related to inflation. In quintessence models, the present acceleration
is due to the presence of a scalar field slowly rolling on its potential, so that
the ideal connection with inflation is evident.
In the case of f(R) theories, a phase of accelerated expansion can also
be caused by modifications to the action of GR. In fact, the possibility of an
early inflationary phase was firstly discussed by Starobinsky in 1980. On
the other hand, since both the early, inflationary Universe and the present
one are accelerating, it is natural to investigate whether the two phenomena
could be related. Although an explanation of both inflation and the present
234 Primordial Cosmology
acceleration has been proposed, the results to date are not yet satisfying.
5.7.1 Dark energy
In the mentioned models, the acceleration is due to the presence of an exotic
component, dubbed "dark energy", with negative pressure (in particular,
Wdc < —1/3). This leads to a repulsive gravity (as it can be naively seen
from the fact that the quantity p + 3P, which is the source of the Poisson
equation for the gravitational field in the weak field limit, becomes negative)
and thus to acceleration. This kind of models invoke a modification of the
right-hand side of the Einstein equations, i.e. of the energy-momentum
tensor. The simplest candidate for a negative-pressure component is the
energy density associated to the quantum vacuum. From a mathematical
point of view, the vacuum energy is equivalent to a cosmological constant
since both give rise to a contribution to the energy-momentum tensor with
equation of state P = —p. However, the computation of the quantum zero-
point energy leads to divergent or anyway very large values. In general, the
vacuum energy density is of the order of k^ &x , where fc ma x is the ultraviolet
cutoff imposed to avoid divergences. Taking the cutoff at the Planck scale,
we get p vac ~ mp ~ 10 112 eV 4 . On the other hand, the present dark energy
density is p dc ~ p c ~ 10 _5 /i 2 GeV/cm 3 ~ 10 _11 eV 4 , so that this simple
estimate is wrong by ~ 120 orders of magnitude. The situation does not
get much better by considering a cutoff at the electroweak scale. In fact,
the cosmological energy density corresponds to an energy of the order of
10 -3 eV, a scale where it is unreliable to invoke any new physics. The very
large value of the vacuum energy compared to the observed density of the
Universe goes under the name of "cosmological constant problem" .
A subset of dark energy models are the so-called dynamical dark energy,
or "quintessence" models, where a scalar field is responsible for the present
acceleration. Differently from the case of vacuum energy, the scalar field is
dynamical and then the equation of state parameter w is expected to vary
with time, i.e. w = w(z). The scalar field has to be homogeneous (at least
to zeroth order) in order to satisfy the requirements of the cosmological
principle. The evolution of the field in a cosmological setting is given by
Eq. (5.21), while its density and pressure are given by (see Sec. 2.2.2)
p^ = y + vtf) ; p * = y~ v{4>) ' (5 - 90)
where cf> 2 /2 is the kinetic energy of the field and V(tfi) is its potential. The
The Theory of Inflation 235
equation of state parameter w<f, for the scalar field is thus given by
P 6 6 2 /2 - V{6)
wa ) = — = ^ 1 — . (5.91)
P4> ^ 2 /2 + V(4>)
When the energy of the field is dominated by its potential energy (i.e. the
field is slowly varying, (f> 2 /V < 1). then w<p ~ — 1, the energy density of the
field is nearly constant (we recall that p<p oc a"^ 1+w *>) and the field mimics
a cosmological constant. This is exactly what happens during the slow-roll
phase of inflation. On the other hand, when the field is rapidly varying,
4> 2 /V 3> 1 and «70 ~ +1, so that p^ oc a -6 . The regime during which the
energy of a field is dominated by its kinetic energy is called kination. In
general, the equation of state parameter can take any value between — 1 and
+1. When w^ < —1/3, this can possibly give rise to a phase of accelerated
expansion (if this actually happens depends on the energy density of the
other components of the cosmological fluid) . By assuming a non-standard
form of the kinetic energy term, it is also possible to obtain the so-called
phantom scenarios when w<p < — i. uicaniu;', that the scalar field density
increases with time. The phantom models, however, are typically unstable
with respect to perturbations.
The evolution of the field and of its equation of state parameter strongly
depend on the form of the potential. In general, two classes of model can be
distinguished, depending on whether the velocity of the field increases with
time or not, i.e. <fi «c 0. When <f> > 0, the field rolls faster with time so that
it starts as a cosmological constant-like component and then evolves away
from w < — 1. These models are called thawing. On the contrary, in models
with <f> < 0, the field rolls more slowly with time, so that the cosmological
constant-like behavior is recovered at late times. These models are called
freezing. A peculiar feature of some freezing models is that they have a
tracking behavior, meaning that they track the dominant component of
the energy density of the Universe at early times, and then they dominate
at late times. This can possibly solve the "coincidence problem", namely
the puzzling fact that although the cosmological densities of matter and
vacuum energy vary very differently with time, nevertheless we happen to
live in a time in which they are just a factor of two apart.
Quintessence models have to face some issues. First of all, they still
suffer from the cosmological constant problem. The minimum V of the
potential has to be very small or exactly zero in order to avoid it. Secondly,
in order to be responsible for the expansion, the effective mass of the field
?7i0 = \/V"(4>) has to be very small, of order H^ 1 = !CT 33 eV (i.e. its
236 Primordial Cosmology
Compton wavelength has to be of order of the Hubble radius), while its
vacuum expectation value (<>) has to be of order of the Planck mass mp.
This also poses a hierarchy problem, i.e. it should be explained why m^ is
60 orders of magnitude smaller than (<j)} .
5.7.2 Modified gravity theory
The geometrical and tensor structure of GR determines the kinematics of
the gravitational field in a very consistent formulation, but its dynamics
admits a wide class of different proposals. In fact, the Einstein-Hilbert
action is only the most simple proposal in agreement with the experimental
data and its most striking feature is the absence in the field equations of
derivatives of order higher than the second. A more general formulation
of the gravitational field dynamics is the replacement of the Ricci scalar in
the Einstein-Hilbert action by a generic function f(R), which reduces to
f(R) ~ R in the weak field limit, when the spacetime curvature is small
enough. In what follows, we will consider a gravitational action of the form
St = -£- I V=9f(R)d 4 x, (5.92)
ZK J M
where the function / corresponds to oo 1 degrees of freedom. Such open
choice for the geometrodynamics allows to interpret the Universe acceler-
ation (which is intrinsically a dynamical "anomaly" of the Universe evolu-
tion) by means of the additional term entering the new Einstein equations
(and hence the modified Friedmann ones too). In fact, the variation of
the action (5.92) with respect to the contravariant metric g i: > implies the
following set of field equations (having order of differentiation greater than
two)
f'Rii - \f(R)9H ~ ViV,-/' + 9ijVlV l f = KT tj , (5.93)
where /' = df/dR. These modified Einstein equations can be recast in the
form
Rii ~ \R#H = « {Tij + 7^ urv ) , (5-94)
= (±F(R) - VjV'/') gij + ViVjf , (5.95)
with the identiiicatio
F(R) = f(R) — R denoting the deviation from the Einstein-Hilbert La-
grangian density. The generalized theory can be rewritten as an Einstein-
like theory with a curvature term as a source. The interpretation of the (hu-
verse acceleration in this modified gravity approaches is that the additional
The Theory of Inflation 237
terms in the field equations do not affect significantly the early Universe
thermal history but, in the late evolution, they are able to mimic a per-
fect fluid contribution, having a dark energy equation of state P < —p/3.
Recalling that for a RW geometry the scalar of curvature reads as
R=-6 \-+ (-) +4 . ( 5 - 96 )
the (synchronous) Friedmann equation (3.46) is deeply modified and takes
the form
3^Rf" - 3^/' - i/ = K p(a) . (5.97)
In Eq. (5.97) p(a) denotes the energy density as a function of the cosmic
scale factor in the standard form p oc 1/a 37 , as it is guaranteed by the
continuity equation (3.35). This equation, like in the Friedmann case, is
the one determining the full system dynamics. Indeed many different ap-
proaches succeeded in deriving an acceleration of the late Universe, offering
a rather consistent cosmological picture. Any significant modification of
the dynamics on cosmological scales must also allow to reconcile the Solar
system data with the deviations due to the non-Einsteinian terms. It is
worth reminding how a modified f(R) theory of gravity must be consistent
with the observations on all the length scales of physical interest. Such re-
quest and the problems of possible degeneracy of different theories provide
the most challenging tasks of these revised dynamical approaches.
When dealing with the generalized gravitational action (5.92), we re-
quire f(R = 0) = to avoid a huge cosmological constant, otherwise a
fine-tuning of the model parameters would be needed. Furthermore, in or-
der to recover GR for low curvature values, we take the representation of
f(R) in the form
f(R) = R + F(R) , limF(R) = 0, (5.98)
where we can address both analytical and non-analytical expressions for
F(R).
The two prescriptions above are not always addressed in the literature
and they must be intended as simplicity requests for the modified action.
Scalar-tensor theory The scalar-tensor representation is based on
translating the scalar degree of freedom related to the function f(R) into
a dynamical scalar field coupled to the Einstein-Hilbert dynamics. This
result is achieved via a suitable conformal transformation on the original
space-time metric.
Primordial Cosmology
By means of the two auxiliary fields (i.e. Lagrange multipliers) A and
B, the action (5.92) can be rewritten as
SsT = ~hJ d * x ^ [B(A -R) + f(A)} .
(5.99)
The variation with respect to B gives R = A, while the variation with
respect to A provides B = —df/dA = —f'(A), corresponding to the so-
called Jordan frame, so that the action (5.99) takes the form
S ST = -^jd 4 xV^lf'(A)(R-A) + f(A)} . (5.100)
Let us now redefine the metric tensor as gij = e^^^gij and obtain
\/=$ = eV¥* v / = $ (5.101)
R = e"^^ (r - ng^d^djA , (5.102)
so that adopting the identification </> = —^3/2k In f'(A), the gravitational
action (5.100) can be restated as the scalar-tensor one
Sst = -^Jd 4 x^g~R + fdfixyfljl^gVdiWjt- V ((/>)] , (5.103)
where the potential term is defined via the relation 9
V{cj>)= f -^, (5.104)
once the field A is expressed in terms of <j) as
i = /'- 1 ( e V^*). (5.105)
This restated scheme is called the Einstein frame because the standard
geometrodynamics is recovered, though the theory is no longer in vacuum
and a real self-interacting scalar field appears.
The scalar-tensor scenario associated to a modified f(R) theory of grav-
ity offers a natural context to solve the acceleration puzzle in the same spirit
traced in the previous subsection, i.e. the scalar field dynamics is respon-
sible for the dark energy density and is minimally coupled to gravity.
We conclude by stressing that the equivalence between the Jordan (origi-
nal) frame and the Einstein one has not been definitely established although
the former approach is commonly preferred despite its high complexity.
9 Our definition of the potential differs by a sign from the definition usually found in the
literature on the subject. This is because we use the (H ) signature of the metric
instead of the ( h +- 1-) signature commonly used in the literature on the scalar-tensor
theory. The two choices can be related by letting f(R) — > —f(R).
The Theory of Inflation 239
5.8 Guidelines to the Literature
The theory of inflation is treated in many books, like those by Kolb &
Turner [290], Linde [323,324], Lyth & Liddle [327] and Mukhanov [357].
A discussion of the standard model paradoxes, treated in Sec. 5.1, can be
found in each of them.
The spontaneous symmetry breaking and the Higgs mechanism, dis-
cussed in Sec. 5.2, are described in most books on quantum field theory,
like those by Mandle & Shaw [335] and Weinberg [463].
The idea behind the inflationary paradigm, as described in Sec. 5.2, was
first proposed by Guth in 1980 [212], in his model now called old inflation.
The idea of an early phase of inflationary expansion had been previously
discussed by Starobinsky in the context of modifications to the action of
GR [426], albeit with no reference to the possibility of solving the short-
comings of the SCM. New inflation and the slow-roll were introduced by
Linde [320] and Albrecht & Steinhardt [2]. Other inflationary models that
have been proposed include the chaotic inflation model by Linde [321], the
double inflation model by Turner ^ Silk [419] and the power-law inflation
by Lucchin & Matarrese [326]. The number of existing inflationary models
is indeed extremely large: we refer the interested reader to the review [328]
and to the book by Liddle & Lyth [327], Chap. 8.
The theory of phase transitions and tunneling, the effective potential
and the coupling to the thermal bath discussed in Sec. 5.3.1 are treated in
more detail in [290,323,324,357].
The theory of reheating, discussed in Sees. 5.4.2 and 5.6.2 is treated
in the books by Linde [323,324] and Mukhanov [357]. In particular, the
latter covers some more recent developments that we have left aside for
pedagogical purposes.
The generation of primordial fluctuations, briefly discussed in Sec. 5.6.4,
is the focus of the above-mentioned book [327].
The Coleman- \\ ■ i.ul •> i p. pm < nl i il < u < n d in Sec. 5.6.3 was introduced
in [124].
The observational evidences and the possible explanations for the
present acceleration of the Universe are discussed in [178]. Quintessence
models, discussed in Sec. 5.7.1 were introduced in [108]. A comprehen-
sive review on dark energy can be found in [379]. For what c
f(R) theories discussed in Sec. 5.7.2 we refer the reader to the i
240 Primordial Cosmology
in [109,109,366,425]. In particular, an interesting scenario able to interpret
the inflation paradigm and the Universe acceleration into a unified picture
by modified gravity, is provided by [365].
Chapter 6
Inhomogeneous Quasi-isotropic
Cosmologies
In this Chapter, we analyze the so-called quasi-isotropic solution, firstly
derived by Lifshitz and Khalatnikov in 1963 (KL). This model is a general-
ization of the FRW cosmology in which a certain degree of inhomogeneity,
and hence of anisotropy, is introduced in dynamics of the Universe. The in-
homogeneity of the space slices is reflected to the presence of free functions
of the coordinates, which are available for the Cauchy problem. In par-
ticular, the original KL solution, corresponding to the radiation-dominated
Universe where such a quasi-isotropic regime corresponds to a Taylor expan-
sion of the metric tensor in time, deals with three physically independent
spatial functions. Their presence ensures that the class of solutions arises
from three independent spatial degrees of freedom able to freely fulfill initial
conditions on a non-singular spatial hypersurface.
The Chapter is devoted to describe the behavior of the quasi-isotropic
solution in different cosmological contexts, characterized by a suitable na-
ture of the matter source. In particular, we analyze the inflationary be-
havior of a quasi-isotropic Universe in the cases of a dominant massless
scalar field (the primordial inflaton) or of a cosmological constant (the
slow- rolling phase). The nature of the quasi-isotropic solution when both a
massless scalar field and an electromagnetic field are present, is compared
with the corresponding context when ultrarelativistic matter replaces the
vector component. The quasi-isotropic model is treated in the last section
of this Chapter in a viscous framework, which generalizes the KL regime
to the presence of out-of-equilibrium features, described by a bulk viscosity
coefficient.
Concerning the inflationary scenario, the presence of the scalar field
allows to deal with an arbitrary spatial distribution of the ultrarelativis-
tic energy density, but the corresponding spectrum of inhomogeneities is
242 Primordial Cosmology
washed out later by the de Sitter phase. This issue clarifies the necessity
of a quantum nature for the perturbations which originated the large-scale
structure observed in the present Universe.
6.1 Quasi-Isotropic Solution
Many interesting results can be derived from the study of the general prop-
erties of the cosmological solution of the Einstein field equations, in par-
ticular regarding the chaoticity characterization (as discussed in details in
Chaps. 7, 8, 9), and the existence of a singularity in a general framework.
In order to describe the present Universe - which appears homogeneous
and isotropic from experimental observations at large scales, see Chap. 4, it
is interesting to investigate its gravitational stability. Hence, the evolution
backwards in time of small density perturbations is of particular relevance
when considering cosmologii al models more general than the homogeneous
and isotropic one, since the assumptions of uniformity and isotropy are
justified only at an approximate level.
In 1963, Lifshitz and Khalatnikov first proposed the so-called quasi-
isotropic solution discussed in this Chapter. This model is based on the idea
that the space contracts maintaining linear distance changes with the same
time dependence order by order (i.e. considering a Taylor-like expansion of
the three-metric).
The Friedman n ilulii n u< u l In L! I u < i.-< pom n to the radia-
tion dominated era, is a particular case of this class of solul ions in which the
space contracts in a quasi-isotropic way, providing a solution which exists
only in a space filled with matter.
6.2 The Presence of Ultrarelativistic Matter
When considering the isotropic solution in a generic reference frame,
isotropy and homogeneity imply the vanishing of the off-diagonal metric
components go a . The decrease of such functions is related to the equation
of state when regarded in a co-moving frame. In fact, in the inhomoge-
neous case we may deal with a synchronous and co-moving frame only in
the presence of a dust fluid (i.e. P = 0).
For the ultrarelativistic matter the equation of state reads as P = p/3
and the metric h a p for the isotropic case is linear in t. Hence, when search-
ing for a quasi-isotropic extension of the Robertson- Walker geometry the
Inhomogeneous Quasi-isotropic Cosmologies 243
metric should be expandable in integer powers of t, asymptotically as t — > 0,
following the Taylor-like expansion
/^(M)=f>sV)(£) ,
<^)^-^\ *o", (6.2)
and to is an arbitrary time (t <C to), while the existence of the singularity
implies a a l = 0. In what follows, we will deal only with the first two terms
of this expansion, i.e.
After a suitable rescaling we get
h a0 = ta a p + t 2 b al3 + ... , (6.4)
whose inverse matrix to lowest order reads as
h aP = t~ 1 a aP -b aP + ... . (6.5)
The tensor a a ^ is the inverse of a a p and is used for the operations of rising
and lowering indices as well as for the spatial covariant differentiation, i.e.
a Q ^a / g 7 = <5" and fro = a" 7 & 7 g is ensured by the scheme of approximation.
We recall how the Einstein equations in the synchronous system (see
Eqs. (2.98)) assume the form
K = ~\dtK-\ kik<$ = K (V ° - It) (6.6a)
R° a = ^(V kP-V a kl) = K T° (6.6b)
Ri = y= dt (/hk£) -P ( i = n (ri - \tA , (6.6c)
where the tensor k a p and its contractions read as
k afj = d t h afj = a a(3 + 2tb a(j (6.7a)
l£ = hf ,s k a s = t- 1 6g + bP (6.7b)
k = d t lnh = 3t~ 1 + b, (6.7c)
where b = 6" and from which we get
h = det(h afi ) ~ t 3 (l + tb) det(a Q/3 ) . (6.8)
244 Primordial Cosmology
Let us note that the notation in terms of k a p is equivalent to the one
introduced in Eq. (2.66), since we have k a p = — 2K a p.
We complete this scheme by observing how this framework is covariant
with respect to a coordinate transformation of the form
t' = t + f(x->), x a ' = x a '(x->), (6.9)
being / a generic space dependent function. Such property will hold for all
paradigms treated throughout the current chapter.
We recall that the energy-momentum tensor for an ultrarelativistic per-
fect fluid takes the form
T lk = 1 (Au iUk -g ik ),
(6.10)
which provides the following relations
T ° = l -p(Aul - 1)
(6.11a)
^o 4
(6.11b)
r^ = -|(4« a «^ + ^)
(6.11c)
T = 0,
(6. lid)
where u@ = h at3 u a . Consequently, the Einstein
equations
reduce to the
partial differential system
\dtK + \kik a p = -«| (Aul - 1)
(6.12a)
5(V / ,*g-V a tf) = |«pu a «°
(6.12b)
2y/h V > 3 v
u fi + €) ,
(6.12c)
Ricci tensor obtained by the metric h a p and m denotes the matter four-
velocity vector field.
Computing the left-hand side of (6.6a), (6.6b) up to zeroth- 0(l/t 2 )
and first-order 0(l/t), we can rewrite them as
" 4^ + Yt = K 3 (_4U ° + 1} ' (6 ' 13a)
l -{V a b-V p bi) = -^npu a u\ (6.13b)
respectively. Let us consider the identity 1 = mu l ~ Uq — t~ 1 a a/3 u a u ! 3.
If we assume that its last term is negligible, so that u ~ 1, a consistent
Inhomogeneous Quasi-isotropic Cosmologies 245
solution can be found for the system (6.13). In fact we get p ~ t~ 2 and
u a ~ t 2 . From Eq. (6.13a), one can find the first two terms of the energy
density expansion, and, from Eq. (6.131)). I be leading term of the velocity
and they read as
Kp= i~l (fU4a)
t 2
u a = — (V a b - V p b p a ) . (6.14b)
As a consequence, the density contrast 8 can be expressed as the ratio
between the first and zeroth-order energy density terms, i.e.
8=-ht. (6.15)
This behavior implies that, as expected in the standard cosmological model,
the zeroth-order term of the energy density diverges more rapidly than the
perturbations and the singularity is naturally approached with a vanishing
density contrast.
Besides the solutions for p and u a , one has to consider the pure spatial
components of the gravitational equation (6.6c). To leading order, the Ricci
tensor can be written as 3 R@ = A^/t, where A^ a is constructed in terms
of the constant three-tensor a a p. The terms of order t~ 2 in Eq. (6.6c)
identically cancel out, while those proportional to t^ 1 give
K + \bi + ^b8i = . (6.16)
Taking the trace of equation (6.16), the relation between the six arbitrary
functions a a p and the coefficients b a p from the next-to-leading term of the
expansion can be determined as
bi = -\Ai + ^A8i. (6.17)
It is worth reminding that, in the asymptotic limit t — > 0, the matter
distribution as in Eq. (6.14a) becomes dominantly homogeneous because p
approaches a value independent of b.
From the tridimensional Bianchi identity V p,A^ a = ^\7 a A, the relation
\7pbP = | d a b can be determined; this gives the final expression for the
three-velocity distribution as
t 2
u a = —d a b. (6.18)
This result implies that, in this approximation, the three-velocity is a gra-
dient field of a scalar function. As a consequence, the curl of the velocity
vanishes and no rotations take place in the fluid.
246 Primordial Cosmology
Finally, it must be observed that the metric (6.3) allows an arbitrary
spatial coordinate transformation while the above solution contains only
6 — 3 = 3 arbitrary space functions arising from a a/ 3. The particular choice
of these functions, corresponding to the space of constant curvature (A^ =
const, x 6& ), can reproduce the pure isotropic and homogeneous model.
6.3 The Role of a Massless Scalar Field
In this Section we discuss the quasi-isotropic Universe dynamics in the
presence of ultrarelativistic matter and a real solf-uiloractuig scalar Held
asymptotically close to the cosujologieal singularity, while in Sec. 6.5 we
will see the opposite limit in the framework of an inflationary scenario, i.e.
far from the singularity when a cosmological constant term arises.
In particular, the presence of the scalar field kinetic term allows the
existence of a quasi-isotropic solution characterized by an arbitrary spatial
dependence of the energy density associated to the ultrarelativistic matter.
To leading order, there is no direct relation between the isotropy of the
Universe and the homogeneity of the ultrarelativistic matter distributed in
it. Indeed the matter energy density enters the equations to first order only.
In the presence of a perfect ultrarelativistic fluid and of a self-interacting
scalar field <f>(t,x) described by a potential term V{<j>), the Einstein equa-
tions reduce to the following partial differential system
\dtK + \*&t = K \-^{W - 1) - i(d t 0) 2 + V(<f>)] (6.19a)
\ (v,jft£ - V a kf) = k ( ^ P u a u + d a 4>d t 0j (6.19b)
-±=dt (Vhk£) + 3 Ri
= K \h fia [^pu a u a + d a cf>d a A + (I + U(0)) d . (6.19c)
The partial differential equation describing the scalar field 4>(t, x) dy-
namics is coupled to the Einstein equations and reads as
1 o dV
dtt4>+-kZdtct>-h al3 \7 a \7pct>+ — = (6.20)
and finally the hydrodynamic equations VjT/ = 0, introduced in Sec. 2.2.1,
sr 29, 2010 11:22
Inhomogeni ous Quasi-isotropic Cosmologies 247
accounting for the matter evolution read explicitly as
-Ld t (Vhp 3/4 uo) + -j=d a (Vhp 3/4 u a ) = (6.21a)
4p I -d t uo 2 + u a d a u + -k a pu a u^ ) = (l — uo 2 ) d t p — u u a d a p (6.21b)
Ap (u d t u a + u^d p u a + ^u^d a h M \ = -u a u d t p + (5^ - u a u^d p p.
(6.21c)
The presence of the scalar field allows to relax the assumption of ex-
pandability in integer powers adopted in (6.1). Indeed, similarly to what
happens for the FRW Universe in the presence of the scalar field, we will
consider an expansion including also non-integer powers. In order to intro-
duce a quasi-isotropic scenario (eventually inflationary, see below Sec. 6.5)
considering small inhomogeneous corrections to leading order, we require a
three-dimensional metric tensor having the following structure
h aP (t, x) = a 2 (t)^ (* 7 ) + b 2 (t)e a0 (xi) + O (b 2 )
= a 2 (t) [^ (x^) + r,(t)9 aP (a-*) + O (r, 2 ) ] (6.22)
where we defined r\ = — and suppose that rj satisfies the condition
lim7?(i) = 0. (6.23)
In the limit of the approximation (6.23), the inverse three-metric reads as
^ (< ' X) = ak) (^ {X1) ~ ^^ {xr) + ° ^ ) ' (6 ' 24)
where £ Q/3 denotes the inverse matrix of £ Qj g and assumes a metric role, i.e.
we have
^T£ Q7 = Si , 9 afi = i ai ^ 5 lS . (6.25)
The covariant and contravariant three-metric expressions lead to the rela-
tions
k1 = 2-5i+r ] e l i => k* = 6- + r]6, = 9%. (6.26)
Since the equality d t (lnh) = k% holds, we get
h = ja & e^ e => Vh= ^/]a 3 e^ e
'\/ja 3 (l + \v0 + O(r, 2 )^ , (6.27)
248 Primordial Cosmology
once j = det ^ a p has been denned.
The Landau-Raychaudhury theorem (see Sec. 2.4) applied to the present
case implies the condition
lima(i)=0. (6.28)
The set of field equations (6.19) is thus solved under the following assump-
(1) the validity of the limit (6.28);
(2) to retain only terms linear in r\ and its time derivatives;
(3) to neglect all terms containing spatial derivatives of the dynamical
variables, in order to obtain asymptotic solutions in the limit t — > 0;
(4) finally checking the self-consistence of the approximation scheme.
Although the possibility to neglect the potential term V(4>) is not en-
sured by the field equations, it is based on the idea that, in an inflationary
scenario, the scalar field potential energy becomes dynamically relevant
only during the slow-rolling phase, far from the singularity, while the ki-
netic term asymptotically dominates. Let us start from Eq. (6.20) to obtain
the following
d t 0=^-e-^ 9 ~^-(l-±i 1 9 + O( V )) (6.29)
{M f = * e -*
-^(l-^ + Ofa)),
ere d is a constant.
In the same approximation, fro
m (6.21a), we get
Vhp 3/4 u = l( X i) =► p ~ -
Z 4 / 3 / 2
(6.31)
being /(a; 7 ) an arbitrary function of the spatial coordinates. Let us analyze
the Einstein equations (6.19). Taking into account (6.30) to first order in
77, Eq. (6.19a) reads as
3 - + ^ + ( h + -V - ^-v) = -4(3 + 4« 2 ) (6.32)
^U a Uji
sr 29, 2010 11:22
Inhomogem ous Quasi-isotropic Cosmologies 249
Furthermore, Eq. (6.19c) reduces to
|(a 8 )"^+(a^)-(?g + i[(o s H-^
= |«P (^ + ^^ 7 «a%) « 3 (l + ^) • (6-34)
In agreement with our assumptions, in Eq. (6.34) we neglected the spatial
curvature term which, to leading order, reads as
W a (t,x) = -± f) AP a (x~<) (6.35)
where j4 aj g(x 7 ) = S.p-yA^ is the Ricci tensor corresponding to £ Q/ 3(a; 7 ). Tak-
ing the trace of Eq. (6.34) we get
2(a 3 )' + (a^y-9 = \k P (3 + Au 2 ) a 3 (l + ±rfi) . (6.36)
The compatibility of Eqs. (6.32) and (6.36) is ensured by the solution to
the following system
nd 2
(a 3 )-+3a 2 a+ — = (6.37a)
Ind 2
3(a 3 ry)- + 3a 3 7? + 2(a 3 )?) + 9a 2 ar7 — 77 = . (6.37b)
Equation (6.37a) admits the solution
in correspondence to the choice d= J^j-.
Substituting the expri ion 6.3 i for a(t) in Eq. (6.37b) we get
3^ + 4?) -2^ = 0. (6.39)
By setting
r,(t) =(£)', (6.40)
the differential equal ion ((i.371)) reduces to the following algebraic equation
3x 2 + :r-2 = => x = -!,-. (6.41)
Since r?(£) must vanish for i — ► 0, we exclude the negative solution x = — 1,
obtaining thus
■G)'
(6.42)
sr 29, 2010 11:22
Primordial Cosmology
From these solutions for a(t) and rj(t), the consistence of the model
provides u a expressed as
In conclusion, we get the following identification regarding the arbitrary
function ^(x 7 )
V3k(3 + 4v 2 )£ 2
with
v 2 = e P v a v p . (6.45)
From these results and from Eq. (6.34) one obtains the tensor 9 a p{x' 1 )
9 °0 = ^-2[(l-2v 2 )t a ( i + lOv a v /) ] =* 9 = C, (6.46)
where C( x7 ) denotes an arbitrary function of the spatial coordinates. The
energy density of the ultrarelativistic matter is found, to leading order, in
the form
p(t,x) = — [ 5C{X ' ( \ + o(f) , (6.47)
3 K [3 + 4« 2 (a; 7 )Ji 2 / 3 i 4 /3 VW
allowing to integrate the scalar field equation (6.20) as
*">-\/sK£)-K£r«" 7)+ 'H +o (£) ,6 - 48)
where a(x" 1 ) is an arbitrary function of the spatial coordinates.
Finally, Eq. (6.19c) yields the expression for the functions v a in terms
of ( and of the spatial gradient d a a as
_ 3(3 + 4v 2 ) =w ^ (g49a)
10CV1H
2±r 2 -l + VT^ (6 _ 49b)
2(1 - 16r 2 )
where r represents the quantity
yl^d a adpc
Inhomogeneous Quasi-isotropic Cosmologies 251
The simple case a = 0, in correspondence to which v 2 = (v a = 0) leads
to the solutions
a p = ^C(z 7 K^ (6.50a)
P ^ = h^)^+0(£) (6.50b)
u a (t, x) = h a In (C(^)) t + o(£)- (6-50d)
Finally we obtain the three-dimensional metric tensor as
On the basis of Eqs. (6.50), the hydrodynamic equations (6.21) reduce to
identities in the approximation considered here.
It is worth noting that, in correspondence to a fixed time t* (t* <C to),
p*(x^) = p(t*,x), u* a (x^) = u a (t*,x), (6.51)
then, in terms of these time independent quantities, Eqs. (6.47) and (6.43)
read in the more expressive form as
_.(if + (i),_<(i)"% (i). «,,,
The solution shown here is completely self-consistent to the first two
orders in time and contains five physically arbitrary functions of the spatial
coordinates: three out of the six functions £ a/ g (the remaining three of them
can be fixed by pure spatial coordinates transformations) , the spatial scalar
C(x 7 ) and finally a(x 1 ).
The independence of the functions £ Q/ g, C an d a implies the existence
of a quasi-isotropic dynamics in correspondence to an arbitrary spatial dis-
tribution of ultrarelativistic matter. The kinetic term of the scalar field
behaves, to leading order, as ~ a~ 6 and therefore, in the limit a — > 0, it
dominates the ultrarelativistic energy density. The latter diverges only as
~ a~ 4 and therefore the spatial curvature term ~ a~ 2 is negligible.
Thus, these behaviors are at the ground of the possibility to deal with
an unconstrained spatial dependence of the ultrarelativistic energy density.
The homogeneity is ensured by the zeroth order contribution from the scalar
field.
252 Primordial Cosmology
6.4 The Role of an Electromagnetic Field
Here we analyze the dynamical behavior, near the cosmological singularity,
of a quasi-isotropic Universe in the presence of an electromagnetic field and
a real massless scalar field. More precisely, we show how the presence, in
spite of its vectorial nature, of an electromagnetic field on a quasi-isotropic
background is allowed due to the dominant character of the scalar field
kinetic term. In fact, as discussed in Sec. 2.2.3, the electromagnetic field
has an anisotropic energy momentum tensor and it would be incompatible
with the quasi-isotropic assumption, when treated to lowest order. On
the other hand, near the singularity the scalar field dominates the quasi-
isotropic metric to leading order.
We outline the complete equivalence existing between the dynamical
effect produced, on a quasi-isotropic Universe containing a real massless
scalar field, by the presence of an electromagnetic field and by an ultra-
relativistic matter component, discussed in detail in Sec. 2.2.3. In both
cases, the metric of the Universe and the scalar field acquire, to the first
two orders of approximation, the same time dependence, and similarly their
corresponding energy densities.
Since close enough to the singularity (i.e. at sufficiently high temper-
ature) the presence of a potential term for the scalar field is dynamically
negligible, then our evolutive scheme can be thought of as an inflationary
scenario, yet far from the later slow-rolling phase (where the potential term
provides the main dynamical contribution).
The general formula! iou of the cosmological problem describing the dy-
namics of a three-dimensional Universe with an electromagnetic field Fik
(see Sec. 2.2) and a real massless scalar one 4> is based on an action of the
form
Sg+EM = ~Y K \ ^ ( R + ^ FtkFtk ~ K 9 ik di<t>d k <l>) d 4 x . (6.53)
By varying the action (6.53) with respect to these three fields, we get the
dynamical equations for F^ as
F lm F lm 5n +g M di(j)di(j)\ (6.54a)
l
■F u l
dFik
dx>
dx k
dF u
+ dx k +
sr 29, 2010 11:22
Inhomogeneous Quasi-isotropic Cosmologies
and for the scalar field <fi m the covariant form as
7=aH^^)= ' (6 ' 54d >
In Eq. (6.54c) the ordinal \ pari ial derivatives can be equivalently r
by the covariant ones.
Since the energy momentum tensor of the electromagnetic field is trace-
less, from Eq. (6.54a) we get
R - Kg ik di(f)d k (t> = . (6.54e)
Due to the real character of the scalar field (i.e. it does not bring any
electric charge), the two matter fields interact only through the space-time
curvature.
In a synchronous reference frame, the dynamical equations (6.54) reduce
to the partial differential system
\d t k* + \kik% = -■%- [E a E a + \B a RB aP ) - n{d t (t>) 2 (6.55a)
2 4 ' 8tt \ 2 /
-{Vpkl - V a k1) = ^-B^E 13 + Kd a <j)d t <p (6.55b)
2y/h
- -?- (E a E !i - -E~EF6i - B ai B^ + \b iS B^5^]
Air \ 2 4 J
+ nh^d a (t>d^(t>, (6.55c)
which is coupled to the Maxwell equations
^=d a (VhE a ) = (6.56a)
Vh
-^= \d t {VhE a ) + dp(VhB afi )\ = (6.56b)
d t B aP + d p E a - d a E fi = (6.57a)
d 7 B aP + d p B ia + d a B M = , (6.57b)
and finally to the equations describing the scalar field
-^= [d t (Vhdt^) - d a {Vhh ap d^)\ = (6.58a)
\d t k% + \kik% + w y_ ' +P + K[(dt<t>) 2 ~ h ali d a 4>d^\ = . (6.58b)
2 4 y/fl
254 Primordial Cosmology
Above, according to Sec. 2.2.3, we adopted the definitions
E a = F 0a E a = h a ' 3 E fj = -F 0a (6.59)
B a p = F afj B'l = h fi '<B ai = -Fj . (6.60)
We are interested to a quasi-isotropic solution of the system (6.55)-
(6.58), i.e. according to a three-dimensional metric tensor of the form as in
Eq. (6.22).
In what follows we shall analyze the field equations (6.55)-(6.58), re-
taining only terms linear in r\ and its time derivatives. The analysis of Eqs.
(6.55) is based on the construction of an asymptotic power-law solution in
the limit t — > 0, and by verifying a posteriori the self-consistence of the ap-
proximation scheme. In particular, since the action of the time derivative
operator on a power-law expression provides terms of lower order in time,
we can neglect the contribut ion of the spatial derivatives.
In the limit of this approximation, from Eqs. (6.56b) we get
E' = ^l + 0(±) (6.61.)
where d a generic constant and £ a {x 1 ) an arbitrary vector field of the
spatial coordinates. By substituting this expression for E a in Eq. (6.56a),
we get the constraint for £ a
1
For what regards the scalar field dynamics, the approximation (6.30) holds.
When it is included in the Einstein Eq. (6.58b), we get the system obtained
by the two (zeroth- and first-order) components as
3a 2 a + a 3 + ^- = (6.62a)
a 3 ij + 2a 2 dr, + (a 3 r,)- - (a 3 + 2^- J r/ = , (6.62b)
where p is a constant of integration related to the scalar field. Repeating the
same steps performed in the previous Section and choosing the constant p =
V3k- ^o ' we £ et ^ e same *™ e dependence for a(i) and ?/(/) as in (6.38) and
(6.42), respectively. Whon sucb expressious uudcr approximation (0.6.1 ! arc
inserted in Eq. (0.57a) we get
B a[i = dV2B afj {x r ) +ol(j-J ) , (6.63a)
V a £ a = -^=d a (Vh£ a ^j = . (6.61b)
i-isotropic Cosmologies
B a , = d a B p - d p B a , (6.63b)
being B a (a; 7 ) an arbitrary spatial vector. So far, it is straightforward to
realize how Eq. (6.57b) is identically verified by Eq. (6.38) and Eq. (6.42).
Using the expressions (6.61), (6.30), (6.38), (6.42), and (6.63), Eqs.
(6.55a) and (6.55c) turn out to be automatically satisfied to zeroth-order
approximation, while, to first order, they require 1 the idculilications
d = ±J-^* (6.64a)
6 = £ a £ a + B a0 B al3 (6.64b)
a0 = -5£ a £p + WB ai B^ + {2£ i r - W^B 1 *)^ (6.64c)
where we set
£ a = £ a0 £f> , Bj = i^B ai . (6.65)
In agreement with our approximation, the spatial curvature term, having
the form as in Eq. (6.35), is still negligible. By integrating the equation
corresponding to Eq. (6.30) we get
*->=^(£H(^) 2/ w^)]. - vv „,
(6.66)
being a{x 1 ) an arbitrary function of the spatial coordinates.
Finally, in terms of all the expressions above obtained, we observe that
the leading order (0(l/t)) of Eq. (6.55b) reduces to the differential con-
straint
d a a = --V^B^ 13 . (6.67)
It is worth noting how an exact evaluation of the terms to next order in
Eq. (6.55c) (©(l/i 1 ' 3 )) involves higher order terms in the expansions of E a
and .Bq/3, which contain contributions from nonlinear terms in 77(f) and its
time derivatives.
From the whole scheme, we see that the spatial tensor £ Q/ 3 can be ar-
bitrarily assigned, while the quantities £ a , B a and a are subjected to the
constraint (6.67) only. By assigning the functions B a (i.e. B12, B23 and
B31) the set of equations in (6.67) reduces to an algebraic inhomogeneous
system in the three unknowns £ a which, being deti3 a ^ = 0, in order to be
256 Primordial Cosmology
solved requires the validity of the following partial differential equation for
the function a
£i2<9 3 ct + £3i<9 2 ct + £23<9ict = 0. (6.68)
Due to its linear homogeneous structure, this equation always admits a
solution in correspondence to any choice of B a . Once solved, the algebraic
system (6.67) allows us to express two of the components £ a in terms of
the third one, and the quantities B a and the spatial gradients in terms of
the function a (solution of Eq. (6.68)).
Taking into account the three allowed general transformations of the
spatial coordinates, which remove three degrees of freedom among the ten
free functions (six from £ a ^, three B a and one from £ a ), the number of
physically arbitrary functions of the spatial coordinates available for the
Cauchy problem reduces to seven.
By comparing the analysis here developed with the one in Sec. 6.3, where
the ultrarelativistic matter replaces the role of the electromagnetic field, the
complete dynamical equivalence between these two cases arises. Indeed, a
quasi-isotropic Universe in which a real scalar field lives (whose dynamics
asymptotically has a dominant character) receives the same dynamical con-
tribution from the ultrarelativistic matter (described by a perfect fluid with
an ultrarelativistic equation of state), as well as from an electromagnetic
field. The solutions for a(t) and rj(t) take, in both cases, the same power-
law expressions together with the correspondence among the two sets of
spatial functions as
C o £ a £ a + B af3 B al3 (6.69)
3(3 + 4v 2 ) (3 + 4*/%
being v 2 as defined in Eq. (6.49b). Similarly, for r defined as in Eq. (6.49c),
we have the correspondence
y/2(£ a £ a + B c
s^yv^
n S-rBp S £ s ■ (6.71)
Here v a denote the spatial distribution of the fluid four-velocity spatial
components, as in Eqs. (6.43) and (6.33). A complete correspondence
exists in the two cases with respect to the form taken by the scalar field
and by the energy densities too. The ultrarelativistic matter, in the absence
of a scalar field, can yet survive on a quasi-isotropic background unlike the
electromagnetic one.
Inhomogeneous Quasi-isotropic Cosmologies 257
The microwave background radiation, whose dynamical contribution
is well described by the ultrarelativistic equation of state (pure radiation
component), indeed corresponds to a completely incoherent (disordered)
electromagnetic field.
6.5 Quasi-isotropic Inflation
In this Section we find a solution for a quasi-isotropic inflationary Universe
which allows to introduce a certain degree of inhomogeneity. We consider
a model which generalizes the flat FRW model by introducing a first-order
inhomogeneous term, whose dynamics is induced by an effective cosmolog-
ical constant. As above, the three-metric tensor consists of a dominant
term, corresponding to an isotropio-likc component, while the amplitude of
the first-order term is controlled by the higher order function r)(t).
In a Universe filled with ultrarelativistic matter and a real self-
interacting scalar field, we analyze the dynamics up to first order in rj, when
the scalar field performs a slow roll on a plateau of a symmetry breaking
configuration and induces an effective cosmological constant.
We show how the spatial distributions of the ultrarelativistic matter
and of the scalar field admit an arbitrary form but nevertheless, due to the
required inflationary e-folding, it cannot play a significant dynamical role in
the process of structure formation (via the Harrison-Zeldovich spectrum).
As a consequence, we reinforced the idea that the inflationary scenario is
incompatible with a classical origin of the cosmological structures.
As seen in Chap. 5, the inflationary model is, up to now, the most
natural and complete scenario able to solve the problems appearing in the
Standard Cosmological Model, like the horizon and flatness paradoxes; in-
deed, such a dynamical scheme, on the one hand is able to justify the high
isotropy of the cosmic microwave background radiation (and in general the
large-scale homogeneity of the Universe) and, on the other hand, provides a
mechanism for the generation of a nearly scale-invariant spectrum of inho-
mogeneous perturbations (via the quantum fluctuations of the scalar field) .
Moreover, as it will be shown in detail in Chap. 8, a slow-rolling phase
of the scalar field allows to connect the Mixmaster dynamics with a later
quasi-isotropic Universe evolution, in principle compatible with the stan-
dard cosmological picture.
In Sec. 6.3, the quasi-isotropic solution has been discussed in the pres-
258 Primordial Cosmology
ence of a kinetic energy-dominated real scalar field, which leads to a power-
law solution for the three-metric, and predicts interesting features for the
dynamics of ultrarelativistic matter.
In this section the opposite dynamical scheme, i.e. when the scalar field
undergoes a slow-rolling phase due to the dominance of the effective cosmo-
logical constant over its kinetic energy, is analyzed. A detailed description
is provided, in a synchronous reference frame, of the three-metric, of the
scalar field and of the ultrarelativistic matter dynamics up to the first two
orders of approximation, showing that the volume of the Universe expo-
nentially expands and induces a corresponding exponential decay (as the
inverse fourth-power of the cosmic scale factor), both of the three-metric
corrections, and of the ultrarelativistic matter. The spatial dependence
of this component is described by a function which remains an arbitrary
degree of freedom, nevertheless there is no chance that, after the de Sit-
ter phase, the relic perturbations survive enough to trace the large scale
structures formation. This behavior suggests that the spectrum of inho-
mogeneous perturbations cannot directly arise by the classical field nature,
but only by its quantum dynamics (see Sec. 5.6.4).
The presence of the scalar field kinetic term, considered negligible here,
induces, near enough to the singularity, a deep modification to the general
cosmological solution, leading to the appearance of a dynamical regime
during which, point by point in space, the three spatial directions behave
monotonically, as discussed in Sec. 8.7.1 for the homogeneous Mixmaster
Uiodol.
6.5.1 Geometry, matter and scalar field equations
As done before, let us describe the matter field as a perfect fluid with
ultrarelativistic equation of state P = — together with a scalar field </>(t, x)
with a potential term V (</>). In what follows, we write the Einstein equations
E[ P
- k fe cnil
where the label m and <fi are adopted to distinguish between the matter
and scalar field energy momentum tensors.
The set of interactions (6.72), similarly to what described in Sec. 6.3,
explicitly reduces to the system (6.19), coupled with the dynamics of the
scalar field <j){t 7 x) described by the partial differential Eq. (6.20). The
Inhomogeni ous Quasi-isotropic Cosmologies 259
hydrodynamic equations, taking into account the matter evolution, in a
synchronous reference and for the ultrarelativistic case, have the structure
of Eqs. (6.21). This scheme is consistent with the invariance under the
transformation as in Eq. (6.9).
6.5.2 Inflationary dynamics
In order to introduce small inhomogeneous corrections to the leading order
in a quasi-isotropic inflationary scenario, wc consider a three-dimensional
metric tensor having the structure outlined in Eqs. (6.22)-(6.27).
Then the field equat ions (6. L9) are analyzed retaining only terms linear
in rj and its time derivatives, and verifying a posteriori the self-consistency
of the approximation scheme.
In the quasi-isoi ropic approach, we assume that the scalar Held dynam-
ics in the plateau region (see Sec. 5.4.1) is governed by a potential term of
the form
V(4>) = pa + K(</>) , p A = const. , (6.73)
where p\ is the dominant term and K{<j>) is a small correction. The role of
K, as shown in the following, is to drive inhomogeneous corrections via the
^-dependence; the functional form of K can be any of the most common
inflationary potentials, as they appear near slow-roll region.
What follows remains valid, for example, in the relevant cases of the
quartic and Coleman- Weinberg potentials introduced in Sec. 5.4.1, where
the corrections to the constant pA term arc
4 '
(6.74)
B,a = const.
In the following, explicit calculations are developed only for the first case.
The inflationary solution is obtained under the usual requirements
l - (d t 4>) 2 « V {<f>) (6.75a)
I d tt <P I < I KM I • (6.75b)
The above approximations and the substitution of Eq. (6.26) reduce the
scalar field Eq. (6.20) to the form
( 3 l + lv0)d t <j>-aj<j> 3 = 0,
260 Primordial Cosmology
where the contribution of the spatial gradient of <*; has been assumed to be
IU'!>li<>j]>].<\
Similarly, the quasi-isotropic approach (in which the inhomogeneities
become relevant only in the next-to-leading order), once the spatial deriva-
tives in Eq. (6.21) have been neglected, leads to
?4/3 / o \
Vh P ^u = W) =* P ~ j2/3a4uQ4/3 (l ~ ~ 3 V0 + Otf)) , (6.77)
where l(x 7 ) denotes an arbitrary function of the spatial coordinates.
Let us now face, in the same approximation scheme, the analysis of the
Einstein Eqs. (6.19). Taking into account Eq. (6.75a), to first order in rj,
Eq. (6.19a) reads as
3^ + (\v +lv)0- np A = -«| (3 + 4u 2 ) , (6.78)
where u 2 and u. a are given by Eq. (6.33). Equation (6.19c) reduces to the
form
|(o 3 )" 6i + (a 3 r,y O a + \[(a 3 y v] <Wg + aA a
= - [^ (^ ~ VO^) \ P u aUl + (f + p A ) 8i ] 2a 3 (l + f ) , (6-79)
where the spatial curvature term is expressed, to leading order, as in
Eq. (6.35). The trace of Eq. (6.79) yields the additional relation
2 (a 3 )' + (a 3 ry)- 9 + aA« a = K [| (3 + 4 U 2 ) + 3p A ] 2a 3 (l + y) • (6-80)
Comparing Eq. (6.78) with the trace Eq. (6.80), via their common term
(3 + 4u 2 )p/3, and equating the different orders of approximation, we get
the following equations
(a 3 ) " + 3a 2 a - 4/«p A a 3 = (6.81a)
A af} = (6.81b)
3 (a 3 7?) " + 3a 3 ?) + 2 (a 3 ) ' f] + 9a 2 7?a - l2np A a 3 7] = . (6.81c)
Since Eq. (6.81b) implies the vanishing of the three-dimensional Ricci tensor
and this condition corresponds to the vanishing of the Riemann tensor, we
can conclude that the Universe described by this solution is fiat up to
leading order, i.e.
Up = S a p =* 3 = 1. (6-82)
sr 29, 2010 11:22
Inhomogem ous Quasi-isotropic Cosmologies 261
Equation (6.81a) admits the accelerating solution
fl (i) = a exp(^pi), (6.83)
where a is an integration constant.
Expression (6.83) for a(t), when substituted in Eq. (6.81c) yields the
following differential equation for r\
4 /
V+ -V3kpa?? = 0, (6.84)
whose only solution, satisfying the condition expressed by Eq. (6.23), reads
r)(t) = ?7o exp f - - ^3kpa t j => rj = t? (—J , (6.85)
and we require 770 <C a .
Equations (6.77) and (6.78), in view of the solutions (6.83) for a(t) and
(6.85) for r](t), are matched by posing
u a (t,x)=v a (x^) + 0(r 1 2 )
(u ) 2 = l + o(^-) wl, (6.86)
respectively. Equation (6.87) implies < for all values of the spatial
coordinates. The comparison of Eq. (6.77) with Eq. (6.87) leads to an
explicit expression for I {x 1 ) in terms of 9 as
/4 \ 3/4
K xl ) = f gPA»7oao 4 J (-6») 3/4 .
Defining the auxiliary tensor with unit trace 6 Q /3(a; 7 ) = 6 a p/6, the abo^
analysis allows, from Eq. (6.79), to obtain the expression
From Eq. (6.76), the explicit form for a, once expanded in powers of 77,
yields the first two orders of a.pproxhiial ion for the scalar field as
2C 2 ui '
262 Primordial Cosmology
where C is an integration constant. Finally, Eq. (6.19c) provides v a in terms
Of 6 as
v a = ~-^=d a ]n\0\. (6.91)
On the basis of Eqs. (6.89)-(6.91), the hydrodynamic Eqs. (6.21) reduce
to an identity, to the leading order of approximation; in fact, such equations
contain the energy density of the ultrarelativistic matter, which is known
only to first order (the higher one of the Einstein equations) so that higher
order contributions cannot be taken into account.
As long as (/ r — /,) is sulUriont l> large, it can be checked that the solution
here constructed is completely self-consistent at least up to the order of ap-
proximation considered here and contains one physically arbitrary function
of the spatial coordinates 6{x 1 ). This function, being a three-scalar, is not
affected by spatial coordinate I ransformations. In particular, the quadratic
terms in the spatial gradients of the scalar field are of order
( ^ )2 "°(^^f) (6 - 92)
and therefore can be neglected with respect to the other inhomogeneous
terms. This solution instead fails when t approaches t r , therefore its validity
requires that the de Sitter phase ends (with the fall of the scalar field in
the true potential vacuum) when t is still much smaller than t T .
6.5.3 Physical considerations
The peculiar feature of the solution constructed above lies in the free char-
acter of the function 6 which, from a cosmological point of view, implies
the existence of a quasi-isotropic inflationary solution together with an ar-
bitrary spatial distribution of ultrarelativistic matter and of the scalar field.
The Universe emerging from such inflationary picture has the appro-
priate standard features, but with the presence of a suitable spectrum of
classical perturbations, due to the small inhomogeneities. The power spec-
trum of fluctuations can be modeled in the form of a Harrison- Zeldovich
spectrum. In fact, expanding the function 6 in Fourier series as
° {xl) = (2^)3 j- " § (*) e< *'* d3 * ' (6 ' 93)
we can impose a Harrison-Zeldovich spectrum by requiring
1 ~ 6 |2 = iTjs ' z = const ' (6 ' 94)
However, let us complete our picture considering that:
Inhomogeneous Quasi-isotropic Cosmologies 263
(i) limiting our attention to leading order, the validity of the slow-
rolling regime is ensured by the natural conditions
O {y/itpK{t - t r )) < 1 , w>0( K 2 p A ), (6.95)
which translate Eq. (6.75b) and Eq. (6.75a), respectively;
(ii) denoting by t{ and if tin b inn i i and the end of the de Sitter
phase, respectively, we should have i r 3> if and the validity of the
solution is guaranteed if
(a) the flatness of the potential is preserved, i.e. uKp 4 <C pj\: such
a requirement coincides with the second inequality of (6.95);
(b) denoting with A the width of the flat region of the potential,
we require that the de Sitter phase ends before i becomes
comparable with i r , i.e.
0(i f ) - <MiO ~ ^ry - 0(A) , (6-96)
where the solution has been expanded to first order in ii,f/i r ;
via the usual position (if — i;) ~ O(10 2 )/y/Kp\, the relation
(6.96) becomes a constraint for the integration constant i r .
(iii) The exponential expansion should last long enough in order to solve
shortcomings present in the SCM. It has been shown in Sec. 5.5
that the minimum number £ of e-folds necessary to solve the short-
comings is typically £ ~ 60 so that a f / ai ~ e 60 ~ O(10 27 ). Thus
any perturbation that could be present before inflation would be
reduced by a factor ~ (Vf/Vi) ~ (<H/a f ) 4 ~ 0(1O- 108 ). Though
these inhomogeneities increase as a 2 when they are at scale greater
than the horizon, they should start with an enormous amplitude
in order to play a role in the process of structure formation. This
result supports the idea that the spectrum of inhomogeneous per-
turbations cannot have a classical origin in the presence of an in-
flationary scenario.
So far, we have estimated pf/pi i.e. the ratio of the inhomogeneous
terms pf and p ; . In fact, after the reheating the Universe is dominated by
a homogeneous (apart from the quantum fluctuations) relativistic energy
density p r to which the relic pi is superimposed after inflation and therefore
we have
P L = P L P_L = (^]Pi i (6 .9 7)
Pr Pi Pr \OiJ Pr
264 Primordial Cosmology
where the inhomogeneous relativistic energy density before the inflation p\
and the uniform one p r , generated by the reheating process, are of the same
order of magnitude.
When referred to a homogeneous and isotropic FRW model, the de
Sitter phase of the inflationary scenario rules out the small inhomogeneous
perturbations so strongly to prevent them from seeding the later structures
formation. This picture emerges sharply within the inflationary paradigm
and it is at the basis of the statement that the cosmological perturbations
arise from the quantum fluctuations of the scalar field.
Though this argument is well settled down and is very attractive even
because the predicted quantum spectrum of inhomogeneities takes the
Harrison-Zeldovich form, nevertheless the question remains open whether,
in more general contexts, it is possible that early classical inhomogeneities
can survive to be relevant for the origin of the cosmological structures.
6.6 Quasi-Isotropic Viscous Solution
In order to generalize the quasi-isotropic solution of the Einstein equations
in the presence of dissipative effects, we consider a power law extension of
the three-metric generalizing Eq. (6.4) as
h a p = t x a aP + t v b afj , h afi = t~ x a afi - t y ~ 2x b afs . (6.98)
Here, the constraints for the space contraction toward the singularity (i.e.
x > 0), and for the internal consistency of the perturbative scheme (i.e.
y > x) have to be imposed for the proper development of the model. In
this approach, the extrinsic curvature and its contractions read as
k afi = xt x ~ 1 a a fi+yt v ~ l b afj , (6.99a)
fcf = x r 1 Si + (y - x) t y ~ x - x b p a , (6.99b)
k =3xt~ 1 + (y-x)*"-*' 1 b. (6.99c)
The following relation also holds
d t In Vh = - k = - xt" 1 + -(y-x) i^" 1 b . (6.100)
The aim is to obtain constraints and relations for the exponents x, y in
order to guarantee the existence of the solutions of this model. We can thus
write down the final form of the components of the Ricci tensor contained in
Inhomogeneous Quasi-isotropic Cosmologies
the Einstein equations (6.6). These new expressions allow us to generalize
the original quasi-isotropic approach and explicitly read as
«8 = - ?E ^ + (»-*)(» -1)2^. (6.101a)
K = (V Q 6 - Vpbi ) ^ZJL. , ( 6 .ioib)
p x(3x - 2) gg (y- ^(2^ + 3; -2) ^
+ &« + # + ^. (-C
We note that in Eq. (6.101c), A& represents the three-dimensional Ricci
tensor built from the metric a a p, as in Sec. 6.2. On the other hand, the
higher-order term B@ denotes the part of P£ containing the three-tensor
b a a-
6.6.1 Form of the energy density
In this Section, we treat the immediate generalization of the non-viscous
LK scheme. We consider the presence of dissipative processes affecting the
fluid dynamics, as it is expected in the early phases of the Universe, es-
pecially at temperatures above O(10 16 GeV). As discussed in Sec. 3.3 this
extension is represented by an additional term in the expression of the en-
ergy momentum tensor (6.10) and it can be derived from thermodynamical
properties of the fluid. The restated tensor reads as
T ij = (P + p)u i u j -pg ij
= | (4u iUj - 9ij ) - (Vm^UiUj - 9ij ) , (6.102a)
P = P-(Viu l , (6.102b)
where P = p/3 denotes the usual Ihcrniostatic pressure in correspondence
of an ultrarelativistic equation of state and £ is the bulk viscosity coefficient,
introduced in Sec. 3.3. In what follows, we neglect the shear viscosity for
consistency with the quasi-isotropic cosmological evolution (see Sec. 6.6.2).
The coefficient £ has to be expressed in terms of the thermodynamical
parameters of the fluid. In particular, as in Sec. 3.3.1, this quantity is
assumed to be a power-law function of the energy density fluid
( = ( p s , (6.103)
where Co is a constant and ,s is a dimensionless parameter whose behavior
in correspondence to large values of p is constrained in the range ^ s ^ \ .
266 Primordial Cosmology
Let us write the expressions of the mixed components of the tensor
(6.102a) up to higher-order corrections as
^0° = |(4«g - 1) - CoP s ViU l (ul - 1) (6.104a)
T = -ZCop^iU 1 (6.104b)
Tj> = -^{Au^ + 8%) - Q p s V l u l (u aU P + 8%) (6.104c)
(6.104d)
Til
= ~ 3 pu c
,«° - Cop'Vi^Uat
the divergence
: of the four-velocity reads
y lU l =
= d t In vh = -xt~
9
l + \iv-
- l b. (6.105)
e valid, as in the non- viscous case, the relation ufj ~ 1, whose
consistence must be verified a posteriori comparing the time behavior of
the quantities involved in the model. Taking into account the expressions
(6.104a) and (6.104b), wc can recast the Einstein equation (6.6a) in the
form
3x(2-x) , ,. b [ 9a: , 3(y-xK ,1
(6.106)
In what follows, as in Sec. 3.3.1, we fix the value s = ^ in order to deal
with the maximum effect that bulk viscosity can have without dominating
the dynamics of the cosmological fluid.
Since we are interested in the asymptotic limit t — ¥ 0, such choice for
s is the appropriate one to include dissipative effects in the primordial
dynamics. Thus, from Eq. (6.106) the energy density p can be expanded as
,=f+^> ^(i+sH> cm")
where the constants e and e\ will be determined combining the —
gravitational equation with the hydrodynamical equations, comparing all
terms order by order. We remark that only for the case s = \ all terms
of Eq. (6.106) have the same time behavior up to first order because of
Eq. (6.107).
6.6.2 Comments on the adopted paradigm
In this Section, we discuss in some details the hypotheses at the ground of
our analysis of the quasi-isotropic viscous Universe dynamics.
Inhomogeneous Quasi-isotropic Cosmologies 267
As introduced in Sec. 3.1.5, the micro-physical horizon, i.e. the Hubble
length, plays a crucial role as far as the thermodynamical equilibrium is
concerned. In the isotropic Universe, this quantity is fixed by the inverse of
the expansion rate, H^ 1 = (a /a) and provides the characteristic scale below
which the particle interactions can preserve the thermal equilibrium of the
system. Therefore, if the mean free path of the particles £ is greater than
the microphysical horizon (i.e. £ > H^ 1 ), no notion of thermal equilibrium
can be recovered at the microcausal scale. If we denote the number density
of particles as n and the average cross section of the interactions as a,
the mean free path of the ultrarelativistic cosmological fluid (in the early
Universe the particle velocity is very close to the speed of light) takes the
form £ ~ 1/na. Interactions mediated by massless gauge bosons are in
general characterized by a cross section a ~ a 2 T~ 2 (a = j-, where g is
the coupling constant of the corresponding interaction) and the physical
estimate n ~ T 3 provides £ ~ 1/ a 2 T . During the radiation dominated era
H ~ T 2 /m P , so that
£ I—H- 1 . (6.108)
Thus, in the case T > a 2 m P ~ £>(10 16 GeV), i.e. during the earliest
epoch of the pre-inflating Universe, the interactions are "frozen out" and
they are not able to establish or to maintain the thermal equilibrium. At
temperatures greater than O(10 16 GeV), the contributions to the estimate
(6.108) due to the mass term of the gauge bosons can be ruled out for all
known and so far proposed perturbative interactions.
As a consequence of this non-equilibrium configuration of the causal re-
gions characterizing the early Universe, most of the well-established results
about the kinetic theory concerning the cosmological fluid nearby equilib-
rium are not directly applicable. Indeed, the kinetic analysis is generally
based on the assumption of a finite mean free path of the particles and, in
particular, the viscosity is characterized by simply retaining pure collisions
among the particles. However, when the mean free path is greater than the
microcausal horizon, £ can be regarded as infinite for any physical purpose.
The original analysis of the viscous cosmology is due to the Landau
school which, aware of these difficulties for a consistent kinetic theory,
treated the problem on the basis of a hydrodynamical approach. A no-
tion of the hydrodynamical description can be provided by assuming that
an arbitrary state is adequately specified by the particle flow vector and
the energy momentum tensor alone. In particular, the entropy flux has
to be expressed as a function of these two dynamical variables without
268 Primordial Cosmology
additional parameters. Thus, the viscosity effects are treated through a
thermodynamical description of the fluid, i.e. the viscosity coefficients are
fixed by the macroscopic parameters which govern the system evolution.
The most natural choice is to take these viscosity coefficients as a power
law of the energy density of the fluid. Such phenomenological assumption
can be reconciled, for some simple cases, with a relativistic kinetic theory
approach, especially in the limits of small and large energy densities.
Considering the hydrodynamical point of view, we retain the same equa-
tion of state which characterizes the corresponding ideal fluid. This is
supported by the idea that viscosity effects provide only small corrections
to the thermodynamical setting of the system. Since we are treating an
ultrarelativistic thermodynamical system, very weakly interacting on the
micro-causal scale, it is appropriately described by the equation of state
P = p/3.
Let us note that the shear viscosity r\ is not included in the present
scheme. Indeed, this kind of viscosity accounts for the friction forces acting
between different portions of the viscous fluid. Therefore, as far as the
isotropic character of the Universe is conserved, the shear viscosity must
not provide any contribution, as discussed in Sec. 3.3.1. On the other
hand, the rapid expansion of the early Universe suggests that an important
contribution comes out from the bulk viscosity as an average effect of a
quasi-cquilibrium evolution.
Our analysis deals with small inhomogeneous corrections to the back-
ground FRW metric. In principle, to first order in our solution, shear
viscosity should be included in the dynamics as well. In that case, if the
bulk viscosity coefficient behaves as ( ~ p s , correspondingly the shear is
r\ ~ p r , where r must satisfy the constraint condition r ^ s + ^. As dis-
cussed in Sec. 3.3.1, we treat the case s = \, thus getting r ^ 1 for the r\
coefficient. Such constraint implies that the shear viscosity is no longer a
first order correction in this solution. For smaller values of s, the shear vis-
cosity can be included without leading to unphysical solutions. In fact, the
shear viscosity provides, among others, an equivalent contribution to the
bulk one, since the energy-momentum tensor of the viscous fluid contains
the term
Ta ~ ... - (C- |»7)V,u'(«iU;+fftf) + - • (6-109)
Let us observe that, to zeroth order, V;u J ~ 0(l/t), while the first-order
correction to the energy density behaves as 0(l/t x ) and we have shown
the relation 1 < x < 2 in Sec. 3.3.1. The request x ^ 1 comes out from
Inhomogeni ous Quasi-isotropic Cosmologies 269
the zeroth-order analysis which, due to the isotropy, is independent of the
shear contribution. Since the estimate
O (v) ~ o ( P r ) ~ o
(±)
holds, we can conclude that the shear viscosity would produce the incon-
sistency associated to the term
V V t u l ~ O (— !+r) ■ (6-Hl)
The request rx + 1 ^ 2 would make the contribution in (6.111) dominant
in the model, despite the basic assumption that the shear viscosity must be
negligible to the leading isotropic order. Thus, to include the shear viscosity
in a quasi-isotropic model, we should consider the case s < i but it is
not appropriate for analyzing the asymptotic limit towards the singularity
because the corresponding contribution vanishes with no influence on the
dynamics.
Let us discuss the implementation of a causal thermodynamics for this
cosmological model. The hydrodynamical theory of a viscous fluid is appli-
cable only when the spatial and temporal derivatives of the matter veloc-
ity are small, i.e. the characteristic rate of the fluid reaction is negligible
with respect to the speed of light. This condition is expectedly violated
in the asymptotic limit near the cosmological singularity. In this way, the
viscous fluid would be described by a relaxation equation similar to the
Maxwell equations in the theory of viscoelast icily. In that scheme, the
energy- momentum tensor assumes the form (3.85). In the very early Uni-
verse, the relation between U L , and the relaxation time To reads as
n„ + ri^ro = cv 4 u 4 . (6.112)
The relaxation time can be expressed as 7o/( ~ 1/p: this physical assump-
tion follows from the transverse wave velocity in matter which has a finite
(non-zero) magnitude in the case of large values of p.
The time dependence of To follows from the fact that p ~ 1/t 2 to leading
order and then, using Eq. (6.103), the relaxation time behaves as t ~ £ 2 ~ 2s .
Since s = \ and thus t ~ t, if we assume a power law dependence for IT,,
(according to the structure of the solution) such as H v ~ H v /t, the relation
(6.112) rewrites as
U v = Co p s \7m l . (6.113)
From this analysis we recover the standard expression for the bulk viscous
hydrodynamics, provided by the reparametrization Co — > Co of the bulk co-
efficient. This is compatible with the paradigm of causal thermodynamics,
270 Primordial Cosmology
since it would affect only qualitative details (i.e. rescaling some coefficients),
without altering the validity of the solution.
6.6.3 Solutions of the OO-Einstein and hydrodynamical
equations
Thus far, we exploited Eq. (6.6a) in order to obtain the qualitative expres-
sion for the energy density p when I Ik- matter (ill i u<>, t he space was described
by a viscous fluid energy-momentum tensor. Let us consider Eq. (6.106)
rewritten as
r 3 . , 9, , 1 o
- -x(2 - x) + kcq - -CaXy/Ke^\ t
- ^(y - x)( ^]bt y - x - 2 = , (6.114)
coupled to the hydrodynamical ones VjT/ =0. In the non- viscous case
(Co = 0), the energy density solution is determined without exploiting the
hydrodynamical equations, since p comes directly from the 00-gravitational
equation. To the order of approximation considered here (u a being neg-
ligible with respect to u ), the energy- momentum tensor conservation law
provides the equation
d t p + d t (\nVhJ \^p-CoP s d t {\nVhj\ =0, (6.115)
which rewrites as
2Ke (a;- 1)- -( x 2 y/ne^lr 3
+ Lei U {y - x - 2) + 2xb - d -C x 2 b (/ce )" 1/2 ")
+ -{y - x)be --x(y- x)Coby / l^\ t v ~ x - 3 = .
(6.116)
When Eq. (6.114) is coupled to Eq. (6.116) it provides a polynomial ex-
pression in t and must be solved order by order in 1/t (in the asymptotic
limit t — > 0). Since for the consistency of the solution y > x (as detailed
when we discussed Eq. (6.98)), applying the polynomial identity principle
we get the unique values
x = 1=— , K e = I x 2 . (6.117)
1 - ^P Co 4
Inhomogeneous Quasi-isotropic Cosmologies 271
The parameter Co is constrained as Co ^ -^ts m order to satisfy the condition
x > 0. In this way, the exponent of the metric power law x runs from
1 (which corresponds to the non-viscous limit Co = 0) to infinity. Such
constraint on Co arises from a zeroth-order analysis and defines the existence
of a viscous Friedmann-like model, in which the early Universe expands with
a power law in time.
Comparing the two first-order identities (involving terms proportional
to t v ~ x ~ 2 and t v ~ x ~ 3 ), we get an algebraic equation for y
y 2 -y(x+l) + 2x-2 = 0, (6.118)
whose solutions are y = 2,y = x— 1. The latter does not fulfill the condition
y > x, thus the first order correction to the three-metric is characterized
by the following parameters
y = 2, Kei = -^x 3 + 2x 2 -2x. (6.119)
In the non-viscous case (Co = 0) we get x = 1, ne = §, nei = —\, which
reproduce the energy density solution (6.14).
The consistency of the model is ensured by constraining the parameter
x to values x < y. Thus, from Eq. (6.117), the quasi-isotropic solution
emerges only for
Co < Co* = ■£= > (6-120)
i.e. for small enough viscosity. When the viscous parameter Co overcomes
the critical value Co, the quasi-isotropic expansion in the asymptotic limit
as t — > cannot be considered, since the perturbations would grow more
rapidly than the zeroth-ordcr terms. The perturbation dynamics in a pure
isotropic picture yields a very similar asymptotic behavior when including
viscous effects. The Friedmann singularity scheme is preserved only if we
deal with limited values of the viscosity parameter, in particular obtaining
the condition Qj"' < Q' } /'■'>: this constraint is physically motivated consid-
ering that the Friedmann model is a particular case of the quasi-isotropic
Soldi !!)!].
The solution of the unperturbed dynamics (>,ives rise to the expression
of the metric exponent x in terms of the viscous parameter Co an d to the
zeroth-order expression of the energy density which reads as
3x 2
K P=772+-- ( 6 - 121 )
272 Primordial Cosmology
In order to characterize the effective expansion of the early Universe, let us
recall the expression of the total pressure P (6.102b) to leading order as
p = \ p + Yt Coy ^ x ' (fU22)
obtained from the four-divergence (6.105) truncated to zeroth order. From
these relations, the condition P > yields the inequality
Co < Co72 , (6.123)
which strengthens the constraint (6.120) and restricts the x-domain to
Ml-
The request of a positive (at most zero) total pressure is consistent with
the idea that the bulk viscosity must not affect too much the standard
dynamics of the isotropic Universe. In this respect, we consider the domain
(6.123) as a physical restriction to the initial conditions for the existence
of a well-grounded quasi-isotropic solution.
Let us rewrite the expression of the energy density to analyze the evolu-
tion of the density contrast. In the presence of the bulk viscosity, p assumes
the form
„.g- f/»-y + »» , ( ,. m)
and, hence, the density contrast 5 (defined in Sec. 3.4) can be written as
'~i(! + H^- (6 ' 125)
Since x runs from 1 to 2 as the viscosity increases towards its critical value,
the density contrast evolution is strongly damped by the presence of dissi-
pative effects which act on the perturbations. In this sense, the viscosity
can damp the evolution of the perturbations forward in time. This behavior
of the density contrast toward the singularity (S — > 0) is characterized by
a weaker power law in time in comparison to Co approaching Q. In corre-
spondence to such threshold value, the density contrast remains constant
6.6.4 The velocity and the three-metric
While the 00-Einstein equation provides a solution for the energy density,
let us complete the dynamical scenario analyzing the quasi-isotropic model,
to verify the consistency of our approximations considering the solutions to
the whole system of gravitational equations.
Inhomogeneous Quasi-isotropic Cosmologies
Imposing the condition s = \, the Einstein equation (6.6b) read:
(y-x)b
Substituting Eq. (6.124) into Eq. (6.126), we get the expression for the
velocity, which to leading order reads as
Cq. (6.126), we get
u a = ^^{d a b-\7pbi)t 3 - x , (6.127)
where we neglected terms of order 0{t~ x ) and 0(t 1 ~ x ). The assumption
Uq ~ 1 is verified since M a ii" ~ £ 6 ~ 3:r and can be neglected in the four-
velocity expression mu l = 1. The approximated hydrodynamical equation
(6.116) is still self-consistent using Eq. (6.127) for u a .
Let us address Eq. (6.6c): the first two leading order terms of the right-
hand side are 0(t~ 2 ) and 0(t~ x ), respectively, only if x < 2, as in our
scheme. Hence, u a u^ can be neglected and terms 0(t~ 2 ) cancel out, while
those proportional to t~ x give the equation
A^+Abi+BbS? +05^ = 0, (6.128)
which generalizes Eq. (6.16), and the quantities A, 23, C are defined as
A = \{A-x 2 ) (6.129a)
B= 1 -{2x-l)(x-2) 2 -\x{x-2) (6.129b)
C = ~(2-x)(x-l), (6.129c)
respectively. Taking the trace of Eq. (6.128), we obtain the relation
{A + 323)6 = -A - 3C (6.130)
which, combined with the Ricci three-tensor relation V ' pA^ = ^VpA, pro-
vides the equation
2AVpb a = (A + B)d a b . (6.131)
Let us write down the three- velocity in terms of the trace of the perturbed
metric tensor b as
u a = 2 ~ xA ~ B t 3 - x d a b. (6.132)
Ax A
The solution constructed here matches the non- viscous solution (6.18) if we
set Co = and it is completely seU'-oousistent up to the first two orders in
time. The present model contains three physically arbitrary functions of
274 Primordial Cosmology
the spatial coordinates only, i.e. the six functions a„/j minus three degrees
of freedom that can be eliminated by fixing suitable space coordinates. The
only remaining free parameter of the model is the viscosity Co-
The quasi-isotropic solution exists for particular values of the bulk vis-
cosity coefficient Co only. When the dissipative effects become too relevant,
we are not able to construct the solution following the line of the LK model.
In fact, when Co approaches the threshold value Co = TJ75, the approxima-
tion scheme fails and the model becomes not self-consistent.
By requiring the viscosity parameter Co to be smaller than its critical
value, the behavior of the density contrast is influenced by the presence of
the bulk viscosity. As far as dissipative effects are taken into account, the
density contrast contraction (5 — > as t — > 0) is damped out, or remains
constant if Co equals its critical value.
6.7 Guidelines to the Literature
With reference to Sec. 6.1, for a discussion of the Einstein equations in a
synchronous reference, as introduced in Sec. 2.4, we refer to the classical
textbook [301].
The original derivation of the quasi-isotropic solution for the radiation
dominated Universe, discussed in Sec. 6.2, was provided by Lifshitz & Kha-
latnikov in [312]. For a generalization of this solution to a generic equation
of state, see [272].
When introducing the scalar field in the quasi-isotropic model, the re-
sulting features near the cosmological singularity, as detailed in Sec. 6.3,
were derived in [349]. An interesting related analysis, starting from the
long- wavelength approximation can be found in [439].
The presentation in Sec. 6.4 of a quasi-isotropic solution in the presence
of a massless scalar field and an electromagnetic component is stated in the
article [350].
The description of the quasi-isotropic inflation in Sec. 6.5 refers to [259]
with reference also to [439].
For a general discussion regarding the inflationary scenario and the ori-
gin of a perturbation spectrum, as introduced in Sec. 5.6.4, we refer to the
following textbooks [155,290,327,370].
The analysis of the quasi-isotropic solution in the presence of the bulk
viscosity presented in Sec. 6.6 is inspired by the derivation of [112]. The
comparison of these results with the behavior of the inhomogeneous per-
Inhomogeni ous Quasi-isotropic Cosmologies 275
turbations in a viscous FRW Universe is allowed by reading [111].
For the original analysis of the influence of the bulk viscosity on the
early Universe dynamics see [61,63,67]. Such line of research treated the
description of bulk viscosity within a hydrodynamical approach, as it is
addressed in Sec. 6.6.
This page is intentionally left blank
PART 3
Mathematical Cosmology
In these Chapters, the evolution of the Universe near the singularity is dis-
cussed under more general hypotheses than homogeneity and isotropy. The
analysis of the homogeneous models of the Bianchi classification is presented
in details, both in the field equations and the Hamiltonian frameworks.
Chapter 7 is dedicated to the investigation of the geometry and the dy-
namics of the homogeneous, but anisotropic cosmologies, as prescribed
by the Einstein geometrodynamics. The chaotic nature of the Belinskii-
Khalat uikov-Liishitz oscillatory regime is outlined.
Chapter 8 concerns the Hamiltonian description of the Bianchi type VIII
and IX models near the singularity (the so-called Mixmaster Universe).
The question inherent the proper characterization of the Mixmaster chaos
and of its covariant nature is analyzed with accuracy.
Chapter 9 presents a picture of the generic cosmological solution in the
asymptotic limit to the singularity (the so-called inhomogeneous Mixmas-
ter Universe) . The piecewise nature of this solution is analyzed by means of
the Einstein equations, as well as of the Hamiltouian representation, i he
parametric role of the space coordinates is outlined and the existence of a
space-time foam is argued.
This page is intentionally left blank
Chapter 7
Homogeneous Universes
In this Chapter we analyze the dynamics and the morphology of the ho-
mogeneous but anisotropic cosmological models. This set of Universes has
been classified by Bianchi in 1898 into nine different types, corresponding
to the independent groups of isometries for the three-dimensional space.
We provide a precise definition of homogeneity for a space manifold and
outline the underlying Lie algebra and the Jacobi identities which lead to
the Bianchi classification. Via the projection technique on the triad set-
ting of the three-manifold, we show how the Einstein equations reduce to an
ordinary differential system in time. In fact, the homogeneity constraint im-
plements the dynamical equivalence of all the spatial points, so that spatial
gradients cannot enter the Universe evolution. The space geometry remains
fixed by the 1-forms describing the specific model, while the dynamics is
summarized by three scale factors, associated to the evolution of the three
independent spatial directions, according to the Einstein prescription.
The analysis of the simple Bianchi I model, derived by Kasner in 1921,
has a peculiar role in the construction of the asymptotic regime to the
cosmological singularity of all the other Bianchi Universes.
Then, we discuss the Bianchi type II and we outline how its evolution is
characterized by two distinct Kasner regimes (Kasner epochs) related by a
specific map which accounts for the effect of the spatial curvature inducing
the transition process.
The Bianchi VII model emerges as consisting of a Kasner era, i.e. a se-
quence of Kasner epochs during which two space directions oscillate toward
the singularity and the third one decays monotonically, plus a final stable
Kasner regime associated to a re-increase of the decaying direction.
The behavior of the Bianchi type VIII and IX models is discussed in
detail as far as the Universe approaches the singularity and in both cases,
280 Primordial Cosmology
we outline the existence of an oscillatory regime of the solution, consisting
of an infinite sequence of Kasner eras. The statistical properties of this
piecewise solution are addressed and the stationary distribution functions
of the metric parameters arc derived. Thus, the most general models (types
VIII and IX) allowed by the homogeneity constraint are still characterized
by a cosmological singularity, but the corresponding asymptotic regime is
characterized by an anisotropic morphology and outlines significant features
of a chaotic dynamics. Such a chaotic solution admits, as we shall see in
Chap. 9 a natural inhomogeneous extension, which provides the evolution
to the singularity of a generic model, i.e. having, in vacuum, four arbitrary
degrees of freedom (four free space functions) to address a generic Cauchy
problem on a non-singular spatial hypersurface.
7.1 Homogeneous Cosmological Models
A space is said to be homogeneous if its metric tensor admits an isometry
group that maps the space onto itself. Such group results to be generated
by the Killing vector fields forming a Lie algebra. The corresponding sim-
ply transitive Lie group can be identified, via its orbits, with the Cauchy
surfaces filling the space-time manifold. Considering the Maurer-Cartan
1-forms (7.28), the homogeneous (and in general anisotropic) metric ten-
sor is written as in Eq. (7.37). The matrix -q a b thus depends on the time
coordinate only, is position independent on each Cauchy surface, and the
Einstein equations reduce to a system of ordinary differential equations.
7.1.1 Definition of homogeneity
Let us consider a group of transformations
x" -> Si" = f" (x,t) = f?(x) (7.1)
on a space E (eventually a manifold), where JT a } a=1 r are r independent
parameters that characterize the group. Furthermore, let To correspond to
the identity
r(x,r ) = x». (7.2)
Let us take an infinitesimal transformation close to the identity t + St so
that
x^x» = r(x,T + ST)
« /" (x, r ) + (?£ J (x, r ) <JT a = s" + ^ (x) Sr a (7.3)
= (1 + 5t%)^.
Here the r first-order differential operators {£ a } are defined by £ a = £^<9 M
and correspond to the r vector fields with components {£,%}, constituting
the generating vector fields. This way, under the infinitesimal transforma-
tion (7.3) all points of the space E are translated by a distance Sx^ = ST a ^
in the coordinates {x^} and thus
x"«(l + 5T a Ca)^«e 5r<, «°^. (7.4)
The finite transformations (which have the structure of a group) may be
represented as
2" -> *" = e'°*"*" (7-5)
where {# a } (a = l,...,r) are r parameters on the group. In particular,
if the group is a Lie group, the generators £ a form a Lie algebra, i.e. a
real r-dimensional vector space where {£„.} is a basis which is closed under
commutation relations
{Za^ h }=U b -tea=C C ab Z c , (7.6)
where C" : ah arc called I he structure constants of the group.
Let us consider a Lie group acting on a manifold E as a group of trans-
formations (7.1), and let us define the orbit of a; as
U (x) = {f T Wire 0} (7.7)
i.e. the set of all points that can be reached from x under the action of the
transformations .
Definition 7.1. The group of isometry at x is given by the subgroup of &
which leaves x fixed, i.e.
©* = {fr (X) = X | T G ©} . (7.8)
Suppose ©a; = {to} and /g (a;) = E, i.e. every transformation of 25
moves the point x and every point in E can be reached from a: by a unique
transformation. Since <5/& x = {t/tq \t G ®}, the group (5 is isomorphic to
the manifold E and one may identify the two objects. The transformations
282 Primordial Cosmology
leaving invariant the metric // Q y are called isometries, particular cases of
diffeomorphisms. In the particular case of an isometry group, the {££}
satisfy the so-called Killing equation
V Q & + V^ Q = , (7.9)
and for this reason, are called Killing vector fields. They satisfy a Lie
algebra and generate the groups of motions via infinitesimal displacements,
yielding conserved quantities and allowing a classification of homogeneous
spaces.
We now introduce the concept of an invariant basis. Suppose {e a } is a
basis of the Lie algebra g of a group ©
and define -f ab = C c ad C d bc = -f ba . This quantity is symmetric by definition
and provides a natural inner product on g
lab = e a ■ e b = 7(e a ,e 6 ) . (7-11)
When det (jab) 7^ 0, Eq. (7.11) is non-degenerate and the group is called
semi-simple. The r vector fields {e a } form a frame, because they may be
used instead of the coordinate basis { -^ } to express an arbitrary vector
field on 0. The basis {e a } is called invariant 1 if
L ?a e 6 =[£ a ,e h ] = 0. (7.12)
This means that e a are vector fields invariant under the action of the Killing
vectors £ a .
Summarizing, a manifold S is said to be invariant under a group if
there are m (where m = dim©) Killing vector fields £ a which satisfy the
commutation relation (7.6). As soon as one identifies to E, the metric
tensor h a p on E is invariant under the action of the group 0. Thus, h a p
corresponds to an invariant tensor on such a group, and it is completely
specified by the inner product (7.11) of the invariant vector fields e a .
Let us consider the case of a space-time whose metric gij is invariant
under a three-dimensional isometry group. Given an invariant basis {e a },
the spatial metric at each moment of time can be specified by the spatially
constant inner products
e a ■ e b = Vab (t) . (7.13)
lr To be more precise, e a are called "left-invariant", while £ a "right- invariant".
Homogeneous Universes 283
Here, the tetradic projection n a b of h a p (see Sec. 2.5) has been identified
with 7 afc of Eq. (7.11).
We can give the following definition of a spatially homogeneous space-
timo:
Definition 7.2. A space-time (M,gij) is spatially homogeneous if a family
of space- like surfaces E, exists sii.c.Ii that for any two points p,q € E t , there
is a unique element r : M — > M of a Lie group © such that r(p) = q.
Because of the uniqueness of the group element r, © is said to act simply
transitively on each E t . Such condi! ion implies also that the Killing vectors
are linearly independent, and that 2 dim© = dimE 4 = 3. Spatially ho-
mogeneous models are those for which the symmetry group acts simply
transitively on each spatial manifold, i.e. the space-time topology is given
by
M=R®©. (7.14)
It is worth noting that, because of the identification of © with E t , the
action on E t of the isometry r corresponds to a left multiplication by r on
0. This way, tensor fields invariant under the isometries correspond to the
loft-invariant ones on 0.
7.1.2 Application to Cosmology
For a spatially homogeneous space-time, one needs only to consider a rep-
resentative group from each class of equivalence of isomorphic Lie groups
of dimension three. The classification of inequivalent three-dimensional
Lie groups is called the Bianchi classification and determines the various
possible symmetry types for homogeneous three-spaces, just as K = 0, ±1
classify the possible symmetry types for homogeneous and isotropic FRW
three-spaces.
The space-time metric gij in the homogeneous models must reflect that
the metric properties are the same in all space points. Under the isometry
r : x — > x', the spatial line element
all 2 = h a p(t,x)dx a dx , (7.15)
has to be invariant (as discussed above it is also left-invariant), which im-
plie
9 {t,x')dx' a dx' d ,
n /i-(lin.u-Dsioual manifold, din
284 Primordial Cosmology
where h a p has the same form in the old and in the new coordinates. The
metric tensor for a homogeneous space-time is obtained by choosing a basis
of dual vector fields ui a which are preserved under the isometries. This basis
is dual to the left-invariant one (7.12), i.e. co a (ei,) = 5%.
In the general case of a non-Euclidean homogeneous three-dimensional
space, there are three independent differential forms which are invariant
under the transformations of the group of motions. These forms, however,
do not represent the total differential of any function of the coordinates.
We shall write them as co a = e a a dx a and hence the spatial line element is
re-expressed as dl 2 = ri a b{e'^dx a )(e b i:i dx 13 ) so that the triadic representation
of the metric tensor reads as
h a p{t,x) = r, ab {t)e a a {x>)e) t {x'), (7.17)
where r\ a h is a symmetric matrix depending on time only. Differently from
the general case discussed in Sec. 2.5, the homogeneity condition allows
one to encode the time dependence of the triads into r] a b- In contravariant
components we have
h a »(t,x) = T,° b {t)eZ(xi)4(xi), (7.18)
where rf h denote the components of the matrix inverse of r\ a \,. All the
results obtained in Sec. 2.5 apply straightforwardly also here.
The relationships between the covariant and contravariant expressions
for the three basis vectors are
e 1 = i[e 2 Ae 3 ], e 2 = \ [e 3 A e 1 ] , e 3 = \ [e 1 A e 2 ] ,
(7.19)
where e a and e a must be formally considered as Cartesian vectors with
components e£ and e", while v represents
v =\ e a a \=e Y ■ [e 2 Ae 3 ] . (7.20)
The determinant of the metric tensor (7.17) is given by h = rjv 2 , being
■q = det(rj a b).
The invariance of the line element (7.16) under the action of the symme-
try group implies that (for easier notation we denote hereafter x = x 1 )
e a a (x) dx a = e a a (x r ) dx' a , (7.21)
where e^ on both sides of Eq. (7.21) are the same functions expressed in
terms of the old and the new coordinates, respectively. The algebra of the
triads permits one to rewrite Eq. (7.21) as
d -^- = el{x')el{x). (7.22)
(7.23)
Homogeneous Universes 285
This is a system of differential equations which defines the change of coor-
dinates x'P(x) in terms of given basis vectors. Integrability of the system
(7.22) is expressed in terms of the Schwartz condition
d 2 x'P d 2 x'P
dx a dx~< ~ dx~<dx a
which, explicitly, leads to
[~dx^ eb {X > ~ ~dx^ €a {X >\ ^ {X) 6a {X}
= efM [^M_^Ml. (7 . 24)
aK J [ dx a dx~f \ { '
Multiplying both sides of (7.24) by e < ^{x)ej(x)e i fi {x') simple algebra pro-
vides the following form for the left-hand side of the equation
d4W) r s (T r, de^x') s }
dx ,s c[ ' dx' s d[ >\
\del(x') deUx')~\
and the right-hand side gives the same expression but in terms of x. Since
x and x' are generic coordinates, such an equality implies that both sides
must be equal to the same set of constants, and Eq. (7.25) reduces to
which gives the relations of the vectors with the group structure constants
C c ab- Let us note that such constants coincide with the three-dimensional
analogous of Eq. (2.111b). Multiplying Eq. (7.26) by ej, we finally have
<£-<£ -<>•-.'■
Such expression states that the homogeneity condition reduces to a con-
straint on the left-invariant 1-forms u" = e" x dx" which have to satisfy the
Maurcr-Cartan structure equation
duj a + ]-C a bc to b A uj c = . (7.28)
By construction, we have the antisymmetry properly (" lf!l = —("),„ from
the formula (7.26). In terms of the vector fields e a = e"<9 Q , Eq. (7.27)
rewrites as Eq. (7.10). Homogeneity is then expressed as the Jacobi identity
[[e„, e b ] , e c ] + [[e b , e c ] , e„] + [[e c , ej , e b ] = (7.29)
286 Primordial Cosmology
and is eventually stated as
cf ab cd cf + C f bc C d af + C{ a C d bf = , (7.30)
or, in the language of forms, as
d 2 uj a = 0. (7.31)
Introducing the two-indices structure constants as C c ab = e a bdC dc (or,
equivalently, C dc = e abd C c ab ), where e abc = e abc is the totally anti-
symmetric three-dimensional Levi-Civita tensor, the Jacobi identity (7.30)
becomes e bc dC cd C ba = 0. We also note that Eq. (7.26) can be restated as
C ah = -- e a Ae b , (7.32)
where the vectorial operations are to be performed as if coordinates x 1
were Cartesian ones. The problem of classifying all homogeneous spaces
thus reduces to finding out all inequivalent sets of structure constants of a
three-dimensional Lie group.
As we will see, by this formalism the Einstein equations for a homoge-
neous Universe can be written as a sa si ( 1 1 o] (lei utial equations
which involve functions of time only, provided the use of projections of 3-
tensors on the tetradic basis.''
7.1.3 Bianchi Classification and Line Element
The list of all three-dimensional Lie algebras was first accomplished by
Bianchi in 1897 such that each algebra uniquely determines the local prop-
erties of a three-dimensional group. Given a homogeneous space-time with
its symmetry group being the "Bianchi Type N" (N =1, ..., IX), the
corresponding structure constants can be written as
C" bc = Sh c . ( yn d " + S"ai, — 6 b a c , (7.33)
or, equivalently,
C ab = n ab + £ abc ac ^ , ? ^
where n ab = n ba and a a
= C\ a . From Eq. (7.33), the Jacobi identity (7.30)
reduces to the condition
n ab a b = 0, (7.35)
3 The explicit coordinate dependence of the basis vectors is not necessary for the equa-
tions ruling the dynamics. In fact, such choice is not unique as e a = Ab a e b yields again
a set of basis vectors, for any o
and a three-dimensional Lie group (algebra) is then determined by assigning
a dual vector a c and a symmetric matrix n ab satisfying the constraint (7.35).
Without loss of generality (i.e. with a global rotation of the triad vectors),
we can set a c = (a, 0,0) and reduce the matrix n ab to its diagonal form
n ab = diag(ni, n 2 ,n 3 ). The condition (7.35) reduces to am = 0, i.e. either
a or m has to vanish. The sub-classification as class A and class B models
refers to the case a c = or a c / 0, respectively. The Jacobi identity can
also be restated in terms of the vector fields e as
[ei,e 2 ] = -ae 2 + n 3 e 3
[e 2 ,e 3 ]=niei (7.36)
[e 3 ,ei] =n 2 e 2 + ae 3
where the set of parameters a > and ni,n 2 ,n 3 can be rcscaled to unity
by a corresponding constant re-scaling of the triad. Therefore, all three-
dimensional Lie algebras can be classified (according to the Bianchi classi-
fication) into nine types, six of class A and three of class B, according to
Table 7.1.
It is worth noting that the Bianchi type I is isomorphic to the three-
dimensional translation group M 3 , for which the flat FRW model is a par-
ticular case (once isotropy is restored). Analogously, the Bianchi type V
contains, as a particular case, the open FRW space. Finally, the closed
FRW spatial line-element can be obtained in the isotropic limit of type IX
model whose symmetry group is SO(3) (see Sec. 8.1). Bianchi IX is then
the most general model in which the topology of the spatial surfaces is given
by the three-sphere S 3 . It has been shown by Lin and Wald in 1990 that
all these models first expand and, after reaching a turning point, start to
collapse. Of course, the closed FRW Universe belongs to such a class.
The metric tensor g rj can be immediately written by considering a basis
of dual vector fields ui a preserved under the isometries. Recalling Eq. (7.17),
the four-dimensional line element is then expressed as
ds 2 = N 2 (t)dt 2 - r) ab (t) uj a uj h , (7.37)
parametrized by the proper time, where the one- forms u a = w a (x 7 ) obey
the Maurer-Cartan equations (7.28). Notice that o>° take value in the Lie
algebra g of the symmetry group & and (in the case of 4 © = GL n ) can be
written as
W ( T ) = T^dr, re®. (7.38)
,r group GL n denotes the set of real nxn matrices with non- vanishing
.urse, SO{n) and SU(n) are subgroups of GL n .
Primordial Cosmology
^responding to the Bianchi c
Type j
a
ni
n 2
n 3
I
II
1
VII
1
1
VI
1
-1
IX
1
1
1
VIII
1
1
-1
V
1
IV
1
1
Vila
a
1
1
m(o = i) 1
VI a (a#l)|
Let us now write the Einstein equations for a homogeneous Universe. In
the tetradic basis u a , the vicrbcin components (see Sec. 2.5) can be written
in the form of a system of ordinary differential equations which involve
functions of time only
- K b n Ki -
■I*
= K c b (C b ca - 5 b a C d dc ) =
1 d.
b Vvdt
where the relation K a {,
Ricci tensor 3 R a b b<
(v«)
(7.39a)
(7.39b)
(7.39c)
— di.'Hab/'Z holds. The triad components of the
(2.112b)
3 R ab = -\(c cd b C cda + C cd b C dca - \c b cd C acd
2 v 2 (7.40)
- C\ d C d + C\ d C h d ) .
The dynamics of the homogeneous (but anisotropic) Universes will be
investigated in detail in the following Sections.
7.2 Kasner Solution
The simplest solution of the Einstein equations (7.39) in the framework of
the Bianchi classification is the so-called Kasner solution, i.e. the vacuum
type I model.
The simultaneous vanishing of the three structure constants and of the
parameter a implies the vanishing of the three-dimensional Ricci tensor
Furthermore, since the three-dimensional metric tensor does not depend
on space coordinates, also Ro a = holds, as confirmed by Eq. (7.39b); we
stress that this model contains the standard Euclidean space as a particular
case. The system (7.39) describes a uniform space and reduces to
K a a + K h a Kl = , (7.42a)
^|(v/^)=0. (7.42b)
From Eq. (7.42b) we get the first integral
yftjK* = ( b a = const. , (7.43)
and contraction of indices a and b leads to
K? = i- = £=, (7.44)
a 27? ^?
and finally
ri = (Cft 2 . (7.45)
Without loss of generality, a simple rescaling of the coordinates x a allows
Substituting Eq. (7.43) into Eq. (7.42a), one obtains the relations among
the constants Ca
CbC = 1 , (7-47)
and lowering index b in Eq. (7.43) one gets a system of ordinary differential
equations in terms of r\ a b
Vab = -CaVcb ■ (7.48)
The set of coefficients Q can be considered as the matrix of a certain linear
transformation, reducible to its diagonal form. In such a case, denoting its
eigenvalues as (pi,p m ,Pn) £ K, and its eigenvectors as l,m, n the solution
of (7.48) writes as
,7 a6 = t 2pi lj b + t 2pm n a n b + t 2pn m a m b . (7.49)
290 Primordial Cosmology
If we choose the frame of eigenvectors as the triad basis (recall that e a a = 5%)
and label the coordinates as x 1 , x 2 , x 3 , then the spatial line element reduces
to
dl 2 = t 2p > (dx 1 ) 2 + t 2p ™ {dx 2 ) 2 + t 2p " {dx 3 ) 2 . (7.50)
Here pi,p m ,p n are the so-called Kasner indices, satisfying the two relations
Pi+p m +p n = l (7.51a)
Pl+P m +P 2 n = l i (7.51b)
and therefore there is only one independent parameter characterizing the so-
lution. Equation (7.51a) follows from relation (7.46), while Eq. (7.51b) lat-
ter comes from Eq. (7.47). Except for the cases (0,0, 1) and (-1/3, 2 /s, 2 / 3 )>
such indices are never equal to each other, and one of them is negative
while two are positive; in the peculiar case pi = p m = 0,p n = 1, the metric
is reducible to a Minkowskian form by the transformation
tsmhx 3 =£, tcoshx 3 =T. (7.52)
It is worth noting that in this particular case, the singularity in t = is a
fictitious one.
Once that Kasner indices have been ordered according to
Pi < P2 < Pz , (7.53)
their corresponding variation ranges are
-\<Pi<®, 0<p 2 <^, -<ps<l. (7-54)
This ordered set of indices admits the following parametrization
, -u l + u u(l + u)
Pl(u) = TT^T^ P2(u) = TT^T^ P3(M) = TT^T^
(7.55)
as the parameter u varies in the range (see Fig. 7.1)
1 < u < +oo. (7.56)
The values u < 1 lead to the same range by following the inversion
property
Pi(^)=Pi(u), p 2 (^)=p 3 (u), pjL\=p 2 (u). (7.57)
The line element from Eq. (7.49) describes an anisotropic space where
volumes linearly grow with time, while linear distances grow along two
directions and decrease along the third one, different from the Friedmann
solution where all distances contract towards the singularity with the same
behavior. This metric has only one non-eliminable singularity in t = with
the single exception of the case p 1 = p 2 = 0, p% = 1 mentioned above,
corresponding to the standard Euclidean space.
)f the parameter 1/u. The domain of
perty (7.57) holds.
7.2.1 The role of matter
Here we discuss the temporal evolution of a uniform disi ribul ion of matter
in the Bianchi type I space near the singularity; it will result in behaving
as a test fluid and thus not affecting the properties of the solution.
Let us take a uniform distribution of matter and assume that its inflti-
i the gravitai tonal Hold can be neglected in a certain region of evolu-
tion. The hydrodynamics equations that describe its evolution i
perfect fluid are (2.18) and the spatial components of (2.19) (se<
i terms of a
Sec. 2.2.1)
S7 k [(p + P)u k ] =u%P. (7.58)
u ^^ = (-^) ( d » P ~ u a u k d k P) . (7.59)
In the neighborhood of the singularity, it, is necessary to use the ultra-
relativistic equation of state P = p/3, and then we get that Eq. (7.58)
becomes
V fc (p 3 /V)=0. (7.60)
By using the metric (7.50), and imposing that all the functions depend on
the time variable only, we get
^ (' 3/v ) - j=p (^ 3,4 »") - \i (" 3,v ) - o <"'»
292 Primordial Cosmology
coupled to Eq. (7.59); the final system stands as
!(t^V")-0. (7.62a)
From Eq. (7.62), we obtain the two integrals of motion
tu p ' = const. , u a p ' = const. (7.63)
From Eq. (7.63) we see that all the covariant components u a are of the
same order of magnitude. Among the contravariant ones, when t — > the
greatest is u 3 = u 3 t~ 2p3 (pi < p 2 < P3). Retaining only the dominant
contribution in the identity utu 1 = 1, we have Uq ks u 2 t~~ 2p3 and, from
Eq. (7.63),
p ~ t- 2 ^+^ = t-w-ri , u a ~ id"P3)/2 _ (7 64)
As expected, p diverges as t — > for all values of p^, except for p% = 1 (this
is due to the non-physical character of the singularity in this case). The
validity of the test fluid approximation is verified from a direct evaluation
of the components of the energy-momentum tensor T^, whose dominant
terms are (2.14)
T ° ~ pul ~ £~( 1+P3 > , Tl ~ p ~ r 2(1 " P3) , (7.65a)
T| ~ pw 2 u 2 - r( 1+2p2 - p3) , T| ~ pw 3 w 3 ~ i-( 1+ P ;i ) . (7.65b)
As i — > 0, all the components grow slower than t~ 2 , which is the behavior
of the dominant terms in the Kasner analysis. Thus, the fluid contribution
can be disregarded in the Einstein equations near the singularity.
This test character of a perfect fluid on a Kasner background remains
valid even in the Mixmaster scenario, both in the homogeneous as well as
in the inhomogeneous case. The reason for the validity of such an extension
relies on the piecewise Kasner behavior of the oscillatory regime.
7.3 The Dynamics of the Bianchi Models
The Kasner solution properly approximates the cases when the Ricci tensor
3 R a p appearing in the Einstein equations is of order higher than 1/t 2 and
thus negligible. However, since one of the Kasner exponents is negative,
terms dominant with respect to t~ 2 appear in the tensor 3 R a p and make
the Kasner solution unstable toward the initial singularity.
(A m 6 2 -A„c 2 ) :
'H
(7.67a)
(w-x nC r
]-0
(7.67b)
(W - x m b 2 y
l-o
(7.67c)
Homogeneous Universes 293
Let us introduce three spatial vectors e a = {l(x 1 ) 1 m{x 1 ),n(x 1 )} sa-
tisfying the homogeneity conditions (7.32) and take the matrix h a p in the
diagonal form
h afj = a 2 (t)l a l fj + b 2 (t)m a m p + c 2 (t)n a n . (7.66)
Such vectors are called Kasner vectors, while a(t), b(t), c(t) are three diffe-
rent cosmic scale factors. Consequently, the Einstein equations in a syn-
chronous reference and for a generic homogeneous cosmological model in
empty space are given by the system
, {abcy 1 r 2 4
abc 2a^b z c z L
_ Rm = (abc)' + 1 r A2 b<
m abc 2a 2 b 2 c 2 l m
(abc)- 1 r 2 4
afcc 2a 2 6 2 c 2 i n
-R° = £ + J! + £ = (7.68)
where the other off-diagonal components of the four-dimensional Ricci ten-
sor identically vanish due to the diagonal form of rj a b as in Eq. (7.66).
Eventually, the 0a components of the Einstein equations can be non-zero if
some kind of matter is present, leading to an effect of rotation on the Kas-
ner axes. The constants (A/, A m , A„) correspond to the structure constants
Cn, C22, C33 respectively, introduced earlier in Sec. 7.1.2. In particular, we
will study in details the cases of the models II, VII, VIII and IX of the
Bianchi classification, which correspond to the triplets (1, 0, 0), (1, 1, 0),
(1, 1, -1) and (1, 1, 1) respectively.
Through the notation
a = \na, /3 = ln6, 7 = lnc (7.69)
and the new temporal variable r defined as
dt = abc dr , (7.70)
Eqs. (7.67) and (7.68) simplify to
2a TT = {\ m b 2 - \ n c 2 ) 2 - X 2 a i (7.71a)
2p TT = (X ia 2 - \ n c 2 ) 2 - X 2 m b A (7.71b)
2 7tt = (\ ia 2 - X m b 2 ) 2 - A 2 / , (7.71c)
Primordial Cosmology
- {a + /3 + 7 ) tt = a T f3 T + a T j T + /3 t7t , (7.72)
where subscript r denotes the derivative with respect to r. After some
algebra on the system (7.71) and using (7.72), one obtains the first integral
a T j3 T + a T j T + /3 T 7 T
= i (A 2 a 4 + A^6 4 + \ 2 n c 4 - 2\ l \ m a 2 b 2 - 2X l X n a 2 c 2 - 2X m X n b 2 c 2 )
(7.73)
involving first derivatives only. The Kasner regime (7.50) discussed before
is the solution corresponding to neglecting all terms on the right-hand side
of Eqs. (7.71). However, such behavior cannot persist indefinitely as t — >
since there are always some terms on the right-hand side of Eq. (7.71) which
g and not negligible up to the singularity.
~^bc~~
26 2 c 2 '
(abc)-
abc
a 2
26V '
(abc)- _
abc
a 2
26V '
7.3.1 Bianchi type II: The concept of Kasner epoch
Introducing the structure constants for the type II model (see Table 7.1),
the system (7.67) and Eq. (7.68) reduce to
(7.74a)
(7.74b)
(7.74c)
- + - + - = 0. (7.75)
abc
In Eqs. (7.74), the right-hand sides play the role of a perturbation to the
Kasner regime; if at a certain instant of time t they could be neglected, then
a Kasner dynamics would take place. This kind of evolution can be stable
or not depending on the initial conditions. As shown earlier in Sec. 7.2,
the Kasner dynamics has a time evolution which differs along the three
directions, growing along two of them and decreasing along the other. For
example, for a perturbation growing as a 4 ~ t 4pi toward the singularity,
two scenarios are possible: if the perturbation is associated with one of the
two positive indices, it will continue decreasing till the singularity and the
Kasner epoch is stable; on the other hand, if pi < 0, the perturbation grows
Homoger,
and cannot be indefinitely neglected. In this case, the analysis of the full
dynamical system is required, and with the use of the logarithmic variables
(7.69)-(7.70) the system (7.71) becomes
/9 TT =
(7.76a)
(7.76b)
Equation (7.76a) can be viewed as the motion of a one-dimensional point-
particle moving within an exponential potential well: if the initial "velocity"
da /(It is equal to pi, then the effect of the potential will result in a slowing
down behavior, stopping and accelerating again the point up to a new
"velocity" —pi- From there on, the potenl ia! will remain negligible forever.
Furthermore, the second set of equations (7.76b) implies that the conditions
a TT + f3 TT = a TT + 7tt = (7.77)
hold. Let us consider the explicit solutions of Eq. (7.77)
a(-r) = - In (cisech (rci + c 2 )) (7.78a)
(3(t) = c 3 + tc 4 - i In ( Cl sech (rci + c 2 )) (7.78b)
7 (t) =c 5 + tc 6 -- In (cisech (rci + c 2 )) , (7.78c)
where c\ . . . , cq are integration constants. Let us analyze how the solu-
tion (7.78) behaves when the time variable r approaches +oo and — oo;
remembering that
dt = abc dr = Adr = exp(a + /? + ^)dr
we have that
• a T = - Cl /2
■ a T
= ci/2
/? T = C 4 + Ci/2
Pr
= c 4 - ci/2
lim ■
7r = c 6 + Cl/2
lim ■
It
= c 6 - Ci/2
t = - exp(Ar)
t
= — exp(A'r)
, A = c 4 + c 6 + ci
/2
. A '
= c 4 + c 6 - ci/2
This m
sans that
Primordial Cosmology
a(t) = t» f a(t) = t*
b(t) = t Pm Jim I b(t) = t p '™
c(t) = t*~ ! "°[ c{t)=tP ' n
^ idcut hied
"c 4 + c 6 +ci/2' m c 4 + c 6 + Ci/2' n c 4 + c 6 + ci/2
(7.81)
ci/g , _ C4-C1/2 , _ ce-ci/2
' c 4 + c 6 -ci/2' m c 4 + c 6 -ci/2' " c 4 + c 6 -ci/2'
(7.82)
These coefficients satisfy the two Kasner relations (7.51): the first fol-
lows directly by (7.81) and (7.82), while the latter is obtained when the
asymptotic behaviors (7.79) are substituted in (7.72).
We see how this dynamical scheme describes two connected Kasner
epochs (a Kasner epoch is defined as the period of time during which the
solution is well approximated by a Kasner metric and the potential terms
are negligible) , where the perturbation has the role of changing the values
of the Kasner indices. Let us assume that the Universe is initially described
by a Kasner epoch for r — > +00, with indices ordered as pi < p m < p n . The
perturbation (which is the term on the r.h.s. of Eq. (7.74a)) starts growing
and the Universe undergoes a transition due to the potential term. Then a
new Kasner epoch begins, where the old and the new indices (the primed
ones) are related among them by the so-called BKL map
p , n = Pn-2\ Pl \
-2M ' ym 1-2M ' ln 1-2M ' ( 7 .83)
A'=(1-2M)A.
The main feature of such a map is the exchange of the negative index
between two different directions. This way, in the new epoch, the negative
power is no longer related to the ^-direction and the perturbation is damped
and vanishes toward the singularity, accordingly to a stable Kasner regime.
7.3.2 Bianchi type VII: The concept of Kasner era
The analysis of Bianchi type VII can be performed analogously, considering
the Einstein equations (7.67) with (A;,A m ,A„) = (1,1,0)
I nhn\- —rA _1_ hi
(7.84a)
abc
2a 2 b 2 c 2 '
(abc)-
abc
a 4 -b 4
2a 2 b 2 c 2 '
(abc)-
{a 2 -b 2 f
abc
2b 2 c 2
ith
the
constraint (7.68) holding m
together "\
Eq. (7.84) with Eq. (7.74) allows a similar qualitative analysis: if the right-
hand sides of Eq. (7.84) are negligible at a certain instant of time, then
the solution is Kasner-like and can be stable or unstable toward the sin-
gularity t — > depending on the initial conditions. If the negative index
is associated with the n direction, then the perturbative terms a 4 and 6 4 ,
evolving as t ipi and t 4pm , decrease up to the singularity and the Kasner
solution turns out to be stable; in all other cases, one and only one of the
perturbation terms starts growing, blasting the initial Kasner evolution and
ending as before in a new Kasner epoch. The main difference between the
types II and VII is that many other transitions can occur after the first
one, and this can happen, for example, if the new negative Kasner index
is associated with the m direct ion. i.e. with b. In this case, the b A term
would start growing and a new transition would occur with the same law
(7.83). The problem of understanding if, when and how this mechanism
can break down is unraveled considering the BKL map written in terms of
the parameter u, i.e.
Pi =Pi(u) ~\ ( p\ =p 2 (u-l)
Pm=P2(u)\^lp' m =pi(u-l) (7.85)
Pn=P3{u)) [p' n =p 3 {u-l)
Let us represent the initial values of u describing the dynamics as
u = k + x , (7.86)
where fcrj represents its integer part while Xq the fractional one (rational or
irrational). In this representation and from the properties (7.55) and (7.57)
we see how the exact number of exchanges between the I- and m-directions
equates k . For the first ko times, the negative index is exchanged between
I and m and only afterwards it passes to the n direction. Now, a new and
298 Primordial Cosmology
final Kasner epoch begins and no more oscillations take place toward the
singularity. In fact, when u < 2, the next value prescribed by the BKL
map corresponds to the fractional part x of u . It is easy to restate the
parameter u in its natural interval [l,oo) by the replacement u ncw = 1/xq
and making use of Eq. (7.57). The collection of the total ko epochs is called
a Kasner era during which one of the three cosmic scale factors (say, for
example, c) decreases monotonically toward the singularity: in this sense
we can say that, in the general case, the type VII dynamics is composed by
one era plus a final epoch.
7.4 Bianchi Types VIII and IX Models
7.4.1 The oscillatory regime
At this point we are going to address the solution of the system (7.67)
for the cases of Bianchi types VIII and IX cosmological models, following
the standard approach of Belinskii, Khalatnikov and Lifshitz (BKL). Al-
though the detailed discussion is devoted to the Bianchi IX model, it can
be straightforwardly extended to the type VIII.
Explicitly, the Einstein equations (7.67) reduce to
2a TT = (b 2 - c 2 ) 2 - a 4 (7.87a)
2/3 TT = (a 2 - c 2 ) 2 - 6 4 (7.87b)
2 7tt = (a 2 - 6 2 ) 2 - c 4 , (7.87c)
together with the constraint (7.72) unchanged, leading to the consequent
first integral
a T f3 T + a TlT + /3 t7t =l(a i + b i + c 4 - 2a 2 b 2 - 2a 2 c 2 - 2b 2 c 2 ) . (7.88)
Let us therefore consider again the case in which, for instance, the negative
power of the pi exponents corresponds to the function a{t): the perturba-
tion of the Kasner regime results from the terms as Xfa 4 (remember that
A; = 1 for both models) while the other terms decrease with decreasing t.
Preserving only the increasing terms on the right-hand side of Eqs. (7.87)
we obtain a system identical to Eqs. (7.76a), whose solution describes the
evolution of the metric from its initial state (7.50). We have that if
a ~ i Pl , b ~ t P2 , c ~ t P3 , (7.89)
where A is a constant, so that the initial conditions for Eq. (7.76a) can be
formulated as
a T = Api , P T = Ap 2 , lr = Ap 3 , (7.91)
The system (7.76) with Eq. (7.91) is integrated to
a 2_ 2 | Pl | A
(7.92a)
cosh (2 | pi | At)
b 2 = b 2 exp [2A (p 2 - | Pl |) r] cosh (2 | Pl | Ar)
(7.92b)
c 2 = c 2 exp [2A (p 3 - | pi |) r] cosh (2 | Pl | Ar)
(7.92c)
whoro /)o and cq are integration constants.
Let us consider the solutions (7.92) in the limit r — > oo: towards the
singularity they simplify to
a ~ exp [-Apir] (7.93a)
6~exp[A(p 2 + 2p 1 )r] (7.93b)
c~exp[A(p 3 + 2pi)r] (7.93c)
^~exp[A(l + 2pi)r] (7.93d)
that is to say, in terms of t,
a~t p ' , b ~ i p ™ , c ~ t p - , a6c = A't , (7.94)
where the primed exponents are related to the un-primed ones by
/ _ I Pi I / _ 2 | pi | -p 2
" 1 - 2 | pi | ' Fm 1 - 2 | pi | '
_ P3 ~ 2 | pi |
(7.<).")a)
Fn -— ^- f , A , = (l-2|pi|)A, (7.95b)
1 - 2 | pi |
which, clearly, are the same as in the type II case (7.83). Summarizing
these results, we see the effect of the perturbation over the Kasner regime:
a Kasner epoch is replaced by another one so that the negative power of t
is transferred from the 1 to the m direction, i.e. if in the original solution
pi is negative, in the new solution p' m < 0. The previously increasing
perturbation Xfa 4 in Eq. (7.67) is damped and eventually vanishes. The
300 Primordial Cosmology
other terms involving A^ instead of A^ will grow, therefore permitting the
replacement of a Kasner epoch by another. Such rules of rotation in the
perturbing scheme can be summarized with the rules (7.85) of the BKL
map, with the greatest of the two positive powers still remaining positive.
The following interchanges are characterized by a sequence of bounces,
with an exchange of the negative power between the directions 1 and m con-
tinuing as long as the integer part of the initial value of u is not exhausted,
i.e. until u becomes less than unity. Hence, according to Eq. (7.57), the
value u < 1 is turned into u > 1, thus either the exponent pi or p m is nega-
tive and p n becomes the smallest of the two positive numbers, say p n = pi-
The next sequence of changes will switch the negative power between the
directions n and 1 or n and m. In terms of the parameter u, the map (7.83)
takes the form
- 1 for u > 2 ,
(7.96)
- for u < 2 .
- 1
The phenomenon of increasing and decreasing of the various terms with
transition from a Kasner era to another is repeated infinitely many times
up to the singularity.
Let us analyze the implications of the BKL map (7.96) and of the prop-
erty (7.57).
If we write u° = k° + x n as the initial value of the parameter u, being k°
and x° its integral and fractional part, the continuous exchange of shrinking
and enlarging directions (7.85) proceeds until u < 1, i.e. it lasts for
k° epochs, thus leading a Kasner era to an end. The new value of u is
u' = l/x a > 1, with the Kasner indices transforming as in Eq. (7.57), and
the subsequent set of exchanges will be I — > n or m — > n. For an arbitrary,
irrational initial value of u the changes in Eq. (7.85) repeat indefinitely. In
the case of an exact solution, the exponents pi, p m and p n lose their literal
meaning thus, in general, it has no sense to consider any exactly defined
value of u, such as for example a rational one.
The evolution of the model towards the singularity consists of successive
eras, in which distances along two axes oscillate and along the third axis
monotonically decrease while the volume always decreases (approximately)
linearly with the synchronous time t. The order in which the pairs of axes
are interchanged and the order in which eras of different lengths (number
of Kasner epochs contained in it) follow each other acquire a stochastic
character. Successive eras 'condense' towards the singularity. Such general
qualitative properties are not changed in the case of space filled with matter
according to the analysis of Sec. 7.2.1.
7.4.2 Stochastic properties and the Gaussian distribution
A decreasing sequence of values of the parameter u corresponds to e"
tb era there. This sequence, from the starting era has the form Mmax,
1, MmL - 2, ... , u^ in . We can introduce the notation
u W n = x {s) < 1 , u« x = fc (s) + x (s) , (7.98)
where Mmix is the greatest value of u for an assigned era and k^ = Umax
(square brackets denote the greatest integer less than or equal to <„).
The number k^ denotes the length of the s-th era, i.e. the number of
Kasner epochs contained in it. For the next era we obtain
If the sequence begins as fc (0) + x (0) , the lengths fc (1) , fc (2) , . . . are the
numbers appearing in the expansion for x^ ' in terms of the continuous
tract ion
fc( 3 ) + ...
which is finite if related to a rational number, but in general it is an infinite
For the infinite sequence of positive numbers u ordered as in Eq. (7.99)
and admitting the expansion (7.100) it is possible to note that
(i) a rational number would have a finite expansion;
(ii) a periodic expansion represents quadratic irrational numbers (i.e.
numbers which are roots of quadratic equations with integral coef-
(icioiits):
(hi) irrational numbers have an infinite expansion.
The first two cases correspond to sets of zero measure in the space of
e initial conditions.
302 Primordial Cosmology
Since it can be easily checked that the sequence of fc-values in the con-
tinuous fraction expansion x° is extremely unstable with respect to the
initial conditions, and taking into account the approximate nature of the
piecewise Kasner representation of this oscillatory regime, we are led to
address a statistical description.
An appropriate framework arises from studying the statistical distribu-
tion of the eras' sequence and from the analysis of the random properties of
the numbers x (0 ) over the interval (0, 1). For the series x^ with increasing
s there exists a limiting, stationary distribution w(x), independent of s, in
which the initial conditions are completely forgotten. In fact, instead of
a well-defined initial value as in Eq. (7.97) with s = 0, let us consider a
probability distribution for x^ over the interval (0. 1), TFo(.r) for a;' ) = x.
Then also the numbers x^ are distributed with some probability law. Let
w s (x)dx be the probability that the last term in the s-th series x^ = x lies
in the interval dx. The last term of the s — 1 series must lie in the interval
between l/(k + 1) and 1/fc, in order for the length of the s-th series to be
k. The probability for the series to have length k is given by
W s (k)=J^
Ws-tWdx. (7.101)
For each pair of subsequent series, we get the recurrence formula relating
the distribution w s +i(x) to w s (x)
^ +1 (^ = | Ws (^)|^|, (7.102)
' s+l{x) £?Ak + xf Ws \k + x)
(7.103)
If, for increasing n, the w s+n distribution (7.103) tends to a stationary c
independent of s, then w(x) has to satisfy
t^iik + xf \ k + x
A normalized solution to Eq. (7.104) is given by
This can be easily verified by a direct substitution of Eq. (7.105) in
Eq. (7.104), giving the identity
= T(— - "i (7-106)
f-~> \k + x k + x + lj
Substituting Eq. (7.105) into Eq. (7.101), we get the corresponding station-
ary distribution of the lengths of the series k
Finally, since in the stationary limit k and x are not independent (i.e.
x -H- l/(k + x)), they must admit a stationary joint probability distribution
v ' ' (k + x)(k + x + l)ln2 v ' '
which, for u = k + x, rewrites as
i.e. a stationary distribution for the parameter u.
The analysis of these chaotic properties of the map (7.96) was firstly
pursued in the work of I Lin kii I I J tniki ad L ifshitz at the end of the
'60s. Let us finally summarize the fundamental properties exhibited by the
Poincare return map associated to the fractional part x of the parameter
which can be easily derived by Eq. (7.96):
• it has positive metric- and topologic-entropy;
• it has the weak Bernoulli properties (i.e. the map cannot be finitely
approximated) ;
• it is ergodic and strongly mixing.
304 Primordial Cosmology
It remains to be discussed separately the case u>l, the so-called small
oscillations regime, whose details will be discussed in Sec. 7.4.3.
In this phase, the Kasner exponents approach the values (0,0, 1) with
the limiting form
Pi*--, P2~~, P3 « 1 2- ( 7 - m )
The transition to the next era is governed by the fact that not all terms on
the right hand side of Eq. (7.87) arc negligible and two terms have to be
simultaneously retained; in such case, the transition is accompanied by a
long regime of small oscillations regarding two directions lasting until the
next era. Only after this period of evolution a new series of Kasner epochs
begins.
The probability tt associated to the set of all possible values of x^
which lead to a dynamical evolution towards this specific case can be easily
recognized to converge to a number tt < 1. If the initial value of x^
is outside such set, the specific case cannot occur; if xS ' 1 lies within this
interval, a characteristic evolution as small oscillations takes place, but
after this period the model begins to regularly evolve with a new initial
value x(°\ which can only accidentally fall again in such an interval (with
probability tt) . The repetition of this situation can lead to these cases only
with probabilities tt, tt 2 , . . ., which asymptotically approach zero. Anyway,
the dynamics associated to this particular behavior will be discussed in the
next Subsection.
7.4.3 Small oscillations
Let us investigate in more detail a particular case of the solution constructed
above. We analyze an era during which two of the three functions a, b, c
(for example a and b) oscillate so that their absolute values remain close to
each other and the third function (in such case c) monotonically decreases,
being negligible with respect to a and b. As before, we will discuss only
the Bianchi IX model, since for the Bianchi VIII case the arguments and
results are qualitatively the same.
Let us consider the equations obtained from Eq. (7.71) and Eq. (7.73)
imposing exp(7) -c exp(a), exp(/3)
a TT + p rT = 0, (7.112a)
a TT - p TT = e Af> - e 4a , (7.112b)
f/3 T ) = -a T /3 T + -(e 2 «-e^) 2 . (7.112c)
The solution of Eq. (7.112a) is
2r7 2
a + /3=-%-T ) + 21n(a ), (7.113)
So
where ao and £o are positive constants. In what follows we conveniently
replace the lime coordinate r with the new one £ defined as
£ = e exp^(r-r )) (7.114)
in terms of which Eqs. (7.112b) and (7.112c) rewrite as
X«+p + ^sirm(2 X ) = 0, (7. 115a)
7« = --£ + § (2x1 + cosh(2 X ) - 1) , (7.115b)
where we have introduced the notation x = a — fl and ( )j = d( )/d£. Since
r is defined in the interval (—00, To], from Eq. (7.114) we have £ G (0, £o]-
Provided that a general analytic solution for the system (7.115) is not
available, we shall consider the two limiting cases £ 3> 1 and £ <C 1 only.
Let us start with the £ 3> 1 region. In this approximation, the solution
of Eq. (7.115a) reads as
x = M s in(£-£ ), (7.116)
A being a constant and therefore leading to 7 ~ A 2 (£ — £0). As we can
see, the name "small oscillations" arises from the behavior of the function
X- The functions a and b, i.e. the expressions of the scale factors, are
straightforwardly obtained as
a, b = aoJj- (l ± "4 sin(£ " Co)) , (7.117a)
= Co ex P [-A 2 (£ -£)].
The synchronous time coordinate t can be obtained from the relation dt =
abcdr as
t = t exp[-^ 2 (£ -0] • (7.H8)
Of course, these solutions only apply when the condition Co -C ao is satisfied.
Let us discuss the region where £ <C 1. In such a limit, the function x
reads as
X = Kln£ + d, 9 = const., (7.119)
306 Primordial Cosmology
where K is a constant which, for consistency, is constrained in the interval
K G (—1,1). We can therefore derive all the other related quantities, and
in particular
a ^(l + K)/2 ; 6 ^(l_A-)/ 2) (712Qa)
c ^ r (l-^)/4, t „ £(3+tf°)/2 _ (712Qb)
This is again a Kasner solution, with the negative power of t corresponding
to the function c and the evolution is the same as the general one. Moreover,
we can easily note how, for such a Kasner epoch, the parameter u becomes
for K > o : u= i±_| , for K < : u = \^ . (7.121)
Summarizing, the system initially crosses a long time interval during which
the functions a and b satisfy (a—b)/a < l/£ and performs small oscillations
of constant period A£ = 2ir, while the function c decreases with t as c =
cot /t . When £ ~ 0(1), Eqs. (7.117a) and (7.117b) cease to be valid,
thus after this period the function c starts increasing. At the end, when
the condition c 2 /(ab) 2 ~ t~ 2 is realized, a new period of oscillations starts
(Kasner epochs) and the natural evolution of the system is restored.
Let us now derive a correlation between the two sets of constants (K, 9)
and (A, £o) allowing also to relate the initial state of the system to the
final one. Let us firstly relate A and £o to the initial conditions (ito,Oo)
that refer to the value of the parameter u in correspondence to the Kasner
epoch just before the small oscillations, and to the value that the functions
a and b have at the end of that epoch. Imposing the continuity of the time
derivatives of the functions (a + (1) and (a — /?), for t = t (t = t ), and
because of the relation £ r = 2o^£,/(o. one can get the following conditions:
(« + /9)U T0 =P2 + pi= 1 , 1 —2 = ^r ( 7 - 122a )
(« " /?)Ur„ =P2-Pi= , | + 2 "° 2 = 4^4^= • (7.122b)
Equations (7.122a)-(7.122b) provide the relations we are looking for
£o = 2og(l+uo + u§) (7.123)
'- 1 + 2U ° l l , K»D. (7.124)
(we note here that in such a scheme the interchange between the indices p\
and p2 with respect to the functions a and j3 has the only effect of changing
A^ -A).
The correlation between the set of constants (K. 0) and (A£ ) can be
obtained noting that, for x *C 1, Eq. (7.115a) becomes
by replacing the hyperbolic sinus term with its argument. The general
solution to this equation is
X = ciJo(£) + c 2 N (0 (7.126)
where Jo and No denote the Bessel and the Neumann functions to zeroth
order, respectively. A solution to Eq. (7.125) admits the two asymptotic
expressions
for £ » 1 : X = -^ci cos (£ - tt/4) + c 2 sin (£ - tt/4) + 0(1/0 (7.127)
forf«l: x= -c 2 ln^+-c 2 ln(l/2 + c) + Ci. (7.128)
The comparison of Eqs. (7.127) and (7.128) with Eqs. (7.116) and (7.119),
provides the following identifications
c 2 = y^L(cos£o + sin£ ) (7.129a)
Cl = y/^A (cos Co - sin Co) (7.129b)
X=-c 2 = ^L(cos£o + sin£o) (7.129c)
7T y/TT
9= -c 2 (lnl/2 + c) + ci. (7.129d)
Finally, by means of Eq. (7.123) and Eq. (7.124), K can be obtained as a
function of u and a as
K _ 1 1 + 2^0
V2TOo(l +Uo+u 2)l/2
x {cos [2a§ (1 + Uo + u§)] + sin [2al (l + w + «g)] } . (7.130)
The substitution of this expression in Eq. (7.121) yields the new value of
U\ as a function of the initial conditions: u\ = u\{u$, clq).
Rigorously speaking, Eq. (7.130) is valid only for small values of the
function x, i-e. its validity is limited to the region K <C 1. The interest in
that relation, however, relies on the peculiar initial conditions required to
realize such a situation (K <C 1). Since K, as provided by Eq. (7.130), is
in general not close to zero for generic (u ,ao), we can conclude that the
existence of a long era uq 3> 1 does not imply that the successive evolution
308 Primordial Cosmology
has a similar behavior, i.e. ui > 1. In other words, the system is expected
to escape small oscillations to recover its natural evolution associated to a
finite value of u.
Although in the Bianchi type VIII case the derivation presented here
is valid in its guidelines, in such model we have to distinguish between the
two cases for the monotonically decreasing function corresponding or not
to the negative constant v = — 1 during the small oscillations phase. If it
does, the description is exactly the same while, if it does not, the system
(7.115b) slightly changes, but with marginal effects on the whole evolution.
7.5 Dynamical Systems Approach
We have discussed the properties and the dynamics of the homogeneous
models by means of the Einstein field equations following the path of the
Landau school. We will now discuss the framework known as Dynamical
Systems Approach, which is based on the fact that the Einstein equations
for a spatially homogeneous model can be written as an autonomous system
of first order differential equations in the time variable only. The evolution
curves of an autonomous system partition R" into orbits, so to obtain a
dynamical system on M n . Such reduction allows one to adopt the standard
techniques of this field so that, for example, the asymptotic behavior as t — >
±oo can be described in terms of asymptotically stable equilibrium points
(sinks), asymptotically stable periodic orbits, or more general attractors of
higher dimensions. A relevant feature of this approach is that it allows a
description in terms of dimensionless variables. This can be done by means
of a conformal transformation of the metric whose conformal factor brings
the unique dimensional unit (which can be chosen to be a length by setting
k = 1), as detailed at the end of Sec. 7.5.1.
We have seen in Sec. 7.1 how homogeneous spaces can be classified
accordingly to a scheme firstly given by Bianchi. This formulation is based
on the introduction of a group invariant time-independent frame, where the
time dependence is all encoded in the matrix r) a b(t) as in Eq. (7.17). At the
same time, it is possible to give a different but equivalent classification by
introducing a group invariant orthonormal frame such that
h a p=S ab e a a (t,x)e b p(t,x), S ab = diag(l, 1, 1) . (7.131)
This classification studies all the inequivalent forms that the Ricci coeffi-
cients jijk (see Eq. (2.111a)) and their linear combinations Xjjk (as in
Eq. (2.111b)) can take. Let us briefly sketch this picture.
We take an ortho-normal tetrad cj = (:\D,, with the time-like vector £q
coincident with the normalized four-velocity u 1
Given any i'uuctio
commutators
/, we have that the following expressions hold for the
[ei,ej]f =
jexf ■
(7.133)
Because of Eq. (7.132), we have that both the vorticity Uy (see Sec. 2.7.2 for
definitions and notation; notice that we adopt the signature (—,+,+,+))
and the acceleration vector u^VjU 1 vanish. This implies that Aq q = A° fc = 0,
and the remaining non-zero components can be decomposed as
-e ab
ar .
o'
(7.134)
where uj is the tetradic projection of the four- velocity Ui. The functions
X a bc can be decomposed by virtue of a symmetric matrix n ab and a vector
a a (in analogy with Eq. (7.33))
A Q fc c = (-bed
) c a b - d b a,
(7.13."))
The quantities 6 a b,n ab ,a c ,tt a completely determine the coefficients \ijk-
Now, if one repeats the same calculations developed in Sec. 7.1, then
it is led to a similar classification not dealing anymore with constants, but
with functions of the time variable; the results are summarized in Table 7.2
in analogy to Table 7.1.
Table 7.2 Inequivalent structure functions corresponding t<
The signs stand for the positive (+) or negative (-) charactei
Type
a
ni
n 2
n 3
I
II
+
VII
+
+
VI
+
IX
+
+
+
VIII
+
+
V
+
IV
+
+
Vila
+
+
+
VI a
+
+
310 Primordial Cosmology
7.5.1 Equations for orthogonal Bianchi class A models
We will consider only orthogonal models, i.e. models in which the vector u l
is parallel to the vector n % normal to the spatial hypersurfaces, and class A
models, i.e. those in which a c = 0, 0, 0.
The quantities n ab {t) determine the Bianchi type of the isometry group,
and the curvature of the group orbits t = const. This curvature can be
described by the trace-free Ricci tensor 3 S ab and the Ricci scalar 3 R of the
metric induced on the group orbits. It can be shown that if
b ab = 2n c a n cb - {n d d )n ab , (7.136)
then
3 S ab = b ab -^(b c c )S ab (7.137)
^ = -1^^ (7-138)
Using Eq. (2.112b) together with the decomposition (7.134) and (7.135),
the 00- and the a&-components of the Einstein equations in the presence of
a perfect fluid tensor yield the Raychaudhun equation for the expansion 8
and an equation for the shear as
9 = --6 2 -2a 2 - -(p + 3P) , (7.139a)
a ab = -Ba ab - 3 S ab , (7.139b)
while the 0a components yield an algebraic constraint for the shear com-
poneul.s
e abc n cd <j b d = 0. (7.139c)
Thus, we obtain the first integral from the trace of the Einstein equations
expressed as
From the Jacobi identities applied to the vierbein vectors ej, in analogy
with Eq. (7.29), we get the evolution equation for n ab as
is possible to show that a ab and n ab can be diagonal) zed simultaneously
ith a rotation, so that one can study only the eigenfunction components
2 and n a .
Homogeneous Universes 311
Equations (7.139) form a six-dimensional autonomous system of differ-
ential equations for the variables y t = (6, a a , n a ) (from the definition of
vorticity one has Tr(cr a b) = 0), of the form
y i = F i (y j ), i,j = l,...,6. (7.141)
The fundamental observation which is at the ground of such a formu-
lation is that, since 3 S a b and p are quadratic expressions of the variables
j/i, the functions Ft are homogeneous of degree 2, which implies that the
system is invariant under a scale transformation
Yi = Lyi, ^ = L, (7.142)
where L is a length of reference and r a new time variable. This allows
us to introduce dimensionless variables, thereby reducing the dimension of
the system by one unit. Another reason for introducing such dimensionless
variables is the fact that the variables yi do not enable one to distinguish
different asymptotic states, since at the singularity these variables typically
diverge, while at late times in ever-expanding models they tend to zero. A
relevant physical scale is given by the Hubble function H, defined as
H = 6/3, (7.143)
so that it is natural to introduce the dimensionless shear E tt , the dimen-
sionless spatial curvature variables N a and the dimensionless density 5 Q
(X a ,N a ) = (a a ,n a )/H, n = p/(3H 2 ) (7.144)
together with the new time variable r
%-U. (7.145)
It is worth noting how the parametrization (7.144) correctly describes the
dynamics for all the type A models. In the case of the type IX, this is true
only for the expanding phase, i.e. when H > 0. Finally we introduce TZ,
proportional to the Ricci scalar
H = ~ 3 R. (7.146)
We will now show how the Einstein field equations can be recast in a set
of dimensionless system of ordinary equations coupled to a single equation
that brings and describes the evolution of the typical length of the system.
5 In the first works, the standard length adopted was the expansion 9 instead of the
Hubble length H.
312 Primordial Cosmology
Let us assume that the equation of state of matter is given by Eq. (2.15)
with 7 > 1, and write down the full system of equations as
¥L = -(l + q )H. (7.147a)
a - = - (2 - q) S a - S a , no sum over a (7.147b)
(h
dNg
(It
(7.147c)
g =i(£2 + £2 + Vj 2) + l(p + 3P) = 2£ 2 + i(p + 3P) _ (7147d)
From the definition of the shear (2.146), we have the constraint
^ + £2 + £ 3 = 0. (7.147e)
The first integral (7.139d), because of Eq. (7.146), becomes
ft = l-£ 2 -ft, (7.147f)
also addressed as the (iouss constraint. The last equation to be added to
this set is the evolution equation for the density 57, obtained from the four-
divergence of the energy-momentum tensor of the perfect fluid which reads
— = (2q-l)n-3P= [2q - (37 - 5)] fi . (7.147g)
From this last equation, together with Eqs. (7.144) and (7.147a), we can
conclude that for all orthogonal perfect fluid models with a linear equation
of state, the energy density diverges toward the singularity (which is at
r — > —00). By integrating the equations for the Hubble parameter and for
the energy density, we obtain that
Oocexpl / (2 g - 3 7 + 5) dr 1 j
r , \\ => P = 3P 2 0«exp(-3( 7 -l)T) .
H<xexp\-J (l + g)dr|J
(7.148)
7.5.2 The Bianchi I model and the Kasner circle
Let us firstly consider the vacuum case. For the Bianchi I model, we have
that N\ = N 2 = N 3 = 0, implying that 1Z = S a = 0. For the vacuum case,
Homogeneous Universes 313
fl = and the system (7.147) reduces to
^- = -(l + q)H (7.149a)
^ = -(2- 3 )E a no sum over a (7.149b)
q =2E 2 (7.149c)
E 2 = l. (7.149d)
This set of equations can be easily integrated as
q = 2, E a = const, H = #oe _3r (7.150)
while the time variable reads explicitly from Eq. (7.145)
1 + to = FTT ex P( T ) ' ( 7 - 151 )
O-Ho
where i is an integration constant.
Since Eq. (7.149d) holds, we can define the so-called Kasner circle K°
that well represents the Bianchi I vacuum subset gj acuum , and is sketched
in Fig. 7.2.
Furthermore, from Eqs. (7.147e) and (7.149d), we can recover the stan-
dard Kasner relation (7.51) by setting
(Ei, E 2 , E 3 ) = (3pi - 1, 3 P2 - 1, 3p 3 - 1) , (7.152)
thus implying that each point on K° represents a Kasner solution. Be-
cause of the permutational symmetries of such a representation, K° can
be divided into six sectors, labeled by a triplet (abc) that represents a per-
mutation of the fundamental triplet (123). This way, for each point in a
particular set, we have that p a < Pb < Pe-
lf we consider the u parametrization of the Kasner exponents (7.55),
each point on a sector is represented by a unique value of u. The bound-
aries of the sectors are six points associated with solutions that are locally
rotationally symmetric:
• Q a are characterized by (E a , E;,, E c ) = (2, — 1, —1) or, equiva-
lently, (p a , p b , p c ) = (-1/3, 2/3, 2/3). They all correspond to the
value u = 1.
• T a are the Taub points given by (E a , Ef,, E c ) = (—2, 1, 1) or, equiv-
alently, (p a , pb, p c ) = (1, 0, 0); they provide the Taub representa-
tion of Minkowski spacclimc. givci) \>\ u = x,.
sr 29, 2010 11:22
World Scientific Book - 9in x 6in
Primordial Cosmology
Figure 7.2 Representation of the Kasner circle K°. The circle can be divided in six
equivalent sectors that correspond to different choices in the ordering the Kasner expo-
nents p a \ each point on the circle represents a different Kasner solution.
In the case of a perfect fluid with a linear equation of state, we have the
Homogeneous Universes
:-!!:>
following set of equations
£--<! + „*.
(7.153a)
£--(»-.)*..
(7.153b)
g = 2S 2 + i(3 7 -5)^,
(7.153c)
l-ft = E 2 < 1,
(7.153d)
^ = (2<Z + 5-3 7 )S7.
(7.153e)
This set can be integrated for a generic linear equation of state. If we take
{n(r ) = £1 , S a (r ) = S a0 , t = 0} as initial conditions, we have that
s (r) = S a0V /6(l-O )
^(fi -l) + fi exp[3T(2-w)] '
From Eqs. (7.154) one has that
that, together with Eq. (7.153d), implies that the solution can be repre-
sented in the E a plane as a straight line from K° ending, for r — > +00, in
the center of the circle (the so-called Friedmann fixed point).
7.5.3 The Bianchi II model in vacuum
In the case of the type II model, only one of the three N a is different from
zero; we can take Ni = N 2 = 0, N 3 > 0. Furthermore, the Gauss constraint
(7.147f) can be used to eliminate N 3 . This way, the system (7.147) reduces
to
^^ = (q - 2) E o(6) + 4 (1 - E 2 ) (7.156a)
Y 1 = (q - 2) E c - 8 (l - E 2 ) . (7.156b)
In the vacuum case fl = there are no fixed points, while the boundary
of this vacuum subset coincides with the Kasner circle K°. The solutions
can be represented as straight lines that connect one point on K° to a
i'rniioi-iiiid Cosmology
Figure 7.3 Plot of the solution (7.154a) for a given set of initial conditions
(r = 0, Q Q = 0.5, Em = 0.751783, S 20 = -0.651783, S 30 = -0.1). It can be seen how
the dimensionless energy density Q evolves from at the singularity to 1 as r grows.
different one, as sketched in Fig. 7.4. This is the standard result we have
already discussed in the BKL approach in the previous sections. An easy
check is the analysis of the evolution of the "reduced" variables E± defined
= E + + V3E_
= E+ - V3E_
= -2E+.
(7.157a)
(7.157b)
(7.157c)
From these definitions, it follows that the constraint (7.147e) is automati-
cally satisfied and E 2 = E 2 . + El. Then, equations (7.156a)-(7.156b) read
E' + = 2 (E 2 - 1) (E+ - 2) , (7.158a)
E'_ = 2(E 2 - l)E_, (7.158b)
which can be implicitly solved and represented in the E-plane as
£+ = A(£_+2), (7.159)
where A is a real parameter that characterizes the single orbit. The solution
(7.159) draws a star of straight lines originating from (E + , E_) = (2, 0) or,
Homogeneous Universes 317
equivalents, (Ei,E 2 ,E 3 ) = (2,2,-4). This is the point M x in Fig. 7.4.
Then the orbit of Bianchi II vacuum model is just a chord starting from a
point on K° and ending in another point on K°. A direct calculation yields
the standard BKL map for the p„ indices.
This solution is also addressed as the Bianchi type II vacuum subset
Si
Figure 7.4 In the figure above, the type II transitions are sketched in the Ei, £2,1)3
plane. A generic transition connects a point on I lie Kasncr circle K° with a different
point on the same circle, and the arrow denote the evolution toward the singularity. The
n focal point M x of these straight lines is at (E, , £ 2 , £3) = (-4, 2, 2).
7.5.4 The Bianchi IX model and the Mixmaster attractor
theorem
The type IX case has A/i > 0, A^ > 0, N 3 > and the corresponding
system of equations results to be given by the full set (7.147). From the
analysis of the type II model we can obtain an equivalent description as
318 Primordial Cosmology
the one given in Sec. 7.4 where we gave a piecewise representation for the
dynamics of the Bianchi IX model. In the dynamical systems framework,
the same approximation corresponds to maintaining in Eqs. (7.147) only
one of the functions N a , as we did there with the scale factors a, b, c. This
way, the piecewise solution can be represented as in Fig. 7.5. We stress
that, in the £ plane, a Kasner epoch corresponds to a point on K°, while
a transition to a chord connecting two points on K°. An era is identified
with the oscillations among two of the six subsets in which the Kasner circle
is partitioned (in Fig. 7.5, an example of era is given by the sequence of
transitions between the points belonging to the circle 1 sectors Qi — T 3 and
n - q 2 ).
It is worth noting that, when dealing with the Bianchi IX n
easy to reconstruct the metric starting by the only functions r
jdel, it is
,. In the
orthonormal frame, the relation between the diagonal terms of r) a b(t) and
n a is given by
n a {t) = -^- , (no summation over a) . (7.160)
For a generic homogeneous case, a specific algebraic technique to recon-
struct the metric exists. As soon as one of the n a vanishes, this technique
envisages the use of other frame variables like H or a a .
There is an important theorem whose proof was given by Ringstronj
(2001), and it can be stated as follows 6
Theorem 7.1. Let (Si, S 2 , S3, Ni, N 2 , N 3 ) be a generic solution of the type
IX model. Then
lim N X N 2 + N 2 N 3 + iV 3 iVi = (7.161a)
lim n = 0. (7.161b)
This theorem is the only exact mathematical result about the dynamics of
the Bianchi type IX model; however, this result does not completely solve
the main question, whether the exact Mixmaster dynamics is chaotic or
not. Indeed, chaos is associated with the Kasner map that is valid only in
the piecewise approximation, and it is commonly believed that this map
reliably represents the exact dynamics.
This theorem can be restated in a different way if we define the Mix-
master attractor Aix as follows
Definition 7.3. The Mixmaster attractor Aix is the set given by the union
of the Bianchi type I and type II subsets. Since the Bianchi type II consists
of three equivalent representations it can be written as
Aix = BjvacuumUBjjvacuum = K JBvacuuui \ ■ UBvacuumjv 2 Ui3vacuum7v 3 .
(7.162)
Then, an equivalent formulation can be the following
Theorem 7.2. Let Y(t) = (S a (r), N a (r)) be a generic solution of the
Bianchi type IX model. Then
lim \Y(t)-Aix\= lim mtn ZeAlx \Y(t) - Z\ = . (7.163)
6 Indeed, different proofs of this theorem exist, approaching the statement from different
points of view.
320 Primordial Cosmology
This theorem definitely states that the attractor of the type IX model
belongs to Aix, but it does not tell if they coincide or it is only a subset,
being still an open issue. Furthermore, the Mixmaster attractor theorem
does not provide any information about the detailed asymptotic evolution.
For a complete discussion of the several implications we recommend the
interested reader to analyze the wide literature on the topic.
7.6 Multidimensional Homogeneous Universes
The question of chaos in higher dimensional cosmologies has been widely
investigated over the last 20 years. In the case of diagonal models (in
the canonical basis), many authors showed that none of higher-dimensional
extensions of the Bianchi IX type possesses proper chaotic features: the
crucial difference is given by the finite number of oscillations characterizing
the dynamics near the singularity. Chaos, however, is restored (up to 9
spatial dimensions) as soon as different symmetry groups are considered,
such as the non-diagonal models. In this Section, we will follow the analysis
proposed by Halpern (1985) of the diagonal, homogeneous models with four
spatial dimensions, and conclude with some remarks on the non-diagonal
case.
Indeed, the work of Fee in 1979 classifies the four-dimensional homoge-
neous spaces into 15 types, named GO — G14, and is based on the analysis
of the corresponding Lie groups.
The line element can be written using the Cartan basis of left- invariant
forms and explicitly reads as (N = 1)
ds 2 = dt 2 - % s (t)uj r ®w s . (7.164)
The 1-forms ui r obey the relation dw r = —C r pq uj p A to q , where the C r pq
are the four-dimensional structure constants. We limit our attention to the
case of a diagonal matrix 4 ij rs
Vs =diag(a 2 ,6 2 ,c 2 ,rf 2 ) . (7.165)
The Einstein equations are obtained from the metric (7.165) by the stan-
sr 29, 2010 11:22
Homogeneous
Univer.^
dard procedure
outlined
in Sec. 7.3 a
' r ''I
a b
= a + b +
i d
- + d
=
R\
(abed)'
abed
4 R\ =
o,
Rl
_ (abed)
abed
4 R\ =
o,
R\
(abed)
abed
4 R\ =
o,
Rj
(abed)
abed
i R i i =
o,
Rl={^-J L )c m mn = ^
(7.166a)
(7.166b)
(7.166c)
(7.166d)
(7.166e)
(7.166f)
= In a
-- In b ,
= In c
--hid,
where x n (n = 1, 2, 3, 4) denote the scale factors a, b, c, d, respectively, and
the 4 R b a are the tetradic components of the spatial four-dimensional Ricci
tensor 4 R? defined as in Eq. (7.40).
Equations (7.166) can be restated by the use of the logarithmic variables
(7.167a)
(7.167b)
(7.168a)
and the logarithmic time r, i.
We thus obtain the system
a TT = - (abed) 2 4 R 1 1 , (3 TT = - (abed) 2 4 R 2 2
1tt = - (abed) 2 4 R\ , S TT = - {abed) 2 4 R 4 4
= 2a T /3 T + 2a T ~/ T + 2a T 5 T + 2/3 T ~/ T + 2/3 T 5 T + 2~/ T 5 T , (7.168b)
R n =0.
(T.KiSc)
The dynamical scheme (7.168) is valid for any of the 15 models using the
corresponding 4 R a b -
The simplest group to be considered is GO. For such model the functions
4 R a b are all equal to zero and the solution simply generalizes the Kasner
dynamics to four spatial dimensions, which corresponds to the line element
1 below in Eq. (7.170) when considering a Kasner epoch as a phase
322 Primordial Cosmology
of evolution in the G13 model.
Among the five-dimensional homogeneous space-times, G13 is the anal-
ogous of the Bianchi type IX, having the same set of structure constants.
The Einstein equations can be written as
2a TT = [{b 2 - c 2 ) 2 - a 4 ] d 2 , (7.169a)
2f3 TT = [{a 2 - c 2 ) 2 - 6 4 ] d 2 , (7.169b)
2 7tt = [(b 2 - a 2 ) 2 - c 4 ] d 2 , (7.169c)
5 TT = , (7.169d)
together with Eq. (7.168b). If we assume that the BKL approximation is
valid, i.e. that the right-hand sides of equations (7.169) are negligible, then
the asymptotic solution for r — > — oo is the five-dimensional and Kasner-like
line element
ds 2 = dt 2 -Y j t 2p r(dx r ) 2 , (7.170)
r=\
with the Kasner exponents p r satisfying the generalized Ivasncr relations
X> r = X> 2 = l. (7.171)
r=l r=l
This regime can only hold until the BKL approximation is satisfied. How-
ever, as soon as r approaches the singularity, one or more of the terms may
increase. Let us assume p\ as the smallest index; then a = exp(a) is the
largest contribution and we can neglect all other terms, thus obtaining
a TT = — exp(4a + 25) ,
/3 TT = 7tt = - exp(4o ! + 25) , (7.172)
5 TT = .
When considering the asymptotic limits for r — > ±oo, from the solution of
(7.172) we obtain the map
P1+P4 , _ P2+ 2pi + Pi
Pi
- l l + 2p!+p 4 '
1,2 1 + 2pi +
, p 3 + 2pi + pi
, Pi
P6 l + 2 Pl +pi '
? ^ 1 + 2 P1 +;
abed = A't , A'
= (1 + 2pi +p 4 )A
P4
(7.173)
(7.174)
Homogeneous Universes 323
In the new Kasner epoch, the leading terms on the right-hand sides of
(7.169) evolve according to
a 4 d 2 „ t 2(2pi+pi) (7.175a)
6 4 d 2 „ t 2(2rf+pi) (7.175b)
c 4 d 2 „ t 2(2pi+ P i) _ ( 7 1?5c)
From Eq. (7.175a), it follows that a new transition can occur only if one
of the three exponents is negative. Nevertheless this is generally not true,
because there exists a region where the other exponents are greater than
zero. After eliminating p' 4 and p' 3 from Eq. (7.175a) with Eq. (7.171), this
region satisfies the following inequalities
3pl + P 2 2 + Pi - P2 - P1P2 >0, (7.176a)
3pj+pl+P2-pi-piP2>0, (7.176b)
3pj +pj- 5pi - 5p 2 + 5pip 2 + 2 > , (7.176c)
plus a reality condition for p$
1 - 2>p\ - 2>p\ - 2 PlP2 + 2 Pl + 2p 2 >0. (7.177)
The region defined by the validity of (7.176) and (7.177) is plotted in
Fig. 7.6. Thus the Universe undergoes a certain number of transitions
of Kasner epochs and eras; as soon as the Kasner indices pi,P2 assume
values in the shaded region, then no more transitions can take place and
the evolution remains Kasner-like until the singular point is reached.
The type G14 case is quite similar to G13: for this model, the structure
constants are the same as Bianchi type VIII and, under the same hypothe-
ses, only a finite sequence of epochs occurs.
This analysis can be repeated for any of the diagonal homogeneous 4+1-
dimensional models manifesting the same behavior: the absence of chaos in
the asymptotic regime toward the singularity. Furthermore, these results
hold even in highe
7.6.1 On the non- diagonal cases
The full BKL-like dynamics can be recovered up to 10-dimensional space-
times as soon as the assumption of diagonal metric (in the canonical basis)
is relaxed. The main difference with respect to the diagonal case is that
now the "Kasner axes" cu^ do not coincide with the time-independent 1-
forms uj b of the space, but are linear and time dependent combinations of
them as
ujl = A a b (t)uj b . (7.178)
i'rniioi-iiiid Cosmology
P2\
/ 0.8
0.6
0.4
0.2
Pl 1
-0.4\
^-0.2./
\-0.2
-0.4
0.2
0.4
0.6
0.8 1
Figure 7.6 The shaded region c
Eq. (7.176b): as soon as a set (pi
breaks down and the Universe ex
•responds to all of the couples (pi. /m) not satisfying
P2) takes values in that portion, the BKL mechanism
eriences the last Kasner epoch till the singular point.
In the basis of the Kasner vectors, although the time dependei
spatial-geometry is non-diagonal, the d-dimensional metric d r) a b c
kept as diagonal
V6 = diag(af(t),ai(t),...,aS(t))
:e of the
a still be
(7.179)
and the same analysis developed in Sees. 7.2 and 7.3 for the Bianchi models
I, II and IX can be generalized to obtain similar results. In particular,
when we neglect the Ricci scalar in the Einstein equations, so recovering
a generalized Kasner- like solution, we obtain also that the functions A%(t)
are constant during each epoch. This means that we need to specify d(d +
Homogeneous Universes 325
1) — 2 integration constants, though remaining with still enough arbitrary
constants to fit assigned initial data. The key difference with the previous
analysis is just in the introduction of the functions A^ and their constant
behavior during the epochs: in the diagonal case, these coincide with the
Kronecker delta 5% and the system loses d 2 — d arbitrary functions. This
restriction does not allow for C l j k ^ for general i,j,k the reason why
diagonal models are not chaotic. The introduction of the A% functions
is equivalent to a rotation of the triad vectors, and when they behave as
constants, their net effect is to generate new non-zero structure constants
of the group as linear combination of the original ones.
It has been explicitly shown that some homogeneous (i-dimensional mod-
els (up to d = 9) possess C l jk ^ in a generic non-canonical basis. This
way chaos is still present in higher dimensional homogenous models
We refer to Sec. 9.7 for a more general and detailed analysis of the per-
turbation terms to a Kasner regime in the inhomogeneous multidimensional
case.
7.7 Guidelines to the Literature
For the analysis of the homogeneous spaces presented in Sec. 7.1 we refer
to the textbooks by Ryan & Shepley [406] and by Stephani et al. [427]. For
formal aspects concerning the geometrical objects introduced here, see the
books of Wald [456] , while a good text on group theory is for example that
by Zhong-Qi Ma [330]. The original derivation of the Bianchi classification
appeared in [87].
In particular, for the application to Cosmology, for the Bianchi classifi-
cation and the corresponding properties we refer to Landau & Lifshitz [301].
The demonstration of the re-collapsing behavior of the type IX model, both
in vacuum and in presence of a perfect fluid with 7 > 1, is given in [318,319].
The derivation of the Kasner solution given in Sec. 7.2 can be found in
the original paper [267]. However, a satisfactory discussion is also provided
by the standard book [301].
For the dynamics of the Bianchi models considered in Sec. 7.3 we refer
to the review article [354] and references therein.
A valuable introduction to the Einstein equations under the homogene-
ity hypothesis is offered by the textbooks [301] and by Misner, Thorne &
Wheeler [347].
The study of the oscillatory regime pursued in Sec. 7.4 is properly ad-
326 Primordial Cosmology
dressed by the original work by Belinskii, Khalatnikov and Lifshitz [64]
and later reviewed in [65]. A more detailed presentation of the stochastic
properties associated to the BKL map is given in [316] and in [314].
The standard textbook on the topics of Dynamical Systems Approach,
discussed in Sec. 7.5 is the one edited by Wainwright & Ellis [454]. A
clear derivation of the Einstein field equations for a homogeneous model
in a group invariant, orthonormal frame can be found in [168,332] while
its application to cosmological settings were firstly reviewed in [125]. A
discussion on the dynamical properties of orthogonal Bianchi model of class
A can be found in [455]. The original demonstration of the Mixmaster
Attractor Theorem is in [393,394], while a different derivation is for example
in [234]. A comprehensive review on the implications of such a result on
the dynamics, as discussed in Sec. 7.5.4, can be found in [235,236].
We did not face the topic of Consistent Potential Method, firstly devel-
oped by Grubisic & Moncrief in [207] with the related topic of [261]. We
refer the reader interested in some applications to [80, 208] and for a wide
review to [77]. The research line that deals with the Painleve analysis of
the Mixmaster equations was firstly proposed in [133] and later developed
for example in [13, 126].
The first works on the homogeneous Mixmaster dynamics in higher di-
mensional cases, as discussed in Sec. 7.6, are [43,182,224] which underline
that chaos is suppressed, at least in diagonal model. The extension to the
chaotic, non-diagonal case, presented in Sec. 7.6.1, is discussed in [145].
Chapter 8
Hamiltonian Formulation of the
Mixmaster
In this Chapter we provide the Eamiltouiau formulation of the Mixmaster
dynamics, describing in detail how the infinite sequence of Kasner epochs
takes the suggestive form of a two-dimensional point particle performing
an infinite series of bounces within a potential well. After specializing the
Einstein-Hilbert action to the case of homogeneous models, we deal with
a three-dimensional system, whose generalized coordinates correspond to
the three independent logarithmic scale factors. Indeed all the dynamical
content is summarized in the time behavior of the three spatial directions,
while the spatial dependence of the three-geometries enters through the
structure constants only (any other space dependence is integrated out).
As far as we perform a Legendre transformation, we are naturally led to
introduce the so-called Misner variables, which allow to diagonalize the ki-
netic term in the Hamiltonian function. Once recognized how the isotropic
component of the metric (summarized by the Misner variable a) plays the
natural role of time for the configuration space of the anisotropies degrees
of freedom (namely the Misner variables /3±), we reduced the Mixmas-
ter model to the very intuitive picture of a bouncing particle within an
equilateral-shaped receding potential in the evolution toward the singular-
ity (a — > — oo). After introducing the Misner-Chitre like variables, we are
able to get a dynamical scheme in which the potential walls are fixed in
time and, asymptotically to the singularity, are modeled by an infinite po-
tential well. This representation of the Mixmaster model shows how it is
isomorphic in a generic time gauge to a well-known chaotic system, i.e. to
a billiard-ball in a two-dimensional Lobacevskij space.
We also address the Mixmaster model in the Misner-Chitre like variables
as viewed in the framework of statistical mechanics. The existence of an
energy-like constant of motion characterizes the corresponding chaos in
328 Primordial Cosmology
terms of a microcanonical ensemble. The stochastic properties of the system
are then summarized by the associated Liouville invariant measure.
The covariance of the Mixmaster chaos with respect to the time choice is
then discussed, comparing and contrasting different results, with particular
attention to the so-called fractal boundaries method.
Finally we characterize the Mixmaster dynamics when it is influenced
by a scalar field, an Abelian vector potential and a cosmological constant.
These studies allow to determine the cosmological implementation of this
model, especially in view of the inflationary paradigm (of which the massless
scalar field and the cosmological constant are a schematic representation),
as presented in Chap. 5.
8.1 Hamiltonian Formulation of the Dynamics
In order to face the Lagrangian analysis of the Mixmaster model, we re-
state for convenience the geometrical scheme associated to the homogeneity
constraint. In what follows, we restrict our attention to the type VIII and
IX models only, but this analysis is indeed valid for any class A model of
the Bianchi classification. The only difference at the kinematical level is in
the value of the structure constants.
Let us start by considering the line element for a generic homogeneous
space-time in the standard ADM form
ds 2 = N{tfdt 2 -h afi dx a dx 15 , (8.1)
where
h aP = e"'l a (x^)lp{x^ + e*»m a (x<)mf i (x<) + e**n a {x>) nfi {x>) , (8.2)
with q a (a = l,m,n) being functions of time only. The three linear inde-
pendent vectors 1, m and n, due to the homogeneity constraint satisfy the
conditions
l-Curll = A; (8.3a)
— mcurlm = A m (8.3b)
--n-curln = A„ , (8.3c)
(8.3d)
where v = 1 • m A n (as in Eq. (7.20)) and the three constants A a =
(A;,A m ,A„) correspond to the structure constants of the Bianchi classi-
fication (ni,n 2 ,n 3 ) given in Table 7.1. For the Mixmaster model they
(8.4)
Hamiltonian Formulation of the Mix-mast rr
explicitly read as
Type VIII: A; = 1, A m = 1, A„ = -1,
TypelX: A; = 1, A ro = 1, A„ = 1 .
The line element for the Bianchi space can also be expressed in terms of
the 1-forms by setting
h afs dx a dx f3 = r] ab co a uj b = e^6 ab uj a Lo b , (8.5)
so that the correspondence to 1, m and n is obtained from
lo 1 = — sinh ip sinh 8d(f> + cosh.ipd8
(VIII) u? = -cosh V> sinh Odcf) + sinh ipdO (8.6a)
to 3 = cosh 9doj + d^
u 1 = simpsmOdp + cosipdO
(IX) u? = -cosipsm6d<j) + sm^dO (8.6b)
lu 3 = cos 6d<j> + dip ,
where e [0,tt), (p € [0,2tt) and ip e [0,4tt) are the Euler angles. The
Einstein-Hilbert action in vacuum (2.11) can be integrated over the spatial
coordinates (involved through the 1-forms) which factorize out providing
the term 1
/ uj 1 A uj 2 A uj 3 = / sin 6d<f> Ad6 AdiP= (4tt) 2 . (8.7)
It is worth noting that this is just the surface of a three-sphere of radius
2: in fact, the closed RW model is a particular case of Bianchi IX, for
Qa = Qb = <7c (see Sec. 3.2.4). Thus, the vacuum dynamical evolution of the
Bianchi types VIII and IX models is summarized in terms of the variational
principle
SS B = 5 J' £ B {q a ,q b )dt = 0. (8.8)
Here t\ and t 2 denote two fixed instants of time (ti < t 2 ), while the La-
grangian jCb reads as
Cb=~ ^^ l^ (Mm + Mn + q m q n ) ~N 3 r] . (8.9)
lr The integration for the Bianchi type VIII is considered over a spatial volume (4n) 2 in
order to have the same integration constant used for the type IX and to keep a uniform
I.O] UKllisiU.
sr 29, 2010 11:22
330 Primordial Cosmology
A direct computation yields (a, b = l,m, n)
■q = det(7? afc ) = exp I ^ q a J .
(8.11)
From the Lagrangian formulation, the Hamiltonian for the Mixmastor
dynamics is obtained by performing a Legendre transformation, i.e. by
calculating the momenta p a conjugate to the generalized coordinates q a as
Pl dq t n N
_ dC _ 4tt 2 ^?
Pm ~ dq m ~ k N '
Pn ~ dq n n N '
and then taking the standard transformation
NHb = Yl Paqa ~ Cb
(q m + q n )
Hn + qi)
(ffl + Qm)
(8.12a)
(8.12b)
(8.12c)
(8.13)
where the q a are obtained from Eqs. (8.12). This way, we get the action
S B = f dt( Pa q a -NH B ) , (8.14)
Ew 2 - 5>
where Hb = is the scalar constraints for these models.
Let us introduce the "anisotropy parameters" , defined as
~£ 6 9 b '
5> = i-
The functions in Eq. (8.16) allow one to interpret the last term on the right-
hand side of Eq. (8.15) as a potential for the dynamics. It can be rewritten
in the form
^ R = -\ (£#> 29 "-£^W* +( H • ( 8 - 17 )
Hamiltonian Formulation of the Mixmaster 331
The main advantage of writing the potential as in Eq. (8.17), arises when
investigating its proprieties in the asymptotic behavior toward the cosmo-
logical singularity (77 — > 0). In fact, the second term in Eq. (8.17) becomes
negligible, while the value of the first one results to be strongly sensitive to
the sign of the Q a . Thus, the potential can be modeled by an infinite well
-V 3 R = T, e oc(Qa)
®oo(x) ■■
+00, if x <
0, if x > .
By Eq. (8.18) we see how the dynamics of the Universe resembles that of a
particle moving in the domain Hq, defined by the simultaneous positivity
of all the anisotropy parameters Q a .
The same Hamihouian (oi:uiu]aUon can also be obtained from the stan-
dard ADM description of GR, as in Chap. 2. In that case, the line element
(8.5) has to be inserted in Eq. (2.72a), where one defines the momenta
conjugate to the three-metric as
n af3 = Pa e- q "d ab e'Ze b , (8.20)
and a direct calculation yields Eq. (8.15). Furthermore, the condition of
space homogeneity implies that the super-momentum constraint T-L a = 0,
as in Eq. (2.72b), is identically satisfied.
8.2 The Mixmaster Model in the Misner Variables
In this Section, we state the Hamiltonian dynamics of the Mixmaster model
in terms of the so-called Misner-Chitre like variables, which diagonalize
the kinetic part of the Hamiltonian and provide the simple scheme for
the dynamics as corresponding to a two-dimensional point particle moving
within a closed potential domain. When expressed in terms of the Misner-
Chitre like variables the potential well becomes a stationary domain and we
can determine the chaotic properties of the system, which are approached
by means of a Jacobi metric representation of the geodesic flow. Indeed, in
the next Section we deal with a billiard-ball on a Lobacevskij plane, i.e. a
dynamically closed domain on a constantly negative curved surface.
332 Primordial Cosmology
8.2.1 Metric reparametrization
The main advantage arising from the variables q a when treating the ho-
mogeneous dynamics is an explicit description of the evolution in terms of
expanding and contracting axes. For the Kasner solution such variables
can be easily related to the Kasner exponents pi,p m ,p n as
q a (t) = 2p a \nt. (8.21)
Let us consider a set of variables that diagonalize the kinetic term
KT = Y J {Paf-\\T,p)j (8.22)
in order to have a description i:esemblm;>, that of a point-particle. The
quadratic form (8.22) has a negative determinant, i.e. it cannot be reduced
to the typical kinetic energy of Hamiltonian systems Yl Pa i since it is not
positive defined due to the presence of one term with a negative sign. Given
the matrix B
the canonical transformation
( Pa\ ( 1l\ ( OL
13+
P-
diagonalizes KT to the form
KT=^(-clpl + 4 P l + clp 2 _). (8.25)
Among all possible choices for ci,C2,c;j, the set (1. 1, 1) corresponds to
the standard form of the so-called Misner coordinates a, f3± defined as
{ qi = 2 (a + (3+ + V3/?_)
q 2 = 2(a + p + -Vip') (8-26)
q 3 = 2 (a - 2/3+) .
The peculiarity of the Misner variables can be outlined through the com-
plementary definition of r\ a b-, expressed as the factorization
Vab = e 2a {e 2(3 ) ab *+ (lnr;) ob = 2a5 ab + 2(3 ab . (8.27)
Hamiltonian Formulation of the Mixmaster 333
In Eq. (8.27), the exponential term containing a is related to the Universe
volume while j3 a b is a three-dimensional symmetric matrix with null trace
representing the Universe anisotropies which can accordingly be chosen as
The exponential matrix :
0n = 0+ + V^/3-
(8.28a)
022 = 0+ " VS/3-
(8.28b)
033 = -20+ .
(8.28c)
s defined as a power ser
ies of matrices, ;
so that
det (e 2 ^) = e 2 tr " = 1 ,
(8.29)
(8.30)
8.2.2 Kasner solution
In terms of the variables (a, 0+,0_) a
solution as
; get the relations for the Ka;
2^3
From Eq. (8.31a) it clearly
Int = %/3(pi —pi)a =
V3 l + 2u
2 1 + n
variable expressing the isotropic
component of the Universe, proportional to the logarithm of its volume,
while 0± are linked to the anisotropy of the space. Furthermore, the first
of the Kasner relations (7.51) is automatically satisfied.
As soon as we change the time variable by means of Eq. (8.31a), we can
define the so-called anisotropy velocity j3'
fdP + rf0_\
f*=\
da da j
(<s.32)
which measures the variation of the anisotropy amount with respect to the
expansion, parametrized by a. The expression (8.32) resembles the second
Kasner relation in Eq. (7.51). The volume of the Universe behaves as e 3a
and tends to zero towards the singularity (a — > — oo), being directly related
to the temporal parameter. Thus, in the Misner variables the Kasner con-
ditions take the simple form corresponding to the unitarity of the velocity
vector 0', i.e.
I0T
i.
!X:«)
334 Primordial Cosmology
8.2.3 Lagrangian formulation
The variational principle (8.15) rewritten under the coordinate transforma-
tion (8.26) stands as
SS B = S J (p a a + p+$+ + P -$- - NH B ) dt = (8.34)
in which Hb is given by
and the potential V takes the form
(8.36)
where Ub is specified for the two Bianchi models under consideration as
Uyui = e" 8 ^ + 4e" 2 ^+ cosh(2V3/3_) + 2e 4/3 + (cosh(4V3/?_) - l)
(8.37a)
U IX = e" 8/3+ - 4e" 2/3 + cosh(2V3/3_) + 2e 4/3+ (cosh(4V3/3„) - l) .
(8.37b)
We can reconstruct the expression of the conjugate momenta by varying
the action (8.34) with respect to p a ,p± and then inverting the relation, on
the basis of the system
Pa = .^l e ^a (8.38a)
P± = ^fe 3 «$ ± . (8.38b)
The variation with respect to the lapse function N generates the super-
Hamiltonian constraint H. = 0, in agreement with the analysis performed
in Sec. 2.3.3.
8.2.4 Reduced ADM Hamiltonian
In order to obtain the Einstein equations, the variational principle requires
5S to vanish for arbitrary and independent variations of (p±,p a , P±,a, N).
As we have seen in Sec. 2.3.3, the ADM reduction procedure prescribes the
choice of one of the field variables as the temporal coordinate and to solve
the constraint (8.35) with respect to the corresponding conjugate variable.
It is convenient to set t = a and solve Hb = with respect to p a as
Hadm = -Pa = Jpl+pl+V . (8.39)
i Formulation of the Mixmaster
Through Eq. (8.39) we express p a in the action integral, so that the
reduced variational principle in the canonical form roads as
5 I (p + d/3 + +p.
SSadm = S (p+d/3 + + p^d/3- - Kadm^) = .
The dynamical picture is completed by taking into account the choice a = 1
which fixes the temporal gauge according to (8.38a), i.e.
^^- (8 . 41)
The dynamical evolution of the Bianchi type VIII and IX cosmological
models using the isotropic variable a, characterizing the Universe volume
as the appropriate time variable, has been established. Correspondingly,
the pure gravitational degrees of freedom are identified to the variables de-
scribing the Universe anisotropy (f3±). Finally, we introduce the anisotropy
parameters Q a , that, in terms of the Misner variables, read as
3 3a
" - 1 -* + -^- (8.42b)
3d
8.2.5 Mixmaster dyi
In this Subsection, we present the approach to the Mixmaster dynamics as
developed by Misner in 1969.
The Hamilton inn introduced so far differs from the typical expression
of classical mechanics for the non-positive definiteness of the kinetic term,
i.e. the sign in front of p\. The potential term is a function of a (i.e. of
time) and of the Universe anisotropy parametrized by /3±.
The Hamiltonian approach as in Eq. (8.38) provides the equations of
motion as
4 _*«<2, fc __*»"■*, ( 8 .43a)
OPa da
P±=N^, p ± = -N d ^. (8.43b)
dp± dfi±
This set, considered together with the explicit form of the potential (see
Figs. 8.1 and 8.2), can be interpreted as the motion of a "point-particle" in
a potential well, where the term V is proportional to the curvature scalar.
Primordial Cosmology
Figur
hquipotcutiaJ line- of the Bianchi type VIII model in the /3 + ,/3 pla.i
In the regions of the configuration space where V can be neglected, the
dynamics resembles the pure Kasner behavior, corresponding to |/3'| = 1.
Asymptotically close to the origin f3± = 0, the equipotential lines for
the Bianchi type VIII are ellipses
E/vin (p+, 0_) = 5 - 16/3+ + 40/3^ + 72/?i + C(/?|) , (8.44)
while for the Bianchi type IX are approximated by circles
U lx (f3 + ,^) = -3 + 24(p + 2 + f3J)+0(f3 3 ± ) . (8.45)
The expressions of the equipotential lines for large values of | /3+ | and small
i Formulation of the Mixmaster
Figure 8.2 Equipotential lines of the Biauehi type IX model in the i . ;1 plan
|/3_| a
urn
line for both types
" [48/3_ 2 e 4 ^ ,
/?+-
> -oo,|/3_ |<1
► +oo,| /3_ |< 1.
(8.40)
Figures 8.1 and 8.2 represent some of the equipol ential lines U (/3) = const.,
for which the potential values increment of a factor e 8 ~ 3 x 10 3 for A/3 ~ 1.
The Universe evolution is described as the motion of a point-like par-
ticle governed by such potential terms, and is characterized by a sequence
of bounces against the potential walls when the system evolves towards the
singularity. Analogously to the BKL approach of Sec. 7.4, the evolution
consists of a series of Kasner epochs when |/3'| = 1, i.e. the point-Universe
338 Primordial Cosmology
moves far from the walls; then a new epoch with different Kasner parame-
ters takes place after a bounce according to the BKL map.
Let us describe in more detail the bounces performed by the billiard ball
representing the Universe. From the asymptotic form (8.46) for the Bianchi
IX potential, we get the equipotential line /? wa n cutting the region where
the potential terms are significant. The condition for the potential to be
relevant near the cosmological singularity is given by e 4( - a ~ 2fi+ ' ~ %adm
or, in terms of /3 wa ii, by
fi + ~ /3 wall = | - \ ln(H| DM ) • (8.47)
As described before, inside the allowed potential domain, the dynamics is
governed by the Kasner evolution, i.e. "Hadm is constant as in the Bianchi I
model. From Eq. (8.47) we get |/3^ all | = 1/2, i.e. the point in the (/3+,/3_)
plane moves twice as fast as the receding potential wall. The point-Universe
will thus collide against the wall and will be reflected from one straight-line
motion (Kasner dynamics) to another one.
A reflection-like relation lays for the bounces. This interval of evolution
for Bianchi IX, i.e. a two-dimensional particle bouncing against a single
wall, is equivalent to the dynamics of the Bianchi II model and it is ana-
lytically integrable (see Sec. 7.3.1). Its ADM Hamiltonian is given by
3+) ) , (8.48)
which is independent of /3_, i.e. p_ is a constant of motion. Let us search
for another first integral of motion: such quantity can be recovered by a
linear combination of p + and %" DM , in particular as
^ = ^adm-P+/2. (8.49)
The reflection law for the incoming and outgoing particle nearby the wall
can be obtained as follows. Let us denote the angles of incidence and of
reflection of the particle off the potential wall as Oi and Of, respectively.
The velocity j3' is parametrized before the bounce as (f3' + )i = — cosflj,
{fi'_)i = sin#i, and as (/?+)/ = cos Of and {J3'_)f = sinOf after the bounce.
Therefore, considering that p_ and Q are constants of motion, as well as
remembering that (3' ± = p±/H, the relation
sm6f-sm0i = -sm(0i + 0f) (8.50)
holds. This represents the reflection map for the bounce for which a limit
angle for the collisions appears. The maximum angle such that a bounce
Hamiltonian Formulation of the Mixmaster 339
against the wall occurs is given by
N<|0 ma x|=arccos(^f) , (8.51)
and hence, since /3^ all //3' = 1/2, the maximum incidence angle is given
by |#max| = 7r/3. In the foregoing bounces the /3-particle will collide on
a different wall and, because of the wall motion, the angle \8f\ > tt/2 is
allowed. Let us observe that, in terms of the parameter u introduced in
Sec. 7.3.2, the relation (8.50) reads as u f = m - 1.
8.3 Misner-Chitre Like Variables
A valuable framework of analysis of the Mixmaster evolution, able to join
together the two points of view of the map approach and of the continuous
dynamics evolution, relies on a Hamiltonian treatment of the equations in
terms of the Misner-Chitre variables, firstly introduced by Chitre in his
PhD thesis (1972). Such formulation allows one to fix the existence of an
asymptotic (en i like) i • instant of motion once an ADM reduction is per-
formed. By this result, the stochasticity of the Mixmaster can be treated
either in terms of statistical mechanics (by the micro canonical ensemble), ei-
ther by its characterizal ion as isomorphic to a billiard on a two-dimensional
Lobacevskij space. Such scheme can be constructed independently of the
choice of a time variable, simply providing very general Misner-Chitre like
(MCI) coordinates. The standard Misner-Chitre variables (r, (,,9) are the
following
a = -e T coshC (8.52a)
P + = e T sinhCcos6> (8.52b)
/?_ = e T sinhCsin(9 (8.52c)
where < Q < oo, < 6 < 2n, and — oo < r < oo. In order to discuss
the results concerning chaoticity and dynamical properties, it is useful to
deal with a slight modification to the set (8.52) via the MCI coordinates
(r(r),£, 8) through the transformations
a = -e r(T ^ (8.53a)
p+ = e^v/^Tcosfl (8.53b)
13- = e r(T V£ 2 -lsin£ ( 8 .53c)
where 1 < £ < oo, and T(r) stands for a generic function of t: the variables
in Eq. (8.52) correspond to setting T(r) = r and £ = coshC- Such modified
sr 29, 2010 11:22
Primordial Cosmology
set of variables permits to write the anisotropy parameters Q a defined in
Eq. (8.42) as independent of the variable V in the form
X
(cos6> + \/3sm(9)
1
^i(„.-^.)
Q 3 = ^ + 2-
3£
3£
(8. 54a)
(8.54c)
The dynamical quantities, if expressed in terms of the relations (8.54), will
be independent of T(t) too.
The variational principle and the Hamiltonian (8.35) in these new vari-
ables read as
8 I (pat + PeO + p T f - NH) dt = 0, (8.55)
" 3(8tt)2 ^
-7W + " !(f, -" + f
, (8.56)
V^=ex P {-3£e r(T) } . (8.57)
The solution to the super-Hamiltonian constraint leads to the expression
involving "Hadm as
-p T = ^Hadm = ^ Ve^+Ve^ , (8.58)
dr dr
^e-i) P e
(8.59)
In terms of this constraint, the principle (8.55) reduces to the form
S f (pd + pe0-tHAim)dt = O, (8.60)
whose variation provides the Hamiltonian equations for £ and 9 a
hadm (e - 1)
* = - r fel*'~«»-ir
i3d'
(8.61a)
(8.61b)
(8.61c)
i Formulation of the Mixmaster
Analogously to the derivation of Eq. (8.41), the time-gauge relation is ex-
pressed as
^m(*)= 2k Wadm ^. ( 8 - 62 )
thus our analysis remains fullj independent of the choice of the time variable
until the form of T and f is lixcd.
8.3.1 The Jacobi metric and the billiard representation
In this Section, we construct the Jacobi metric associated to the dynamics
of the billiard ball discussed above, relying on a formulation independent
of the choice of a specific gauge. In fact, all the analyses can be restated
in terms of the time variable T without specifying the form of the lapse
function but, for the sake of convenience, we will take the restriction f = 1.
The variational principle (8.60) can be rewritten as
,1V ' ^^" H admW = 0. (8.63)
Nevertheless, for any choice of the time variable r (for example r = t),
there exists a corresponding function T (r) (i.e. a set of MCI variables
leading to the- scheme (8. ()."'>)) defined by the invertible relation
"aW JH e ■ ( }
The asymptotically vanishing of ^/rj near the initial singularity is en-
sured by the Landau-Raiehaudhurv theorem (sec Sec. 2.4), which stands
in this general scheme too, as far as T(t) is an increasing and unbounded
function of t
07-s-O => r (<)-»• oo. (8.65)
Approaching the initial singularity, the limit ^/j\ — > for the Mixmaster
potential (8.17) implies an infinite potential well behavior, as discussed in
Sec. 8.1. In this reduced Hamiltonian formulation, the term T(t) plays
the role of a parametric function of time and the anisotropy parameters
Q a are functions of the variables £,6 only (see Eq. (8.54)). Therefore in
the dynamically allowed domain IIq (see Fig. 8.3) the ADM Hamiltonian
becomes asymptotically an integral of motion as
'/(*
^ ADM = Ve 2 + e 2r V = e
V{£,0}en Q t q Uabm (8.66)
1 — — — = ^ e = E = const .
Primordial Cosmology
gure 8.3 The region 11q(£, 9) of the configuration space where the conditions Q a >
e fulfilled. The dynamics of the point Universe is restricted by means of the curvature
rm which corresponds to an infinite potential well.
The variational principle (8.63) reduces to
$ I (pt<% + pedd - edT) =5 f (p^ + Pe d6) = , (8.67)
which holds since the third term of the integral on the left-hand side behaves
as an exact differential (e = E).
By following the standard Jacobi procedure to reduce the variational
principle to a geodesic one in terms of the configuration variables x a , we set
x al = dx a /dT = g ab pb and, by the Hamiltonian equations (8.61) expressed
in terms of T, we obtain the metric
5 « = |(e 2 -i), ^ = |^r- (^8)
By Eq. (8.68) and using the fundamental constraint relation obtained
rewriting Eq. (8.59) as
^-l)pt 2
= E\
(*.(><))
i be shown that
Hab-'-"'-l-
[(« 2 -l>* 2 + ^
Using the definition
Eq. (8.70) is rewritten a
leading to the relatio
i Formulation of the Mixmaster
dx a ds a ds
~~ ~ds"dT =U rff '
lg a bu a u b
-y— — ds -
Indeed Eq. (8.73) together with p^' + pe9' = E allows us to put the vari-
ational principle (8.67) in the geodesic form
S J E dT = 5 J \/g ab u a u b E ds = S f \]G ah u a u h ds = , (8.74)
where the metric G a b = Eg a t, satisfies the normalization condition
G a (jV"u b = 1 and therefore
% = E. (8.75)
In Eq. (8.73) we adopted the positive root, according to the requirement
that the curvilinear coordinate s increases monotonically with increasing
values of V , i.e. approaching the initial cosmological singularity.
Summarizing, the dynamical problem in the region IIq reduces to a
geodesic flow on a two-dimensional Riemannian manifold described by the
line element
rf 5 2 = £ 2 [^! T + (C 2 -l)^ 2 ] . (8.76)
The above metric has negative curvature, since the associated curvature
scalar is R = — 2 /e 2 . Therefore the point-Universe moves over a negatively
curved bidimensional space on which the potential wall (8.18) cuts the
region IIq, depicted in Fig. 8.3. By a way completely independent of the
time gauge, a full representation of the system as isomorphic to a billiard
ball on a Lobacevskij plane has been provided. Indeed, the freedom of the
gauge choice relies on the possibility to express T(t) via a generic lapse
function (8.62) which, for V = 1, reads as
3(8^) 2 y^ 2r
JVADM(r) = — — • (8.77)
zk Hadm
344 Primordial Cosmology
8.3.2 Some remarks on the billiard representation
From a geometrical point of view, the domain defined by the potential
walls is not strictly closed, since there are three directions corresponding
to the three corners in the traditional Misner picture from which the point
Universe could in principle escape (see Fig. 8.3). However, as discussed
in Sec. 7.4 for the Bianchi models in the BKL framework, the only case in
which an asymptotic solution of the field equations shows this behavior cor-
responds to having two scale factors equal to each other (i.e. 6 = 0). Nev-
ertheless, these cases, corresponding to the Taub Universe (see Sec. 10.10.1
are dynamically unstable and correspond to sets of zero measure in the
space of the initial conditions. Thus, in this sense we can neglect the prob-
ability to reach such configurations and the domain is de facto dynamically
closed.
The bounces (in the billiard configuration) against the potential walls
together with the geodesic flow instability on a closed domain of the
Lobacevskij plane imply the Mixmaster point-Universe to have stochastic
features. Indeed, types VIII and IX are the only Bianchi models having a
compact configuration space, hence the claimed compactness of the domain
guarantees that the geodesic instability is upgraded to a real stochastic be-
havior. On the other hand, the possibility to deal with a stochastic scatter-
ing is justified by the constant negative curvature of the Lobacevskij plane
and therefore these two notions (compactness and curvature) are necessary
for these considerations.
8.4 The Invariant Liouville Measure
In this Section the derivation of the invariant measure of the Mixmaster
model is provided in a generic time gauge. Indeed, the ADM reduction of
the variational problem asymptotically close to the cosmological singularity
permits to model the Mixmaster dynamics by a two-dimensional point-
Universe randomizing in a closed domain with fixed "energy" (just the
ADM kinetic energy), as in Eq. (8.66).
From the statistical mechanics point of view, such stochastic motion
within the closed domain Uq induces in the phase-space a suitable micro-
canonical ensemble representation in view of the existence of the "energy-
like" constant of motion. The stochasticity of this system can then be
Hamiltonian Formulation of the Mixmaster 345
described in terms of the Liouville invariant measure
dg = const x 5(E-e) d^dOdp^dpe (8.78)
characterizing the microcanonical ensemble. The particular value taken
by the variable e (e = E) does not influence the stochastic properties of
the system and must be fixed by the initial conditions. This redundant
information for the stat is1 ical dynamics is removable by integrating over all
admissible values of e. Introducing the natural variables (e, 4>) in place of
(PtiPe) by the transformation
>S<f>, p B =e\f£ 2 - Ismcj), 0<<P<2tt (8.79)
ye -i
the Dirac distribution is integrated out, leading to the uniform and i
malized invariant n
dfj, = d^d0d<f> — - . (8.80)
The approximation on which this analysis is based (i.e. the potential
wall model) is reliable since 1 il is dynamically induced, no matter what time
variable T is adopted. Furthermore, such invariant measure turns out to be
independent of the choice of the temporal gauge, as shown by Eq. (8.77).
It can be shown that, by virtue of the system (8.61), the asymptotic
functions £ (r) , 6 (T) , <fi (T) during the free geodesic motion are governed
by the equations
# _
(8.81a)
(8.81b)
(8.81c)
ye
which admit a parametric solution. However, the global behavior of £ along
the whole geodesic flow is described by the invariant measure (8.80) and
therefore the temporal behavior of T (t) acquires a stochastic character. If
we assign one of the two functions T (t) or N (t) with an arbitrary ana-
lytic functional form, then the other one will exhibit a stochastic behavior
by virtue of the ^-dependence for the quantity ^/rj. Finally, by retaining
the same dynamical scheme adopted in the construction of the invariant
measure, the one-to-one correspondence between any lapse function and
the associated set of MCI variables (8.53) guarantees the covariance with
respect to the time gauge.
346 Primordial Cosmology
8.5 Invariant Lyapunov Exponent
The application of standard methods to characterize the chaotic behavior
of the Mixmaster model has taken a large amount of the efforts made over
the last two decades on this cosmological model. Despite this, a defini-
tive assertion fully based on an exact dynamics is still lacking because the
standard chaos indicators, typically adopted for classical systems, are not
straightforwardly extendible to relativistic systems. Indeed two main fea-
tures make the Mixmaster model challenging to be treated as a dynamical
system:
• the vanishing of its Hamiltonian
• the kinetic term is not positive definite.
A detailed discussion of these aspects, as well as a brief review of suc-
cesses and failures concerning various attempts made over the years, is given
in Sec. 8.6. Here we show that the possibility to reduce the Mixmaster dy-
namics to a two-dimensional one, endowed with an energy-like constant of
motion e, allows us to apply the standard notion of Lyapunov exponents.
Furthermore, the gauge-free nature of this representation implies a covari-
ant characterization of the chaotic feature associated to a positive Lyapunov
exponent for a compact configuration space.
This approach relies on a billiard configuration, result in; 1 , from the dy-
namical evolution of the real system when the singularity is approached.
In fact, the infinite walls schematization of the potential picture comes out
from the asymptotic vanishing behavior of the metric determinant. De-
spite its viability, the treatment we are going to describe replaces a precise
dynamical system (the exact Mixmaster) with an approximated scheme.
The dynamical instability of the billiard in terms of an invariant treat-
ment (with respect to the choice of the coordinates (£, 0)) emerges intro-
ducing the orthonormal tetradic basis
* ' (8.82a)
(8.82b)
Indeed, the vector v' is nothing but the geodesic field, i.e. it satisfies
^+r>V=0, (8.83)
Hamiltonian Formulation of the Mixmaster 347
while the vector »;' is parallel transported along the geodesies, according
to the equation
^ + I>V=0, (8.84)
where V kl are the Christoffel symbols constructed by the reduced metric
(8.76). Projecting the geodesic deviation equation along the vector 2 w l
iii." corresponding connecting vector (tetradic) component Z satisfies the
equivalent equation
d 2 Z Z ,
This expression, as a projection on the tetradic basis, is a scalar one and
therefore completely independent of the choice of the variables. Its general
solution reads as
Z (s) = Cl e s ' E + c 2 e- s/E , ci, 2 = const. , (8.86)
and the corresponding invariant Lyapunov exponent is defined as
-(f) 2 )
which, in terms of Eq. (8.86), takes the value
A„ = — >0. (8.88)
The limit (8.87) is well defined as soon as the curvilinear coordinate .s
approaches infinity. In fact, from Eq. (8.75) the singularity corresponds to
the limit T — > oo, and this implies s — ¥ oo.
When the point-Universe bounces against the potential walls, it is re-
flected from a geodesic to another one, thus making each of them unstable.
Though with the limit of the potential wall approximation, this result shows
that, independently of the choice of the temporal gauge, the Mixmaster
dynamics is isomorphic to a well-known chaotic system. Equivalently, in
terms of the BKL representation, the free geodesic motion corresponds to
the evolution during a Kasner epoch and the bounces against the potential
walls to the transition between two of them. The positivity of the Lya-
punov exponent (8.88) is not enough to ensure the system chaoticity, since
its derivation remains valid for any Bianchi type model. The crucial point
is that for the Mixmaster (types VIII and IX) the potential walls reduce
2 Its component along the geodesic field v % does not provide any physical information
about the system instability.
348 Primordial Cosmology
the configuration space to a compact region (IIq), i.e. the geodesic motion
fills the entire configuration space.
Furthermore, it can be shown that the Mixmaster asymptotic dynamics
and the structure of the potential walls fulfill the hypotheses at the basis of
the Wojtkowsky theorem, thus ensuring that the largest Lyapunov exponent
has a positive sign almost 3 everywhere.
Generalizing, for any choice of the time variable, one gets a stochastic
representation of the Mixmaster model, provided the factorized coordinate
transformation in the configuration space
a =- e T{T) a{9,i) (8.89a)
/?+= e r ^b+(6,0 (8.89b)
13- = e r < r) 6_(0,O , (8.89c)
where T, a, b± denote generic functional forms of the variables r, 6, £.
The present analysis relies on the use of a standard ADM reduction of
the variational principle (which reduces the system by one degree of free-
dom) and overall on adopting MCI variables, ensuring that the asymptotic
potential walls are fixed in time.
8.6 Chaos Covariance
We have discussed the oscillatory regime in the Hamiltonian framework
characterizing the behavior of the Bianchi types VIII and IX cosmological
models as discussed in Chap. 7 within the BKL formalism near a physical
singularity, outlining their chaotic properties: firstly, the dynamical evo-
lution of the Kasner exponents characterized the sequence of the Kasner
epochs, each one described by its own line element, with the epoch sequence
nested in multiple eras. Secondly, the use of the parameter u and its rela-
tion to dynamical functions offered the statistical treatment connected to
each Kasner era, finding an appropriate expression for the distribution over
its domain of variation: the entire evolution has been decomposed in a dis-
crete mapping in terms of the rational/irrational initial values attributed
to BKL.
3 It can be shown that, given a dynamical system of the form dx./dt = F(x), the pos-
itivity of the associated Lyapunov exponents are invariant under the diffeomorphism:
y = <j>(x,t),d.T = X(x.,t)dt, as soon as several requirements bold: these requirements are
fulfilled by the present approximation.
Hamiltonian Formulation of the Mixmaster 349
8.6.1 Shortcomings of Lyapunov exponents
We can outline two conceptual limits for the said approaches:
• the BKL formalism corresponds to a non-continuous evolution to-
ward the initial singularity
• the Hamiltonian approach lacks a proper definition of chaos accord-
ing to the indicators commonly used in the theory of dynamical
systems, i.e. of Lyapunov exponents.
A wide literature faced over the years this subject in order to provide
the best possible understanding of the resulting chaotic dynamics.
The research activity developed overall in two different, but related,
directions:
(i) on one hand, the dynamical analysis was devoted to remove the
limits of the Bi\ L approach related to its discrete nature (by ana-
lytical treatments and by numerical simulations)
(ii) on the other hand, to get a better characterization of the Mixmaster
chaos (especially in view of its properties of covariancc).
The first line of investigation provided satisfactory representations of the
Mixmaster dynamics in terms of continuous variables, mainly studying the
properties of the BKL map and its reformulation in the Poincare plane.
In parallel to these studies, detailed numerical descriptions have been
performed aiming to test the validity of the analytical results.
The efforts to develop a precise characterization of chaos relies on the
ambiguity to apply the standard indicators to relativistic systems. In fact,
the chaotic properties summarized so far were questioned when numerical
evolution of the Mixmaster equations yielded zero Lyapunov exponents.
Nevertheless, other numerical studies found an exponential divergence of
initially nearby trajectories with positive Lyapunov numbers. This discrep-
ancy was solved when considering that, both numerically and analytically,
such calculations depended on the choice of the time variable and, in par-
allel, on the failure of the conservation of the Hamiltonian constraint in the
numerical simulations.
In particular, the first clear distinction between the direct numerical
study of the dynamics and the map approximation, stating the appearance
of chaos and its relation with the increase of entropy, has been introduced
by Burd, Buric and Tavakol (1991). The puzzle consisted of simulations
providing zero Lyapunov numbers, claiming that the Mixmaster Universe
350 Primordial Cosmology
is non-chaotic with respect to the intrinsic time (associated with the func-
tion a introduced for the Hamiltonian formalism) but chaotic with respect
to the synchronous time (i.e., the temporal parameter t). The non-zero
claims about Lyapunov exponents, using different time variables, have been
obtained reducing the Universe dynamics to a geodesic flow on a pseudo-
Riemannian manifold. Moreover, a geometrized model of dynamics defining
an average rate of separation of nearby trajectories in terms of a geodesic
deviation equation in a Fermi basis has been interpreted for detection of
chaotic behavior. A non-definitive result was given: the principal Lya-
punov exponent results always positive in the BKL approximation but, if
the period of oscillations in the long phase is infinite (corresponding to the
long oscillations when the particle enters the corners of the potential) , the
principal Lyapunov exponent tends to zero.
For example, Berger in 1990 reports the dependence of the Lyapunov
exponent on the choice of the time variable. Through numerical simu-
lations, the Lyapunov exponents were evaluated along some trajectories
in the (/3+, /?_) plane for different choices of the time variable, more
precisely r (BKL), Q (Misner) and A, the "mini-superspace" one, i.e.
dX = \— yfi+ y\ + r p 1 _ ] \-l' 1 dT. The same trajectory giving zero Lyapunov
exponent for r or f2-time, fails for A.
Such contrasting results are explained by the non-covariant nature of
the indicators adopted due to their inapplicability to hyperbolic manifolds.
This feature prevented, up to now, to say a definitive word about the general
picture concerning the covariance of the Mixmaster chaos, with particular
reference to the possibility of removing the observed chaotic features by a
suitable choice of the time variable, apart from the indications provided in
the next subsection.
The ambiguity which arises when changing the time variable depends on
the vanishing of the Mixmaster Hamutoiijai] and its non-positive definite
kinetic term (typical of a gravitational system). These features prevent the
direct application of the most used criteria for characterizing the chaotic
behavior of a dynamical system.
Although a whole line of research opened up, the first widely accepted
indications in favor of covariance were derived with a fractal formalism by
Cornish and Levin (1997). Indeed, a complete covariant description of the
Mixmaster chaos, in terms of continuous dynamical variables, is lacking due
to the discrete nature of the fractal approach.
Hamiltonian Formulation of the Mixmaster 351
8.6.2 On the occurrence of fractal basin
In order to give an invariant characterization of the dynamics chaoticity,
many methods along the years have been proposed, but not all approaches
have reached an undoubtable consensus. An interesting one, as introduced
above, relying on the fractal basin of initial conditions evolution has been
proposed in 1997 by Cornish and Levin and opened a debate. The conflict
among the different approaches has been tackled by using an observer-
independent fractal method, though leaving some questions open about
the conjectures lying at its basis.
The asymptotic behavior towards the initial singularity of a Bianchi type
IX trajectory depends on whether or not one has a rational or irrational
initial condition for the parameter u in the BKL map. The numerical
treatment on which the fractal basin boundary method is based necessary
deals with rational values for the initial conditions. In such a scheme, the
effect of the Gauss map has been considered together with the evolution of
the equations of motion, in order to "uncover" dynamical properties about
the possible configurations varying with the initial conditions. Nevertheless,
such approach led to some doubts regarding the reliability of the method
itself.
In fact, let us observe that initial conditions with rational numbers are
dense but yet constitute a set of zero measure and correspond to fictitious
singularities. The nature of this initial set needs to be compared with the
complete set of initial conditions given by the whole real set, with finite
measure over a finite interval: the condu io i mm (.he dynamical
evolution are not complementary between the two domains of initial condi-
tions. In this sense, predictions coming out from the set of rational initial
conditions only cannot be extrapolated to the general case.
Cornish and Levin used a coordinate-independent fractal method to
show that the Mixmaster Universe is indeed chaotic. By exploiting tech-
niques originally developed for the chaotic scattering, they found a fractal
structure, namely the strange repellor (see Fig. 8.4) for the Mixmaster cos-
mology that indeed well describes chaos. A strange repellor is the collec-
tion of all Universes periodic in the space of the model parameters while
an aperiodic one experiences a transient age of chaos if it brushes against
the repellor. The fractal pattern was exposed in the numerical integration
of the Einstein equations and in the discrete map used to approximate the
solution. The fractal approach would be independent of the time coordi-
nate and the chaos reflected in the fractal weave of Mixmaster Universes
Primordial Cosmology
would be unambiguous.
Figure 8.4 The numerically generated basin boundaries in the (u, v) plane are built of
Universes which ride the repellor for many orbits before being thrown off. Similar fractal
basins can be found by viewing alternative slices through the phase space, such as the
(/3, 0) plane. The overall morphology of the basin is altered little by demanding more
strongly anisotropic outcomes (Reprinted figure with permission from N.J. Cornish and
J.J. Levin, Phys. Rev. Lett, 78, 998 (1997). Copyright (1997) by the American Physical
Society. http://link.aps.org/doi/10.1103/PhysRevLett.78.998).
This work is widely accepted in the literature but it is worth noting the
following points:
(1) the chosen points representing this framework are the ones whose
dynamics never reaches the singularity due to the intrinsic numer-
ical limit;
Hamiltonian Formulation of the Mixmaster 353
(2) in the exact Mixmaster dynamics, the natural dynamical evolution
predicts that the point particle representing the Universe evolution
enters the corner with the velocity not parallelly oriented towards
the corner's bisecting line and, after some oscillations, it is sent
back to the middle of the potential. This effect is altered when
opening the potential corners, as requested by the numerics.
(3) The artificial opening up of the potential corners adopted in the
basin boundary approach could induce itself the fractal nature.
8.7 Cosmological Chaos as a Dimensional and Matter De-
pendent Phenomenon
8.7.1 The role of a scalar field
Here we face the influence of a scalar field when approaching the cosmolog-
ical singularity showing how it can suppress the Mixmaster oscillations.
Let us consider the Mixmaster Universe in the presence of a self-
interacting scalar field </>. The Einstein equations are obtained from the
variation of the action in the Hamiltonian form associated to the constraint
H = 0. In particular, the scalar field is rescaled in order that the relative
factor between p 2 ^ and p 2 a equals the unity and we choose the gauge N oc e 3 "
in order to simplify the form of the super-Hamiltonian (8.35) which reads
as
H = KT + PT, (8.90)
K T and PT being the kinematic and potential parts of the Hamiltonian,
kt = -pI+p 2 + + p 2 _+pI, ( 8 - 91 )
PT = ^T-e 4Q t/ix + e 6a V(4), (8.92)
where V(4>) denotes a generic potential of the scalar field. In the Misner
variables the cosmological singularity appears as a — > — oo. Therefore,
unless V(4>) contains terms growing enough with | a |, the very last term in
Eq. (8.92) can be neglected at early times, i.e. e 6a V(<j)) -> as a -> -oo.
Assuming that the spatial curvature can be neglected, i.e. dealing only
with KT , then it is easy to verify that the equations of motion admit the
following solution expressed in terms of a
ffa-* + *±H (8.93)
354 Primordial Cosmology
where tt± = p±/\p a \ and n^ = p<f,/\p a \- The constraint KT = then
becomes
ttI+tt 2 . +nl = 1. (8.94)
Let us study the behavior of the potential (8.92) when the scalar field
is not present. From Eq. (8.94) evaluated for n^, = 0, we can parametrize
tt + =cos6 (8.95a)
7r_=sin(9. (8.95b)
Through Eq. (8.93), the potential (8.92) rewrites as
PT ~ e -4M(i+2c OS e)
+ e -4H(l-cos0-V3si„0) +e -4M(l-cose+v^sin^ ( '' V%)
where we retained the dominant terms only, i.e. the first three in Eq. (8.92).
Except for the set of zero measure 6 = (0, 27r/3, 4-7t/3), any generic value of
6 will cause the growth of one of the terms on the r.h.s. of Eq. (8.96) as
a — ¥ — oo; thus the Kasner regime is not stable toward the singularity. Let
us consider the case (j> 7^ and hence 7r^ > 0. Equation (8.94) is restated
by
Tr 2 + + ir 2 _ = 1 - ttI < 1 , (8.97)
thus none of the terms in Eq. (8.96) grows if the following conditions are
satisfied
2tt + > ,
7T+ - V3tt_ > , (8.98)
7T+ + V3tt_ > ,
situation which realizes for 7r^ < 1/2 and irl < 1/12, that is 2/3 < 7r| < 1.
It can be shown that p a decreases at each bounce and therefore, for any
initial value of p<p, the condition above will be satisfied.
As we have seen, the approach to the singularity of the vacuum Bianchi
IX model is described by a particle moving in a potential with exponen-
tially closed walls bounding a triangular domain. During the evolution, the
particle bounces against the walls providing an infinite number of oscilla-
tions toward the singularity. The scalar field influences such dynamics so
that for values of tt± satisfying the conditions (8.98), there are not further
bounces and the solution approaches a final stable Kasner regime. In other
words, there will be an instant of time after which the point-Universe will
never reach the potential walls again and no more oscillations will appear.
In this sense the scalar field can suppress the chaotic Mixmaster dynamics
toward the classical cosmological singularity.
Hamiltonian Formulation of the Mixmaster 355
8.7.2 The role of a vector field
In this Section, the effects of an Abelian vector field on the dynamics of
a generic (n + l)-dimensional homogeneous model in the BKL scheme are
investigated.
A generic (n + l)-dimensional space-time coupled to an Abelian vector
field A^ = (ip, A a ), with a = 1, 2, . . . , n, in the ADM framework is described
by the action
= / d n xdt ( n
a ^A a + fV a E a
- NH - N a H c
u = — ]maP a - — - (n^) 2 + -h aP E a E< 3
-F a pF a >-
(8.100a)
U a = -2V [3 W a + E^F a0 , (8.100b)
where F a p is the spatial electromagnetic tensor, and the relation V a =
d a + A a holds (sec- Six:. 2.2.4). Moreover, E" and IT" 9 arc the conjugate
momenta to the electromagnetic field and to the n-metric tensor, respec-
tively, which are a vector and a tensorial density of weight 1/2. The vari-
ation with respect to the lapse function A^ yields the super-Hamiltonian
constraint H = 0, while the one with respect to if provides the Gauss
constraint V a E a = 0.
We deal with a sourceless Abelian vector field and thus consider the
transverse (or Lorentz) components for A a and E a only. Therefore, we
choose the gauge condil ions if = and T> a E a = 0, in order to prevent the
longitudinal components of the vector field from taking part to the action.
In the general case, i.e. either in the presence of the sources or in the case
of non-Abelian vector fields, (his simplification can no longer take place in
such explicit form and the terms ip(O a + A n )E a must be considered in the
action principle.
A BKL-like analysis can be developed: after introducing a set of Kasner
vectors 4 l a and the Kasner- like scale factors exp(g a /2), the dynamics is
4 We recall that this notation implies that the vectors have to be formally treated as
bavin;;, Eiuii. Iran components.
356 Primordial Cosmology
dominated by a potential of the form J2 e 2qa \ 2 a , where A a are the projections
of the momenta of the Abelian field along the Kasner vectors. With the
same spirit of the Mixmaster analysis developed in Sec. 7.3.1, an unstable
n-dimensional Kasner-like evolution arises; nevertheless the potential term
inhibits the solution to last up to the singularity and induces the BKL-like
transition to another epoch. Given the relation exp(q a ) = t 2pa , the map
that links two consecutive epochs is (a = 2, . . . , n)
A'i = Ai , X' = X a ( 1 - 2 , ( - n ~ l > Pl ) . (8.101b)
An interesting new feature, resembling that of the inhomogeneous Mixmas-
ter, is the rotation of the Kasner vectors, expressed as
r a = l a + a a h, (8.102a)
o a = ^i = -2 , {n ~ 1)Pl h , (8.102b)
A : (n-2) Pa + np 1 \ 1
which completes the dynamical scheme.
The homogeneous Universe approaches the initial singularity described
by a metric tensor with oscillating scale factors and rotating Kasner vec-
tors. Passing from one Kasner epoch to another, the negative Kasner index
P\ is exchanged between different directions (for instance li and I2) and,
at the same time, these directions rotate in space according to the rule
(8.102b). The presence of a vector field, independently of the considered
model, induces a dynamically closed domain on the configuration space. In
correspondence to these oscillations of the scale factors, the Kasner vectors
l a rotate and the quantities a a remain constant during a Kasner epoch to
lowest order in q a . The vanishing of the determinant h approaching the
singularity does not significantlj affect the rotation law (8.102b). The re-
sulting dynamics provides a map exhibiting a dimensional- dependence, and
it reduces to the standard BKL one for n = 3.
8.8 Isotropization Mechanism
The isotropic FRW model is accurate to describe the backward evolution of
the Universe up to the decoupling time, i.e. 3 x 10 5 years after the Big Bang.
Moreover there are well-established indications that the isotropic dynamics
Hamiltonian Formulation of the Mixmaster 357
is the natural scenario for the primordial nucleosynthesis process, i.e. the
validity of the RW geometry up to 10~ 2 — 10~ 3 seconds after the Big Bang.
There is no argument against the idea that the isotropic Universe can be
extrapolated up to the inflationary age, i.e. t ~ 10~ 34 s. The comparison
of predictions from inflation with the CMB data (see Sec. 4.4) provides a
significant evidence that after the inflationary process our Universe retains
an isotropic morphology up to a very high degree of precision on a scale
depending on the model parameters (see Chap. 5). On the other hand, the
description of the very early stages requires more general models, like as
the homogeneous ones, as suggested by the instability shown in the back-
ward evolution of the FRW Universe with respect to tensor perturbations.
Therefore it is interesting to investigate the mechanisms allowing a transi-
tion between these two cosmological epochs. When the anisotropy of the
Universe is sufficiently suppressed, one deals with a quasi-isotropization of
the model and such configuration can be regarded as a "bridge" between
the two stages. In this paragraph we discuss the origin of a background
space 5 when a real self-interacting scalar field <f> is taken into account. We
adopt the same rescaling for <fi as in Sec. 8.7.1.
Let us restate the Misner-like variables to include the scalar field as
p+ = /3 1 , /3_ = p 2 , cp = V3/3 3 . (8.103)
The action describing the Universe in such scheme reads as
=/4'^'+- a '"-w^(? p '-
Pa 4
(8.104)
where r = 1,2,3, p r and p a are the conjugate momenta to f3 r and a,
respectively. The potential term is defined as U = e 6a W(<fi) + V, V follows
from Eq. (8.36) and W((f>) = ^[h^d^d^] + V((f>). From Eq. (8.104) an
inflationary solution comes out imposing the constraint
e~ 6a U ~ V(4>) ~ const. > e- 2a Ui X (8.105)
which can be realized by an appropriate process of spontaneous symmetry
breaking, exhaustively studied in Chap. 5.
Let us consider the situation when U = e 6a p\, where p\ = const. The
Hamilton- Jacobi equation then takes the form
£(f) 2 -( gy + ^°>-'°°- < 8 -™>
5 The chaotic nature of the evolution toward the singularity implies that the geometry,
and therefore all the geometrical quantities, should be described in an average sense only.
With this respect, during the vacuum Mixmaster, the Uni
hack;;] omul near I lie singularity.
358 Primordial Cosmology
whose solution can be expressed as
S(f3 r , a) ~ J2 K rP + \K a + y In I -^
-K\
where K a (K r ,a) = ±yj^ ^r + PA exp(6a), with some generic constants
-K" = ^Er^r an< ^ ^- The equation of motion for a is readily obtained
from Eq. (8.104) as
da 2Nk ,„
Choosing a as the time coordinate, i.e. da/dt = 1, the time gauge condition
becomes TV = — 3(87r) 2 exp(3a)/(2Kp Q ). Since the lapse function is positive
defined we must also have p a < 0.
According to the Hamilton- Jacobi method, one has firstly to differenti-
ate with respect to K r and then to equate the result to arbitrary constants
/?g, so getting
».-« . .X^J^mII^H. , 8 ,09)
Let us consider the two limits of interest. First, of all, for a — > oo (K a — > oo)
the solution (8.109) transforms into the inflationary one corresponding to
the quasi-isotropization of the model as the functions /3 r approach the con-
stants /3q- Such values are reabsorbed in the Kasner vectors and therefore
this limit is equivalent to getting a vanishing Universe anisotropy.
On the opposite limit, i.e. for a -> -oo (K a -> K), the solution (8.109)
provides the generalized Kasner one as expected, simply modified by the
presence of the scalar field
P r (a)=ft-^-a. (8.110)
The existence of the solution (8.109) shows how the inflationary scenario
can provide the necessary dynamical "bridge" between the fully anisotropic
and the quasi- isotropic epochs during the Universe evolution. In fact, dur-
ing that time, the anisotropics p± are dumped away and the only effective
dynamical variable is a, i.e. the isotropic volume of the Universe. This
shows how the dominant term during the inflation is p\e^ a and any term
involving the spatial curvature becomes more and more negligible although
increasing like (at most) e 4a .
Hamiltonian Formulation of the Mixmaster 359
8.9 Guidelines to the Literature
For an introduction to the original formulation of the Hamiltonian dynam-
ics, presented in Sec. 8.1, associated to the Mixmaster model see [345]. An
analysis of the Bianchi models in the variables Q a can be found in [286] .
A reference textbook providing a satisfactory and advanced description
of analytical mechanics topics relevant to the presented approach is that of
Arnold [15].
The Hamiltonian analysis in the variables, presented in Sec. 8.2 diago-
nalizing the super-Hamiltonian was firstly introduced by Misner in [345].
For a comprehensive discussion of the reduced ADM-procedure of the
Bianchi models dynamics, given in Sec. 8.2.4 in terms of the Misner vari-
ables, we refer to the textbook of Misner, Thorne & Wheeler [347], Ch.
30. The general dynamical scheme underlying such reduction was firstly
provided in [19].
The restatement of the reduced Hamiltonian dynamics in terms of gen-
eralized Misner- ( 'hit re -like variables, provided in See. 8.2.5 can be recovered
in [258]. The first proposal for this type of variables is due to Chitre in [119].
A textbook on the general features of the ergodic theory, as arising
for the Mixmaster model in the Misner-Chitre like variables, is the one by
Cornfeld, Fomin & Sinai [130].
A valuable textbooks on the Jacobi metric associated to a geodesic flow
and on the corresponding ergodic properties (Sec. 8.3), are the ones by
Anosov [14] and by Arnold & Avez [16]. For the specific application to the
Bianchi models, see [119] and [258].
The formula! ion of l bo invariant LioiLvlllo measure, presented in Sec. 8.4,
had two fundamental steps. For the first characterization of a measure in
the configuration space of the Mixmaster, see [118] and for its extension
to the phase-space see [286] (for a discussion of the Artin theorem adopted
in such approach, see [20]). For a discussion of the covariance of the Li-
ouville measure, see [257]. The analysis of non-stationary corrections to
the Mixmaster invariant measure can be recovered in [352]. An additional
interesting literature on the stochastic properties of the Mixmaster can be
found in [39,40,429].
A general analysis of the proper) ios of classical chaotic systems can be
found in [369] . A collection of interesting reviews on the problem of chaos
in General Relativity is provided by the proceedings volume [211].
The debate of invariant Lyapunov exponents, introduced in Sec. 8.5 was
characterized by different conceptual stages. A demonstration of the co-
360 Primordial Cosmology
variance of the Mixmaster chaos in the billiard-ball representation is given
in [258] . For a discussion of the notion of Lyapunov exponents in relativistic
cosmological systems, see [105] (the original definition of Lyapunov expo-
nents can be found [421]). For a derivation of the Wojtkowski theorem,
see [468]. A generic result that links the Lyapunov exponents in differ-
ent frames is given in [356]. There is a wide literature in this subject in
order to provide the best possible understanding of the resulting chaotic
dynamics. The research activity developed overall in two different, but re-
lated, directions: on the one hand the removal of the limits of the BKL
approach due to its discrete nature (by analytical treatments [40, 106, 118]
and by numerical simulations [74,75,79,88,405]); or getting a better char-
acterization of the Mixmaster chaos (especially in view of its properties of
covariance [171, 172,238,429]). For a review on such a topic, see [77,354].
For a discussion of the main features concerning the fractal boundary
approach and its implementation on the Mixmaster dynamics, in Sec. 8.6.2,
see the following literature: [89,131,132,141,142,210,238,410,430].
The first derivation of the chaos removal when the Mixmaster has a
massless scalar field source, describe in Sec. 8.7.1, was provided in [59,60].
For a discussion in the Hamiltonian formalism of the same features, see [76].
A discussion of the role that an Abelian vector field plays in the Mix-
master dynamics in a multi-dimensional space-time, as in Sec. 8.7.2, is
introduced in [69] . This paper contains the details of the calculation repro-
duced in this Section.
An analysis of the role that a cosmological constant can have in
isotropizing the Mixmaster dynamics, discussed in Section 8.8, can be found
in [287] and in the review article [354].
Chapter 9
The Generic Cosmological Solution
Near the Singularity
In this Chapter we will analyze the extension of the Mixmaster dynamics
to the inhomogeneous sector by constructing the generic cosmological solu-
tion near the initial singularity. After a discussion on the Bianchi IX model
instability towards the singularity, we will outline how the Kasner evolu-
tion introduced in Chap. 7 can be upgraded to describe an inhomogeneous
regime, approaching a singular point and having three physically arbitrary
functions available to specify the Cauchy problem on a non-singular space-
like hypersurface. Such generalized Kasner solution can be stable up to the
vanishing of the space volume only if a given condition holds for its metric
functions, which prevents to deal with four physical degrees of freedom, as
required by the generality of the Cauchy problem.
By relaxing such restriction, we are naturally led to represent the generic
cosmological solution as a piecewise Kasner approximation by an iterative
scheme fully equivalent to that singled out for the homogeneous Mixmaster.
Indeed the spatial points dynamically decouple toward the singularity and
play only a parametric role in the Einstein equations. The BKL map retains
exactly the same form as in Chap. 7, but its point-like nature induces a
coupling between the chaotic time dependence of the Mixmaster and the
spatial morphology of the three-hypersurfaces. In particular near enough
to the singularity the space-time takes the structure of a real foam, with a
classical statistical nature.
The Hamiltonian formulation of the inhomogeneous Mixmaster is con-
sidered in the Misner variables with stationary Kasner axes, and also in a
more general gauge-independent framework, when the ADM reduction of
the system is fully developed. Such general scheme allows to extend the
co- variant study performed in Chap. 8, of the Mixmaster chaos covariance
to the generic inhomogeneous sector. An estimate of the spatial gradient
362 Primordial Cosmology
behavior shows how the BKL conjecture stating the local validity of the
Mixmaster (say, within each causal horizon) is statistically well-grounded.
Finally, the Mixmaster dynamics is re-analyzed by introducing a set of
variables appropriate to deal with a dynamical system approach.
This Chapter is concluded with a multidimensional analysis of the in-
3 Mixmaster model which outlines how, up to ten space-time
is, the chaotic features are preserved. However, for higher dimen-
sional cases, we outline the existence of an open region of the Kasner sphere
where the Kasner regime is stable (known as the Kasner Stability Region).
The attractive character of this region in the parameter space can be in-
ferred by the properties of the multidimensional BKL map. For space-times
with more than ten dimensions, the chaotic features of the inhomogeneous
Mixmaster are suppressed in favor of a stable Kasner epoch reaching the
initial singularity.
9.1 Inhomogeneous Perturbations of Bianchi IX
In this Section we describe the inhomogeneous perturbations to a homoge-
neous Mixmaster Universe. The interest in such topic is twofold:
(i) it represents a first step towards introducing more degrees of free-
dom than those available for the homogeneous sector of GR, thus
linking the gravitational (toy) models with the full field theory.
(ii) The dynamics of the perturbations should probe some insight into
the BKL conjecture (see Sec. 9.2) about the generic (namely in-
homogeneous) cosmological singularity. This topic has been firstly
studied by Regge and Hu in 1972.
Dealing with a homogeneous space allows to simplify the usual con-
struction of perturbations. In such spaces every point is equivalent to any
other under the action of an isometric group (see Sec. 7.1.1) and one can
perform all computations at one specific point in space. The general form
of the equations are then generated by simple group invariant operations
on the manifold and the set of tensors are composed by the direct product
of the basis invariant forms operating on the representation function of the
group. In this case, one does not have to construct basis tensor harmonics
as functions of the whole space (like the hyper-spherical tensor harmonics
in the FRW metric; see Sec. 3.5), but rather one can evaluate the product
at one point and use the invariant operators to generate the complete set.
The Generic Cosmological Solution Near the Singularity 363
Any tensor field in a homogeneous space can be expanded in terms of
these tensor harmonics. The perturbations can in general be expressed
in terms of the (left) invariant 1-forms (7.28) with time-dependent expan-
sion coefficients coupled to the representation functions of the underlying
symmetry group of the manifold. For a S0(3)-homogeneous space-time
(namely the Bianchi IX model), the representation functions are the so-
called Wigner Z}-functions D 3 m , m (g), labeled by the spin number j and the
magnetic numbers (m', m). We remind that, because of the identification of
the (topological) three-sphere S 3 with the group manifold SU(2) ~ SO(3),
we can use group elements g = gix 1 ) to coordinate the physical space of
Bianchi IX (which has the S 3 topology).
Let us firstly recall the construction of the Wigner D-functions, obtained
from their definition in terms of matrix elements of the rotation operator
K(cf>, 9, V) = exp(-»0j x ) exp(-idj y ) expHVJ J , (9.1)
in which j x ,j y ,j z are generators of the su(2) Lie algebra and ((j>,9,tp) are
the Euler angles parametrizing the SO(3) left-invariant 1-forms (8.6b). The
Wigner functions can be explicitly expressed as
D j m , m = (jm'\K((l>,9,ip)\jm) = e~ im ' * <$ m , JQ) e - *" 1 * , (9.2)
whore
CmW = <^ m X)Ci'r>)(
(9.3a)
C 3 m , m = ^{j + m>)\(j-m')\{j + m)\{j-m)\ (9.3b)
{ _ l)m '-m + s
CL'mis) =
[{j + m- S y.s\(m' -m + s)\(j - r.
where the sum is over the values of s for which the factorials are non-
negative. For j = I G N, these functions are simply related to the spherical
harmonics Yi tm {6,4>) as
DLm(9) = ("l)" m V^/(2l + l)Yi,-m'(9,4>)e m ^. (9.4)
D : L
364 Primordial Cosmology
The Wigner D-functions can be obtained requiring to satisfy the differential
equations (expressed in terms of the Euler angles)
L 2 Dl, m
\d 2 d 1 / d 2 d 2 d 2 '
= [^ + COt ^ + s]n^l^- 2COS '^^ + ^,
=jU + l)D J m , m , (9.5a)
L3D j m , m = -i^-D j m , m = m'D j m , m , (9.5b)
mm Q^ mm mm
L z Dl, m =-i4-Dl, m =mD J m , m . (9.5c)
In Eqs. (9-5) wc have introduced the two bases {L\, L 2 , £3} and
{L x , L y , L z } of generators for the su(2) algebra and their common Casimir
operator
L 2 = L\ + l\ + l\ = L 2 X + L 2 y + L\ , (9.6)
which is an invariant of the group. The angular momentum operators
{L\i L2, L3} of the three-dimensional rotation group in quantum mechanics
(which are the intrinsic angular momentum operators of a rigid body) are
related to the (left- invariant) vector fields e" (7.19) via the relations
l x = ief 8 a , L 2 = ie^ d a , L 3 = ie* d a . (9.7)
The angular momentum operators {L x , L v , L z } are in turn related to the
(right-invariant) Killing vector fields £" (see Sec. 7.1.1) by the formulae
L x = -i£ d a , L y = -i& d a , L z = -i£% d a . (9.8)
Let us describe the inhomogeneous perturbations. Be h a p(x,t) the (unper-
turbed) spatial metric of the Bianchi IX model. A generic perturbation
-y a p(x, t) = h a p{x, t) - h a p{x, t) (9.9)
to the unperturbed three-metric can be translated into a matrix of space-
scalars "f a b{x,t) by projecting it on the invariant 1-forms, that is
lafl {x,t) = lab {x,t)u a a {x)^{x). (9- 10 )
According to the previous discussion, these scalars can be decomposed
in terms of definite angular-momentum components of the three-metric
7^™ (#,£), labeled by spin and magnetic numbers (j, m)
7ab(x,t)=J2rir(^t)- (9-11)
The Generic Cosmological Solution Near the Singularity 365
turns, can be expressed in terms of Wigner D-functions (9.2) as
and therefore we are expanding uihomogcncous perturbations (intended
as scalar harmonic functions on S 3 ) as a linear combination of Wigner
-D-functions. The time dependent amplitudes 7^™ (t) represent, at fixed
7, 6 (2j + 1) inhomogeneous degrees of freedom (in the case of diagonal
matrices j a b there are 3 (2j + 1) inhomogeneous components), governed by
a set of coupled differential equations (see below).
At a first sight, the dependence of the scalar harmonic functions
-y J a ™{x, t) on only the modes j, to in Eq. (9.11) and Eq. (9.12) seems incom-
patible with a generic decomposition of ^ a b(x, t). In fact, a scalar function
7(5) (for the moment we drop the internal indices a, 6, irrelevant for the
discussion) on SU(2) can be expanded, by the Peter-Weyl theorem, as
7(5)= E -y jm ' mD Lm(9)- (9.13)
The components, at fixed j, will be (2j + l) 2 and no longer (2j + 1) because
they are label by both magnetic numbers (to', to). However, the choice of
not summing over the j, to labels contracted with the Wigner D functions
relies on the fact that the Einstein equations allow to decouple j, to states
from to' states, allowing to fix the perturbations to the metric with definite
j, to. Indeed, only the to' states are mixed by the action of the derivative
operators and thus by the linearized Einstein tensor. In fact, from Eqs.
(9.5a)-(9.5c), it follows that
L + D 3 m , m = (Li+*L 2 ) D 3 m , m
= i^(j + m')(j-m' + l)D j (m ,_ 1)m (9.14a)
L_ D j m , m = (li-i L 2 ) D j m , m
= WU + m' + l)(j-m')Dl ml+1)m (9.14b)
L 3 D 3 m , m = to' D j m , m . (9.14c)
Any invariant operator can be rewritten in terms of Cartesian coordinates
xa = {xi, x 2 , x 3 , x 4 } in the Euclidean space 1 E l in which the three-sphere
S 3 is embedded. Instead of the three co-frames uj a in the Euler angles
chart, the invariant basis in the Euclidean space is given by the coordinate
differentials dx A , related to u a by the transformation matrices S a (xa) as
uj a = 2S A (x A )dx A . (9.15)
366 Primordial Cosmology
Conversely, the coordinate differentials of E 4 can be expressed in terms of
uj a as dx = S^co a /2. The coordinate derivatives, which are the vector
fields of E 4 , are given by
J- = 2S a A (x A )e a . (9.16)
dx A
It turns out to be much easier, using the homogeneity of the S 3 spatial
slices, to evaluate the Cartesian derivatives at the pole xp = {x^ = \,X\ =
X2 = X3 = 0}, where the transformation matrices reduce to S A = —S A , and
yielding the relations
£L~ S - ,9 - 17)
Equation (9.17) allows us, by means of Eqs. (9-7), to express the deriva-
tives in terms of invariant operators. On the other hand, Eqs. (9.14) specify
the actions of the invariant operators on the Wigner D-functions. It fol-
lows that, once the Einstein equations for the Bianchi IX model have been
rewritten in terms of Cartesian coordinates x , the perturbations to homo-
geneity contain states with definite j, m, thus only perturbations labeled
by m! states are mixed.
The above construction allows us to siraj<>,lii [onvardly obtain the per-
turbation equations to Bianchi IX. In particular, the Christoffel symbols
and their derivatives can be computed at the pole xp. Also the (generic)
Regge- Wheeler perturbation equations on an empty background metric
2 SRij = V fc V fc 7y - Vj V fc 7lfe - ViV fe 7 Jfe + ViVj7fc = (9.18)
can be evaluated at xp. This leads to ordinary (in the time coordinate) dif-
ferential equations for 7^™ (t) in which the magnetic number m' is mixed by
the spatial derivatives. As a result of the numeric integration (at the lowest
mode j = 1/2) the perturbations decrease as the volume of the Universe
increases from the singularity and vice-versa. Thus, the Bianchi IX model
is stable in the expanding picture but is unstable when the cosmological
singularity is approached from a non-singular hypersurface, suggesting to
abandon the symmetry requirements when treating the initial singularity.
9.2 Formulation of the Generic Cosmological Problem
From the '60s, the Landau school started to investigate the properties and
the behavior of the generic cosmological solution of the Einstein equations.
A generic solution of the field equations corresponds to a metric gij that
The Generic Cosmological Solution Near the Singularity 367
possesses the correct number of free functions to formulate any Cauchy
problem on some non-singular hypersurface. Prom the study the Cauchy
problem for the Einstein equations can be recognized that in vacuum we
need four unknown functions to specify the physical degrees of freedom for
the gravitational field, and eight functions if a perfect fluid is included into
the dynamics.
The first results in this direction were obtained by Khalatnikov and
Lifshitz in 1963 who extended the Kasner solution to the case when the
homogeneous hypothesis is relaxed. Such solution is stable when reaching
a singular point in the past as hood as a particular condition is imposed, so
reducing the number of arbitrary functions to three (treated in Sec. 9.2.1).
In the following years, by relaxing such condition, this solution was gen-
eralized by Belinskii, Lifshitz and Khalatnikov outlining a very complex
behavior which resembles that of the homogeneous Mixmaster model stud-
ied in Chaps. 7 and 8. This is now called the BKL conjecture and, even if a
rigorous mathematical proof does not exist yet, it is commonly used to de-
scribe the detailed evolution of the Einstein equations in the neighborhood
of a cosmological singularity (see Sec. 9.2.2).
The construction can be achieved firstly by considering the inhomoge-
neous solution for the individual Kasner epochs and then providing a gen-
eral description for the alternation of two successive epochs. The answer to
the first question is given by the so-called generalized Kasner metric, while
the solution to the latter is in close analogy to the replacement rule for the
homogeneous indices.
9.2.1 The Generalized Kasner solution
Lifshitz and Khalatnikov showed that the Kasner solution can be general-
ized to the inhomogeneous case, near the singularity, as
(9.19)
{d,l 2 = h a f)dx a dxP ,
h a p = a 2 l a lp + b 2 m a mp + c 2 n a np ,
where
a ~ t Pl , b ~ t p ™ , c ~ t p " , (9.20)
and pi t , p m , p n are fund ions of spatial coordinates subjected to the condi-
tions
Pi{x^)+p m {x r ) + Pn (x~<) =p 2 (x~<)+p 2 m (xi) +p 2 n (x^) = 1 . (9.21)
368 Primordial Cosmology
Differently from the homogeneous case, the reference vectors I, m, n are ar-
bitrary functions of the coordinates (subjected to the conditions associated
with the Oa-components of the Einstein equal ions).
The behavior in Eq. (9.19) cannot last up to the singularity, unless a
further condition is imposed on the vector corresponding to the negative
index p\\ without loss of generality, we can take pi = p\ < through the
whole hypersurface, and such a condition reads as
Z-VAZ = 0. (9.22)
This restriction ensures that all terms in the three-dimensional Ricci ten-
sor can be neglected toward the singularity (see below Sec. 9.2.2). This
condition reduces the number of arbitrary functions to three, i.e. one less
than the number required to deal with the general case. In fact, the metric
(9.19) possesses 12 arbitrary functions of the coordinates (nine components
of the Kasner axes and three indexes /;,■(. )■'■ )). and must satisfy the two
Kasner relations (9.21), the three Oa Einstein equations, three conditions
arising from three-dimensional coordinate transformations invariance, and
Eq. (9.22).
9.2.2 Inhomogeneous BKL solution
Let us now generalize our scheme by investigating the implications of re-
moving the condition (9.22). This analysis leads to the inhomogeneous
BKL replacement map.
Taking the three-metric tensor in the form (9.19), we are summarizing
the dynamical evolution of the model into the behavior of the scale factors
a, b, c, while the Kasner vectors I, m and n fix generic directions. In gene-
ral, the three-metric associated to a generic inhomogeneous model can be
written, in analogy to Eq. (7.17), as
Kp = Ti ab (t, x)e a a {x 1 )e b p (x 1 ) , (9.23)
where the matrix r] a b depends on the space coordinates because of inhomo-
geneity. In fact, the three linear independent vectors e^ no longer define
an isometry group but simply correspond to a generic choice of their com-
ponents, and therefore the corresponding quantities X a b c (2.111b) do not
behave as constant terms. The metric (9.23) is mapped by simple identi-
fications into metric (9.19) as far as r\ a b = diag(a 2 , b 2 ,c 2 ), e x a = l a ,e 2 a =
m a , e^ = n a and (a, b, c) = (I, m, n).
The solution (9.19)-(9.21) is obtained neglecting the triadic projection of
the three-dimensional Ricci tensor 3 R b a into the vacuum Einstein equations
The Generic Cosmological Solution Near the Singularity 369
which read as
-ij° = -d t K a a + K£K b a = , (9.24a)
-R° a = \ b f a K f b - X s fd K d a + V f9 d f K ga + V fg d f v gb K b a
- r] fB d a Kf g ^ri» b d d ri gb KZ - h(^d a n fg = (9.24b)
-R b a = ]=9t {VvK) + 3 R a = , (9.24c)
Vv
where \ a t, c are the Ricci coefficients given in Sec. 2.5. The validity of the
Kasner behavior can be conveniently formulated in terms of the projections
along the directions I, m, n, satisfying the conditions
3 R\, 3 R™, 3 i?" < t~ 2 , 3 R\ > 3 i?™, X . (9.25)
In fact, the dominant terms should be the ones associated to the time deriva-
tives oir]ab which, for the Kasner behavior, identically vanish but are poten-
tially of order t~ 2 . In the Kasner-like behavior, the off-diagonal projections
ofEq. (9.24c) determine the off-diagonal projections (r/;,„, i]i n . rj mn ), which
result to be small corrections to the leading diagonal terms of the metric.
In this regime, the only non-vanishing projections are the diagonal terms
(Vih Vmm, Inn) and satisfy
Vim < VVllVmm , Vln < y/VllVnn , Vmn < VVmrnVnn ■ (9.26)
The tbree-dimensioiia.] Ricci leuso]' associated to the three-metric (9.19)
reads in the form (all the vectorial operations are performed as in the
Euclidean case)
a 2 [ 1 1 1
3 R n = —! - (alV A al) 2 - - {bmV A bm) 2 (cnV A cnf
- (cnV A bm) 2 - {bmV A en) 2 - (ftmV A al) 2 - (cnV A al) 2
+ (cnV A en) (6mV A bm) + (cnV A al) (a/V A en)
+ (a/V A bm) {bmV A al)
1 ZcnVAaA l/cnVA6m\
b{ A ) >m 'a{ A )
1 (bmSI Acn\ 1 Z&mVAaA 1
Primordial Cosmology
3 R lm = JL. J (al\7 A a/) (ftmV A al) + (bm V A 6m) (aZV A 6m)
+ (a/V A en) (6mV A en) - ^ (cnV A en) [(a/V A 6m) + (6mV A al)}
+ - (6mV A en) (cnV A ai) + - (a/V A en) (cnV A 6m)
ab l/6mVAcn\ l/aZVAcn\
2 [6 V A Am a V
1 fbmV Abm\ 1 f alV A
c V a / n c V A y in I '
(9.27b)
where we have introduced the quantity A = \fh = abc(l-mAn) and the let-
ters I, m, n, following the comma in the indices, denote differentiation along
the corresponding direction. The other components may be obtained from
those given by cyclic permutation of the letters I, m, n and, correspondingly,
ofa,6,c.
In view of these expressions, the condition (9.26) leads to the following
inequalities
3 Ri m ^ab/t 2 , 3 R ln < ac/t 2 , 3 R mn < 6c/t 2 . (9.28)
The diagonal projections 3 R\, 3 R^, 3 R„ contain the terms
aiVA(a/) \ k 2 a 2 _ k 2 a 4
l_(_ar
2\abc(
i<i.2«S!
c(l-[mAn])J b 2 c 2 A 2 t 2 '
and analogous terms with al replaced by bm and cm here 1/fc denotes the
order of magnitude of spatial distances over which the metric significantly
changes and, dealing with a Kasncr regime, A = abc = A(t,x). According
to conditions (9.25), we get the inequalities
a^/k/A < 1 , b^/k/A < 1 , c^/k/A < 1 , (9.30)
which are not only necessary, but also sufficient conditions for the existence
of the generalized Kasner solution. As soon as the conditions (9.30) are
satisfied, all other terms in 3 R\, 3 R™, 3 #™, as well as in 3 R lm , 3 R ln , 3 R mn ,
automatically satisfy Eq. (9.25) and Eq. (9.28) as well. In fact, as soon as
one estimates the Ricci tensor projections, the inequalities (9.28) lead to
the conditions
^(a 2 6 2 ,...,a 3 6,...,a 2 6 C ,...)«l, (9.31)
The Generic Cosmological Solution Near the Singularity 371
containing on the left-hand side the products of powers of two or three of
the quantities which enter in Eq. (9.30). The inequalities (9.31) can be
seen as the generalized version of the condition imposed when addressing
the homogeneous Mixmaster model.
As t decreases, an instant t tr may eventually occur when one of the
conditions (9.30) is violated. 1 Thus, if during a given Kasner epoch the
negative exponent refers to the function a(t), i.e. pi = p\, then at t tr we
have
2tr VX~
(9.32)
Since during that epoch the functions b(t) and c(t) decrease with t, the
other two inequalities in (9.30) remain valid and at t ~ t tr we shall have
6 tr < Otr , C tr < Otr • (9.33)
At the same time, all the conditions (9.31) continue to hold and all the
off-diagonal projections of Eq. (9.24c) may be disregarded. In the diagonal
projections (9.27), only the terms containing a 4 /t 2 become relevant. In
such surviving terms we have
(al-VA (ai)) = B (!-[VoxI]) + o 2 (i-VA !) = o 2 (l-VAl) , (9.34)
i.e. the spatial derivatives of a drop out. As a result, we obtain the following
equations for the replacement of two Kasner epochs
-«i = ^ + ^ = »' ("^
-«™ = ^-^ = ' <" 5b >
-HO = - + I + - = , (9.35d)
which differ from the corresponding ones of the homogeneous case (7.07-
7.68) only for the quantity
which is no longer a constant, but a function of the space coordinates.
Since Eq. (9.35) is a system of ordinary differential equations with respect
lr The case when two of these are simuli am ously violated can happen when the exponents
pi and p2 are close to zero, corresponding to the case of small oscillations (see Sec. 7.4.3).
372 Primordial Cosmology
to time where space coordinates enter only parametrically, such difference
does not affect at all the solution of the equations and the resulting BKL
map, retaining in each space point the form as in Eq. (7.95). Similarly,
the law of alternation of exponents derived for the homogeneous indices
remains valid in the general inhomogeneous case.
9.2.3 Rotation of the Kasner axes
Even if the point-like dynamics is quite similar to that of the homogeneous
case in vacuum, the new feature of the rotation of the Kasner axes emerges.
If in the initial epoch the spatial metric is given by Eq. (9.19), then in the
final one we have
h a p = a 2 l% + b 2 m' a m' p + c 2 n' a n' p , (9.37)
with a, b, c characterized by a new set of Kasner indexes, and some vectors
I', in' , n' . If we project all tensors (including h a p) in both epochs onto the
same directions I, in, n, the turning of the Kasner axes can be described as
the appearance, in the final epoch, of off-diagonal projections rji m , rji n , Vmn,
which behave in time as linear combinations of the functions a 2 , b 2 , c 2 .
The main effects can be reduced to a rotation of the in- and n-axis by
a large angle, and a rotation of the /-axis by a small one which can be
neglected. The new Kasner axes are related to the old ones as
/' = /, m , = m + a m l, n' = n + a n l , (9.38)
where the cr m , a n are of order unity, and are given by
j[/Am]-Vy + ^m-VAi}— ^ (9.39a)
P2 + 3pi
a n = <f[nAZ]-V^-%in.VA/)— ^ r. (9.39b)
Vi + 3pi I L J A A J I ■ [to A n] v '
These expressions can be inferred by linking the two Kasner epochs and
by the Oa components of the Einstein equations which play the role of
constraints over the space functions.
The rotation of the Kasner axes (that appears even for a matter-filled
homogeneous space) is inherent in the inhomogeneous solution already in
the vacuum case. The role played in the homogeneous case by the matter
energy-momentum tensor can be mimicked by the terms due to inhomo-
geneities of the spatial metric. Furthermore, in analogy to the discussion
presented in Sec. 7.2.1, the presence of matter does not influence the gen-
eralized Kasner solution to the leading order. Repeating in the inhomoge-
neous case the same analysis as in Sec. 7.2.1, it is possible to show that,
The Generic Cosmological Solution Near the Singularity 373
near the singularity, the perfect fluid exhibits a test-like behavior point by
point. Its effect is mainly exhibited in modifying the relations between the
arbitrary spatial functions which appear in the solution, now containing
even matter degrees of freedom.
9.3 The Fragmentation Process
We will now qualitatively discuss a further mechanism that takes place
in the inhomogeneous Mixmaster model in the limit towards the singular
point: the so-called fragmentation process.
The extension of the BKL mechanism to the general inhomogenous case
contains the physical restriction of the "local homogeneity": in fact, the
general derivation is based on the assumption that the spatial variation
of all spatial metric components possesses the same characteristic length,
described by a unique parameter k. which can be regarded as an aver-
age wave number. Nevertheless, such local homogeneity could cease to be
valid as a consequence of the asymptotic evolution towards the singular-
ity. The conditions (9.21) do not require that the functions p a (x 7 ) have
the same ordering in all points of space. Indeed, they can vary their or-
dering throughout space an infinite number of times without violating the
conditions (9.21), in agreement with the oscillatory-like behavior of their
spatial dependence. Furthermore-, the most important property of the BKL
map evolution is the strong dependence on initial values, which produces
an exponential divergence of the trajectories resulting from its iteration.
Given a generic initial condition p°(x 7 ), the continuity of the three-
manifold requires that, at two nearby space points, the Kasner index func-
tions assume correspondingly close values. However, for the mentioned
property of the BKL map, the trajectories emerging from these two val-
ues exponentially diverge and, since the p„(x" ; ) vary within the interval
[—1/3, 1] only, the spatial dependence acquires an increasingly oscillatory-
like behavior.
In the simplest case, let us assume that, at a fixed instant of time
to, all the points of the manifold are described by a generalized Kasner
metric, the Kasner index functions have the same ordering point by point,
and pi(x~<), p2{x' ( ), pz{x J ) are described throughout the whole space, by a
narrow interval of M-values, i.e. u € [K, K + 1] for a generic integer 2 K .
We refer to this situation as a manifold composed by one "island" . We can
2 For the sake of simplicity, here we adopt X and K instead of x and k used in Sec. 7.4.2.
374 Primordial Cosmology
introduce the remainder pari of ii(x~) as (see also Sec. 7.4.2)
X(x^) = u(x~<) - [m(x 7 )] , l£[0,l) Vz 7 e£, (9.40)
where the square brackets indicate the integer part. Thus, the values of the
narrow interval can be written as u°(x^) = K° + X°(x~ t ). As the evolution
proceeds, the BKL mechanism induces a transition from an epoch to an-
other; the nth epoch is characterized by an interval [K — n, K — n+1], until
K — a = 0, when the era comes to an end and a new one begins. The new
u 1 (x 7 ) starts from u 1 = 1/X°, i.e. takes value in the interval [1, oo). Only
very close points can still be in the same "island" of u values; distant ones
in space will be described by very different integer K 1 and will experience
eras of different lengths. As the singular point is reached, more and more
eras take place, causing the formation of a greater and greater number of
smaller and smaller "islands" , providing the "fragment at ion" process. Our
interest is focused on the value of the parameter K, which describes the
characteristic wave number of the metric that increases as the islands get
smaller. This implies the progressive increase of the spatial gradients and
in principle could deform the BKL mechanism. By a qualitative analysis
we can argue that this is not the case the progressive increase of the spatial
gradients produces the same qualitative effects on all the terms present in
the three-dimensional Ricci tensor, including the dominant ones. In other
words, for each single value of K and in every island, a condition of the
form
inhomogeneous term k 2 t 25i f(t)
dominant term k 2 t 4pi
Si = l-Pi>0, f{t) = 0{lnt, In 2 *), i«l,
is still valid, where the inhomogeneous terms contain the spatial gradients
of the scale factors, which are evidently absent from the dynamics of the
homogeneous cosmological models. The possibility to neglect such gradients
towards the cosmological singularity is equivalent to state that each space
point evolves independently in agreement with the Mixmaster dynamics. It
is indeed this feature to allow to extrapolate the notion of chaotic behavior
in a local sense, when the generic cosmological solution is concerned.
From the analysis above, the fragmentation process does not produce
any behavior capable of stopping the iterative scheme of the oscillatory
regime.
The Generic Cosmological Solution Near the Singularity 375
9.3.1 Physical meaning of the BKL conjecture
The condition to deal with a Kasner-like regime even in the generic inho-
mogeneous case corresponds, as discussed above, to the inequality
If we introduce the co-moving iuhouiogoncous scale I = \/k and replace
each scale factor by the geometrical average R(t) = \/abc, the inequality
(9.41) rewrites as
R{t)l->t. (9.42)
Let us observe that R(t)l gives the typical physical inhomogeneous scale l m
of the Universe, and t stands for the order of magnitude that the cosmo-
logical and Hubble horizons take in correspondence to R(t). Therefore, the
BKL conjecture holds if we can state, up to the singularity.
/ in > rf H • (9.43)
In other words, we can say that the homogeneous Mixmaster behavior is
recovered when the physical inhomogeneous scale is super-horizon sized.
The mathematical notion of independent space points which dynamically
decouple towards the initial singularity must be replaced, on a physical
level, by independent causal regions evolving accordingly to the local BKL
map.
9.4 The Generic Cosmological Solution in Misner Variables
When considering a cosmological solution containing a number of space
functions such that a generic inhomogeneous Cauchy problem is satisfied
on a non-singular hypersurface, we refer to it as a generic inhomogeneous
model. In the ADM formalism, the corresponding line element reads as
ds 2 = N(t, xfdt 2 - h aP {dx a + N a dt){dx^ + N p dt) , (9.44)
h a p = e^' x H%l a p , (9.45)
where N is the lapse function, N a the shift- vectors and q a (t, x) three scalar
functions; the vectors 1% have components which are generic functions of
the spatial coordinates only, available for the Cauchy data. The general
case, in which ?° are time-dependent vectors, is addressed in the following
Section to simplify the variational principle. It is convenient to introduce
376 Primordial Cosmology
also the reciprocal vectors Z", such that l^l£ = <5£ and l^J.% = 5^. The
dynamics of this system is summarized by the action
JY. ■ J.
■xdt (pad t q a -NH- N a U a ) (9.46)
"Ha = -2/10/3 XI 'mCPm ~ ^6^a9b , (9.48)
p a being the conjugate momenta to the variables q a . By varying the ac-
tion (9.46) with respect to the functions N and N a , we get the super-
Hamiltonian and super-momentum constraints % = and H a = 0, respec-
tively. Let us introduce the Misner-like variables a(t,x), fi±{t,x) via the
transformation (8.26). In terms of this set of configurational coordinates,
the Hamiltonian constraints rewrite as
n = ^e- 3a (-p 2 a + P 2 + + P 2 _+V) (9.49)
H 7 = -J. 7 (f + ^ + ^ f ) + ^[ziZ3(, + - 2 V3 P _)
k (9.50)
-2^3 l 5 2 l*pJ\ I - K[(d 7 a)p a + (d 7 (3 + ) P+ + (9 7 /3_)p_]
V = --^h 3 R. (9.51)
A detailed analysis of the potential term V leads to
V = --L-e^hKx^e-W* + A|(^)e 4 ^+^-)
4k L _ (9.52)
+Ai(^)e 4 ^-^-) + W^, a, /3 ± , d,a, d s 0±j\ ,
where A a refers to the space quantities (related to v in Eq. (9.36))
\ a (x r ) = l a - VAZ a . (9.53)
To outline the relative behavior of the two terms in the potential as the
singularity is approached for a — > — oo, let us consider the quantities D =
exp(3a) and Q a defined in Eq. (8.42). Taking into account these definitions,
the potential V rewrites as
v~Y,( x2 b D2Qb ) + w ( 9 - 54a )
b
w~Y,°( DQb+Qc ) ■ ( 9 - 54b )
The Generic Cosmological Solution Near the Singularity 377
Near the cosmological singularity D — > 0, so that the term W becomes
negligible. Indeed this conclusion is supported by the behavior of the spatial
gradients, which do not destroy the features outlined above (see Sec. 9.5.2).
Through the canonical replacements
P„ = |f (9-55)
>±-m- (9 ' 56)
the classical evolution is summarized by the Hamilton- Jacobi system
+*[**(£-**£)-****&:]} (9 - 57b)
+ { "
Since sufficiently close to the cosmological singularity the potential term
becomes step by step negligible, then, deep inside the potential well, the
solution of Eq. (9.57a) reads as
S = - yjk\ + hi a + k+p + + fc_/3_ , (9.58)
where k± = k± (x 7 ) are arbitrary functions of the coordinates and the minus
sign in front of the square root has been taken considering an expanding
Universe. According to the Jacobi prescription, the functional derivatives of
the above action (9.58) with respect to k± have to be set equal to stationary
quantities c±{x 1 ) and therefore we get the following expressions for (3± in
terms of a as
/?±=7r±(xT)a + c±(xT), (9.59)
= . (9.60a)
yjkl + k 2 _
tt 2 _ = 1. (9.60b)
378 Primordial Cosmology
Substituting the solution (9.58) with (9.59) in the Hamilton-Jacobi equa-
tion (9.57b) corresponding to the super-momentum and taking into account
the relations (9.60), the last sum cancels out leaving the equations
k + + k ~ d 7 (k + + £;_) + d y k + + V3d 7 fc_
k + + k - (9.61)
4-
+d s fe (k+ - 2V3k\ - 2\/3Z^fc_] =0,
which are constraints on the spatial functions only. The above mentioned
functions c±{x 1 ) have been set equal to zero because their presence would
simply correspond to a rescaling of the vectors l®(x ). Thus, our solution
contains ten arbitrary functions of the spatial coordinates, namely the nine
components of the vectors Vt and one of the two functions 7r±. Such ten
free functions have to satisfy the three constraints (9.61); the choice of
the coordinate frame eliminates the arbitrariness of three more degrees of
freedom so that the solution is characterized by four physically arbitrary
functions of the spatial coordinates and, in this sense, it is a generic one.
So far, we have neglected the role of the potential because it influ-
ences the point-Universe evolution only via the bounces producing the
establishment of a new free motion (for a detailed discussion about the
chaotic properties of the random behavior that the point-universe performs
in the potential, see Chap. 8). This effect of the potential is summarized
by the reflection law (8.50) of the point-Universe for a bounce on one of
the three equivalent walls of the triangular poteulial V(a,/3+,/3_). This
Section shows how the generic cosmological solution toward the Big Bang
is isomorphic, point by point in space, to the one of the Bianchi types VIII
and IX models because the spatial coordinates are involved in the problem
only as parameters.
9.5 Hamiltonian Formulation in a General Framework
Let us now extend the analysis of Sec. 9.4 by dealing with a more general
representation of the three-metric tensor h a p- In the ADM formalism, a
generic set of three-vectors on the spatial surface of the splitting can be
defined as
e a a = e qj2 Old a y b (9.62)
where y b denotes three scalar functions. ()" : = Ojj(.n 7 ) a 50(3) matrix on
the hypersurface, and q a three scale factors. This representation of the
The Generic Cosmological Solution Near the Singularity 379
tetrads is equivalent to the following three-metric tensor
h a p = e q »5 ad OlO d c d a y b dpy c . (9.63)
Thus the action for the gravitational field is
S e = f dtd 3 x ( Pa d t q a + U d d t y d -NU- N°"H a ) , (9.64a)
H = %(&*-]>•*>- ±*R} 0.64b)
ri a = U a d a y a + p a d a q a + 2p a {0~ 1 ) a d a Ol , (9.64c)
where p a and II^ are the conjugate momenta to the variables q a and y d ,
respectively. The ten independent components of a generic metric tensor
are represented by the three scale factors q a , the three degrees of freedom
y a , the lapse N and the shift- vector N a ; by the variation of the action
(9.64a) with respect to p a , U a , the relations
d t y d = N a d a y d (9.65)
hold. In particular, Eq. (9.65) states that the functions y a are strictly
connected to gauge transformations. In fact, if we set N a = 0, we obtain
that y = const, i.e. they are not true dynamical degrees of freedom of
the theory. So we can try to remove them from the dynamics, by fixing
the form of N a or by solving the super- momentum constraint. This can be
explicitly done taking r\ = t, y a = y a (t,x) (which is equivalent to perform
a coordinate transformation) and getting
n b = - Pa d ^ b -2 Pa (0-^ (9-67)
and furthermore
q a (t,x) -> q a (ri,y) (9.68a)
Pa (t, x) -» p' a (r,, y) = pai-q, y)/\J\ (9.68b)
d dy b d d ,
m^lnw + lTr,
d dy b d
dx a dx a dy b '
where \J\ denotes the Jacobian of the transformation. The relation (9.68a)
holds in general for all the scalar quantities, while Eq. (9.68b) for all the
scalar densities. The action (9.64a) rewrites as
S= drid z y(p a d ri q a + 2p a {0- l ) c a d ri O a c - NU) . (9.69)
JSxR
(<).6<sd)
(9.70)
where D = exp J2 a Q a i Qa are the anisotropy parameters (8.16) and A a are
the functions
380 Primordial Cosmology
9.5.1 Local dynamics
In this scheme, the potential term appearing in the super-Hamiltonian
(9.51) in these new variables explicitly reads as
v = -iw 3r to »°) = £ x « D2Qa + E DQb+Qc ° K (^) 2 - y> v)
(9.7(
q a , Q a are the anisotropy parameters (8.16) and A a a:
*l = E [°6 V°c (Vy c A Vy 6 ) 1 . (9.71)
k,j
Assuming the y a (t, x) smooth enough (which implies the smoothness of the
coordinates system as well), all the gradients appearing in the potential V
are regular, in the sense that their behavior is not strongly divergent to
destroy the billiard representation. It can be shown (see Sec. 9.5.2) that
the spatial gradients logarithmically increase with the proper time along the
billiard's geodesies and are of higher order. As D — > the spatial curvature
3 R diverges and the cosmological singularity appears; in this limit, the first
term of V dominates all the remaining ones and can be approximated by
the infinite potential well
V = 5>~(Qa), ( 9 - 72 )
resembling the behavior of the Bianchi VIII and IX models (see Eq. (8.18)).
By Eq. (9.72), the Universe dynamics evolves independently in each space
point; the point-Universe moves within the dynamically-closed domain IIq
(see Sec. 8.3) and near the singularity, by virtue of the super-Hamiltonian
vanishing, we have dp a /drj = 0. Finally, the term 2p a (0~ 1 ) c a d rj O'^ in
Eq. (9.69) behaves as an exact time-derivative and can be ruled out from
the variational principle.
Henceforth, the same analysis developed for the homogeneous Mixmas-
ter model in Sees. 8.1 - 8.3 can be straightforwardly implemented in a
covariant way (i.e. without any gauge fixing for the lapse function or for
the shift vector).
Introducing the MCI variables (t,£,6) (see Eq. (8.53) with T(t) = t),
the super-Hamiltonian constraint is solved in the domain Uq as
-=HABM = \l(e-l)pl + -^P- [ , (9-73)
The Generic Cosmological Solution Near the Singularity 381
and the reduced action reads as
SSn Q = 5 [ d V d 3 y (p £ 0„f + Pe d n - Hadm^t) = . (9.74)
By the asymptotic limit (9.72) and by the Hamilton equations associated
with Eq. (9.74) we get de/dr] = de/dr] = and therefore Hadm = e(y a ) is,
point by point in space, a constant of motion even in the non-homoger
9.5.2 Dynamics of inhomogeneities
We will discuss how the spatial gradients of the dynamical variables evolve
toward the singularity. The result is that they increase only logarithmically
and thus are not able to destroy the BKL mechanism (we remind the reader
that the time derivatives increase as the power law t~ 2 ).
Let us introduce a different representation of the Lobaccvskij plane
through the "isotropic" variables r
r= (ri,r 2 ) = 4= (cos 6, sin 6) . (9.75)
In terms of such variables, given the matrix A a = (A^A 2 ), A\
(-\/3, \/3,0) , A\ = (1, 1, -2), the anisotropy parameters (8.42) read as
Q a =[(r + A a )\i-(A a y],
ic limit tow:
r- "HadmJ ,
and the action (9.74), in the asymptotic limit toward the singularity,
rewrites {dr/drj = 1) as
Hadm = c(r, P) = - (1 - r 2 ) \P\ , (9.77b)
P being the momentum conjugate to r. The Jacobi metric associated to
such variational principle rewrites as
Adr 2
ds = (1 _ r2)2 . M < 1 ( 9 - 78 )
and could also be derived by a direct transformation of Eq. (8.70).
We have discussed in details how the asymptotic behavior of the grav-
itational field can be reduced to the direct product of infinite equivalent
382 Primordial Cosmology
and decoupled dynamical systems, each of them described as the geodesic
motion on the Lobacevskij plane where such flow is characterized by expo-
nential instability due to the negative curvature of the manifold (the norm
of the vector connecting two nearby geodesies behaves as oc exp(s)). Then,
the scale of inhomogeneity l m decreases as
/in ~(I;) ~ l ° e M-s), (9-79)
where Iq is an mil ial Lnhomogeneous scale for which, in vacuum and in the
asymptotic limit toward the singularity, s is given by (see Sec. 8.3.1 and
Eq. (8.75))
■■EAt. (9.80)
E I ds--
Thus, for t — ¥ oo the dynamical variables r(j/ 7 ) and P(y 7 ) become ran-
dom functions of the spatial coordinates, in accordance with the point-like
Mixmaster evolution. This means that the invariant measure of the whole
system is given by the direct product of the infinite "point" measures
dn = n y ->dn{y~<,r,P), (9.81)
where 1 dfi reads in these new variables (from Eq. (8.80)) as
dn(y\ r, P) = const x d ^J" , m = j . (9.82)
Furthermore, one can use the following n-point distribution function (eva-
luated either on an init ial (list ribul ion or on a volume of the space) in order
to calculate different mean values, i.e.
= jj{( r ,-% 1 ))%,-ra(a ,
(9.83)
so obtaining that, for |y 7 — y s \ 3> la exp(— s), the averaging and correlating
functions of the dynamical variables take the forms
(r(y-')) = (P(y^)=0, (9.84)
(r„(2/> 6 (2/ 4 )> = r a r b &{yi - y S ) . (9.85)
In order to estimate the growth of the inhomogeneities, we can set r = 0.
In the synchronous time t, the variation of the time variable r can be
The Generic Cosmological Solution Near the Singularity 383
estimated by Vh ~ exp(— 3/2e~ T ) ~ t. By means of Eq. (9.79), the time
dependence of the scale of inhomogeneity takes the form
' in ~ /0 ln7W' ( }
where ho corresponds to an initial condition for the three-metric deter-
minant. This inhomogeneous scale decreases towards the singularity but,
being a coordinate length, its physical behavior is fixed also by the statis-
tical properties of the typical scale factor contained in the metric, i.e. we
have to evaluate l p h ys ~ {h®' hn)Q ~ kn(h® a ' 2 )Q, where the last relation
stands because l- ln does not depend on Q a . The quantity (h^ a ' 2 )Q is given
by the integral
h Q.,/2
p(Q a ) being the distribution function resulting from the invariant measure.
The main contribution is given by Q m ; n = and the explicit form of the
distribution reads as
p(Qa) = 2 . (9.88)
7rVQa(l-Qa)(l + 3Q a )
For very small values of Q a , we finally get the estimate of the growth of
the spatial lengths / p h ys , as
(9.89)
This means that the physical inhomogeneous scale decreases towards the
singularity (h — > 0), but only logarithmically and the inhomogeneities be-
come over-horizon-sized when approaching the singularity.
9.6 The Generic Cosmological Problem in the Iwasawa
Variables
A different formulation 3 of the problem of generic cosmological singularity
has been recently given in terms of the so-called Iwasawa variables. This
particular parametrization of the spatial metric allows a unique decompo-
sition of h a p in the product of two triangular matrices and a diagonal one,
i.e.
h = Af T D 2 Af , (9.90a)
3 In this Section we adopt the signature (-,+,+,+) and k = 1.
sr 29, 2010 11:22
Primordial Cosmology
/l ri\ Tl2
M = 1 n ;
\0 1
-P o
e- fj2 I . (9.90c)
This representation corresponds to the Gram-Soli midl orthogonalization of
the initial coordinate coframe dx a , that is
h a pdx a dx = e - 2/3 Vf9 a (9.91)
where 4 8 a = J\f a a dx a . The explicit form for h a g is
h a0 = e~ 2pa N a a N a p. (9.92)
The Lagrangian C g (2.11) in vacuum can be rewritten as
£ g = Y7f [**^ + \ E e2ipb ~' n ^ a ^af] + ^ 3 i? , (9.93)
K L a<fc J K
where Af a a denotes the inverse of Af a a and we introduced the "metric" g b
with signature (-,+,+) in the /3-space defined as
'°- 1 " 1 \ h (\-?-\\
10-1, g ab = -| § -| • (9.94)
1-10/ \-5-3 5/
A standard Logcudn' i rausforiualion yields the canonical formulation we
are interested in. Let p a denote the momentum conjugate to (3 a and P a a a
lower triangular matrix whose components P a are the momenta conjugate
to the Iwasawa variables n a , i.e.
/ 0\
P a a = p 1 . (9.95)
\p2 p3 q)
For N a = 0, Eq. (2.71) explicitly rewrites as
S g = J dt J d 3 xl Pa d t (i a + P a d t n a
\z ab Pa Pb + \ £ e^~^ (P\M\) 2 - A s J 1 .
a<6 J J
(!).<)())
2k7V I 1
"7T
In Eq. (9.96) the super-Hamiltoniarj can be idem, Lfied as the term in square
brackets.
4 It follows also that the frame vectors are given by e a = J\f a a d a .
The Generic Cosmological Solution Near the Singularity 385
9.6.1 Asymptotic freezing of the Iwasawa variables
The asymptotic dynamics of the Iwasawa degrees of freedom can be studied
in the simple case of a Kasner like solution in vacua. The details of such
a model are discussed in Sec. 7.2 where, in particular, the line element is
of the form (7.49). Adopting the parametrization as in Eq. (9.90), one can
obtain the relation of the new variables (/3 a , n ) with the Kasner parameters
p a and with the frame vectors (denoted as l^ in order to avoid confusion
with n a ), i.e.
p\t) =
-5'
iX ,
P 2 {t) =
io.
X -
InY) ,
I3 3 (t) =
i,„
i,:^
■) • (lW
Al^)] :
t 2p lZ (D Z (D + t 2p a j(2)j(2) + ^(^(3)
ni(t) = ,
t 2 Pll W l W +t 2 PH (2) l{ 2) +t 2 P3| (3) | (3)
n 2 (t) = 1 —^ jL_3 ^- ,
n 3 (t) = 1 (t^+^aw - W'xW - 4 X M 2) )
+ i 2 Pl+ 2 P 3 (/ (l) / (3) _ 4D / (3) )(/ (D / (3) _ / (1) Z (3) ) (997b)
+t 2 Pa+ 2 P3(| (2) | (3) _ z (2) z (3) )(| (2) z (3) _ ,(2)j(3)^
where the functions X and V read as
X(t) = i 2 ' 1 ^) 2 + i 2 »*(Z< 2) ) 2 +t 2p3 (/f } ) 2 , (9.97c)
Y (t) = t 2 » +2 *>(i[% 2) - 4 1} 4 2) ) 2 +t 2p i +2p °(i[ 1) ii 3) - 4 1} 4 3) ) 2
+ f 2 P2+ 2 P3(Z (2) z (3)_ ; (2) z (3) )2 ^ (997d)
Let us assume that the Kasner exponents p a are ordered as p\ < p 2 < Vz-
When approaching the singularity (t -> 0), on one hand we have that
the (3 a variables behave as combinations of the scale factors of the metric
a(t), b(t), c(t) (i.e. the power law in Eq. (7.49)); on the other hand, the
Iwasawa degrees of freedom asymptotically "freeze out" of the dynamics,
being
j(i) j(i) z (i) z (2) _ z (i) z (2)
limni = ^-, limn 2 = -7TT, limn 3 = ) 3 3 ) , (9.98)
t->0 i 1 -. 1 ) t-yO jW t-yO I 'I — I 'I
386 Primordial Cosmology
i.e. they assume constant values.
This result is in agreement with the one obtained in Sec. 9.2.3, where
we showed how the Kasner axes rotate when approaching the singularity.
That law states that two of the three Kasner axes rotate of an angle of
order of unity, even when getting closer and closer to the initial singularity,
implying that the Kasner frame does not admit a stationary limit although
the Iwasawa variables remain unchanged.
Let us consider the case of a BKL piecewise solution: in two consecutive
epochs, the line element has the same functional form as in Eq. (9.19) with
coefficients given by Eq. (9.20), i.e. in one epoch we have that
Kp =Y,t 2 ^H£lf , (9.99a)
while in the other one
3
h a p = J2 t^'^Hflf . (9.99b)
It can be shown that the asymptotic relation n a = n a (l^) and n' a =
n' a (l^') is the same as in Eq. (9.98), thus a direct substitution of Eq. (9.38)
in Eq. (9.98) yields n a = n' a .
9.6.2 Cosmological billiards
Let us now formulate the BKL dynamics in this framework. The starting
point of the analysis of system (9.96) is that (3 a is expected to become a
time-like vector in the vicinity of the initial singularity, i.e. g a bP a P h < 0.
This condition allows one to introduce a new set of configurational variables,
the hyperbolic planar ones (p, 7*) defined as follows
p 2 = -gab(3 a (3 b , (9.100a)
/3 a = p7 a , 7 Q 7a = -l. (9.100b)
In Eq. (9.100), p is a "radial" coordinate that diverge for t — > 0, while 7 a
are the coordinates on the hyperbolic space II. The line element associated
to g a b takes the form
da 2 = -dp 2 + p 2 dU 2 , (9.101)
dU 2 being the standard metric on the hyperbolic space. Rescaling further
the radial variable p as A = In p, and taking the lapse function in the form
N = p 2 Vh, we obtain the super-Hamiltonian as
NU = J [-ir\ + tt 2 ] + V s + V G • (9.102)
The Generic Cosmological Solution Near the Singularity 387
Here, tt\ is the momentum conjugate to A, 7r 7 denotes the collection of
momenta conjugate to 7", Vs is related to the kinetic terms of the off-
diagonal components (the second term in the square brackets in Eq. (9.96),
also called the symmetric potential), while Vg = —h 3 R/4:K 2 is the standard
gravitational potential.
In such formulation, the asymptotic dynamics is governed by the scale
factors f3 a , while the remaining Iwasawa degrees of freedom n a freeze to
constant values. The two potential terms in Eq. (9.102) become (asymp-
totically) functions of the 7° variables only and take the form of the infinite
wells encountered in Sec. 9.5. Indeed, we can model any of the potentials
V{p a ,n a )=Y J CAe- 2wA{fia \ (9.103)
where ca are some functions of all the variables except f3 a and their conju-
gate momenta n a , and WA((3 a ) are linear combinations of the /3 variables.
We can now recover the asymptotic behavior discussed previously. In
the limit t — > 0, i.e. p — > 00, we have that Eq. (9.102) takes the form 5
Hoo = \ (n 2 x + ^)+Y. ©oo(-2^(7)) , (9-104)
see Eq. (8.18). In Eq. (9.104), the sum is over the restricted set of the
so-called "dominant walls" , that is the minimal collection of walls sufficient
to define the billiard table, and obtainable from the following condition
W(7)>0} =► {«u(7)>0}. (9.105)
The asymptotic picture can be summarized as follows. Since the poten-
tials V depend only on the variables on the hyperbolic space, the remaining
degrees of freedom become asymptotically constants of motion, together
with T-Lca. The dynamics is described by the free motion of a non-relativistic
point particle within a billiard, whose surrounding walls are given by the
relations WA'{l a ) = 0. These walls are time-like hvpcrplauos. with space-
like normal vectors (in the /? space).
The scheme described above can be extended to include any kind of
matter (and also to any number of dimensions, see Sec. 9.7), and the re-
sulting billiard can have finite or infinite volume. This is a difference of
crucial importance because, from the standard theory of geodesic motion
on hyperbolic billiard, it is well known that the motion is chaotic if the
5 A key point in the reduction of the potential to a well is that c A i > 0.
388 Primordial Cosmology
volume is finite, while it is not chaotic if the volume is infinite. In the first
case we have an infinite sequence of bounces, like those characterizing the
BKL evolution; in the latter, only a finite number of bounces takes place.
It is worth noting that in the case of standard gravity in vacua, the volume
of the billiard is finite, as we will discuss in detail in Sec. 10.9.
9.7 Multidimensional Oscillatory Regime
Let us consider a (d + l)-dimensional space-time (// > 3), whose associa-
ted metric tensor obeys a dynamics described by the generalized vacuum
Einstein equations
d+1 R lk = 7 (i,k = 0,1,. .., d) , (9.106)
where the {d + l)-dimensional Ricci tensor takes its natural form in terms
of the metric components gik{x l ). It can be shown that the inhomogeneous
Mixmaster behavior finds a direct generalization in correspondence to any
value of d. Moreover, in correspondence to d > 9, the generalized Kasner
solution acquires a generality character, in the sense of the number of arbi-
trary functions, i.e. without a condition analogous to the one in Eq. (9.22).
In a synchronous reference (described by the usual coordinates (t,x)), the
time-evolution of the (i-dimensional spatial metric h a p{t,x) singles out an
iterative structure near the cosmological singularity (t = 0). Each single
stage consists of intervals of time (Kasner epochs) during which h a p takes
the generalized Kasner form
h a0 (t, x)=J2 t 2pa Ol . (9.107)
a=l
where the Kasner indexes p a (x J ) satisfy
d d
5>a(* 7 ) = £Pa(* 7 ) =1, (9-108)
o=l o=l
and l^x 1 ), . . . , laix 1 ) denote d linear independent vectors whose compo-
nents are arbitrary functions of the spatial coordinates. In each point
of space, the conditions (9.108) define a set of ordered indexes {p a }
(pi < P2 < ••• < Pd) which, from a geometrical point of view, fixes one
point in K d , lying on a connected portion of a (d — 2)-dimensional sphere.
We note that the conditions (9.108) require p\ < and pd-i > 0, where
the equality takes place for the values p\ = . . . = pd-i = and only p d = I.
The Generic Cosmological Solution Near the Singularity 389
For two consequent Kasner epochs, the following d-dimensional BKL map,
linking the old Kasner exponents p a to the new ones q a , holds
_ -pi - P _ P 2 _ Pd-2
qi ~l + 2 Pl +P' q2 ~ l + 2 Pl + P''"' qd ~ 2 ~l + 2 Pl +P
(9.109)
p d -i + 2gi + P Pd + 2 Pl + P
qd - 1= l + 2 Pl + P ■ qd= l + 2 Pl + P (9 ' 110)
where P = J2t=2 Pa-
It can be shown that each single step of the iterative solution is stable,
in a given point of the space, if
limt 2d RP=0. (9.111)
The limit (9.111) is a sufficient condition to disregard the dynamical effects
of the spatial curvature in the Einstein equations. An elementary compu-
tation shows how the only terms capable to perturb the Kasner behavior
in t 2 d R contain the powers t 2aabc , where a a \, c are related to the Kasner
exponents as
a abc = 2 Pa + ]T Pd , (a^b,a^c,b^d), (9.112)
d^a,b,c
and for generic l a , all possible powers t 2aabc appear in Eq. (9.111). This
leaves two possibilities for the vanishing of t 2 d R^ as t — > 0. Either the
Kasner exponents can be chosen in an open region of the Kasner sphere
defined in (9.108), such to have a a bc positive for all triples a,b,c, or the
conditions
a abc {x~ 1 )>Q yx\...,x d (9.113)
are in contradiction with Eq. (9.108), and one must impose extra conditions
on the functions l a and their derivatives. The second possibility occurs, for
instance, for d = 3, since a a b c is given by 2 Pa , and one Kasner exponent
is always negative, i.e. ai,d-i,d- It can be shown that, for 3 < d < 9, at
least the smallest of the quantities (9.112), i.e. ai t d-i,d is always negative
(excluding isolated points { P i} in which it vanishes). Thus, Eq. (9.107) is
a solution of the vacuum Einstein equations to leading order if and only if
at least the vector I 1 = I, associated with the negative ai,d-i,d, obeys the
additional condition
Z-VA/ = 0, (9.114)
and this requirement kills one arbitrary function of the space coordinate,
as we have seen in details in Sec. 9.2.2.
390 Primordial Cosmology
Finally, for d > 10 an open region of the (d — 2)-dimensional Kasner
sphere where a\ t d-i,d takes positive values exists, the so-called Kasner Sta-
bility Region (KSR). For 3 < d < 9, the evolution of the system to the sin-
gularity consists of an infinite number of Kasner epochs, while for d > 10,
the existence of the KSR, implies a deep modification in the asymptotic
dynamics. In fact, the indications presented by Demaret in 1986 and by
Kirillov and Melnikov in 1995 in favor of the "attractivity" of the KSR, im-
ply that in each space point (excluding sets of zero measure) a final stable
Kasner-like regime appears.
In correspondence to any value of d, the considered iterative scheme
contains the right number of (d+l)(d—2) physically arbitrary functions of
the spatial coordinates, required to specify generic initial conditions (on a
non-singular space-like d-hypersurface) . In fact, we have d 2 functions from
the d vectors I and d — 2 Kasner exponents; the invariance under spatial
reparametrizations allows to eliminate d of these functions, and other d
because of the 0a Einstein equations (which play also the role of constraints
for the space functions). This piecewise solution describes the asymptotic
evolution of a generic inhomogeneous multidimensional cosmological model.
9.7.1 Dilatons, p-forms and Kac-Moody algebras
We summarize some properties about the insertion of p-forms and dilatons
in the gravitational dynamics in the multidimensional case.
The inclusion of massless p-forms in a generic multi-dimensional model
can restore chaos when it is otherwise suppressed. In particular, even
though pure gravity is non-chaotic in d = 10 space-times, the 3-forms of
d+ 1 = 11 supergravity make the system chaotic. The billiard description
in Iwasawa variables given in Sec. 9.6 in the four-dimensional case is quite
general and can be extended to higher space-time dimensions, with p-forms
and dilatons. If there are n dilatons, the billiard is a region of the hyper-
bolic space Ud+n-i, and in the Hamiltonian each dilaton is equivalent to
the logarithm of a new scale factor fi a . The other ingredients that enter
the billiard definition are the different types of the walls bounding it: in
addition to the symmetry and to the gravitational walls Vs , Vq , in the gen-
eral case p-form walls are also present (that can be divided in electric and
magnetic walls). All of them are hyper-planar, and the billiard is a convex
polyhedron with finitely many vertices, some of which are at infinity. We
have seen in Sec. 8.7.2, how an Abelian 1-form can restore chaos in higher
dimensional homogeneous models. The analysis performed in the case when
The Generic Cosmological Solution Near the Singularity 391
p-forms are present can be thought as the maximal generalization of this
scheme.
Finally, we want to stress that some of the billiards we have described
can be associated with a Kac- Moody algebra; in this framework, the asymp-
totic BKL dynamics is equivalent to that of a one-dimensional non-linear
cr-model based on a certain infinite dimensional space.
9.8 Properties of the BKL Map
In this Section we discuss from a mathematical point of view the main
properties of the BKL map in a generic number of dimensions d as outlined
by Elskens and Henneaux in 1987.
A set of Kasner indexes is a set of parameters p a (a = 1 , . . . , d) that
satisfy the conditions
X> = X>a = l, (9-115)
Pi <P2 <-..<Pd-i<Pd- (9-116)
The constraints in Eq. (9.115) define the so-called Kasner sphere (in d — 2
dimensions), while the inequalities (9.116) coincide with the request of an
ordered set of Kasner parameters.
We call BKL map the application T : {p a } e R d -»■ {p' a } G R d such
that
p' a = ordering of q a (9 . 1 1 7a)
(9.117b)
(9.117c)
(9.117d)
(9.117e)
(9.117f)
1 + 2 Pl + E '
1 + 2pi + E '
Pd-i
1 + 2pi + E '
p d _i + 2pi + E
1 + 2pi + E '
p d + 2pi + E
l + 2p! + E '
392 Primordial Cosmology
where £ is defined as
= £*-
We have seen in the multidimensional case that the occurrence of such
transition is possible if and only if at least one of the quantities a a bc is
negative, which are defined as
a abc = 2 Pa + Yl Pd- (9-119)
d?{a,b,c}
This situation always happens in a number of space-time dimensions n =
d + 1 < 10. On the other hand, when n > 11 a region KSR of non-zero
measure exists where all the a aoc are greater than zero, so that the BKL
oscillations will stop as soon as the transit ion mechanism brings to a set of
Kasner indexes in this region.
We have already studied in detail the three-dimensional case of t his map
(see Sec. 7.4) that exhibits stochastic features. In the higher dimensional
case the following two properties hold:
(i) The BKL map is chaotic when the number of spatial dimensions d
is smaller than 10, and almost every set of initial p a evolves visiting
any parameter region of non-zero measure,
(ii) The map is not chaotic in d > 10 and, for almost every set of initial
p a , the evolution reaches the KSR.
Indeed, there is no contradiction between i) and ii). In fact, even if one
can pass from d > 10 to d < 9 by a dimensional reduction, i.e. by taking
some of the Kasner indices equal to zero, such sets correspond to regions
of zero measure. This phenomenon ensures that in d > 10 some subsets
of points of zero measure exist with never ending oscillations of the scale
(.actors.
9.8.1 Parametrization in a generic number of dimensions
We will now discuss a parametric representation of the Kasner exponents
that generalizes the one expressed in terms of u for the three-dimensional
case (see Sec. 7.2).
sr 29, 2010 11:22
Tin (U a ( , i the Singularity
Pd < 1- The parametrization is then given by
Pa = Y> a=l,...,d-2,
d-2
= a ~ X T , (9.120c)
(9.120d)
r = £k-^5>
This parametrization is valid in the range defined by the inequalities coming
from the conditions pd-2 < Pd-i < Pd and then
2v d - 2 < 1 - £ v a , (9.123a)
d-2 /d-2 \ 2
2Z = J^vl+ \^2vb\ >2. (9.123b)
o=l \6=i /
This new set of parameters v a is ordered by construction. The correspond-
ing inverse relations read as
v a = T Pa = Pa a = l,...,d-2, (9.124)
1 -Pd
which provides the new parametrization plus the additional relations
d-2
v d -i = l-^Tv a , (9.125a)
UX>6-1) . (9.125b)
Vd =
6 The case Pd = 1 corresponds to the fixed set {0, 0, ... , 0, 1} and will be recovered a
the point at infinity.
394 Primordial Cosmology
Case d — 3. We have only d — 2 = 1 independent parameters, that is
v = V\. From Eq. (9.124) we get
v = -^— . (9.126)
1 ~P3
From Eq. (9.121) we have T = v 2 - v + 1, so that from Eq. (9.120) we get
This set of relations is constrained by those in Eq. (9.123) that now read
w<l/2, v 2 >l 7 (9.128)
that mean v < — 1. This way, Eq. (7.55) is recovered by setting v = —u.
Case d — 4. We have 4 — 2 = 2 independent parameters, 5\ = v\ and
(5 2 = t>2, so that
pi = Y> P2 = |' ( 9 - 129a )
P3 = ^P^, Pi = ^±, (9.129b)
T = <5? + 5l + 6x62 -Si-S 2 + 1- (9.130)
The range of validity of this parametrization is bidimensional and is given
by the inequalities
(?-i~\ 1 (9.131)
\6l + 6% + 6x62 > 1
together with the condition <5i < (52. The solution is sketched in Fig. 9.1,
while in Fig. 9.2 the tridimensional representation of Pi,P2,P3 and pi is
presented in the allowed domain.
9.8.2 Ordering properties
In this Subsection we will study the rein I ion existing between an ordered
set of Kasner exponents and an unordered one.
Let us denote q a as the coordinates of a generic point on the d-
dimensional Kasner sphere, i.e. {q a } is a generic unordered set of exponents.
Let Wf be the corresponding paramclrizaUon via (9.122) and (9.124). If
the q a are not ordered, then one of the inequalities ( 9. ! 23 1 will be violated.
At the same time, let p a be the ordered set associated to q a
{Pa} =i,...,d = ordering oi{q b } b=1A (9.132)
sr 29, 2010 11:22
World Scientific Book - 9in x 6in
The Generic Cosmological Solution Near the Singularity 395
W^
[vastier p
e 9.1 Domain of validity of the (^ . 5 2 ) p
ition for the four-dirr
and {v g } _ 1 d the corresponding parametrization. This way, the relation
between the qb and p a is straightforward.
From an analysis of the parametrization (9.124) and (9.122), the rele-
vant quantities are T, Z, v d -i and v d , (defined in Eqs. (9.122), (9.123b),
(9.125a) and (9.125b), respectively) which depend on qd-i,qd and on the
relation with the other d — 2 exponents. This implies that there are three
different cases only:
(i) qd and qd-i are the largest indexes;
(ii) qd is the highest but qd-i is not the second highest;
(iii) qd is not the largest index.
I'miionliid Cosmology
Figure 9.2 Three-dimensional representation of the ordered (pi < P2 < P3 < Pa) Kasner
parameters in the four-dimensional case as a function of <5i and 02 • One can see how P4
is always greater than zero, while pi is always negative.
Let us discuss each case in more details.
(i) Qd > <7d-i > q a Va = 1, . . . , d — 2
In this case, pd = qd and pd—i = Qd—i, while (pd-2, ■ ■ ■ ,Pi) =
ordering of (r/,;_2, • • • ,<Zi)- These relations imply that
{v g } = ordering of {w f } (9.133)
and leave T, Z, Vd-i and Vd unchanged. Finally, the transforma-
tion (9.133) is a permutation, and thus has unitary modulus,
(ii) qd ^ 9a and qd-i ^ q a Va = 1, . . . , d — 2
In this case, we have that
T( W ) = T», (9.134)
so that we have to order the first d — 1 elements only; this can be
done in three steps by applying (9.133) as
w'j = ordering of w/, / = 1, . . . ,d — 2
(9.135a)
w'd-l = Wd-l
(9.135b)
w d = w d
(9.135c)
by exchanging Wd-i with Wd~2
w" f =w' fi / = l,...,d-3
(9.136a)
W "d-2 = W d-1 = l ~ Z^ W 'f
(9.136b)
w'd =w' d =w d
(9.136c)
The Generic Cosmological Solution Near the Singularity 397
and finally by applying (9.133) we get
v g = ordering of w" g , g=l,...,d-2 (9.137a)
u d _i = «#_! (9.137b)
v d = w" d (9.137c)
It has been shown by Elskens and Henneaux (1987) that these
combined transformations have unit modulus, thus implying that
the volume in the v-space is preserved.
(in) qdjtqa Vo = l,...,d-l
Without loss of generality, we can assume qd-i to be the greatest
exponent (indexes can be rearranged this way following the previ-
ous procedure). If qd-i > qd then inequality (9.123b) is violated.
In this case, T(w) is given by
T(«;) = — !— = — ^ . (9.138)
1 - qd 1 - Pd-i
Because Wi = Ygj, to obtain the ordered set {v} we have to rescale
T by a factor (1 — pd-i)/(l — Pd), which is equal to
1 -Pd-T,
l-Pd
because Z{v) = 1 + v d — fd-i- From (9.139) follows that
Z ^ = 7TT ( 9 ' 14 °)
and so the correct transformation is given by
v S=^f-s- (9- 141 )
In this case the transformation is not unitary, but has modulus
1/Z> 1.
9.8.3 Properties of the BKL map in the v space
The BKL map assumes a simple expression in terms of the parametrization
(9.120). A direct substitution yields that, if taking the q a as in Eq. (9.117)
where q\ and qd-i have been exchanged, 7 the map Tr for the reduced
7 This re-statement of the map leads to the same set of Kasner exponents si
a permutation.
Primordial Cosmology
variables is given by
wi=v 1 + l, (9.142a)
w 2 = v 2 , (9.142b)
w d _ 2 = v d „ 2 . (9.142c)
Together with Eq. (9.142), the following relations hold
tOd-i = u«i_i - 1 , (9.142d)
w d = Z(v) + Vl , (9.142e)
Z(w) = Z(v)-v 1 +v d - 1 . (9.142f)
The map Tr : {v} — > {w} defined by Eq. (9.142) exhibits several prop-
erties which are summarized below but we shall not prove here. Given a
set of ordered {«}, then
• at least one set {q} such that {v} is the image of {q} through Tr,
i.e. Vf = Tnq a , always exists.
• If Wf = TftV a , then the new parameters need at least one rear-
rangement for d < 9 if the exchange q\ <-> q d -i is not performed.
• Such rearrangement necessarily involves q d -\ which cannot remain
at the next-to-last place.
• The Kasner Stability Region KSR coincides with the following set
of values available for the Kasner indices:
(1) if d < 8, KSR= {oo} = {(0, 0, . . . , 0, 1)}
(2) if d=9, KSR = {oo, c, c'} where c = (-1, -|, -|, |, . . . ,|)
and d = (- ^ , -\, -\, -\, \, \, \, |, |)
(3) for d > 10, the KSR has non-zero measure.
9.9 Guidelines to the Literature
The original analysis on the Bianchi IX model stability presented in Sec. 9.1
was given by Regge & Hu in [245]; some developments can be found in
[243,244].
The first derivation of the generalized Kasner solution discussed in
Sec. 9.2.1 was given by Lifshitz <V Khalatnikov in [312]. For a complete
presentation of the inhomogeneous BKL solution, as in Sec. 9.2.2, see [65]
and for the specific case of the small oscillations, see [58]. A critical re-
analysis of the BKL solution can be found in [44] . For a more recent study
The Generic Cosmological Solution Near the Singularity 399
on the inhomogeneous Mixmaster, see [236], while for numerical support to
the Conjecture, see for example [78,191,459]. The mechanism of Kasner
axes rotation, as presented in Sec. 9.2.3, is provided in [65]. More recently,
Belinskii reviewed the main problems in the topic of cosmological singular-
ity in [57].
A discussion of the fragmentation process, illustrated in Sec. 9.3, is
addressed by [284] and [348].
The Hamiltonian formulation of the inhomogeneous Mixmaster model in
terms of Misner variables, as in Sec. 9.4, can be found in [260]. For analysis
of the inhomogeneous Mixmaster model when the Kasner vectors are time-
dependent quantities as discussed in Sec. 9.5, see [283]. An extension of the
formalism presented in Sec. 9.5.1, addressed in view of a generic choice of
the gauge can be found in [70] where the covariance of the inhomogeneous
Mixmaster chaos is outlined. A proof concerning the negligibility of the
spatial gradients presented in Sec. 9.5.2 in the inhomogeneous Mixmaster
(in more than 4 dimensions and also in the presence of a scalar field) can
be found in [285].
For a complete discussion of the multidimensional inhomogeneous Mix-
master (in vacuum as well as in presence of matter) adopting Iwasawa
variables as presented in Sec. 9.6, see the review [137] or the lecture [135].
A very recent result about the structure of the space-time in the BKL limit
is given in [136]. For a discussion on the implication of such a framework
on the de-emergence of space-time near the singularity, see [138].
The Dynamical System approach in the non- homogeneous case, that we
did not present here, is reviewed in [236], A rigorous attempt to formulate
the BKL Conjecture in such framework was firstly given in [441] and de-
veloped for example in [12,191]. For what concerns the BKL Conjecture in
the connection formalism for GR, see [27].
Tlii" original discussion of the mult idimensioual oscillatory regime, pro-
vided in Sec. 9.7, can be found in [146,147]. The ergodic theory of the
multidimensional BKL map, as discussed in Sec. 9.8, is analyzed in [169].
This page is intentionally left blank
PART 4
Quantum Cosmology
In these Chapters, we provide a characterization of the quantum behavior
of the cosmological models, as described by the implementation of the most
viable quantum gravity approaches.
Chapter 10 concerns the derivation of canonical quantum gravity in the
metric approach and its application to various cosmological contexts. We
formulate the quantum cosmological problem and compare the Dirac quan-
tization procedure with the path integral formalism.
Chapter 11 provides a discussion of generalized Heisenberg algebras, re-
lated to cut-off features of space-time. Particular attention is devoted to
the so-called polymer quantization approach ( which mimics Loop Quantum
Cosmology features) and the generalized uncertainty principle prescription
(related to the String theory paradigm).
Chapter 12 is focused on the derivation of the Loop Quantum Gravity
theory. The quantum dynamical implications are developed on a cosmo-
logical setting, outlining successes and shortcomings of the minisuperspace
formulation. Quantum cosmological model based on the extension to the
minisuperspace of the generalized Heisenberg algebras are also presented
for some relevant cases.
This page is intentionally left blank
Chapter 10
Standard Quantum Cosmology
In this Chapter we face the analysis of the quantum gravity problem in
the framework of the metric approach and we implement this theory to the
study of the quantum Universe morphology. Indeed, the functional nature
of this scheme of quantization makes solvable the quantum dynamics prob-
lem just in correspondence to highly symmetric models, as those concerning
the cosmological setting. In this respect, when treating the quantum Uni-
verse, we will be able to analyze systems with a finite number of degrees of
freedom, i.e. the so-called minisuperspace models.
We start by deriving the Wheeler-DeWitt (WDW) equation describing
the quantum dynamics of the gravitational field, as a direct consequence of
implementing to operator level the classical Hamiltonian constraints. The
resulting physical states are annihilated by the super-Hamiltonian and the
super-momentum operators and by the momenta conjugate to the lapse
function and the shift vector as well. The implications of these quantum
constraints are clearly outlined with particular reference to the equivalence
between the super-momentum conditions and the invariance of the state
function under spatial diffeomorphisms.
The link between this Dirac approach to quantum gravity and the path
integral formalism, as heuristically extended to the gravitational sector, is
properly addressed aiming to show that the WDW scheme can be recovered
by the path integral approach in some limit.
The annihilation of the state function by the super-Hamiltonian opera-
tor implies the absence of a time evolution of the quantum gravity configu-
rations. We face this problem of the canonical quantum gravity, known as
the frozen formalism, within two different frameworks:
(i) fixing a time variable before implementing the quantization proce-
dure:
404 Primordial Cosmology
(ii) recognizing a physical time at quantum level, when the Dirac con-
straints have already been implemented as operators.
Finally we outline as main point the possibility to deal with a timeless
quantum gravity, in which evolution is essentially a relational property
between different system components.
We then face the quantum gravity problem under the cosmological hy-
potheses, leading to what is commonly called quantum cosmology. We
discuss the form that the general theory takes in correspondence to the
symmetry restrictions of the primordial Universe. The nature of the mini-
superspace and how the cosmological singularity could be removed by the
quantum evolution are analyzed in some detail.
The path integral representation of the isotropic Universe is given to de-
rive the precise equivalence we mentioned above with the Dirac quantization
procedure. The full mathematical equipment to achieve this equivalence
statement is provided and the most relevant steps of the proof outlined.
The possibility to use a real scalar field as a good relational time is suc-
cessfully explored, especially in view of the implementation we will make of
this scheme in Chap. 12, when discussing the Big Bounce in Loop Quantum
Cosmology.
Wide space is dedicated to the interpretation of the Universe wave func-
tion in the semiclassical approximation. According to the Vilenkin analysis,
we show that a proper separation of the system into a classical part and
a small quantum portion allows to recover a Schrodinger dynamics for the
latter system component. An elucidating example concerning the Universe
isotropization is presented, where the Universe volume is the classical time
coordinate and the small Universe anisotropies evolve in a full quantum
scheme.
We discuss how boundary conditions for the WDW equation can be
properly fixed in order to make predictions from the theory. In particular,
we analyze the important case of the no-boundary proposal and of the
tunneling boundary problem, outlining and comparing their implications
on the quantum Universe morphology.
After the setting of the full quantum cosmology theory, the Chapter ends
with a series of important and meaningful applications, i.e. the isotropic
Universe in the presence of a scalar field, the Taub Universe and the Mix-
master model, both in the Misner and Misner-Chitre variables representa-
tion. In particular, the quantum Mixmaster in the half Poincare plane is
finely described to fix some relevant features as the spectrum discreteness
Standard Quantum Cosmology 405
and the existence of a point-zero energy.
10.1 Quantum Geometrodynamics
In this Section we will give an overview of canonical quantum gravity in
the metric formalism. In this framework, the three-metric of the Cauchy
surfaces is adopted as the configuration variable. This approach, known as
the WDW theory, will also be related to the path integral formulation of
quantum gravity.
10.1.1 The Wheeler-DeWitt Theory
As we have seen in Sec. 2.3, the Einstein theory of gravity can be written as
a dynamical system subjected to first-class constraints with a Dirac algebra
(2.80)-(2.80c). The configuration space of canonical gravity, on which the
constraints are defined, is the space of all the Riemannian three-metrics
Riem(E) modulo the spatial diffeomorphisms group Diff(E) of the slicing
surface E. Explicitly, it reads as
^ = Wr- (10 - 1 »
This is the space of all the three-geometries and is known as the Wheeler su-
perspace which is infinite-dimensional, but of course there is a finite number
of degrees of freedom at each space point.
To implement the quantization of such a constrained system there are
essentially two ways. The first one relies on solving the constraints (2.77) at
a classical level in order to deal with a formulation based on unconstrained
physical variables only. This approach is the so-called reduced quantization
procedure and has several faults even in more simple frameworks, as for
example in quantum electrodynamics it is consistent in the non-interacting
case only.
Thus, one usually follows the second approach to quantize first-class con-
strained systems, known as the Dirac scheme (discussed in more details in
Sec. 12.1). In this scheme the quantum theory is constructed without solv-
ing the classical constraints. The Poisson brackets are then implemented
as commutators and the constraints select the physically allowed states.
In particular, given a (first-class) constraint C = 0, a physical state must
remain unchanged when one performs (gauge) transformations generated
by C. This consideration has to be implemented at a quantum level. The
406 Primordial Cosmology
physical states are thus the ones annihilated by the quantum operator con-
straints, i.e. by imposing the relation
C|*)=0. (10.2)
It is worth noting that the reduced and the Dirac quantizations are formally
equivalent to each other but, in general, it may break down because of
fa< I or-ordcring problems.
The first step of the canonical quantization a la Dirac of GR in the
metric formulation relies on implementing the Poisson algebra (2.74a) -
(2.74c) in the form of the canonical commutation relations
]fi a p(x,t),h jS (x',t)^ =0 (10.3a)
[n a0 (x, t), rr 7< V, *)] = o (io.3b)
\h lS (x,t),U a>:) (x',t)\ = i5^ ) 6 3 {x-x'). (10.3c)
This is only a formal prescription and requires some remarks:
(i) Eq. (10.3a) is a kind of microcausality condition for the three-
metric field, though the functional form of the constraints is in-
dependent of any foliation of space-time. This confirms that the
points of the three-manifold E are space-like separated,
(ii) The above relations are not compatible with the requirement that
the operator h a p(x, t) has a positive definite spectrum. In fact, the
classical quantity h a ;(•''■ ' ) ' s a Riemannian metric tensor, i.e. it is
positive definite. Such property should also be implemented at a
quantum level. More precisely, for any (non-vanishing) vector field
£ a {x), the classical relation
h(£ ® = f d 3 xC^h a/3 > (10.4)
holds. It is reasonable to require that this feature is implemented
at a quantum level as
M£<g>O>0. (10.5)
However, we know that if II a/3 is a self-adjoint operator it can be ex-
ponentiated as an unitary operator. The spectrum of this operator
takes negative values, similarly to the spectrum of the translation
operator in quantum mechanics being the entire real axis. The
problem is to give a physical meaning to these negative values.
The self-adjoint property of the momentum operator is therefore
Standard Quantum Cosmology 407
not compatible with the positive definite requirement of the three-
metric operator. The positiveness of the configuration operator can
be recovered by restricting the Hilbert space (for an explicit exam-
ple, see Sec. 10.8), but this implies that the momentum operator
is no longer self- adjoint.
Proceeding in a formal way, one imposes the constraint equations (2.77)
as operators to select the physically allowed stales, thai is
H (h,fl) q = , (10.6a)
# a (£,n)* = 0. (10.6b)
Here, ^ is known as the wave function of the Universe. As we have seen,
the Hamiltonian for the Einstein theory (2.75) reads as
H= f d 3 x{NH + N a H a ), (10.7)
and therefore, considering Eqs. (10.6) in a putative Schrodinger-like equa-
tion such as
i— * t = H* t = 0, (10.8)
dt '
the state functional \P t results independent of "time" . This is the so-called
frozen formalism because it apparently implies that nothing evolves in
a quantum theory of gravity. Loosely speaking, an identification of the
quantum Hamiltonian constraint as the zero-energy Schrodinger equation
Hv]> = holds. Such feature is known as the problem of time and deserves
to be treated in some details in Sec. 10.2. It is worth noting that, by the
expression (10.7). we assumed the primary constraints
C{x,t) = U(x,t)=0, C a (x,t) = U a (x,t) = 0, (10.9)
to be satisfied. The wave functional Vl/ = ty(h a p, N, N a ) then becomes
functional of the three-metric only, i.e. \& = ^(h a p)-
Let us now explicitly discuss the meaning of the quantum constraints
(10.6). First of all, a representation of the canonical algebra can be chosen
h aP y = h al3 ^ , n Q/3 * = -i— — , (io.io)
0h a ji
which is the widely used representation of the canonical approach to quan-
tum gravity in the metric formalism. However, the above equations do not
define proper self-adjoint operators because of the absence of any Lebesgue
408 Primordial Cosmology
measure on E and moreover they are not compatible with the positivity
requirement h a p > 0. Ignoring at this level such problems, we proceed
further in a formal way.
The easiest constraint to be addressed is (10.6b), which is the so-called
diffeomorphism (or kinematic) one. Considering Eqs. (2.72b) and (10.10),
it reads as
H a i lt = 2ih on Vf } —— = 0. (10.11)
dh^p
As shown in Sec. 2.3, the functional H (/) generates the Lie algebra diff(E)
and this feature must also be implemented at a quantum level. The relation
(10.11) implies that the state functional \P is a constant on the orbits of
the spatial diffeomorphism group Diff(E). The functional ^>(h a p) is thus
defined on the whole class of three geometries {h a p} (invariant under spatial
diffeomorphisms) and not only on Riem(E), i.e. 'J 7 = ^({h a p}). More
explicitly, under the infinitesimal transformation
x a -»• x a + SN a , (10.12)
the three-metric h a p becomes
h a p -)• h a p - (\7 a SN p + \7p5N a ) . (10.13)
The wave functional ^(h a p) thus transforms as
V(h a p) -» *{Kp) - 2 f d 3 x 1 ^-\7 a 6Np . (10.14)
7s °h a p
By integrating by parts 1 (assuming that 6N a vanishes at infinity) this ex-
pression, we obtain that the condition
V«-— = (10.15)
dn a p
implies that ty is invariant under infinitesimal coordinate transformations,
i.e. the constraint (10.11) holds. The wave functional Vl/ does not depend
on the particular form of the three-metric, but on the three-geometry only
(namely all the three-metrics related by a coordinate transformation). We
have thus recovered that the configuration space of the canonical quantum
gravity is exactly the Wheeler superspace defined as in Eq. (10.1).
The dynamics is generated via the scalar constraint (10.6a), provid-
ing the famous WDW equation obtained by DeWitt (following the idea of
\\ heeler) in 1967, which explicitly reads as
™=- 5 °™i»3^- : £ 3r * =o > (io - i6)
ce of this integral is ensured by the fact that W* 13
\fh in its definition (2.69a).
Standard Quantum Cosmology 409
where Ga/3-yS is the supermetric (2.72c). The factor-order ambiguity is not
addressed at this stage. We have chosen the simplest factor ordering, i.e.
the one with the momenta placed to the right of h a p. This ordering how-
ever is not self-adjoint in the kinematic Hilbert space associated with the
representation (10.10).
Equation (10.16) is not a single equation, but actually one at each space
point x e S. It is a second-order hyperbolic functional differential equa-
tion which is not defined on the space-time, but on the configuration space
(10.1). At a quantum level the space-time itself has disappeared as the par-
ticle trajectories are absent in quantum mechanics. In fact, in GR the space-
time is the analogous to a particle trajectory in classical (non-relativistic)
mechanics.
The WDW equation is at the heart of the Dirac constraint quantization
approach and the key aspects of the canonical quantum gravity are all
connected with it. Mathematical and conceptual problems emerge in the
WDW approach to quantum gravity, and the most relevant ones can be
summarized as:
(i) The WDW constraint (10.6a) is not polynomial, neither analytical
in the three-metric. Moreover, since Eq. (10.16) contains products
of functional differential operators evaluated at the same spatial
point, it is hopelessly divergent. Distributions in the denominator
are also not clearly defined,
(ii) Although ignoring the problems at point i), it is not possible to
find a formal solution to the WDW equation. As a matter of fact,
not even the constant state
V(h a p) = const (10.17)
is a solution. The WDW equation implies that a physical state \&
should be an eigenvector of the operator H(x,t) with eigenvalue
and thus some sort of boundary conditions have to be imposed on
\]/, but the theory does not give any information about how to set
them,
(iii) Understanding the physical meaning of the WDW equation is one
of the most challenging problems, as for example it requires a notion
of "time" (or "time-evolution") at a quantum level. Such feature
can be eventually related to the fact that the classical slicing is
performed before the quantization procedure.
(ti a0 ,x'\h a0 ,x) = J2 J '*
410 Primordial Cosmology
10.1.2 Relation with the Path Integral Quantization
It is interesting to analyze the relation between the canonical (a la Dirac)
quantization of gravity and the path integral quantization framework. The
latter approach is also known as covariant quantum gravity since the space-
time covariance is manifestly preserved. In fact, one integrates over the
whole space-time metric in analogy with the path integral in quantum me-
chanics. The main ideas of this scheme can be summarized as follows and
its implementation in the minisuperspace arena is given in Sec. 10.4.
The Feynman amplitude between an initial configuration (a state with
an intrinsic metric h a g on £) and a final configuration (a state with an
intrinsic metric h' a/3 on £') is given by
~* ' (10.18)
Here M. denotes the space-time manifold, S g denotes the Einstein-Hilbert
action (2.11) and the integration over T>g includes an integration over the
three- metric h a p, the lapse function N and the shift vector N a . In anal-
ogy with ordinary Quantum Field Theory (QFT), one performs the Wick
rotation t — > —it and takes into account the Euclidean action I g = —iS g ,
with the sum over metrics with signature (+ + H — \-). This way one deals
with the so-called Euclidean approach to quantum gravity, mainly due to
Hawking and his group in Cambridge.
However, also this elegant approach to the quantum gravity problem
suffers significant drawbacks as, in particular
• the gravitational action is not positive definite. Thus, dhTorouily
from the Yang-Mills theory, the path integral does not converge if
the sum is considered only on real metric with Euclidean signature.
To deal with this feature a complex metric in the sum have to be
added, though a unique prescription does not exist.
• The measure Vg in equation (10.18) is ill-defined and up to now
there is no rigorous definition.
Disregarding these shortcomings, the wave function of the Universe * (on
the surface £ with three- metric h a p) can be defined by the functional path-
integral as
9(h a fi, S) = V [ Vg e- J « , (10.19)
where the sum is over a class of four-metrics gij taking values h a p on
the boundary S. The convergence of this integral is ensured by including
Standard Quantum Cosmology 411
complex manifolds in the sum, i.e. by imposing boundary conditions which
restricts the manifold where the integration is performed.
The Euclidean theory can then be considered as the quantum gravity
sector where the "initial boundary conditions" on * should be imposed.
In particular, in order to evaluate the path integral (10.19), a saddle-point
approximation has to be taken into account. By means of this approxima-
tion, the action I g is described by the dominating classical solutions only.
The Euclidean world is thus considered as the fundamental one, while the
Lorentzian world is regarded as an emergent phenomenon where the saddle
point is complex. This argument implies that Eq. (10.19) can be regarded
as the "starting point" and boundary conditions have not been foisted (no-
boundary proposal). In fact, one integrates over such metrics where the
only boundary is given by that corresponding to the actual Universe. The
question of boundary conditions is of fundamental importance in primordial
cosmology and will be discussed in Sec. 10.7.
The Euclidean wave function (10.19) is consistent with the canonical
framework. In fact, it is possible to show that it satisfies the WDW equa-
tion (10.16) provided that the action, the measure and the class of paths
summed over are invariant under four-diffeomorphisms. This way a con-
nection between the covariant and the canonical approaches to quantum
gravity is established, although making these formal arguments rigorously
defined is far from being trivial.
As a last point, a boundary term in the Einstein-Hilbert action (2.11)
has to be taken into account. In fact, the Ricci scalar contains terms which
are linear in the second derivatives of the metric. The path integral ap-
proach requires an action which depends on the first derivatives only, which
can be accomplished by removing the second derivatives by means of inte-
gration by parts. More precisely, the tuusleiu-Hilberl action is consistent
only when the underlying space-time manifold is closed, i.e. it is compact
without boundary. In the event that the manifold has a boundary, the
action should be supplemented by a boundary term so that the variational
principle is well-defined. Such term, known as (lie Gibbons-Hawking- York
boundary term, reads as
Sgkk = T^[ d 3 xVhK,
where =F refer to a space-like or time- like boundary dA4, respectively, and
K is the trace of the extrinsic curvature (2.66).
412 Primordial Cosmology
10.2 The Problem of Time
One of the major conceptual problems in quantum gravity is: what is time?
Indeed, as we have previously seen, in the canonical formulation of quantum
gravity the Schrodinger equation is replaced by the WDW one. An external
time coordinate does not explicitly appear in the formalism. This feature
distinguishes the quantization of a diffeomorphism-invariant field theory
(like non-perturbative canonical quantum gravity) from an ordinary quan-
tum field theory (regarded as a quantum theory over a fixed background
metric structure). In fact, GR is a fully parametrized theory, recognized in
the canonical framework via the appearance of infinitely many constraints
as standing in the structure of the Hamiltonian (see Sec. 2.3).
The task of appropriately define the notion of time at a fundamental
level is deeply connected with the role assigned to temporal concepts in all
theories of physics different from GR. For example, in Newtonian physics,
as well as in non-relativistic quantum mechanics, time is an external param-
eter to the system itself and is treated as a background degree of freedom.
In ordinary QFT the situation is conceptually the same since a Minkowski
background is fixed and the Newtonian time is replaced by the time mea-
sured in a set of relativistic inertial frames. On the other hand, the key
aspect of GR is the difiVomorphisnis in variance of the phj sical laws. There
is no background space-time metric (flat or curved) over which phenomena
happen. The space-time metric itself is a dynamical entity (it is the gravi-
tational field) and a space-time location can thus be only relational. In the
non-general-rolai i\ i; I u physics, the location is given in terms of reference-
system objects which, defining a fixed background, are decoupled from the
field under consideration. In GR, objects (fields) are localized only with
respect to each other and points of the space-time are not a priori distin-
guishable.
The notion of "time" (or a fixed background metric) plays a crucial role
in the formulation of the quantum theory. In fact, the conventional Copen-
hagen interpretation of quantum mechanics, as well as the whole framework
of QFT, breaks down as soon as the metric is no longer fixed. Concepts
like probability and measurement are highly non-trivial in a timeless physics
since, for example, the inner product in quantum mechanics is a quantity
conserved in time. Moreover, the Wightman axioms of QFT (which lead to
the notions of canonical commutation relations, microcausality, propaga-
tors, etc.) are based on a fixed causal structure which is no longer available
in a diffeomorpliism-invarianl theory like as GR. Finally, as showed by Un-
Standard Quantum Cosmology 413
ruh and Wald in 1989, a perfect clock (in the sense of a quantum observable
T whose values monotonically grow with abstract time t) is not compatible
with the physical requirement of an energy positive spectrum. This behav-
ior, absent in classical physics, can be understood as a peculiar feature of
the quantum theory.
Keeping in mind all these considerations when addressing the problem
of time in quantum gravity, there are essentially three ways to face it:
introduce time before the quantization, after the quantization, or dealing
with a timeless framework.
10.2.1 Time before quantization
This approach is essentially based on three steps:
(i) identify a time coordinate as a functional of canonical variables,
(ii) rewrite (solve) the classical scalar constraint in the form
(iii) quantize the new expression leading to a Schrodinger-like equation
i = /ia*. (10.22)
The evolution of the so-called "physical Hamiltonian" tiA is therefore
described with respect to the "time" qA, i.e. the variable canonically con-
jugate to Pa- This way a notion of time can be implemented before the
quantization procedure, by two possible formulations.
In the first one a time variable is constructed from the phase space
gravitational ones. The true degrees of freedom of GR (up to gauge fixing)
are recovered by a canonical transformation, the constraints are classically
solved and then they are quantized leading to Eq. (10.22). This is the so-
called internal time approach and it is noting but the implementation at
quantum level of the ADM reduction of the dynamics described in Sec. 2.3.
This scheme is also known in its general form as the multi-time approach
since one deals with an infinite set of Schrodinger equations, one for each
space point.
The second possibility relies on adding matter fields to the gravitational
dynamics and then regarding the evolution with respect to these matter
clocks. The seminal work in this direction was made by Brown and Kuchar
in 1995 in which an incoherent dust, i.e. a dust with the gravitational
interaction only, is included in the dynamics. This procedure leads to the
414 Primordial Cosmology
new constraints r H{x l ) and H a (.v') in which the dust plays the role of time
and the true Hamiltonian does not depend on the dust variables.
Let us introduce the variables (T, Z a ) and the corresponding conjugate
momenta (M, W a ), so that the values of Z a be the co-moving coordinates
of the dust particles and T be the proper time along their worldlines. In
this scheme the new constraints read as
H = P(x l ) + h(x\h af37 n a P) = (10.23a)
U a = P a (x l ) + h a (x\T,z a ,h a(3 ,n af3 ) = (10.23b)
where
h = -y/G(x*) , G(x t )=H 2 -h afi n a H , (10.24a)
h a = Z^Up + ^fG{X)dpTZ'i , (10.24b)
where P is the projection of the rest mass current of the dust onto the
four- velocity of the observers and P a = —PW a . This way the Hamiltonian
h does not depend on the dust. As we can immediately recognize, the form
of Eq. (10.23a) is exactly the desired one. Therefore, when such constraint
is implemented at a quantum level, a Schrodinger equation for the wave
functional * = *(f, h) as
*ST =h * (1 °- 25)
is recovered. The central point of this procedure is the independence of the
effective Hamiltonian h on the dust because this allows a well-posed spectral
analysis formulation. In fact, h commutes with itself and furthermore the
Schrodinger equation can be split into a dust-(time-)dependent part and a
truly gravitational one.
It is worth stressing that the Brown-Kuchaf mechanism relies on a du-
alism between time evolution and matter fields (in particular a dust fluid).
Let us analyze in some details this feature. As a starting point we sup-
pose that the state functional * is defined on the Wheeler superspace of
the three-geometries {h a p} (i.e. it is annihilated by the super-momentum
operator H a ) and that the theory evolves along the space-time slicing so
that \P = ^{t,{h a p}). Thus, the quantum evolution of the gravitational
field is governed by a smeared Schrodinger equation
id t ^ = H* = / d 3 x (nh) * . (10.26)
=-J^x(nh)
As usual, H is the super-Hamiltonian operator, N = N(t) the lapse function
and E t the one-parameter family of Cauchy surfaces which fills the space-
time. Due to the evolutionary character of the dynamics, this approach is
known as evolutionary quantum gravity. Considering the expansion of the
wave functional as
* = J De4>(e,{hij})exp{-i(t-to) J d 3 x(Ne)\ , (10.27)
where De is the Lebesgue measure in the space of the functions e(x l ), an
eigenvalue problem for the stationary wave function ip appears. Explicitly,
n X = e X , n aX = 0, (10.28)
which outline the appearance of a non-zero supcr-Eainillouian eigenvalue,
differently from the WDW framework.
In order to address the meaning of the time-variable t in Eq. (10.26),
we have to analyze the classical limit of the theory. This step is formulated
by means of the WKB paradigm, i.e. the wave functional "§> is replaced
by its corresponding zero-order WKB approximation \P ~ e lS . Thus the
eigenvalues problem (10.28) reduces to the following classical counterpart
fIJS = e = -2VhT 00 , HJaS = (10.29)
where H J and HJ a denote operators which, acting on the phase S,
reproduce the super-Hamiltonian and super-momentum Hamilton- Jacobi
equations, respectively (see Sec. 2.3). The classical limit of the adopted
Schrodinger quantum dynamics is then characterized by the appearance
of a new matter contribution (associated with the non-zero eigenvalue e)
whose energy density reads as
(10.30)
The quantity T 00 refers to the 00-component of the induced matter energy-
momentum tensor T^. Since the spectrum of the super-Hamiltonian has
in general a negative component, we can infer that, when the gravitational
field is in the ground state, such matter arising in the classical limit can
have a positive energy density. The explicit form of Eq. (10.30) is that of
a dust fluid co-moving with the slicing three- hypersurfaces, i.e. the field n %
coincides with the four- velocity normal to the Cauchy surfaces. In other
words, we deal with an energy-momentum tensor T^- = pninj. However,
such a matter field with a negative energy density cannot be regarded as an
ordinary one since it does not satisfy the strong energy condition (2.153).
A dualism between time evolution and matter fields can now be estab-
lished analyzing the problem from the opposite perspective. We consider
416 Primordial Cosmology
a gravitational system in the presence of a macroscopic matter field, de-
scribed by a perfect fluid having a generic equation of state (2.15). The
energy-momentum tensor associated to this system reads as
Tij = -ypu lUj - (7 - l)p9ij . (10.31)
To fix the constraints when matter is included in the dynamics, let us make
use of the relations
<MV— .j*. ,10.32.)
G l3 n l d a y d = k^= , (10.32b)
2Vh
where Gij is the Einstein tensor (2.10) and d a y l are the vectors tangent to
the Cauchy surfaces, i.e. nid a y l = 0. Equations (10.32), by (10.31) and
identifying m with rn (i.e. the physical space is filled by the fluid), rewrite
p = ^=, H a = 0. (10.33)
Furthermore, we get the equations
Gijdatfdpy* = G aP = k(j - l)ph a p . (10.34)
We now observe that the conservation law VjT/ = implies the following
two conditions
7 V, (pu l ) = (7 - l)u l d lP (10.35a)
u^VjUi = (l - - J (d t lap- UiU j dj lap) . (10.35b)
With the space-time slicing, looking at the dynamics in the fluid frame (i.e.
n l = <5q), by the relation n l = (1/N, —N a /N), the co- moving constraint
implies the synchronous nature of the reference frame. Since a synchronous
reference is also a geodesic one, the right-hand side of Eq. (10.35b) must
identically vanish and, for a generic inhomogeneous case, this implies 7 = 1.
Hence, Eq. (10.35a) yields p = -e(x l )/2v r h and substituted into (10.33),
we get the same Hamiltonian constraints as above in Eq. (10.28), as soon
as the function e is turned into the eigenvalue e. In this respect, while e
is positive by definition, the corresponding eigenvalue can also be negative
because of the structure of H.
We can conclude that a dust fluid is a good choice to realize a clock in
quantum gravity, because it induces a non-zero super-Hamiltonian eigen-
value into the dynamics. Moreover, for vanishing pressure (7 = 1),
Standard Quantum Cosmology 417
Eq. (10.34) reduces to the proper vacuum evolution equation for h a p, thus
outlining a real dualism between time evolution and the presence of a dust
fluid.
Both approaches described above (the multi-time and the dust-clock)
give an evolutionary quantum dynamics for the gravitational field. Al-
though they could seem to overlap each other, this is not the case. The
latter framework is based on a full quantization of the system while the
multi-time scheme relies on a quantization of some degrees of freedom only.
In fact, the constraints are classically solved before implementing the quan-
tization procedure, thus violating the geometrical nature of the gravita-
tional field in favor of real physical degrees of freedom. This fundamental
difference between the two approaches is evident, for example, in a cosmo-
logical context. When a minisupcrspacc model is quantized in the ADM
formalism (see below), the scale factor of the Universe is usually chosen as
an internal time coordinate. On the other hand, in a dust-like approach,
the scale factor is treated on the same footing of the other variables (for
example the anisotropies) and the evolution of the system is considered
with respect to a privileged reference frame (i.e. the dust one).
Let us point out some problems that emerge when identifying a time be-
fore quantization. Firstly, it should be noted that the resulting Schrodinger-
like Eq. (10.22) is in general inequivalent to the original WDW equation.
In particular, the choice of the time variable is not unique and the con-
ditions a variable has to satisfy in order to stand as a good time are not
univocally defined. Moreover, it is well known that different choices of
time lead to different (unitarily inequivalent) quantum descriptions and it
is unclear how these predictions can be related to each other. In fact, in
non-linear systems, most of the (classical) canonical transformations cannot
be represented as unitary operators while maintaining the irreducibility of
the canonical commutation relations. A second, more technical, question is
due to the impossibility to obtain a global solution of the constraint in GR
(this result was obtained by Torre in 1992). This feature forces to privilege
the choice of a matter field as time coordinate. However, such a matter
clock has to satisfy two basic requirements: (i) its Hamiltonian has to be
linear in the momentum variables and (ii) it has to describe physical clocks,
i.e. they should run forward. A natural (and simple) matter field which
accomplishes these basic assumptions is represented by a massless scalar
field. Its role as a clock time will be analyzed in Sec. 10.5 and implemented
in different frameworks below.
418 Primordial Cosmology
10.2.2 Time after quantization
This approach is exactly the Dirac scheme to quantize a constrained system,
i.e. the theory we have described in the previous Section (together with
all its connected problems). The result is then the frozen formalism of the
WDW equation in which it seems that no evolution takes place. To recover
a time notion at this level (and what it implies) there are mainly two ways.
The first idea relies on the observed similarity between the WDW
Eq. (10.16) and the Klein-Gordon one (2.22). In fact, the WDW equation
can be seen as a Klein-Gordon equation with a varying mass. The Hilbert
space can then be naturally obtained by constructing a Klein-Gordon-like
inner product for quantum gravity. It is worth noting that, as usual, the
corresponding probability can be negative. A priori this feature can be over-
come by looking at the hyperbolic features of the WDW equation: it allows
to characterize an internal time variation and, in some cases, to pursue a
separation between positive and negative frequencies. There are however
some crucial differences with respect to the scalar field theory. The mass-
like term in Eq. (10.16), i.e. Vh, A Ii. can take both positive and negative
values, while the standard potenl ials for I lie scalar field are only positive by
definition. This feature makes it impossible to prove the positivity of the
Klein-Gordon scalar product even when positive-frequency solutions are se-
lected. Moreover, a suitable Killing vector field on Riem(S) (which permits
the frequency decomposition) is not in general available. From this point
of view, a general prescription able to define a Hilbert space for the WDW
theory is far from being stated.
The second possibility to recover a time notion after the quantization is
based on a semiclassical interpretation. In this approach, time is a meaning-
ful concept only in some semiclassical sectors of the full WDW theory. The
main idea is that time, and thus space-time, does not exist at fundamen-
tal level but emerges as an approximate feature only under some suitable
conditions. In practice, one usually expands the wave functional $ in a
WKB-like form from which a time variable is extracted. To leading order
of this approximation, the WDW equation is replaced by a Schrodinger one
and the system can be probabilistically described using the associated inner
product. This paradigm appears to be very useful in quantum cosmology
and will be described in detail in Sec. 10.6. Of course this approach suffers
the limit of choosing by hand a preferred state and it is not fully defined
how to describe the system once approaching the real Planck regime.
10.2.3 Timeless physics
The last approach to solve the problem of time in quantum gravity is based
on the idea that there is no need of time at a fundamental level. The quan-
tum theory of gravity can be constructed without a notion of time and such
concept may arise only in some special situations, i.e. in specific approx-
imations of the theory. The physical point of view behind this reasoning
relies in taking seriously some lessons from GR:
• in (general) relativistic physics there is not an independent observ-
able quantity which plays the role of parameter for the evolution;
• any motion is the relative evolution between observables. Such a
situation is also true in the pre(general)-relativistic context.
In fact, what we measure in Newtonian physics are the relative changes
of a system quantity (e.g. the elongation of a pendulum) and of a clock
one (e.g. the motion of the second hand). It is then assumed (because it is
more convenient) that there is a background quantity (time) with respect
to which the former can be evolved.
It turns out that a timeless classical mechanics can be univocally for-
mulated. This framework is based on observables and states which are
meaningful also in a general relativistic scheme. Let us give only the main
ideas of this approach without entering the details. Any classical system
can described by a triple (C, T, /) as follows.
• C is the configuration space: the space spanned by the (partial)
observables which we are interested in. The physical relevant ob-
ject is the relational measurement of these observables (e.g. the
elongation of the pendulum with respect to the second hand clock
• r is the phase space: the space spanned by quantities which coor-
dinatize the relative motion (e.g. the initial conditions).
• / : C x T — > V (V being a linear space) gives the evolution via the
equation / = 0: the motion is a relation between partial observ-
ables in C with suitable boundary conditions in I\
No usual notion of time has been used. Kinematics is here given by C, while
dynamics is contained in T and /. A non-relativistic system appears as a
peculiar case of this framework when a partial observable plays a special
role. Such a variable is called time t and the Hamiltonian H takes the
Primordial Cosmology
particular form
H=pt+Ho(q t ,p\t) = 0, (10.36)
where p t is the momentum conjugate to t. A timeless quantum mechanics
(or a quantum version of a relativistic classical mechanics) can also be con-
structed although this framework is probably not yet complete. However,
the ordinary quantum mechanics is, by construction, not complete as it can
describe non-relativistic systems only.
Let us now analyze these issues in quantum gravity. The simplest way
to deal with a timeless interpretation of the WDW theory relies on con-
structing the inner product as
(*|$)=/ Vh¥(h)$(h), (10.37)
where Vh is a formal integration measure over the three-geometries. This
choice seems to be the more natural one and the Hilbert space, on which the
operators (10.10) are self-adjoint, is given by L 2 (Riem(£),r>/i). However,
these relations are completely ill defined, purely formal and any probabilis-
tic interpretation based on Eq. (10.37) is meaningless.
A more interesl ing framework i o be analyzed is the so-called evolution of
constants of motion. This approach is mainly due to Rovelli. We have seen
that in a generally covariant system, like GR, observables of the theory (not
to be confused with the partial observables mentioned above) are constants
of motion.
Let us consider a toy quantum gravity model described by only one
scalar constraint r H{q,p) = de-lined on <i liui.te-dimensi.on.a.] phase space S.
A (classical) physical observable O is then a function Poisson-commuting
with all the constraints, i.e. in this case {0,TL} =0.0 remains defined as
a constant of motion (this is the frozen formalism of classical gravity). We
introduce a function T = T{q,p) such that for any i£l the surface
S t = {(q,p)eS\T(q,p)=t} (10.38)
intersects any dynamical trajectories (generated by %) only once. In other
words we require that
{T,H}^0, (10.39)
which means that T is not an observable of the theory. The key idea is then
to associate any function F on the phase space S a one-parameter family
of observables F t (i.e. {F t ,H} = 0) such that F t = F on the subspace St-
Evolution is then described by the dependence of the observables F t on the
Standard Quantum Cosmology 421
parameter t. We then deal with the classical analogous of the Heisenberg
picture in the quantum mechanics. An explicit computation shows that the
dynamics of these functions is given by the equation
{T,H}^ = {F,H} . (10.40)
In the particular case as {T,H} = 1 (when T is called a perfect clock),
the Hamiltonian H decomposes as H = pr + Ho, i.e. like (10.36). Thus
{F t , Ho} = {F, Ho} and therefore we obtain the standard equation of mo-
§ = {*,*>} (10.41)
for the one-parameter family of observables F t . We have then formulated
a solid implementation of the general framework discussed above.
The following step is then to quantize such a system. More precisely,
the algebra generated by the classical functions F t has to be represented in
a suitable Hilbert space. Although this task is well posed, some problems
still arise. In particular, an operator formulation of the observables F t is
far from being trivial and it is not clear if a single Hilbert space can account
for all the possible choices of the internal time function T or not.
10.3 What is Quantum Cosmology?
Quantum cosmology denotes the application of the quantum theory to the
entire Universe. However, the following question arises: why implementing
the quantum physics to the Universe as a whole? At first sight, quantum
physics seems to be applicable and relevant only at microscopical scales.
On the other hand, near enough to the Big Bang, the Universe should be
treated like a quantum object as a whole. In fact, it is the only state which
entangles system and environment, i.e. we can say that the system itself
is coupled to its environment. Strictly speaking, the whole Universe is the
only closed quantum system in Nature. Indeed, the so-called decoherence
is a possible quantum mechanism able to lead to the manifestation of the
Universe as a macroscopic classical object. Loosely speaking decoherence
describes the process of entanglement of a system with its natural environ-
ment. A system assumes classical features through the unavoidable and
irreversible interaction with the environment. From this perspective, as
any macroscopic object, the Universe is at the same time of quantum na-
ture and of classical appearance in most of its stages. However, there exist
422 Primordial Cosmology
regimes where the latter does not hold and the quantum nature is revealed.
It is expected that the primordial cosmological regime (namely near the
Big Bang) resembles such peculiar situation.
Quantum cosmology is not necessary related with a quantum gravity.
In fact, quantum gravity (intended as the quantum formulation of the grav-
itational field only) is the theory of one field among the many degrees of
freedom of the entire Universe and, in this aspect, it is not different from
the quantum theory of the electromagnetic field. However, since gravity is
the dominant interaction at large scales, any realistic formulation of quan-
tum cosmology should be based on a quantum theory of gravity. Quantum
cosmology is therefore a natural arena to investigate quantum gravity as
part of a more general context.
10.3.1 Minisuperspace models
Let us apply from an operative point of view the quantum framework to cos-
mological models. This way the fields are restricted to a finite dimensional
subspace of the (infinite dimensional) Wheeler superspace. In fact, the cos-
mological models arise as soon as spatially homogeneous (or also isotropic)
space-times are taken into account and. as we said, for each point x a e S
there is a finite number of degrees of freedom in superspace. All but a finite
number of degrees of freedom are frozen out by imposing such symmetries
and the resulting finite dimensional configuration space of the theory is
known as minisuperspace. From this perspective, quantum cosmology is
the minisuperspace quantization of a cosmological model. The diffeomor-
phism constraint H a = is automatically satisfied and one deals with a
purely constrained quantum mechanical system (no longer a field theory)
described by a single WDW equation for all the spatial points.
However, it is not yet demonstrated that the truncation to minisuper-
space can be regarded as a rigorous approximation of the full superspace.
Strictly speaking, setting most of the field modes and their canonically
conjugate momenta to zero violates the uncertainty principle, thus can be
considered as an ad hoc procedure. On the other hand, classical cosmology
is based on these symmetric models and their quantization should give an-
swers to the fundamental questions like the fate of the classical singularity,
the inflationary expansion and the chaotic behavior of the Universe toward
the singularity. Moreover, as we have seen in Chap. 9, in the general context
of inhomogeneous cosmology, the spatial derivatives of the Ricci scalar are
negligible with respect to the temporal ones, toward the singularity (BKL
Standard Quantum Cosmology 423
conjecture). A minisuperspace model can be relevant for the description of
a generic Universe toward the classical singularity when restricted to each
cosmological horizon. Such quantum cosmology can be regarded as a toy
model which hopefully may capture some of the essence of the full quantum
Let us now define the model. A generic n-dimensional homogeneous
minisuperspace system involves the following assumptions:
(i) the lapse function is taken to be space-independent, i.e. N = N(t);
(ii) the shift vector is taken to be zero, i.e. N a = 0. The line element
(2.64) then reads as
ds 2 = N 2 (t)dt 2 - h a fi(x, t)dx a dx li . (10.42)
(iii) The three-metric h a p is described by a finite number n of homo-
geneous coordinates q (i). This is the crucial assumption. In
the Hamiltonian framework their conjugate momenta are given
by PB(t), where A,B = 1, . . . ,n. This way we deal with an n-
dimensional mechanical system.
We will discuss here only the vacuum cast 1 which can be straightforwardly
generalized if matter fields are included into the dynamics, i.e. q A should
include matter variables also. Of course, the FRW and the Bianchi models
are particular cases of such a framework. The action for this model is given
by
S s = f dt( PA q A -NH)= I dt [ PA q A - N (g AB PA PB + U(q))] , (10.43)
where Q AB is called the minisupermetric. The variation with respect to the
lapse function leads to the scalar constraint
H(q A ,p A ) = g AB pAPB + U(q) = 0, (10.44)
and the equations of motion read as
q A = N{q A , H}, VA = N{ PA , H}. (10.45)
The minisupermetric Gab is the reduced version of the supermetric Ga/3-yS,
where the indices A, B = {af3},{jS} run over the independent components
of the three-metric h a p. It has Lorentzian signature (+, —,—,—,—,—) and
explicitly defined as
g AB dq A dq B = j d 3 x g a0l5 5h aP 5h lS . (10.46)
424 Primordial Cosmology
In action (10.43) U(q) denotes the potential term given by
U = ~y I d 3 xVh 3 R. (10.47)
A minisuperspace model can be regarded as a rdal.ivinl.ic particle moving
in an n-dimensional curved space-time with metric Gab subjected to a po-
tential U{q). The Hamiltonian constraint (10-44) reflects the parametriza-
tion invariance of the theory. This symmetry is the residual of the four-
dimensional diffeomorphisms invariance of the full theory.
The canonical quantization a la Dirac of this model is straightfor-
ward (the path integral quantization of a minisuperspace model is given
in Sec. 10.4). The WDW equation of such a system reads as
WS = (-V 2 + 17)* = 0, (10.48)
where * = ^(q) denotes the wave function of the Universe. Here Va is the
covariant derivative constructed from the metric Gab and the Laplacian
V 2 = V a^ A is given by
V 2 = -j=d A (VGG AB d B ) , (10.49)
where G = | det Gab\- The factor ordering in Eq. (10.48) has been fixed by
Eq. (10.49) and this choice is peculiar because the WDW equation has the
same form in any (minisuperspace) coordinate systems and it is invariant
under the redefinitions of the three-metric fields q A — ¥ q' A (q A ).
10.3.2 Interpretation of the theory
Let us discuss the important feature of quantum cosmology regarding the
interpretation of the wave function of the Universe * for the extraction of
physical properties and considering the differences with respect to ordinary
quantum mechanics. Let us firstly list the assumptions at the basis of the
standard interpretation of quantum mechanics, i.e. the Copenhagen School
QM1 A clear distinction between the classical and the quantum world is
assumed. In particular, there exists an external (classical) observer
to the quantum system. The model under investigation is not ge-
nuinely closed.
QM2 Predictions are probabilistic in nature and performed by measure-
ments of an external observer. These measurements are performed
on a large ensemble of identical systems or many times on the same
system (in the same state).
QM3 Time plays a central and peculiar role (see Sec. 10.2).
On the other hand, quantum cosmology is defined up to the following as-
sumptions.
QC1 There is no longer an a priori splitting between classical and quan-
tum worlds. The analyzed quantum model is the Universe as a
whole, i.e. it is closed and isolated without external classical ob-
QC2 No external measurement, crutch is available, and an internal one
cannot play the observer-like role due to the Planck conditions to
which a very early Universe is subject. The Universe is unique by
definition and it is not possible to perform many measurements on
it arranged in the same state.
QC3 The time coordinate is not an observable in GR and at a quantum
level the problem of time appears.
The most accepted idea to face these features relies in accepting that a
meaningful interpretation of the wave function of the Universe can be re-
covered at a semiclassical level only. A quantum-mechanical interpretation
is possible only for a small subsystem of the entire Universe, i.e. in the
domain where at, least, some of the minisuperspace variables can be treated
as semiclassical in the sense of a WKB approximation. This framework will
be analyzed in Sec. 10.6.
10.3.3 Quantum singularity avoidance
An expected natural result of any quantum cosmology should be to tame
the classical cosmological singularities. In fact, as the classical fall of the
electron on the nucleoli is tamed by quantum effects, it is widely expected
that a quantum Universe should be singularity-free. However, to examine
the behavior of a classical singularity at a quantum level, a general criterion
for determining whether the quantized model actually collapses or not has
still to be fixed, constituting a non-trivial task due to the lack of a complete
quantum theory of gravity.
As we have seen in Sec. 2.7, a space-time singularity in GR can be
defined using two criteria:
(i) the causal geodesic incompleteness (global criterion);
(ii) the divergence of the scalars built up from the Riemann tensor
(local criterion).
426 Primordial Cosmology
Although the second one is useful to characterize a singularity, it is un-
satisfactory since a space-time can be singular without any pathological
character of these scalars. Furthermore, not all singularities have large
curvature and, most importantly, a diverging curvature is not the basic
mechanism behind the singularity theorems.
At a quantum level, the task of defining a quantum singularity is more
challenging. In fact, the classical (smooth) space-time can only be ap-
proached as a "low-energy" limit of the quantum theory "far enough" from
the singularities. In this sense, criterion (i) cannot be a valid measure for a
singularity in quantum gravity, since the space-time itself cannot be clearly
defined at this level, i.e. differential geometry is expected to hold only
to the classical approximation of the full quantum theory of gravity. In
quantum cosmology, the original idea (proposed by DeWitt) to deal with
a singularity-free Universe is to impose that the wave function of the Uni-
verse vanishes in correspondence to the singularity. For example, in the
FRW case, where the Big Bang singularity appears for a = 0, this prescrip-
tion is realized by demanding
*(o = 0) = 0. (10.50)
Unfortunately, this is a boundary condition that does not guarantee the
quantum singularity avoidance since it does not bring any physical infor-
mation on the Universe dynamics. It seems better to study the expectation
values of the observables which classii ally vanish at the singularity.
Let us suppose to construct a Hilbert space for the theory and that
\^(q, t)\ 2 represents merely a probability density. In this way one might
have an evolving state that "bounces" even if |\I/(a = 0, t)\ ^ for all t. A
bouncing state clearly describes a nonsingular quantum Universe dynam-
ics. On the other hand, if one were able to construct a wave packet with
probability
P S = J \y{q,t)\ 2 dq~0, (10.51)
where 5 is a very small but finite quantity, then he could reasonably claim
to have a no-collapse scenario.
Such an approach is in agreement with the so-called principle of quan-
tum hyperbolicity recently formulated by Bojowald. This principle pos-
tulates that a quantum state which evolves in a unique and well-defined
manner through a (classical) singular configuration can be considered as
an evidence of the singularity avoidance. The persistence of a singularity
at a quantum level is then manifest if the quantum dynamics brakes down
Standard Quantum Cosmology 427
without extending the domain of applicability toward the classical singular
regime. These arguments will be clearer below when specific models will
be analyzed.
It is worth noting that not all the approaches to quantum cosmology
lead to a singularity-free Universe. In particular, the WDW theory is not
able to solve the cosmological singularity even in the simpler models. This
task is successfully accomplished by loop quantum cosmology in which the
Big Bang is replaced by a Big Bounce. Both the WDW and the LQC
frameworks will be discussed in details below.
10.4 Path Integral in the Minisuperspace
As we have seen in Sec. 10.3, the minisuperspace quantization corresponds
to freeze out all dynamical degrees of freedom but a finite number, as al-
lowed by the symmetries of the model. The invariance of the full theory
under iour-diuK'tisumal diii.comorpliisms is then translated into an invari-
ance under some specific reparametrizations. At the level of canonical the-
ory, the diffeomorphism invariance of GR is guaranteed by the appearance
of four constraints. The three super-momentum constraints (2.72b) are
linear in the momenta and generate diffeomorphisms within the Cauchy
hypersurfaces. This symmetry resembles one of the ordinary gauge theo-
ries. On the other hand, the scalar constraint (2.72a) is quadratic in the
momenta, a feature not present in the usual gauge theories. Such constraint
expresses the invariance of the theory under time reparametrizations, but
it also generates the dynamics. Therefore, in the Einstein theory symmetry
and dynamics are unavoidably entangled, a feature usually paraphrased as
the background independence of the theory.
The objective of this Section is to analyze the relation between the
canonical (a la Dirac) and the covariant (a la Feynman) quantization meth-
ods in reparametrization-invariant theories described by the action (10.43),
a task firstly accomplished by Halliwell in 1988.
Let us consider an arbitrary function e = e(t) and the transformations
generated as
5 e q A = e{q A i n} = e^- (10.52a)
S £PA = e{ PA ,H} = -eg- I
S t N = e, (10.52b)
428 Primordial Cosmology
where H is explicitly given in Eq. (10.44). Let the time interval be i £
[io>*i]) tnus the action (10.43) changes under the transformation (10.52) by
the amount
8 e S s = J' dt (q A S ePA + p A S e q A - V.S e N - N5M) ■ (10.53)
The last term is zero since S e H = €{%,%} = 0. Performing a partial
integration of the second term and considering the transformations (10.52)
we obtain
*■*-['("£-*)]»■ (ia54)
The term in the round bracket s gives
G AB PAPB ~ U(q) ± 0. (10.55)
The action (10.43) thus remains unchanged if and only if we impose
e(i ) = = e(ii), (10.56)
that is the boundaries must not be transformed. In fact only in this case the
term in Eq. (10.54) vanishes. The transformations (10.52), with the appro-
priate boundary conditions (10.56), are thus the reparametrizations under
which the minisuperspace action is invariant. Note that this condition is
not imposed in gauge theories where one deals with linear constraints of
the form a(q)p = (e.g. the Gauss constraint). In such a case the term
in round brackets in Eq. (10.54) vanishes. The main difference between
gravitation and gauge theories is that the constraint H = is quadratic in
the momenta.
In order to construct the path integral, one has to note the nature of the
constrained system. In particular, a gauge fixing ensures that equivalent
histories are counted only once. The symmetry of the theory is broken by
the gauge fixing condition
G = N- f( PA ,q A ,N) = 0, (10.57)
where / is an arbitrary function. This kind of gauge is often called "non-
canonical" since it depends on N (otherwise is called "canonical"), and
is the analogous of the Lorentz gauge diA 1 = used in electrodynam-
ics. In fact, .4° behaves as a Lagrangian multiplier in gauge theories (see
Sec. 2.2.4), i.e. plays the role of N. On the other hand, the Coulomb gauge
d a A a = is a familiar example of a "canonical" gauge.
We are now able to write down the path integral for the model. Let
us consider the paths {q A (l ). ))_\ (I). A r (/)} such that p/i's and N are free at
the end points to and t\. On the other hand, the q s satisfy the boundary
conditions q A {t\) = q A . The wave function of the Universe thus reads as
V(q A )= fv PA Vq A VNS(G)A G e iS ^ g ' p ' N \ (10.58)
where T> denotes the usual functional integration measure and Aq is
the Faddeev-Popov determinant associated to the gauge-fixing condition
(10.57). In particular, the latter ensures the path integral to be indepen-
dent of the gauge-fixing functional G. It turns out that it is more convenient
to work in the gauge
N = Q => / = 0. (10.59)
In fact, skipping technical details, Aq is the determinant of the operator
S^G/Se. In the gauge (10.59) such operator becomes d 2 /dt 2 , whose de-
terminant is a constant and the Faddeev-Popov measure Aq is indeed a
constant.
As a result, the functional integration over N reduces to an ordinary
integration leading to the path integral
■qj(q A ) = J dN fv PA Vq A e iS ^ q - p ' N ^ = f dN ip(q A , N) . (10.60)
This formula is exactly the time-integration of an ordinary quantum me-
chanical propagator in which the lapse function TV plays the role of time
coordinate. As in quantum mechanics, the function tp(q A ,N) satisfies the
Schrodinger equation with N as time coordinate, that is
i d ^ = m- do.61)
Let us now act on Eq. (10.60) with the operator % defined in Eq. (10.48)
(this is the minisuperspace version of the scalar constraint operator (10.6a)).
Then, using Eq. (10.61), we obtain
HV(q A ) = JdNi^ = i^(q A ,N)\ h , (10.62)
where ip(q A , TV)|b stands for ip(q A ,N) evaluated at the end points (bound-
ary) of the N integral. Obviously, the wave function of the Universe satisfies
the WDW Eq. (10.48) if the condition
,[,(q A ,N)\ h = (10.63)
is satisfied. These end points (or equivalently the contour on which the
wave function is integrated over) are chosen to ensure such a condition.
430 Primordial Cosmology
We have shown the existence of a precise relation between the canonical
and the covariant quantization frameworks, for a minisuperspace model,
providing a specific implementation of what discussed in Sec. 10.1.
As a last point, we mention that one usually performs a rotation to
the Euclidean time. However, differently from the case of ordinary matter
fields, the minisuperspace action (10.43) is not positive definite. Complex
integration contours are then necessary to give a precise meaning to the
path integral. This feature will be discussed in Sec. 10.7.
10.5 Scalar Field as Relational Time
In this Section we analyze in some details the role of a matter field as a
time clock. In particular, we focus on a massless scalar field <fi, employed as
a time-like variable for the quantum dynamics of the gravitational field. As
we have seen, when the canonical quantization procedure is applied to GR,
the usual Schrodinger equation is replaced by a WDW one in which the time
coordinate is dropped out from the formalism. One possible solution to this
problem can be provided by including in the dynamics a matter field and
letting evolve the physical degrees of freedom of the gravitational system
with respect to it. This way the dynamics is described from a relational
point of view, i.e. the matter field behaves as a relational clock.
In quantum cosmology, the choice of a scalar field appears as the most
natural one. In fact, near the classical singularity, a monotonic behavior
of eft as a function of the isotropic scale factor always appears. It is worth
remembering (see Chap. 5) that the behavior of a massless scalar field well
approximates the one of an inflaton field when its potential is negligible at
high enough temperature.
Let us consider the case of the Bianchi IX model in the presence of
such a field. By considering the (m = 0) Lagrangian density (2.21) over a
homogeneous space-time, it is immediate to show that the energy density
of (p = <f>(t) is given by
P^=v\la\ (10.64)
where p^ = p<p(t) denotes the momentum canonically conjugate to <p. Here-
after 4> and V4> have been rescaled, with respect to Eq. (2.21), of a factor
V327T 2 . The scalar constraint in the Misner variables (a = e",/3±) has the
form (see Eq. (8.35))
Wix + H = -^ \- ^ + 1 (^ + p 2 _)
«C/ix(/3±)4
(!().().■))
where £/rx(/3±) is the potential term given by the curvature scalar as in
Eq. (8.37b). The phase space of this system is eight-dimensional with co-
ordinates (a, p a , P±,p±,(/),P4>) where p^ is a constant of motion because of
the absence of a potential term V{4>). Thus each classical trajectory can
be specified with respect to <p, i.e. the scalar field <f> can be regarded as an
internal clock for the dynamics. This condition can be imposed requiring
the time gauge to be
= jV^ = l, (10.66)
<JP4>
namely fixing the lapse function as
N = a 3 /2 P< p . (10.67)
By adopting such a gauge, we deal with an effective Hamiltonian H e in the
4> time that explicitly stands a
P^ = H e
3(4tt) 4 _
/3(8tt) 2
When this model is canonically quantized, the associated WDW equa-
tion describes the wave function vf = <5(a, (3±, <f) evolution with respect to
4>. More precisely, from Eq. (10.65) it follows that 2
{&l + e)« = o,
e - « - ^ [-' a * + <* + * - ^«*H - (10 - 69 '
As usual the WDW equation can be thought of as a Klein-Gordon like
equation where <p plays the role of (relational) time and 9 of the spatial
Laplacian. In order to have an explicit Hilbert space, the natural frequen-
cies decomposition of the solutions of Eq. (10.69) is performed and the
positive frequency sector is considered. The wave function
y(a,p ± ,4>) = e l "U(a,P±) (10.70)
corresponds to positive frequencies with respect to <j> and to 2 denotes the
spectrum of 6. The function in Eq. (10.70) satisfies the positive frequency
2 Since the normal ordering doesn't affect what follows, we adopt the simplest one.
432 Primordial Cosmology
(square root) of the quantum constraint in Eq. (10.69) and we deal with a
Scl i rodinger-like equation
-id+V = Ve® , (10.71)
with a non-local Hamiltonian V9.
We now analyze how the massless scalar field can be regarded as an
appropriate time parameter for the gravitational dynamics. Let us con-
sider the dynamics toward the cosmological singularity, i.e. in the purely
quantum era described by
««^-0(!p). (10.72)
In this region the potential term in Eq. (10.68) a 4 frx(/3±) can be neglected.
Notably it is possible to show that, by means of a WKB expansion, the
quasi-classical limit of the Universe dynamics is reached before the potential
term becomes relevant.
The classical equations of motion are obtained from Eq. (10.68) and are
given by
~^ = \U(a„\2 I = = ( 10 ' 7
d 4> V '"'"" ' , i,r-n A -p<
dp+
where p| = p\ + p 2 _- A solution to the system (10.73) has the form
Pa (<f>) = ^exp
A and B being integration constants and p\ = const., providing a mono-
tonic dependence of the isotropic variable of the Universe a with respect to
the scalar field <fi. The field </> shows to be a satisfactory (relational) time
for the gravitational dynamics, a feature which remains valid for isotropic
models, i.e. when /3± = 0. This way a massless scalar field is largely used in
quantum cosmology as matter clock and we will see below some applications
of this framework.
10.6 Interpretation of the Wave Function of the Universe
In this Section we will discuss in details the semiclassical approximation of
quantum cosmology. It deserves interest because it leads to a probabilistic
interpretation of the theory, i.e. some of the problems previously addressed
(see Sec. 10.3) can be solved. In particular, a meaningful wave function
of the Universe is constructed, although probability and unitarity result
approximate concepts only.
Let us analyze the definition of probability in minisuperspace outlining
the differences with respect to ordinary quantum mechanics. As we have
seen above, a probabilistic interpretation of quantum cosmology cannot be
clearly formulated due to the nonexistence of external or internal classical
(or at least semiclassical) observers and furthermore the probability den-
sity in minisuperspace is ill defined. In quantum mechanics, given a wave
function ^(x,t) describing a system, the probability to find the system in
a configuration-space element d£l x at time t is given by
dP=\V(x,t)\ 2 dil x , (10.75)
providing a positive semidefinite probability, i.e. dP > 0. On the other
hand, the wave function of the Universe ^ generically depends on the three-
metric, on the possible matter fields and no dependence on time explicitly
appears. In analogy to quantum mechanics, the associated probability in
quantum cosmology can be defined as
dP= \^(q)\ 2 VGd n q, (10.76)
an expression that is however not normalizable since its integral over the
whole minisuperspace diverges. Such behavior can be considered as the
analogue of the quantum mechanical feature
[\V(x,t)\ 2 dn x dt = oo. (10.77)
In fact, in quantum cosmology time is included among the set of variables
qA and the element \/0d n q corresponds exactly to dD, x dt. This way, it
remains unclear how the probability conservation can be recovered.
To avoid these undesirable features, an alternative definition of the Uni-
verse probability can be formulated in terms of conserved currents j such
S7 A j A = 0, j A = - l -g AB {^X7 B m -Wb**). (10.78)
434 Primordial Cosmology
This approach arises from the analogy between the WDW and the Klein-
Gordon theories. In fact . the WDW equation (10.48) can be seen as a Klein-
Gordon equation with a variable mass U(q). The corresponding probability
to find the Universe in a surface element cE>a is given by
dP = j A dZ A (10.79)
and the conservation of the current j ensures the conservation of prob-
ability. The main problem relies on the fact that it can still be nega-
tive, similarly to the problem of negative probabilities in the Klein-Gordon
framework. A possible route, following again the analogy, can be a second
quantization of the system (note that this case would actually correspond
to a third quantization), but such approach leads to several difficulties and
it will not be discussed here.
The semiclassical framework is introduced exactly to solve this puzzle.
The underlying idea is that a correct definition of probability (positive
semidefinitc) in quantum cosmology can be formulated by distinguishing
between semiclassical and quantum variables. In particular, the variables
which satisfy the Hamilton- Jacobi equation are regarded as semiclassical.
It is also assumed that the quantum variables do not affect the dynamics
generated by the semiclassical ones. In this respect, the quantum variables
describe a small subsystem of the Universe while the semiclassical variables
play the role of an external observer for the purely quantum dynamics.
Such approach allows one to match the assumptions underlying quantum
cosmology (Q( ' I . QC2, ( JC3) with the ones of ordinary quantum mechanics
(QM1, QM2, QM3).
We will hislh lisoi ll i ' u i ud tli cod uk i a specific
implementation.
10.6.1 The semiclassical approximation
For pedagogical reasons we firstly analyze a purely semiclassical model,
where all the configuration variables q A are semiclassical and the wave
function &(q) is given by
* = A{q)e iS{q) . (10.80)
This state admits a WKB expansion and to lowest order it leads to the
Hamilton- Jacobi equation for S
G AB (\7 A S)(\7 B S) + U = 0. (10.81)
Considering the expansion to next order, one obtains the continuity equa-
tion for the amplitude A and it leads to the conserved current
f = \A\ 2 V C S. (10.82)
As usual, the classical action S(q) describes a congruence of classical trajec-
tories and a probability distribution on the (n — l)-dimensional equal-time
surfaces can be defined. More precisely, considering thai p\ = S7aS, the
vector tangent to the classical path is given by
q A = N^- = 2NV A S. (10.83)
OVA
By requiring only single crossings between trajectories and such equal-time
surfaces, formulated as
q A dS A > 0, (10.84)
the probability (10.79) results to be positive semidefinite. The wave func-
tion ty can eventually be rescaled so that the probability is normalized to
unity. It is worth stressing that, as showed by Halliwell in 1987, a wave
function of the form e tS corresponds to a classical space-time which can be
predicted when a wave function of the Universe is peaked on a classical con-
figuration. A correlation of the form pa = dS/dq , where S is a solution
of the Hamilton-Jacobi Eq. (10.81), is exactly expected when considering a
wave function as e lS .
Let us now consider the case in which not all the minisuporspaoo
variables are semiclassical. We assume that there are m quantum vari-
ables labeled by pi (I = 1, . . . , m) and n — m semiclassical variables qA
(A = 1, . . . , n - m). We also demand that the effect of the quantum vari-
ables on the dynamics of the semiclassical ones can be neglected, similarly
to considering negligible the effect of electrons on the dynamics of nuclei in
the Born-Oppenheimer approximation. The semielassica] degrees of free-
dom are thus treated as the "heavy" nuclei and the quantum ones as the
"light" electrons. This way, the WDW Eq. (10.48) can be decomposed in
a semiclassical and in a quantum part. The semiclassical operator
Ho = -Vl + U(q) (10.85)
is obtained neglecting all the quantum variables pi and the corresponding
momenta tt 1 , corresponding to the part previously analyzed. The quantum
operator is denoted by T-L p and the smallness of the quantum subsystem can
be formulated requiring that its Hamiltonian Tip be of order 0(e~~ 1 ), where e
is a small parameter proportional to h. Since the action of the semiclassical
436 Primordial Cosmology
Hamiltonian operator H on the wave function ^ is of order 0(e -2 ), the
idea that the quantum subsystem does not influence the semiclassical one
can be formulated as
¥— = <D(e). (10.86)
Ho*
Such requirement is pi i > < IK u i 01 bl< iuc< Ik unci -deal properties
of a cosmological model as well as the smallness of a quantum subsystem
are both expectably linked to the fact that the Universe is large enough.
The minisupcTspace metric can consequently be expanded in terms of e
GAB(q,p) = G° AB (q) + 0(e), (10.87)
and the Universe wave function ty(q,p) is assumed (notice that this is an
ansatz for the solution) to be
* = *o(«)x(«, P) = A(q)e iS{q) x(q, p)- (10.88)
The wave function is WKB-like in the q coordinates, i.e. the amplitude A
and the phase S depend on the semiclassical variables only. On the other
hand, the additional function \ describes the quantum subsystem and it
depends on p and only parametrically on the q variables, in the sense of
the Born-Oppenheimer approximation. The function \& satisfies the WKB
equations analyzed above and the function x h as to be a solution of
[vjj + 2(V (ln A))V + 2i(V 5)V -H p ]x = 0, (10.89)
where the operator Vo is built using the metric Q\ B {q) as before.
Such an equation describes the evolution of the quantum subsystem. It
is worth noting that the first two terms are of higher order in e with respect
to the third one and can be neglected, resulting in
2z(V S)VoX = HpX- (10.90)
In order to obtain a purely Schrodinger equation for the wave function \
we need to redefine a time variable and using the classical relation (10.83)
we obtain
i^=NH pX . (10.91)
A time parameter thus arises only at a semiclassical level where the wave
function is oscillatory, i.e. it is a consequence of the initial assumption on
* as in Eq. (10.88). From this perspective such an approach represents a
i of the idea that time is an emerging feature on a
Standard Quantum Cosmology 437
classical background. This way the problem of time is solved in the spirit
of the ordinary quantum theory.
The last point to address is to express the probability distribution. From
the peculiarity of this framework, two different probability currents can be
obtained, one for the semiclassical set and the other for the quantum one.
The (total) probability distribution can be written as
a(q,p,t) = a (q,t)a x (q,p,t), (10.92)
where <jq is the probability distribution for the semiclassical variables and
o x = \ X (q,p,t)\ 2 (10.93)
denotes the probability distribution for the quantum variables on the clas-
sical trajectories <za(£) where the wave function \ can be normalized. Let
us consider the surface element on equal-time surfaces cE = dY,odfl p , cEq
remaining defined from the metric G°(q)- The probability distribution oq
is normalized as J a^dY^o = 1 and therefore x(q->P->t) can be normalized as
/ \ x \ 2 dn p = i, dn p = x /\detg IJ \d m p. (10.94)
The whole probability distribution in Eq. (10.92) is normalizable.
Summarizing, we have recovered the standard interpretation of the wave
function for a small subsystem of the Universe (only) in agreement with the
intrinsic approximate interpretation of the Universe wave function. In fact,
in the interpretation of quantum mechanics, all realistic measuring devices
have some quantum uncertainty. The bigger the apparatus, the smaller
the quantum fluctuations. In this sense, we are able to give a meaningful
interpretation of the wave function of the Universe only in a semiclassical
domain where the conventional law of physics apply.
In conclusion, we recall the two assumptions underlying this model:
(i) the analysis has been developed within the minisuperspace regime;
(ii) the fundamental requirement of existence of a family of equal-t hue
surfaces is taken as a general feature.
10.6.2 An example: A quantum mechanism for the
isotropization of the Universe
We now discuss how the scenario above described can be implemented to
find a mechanism able to isotropize a quantum Universe which is weakly
anisotropic. The minisuperspace cosmological model we consider is the
438 Primordial Cosmology
quasi-isotropic Mixmaster Universe with a cosmological constant, a quite
general system exactly solvable for which an isotropization mechanism nat-
urally arises.
Such dynamics is summarized by the scalar constraint (see Sec. 8.2)
where k = k/3(87t) 2 and the quadratic /3-term is the first-order expansion
given in Eq. (8.45). The isotropic potential U(a) explicitly reads
^»=iH-i + f° 2 )- (io - 96)
We remember that the Misner variables a = a(t) and j3± = j3±(t) describe
the isotropic expansion and the shape changes (anisotropies) of the Uni-
verse, respectively. The phase space of this model is six-dimensional and
the cosmological singularity appears for a — > 0. As usual pa = A/k is the
energy density associated with a cosmological constant 3 and, as discussed in
Sec. 5.4.1, far enough from the singularity this term dominates the ordinary
matter fields, a necessary condition for the emergence of the inflationary
scenario.
In order to consider the semiclassical scheme, it is natural to regard the
isotropic expansion variable a as the semiclassical one while considering
the anisotropy coordinates /3± (the two physical degrees of freedom of the
Universe) as the purely quantum variables. We are assuming ab initio
that the radius of the Universe plays a different role with respect to the
anisotropies. The wave function of the Universe '5 = \&(a,/3±) then reads
as (see Eq. (10.88))
* = *oX = A{ay s ^x(a, 0±). (10.97)
The Hamilton- Jacobi equation for S and the continuity equation for the
amplitude A are respectively given by
-RA{S'f +aUA + V q = (10.98)
j(A 2 S')' = 0, (10.99)
where the prime denotes differentiation with respect to the scale factor a
and V q = RA" is the so-called quantum potential, which in this model is
negligible far from the classical singularity even if the h — > limit is not
taken into account (see below). The evolutionary equation (10.90) for the
3 The cosmological constant A has dimension [A] = 1/re.
quantum state x (i-e. neglecting higher order correction terms in e) reads
-2ia 2 S'd aX = U P X, U p = pl+p 2 _ + ^{pl+pl). (10.100)
The Schrodinger Eq. (10.91) for the wave function \ is obtained by taking
into account the vector tangent to the classical path. Using p a = S' , the
equations of motion (10.98) and considering the time gauge da/dt = 1, it
is possible to define the new time variable r such that
dr = N-^da. (10.101)
Far from the singularity (namely in the asymptotic interesting region a 3>
l/\/A) the evolution equation (10.100) rewrites as
id TX =-[-Ap + uj 2 (T)(Pl + p 2 _]
~~ 12yAa 3+ °(U a 2j
(10.103)
and w 2 (r) = C/t 4 ^ 3 , C being a constant given by 2C = 1/6 4 / 3 (kA) 2 / 3 . The
dynamics of the Universe anisotropies subsystem can then be regarded as
a time-dependent bi-dimensional harmonic oscillator with frequency w(r).
The construction of a quantum theory for a time-dependent, linear,
dynamical system has remarkable differences with respect to the time-
independent one. If the Hamiltonian fails to be time- independent, solu-
tions which oscillate with purely positive frequency do not exist at all, i.e.
the dynamics of the wave function is not carried out by a unitary time
operator. In particular, in the absence of a time translation symmetry,
no natural preferred choice of the Hilbert space is available. However, in
the finite-dimensional case, the Stone- Von Neumann theorem holds (see
Sec. 11. 1). This way, the theory is unitarily equivalent to the standard
(namely Schrodinger) one for any choice of the Hilbert space.
The quantum theory of the harmonic oscillator with time-dependent fre-
quency is well known and the solution of the Schrodinger equation (10.102)
can be analytically obtained. The analysis is mainly based on the use of the
"exact invariants method" and on some time-dependent transformations.
An exact invariant J(t) is a constant of motion, namely
J=^- = d T J-i[J,H p } = 0, (10.104)
440 Primordial Cosmology
and is Hermitian (J^ = J). For the Hamiltonian H p as in Eq. (10.102) it
explicitly reads as
J± = \ (C 2 PI + {&± - ^±) 2 ) , (10.105)
where £ = £(r) is any function satisfying the auxiliary nondinear differential
equation
£ + w 2 £ = r 3 - (10.106)
The goal for the use of the invariants (10.105) relies on the fact that they
match the wave function of a time- independent harmonic oscillator with the
time-dependent one. Let c6„ (p, t) be the eigenhmctions of J forming a com-
plete orthonormal set corresponding to the time-independent eigenvalues
k n = n+ 1/2. These states are related to the eigenfunctions (f>„ = n (/3/O
of a time-independent harmonic oscillator via the unitary transformation
T = ex P H£/3 2 /2£) (10.107)
as <fi n = £}' 2 T({) n . The non-trivial (and in general non-available) step in
this construction is an exact solution of the auxiliary equation (10.106).
However, in our ease it, can be explicitly constructed as
« = V7s( 1 + I ^)- (1 °' 108 '
Finally, the solution to the Schrodinger equation (10.102) is connected to
the J-eigenfunctions 4> n by the relation
X„(/3,T)=e M "( T Vn(/3,T). (10.109)
The general solution to (10.102) can thus be written as the linear com-
bination x(/3, t) = J2 n c nXn(f3,T~), c n being constants. Here, the time-
dependent phase a„(r) is given by
a —(" + i)L'wm- (10 - 110)
The wave function x ls given by \ n = X+X-i where
X± = Xn(P±, r) = C *-^-H n {fi ± /(-) exp [l (ft" 1 + *C 2 ) Pl\ ,
(10.111)
in which H n are the usual Hermite polynomials of order n. It is immediate
to verify that, when ui(t) — > uj = const, and £(t) — > £ = 1/^/aJp (namely
a(r) — > —u)o(n+l/2)T), the wave function of a time-independent harmonic
oscillator is recovered.
Standard Quantum Cosmology 441
Let us investigate the probability density to find the quantum subsystem
of the Universe in a given state. As a result, the anisotropies appear to be
probabilistically suppressed as soon as the Universe expands enough far
from the cosmological singularity (it appears for a — > or r — > oo). Such
a feature can be realized from the behavior of the squared modulus of the
wave function (10.111) which is given by
\Xn\ 2 ^ ^\H n+ ((3 + /t)\ 2 \H n _(f3-/t)\ 2 e- p2/e , (10.112)
where 1 = f3+ + 0i_ ■ Notice that such a probability density is still time-
dependent through £ = £(r) since the evolution of the wave function \
is not traced by a unitary time operator. As we can see from (10.112),
when a large enough cosmological region (namely as soon as a — > oo or
r — > 0) is considered, the probability density to find the Universe is sharply
peaked at the isotropic configuration, i.e. for \f3±\ ~ 0. In this limit (which
corresponds also to £ — > 0) the probability density |x„ = o| 2 of the ground
state (n = n + + n_ = 0) is given by
\Xn= \ 2T -^5{P)- (10-113)
The probability density is then proportional to the Dirac (^-distribution
centered on (/?+,/?_) = (0,0) (see Fig. 10.1).
Summarizing, when the Universe moves away from the cosmological sin-
gularity, the probability density is asymptotically peaked around the closed
FRW configuration. Near the initial singularity all values of anisotropies
P± are almost equally favored from a probabilistic point of view while,
as the radius of the Universe grows, the isotropic state becomes the most
probable one. This result relies on considering the isotropic scalar fac-
tor a as a semiclassical variable. Furthermore, we can write a positive
semidefinite probability density and provide a clear interpretation of the
model. The validity of such assumption can be verified from the analysis of
the Hamilton-Jacobi equations (10.98). In particular, the WKB function
^o = exp(iS + In A) approaches the quasi-classical limit e as soon as the
limit a > l/\/A is considered.
10.7 Boundary Conditions
Boundary, or initial, conditions are usually regarded in the context to ar-
range a physical system to perform an experiment. If the system we are
analyzing is the entire Universe, boundary conditions can be arbitrarily
i'rniioi-iiiid Cosmology
l*n=0l
Figure 10.1 The wave function of the ground state Xn=o {P± ,
ical singularity, i.e. in the r — > limit. In the plot we take C -
-) far from the cosmolog-
chosen. Thus, a proper choice (together with the removal of the cosmo-
logical singularity) can be considered as the main goal of any satisfactory
quantum cosmology. Initial conditions are fundamental in cosmology since
they determine the further evolution of the Universe as a whole. For ex-
ample, whether the Universe has been subjected to an inflationary phase
consistent with the observations is one of the questions that can be ad-
dressed. At a quantum level, in principle, initial (boundary) conditions
have to select just one wave function of the Universe from the many al-
lowed by the dynamics, in order to select a particular solution of the WDW
equation. A priori, such state should contain all the information to describe
our Universe. However, there is not a physical hint to choose appropriate
boundary conditions and only motivations like mathematical consistency
and simplicity may be invoked.
The two most studied boundary conditions are the no-boundary and the
tunneling proposals, discussed here at a pedagogical level.
10.7.1 No-boundary proposal
The no-boundary proposal has been formulated by Hartle and Hawking in
1983 and it is essentially of a topological nature, based on the Euclidean
path integral wave function of the Universe (10.19). This proposal consists
of two parts: (i) the sum in Eq. (10.19) is restricted to include only com-
pact Euclidean four-dimensional manifolds M and (ii) the Cauchy surface
S, on which '3/ is defined, forms the only boundary for these geometries.
Therefore, no additional boundary conditions need to be imposed.
From an operative point of view, one usually works with the lowest-order
WKB wave function (see Sec. 10.6)
* = Ae^, (10.114)
where J g = — iS g is the classical action evaluated along the solution to
the Euclidean field equations. The task is thus to find appropriate initial
conditions which correspond to the no-boundary proposal at the classical
level. This way one imposes that: (i) the four-geometry closes, (ii) the
saddle points in the fund tonal int egral (10.19) correspond to regular metrics
which are solution to the field equations.
In the simplest case the geometry is described by the Hartle-Hawking
instanton, see Fig. 10.2. Half of the Euclidean four-sphere S 4 (for small
scale factor) is matched with a de Sitter space as the analytical continua-
tion of S 4 . The three-geometry matching of the two spaces has vanishing
extrinsic curvature. Since S 4 is compact, there is no boundary at the pole
r = (t is the imaginary time obtained by the Wick rotation t — > —it).
This is a non-singular four-geometry in which the Euclidean regime (imagi-
nary time) describes the epoch where the scale factor is small while, as soon
as the Universe expands enough, the dominating regime is the Lorentzian
one (real time). The Lorentzian world can be regarded as an emergent
phenomenon, i.e. there is a transition between the imaginary time and the
standard one once the Lorentzian regime is approached from the Euclidean
one.
As we mentioned in Sec. 10.4, the path integral to convergence requires
a complex contour of integration in Eq. (10.19). The main problem is
that, although convergent contours can be found, they are not univocally
fixed. The wave function depends on which contour is considered and the
no-boundary proposal is not able to give a unique physical prediction and
therefore some extra information to determine the contour must be added.
Let us now show the results of this scheme when implemented in a simple
minisuperspace model, describing a closed FRW Universe with a scalar field
:r 29, 2010 11:22
World Scientific Book - 9in x 6in
PrimordialCosmology |
i'rniioi-iiiid Cosmology
e 10.2 The Hartle-Hawking instant.
with potential V = V(<j)). Such computation is performed considering the
semiclassical (namely saddle-point) approximation of the functional integral
and choosing a particular integration contour. Introducing the quantity
(10.115)
e function
s _£ (s .v-ir-i,
s usual, denotes the scale factor, the no-boundary v
3 ( + 3^) (etS + e ~ tS) ' (1 ° J16)
which is real being a sum of the WKB zeroth order wave function e tS and
its complex conjugate.
Before analyzing some of its cosmological implications, let us discuss the
other boundary proposal, showing at the end a comparison between these
10.7.2 Tunneling proposal
The tunneling proposal, in its final version, has been formulated by Vilenkin
in 1988 and predicts the existence (by a tunneling mechanism) of 1 Ik ■ [.in-
verse from nothing. In analogy with the Klein-Gordon theory, it is proposed
to divide the solutions of the Y\ D\V equal ion in positive and negative fre-
quency solutions. Then only the wave fund ion consisting in outgoing modes
has to be considered. More precisely, this proposal can be formulated as: (i)
the wave function is everywhere bounded and (ii) it consists solely of outgo-
ing modes at singular boundaries of superspace, except for the boundaries
corresponding to vanishing three-geometries.
The problem now is to define the meaning of outgoing. In quantum
mechanics, having the reference phase e~ tult , an outgoing plane wave is
e tkx . In general, such a paradigm is of course vague due to the absence
of Killing vectors in the superspace. In fact, it is not always possible to
clearly define incoming and outgoing modes. On the other hand, in a WKB
minisuperspace context, such a decomposition can be exactly formulated.
As we have seen, a WKB oscillatory wave function ^ ~ e leads to a
conserved current (see Eq. (10.82))
j~VS. (10.117)
The incoming and outgoing modes can be defined with respect to the di-
rection of VS = p on the considered surface. In particular, a mode is
defined to be outgoing at the boundary if the quantity VS points outward.
Intuitively, the tunneling proposal reduces the possible ensemble of the Uni-
verses described by ^/, retaining only those with positive momentum VS,
i.e. able to emerge from nothing.
In the minisuperspace model considered above (FRW closed Universe
plus a massive scalar field) the tunneling wave function reads as
*T~expf-^M e lS , (10.118)
where S is given by (10.115). Such solution is complex, different from the
Hartle-Hawking one (10.116), consisting of just one WKB component.
446 Primordial Cosmology
10.7.3 Comparison between the two approaches
Let us discuss some physical implications of the above two proposals. The
main difference between the tunneling wave function (10.118) and the no-
boundary solution (10.116) is the sign in the non-oscillating exponential
term
^aw)' (10 ' 119)
which has deep implications on the inflationary dynamics. Since we are
interested in discussing such cosmological era, we will focus on the slow-roll
approximation (sec Sec. 5.4.1). i.e. we consider the case
cj)(t) ~ <j) = const . (10.120)
Both wave functions are strongh peaked around the classical solutions, i.e.
those satisfying the first integral p = VS*. These solutions are of inflationary
type
a^^e^K (10.121)
However, the strength of the inflationary phase depends on the value <fio
which can, or cannot, lead to a sufficient inflation. The two frameworks
favor different values of <fio, i.e. different inflationary scenarios.
Let us now discuss the two proposals. The component e lS of the no-
boundary wave function must be compared with the tunneling one, focusing
our attention on the real exponential terms (10.119) only. The task is to
show which wave function inflates for the correct amount. This can be
realized by integrating the probability flux (10.79) on the surface separating
the oscillating and the tunneling regions. From (10.115), this surface is
defined by
a 2 V(<p) = l. (10.122)
Because during the slow-rolling phase we have <j> ~ const., the conserved
current (10.82) points to the a-direction. A natural choice for the surface
S is then a = const., with a current j given by
j ~ \A\ 2 = exp (±^f) . (10-123)
and therefore the probability measure reads as
dP = j-d£~exp(±-^)d(l>. (10.124)
Standard Quantum Cosmology 447
The ± signs refer to the no-boundary and the tunneling wave functions,
respectively.
Let the range of the initial values of the scalar field (fr ~ (fro be (fro e
[4>m, <fru\- Let then (fr s G [<p m ,4>M] be the value for a sufficient inflation,
i.e. the right amount appears if (fro > <fr s , while if <fr < <fr s the inflation
will not be strong enough. Therefore, the probability to obtain a sufficient
inflation is given by the probability to have the right behavior over all the
possibilities, explicitly as
ft M dP ff M exp(±^ 7 )d6
P (0o > <fr s / ^0 G [0 m , <fru\) = % = ~7h 2 • ( 10 - 125 )
Such a quantity shows that an inflation strong enough seems to be favored
by the tunneling wave function (— ) , which favors large values of (fro over the
small ones. On the other hand the no-boundary wave function (+) seems to
disfavor it. A sufficient inflation seems to be a prediction of the tunneling
wave function (10.118) only. However, no definitive answer on this issue
has been given yet.
10.8 Quantization of the FRW Model Filled with a Scalar
Field
In this Section we discuss the canonical quantization of the FRW Universe
with a scalar field in the WDW framework. The general prescription to
quantize a minisuperspace model has been previously described and is now
implemented to the isotropic models analyzed in Sec. 3.2. The FRW models
are described by the line element (3.77) and have a two-dimensional phase
space where the only non-vanishing Poisson brackets are
{a, Pa } = l.
(10.126)
dynamics
; is summarized by the scalai
: constraint (3.81)
which we
ivenience
HRW = ~2^^~^T~ a + 27r2pa3 = °' (10.127)
We are interested to analyze a flat model (K = 0) filled with a massless
scalar field (fr whose energy density reads as p^ = p^/a 6 (in our conven-
tions (fr has dimension of an energy). This is an interesting model since
it is exactly solvable and represents for quantum cosmology what the har-
monic oscillator is for quantum field theory. The scalar constraint (10.127)
Primordial Cosmology
"HrW+0 = -~
|-2tt
- M.
(10.128)
24tt 2 a
Before quantizing this model we have to define a time coordinate for the
dynamics. As discussed in detail in Sec. 10.5, an appropriate choice is to
take <j) as a relational time variable, i.e. the time gauge <f> = 1 by fixing the
lapse function as
47T 2 P0 '
In this case the Friedmann Eq. (3.46) is given by
(10.129)
and its solutions reads as
a{4>) = 0o e ±B (*-*o)
(10.131)
where clq and <pQ are integration constants. The plus sign describes an
expanding Universe from I lis Big Bang, while the minus sign a contracting
one into the Big Crunch. The classical cosmological singularity is reached
at 4> = ±oo by every classical solution.
Fixing the lapse function as said, one deals with an effective Haniiltonian
Tie with respect to cj>, given by
p^ = ±B\ Pa a\=H e .
(10.132)
The momentum p^ plays the same role as the constant energy in classical
mechanics and the ± signs select the direction of time. Such a Hamilto-
nian H e is known as the Berry-Keating-Connes Hamiltonian. Although the
direct quantization of this Hamiltonian is not trivial due to the presence
of the absolute value function, we focus only on the positive term p a a. In
fact, since 7i e is a conserved quantity, it is sufficient to analyze initial states
which are superposition of positive eigenstates only.
The main quantum features of this model can be immediately obtained
in the Heisenberg picture. The time evolution of any observable O can
be realized with respect to the Hamiltonian (10.132), i.e. the equation of
motion for the expectation value
d .
do
(<D) =
-i([0,H e ]
(10.133)
Standard Quantum Cosmology 449
holds. Equation (10.133) for the scale factor a and its conjugate momentum
p a read as
£<«>=*<«>, £<*> = -*<*>■ (10-134)
These trajectories are in exact agreement with the classical ones. In order
to discuss the fate of the oosmologica] singularity at quani urn level, we have
to analyze the evolution of a semiclassical initial state. In particular, we
refer to the requirements that
(i) its expectation value is close to the classical one
(ii) the fluctuations (AC) 2 are small enough, that is
(AC) 2 < (Of. (10.135)
The scale factor fluctuations (Aa) 2 = (a 2 ) — (a) 2 obey the equation
and therefore they are neither constants nor bounded during the evolu-
tion. On the other hand, the relative fluctuation (Aa) 2 /(a) 2 is a conserved
quantity which satisfies the equation of motion
Such a behavior ensures that an initial semiclassical state r
Let us consider the dynamics backward in time (toward the singularity)
of an initial state sharply peaked on the expanding (plus sign) classical
trajectory (10.131). By an initial state we refer to a state which is peaked
at late times, i.e. at an energy much smaller than the Planck one. Roughly
speaking it can be considered peaked around the observed classical Uni-
verse configuration. By means of the equations of motion (10.134) and
(10.137), it remains sharply peaked along the whole classical trajectory un-
til the unavoidable fall into the classical Big Bang singularity. This kind of
dynamics undoubtedly indicates that the classical singularity is not tamed
by the WD W formalism.
It is now instructive to analyze this model in the Schrodinger represen-
tation. Regarding the massless scalar field as a relational time, the WDW
associated to the constraint (10.128) takes the form
(d 2 , + 6)# = , (10.138)
450 Primordial Cosmology
where the action of the operator 6 = ti 2 reads as
e* = -Badl{aV) (10.139)
for the wave function of the Universe 'A 7 = ^(a, 4>). The operator is self-
adjoint even if p a — > — id a is not. In fact, since the classical range of a is
(0, oo), the natural choice for the Hilbert space in the quantum theory is
L 2 (M. + ,da) where the symmetric operator — id a has no self-adjoint exten-
sions. This feature (which arises also in the ordinary quantum mechanics
of a particle confined in the semi-axis) is exactly the minisuperspace remi-
niscence of the problem (ii) regarding the commutation relations (10.3). As
in Sec. 10.5, considering the positive frequency modes
*(o, (p) = e ; ^i(o) , (10.140)
we obtain the eigenvalues problem
(6 - w 2 )V^ = 0. (10.141)
The solution of this equation is given by
%j} u {a) = A_a~ (1 " 7)/2 + A + a" (1+7)/2 , (10.142)
where "f 2 = 1 — 4u 2 /^/B and the spectrum uJ 2 is purely continuous. As be-
fore, ^{a,4>) satisfies the Schrodinger-like equation —id^ = VQ^ . Such
wave function is diverging at the singularity (a — > 0) and the probability
(10.51) is also diverging. It is now possible to construct a wave packet
peaked at late times and analyze its dynamics toward the classical singu-
larity. A wave packet is a superposition of the eigenfunctions
V(a,(j>) = f dwA{w)e iu ' t 'il) u (a) (10.143)
and, for example, we can take A(w) as a Gaussian weighting function cen-
tered in w > 'p- As a result, these wave packets remain localized around
the classical trajectory and fall into the cosmological singularity. The sin-
gularity non-avoidance is recovered in the Schrodinger framework too.
Let us conclude by stressing the fate of the cosmological singularity at
a quantum level. As showed by Gotay and Demaret in 1983, the avoidance
of the singularity in the WDW framework crucially depends on the choice
of time. In general, a clock can be of two different kinds. It can be slow (its
corresponding classical dynamics is incomplete) or fast (its corresponding
classical dynamics is complete). More precisely, a time variable t is a fast
time if the singularity occurs at either t = — oo or t = +oo. If this is not the
case, t is called a slow time. In this terminology, the massless scalar field we
Standard Quantum Cosmology 451
have used above is a fast time since the Big Bang singularity appears at <j> =
±00. The conjecture is the following: any quantum dynamics in a fast-time
clock is always singular. This is exactly what we obtained. On the other
hand, as we will discuss in Sec. 12.2, the LQC framework clearly avoids a
(fast-time) Big Bang singularity replacing it by a Big Bounce dynamics.
Although such behavior seems to contradict this conjecture, the conflict is
solved since in LQC we deal with a unitarily inequivalent representation,
with respect to the Schrodinger one, of the canonical commutation relations.
10.9 The Poincare Half Plane
This Section is devoted to introduce the Poincare half plane. By means
of this framework we will discuss the quantum dynamics of the Taub cos-
mological model in Sec. 10.10 as well as of the Mixmaster Universe (see
Sec. 10.12). In particular, we here introduce a suitable form of the MCI
variables (characterized by static potential walls) known as the Poincare
half plane representation. The choice of such a parametrization of the
Lobacevskij plane allows to deal with a simple geometry which reduces
the differences between the Bianchi I model and the Mixmaster type to a
problem of boundary conditions.
The so-called Poincare variables (u, v) are defined as
H
V3v
( y/Z{\ + 2u
(10.144a)
(10.144b)
In the vicinity of the initial singularity, we have seen that the potential
term behaves as a potential well and as soon as we restrict the dynamics to
IIq, Hadm = e and we can rewrite Eq. (8.60) and Eq. (8.58) respectively
<^n = S f gIt( Pu u + p v v - Hadm) = (10.145a)
Hadm = v^pl+pl . (10.145b)
The asymptotic dynamics is defined in a portion Hq of the Lobacevskij
sr 29, 2010 11:22
452 Primordial Cosmology
plane, delimited by inequalities
Q 1 (u,v) = -u/d>0
Q 2 (u,v) = (l + u)/d>0
Q 3 (u,v) = (u(u+l)+v 2 )/d>0
d=l + u + u 2 + v 2 ,
whose boundaries are composed by geodesies of the pla
lines and one semicircle centered on the absolute v =
sketched in Fig. 10.3.
(10.146a)
(10.146b)
(10.146c)
(10.146d)
ie, i.e. two vertical
0. This region is
Figure 10.3 U.q(u,v) is the available portion of the configuration space in the Poincare
upper half plane. It is bounded by three geodesies u = 0, u = -1, and (u + 1/2) 2 + v 2 =
1/4, and has a finite measure /i = n.
The billiard has a finite measure and its region open at infinity together
with the two points on the absolute (0,0) and (—1,0) correspond to the
three cuspids of the potential in Fig. 8.3. It is easy to show that, in the
(u, v) plane, the measure in Eq. (8.80) becomes
1 dudv .
dfi= — . (10.147)
Standard Quantum Cosmology 453
10.10 Quantum Dynamics of the Taub Universe
In this Section we focus on the quantum features of the Taub Universe in
the WDW framework. Such a cosmological model is a natural step toward
the quantization of the more interesting case of Bianchi IX Universe. We
will firstly analyze the classical model and then quantize it.
10.10.1 Classical framework
The Taub cosmological model is homogeneous and its symmetry group is
50(3), i.e. the same as for Bianchi IX. However, this Universe is rotation-
ally invariant about one axis of the three-dimensional space. The case of
Taub is thus the natural intermediate step between FRW (which is invari-
ant under rotations about any axis) and the Bianchi IX Universe, in which
the rotational invariance is absent due to the presence of three intrinsically
different scale factors.
The line element of the Taub space-time reads as
ds 2 = N\t) dt 2 - e 2a (e 2 P) ab Lj a Lo b , (10.148)
where the left-invariant 1-forms lj" = uj" x dx a satisfy the Maurer-Cartan
Eq. (7.28). The variable a(t) describes the isotropic expansion of the Uni-
verse and Pab{t) is the traceless symmetric matrix
/3 a6 = diag (/?+,/?+, -2/3+) (10.149)
which determines the anisotropy via /3+ only. The Taub model then cor-
responds to the particular case of Bianchi IX as soon as /?_ =0. The
determinant of the matrix r/ a & corresponds to -q = e 6a and the classical
singularity appears for a — ¥ — oo.
Performing the usual Legendre transformation we obtain the Hamilto-
nian constraint for this model. The complete Hamiltonian framework can
be recovered from the Bianchi IX one imposing /?_ = 0, and hence p_ = 0.
In particular, we are interested to the analysis of the Taub model in the
Poincare plane (see Sec. 10.9). The obtained dynamics is equivalent to the
motion of a particle in a one-dimensional half-closed domain. As we can
see from Eqs. (8.53) and (10.144), this particular case arises for
= O^ U = -i, S = t + *1±. (10.150)
The ADM Hamiltonian of the Taub Universe is obtained from Eq. (10.145b)
and reads as
454 Primordial Cosmology
where v e [1/2, oo). The above Hamiltonian (10.151) can be further sim-
plified defining a new variable
x = \nv (10.152)
and becomes
Hl BM = Px =p. (10.153)
Within this framework, the Taub model is described by a two-
dimensional system in which the variable r is considered as time, while
the variable x describes the single degree of freedom of the Universe, i.e.
the change of shape. The classical singularity arises for t — > oo. The
configuration variable x is related to the Universe anisotropy /3+ via the
expression (8.53), for (10.150), as
By this equation, a monotonic relation between /3+ and the configuration
variable x G [a;o,co), where xq = ln(l/2), appears, measuring the degree of
anisotropy of the Universe. The isotropic shape of the Taub model (which
corresponds to (i + = 0) comes out for the particular value x = ln(V3/2),
leading to the closed FRW Universe.
Let us analyze the correspond imi dynamics. The equations of motion
follow from Eq. (10.153) and the system describes a free particle (the point-
Universe) bouncing against the wall at x = x$. The Taub model can be
interpreted as a photon in the Lorentzian plane (t. x) and the classical
trajectory is on its light-cone. The incoming particle (r < 0) bounces on
the wall (x = xq) and then falls into the classical cosmological singularity
(t -> oo) (see Fig. 10.4).
The quantum dynamics of this cosmological model is discussed below in
the WDW framework and, in a different quantization scheme, in Sec. 12.6.
10.10.2 Quantum framework
The quantum dynamics of the Taub model is here analyzed in the context
of the ADM reduction of the dynamics. For later purposes (see Sec. 12.6)
we choose the wave function in the momentum representation. The variable
r is then regarded as a time coordinate and a Schrodinger-like equation
id T V(T,p)=nl BM *(T,p) (10.155)
holds. Therefore we obtain the eigenvalue problem
k 2 Mp)=P 2 Mp), (10.156)
:r 29, 2010 11:22
World Scientific Book - 9in x 6in
PrimordialCosmology |
Standard Quantum Cosmology
■s of the Taub Uni
¥|
Jo
dkA(k)ip k (x)e
(10.157)
We have to square the eigenvalue problem in order to correctly impose
the boundary condition. In order to proceed forward, we assume that the
functional form of the eigenfunctions be the same either with or without
the square root. Correspondingly, we assume that the eigenvalues are the
square of the original problem. The solution to Eq. (10.156) is the Dirac
^-distribution
xp k (p) =5(p 2 -k 2 ). (10.158)
The wave functions of the model in the coordinate space are given by
k[X) 2 7_ 00 27T
- (A6(p - k) + B6(p + k))
(10.159)
Primordial Cosmology
where A and B are integration constants. This way, the boundary condition
ij>(x = xo) = (10.160)
fixes one integration constant providing the eigenfunctions
2k ^
(10.161)
Let us now investigate the fate of the classical singularity at a quantum
level. In particular, we will construct and examine the motion of wave
packets leading to a precise description of the evolution of the Taub model.
The wave packets are superposition of the eigenfunctions (10.161) as in
Eq. (10.157). Similarly to the FRW case, we can take A(k) as a Gaussian-
like function
A(k) = ke 5^~
peaked at energies much smaller than the Planck c
at the numerator in Eq. (10.162) simplifies the om
(10.162)
. Let us note that k
i Eq. (10.161). The
1*1 WDW
Figure 10.5 The evolution of the probability density of the wave packets |\£(t, x)\ in
the WDW case for the Taub model. The wave packets are peaked along the classical
trajectories previously described. The x variable is in the [x Q = ln(l/2), 5]-interval.
plot resulting from the superposition of the eigenfunctions in Eq. (10.161)
Standard Quantum Cosmology 457
with the Gaussian-like weight function in Eq. (10.162), is given in Fig. 10.5.
The wave packets are peaked along the classical trajectories analyzed above.
The probability amplitude to find the particle (Universe) is peaked around
the whole trajectory, thus no privileged region arises in the (t, x)-plane. As
a matter of fact, the "incoming" Universe (r < 0) bounces to the potential
wall at x = xq and then falls into the classical singularity (t — > oo).
Also in this case, the WDW formalism is not able to shed light on the
necessary quantum resolution of the classical cosmological singularity. We
will discuss in Sec. 12.6 how this picture changes in generalized quantization
schemes.
10.11 Quantization of the Mixmaster in the Misner Picture
In this Section we provide a first insight into the quantum dynamics of the
Bianchi IX cosmological model (for the classical description see Sec. 8.2).
We discuss the approach relying on an adiabatic approximation ensured by
the behavior of the potential term toward the oostnological singularity. In
this scheme, which has to go back to the seminal work of Misner in 1969,
the potential is modeled as an infinite square box with the same measure
as in i he original triangular picture. We will see that the wave function
oscillates with a frequency increasing with the growth of the occupation
number. A more complete analysis of the quantum dynamics of the Bianchi
IX Universe will be the subject of Sec. 10.12.
As we have seen, by replacing the canonical variables with the cor-
responding operators and implementing the Hamiltonian constraint as a
condition for the physical states, the system is described by the function
ty = ^(a, j3±). Here we adopt a = In a, where a is the isotropic scale factor
of the Universe, and the classical singularity appears as a — > — oo. Adopt-
ing the standard representation in the configuration space we address the
WDW equation corresponding to Eq. (8.35) as
#ix* = - ^£^ e ' 3a (~9l + d 2 + + d 2 _-V)* = 0. (10.163)
A solution to this equation can be searched in the form
*(a,/3±) = ^r„(a)V>„(a,/3±), (10.164)
where the coefficients T n are a-dependent amplitudes. In particular, we
require that the evolution in a is mainly contained in these coefficients
458 Primordial Cosmology
and that the functions ip n ((x,/3±) depend on a parametrically only. This
condition, known as the adiabatic approximation, reads as
|<9 Q r n | > \d a ip n \. (10.165)
Equation (10.163) reduces to the ADM eigenvalue problem
(-d 2 + -d 2 _+V)il) n = El{a)4> n . (10.166)
Note that the "energy" eigenvalues E n arc a -dependent. We now approxi-
mate the triangular domain of the potential U(/3±) by a two-dimensional
rectangular box centered in /3± =0. Such an approximation is valid only
asymptotically to the singularity where U(/3±) becomes an infinite poten-
tial well on a triangular basis. The two different domains are required to
have the same area. The eigenvalues E n (and also the eigenfunctions) are
assumed to be the same as those in the square box, i.e. E 2 = (rnr/L) 2 .
Here L 2 denotes the area of the box which we demand to be equal to
L 2 = 3\/3^ all . (10.167)
In fact this is the area of the triangular domain as /3 wa ii = —a/2. The
eigenvalues E n are thus given by
^KsKfVa' (10J68)
where n 2 = n\ + n 2 _ and n± G N are the two independent quantum numbers
corresponding to the variables (3±, respectively. Substituting the expres-
sion for the wave function (10.164) in Eq. (10.163) we get the differential
equation for T n
E(^ r «)^+E r «(^«)
+ 2 ^(<9 Q r„)(d a Vn) + Y, E l T n^n = , (10.169)
which, in the limit of the adiabatic approximation (10.165), simplifies to
rl 2 r n 2 Att 2
^ + %„ = 0, a 2 n = §-n 2 . (10.170)
da 2 a 2 3 3 / 2
The equation above is solved by trigonometric functions in the form
r n (a) = Civ^sin Q^lna) + C 2 ^cos Q^lna) , (10.171)
where p n = o? n — 1. From Eq. (10.171), the self-consistence of the adiabatic
approximation is ensured. Figure 10.6 shows the behavior of T n (et) for
Standard Quantum Cosmology 459
various values of the parameter a n . Such wave function behaves like an
oscillating profile whose frequency increases with occupation number n and
approaching the cosmological singularity, while the amplitude depends on
the a variable only.
By this treatment, Misner himself obtained the interesting result that
the occupation number n, on average, is constant toward the singularity.
More precisely, taking an average over many runs and bounces, it is possible
to get the relation
(H ADM a) = const. (10.172)
where -Hadm is the Hamiltonian with respect to the time variable a (see
Sec. 8.1). Replacing -Hadm with the energy eigenvalues (10.168) the result
(n) = const. (10.173)
is obtained, i.e. n can be regarded as an adiabatic invariant. Let us consider
an initial semiclassical state (in the sense of n > 1) and extrapolate its
backwards evolution toward the cosmological singularity. The semiclassical
character of this state is then preserved during the whole dynamics although
the Universe reaches a full Planck regime. Such a behavior is in agreement
to what obtained in the FRW case (Sec. 10.8) as well as in the Taub model
(Sec. 10.10).
10.12 The Quantum Mixmaster in the Poincare Half Plane
The Misner representation (see Sec. 10.11) provided a good insight in some
qualitative aspects of the Mixmaster model quantum dynamics and allowed
some physical considerations on the evolution toward the singularity. How-
ever, in this picture the potential walls move with time providing an obstacle
toward a full implementation of a Schrodinger like quantization scheme. In
this Section we perform a further description of the quantum properties
associated to the Mixmaster dynamics when addressed in the canonical
metric approach. Indeed we will consider the MCI variables (characterized
by static potential walls) of the Poincare half plane representation (intro-
duced in Sec. 10.9) . The choice of such a parametrization of the Lobacevskij
plane allows one to deal with a simple geometry which reduces the differ-
ences between the Bianchi I model and the Mixmaster type to a problem
of boundary conditions. Such an improvement of the quantization scheme
permits to refine the Misner analysis outlining for instance the discreteness
of the energy spectrum and the existence of a zero point energy.
Primordial Cosmology
Figure 10.6 Behavior of the solution T n (a) for three different values of the parameter
k n = 1, 15,30. The bigger k„, the higher the frequency of oscillation is.
10.12.1 Continuity equation and the Liouville theorem
Since the Mixmaster provides an energy-like constant of motion toward the
singularity, the point Universe randomizes within a closed domain and we
can characterize the dynamics as a microcanonical ensemble, as discussed
in Sec. 8.3.
The physical properties of a stationary ensemble are described by a dis-
tribution function p = p(n, j'.p,,,/),,), representing the probability of finding
the system within an infinitesimal interval of the phase space (u,v,p u ,p v ),
and it obeys the continuity equation
d{up) d{vp) d(p u p) d{p v p) _
where the dot denotes the time derivative and the Hamilton equations a
sociated to Eq. (10.145b) read a
p u = , (10.175a)
p v = --. (10.175b)
(10.176)
From Eq. (10.174) and Eq. (10.175) we obtain
v^Pudp_ + v^p 2L dp__ e dp =Q
e du e dv v dp v
The continuity equation provides an appropriate representation sufficiently
close to the initial singularity only, where the infinite potential wall ap-
proximation properly works. Such a model corresponds to deal with an
energy-like constant of motion e and fixes the microcanonical nature of
the ensemble. Since we are interested to the distribution function in the
(u, v) space, we will reduce the dependence on the momenta by integrating
p(u,v,p u ,p v ) in the momentum space. Assuming p to be a regular, vanish-
ing at infinity in the phase-space, limited function, we can integrate over
Eq. (10.176) getting the equation for w = w(u, v; k) as
dw f E\ 2 dw E 2 -2C 2 v 2 w
^ + iU)- i ^ + cv 7^m =0 - (I0177)
where the constant C appears, due to the analytic expression of the HJ
solution to (2.84a) for this model, i.e.
2 E2v \+D, (10.178)
where D is an integration constant and we have taken e = E. In other
words, we expressed the time derivative of u, v in terms of the momenta
by the Hamilton equations (10.175) and, in turn, such momenta via the
Hamilton function S(u.v).
However, the distribution function cannot depend on the initial condi-
tions that fix the constants C and E, and they must be ruled out from the
final result. We obtain the following solution in terms of a generic function
^,v;C)= \_ _ L . (10.179)
462 Primordial Cosmology
The distribution function cannot contain the constant C and the final result
is obtained after the integration over it. We define the reduced distribution
w(u, v)= I w{u, v- k)dk , (10.180)
where the integration is taken over the classical available domain for p u = C
expressed as A = [-E/v,E/v]. In Eq. (10.147) we demonstrated that
the measure associated to it is the Liouville one; the measure w mc (after
integration over the admissible values of </>) corresponds to the case g =
const.
U>mc(«, V)= [' / dC = J . (10.181)
Summarizing, we have derived the generic expression of the distribution
function fixing its form for the microcanonical ensemble. This choice, in
view of the energy- like constant of motion "Hadm, is appropriate to describe
the Mixmaster system restricted to the configuration space. This analysis
reproduces in the Poincare half plane the same result as the stationary
invariant, measure described in Sec. 8.4.
10.12.2 Schrodinger dynamics
The Schrodinger quantum picture is obtained in the standard way, i.e.
by promoting the classical variables to operators and imposing Dirichlet
boundary conditions onto the wave function as
m(dU Q ) = 0. (10.182)
The quantum dynamics for the state function ty = ^(u,v,t) is governed
by the Schrodinger equation
i* = *a D m* = ^- 2 - *-.- (*.-j * . (10.183)
We have to address two main problems: the operator-ordering for the
position and momentum (here parametrized by the constant a) and the
non-locality of the Hamiltonian operator. Indeed, when solving the super-
Hamiltonian constraint with respect to p T , the ADM Hamiltonian contains
a square root and consequently it might define a non-local dynamics.
The question of the correct operator-ordering is addressed in the next
Section comparing the classic evolution versus the WKB limit of the
Standard Quantum Cosmology 463
quantum-dynamics and requiring a proper matching. On the other hand,
we will assume the operators %adm and ?i ADM to have the same set of
eigenfunctions with eigenvalues E and E 2 , respectively. It is worth noting
that in the domain Uq, HADM has a positive sign (the potential vanishes
asymptotically). Under these assumptions, we will solve the eigenvalue
problem for the squared ADM Hamiltonian given by
*ioM*B - p£ - v^l (.«!)] * E - EH E , (10,84,
where * E = Ve(u,v,E).
In order to study the WKB limit of Eq. (10.184), we separate the wave
function into its phase and amplitude
*e(«, v, E) = y/r(u,v,E)e t,T ^ v - E) . (10.185)
In Eq. (10.185) the function r(u,v,E) represents the probability density
and the quasi-classical regime appears in the limit h — > 0; substituting
Eq. (10.185) into Eq. (10.184) and retaining only the lowest order terms in
itain the system
= E , (10.186a)
dr da dr da ( a da d 2 a d 2 a\
dudu dv dv \v dv dv 2 du 2 )
In view of the HJ equation and of Hamiltonian (10.145b), we can identify
the phase a as the functional S defined in Eq. (10.178). Because of this
identification, Eq. (10.186b) reduces to
Comparing Eq. (10.187) with Eq. (10.177), we see that they coincide for
a = 2 only. This correspondence is expected for a suitable choice of the con-
figurational variables; however, it is remarkable that it arises for the chosen
operator-ordering only, allowing to fix a particular quantum dynamics for
the system. Summarizing, we have demonstrated that it is possible to get a
WKB correspondence between the quasi-classical regime and the ensemble
dynamics in the configuration space, and we provided the operator-ordering
to quantize the Mixmaster model
^-sW)' (10 ' 188 »
464 Primordial Cosmology
10.12.3 Eigenfunctions and the vacuum state
Once fixed the operator ordering by a = 2, the eigenvalue equation (10.184)
rewrites as
kjj + v2 ^2 + 2v ^ + E2 \ *e(«, v,E) = 0. (10.189)
By redefining ^ E (u,v,E) = ip(u,v, E)/v, we can reduce (10.189) to the
eigenvalue problem for the Laplace-Beltrami operator in the Poincare plane
V L bV(«, v, E) = v ^— + —j ^(u, v, E) = E s ^{u, v, E) , (10.190)
which is central in the harmonic analysis on symmetric spaces and has been
widely investigated in terms of its invariance under SL(2, C). Its eigenstates
and eigenvalues are
M% v) = av s + by 1 '* + ^Y, a n K s _ 1/2 (2n\n\v)e 2 * mu (10.191)
VlB^(«, V) = 8(8 ~ 1)^ 8 (U, V) (10.192)
where a,b,a n G C, K s _i/ 2 (2imv) are the modified Bessel functions of the
third kind and s denotes the index of the eigenfunction. This is a continuous
spectrum and the sum runs over every real value of n. The eigenfunctions
for this model read as
* E (u, v, E) = av'- 1 + bv-° + ]T a n K *-^\n\v) ^ lnu ^ Qm)
y/t
with eigenvalue
E 2 = s(l-s). (10.194)
To impose Dirichlel boundary conditions for the wave functions, we will re-
quire a vanishing behavior on the edges of the geodesic triangle of Fig. 10.3.
Let us approximate the domain with the one in Fig. 10.7; the value of the
horizontal line v = 1/ir provides the same measure for the exact as well as
for the approximate domain
L^T=L ^= n - (io - i95)
The difficulty to deal with the exact boundary conditions relies on the
mixing of solutions with different indices s due to the semicircle that bounds
the domain from below. The Laplace-Beltrami operator and the exact
:r 29, 2010 11:22
World Scientific Book - 9in x 6in
PrimordialCosmology |
Standard Quantum Cosmology
.7 The approximate domain where v,
- 1/ir for the straight line preserves th
mpose the boundary conditions. The
boundary conditions are invariant under a parity transformation u — > — u;
however, the full symmetry group has two one-dimensional irreducible
representations and one two-dimensional representation. The eigenstates
transforming accordingly to one of the two-dimensional representations are
twofold degenerate, while the remaining are non-degenerate. The latter can
be divided into two classes, satisfying either Neumann or Dirichlet bound-
ary conditions. We focus our attention on the second case. The choice of
the line v = l/ir approximates the symmetry lines of the original billiard
and corresponds to the one-dimensional irreducible representations. The
conditions on the vertical lines u = 0, u = — 1 require to disregard the first
two terms in Eq. (10.193); furthermore, we get the condition on the last
J2 e 2™u _> £ sin(7rnu ) , (10.196)
n^0 n=l
for integer n. As soon as we restrict to only one of the two one-dimensional
466 Primordial Cosmology
representations, we get
Y^ e 2™»« _j. ^ S i n (27rnu) , (10.197)
while the condition on the horizontal line implies
Y^ a n K s _ l/2 (2n) sin(2n7ru) = , Vm £ [-1, 0] , (10.198)
n>0
which in general is satisfied by requiring K s _i/ 2 (2n) = only, for every
n. This last condition, together with the form of the spectrum (10.194),
ensures the discreteness of the energy levels, because of the discreteness of
the zeros of the Bessel functions. The functions K v {x) are real and positive
for real argument and real index, therefore the index must be imaginary,
i.e. s = l J2 + it. In this case, these functions have (only) real zeros, and
the corresponding eigenvalues turn out to be real and positive.
E 2 =t 2 + -. (10.199)
The eigenfunctions (10.193) exponentially vanish as infinite values of v are
approached. The conditions (10.198) cannot be analytically solved for all
the values of n and t, and the roots must be numerically worked out for
each n. There are several results on their distribution that allow one to find
at least the first levels: a theorem by [373] on the zeros of these functions
states that K lv (yx) =0o-0<a;<l; furthermore, the energy levels
(10.199) monotonically depend on the values of the zeros. Thus, one can
search the lowest levels by solving Eq. (10.198) for the first values of n; we
will discuss below some properties of the spectrum, while now we analyze
the ground state only.
A minimum energy exists, as follows from the quadratic structure of
the spectrum and from the properties of the Bessel zeros; its value is E 2 =
19.831, and the corresponding eigenfunction is plotted in Fig. 10.8. The
eigenstate is normalized through the normalization constant TV = 739.466.
The existence of such a ground state has been numerically derived, but it
can be inferred on the basis of general considerations about the Hamilto-
nian structure. The Hamiltonian, indeed, contains a term v 2 p 2 l which has
positive definite spectrum and does not admit vanishing eigenvalues.
10.12.4 Properties of the spectrum
The study of the distribution of the highest energy levels relies on the
asymptotic behavior of the zeros for the modified Bessel functions of the
sr 29, 2010 11:22
Figure 10.8 The ground s
wave function of the Mixm
third kind. We will discuss the asymptotic regions of the (t, n) pla:
two cases t > n and t ~ n » 1 .
(i) For t> n, the Bessel functions admit the representation
aflE i
~ ( _ 1)fc
,sa Z^72feTr u 2fc+i
where a = x /4 — -\A 2 — n 2 + iarccosh(i/n), p = n/£ and u^ are the
polynomials
Ufc+ i(t) = i* 2 (l - t 2 K(*) + J / (1 - M 2 )u k (t)dt .
1 ° Ja
(10.201)
Retaining in the expression above only terms of order 0( n /t), the
zeros are fixed by the relation
468 Primordial Cosmology
In the limit n/t <C 1, Eq. (10.202) can be recast as
tlog(t/n) = ln => t= ^ , (10.203)
productlog (^J
where productlog(,z) is a generalized function giving the solution
to the equation z = we w and, for a real and positive domain, is a
monotonic function of its argument. In Eq. (10.203) I is an integer
number much greater than unity in order to verify n/t <C 1.
(ii) In case the difference between 2n and t is 0(n 1 ^ 3 ) for t,n>l, we
can evaluate the first zeros k StU by the relations
k s , v r.u + J2(-iy Sr (a s )[-) , (10.204)
r=0
where a s is the s-th zero of Ai((2/z) 1 / 3 ), Ai(z) is the Airy function
and si are appropriate polynomials. From llii.s expansion it results
that, to lowest order
t = 2n + 0.030n 1 / 3 . (10.205)
Equation (10.205) provides the lowest zero (and therefore the en-
ergy) for a fixed value of n and also the relation for the eigenvalues
for high occupation numbers as
E 2 ~4n 2 + 0.12n 4/3 . (10.206)
Let us discuss the completeness of the spectrum and the definition of
a scalar product. The problem of completeness can be faced by studying
firstly the sine functions and then the Bessel ones. On the interval [—1,0],
the set sin(27rnu) is not a complete basis, but as soon as we request the
wave function to satisfy the symmetry of the problem, it becomes com-
plete. Let us take a value n > 0, thus the functions (10.193) have the
form V E {u,v) = sm(2Tr nu)g(v), which substituted in Eq. (10.189) provides
v 2 (d 2 + (27m) 2 )) g(v) = s(l-s)g(v), whose solutions are exactly the Bessel
functions. This property together with the condition on the line v = 1/tt
forms a Sturm-Liouville problem with a complete set of eigenfunctions.
Therefore, such eigenfunctions define a space of functions where we can
introduce a scalar product, naturally induced by the metric of the Poincare
U>,4>)=JMu,v)4>Hu,v)^. (10.207)
Standard Quantum Cosmology 469
Now we briefly discuss if the presence of a non-local function, like the
square-root of a differential operator, can give rise to non-local phenomena.
A wavepacket which is non-zero in a finite region of the domain (v < M) and
far from infinity fails to run to infinity in a finite time, i.e. the probability
P(v > M) to find the packet far away exponentially vanishes. Indeed,
P(v > M)
./°/-|.v
M v 7
< AJ- (sup Vf Me~ 2M (10.208)
where sup('I') is the maximum value of the wavepacket in the domain v < M
and Ei(z) = — J_ e~' /idl is the exponential integral function. We can
conclude that nevertheless the square root is a non-local function, non-
local phenomena do not appear (like the case of a wavepacket starting from
a localized zone and falling out to infinity).
10.13 Guidelines to the Literature
Quantum geometrodynamics, discussed in Sec. 10.1, has been firstly ana-
lyzed by DeWitt in [149-151]; for a review see for example [32,262,281,297]
while a good textbook is that of Kiefer [279]. The Euclidean approach
to quantum gravity (Sec. 10.1.2) has been proposed mainly by Gibbons,
Hawking and collaborators in [194,195,225,229]; for a detailed discussion
see for example the book edited by Gibbons & Hawking [196]. For the
Gibbons-Hawking- York boundary term see also [470] . For a general review
on quantum gravity, see also [113].
Discussions about the problem of time (Sec. 10.2) can be found in [262,
442] while for a more recent account see [186,280,399,416]. The role of a
quantum perfect clock is discussed in [443] . The Brown-Kuchaf mechanism
has been proposed in [104,299]. The evolutionary approach to quantum
gravity has been proposed in [353] and developed in [52,342,355]. The
result of Torre is in [440]. The relational point of view, introduced by
Rovelli, has been elaborated in [154,397,398].
The first formulation of a minisuperspace theory as described Sec. 10.3
470 Primordial Cosmology
can be found in [344. 346. 465]. Recent reviews on quantum cosmology
are [170,218,282,467]. A more detailed list of classical papers is given
in [217]. The interpretation of the theory given in Sec. 10.3.2 is discussed
in [219,220,387,451], while quantum cosmological singularities (Sec. 10.3.3)
are analyzed in [90, 149, 199]. The principle of quantum hyperbolicity can
be found in [93].
The path integral quantization of a minisuperspace model discussed in
Sec. 10.4, has been formulated in [216] and developed in [221-223,276];
for some reviews see [218,360,417], while for an explicit application to the
Kasner Universe see [81].
A clear and complete discussion of matter fields as relational times, is
in [262]. In particular, the role of the scalar field, described in Sec. 10.5, is
analyzed in [90,199].
The problem of interpreting the wave function of the Universe (Sec. 10.6)
is discussed in [451] and developed in [45,110,277,310] (the analysis of the
wave function correlation is in [215]). The application to the quasi- isotropic
Mixmaster Universe is formulated in [47].
Boundary conditions, discussed in Sec. 10.7, are reviewed for example
[218,278,452]. The no-boundary one has been proposed in [225,230] (a
recent development can be found in [179,226,227]), while the tunneling one
in [322,449,450]. The comparison between these two schemes is proposed
in [206,293,372] (see also [218]).
The quantization of the FRW model presented in Sec. 10.8 has been
formulated in [90,308]. The singularity avoidance conjecture is in [199,200].
A discussion on the Poincare half plane presented in Sec. 10.9 is in
[286,354,435].
The original work that introduces the Taub cosmological model
(Sec. 10.10) is [431]; a complete discussion is proposed in 1400]: the quanti-
zation of the model is in [114,330].
The scheme presented in Sec. 10.11 is mainly based on the original
work [340]. This model has been developed, for example, in [200].
The quantization of the Mixmaster model (Sec. 10.12) in the half plane
is presented in [71]; for a different approach see [201], while for a discussion
on some problems related to the sign of the Bianchi I X potential see [275].
A clear analysis of the Laplace-Beltrami operator can be found in the book
of Terras [435] while an application to quantum gravity is in [389] .
The Mixmaster Universe has been investigated in the framework of
what is called quantum chaos, searching the possible link with the classical
behavior. This topic, not addressed here, has been faced for example
in [73, 180, 181,202]; for a different but related analysis, see [134,203].
This page is intentionally left blank
Chapter 11
Generalized Approaches to Quantum
Mechanics
This Chapter presents some non-standard approaches to quantum mechan-
ics which will be relevant in modern formulations of quantum cosmology
analyzed in Chap. 12. We devote particular attention to the polymer repre-
sentation of quantum mechanics because of its relation with Loop Quantum
Cosmology. Deformed Heisenberg algebras are also analyzed paying atten-
tion to their connection with non-commutative geometries as well as with
String Theory.
We start with a concise introduction to the algebraic approach to quan-
tum physics. This scheme is relevant in order to investigate singular repre-
sentations of the canonical commutation relations, being the polymer one
of these. Attention is devoted to the uniqueness theorem of quantum me-
chanics and to the Gelfand-Naimark-Segal construction.
Starting with the relation with the standard representation of quantum
mechanics (namely the Schrodinger one), we analyze the structure under-
lying the polymer quantum mechanics. This framework is the quantum
mechanical scheme behind Loop Quantum Gravity (see Sec. 12.1) once a
system with a finite numbers of degrees of freedom is taken into account. In
this respect, Loop Quantum Cosmology (see Sec. 12.2) can thus be regarded
as the implementation of this quantization technique in the minisuperspace
dynamics.
We then discuss the notion of the Planck scale as a fundamental mini-
mal length in quantum gravity. A simple derivation of such a scale will be
showed in the context of String Theory stressing the differences with re-
spect to the particle framework. The main implication of a minimal length
results to be a modification (or deformation) of the Heisenberg uncertainty
principle-.
The modifications of the Heisenberg algebra in a specific non-
474 Primordial Cosmology
commutative space-time will also be analyzed. We discuss the relation
with the String Theory uncertainty principle and we then analyze the for-
mulation of quantum mechanics in the presence of a minimal scale. The
implementation of these approaches in quantum cosmology will be given in
the second part of Chap. 12.
11.1 The Algebraic Approach
In this Section, we will discuss in a pedagogical manner some elementary
aspects of the so-called algebraic approach to quantum physics. The main
results will be used to describe the polymer representation of quantum
The main idea of the algebraic approach is to consider the observables
as the relevant objects of the theory. This procedure is the opposite to the
usual construction where observables are "secondary" objects only. In the
standard formulation of quantum mechanics, the first step is the construc-
tion of a Hilbert space T and the definition of vectors living in it. The
vectors in the Hilbert space are the states of the theory and one defines
the observables as operators which act upon the states. In this sense, the
observables are secondary objects of the theory, while the states are the
elementary building blocks. In particular, a self-adjoint operator, defined
in J 7 , corresponds to each measurable quantity of the classical theory. The
spectrum of this operator defines the possible values which may be mea-
sured during an experiment. For physical purposes, the description of the
Hilbert space and the choice of basis are irrelevant. From a physical point
of view, the relevant descriptors are the observables (namely the operators
corresponding to measurable quantities). To be more precise, let us take
two different formulations of a theory described by two Hilbert spaces T\
and Ti. Furthermore, let A\ and A 2 be two operators defined in the rela-
tive Hilbert spaces T\ and Ti corresponding to the same observable. These
formulations are equivalent if there exists a unitary map U : T\ — > Ti such
that
A 2 = UA 1 U~ 1 . (11.1)
It is worth noting that the quantization procedure is far from being
unique. As is well known, a classical theory is invariant under canonical
transformations, while the quantum approach is invariant under unitary
transformations. The group of (classical) canonical transformations is how-
ever not isomorphic to the unitary group of the Hilbert space in which (q,p)
Generalized Approaches to Quantum Mechanics 475
are irreducibly represented. This way, the results of quantization depend on
the choice of the classical canonical variables and the empirical evidence
guides in the construction of the quantum theory.
As we mentioned above, the algebraic approach inverts the roles played
by observables and states. In such a framework, one begins by constructing
an abstract algebra whose elements are the observables. The states are
defined in a second moment as the objects which act upon observables by
associating a real number to each observable, corresponding to taking the
expectation values as in the standard way. The main goal of this approach
is that all states, in particular those arising in unitarily inequivalent rep-
resentations, are treated on an equal footing and one can define a theory
without the need to select a preferred construction.
In the first step we will introduce some basic concepts and the unique-
ness representation theorem of quantum mechanics, and thereafter we will
face the Gelfand-Naimark-Segal (GNS) construction and the Fell theorem.
11.1.1 Basic elements
Let us start by introducing the notion of a C*-algebra. A vector space over
C is defined as a set on which are defined the operations of addition and
scalar multiplication. An algebra A over C is a vector space over C with
an additional multiplication map
x:AxA->A, (11.2)
which is bilinear (i.e. linear in each variable) and associative. From now
on the "x" operator will be dropped. If we add the involution map (also
called the *-operation) * : A — > A satisfying
A** = A, (11.3)
(AB)* = B*A*
for any A,B G A, then the algebra A will be said a *-algebra. The last
element needed to construct a C*-algebra is the introduction of a topology,
i.e. a definition of a neighborhood of an element of A. The most evident
topology for the algebra is the norm map ||.|| : A — > R + such that:
IMII = MPII, (ii.4)
|L4 + B||<|L4|| + ||B||,
\\AB\\<\\A\\\\B\\,
476 Primordial Cosmology
where a e C. Such norm induces the metric d(A, B) = \\A — B\\. Finally,
if we require that
U*\\ = \\A\\, (11.5)
\\A*A\\ = \\A\\ 2
for all A G A, then the algebra A will be a C*-algebra.
The other key ingredient in the algebraic approach is the definition of
an algebraic state, not to be confused with the definition of a state in the
ordinary formulation of quantum mechanics. To avoid any confusion, we
call the latter a physical state. An algebraic state w of the quantum theory
is defined to be a linear map
uj:A^C : (11.6)
from a C*-algebra A to C. This object is positive definite
lo(A*A) > MA e A (11.7)
as well as normalized
W (I) = |M| = 1, (11.8)
I being the identity element of A. Let us now assume that:
(i) the self-adjoint elements of A correspond to observables. This
means that the ^-operation corresponds to taking the adjoint.
(ii) The unit element I is the trivial observable having the value 1 in
any physical state (which arc vectors in the Hilbert space).
This way, an algebraic state (a normalized positive linear form u>) can be
interpreted as an expectation value over the observables, explicitly
lo(A) = {A). (11.9)
This observation clarifies the physical interpretation of the algebraic ap-
proach and its precise formulation will be given below in terms of the so-
called GNS construction.
Let us illustrate such statement considering the simplest mechanical
system, i.e. a particle on the real axis R. The phase space of this model is
R 2 with coordinates (q,p) satisfying the Poisson brackets
{q,p} = l. (11.10)
The quantum kinematic correspondence to this system is described by oper-
ators, represented on a Hilbert space, satisfying the canonical commutation
relations
[q,p]=ii. (11.11)
Generalized Approaches to Quantum Mechanics 477
Consider the algebra generated by the exponentiated versions of the basic
operators (q,p) which are denoted by
where a and 13 have dimensions of momentum and length, respectively. The
canonical commutation relation (11.11) becomes 1
U(a)V(p) = e- ial3 V(p)U(a). (11.13)
The two quantities in Eq. (11.12) generate the so-called Weyl algebra W
which is obtained by considering the linear combinations of the generators
(11.12). A generic element W = W(a,/3) of W can in general be expressed
W = Y, 0W«i) + BiV(0i)) , (11.14)
and the Weyl algebra has the natural structure of a C*-algebra.
From this perspective, the quantization of a mechanical system consists
of finding a unitary irreducible representation of the Weyl algebra W on
a Hilbert space J ' . It is natural to ask which is the role of the usual
Schrodinger representation of quantum mechanics and to investigate about
other possible representations of the canonical commutation relations. The
Weyl algebra is exactly the way to answer this question.
The ordinary Schrodinger construction is based on the choice of the
Hilbert space
T = L 2 (R,dq), (11.15)
which is the space of the square integrable functions with respect to the
Lebesgue measure dq on R. The basic operators are then represented as
M{q) = q1>(q), (11-16)
Pi>(q) = -id q ip{q).
The Stone- von Neumann theorem ensures that this is the unique irreducible
representation of the Weyl algebra (namely, of the canonical commutation
relations) if the operators (11.12) are weakly continuous in the parame-
ters a and /?. The Schrodinger representation is unique up to unitarily
equivalence. There are however many irreducible representations where
i i I i i 1 1 I
478 Primordial Cosmology
the continuity condition is not satisfied (such kind of representations are
often called singular representations) such as, for example, the polymer
representation (see Sec. 11.2) belongs to the latter class.
It is worth noting that such a fundamental theorem is valid only for
systems with finite degrees of freedom, i.e. for quantum mechanics. For
system with infinite degrees of freedom (field theories) there exists a host
of inequivalent, irreducible representations of the canonical commutation
relations which defies a useful complete classification.
11.1.2 GNS construction and Fell theorem
The relation between the algebraic and the ordinary approach to the quan-
tum theory can be formulated through the celebrated construction given
by Gelfand, Naimark and Segal (GNS) which can bo stated as follows.
Theorem 11.1 (GNS Construction). Let A be a C* -algebra with unit
and let u : A — )• C be a state. Then there exist a Hilbert space J- ', a
representation 2 n : A — > L^) and a vector Q, € T such that
U (A) = (£l\n(A)\Sl)r . (11.17)
These objects satisfy the additional property that f2 is cyclic, i.e. the vectors
tt(A)Q for all A G A comprise a dense 3 subspace of T . The triplet (J ", n, Q)
is uniquely determined (up to unitary equivalence) by these properties.
Each positive linear form to over a C*-algebra defines a Hilbert space
as well as a representation of the algebra by linear operators acting on the
Hilbert space. One key aspect of the GNS construction is that one can
have different, but unitarily equivalent, representations of the Weyl algebra
which yield equivalent theories.
Let us now sketch some details of this construction. As we have seen,
the algebra A is a linear space (over C) and the state to defines a Hermitian
scalar product on A by
(A\B)=w(A*B) (11.18)
for A, B € A. However, due to the positivity condition (11.7), this scalar
product is semi-definite positive. In fact, it can occur that, for some X €
A, we have ui(X,X) = 0. In order to have a (properly) positive definite
2 We denote by L{T) the collect ion of all bounded linear maps onT. It has the s1
of a C" algebra.
3 A subspace Y of a topological space T is said to be dense in T if the closure of Y is
equal to T.
Generalized Approaches to Quantum Mechanics 479
inner product, we have to factor out the contributions given by the set
J of elements X. The inner product will be taken in the corresponding
factor space A/ J. An element of the factor space is denoted by [A] and
corresponds to the equivalence class
[A} = {A + X} with AeA,XeJ. (11.19)
The Hilbert space T is thus defined by the completion of A/ J with respect
to the norm (11.7). The product in A then defines the representation
7T : A -> L{T) by
ir(A)[B] = [AB], (11.20)
for all A G A. Finally, the cyclic vector fl corresponds to the identity
element of the algebra A.
This construction can be inverted. In general, any vector $e J defines
an algebraic state
w 9 (A) = (9\n(A)\V)j:, (11.21)
and furthermore, since ft is a cyclic vector, the vector ^ can be approxi-
mated by ir(B)tt. The state (11.21) can be given as
u*(A) ~u(B*AB) = (B\AB) , (11.22)
with B € A.
The GNS construction shows that states over a C Y * -algebra come in fam-
ilies. In fact, a single (algebraic) state u determines a family of (physical)
states by means of Eq. (11.21). More generally, it is possible to consider
the states
iu p (A) = Tr[p7r(A)], (11.23)
where p is a density matrix. The collection of all the states (11.23) is the so-
called folium of the representation ir. The notion of a folium is fundamental
in order to enunciate the Fell theorem, one of the most important ones in
the algebraic approach to the quantum theory.
Theorem 11.2 (Fell Theorem). The folium of a faithful representation 4
of a C* -algebra is weakly dense in the collection of all states.
For a better understanding, let us reformulate this theorem. Let (.Fi,7ri)
and (J2, 7r 2 ) be (possibly unitarily inequivalent) representations of the Weyl
algebra W in the sense of the GNS construction. Let A\, . . . , A n G W and
4 A representation 7r is said to be faithful if ir(A) ^ for A ^ 0.
480 Primordial Cosmology
£i,...,e„ > 0. Let uji be an algebraic state corresponding to a density
matrix on the Hilbert space T\ . The Fell theorem ensures that there exists
a state u>2, corresponding to a density matrix on Ti, such that
\uniAi) - w 2 (Ai)\ < *, (11.24)
for all i = l,...,n. The theorem shows that, although two representa-
tions of W can be inequivalent, the determination of a finite number of
expectation values of observables in W, made with finite accuracy, cannot
distinguish between different representations. In physics it is not possible
to perform infinitely many experiments and furthermore each experiment
has a finite accuracy. This way, by monitoring a state, we can at most
determine a weak neighborhood in the space of all states. The Fell theorem
states that we cannot find out in which folium the state lies.
Let us put forward this consideration by assuming that the observables
in an algebra A are the only measurable quantities of a quantum field
theory. Thus, because of the physical realistic limitation of finitely many
measurements with finite accuracy, different (namely inequivalent) repre-
sentations of the algebra are "physically equivalent" and the choice of the
representation is physically irrelevant. Such a (fascinating) statement is
however not valid in general as, in fact, there are additional observables in
the theory which cannot be represented in A, as for example the energy-
momentum tensor. Two representations should not be "physically equiva-
lent" with respect to these additional observables and, in those cases, not
treated in this book, the so-called Badamanl condition has to be invoked.
11.2 Polymer Quantum Mechanics
The polymer representation of quantum mechanics is based on a singular
(non-standard) representation of the canonical commutation relations. In
particular, in a two-dimensional phase space, it is possible to choose a dis-
cretized operator, whose conjugate variable cannot be directly promoted as
an operator. From a physical point of view, this scheme can be interpreted
as the quantum-mechanical framework for the introduction of a cutoff. Its
continuum limit . which corresponds to the removal of such a cutoff, has
to be understood as the equivalence class of theories modified at different
microscopical scales.
This framework is relevant to make a bridge with the Planck scale
physics. In particular, it is interesting when treating the quantum-
mechanical properties of a background-independent canonical quantum the-
Generalized Approaches to Quantum Mechanics 481
ory of gravity. More precisely, the holonomy-fiux algebra used in Loop
Quantum Gravity (LQG, see Sec. 12.1) reduces to a polymer-like algebra,
when a system with a finite number of degrees of freedom is considered.
Loop Quantum Cosmology (LQC, see Sec. 12.2) can be regarded as the im-
plementation of this quantization technique in the minisuperspace dynam-
ics. Finally, from a quantum-field theoretical point of view, the polymer
representation is substantially equivalent to introducing a lattice structure
on the space.
In this Section we analyze the polymer quantum mechanics starting
from its relation with the Schrodinger representation. The kinematics and
the dynamics of the polymer particle will be discussed later.
11.2.1 From Schrodinger to polymer representation
As we have seen in Sec. 11.1.1, the Stone- Von Neumann uniqueness theorem
ensures that the Schrodinger representation is (up to unitary equivalence)
the only irreducible representation of the Weyl algebra W in which the
operators (11.12) are continuous functions of a and fi. The polymer case is a
particular representation in which this condition is not satisfied, providing a
unitarily incquivalont representation of the canonical commutation relations
and then the physical predictions of the two frameworks will differ. The
link between the Schrodinger and the polymer representations is implicitly
given by the Fell theorem (11.2) since, as we have seen, it ensures that it is
possible to approximate states in the standard representation by states in
a singular representation of the Weyl algebra. However, the Fell theorem
is not constructive because it does indicate how to recover a non-standard
representation. We will show how il is possible to obtain an explicit singular
representation of the Weyl algebra and its manifest link to the Schrodinger
A classical system is described, in the Bamiltouiau formalism, in leru.is
of a symplectic manifold (r, w) where T is the phase space and vj is the
symplectic 2-form which defines the Poisson brackets as
{f,g} = m ab V a fV b g. (11.25)
The quantization aims to find a representation of the canonical commuta-
tion relations (11.13) in a Hilbert space. To analyze the representations of
the Weyl algebra it is useful to introduce the complex structure J : T — > T
such that J 2 = — 1. We focus on one-dimensional mechanical systems and
482 Primordial Cosmology
thus J can be denned by a length scale d only, so that explicitly we have
and thus J : (q,p) — > (—d 2 p, q/d 2 ). This has to be compatible with the
symplectic structure and thus it induces a positive definite, real, inner prod-
uct on T by
g{v,v') = w{v,Jv'), (11.27)
where v denotes a vector in the phase space T = R 2 , namely v = (q,p).
By means of the complex structure J, the Hilbert space can be viewed as
a real vector space. In particular, the hermitian (complex) inner product
is given by
( V \ v ')= l -g( v y )+ l - w {v,v') (11.28)
and then it explicitly decomposes into a real and an imaginary part. No-
tably, the triple (J, g, w) equips the Hilbert space with the structure of
a Kahler space, providing the starting point of the so-called "geometric
formulation of quantum mechanics" .
A relation with the Schrodinger represent ai iou ( namely with the Hilbert
space (11.15)) is recovered by the GNS construction. From Eq. (11.9), the
complex structure (11. 2d) uniquely deiines l ho algebraic state u that yields
(U(a))=e-i d2a2 (11.29a)
(V(/3)) = e --^ 2/d \ (11.29b)
The Hilbert space undoriviti" ibis I'ramowoi'k. i.e. the one equipped with
the extra structure J (or d), is given by
T d = L 2 {R 1 dq d ) (11.30)
where the measure dqd is no longer trivial and reads as
dq d = -^=e-fl d2 dq. (11.31)
aV 7r
The relation with the standard representation is given in terms of a map
between the two frameworks. More precisely, the Hilbert space T d can be
mapped into the Schrodinger one (11.15) by means of an isometric isomor-
phism /C : J-d — > T which is explicitly given by
K= (11.32)
All the ^-representations are unitarily equivalent and this is nothing but
an explicit manifestation of the Stone- Von Neumann uniqueness theorem.
However, in the limiting cases d — > and d — > oo, the map (11.32) is ill
defined and indeed the polymer representation arises in these "regimes" .
Generalized Approaches to Quantum Mechanics 483
11.2.2 Kinematics
The polymer representation of quantum mechanics is constructed so far in
an abstract way, i.e. without addressing any relation with the Schrodinger
one. We start by considering abstract kets |/x), where /jgM, and a suitable
finite subset defined by /i; G 1 with i = 1, 2, . . . , N. The polymer inner
product between these kets and bra is assumed to be
(MH = <V, (11.33)
which is a Kronecker-delta rather than the usual Dirac-delta distribution.
The kets are then assumed to be an orthonormal basis along which any state
\ip) can be projected. Given two states \ip) = J2 t a -i\f l -i) anf l \ ( P) = J2j bjWj),
the inner product between them is given by
(#> = EDWjW=E^, (11-34)
i j k
where k labels the available intersection points. This defines a Hilbert space
-Fpoi.
In quantum mechanics there are two basic operators, the multiplica-
tion and the displacement: let us investigate how they act on the polymer
Hilbert space. The symmetric "label" operator e is such that
e» = n\n) . (11.35)
The second group of operators is given by a one-parameter family of unitary
operators, s(A), such that
S(A)|A*) = |/i + A). (11.36)
Because all kets are orthonormal by means of Eq. (11.33), s(A) is (weakly)
discontinuous. As a result, it cannot be obtained from any hermitian op-
erator by exponentiation. In this sense, the continuity hypothesis of the
Stone- Von Neumann theorem has been relaxed and therefore the polymer
quantum dynamics turns out to be a non-standard representation of the
Weyl algebra. It is worth noting that the polymer Hilbert space is not
separable. 5
For the toy model of a one-dimensional system, whose phase space is
spanned by the variables p and q, the polymer representation techniques
find interesting applications when one of the two variables is supposed to
be discrete. This discreteness will affect both wave functions, obtained by
projecting the physical states on the p or q basis (polarization), together
5 A Hilbert space is separable if and only if it admits a countable orthonormal basis.
484 Primordial Cosmology
with the operators associated to the canonical variables, acting on them.
We will discuss only the case of a "discrete" position variable q, and the
corresponding momentum polarization. In this case, the wave functions are
given by
VV(P) = (P|A*> = e w - (11.37)
Accordingly to the previous discussion, the label operator e is easily iden-
tified with the position operator q, i.e.
gW = -*d P Vv = MVv- ( 1L38 )
On the other hand, the V(A) multiplicative operator in (11.12) corresponds
exactly to the "shift" operator s(A), and in fact one has
V(A) tp„ = e iXp e ip » = W+A) ■ (11-39)
This way, since such operator is discontinuous in A, the variable p cannot
be directly implemented as the operator p in the Hilbert space and only
the operator V(A) is well defined.
The position operator q is discrete, but in a weaker sense with respect
to have a discrete spectrum. More precisely, although its eigenvalues are
continuous (// € M), all the eigenvectors are normalizable. Hence, this
Hilbert space can be expanded out as a direct sum, rather than as a direct
integral, of the one-dimensional eigenspaces of q. This clarifies the difference
with respect to the ordinary representation.
It can be shown that, from generic representation theory arguments,
the corresponding Bilboil space is given by
Jp ol = L 2 (R s ,d M ), (11.40)
which is the set of square- integrable functions defined on the Bohr compact-
ification of the real line Rb, with a Haar measure d/i. Since the kets \/j.) are
arbitrary but finite, the wave functions can be interpreted as quasi-periodic
functions, with the inner product
«vIVa>= / d^l{p)^ x {p)
JR B
= lim -L f dpi;l(p)Mp)=^x. (11.41)
Such construction corresponds to the limit d — ¥ oo case of the previous
discussion (in which p and q are interchanged) . It is worth noting that the
Hilbert space of polymer quantum mechanics -F P oi is exactly the same of
LQC (see Sec. 12.2). This concludes our analysis on the kinematical aspects
of the polymer quantization procedure.
Generalized Approaches to Quantum Mechanics 485
11.2.3 Dynamics
The Hamiltonian H describing a quantum mechanical system is usually a
function of both position and momentum, i.e. of the form H = p 2 + V(q).
As we have seen, in the particular case of a discrete position variable in the
momentum polarization, p cannot be implemented as an operator, so that
some restrictions on the model are still required.
First of all, a suitable approximation of the kinetic term is needed. For
this purpose, it is useful to restrict the arbitrary kets |/l*j), with ieKto
\fii) with ieZ. In other words, we introduce the notion of regular graph
7 Mo , defined as a numerable set of equidistant points, whose separation is
given by the parameter fi expressed as
-/ Ho ={qeR\q = n f i 0l \/neZ}. (11.42)
The associated Hilbert space T lfi is now separable. Because of the regular
graph [i , the eigenfunctions of p Mo must be of the form
exp(im/xoP) ) (11.43)
with m G Z, which are Fourier modes of period 2tt/hq. The inner prod-
uct (11.41) is equivalent to the inner product on a circle S 1 with uniform
{Hp)Mp))u = £ r M ° d P^(pMp) ■ (ii-44)
27r ./-ir/w
The "dynamical" Hilbert space reads as
J- 7mo =L 2 (S\dp) (11.45)
and there it is possible to construct an approximation for the displacement
operator V(X) whose action is the shift of a ket \fj, n ) to the next one |/x n+ i).
Thus, the parameter A has to be fixed to the lattice scale /zo leading to the
desired result
V{no)W) = |/i„ + no) = |/i„ +1 ) . (11.46)
Thereafter, we can build a regulated operator p Mo to implement the usual
incremental ratio in a discrete manner and is defined as
Pu \Pn) = TT— (V(jM)) ~ V(-/Uo)) \Hn)
2tfi a
= J-(\^ n+1 )-\ lin _ 1 )). (11.47)
2tfi a
This basic shift operator will be of fundamental importance when construct-
ing (approximating) any function of p, e.g. the Hamiltonian H itself. From
486 Primordial Cosmology
the relation (11.47), the polymer paradigm can be recovered by the formal
substitution
p-> — sin(^oP), (11.48)
Mo
where the incremental ratio (11.47) has been evaluated for exponentiated
operators.
The Hamiltonian operator in the polymer Hilbert space includes the
pfi °P era tor which can be defined in (at least) two ways. The first one is
to apply the operator (11.47) twice, i.e. p 2 n = p IM) ■ p IM] , leading to two-
step shifts in the graph. The second possibility is to define p 2 from its
approximation in terms of a cosine function, i.e.
pl,\Hn)*\[l-COS(pM>)]\Hn)
Mo
= \{2-V(no)-V(-n ))\n n ), (11.49)
Mo
leading to a one-step shift and, for such reason, the second possibility ap-
pears more suitable. The Hamiltonian operator H^ , which lives in J-" 7 ,
K»o = lT L + V(q), (11-50)
2m
where p 2 ^ is given by Eq. (11.49) and the differential operator q is well
defined in the Hilbert space.
To conclude, we will discuss how to remove the regulator fi which was
introduced as an intermediate step when constructing the dynamics. The
physical Hilbert space can be defined as the continuum limit of effective
theories at different scales and can be shown to be unitarily isomorphic to
the ordinary one J"s = L 2 (K, dp). Indeed, it is impossible to obtain F$
starting from a given graph 70 = {qk € M.\qk = kao, Vfc G Z} by dividing
each interval ao into 2™ new intervals of length a n = a /2 n , because Ts
cannot be embedded into T vo \. It is however possible to go the other way
round and to look for a continuous wave function that is approximated by
a wave function over a graph, in the limit of the graph becoming finer. In
fact, if one defines a scale C n , i.e. a decomposition of R in terms of the
union of closed-open intervals that have lattice points as end points and
cover R without intersecting, then can approximate continuous functions
with functions that are constant on these intervals. As a result, at any given
scale C n , the kinetic term of the Hamiltonian operator can be approximated
Generalized Approaches to Quantum Mechanics 487
as in Eq. (11.49), and effective theories at given scales are related by coarse-
graining maps. In particular, it is necessary to regularize the Hamiltonian,
treated as a quadratic form, as a self-adjoint operator at each scale by
introducing a normalization factor in the inner product. The convergence
of microscopically corrected Hamiltonians is based on the convergence of
the energy levels and on the existence of completely normalized eigenvectors
compatible with the coarse-graining operation.
11.3 On the Existence of a Fundamental Scale
The combination of quantum mechanics and GR leads to a fundamental
minimal length, naturally related to the Planck scale. The argumentations
for this intuition are very general and any approach to quantum gravity 6
predicts such a scale. In this Section we describe in a very simple man-
ner how such minimal scale appears as soon as we deal with energy regimes
where both quantum effects and gravity are relevant. This argument has to
be considered as an attempt to loosely explain why the idea that quantum
gravity implies an absolute limit on the localization of events is commonly
accepted in the scientific community. A rigorous description of quantum
space time is however not possible at the present stage of the theories, be-
cause there is not yet a complete quantum theory of gravity. Nevertheless,
a minimal scale appears from a very general overlap between the existing
physical theories.
A maximal localization scale is already present in relativistic quantum
mechanics and historically lead to the birth of quantum field theories. The
Heisenberg uncertainty relation implies that the position uncertainty is pro-
portional to the inverse of the momentum uncertainty, i.e. AqAp > 1 (nu-
merical factors have been ignored) . For a relativistic particle (E ~ p) this
relation stands as AqAE > 1 and, if we consider a position uncertainty
smaller than its Compton length (Aq < 1/E), we get the key relation
AE>E. (11.51)
The energy uncertainty of a relativistic particle is larger than its rest mass,
i.e. the concept of a single particle is rather unclear. We have thus to turn
6 The two main approaches for a quantum theory of the gravitational field are Loop
Quantum Gravity and Sinn;; Theory . The existence of a minimal length is a prediction
of both these (very different) theories. In the former, the physical space appears to be
granular (see Sec. 12.1), while in the latter a minimal length is a direct consequence of
considering strings instead of particles (see Sec. 11.4).
488 Primordial Cosmology
on a multi-particle theory which can be formulated in terms of a quantum
field theory.
Let us now introduce GR effects. The space time is here dynamical and
the metric evolves in tandem with matter according to the Einstein equa-
tions. This way, a quantum uncertainty in the position of a particle affects
(by Heisenberg) the uncertainty in momentum, which leads (by Einstein)
to an uncertainty in geometry which, in turn (again by Einstein), induces
an additional uncertainty in the position of the particle. Furthermore, to
get a high resolution we need high energy photons which strongly curve the
space-time which in turn increases the disturbance on the measure. The
geometry itself is now subjected to quantum fluctuations and, as already
argued by Wheeler in 1963 the quantum space time should appear as a
foam.
These considerations can be better explained by the following simple
example. Suppose that we want to resolve a spherical region of radius I
and then a photon of wavelength less than I (its energy will be greater than
I /I). Such photon carries out an energy density p greater than l/l 4 and the
corresponding Einstein equations for this system can be roughly written as
d 2 g~np>^, (11.52)
where g is the amplitude of the space-time metric. Therefore, the photon
generates a gravitational potential g > k/1 2 and the length that is being
measured will have an uncertainty equal to
V^>/p- (11-53)
A minimal uncertainty of order of the Planck length is thus predicted in
any quantum gravity framework. Such a basic example illustrates a result
which can be obtained in a large variety of more sophisticated and rigorous
schemes.
11.4 String Theory and Generalized Uncertainty Principle
A general prediction of String Theory is the existence of a minimal ob-
servable physical length, leading to the Generalized Uncertainty Principle
(GUP), i.e. a modification of the Heisenberg uncertainty relation. Such
result is expected to be quite general since it can be derived from model-
independent arguments, like the analysis of the transition amplitudes. This
result was also obtained in high-energy Gedanken string scattering experi-
ments.
Generalized Approaches to Quantum Mechanics 489
In what follows we briefly review how a minimal length naturally arises
in string theory using the path integral formulation.
String theory assumes as fundamental objects one dimensional entities,
i.e. strings. The world line of a string (which is called the worldsheet) is
a two-dimensional object rather than one-dimensional as in the standard
particle framework. The starting point of the theory is the Polyakov action
for a bosonic string. It is assumed that strings propagate in the Minkowski
space-time with the worldsheet given by X l (<j a ), where <r° = r and a 1 = a
are the (time-like and space-like, respectively) coordinates on the world-
sheet. The Polyakov action reads as
SrtringK-i / 'drday/GG^daX'dpXi, (11.54)
where G a p = G a /s(a,r) is the intrinsic metric of the worldsheet, G =
\detG a/3 \ and
Z s oc/ P (11.55)
is the fundamental lengl h characterizing the string theory (the proportion-
ality constant is a parameter model-dependent).
Let us consider the quantum mechanics of a non-relativistic particle.
The usual Feynman path integral is obtained by a regularization of the
world line of a particle x(t) with parameter e and the generating functional
is given by
Vrtidc- fvxexp -^^ e f^j I (11.56)
where A is a dimensional constant. From this expression, the particle trav-
els, at each step, a dist aiicc
({5xf)~e\\ (11.57)
which means that taking e — > the resolution can be arbitrarily high, i.e.
also infinite. Such a divcr<>,u)!>, rcsolui ion allows us to introduce the concept
of particle state in quantum mechanics and the local-operator formalism,
as well as equal-time commutators, in quantum field theory. On the other
hand, as is well known, the infinite resolution is at the basis of ambiguities in
non-linear systems like the operator-ordering ambiguity, the UV divergences
and so on.
This picture drastically changes in striu<>, theory. Consider a discretiza-
tion of the worldsheet coordinates as regularization. The path integral for
the action (11.54) reads as
i'rniioi-iiiid Cosmology
in which the e factor in front of the square bracket arises since the surface
we are regularizing (i.e. the worldsheet) is two-dimensional. This way,
analogously to Eq. (11.57), the resolution in string theory reads as
((SX) 2 ) ~ ll .
(11.59)
The e factor is ruled out and, no matter how finely the wordsheet i;
cretized (e —> 0), the geodesic distance between two adjacent points
l s . The appearance of the minimal length l s is the direct consequence of
dealing with two-dimensional objects.
The presence of such a scale modifies the Heisenberg uncertainty rela-
tions. In particular, the ordinary uncertainty relation is replaced by the
GUP one given by (see Fig. 11.1)
M
"2 \Ap
- K Ap
(11.60)
Plot of the uncertainty Aq vs.
huh r( fiint.y for the q variable.
Ap (Eq. (11.60)), showing the exisl
We conclude by stressing that an ultra-violet/infra-red divergence mix-
ing is explicitly manifest in the relation (11.60). When the uncertainty in
momentum Ap is large (UV) , the uncertainty in the position Ax is propor-
tional to Ap and therefore is also large (IR). This is an evidence that, in
string theory, the physics at short distances (in contrast to local quantum
field theory) is not clearly separated from the physics at large scales.
Generalized Approaches to Quantum Mechanics 491
11.5 Heisenberg Algebras in Non-Commutative Snyder
Space-Time
The analysis of non-commutative space-time geometries has recently at-
tracted a significant attention by the theoretical physics community. Such
theories are in fact believed to be candidates for an effective limit of quan-
tum gravity as soon as the degrees of freedom of gravity are integrated
out. This way, the only remaining trace of (quantum) gravity should be
the presence of an energy scale (related to the Planck one). Such scale is
the responsible for the non-commutativity of the space-time coordinates.
This intuition has been corroborated by different results. In particular,
a non-commutative space-time exactly arises as the flat limit of a three-
dimensional quantum gravity model; furthermore, non-commutative ge-
ometries have a natural connection with string theory and another motiva-
tion comes from the so-called doubly special relativity, which corresponds to
the framework of special relativity where two observer-independent scales
are taken into account. The first quantity is obviously the speed of light,
while the second one is an energy scale coming out from a (quantum) gravity
reminiscence and the presence of two scales justifies the definition "doubly" .
Interestingly, the space-time underlying such an extended (often also called
deformed) special relativity is non-commutative.
In this Section we will consider the Snyder space-time, a particu-
lar type of non-commutative space-time geometry, briefly describing the
modifications induced on the Heisenberg uncertainty relation. The non-
commutativity of the Snyder space-time is encoded in the commutators be-
tween the coordinates. These are proportional to the (undeformed) Lorentz
generators and the (algebraic) Poincare symmetry underlying this space-
time is undeformed, mentioning only that the symmetry deformations ap-
pear at the co-algebraic sector level only. Such a geometry will be imple-
mented in quantum cosmology in Sec. 12.5, showing a quantum Big Bounce
for the Universe dynamics.
Let us consider an n-dimensional non-commutative (deformed)
Minkowski space-time such that the commutators between the coordinates
have the non-trivial structure ({i,j, •••} G {0, . . . , n})
[q i ,q J ] = sM lj , (11.61)
where qi denote the non-commutative coordinates and s 6 R is the de-
formation parameter with dimensions of a squared length. For s = we
recover the ordinary Minkowski framework. Let us demand that the sym-
metries of such a space are described by an undeformed Poincare algebra
492 Primordial Cosmology
and therefore the Lorentz generators
M tj = -M jt = i{q lPj - qj Pi ) (11.62)
satisfy the ordinary SO(n, 1) algebra
[My , M k i] = Vjk M tl - mk M 3 i - Vjl M lk + m M jk (11 .63)
and the translation group is not deformed, i.e.
]pi, Pj ]=0. (11.64)
We also assume that the momenta pi and the non-commutative coordinates
Qi transform as undeformed vectors under the Lorentz algebra, i.e. the
commutators
[MijAk] = VjkQi ~ mkQj, (11.65)
[Mij,p k ] = rjjkPi - rn k pj (11.66)
hold. The quantity p 2 = rf^PiPj is then a Lorentz invariant.
Tin 1 relations (11.61)-(11.66) define the non-commutative Snyder space-
time geometry but, however, they do not uniquely fix the commutators
between qi and pj. In fact, there are infinitely many commutators which
are all compatible (in the sense that the algebra closes by virtue of the
Jacobi identities) with the above natural requirements. This feature can
be understood by means of the concept of realization.
A realization on a non-commutative space is defined as the rcscaling of
the deformed coordinates g, in terms of the ordinary phase space variables
{quPj) as
& = *«(p)<&. (11-67)
The most general SO(n, 1) covariant realization for q^ is given by
q t = Qi Vl (A) + s (qjp^pi y 2 {A) , (11.68)
in which <pi and ip 2 are two functions of the dimensionless quantity A =
sp 2 . The boundary conditions one has to impose in order to recover the
commutative framework read as
^(* = 0) = 1. (11.69)
The rescaling (11.68) depends on the adopted algebraic structure, but the
two functions tpi and ip 2 are not uniquely fixed. Indeed, given any function
ip i satisfying (11.69), the function ip 2 is determined by inserting (11.68)
into the commutator (11.61), that is
l + 2y>iyi
V2 = ~n oTTT' (11.70)
Generalized Approaches to Quantum Mechanics 493
where the dot denotes differentiation with respect to A. On the other hand,
the realization (11.68) inserted into Eq. (11.65) provides an identity and
therefore only a single condition on ipi and ip 2 is required. The generic
realization (11.68) is completely specified by the function <pi and there are
infinitely many ways to express, via ipi, the non-commutative coordinates
(11.61) in terms of the ordinary ones without deforming the original sym-
metry.
The commutator between g$ and pj arises from the realization (11.68)
and can be expressed as
[qi,Pj] = i (Sijcpi + sp tPj ip 2 ) . (11.71)
This relation describes a deformed Heisenberg algebra and from it we obtain
the (generalized) uncertainty principle underlying the Snyder geometry as
AqiA Pj > i|M^i) + *<PiPm)\- ( n - 72 )
The ordinary framework is recovered in the s — ¥ limit. In other words,
the deformation of the unique commutator between the spatial coordinates
defined in Eq. (11.61) leads to infinitely many realizations of the algebra,
and thus of generalized uncertainty relations (11.72), all of them consistent
with the assumptions underlying the model. Notice that, unless ip 2 =
0, compatible observables no longer exist but they are coupled to each
other and an exactly simultaneous measurable couple (q~i,Pj) is no longer
allowed. A measure of the f-component of the (non-commutative) position
will always affect a measure of the j(^ i)-component of the momentum by
an amount Apj ~ \s(piPjip 2 )\/Aq~i.
Let us now consider the Euclidean subspaee of the Snyder geometry, i.e.
a Snyder space. This case can be recovered from the above framework by
considering SO(n) generators (instead of the Lorentz ones), deformed co-
ordinates and momenta invariant under rotations and the Euclidean metric
instead of the Lorentzian one. Also in this case infinitely many realizations
of such non-commutative geometry exist. However, for one-dimensional
mechanical systems, this picture is (almost) uniquely fixed. In this case the
symmetry group is trivial (50(1) = I) and the most general realization is
given by
q = q<p(A) = qy/l-sp i . (11.73)
The commutation relation (11.71) then reduces to
[q,p] = iy/l-sp 2 (11.74)
494 Primordial Cosmology
and the only freedom relies on the sign of the deformation parameter s. In
such a case, the uncertainty relation (11.72) rewrites as
AqAp> I|(^/i- ap 2)| (11.75)
and, if s < 0, the minimal observable length
Ag min = ^£ (n . 76)
is predicted. On the other hand, if s > 0, a natural cut-off on the momen-
tum arises given by \p\ < y/l/s.
Expanding inequality (11.75) to first order in s, the generalized uncer-
tainty relation predicted by String Theory (11.60) holds. In this case, the
string length l s is related to the deformation parameter s by the relation
l s = \/—s/2 (thus s < 0). On the other hand, if s > 0, a zero uncertainty in
the non-commutative coordinate appears as soon as Ap = yj{l — s(p))/s.
We can conclude that a maximum momentum or a minimal length are
predicted by the Snyder-deformed commutator (11.74) if s > or s < 0,
respectively.
11.6 Quantum Mechanics in the GUP Framework
Let us now discuss some aspects and results of a non-relativistic quantum
mechanics with non-zero minimal uncertainty in position. In one dimen-
sion, we consider the Heisenberg algebra generated by q and p obeying the
commutation relations
[q,p] = i(l + sp 2 ), (11.77)
where s > is a deformation parameter. This commutator can be seen
as the first order term of the Snyder relation 7 (11.74). The commutation
relations (11.77) lead to the uncertainty relation
AqAp > - (1 + s(Apf + s(p) 2 ) , (11.78)
i.e. the same form as for String Theory (11.60). The canonical Heisenberg
algebra can be recovered in the limit s = and the generalization to higher
dimensions will be discussed below. We start our analysis from the modified
algebra (11.77) and thus we do not take the deformal ion parameter s to be
directly related to the string scale l s .
7 The deformation parameter used in this Section is minus one half of the one of
Eq. (11.74). For convenience, we denote it again with s.
Generalized Approaches to Quantum Mechanics 495
The generalized uncertainty principle (11.78) implies a finite minimal
uncertainty in position as
Aq min = yfs (11.79)
which is considered as a fundamental minimal scale of the theory.
Let us now consider the quantization of this system. The existence of
a non-zero uncertainty in position implies some relevant differences with
respect to the ordinary quantum theory. In particular, the physical states
that correspond to position eigenstates are no longer allowed. In fact, an
eigenstate of an observable necessarily has vanishing uncertainty. To be
more precise, let us assume the commutation relations to be represented on
some dense domain D C J 7 , T being a Hilbert space. In the ordinary case,
it is always possible to find a sequence of physical states \ip n ) G D with po-
sition uncertainties decreasing to zero. Therefore, the position eigenstates
can usually be approximated by arbitrary precision by \ip n )- On the other
hand, in the presence of a minimal uncertainty Ag m ; n > 0, it is not possible
any more to find some \ip n ) e D such that
JKm {Aq min ) Hn) = JGm M(« - <<7» 2 |V'> = . (11.80)
Thus, although it is possible to construct position eigenvectors, they are
only formal eigenvectors and not physical states. This feature comes out
from the corrections to the canonical commutation relation (11.77). In
the GUP framework we have lost direct information on the position itself
and, in particular, we cannot directly work in the configuration space but
a notion of quasiposition has to be introduced. Notice that, in general,
a non-commutativity of the coordinates does not imply a finite minimal
uncertainty in position (see discussion in Sec. 11.5).
The algebra (11.77) can be represented in the momentum space, where
the q, p operators act as
pi/>(p)=ptl>(p), (11.81)
qiP(p) = i(l + sp 2 )d p i,(p) ,
on a dense domain S of smooth functions. The measure in the scalar prod-
uct, with respect to which the operators q and p are symmetric (namely
{(4>\q)\<f>) = (ip\(q\</>)) and ((ip\p)\4>) = (ip\{p\(f>))) , is deformed and, in par-
ticular, the scalar product reads as
./-co 1+V
<</#>= / t-t^^W^)- ( 1L82 )
496 Primordial Cosmology
Information on the position can be recovered by studying the states
which realize the maximally- allowed localization. The states of maximal
localization \ip™ l ) 1 which are proper physical states around a position Q,
satisfy the properties
(Vf'MC') = C, (Ag)|^ r .> = A 9min (11.83)
and are called maximal localization states because they obey the minimal
uncertainty condition
AqAp= \(Mll. (11 . 84)
Therefore l.lio [bllowiug equation holds
(^-^ + f^-^))ic'> = °> ( 1L85 )
whose solution, in the momentum space, is given by 8
^? 1 (P) a (1 + s 2)1/2 exp f-i-/^arctan(Vsp)j , (11.86)
where the proportionality factor is given by a normalization constant. As
we can see, these states reduce, in the s = case, to the ordinary plane
waves. However, differently from the canonical case, the states (11.86) are
normalizable and their scalar product is a function rather than the Dirac
distribution.
Furthermore, we can project an arbitrary state \ip) on the maximally
localized states \ip 1 Q l ) 1 giving the probability amplitude for a particle to be
maximally localized around the position £, i.e. with standard deviation
Ag m in- We call these projections the quasiposition wave functions V>(C) =
(?/:"'' | ijj). explicitly given by
^(0 oc J + J exp U-L arctan(^p) J ^(p). (11.87)
This is nothing but a generalized Fourier transformation and in the s =
limit the ordinary position wave function V(C) = (Cl^) is recovered.
As last point we will analyze the generalization of relation (11.77) to n
spatial dimensions. Let us consider the following generalization
[Qi,Pj] = iSijQ. + sp 2 ) + is' Vl p 3 , p 2 = p t p l , (11.88)
s' > being a new parameter. Furthermore, assuming that the translation
group is not deformed (\pi,Pj] = 0) the commutation relations among the
as Aq min = y/s, and then the
Generalized Approaches to Quantum Mechanics 497
coordinates are almost uniquely determined by the Jacobi identity. The de-
formed classical dynamics is summarized in the modified symplectic geome-
try arising from the classical limit of the quantum-mechanical commutators,
as soon as the parameters s and s' are regarded as independent constants
with respect to h. It is possible to replace the quantum-mechanical com-
mutators — i [•, •] via the Poisson bracket {-, •}, thus dealing with a phase
space algebra given by the commutators
{Qi,Pj} = M 1 + *P 2 ) + s'PiPh (H-89)
{Pi,Pj} = 0,
Ui,qj} = YTT^ (k * " mi) ■
From a String Theory point of view, keeping the parameters s and s' fixed
as h — > corresponds to maintaining the string momentum scale fixed as
the string length scale shrinks to zero.
The GUP Poisson brackets are immediately obtained from Eq. (11.89)
and, for any phase space function, they read as
dFdG 0FdG\ . . dFdG .
WtWi - op-WJ {9i ' Pi} + W^ {q ^ } ' (1L90)
It is worth noting that, for s' = 2s, the coordinates qi become commutative
up to higher order corrections, that is
{ qi , q] } = + O( S 2 ). (11.91)
This can be considered a preferred choice of the parameters. However,
although terms like 0(s 2 ) can be neglected, the case in which sp 2 3> 1 is
allowed due to the absence of restrictions on the p-domain, i.e. pGM.
Let us conclude by discussing the application of the deformed Heisenberg
algebras in quantum cosmology (see Sec. 12.5, Sec. 12.6 and Sec. 12.7). As
we have seen in Sec. 10.3, in the minisuperspace theory only a finite number
of the gravitational degrees of freedom are invoked at quantum level (the
remaining are set to zero imposing some symmetry on the spatial metric).
In this respect, the implementation of deformed Heisenberg algebras in
quantum cosmology seems to be physically grounded, also in the light of
the analogy existing between the point Universe in the minisuperspace and
a relativistic particle moving on a curved background. Indeed, the GUP
scheme relies on a modification to the canonical quantization prescriptions
and thus can be reliably applied to any dynamical system.
{F,G}=(
498 Primordial Cosmology
11.7 Guidelines to the Literature
The algebraic approach to quantum physics presented in Sec. 11.1 is devel-
oped in the textbook of Haag [214]; for an introduction to the basic aspects,
see Wald [457]. A good textbook on C*-algebra theory, is for example, that
of Brattelli & Robinson [102], while a pedagogical introduction is given in
the review [302]. The uniqueness theorem of quantum mechanics, the GNS
construction and the Fell theorem are described in the textbooks [214,457].
The polymer quantum mechanics analyzed in Sec. 11.2 has been pro-
posed for example in [56, 115]; recent developments can be found in
[26,128,129,174].
A clear review on a minimal length in quantum gravity (Sec. 11.3) is
[190].
A simple exposition about the prediction of a minimal length in String
theory, as discussed in Sec. 11.4, can be found in [8,268,292,447].
The Snyder non-commutative space-time (Sec. 11.5) has been proposed
in [424] and developed in [35,50,51,209]. A relation between quantum
gravity and non-commutative geometry is in [175]. The connection with
String theory has been formulated in [411]. Doubly special relativity has
been proposed by Amelino-Camelia in [9,10]; see also [333]. For a recent
The quantum mechanics in a GUP framework, discussed in Sec. 11.6,
has been studied in [36, L03,269 271 1. Generalizations to more spatial di-
in [68,117].
Chapter 12
Modern Quantum Cosmology
This Chapter is devoted to the analysis of modern approaches to quan-
tum cosmology developed in the last ten years. Motivated by the Ashtekar
discovery of a gauge-like formulation of GR, a great improvement of the
Wheeler-de Witt quantum theory of gravity has been pursued. This work
culminated in the Loop Quantum Gravity theory of which Loop Quantum
Cosmology is the main application. In such framework the classical Big
Bang singularity is replaced by a quantum Big Bounce opening the way
for a Planck scale physics. Notably, the chaotic dynamics of the Mixmas-
ter Universe is tamed by loop quantum effects and a relation with non-
commutative geometries can be outlined as well.
In the first part of the Chapter, we address the loop approach to quan-
tum cosmology, while in the second one we discuss non-commutative in-
spired quantum cosmological models.
As a starting point, we analyze Loop Quantum Gravity posing our at-
tention on the novel tools, developing in some details the basic steps in
the construction of a kinematic Hilbert space based on the holonomy-flux
algebra and the prediction of a granular structure of space. A particular
attention is also devoted to the way the new constraints are imposed at a
quantum level, as well as to the closure of the quantum constraints algebra
in view of an anomaly free construction.
The implementation of the loop techniques in the minisuperspace arena
leads to a new quantum cosmology theory: Loop Quantum Cosmology.
While its predictions are very close to those of the old quantum geometro-
dynamics theory in the low curvature regime, there is a drastic difference
once the energy density of the Universe approaches the Planck scale: a
Big Bounce is predicted by means of repulsive quantum geometrical ef-
fects. We analyze in details this theory in the isotropic (FRW models) and
500 Primordial Cosmology
anisotropic (Bianchi IX) regimes. The problems regarding the relation of
Loop Quantum Cosmology with the full theory are also investigated.
We then discuss a very recent approach to quantum cosmology whose
aim is to relate Loop Quantum Gravity with its minisuperspace theory.
This approach, known as Triangulated Loop Quantum Cosmology, is based
on a truncation of Loop Quantum Gravity down to a graph with a finite
number of links. The simplest choice is based on a "dipole" graph formed by
two nodes connected by four links. This framework determines a Hilbert
space which describes the Bianchi IX Universe plus few inhomogeneous
degrees of freedom. The dynamics of Loop Quantum Cosmology is there
recovered without heuristic arguments.
The recent interest in non-commutative space-time geometries as a "low
energy limit" of quantum gravity has suggested their implementation in
primordial cosmology. We analyze a quantum cosmological model based on
a particular type of non-commutative space-time in which the commutators
between the four-coordinates are proportional to the Lorentz generators. In
this specific framework, a quantum cosmological bounce d la Loop Quantum
Cosmology takes place, setting a bridge between these two different theories.
Non-commutative geometries are naturally related to generalizations of
the uncertainty principle. In fact, the underlying Heisenberg algebra is
deformed and results in a modification of the uncertainty relations. We
then discuss the implementation of this scheme towards the Taub Universe
as well as towards the Bianchi IX model. In particular, the Taub Universe
offers a nice framework to contrast the polymer loop quantization scheme
with the one behind deformed algebras. The removal of the cosmological
singularity appears to be dependent on the adopted quant izatiou scheme
and also the choice of the variables to be quantized plays a crucial role.
12.1 Loop Quantum Gravity
In this Section we will analyze some of the basic aspects of Loop Quantum
Gravity (LQG) in a pedagogical manner for a non-expert reader.
LQG is a conceptually clear and mathematically rigorous attempt to
quantize GR in a background independent scheme. It aims to define a quan-
tum field theory just on a differential manifold M and not on a background
space-time (M,go), i.e. independently of the choice of a fixed background
metric g ■ In this respect, LQG follows an approach opposite to that of
String Theory. In fact, in String Theory a target space background metric
Modern Quantum Cosmology 501
(mainly Minkowski or Anti de Sitter) is fixed and strings propagate on it.
The construction of a quantum field theory in the absence of a fixed
background (namely a diffeomorphism invariant QFT) is a highly non-
trivial task. We remind that the construction of the ordinary QFT is en-
tirely based on the Wightman axioms, which are in turn based on a fixed
background metric go, usually the Minkowski one. Such fixed background
structure implies a preferred notion of causality (locality) and the Poincare
symmetry group. The only quantum field theories in four dimensions that
are fully understood are the ones related to the behavior of free or pertur-
batively interacting fields. On the other hand, in GR the metric itself is
a dynamical entity which has to be quantized and there is not anymore a
background on which (quantum) physics happens.
LQG is the most advanced implementation of canonical quantum gravity
and it is able to overcome some of the problems of the previous geometro-
dynamical approach (described in Chap. 10) by adopting new conceptual
and technical ingredients. The main result of LQG is the discreteness of the
spectrum of geometric operators like as the area and volume. The quantum
three-geometry can be viewed as composed by quanta of space.
The starting point of LQG is the Hamiltonian formulation of GR in
terms of connection variables (see Sec. 2.6). The Einstein theory, in this
framework, has the form of a background independent SU(2) Yang-Mills
theory. The new phase space is endowed with an SU(2) connection A a a and
its conjugate (gravitational electric) field E". The symplectic geometry is
determined by the only non-trivial Poisson brackets
{A a a (x,t),E^(x',t)} =k6^S^6 3 (x-x / ). (12.1)
The diffeomorphism (vector) and scalar constraints of the theory (2.77) are
rewritten in terms of these new variables respectively as in Eq. (2.139)
n a = E^F^ [3 = (12.2)
H = ^W\ KE " ( £a&c K& ~ 2(1 + ,y2)K ^) = ° ' (12 - 3)
where the parameter 7 is the Immirzi parameter. In comparison with the
metric approach, the additional Gauss constraint (2.138) appearing in the
connection formalism
G a = V a E% = d a E* - -f£ab c A b a E* = (12.4)
holds in addition to Eqs. (12.2). It gets rid of the SU(2) degrees of freedom.
These seven constraints arise from requiring a theory with a whole gauge of
502 Primordial Cosmology
freedom and that any observable (a gauge invariant phase space function)
has to commute with all these quantities. The theory is now described
(in the phase space) by 18 variables (A%, E%) subjected to seven first-class
constraints, each of them eliminating two degrees of freedom. The four
phase space degrees of freedom of the gravitational field are then recovered.
Let us now summarize the conceptual path of the canonical quantization
of gravity a la Dirac, consisting basically of four steps:
a) Find a representation of a set of phase space functions, generating
a Poisson algebra, as operators in a kinematic Hilbert space Jki n -
b) Implement the constraints, here the seven ones in Eqs. (12.2)-
(12.4), as (self-adjoint) operators in Jkin-
c) Describe the space of solutions of the constraints and define the
corresponding inner product. This determines the physical Hilbert
space J-'phys allowing for a probabilistic interpretation.
d) Find a complete set of gauge invariant observables.
The WDW theory (see Sec. 10.1) does not satisfy the first assessment (a)
of the previous list, since no kinematic Hilbert space can be obtained there.
The step (b) is solved only formally, and the remaining two points (c and
d) are even not addressed. The main problem in the geometrodynamics
approach can be drawn back to the choice of h a /3 and TP S as basic variables
of the theory. In fact, it is not possible to find a meaningful representation
of these functionals which satisfies the constraints. LQG is able to overcome
these limitations basically by using a more suitable phase space algebra.
12.1.1 Kinematics
A crucial step to construct a suitable Hilbert space for LQG is to appro-
priately smear the fields A 1 ^ and E^ . This procedure is inspired by what
is usually done in gauge theories. A natural quantity associated to the
connections consists of the holonomies which will be regarded as the basic
variables for the quantum theory. Given a curve on the three-dimensional
surface X, i.e. an edge £, a holonomy hi[A] is defined as
hp\A] = ;Pexp I / A a a T a dx a J . (12.5)
Here V denotes the path ordering and the (anti-hermitian) matrices r a are
generators of SU(2). i.e. they satisfy the commutator relation
[Ta,T b }=e abc T C . (12.6)
Modern Quantum Cosmology 503
This basis is related to the Pauli matrices a a by the equality
Ta = Yi ■ (12 - 7)
Notice that r a are proportional to the T a used in Sec. 2.2.4 being r a =
—iT a . The holonomies he are elements of SU(2) and define the parallel
transport of the connection A a a along the edge £. They contain SU{2) gauge
invariant informations (their trace is gauge invariant) of the connection in
the restriction to a graph T (see below). It is worth noting that they have a
one-dimensional support rather than being smeared over all S. Taking the
trace of the holonomy (12.5) for a closed edge leads to the so-called Wilson
loop which is at the ground of LQG and its name.
On the other hand, the densitized triad E% induces an S£/(2)-valued two
form E" e Qi a 7 . It is then natural to smear the electric field E" on a two-
dimensional surface ScS. This defines the flux electric vector (Ps[E]) a
as
P S [E] = (P s {E}) a T a = J E a a r a t aM dx^ A dx\ (12.8)
In order to compute the Poisson brackets between the functionals hg[A] and
(Ps[E]) a , let us consider a surface S and an edge £ intersecting S in one
point p. Then we divide the edge into the two sub-edges £\ and £2 such
that p is the source of £\ and p is target of £2- The associated holonomies
are respectively denoted as he x and he 2 and lead to
{h t [A],(P s [El\) a } = K-ya(t,S)h tl [A]Tah ta [A]- ( 12 -9)
In Eq. (12.9), a = refers to the case of the edge not intersecting the
surface and a = ±1 when the edge and surface orientation are the same or
the opposite, respectively. We stress that the commutation relation (12.9)
is non-canonical for the presence of hg[A] on the right-hand side.
The quantum kinematics can be constructed by promoting such vari-
ables to quantum operators obeying appropriate commutation relations.
The essential feature of LQG is to consider the holonomy (12.5) as the
configuration variable of the theory. The holonomies he are thus promoted
to operators rather than the connections A a a themselves. The Poisson al-
gebra of holonomies and fluxes is well defined and the resulting Hilbert
space is unique. More precisely, requiring the three-diffeomorphism invari-
ance (there must be a unitary action of such diffeomorphism group on the
representation by moving edges and surfaces in space), there is a unique rep-
resentation of the holonomy-flux algebra that defines the kinematic Hilbert
504 Primordial Cosmology
space ^kin- This is an important theorem ensuring the self-consistency of
the theory.
Let us investigate the kinematical Hilbert space Jki n - This is known as
the spin networks Hilbert space. Spin network states \S), expressed as
\S) = \T,j t ,i n ), (12.10)
are defined by three ingredients: (i) a graph r C E consisting of a finite
number of edges I and nodes n; (ii) a collection of spin quantum numbers
jn = 1 /2, 1, 3 /2, . . ., one for each edge; (hi) other quantum numbers i n , the
intertwiners, one for each node n. Spin networks are colored graphs. Notice
that although spin networks are defined on a three-dimensional manifold
(the edges and the nodes are in fact defined on E), no physical metric is
carried out. The holonomies defined in Eq. (12.5) are taken to transform
in an SU(2) representation of arbii rarj spin Such spin jg- valued holonomy
is denoted as pj{hi[A\).
The wave functional on the spin network is thus given by
^ r , i! [A] = ^(p jl (he 1 [A}),..., Pjn (hiAA})) (12.11)
where the wave function ip is SU(2) gauge invariant and satisfies the Gauss
constraint. More precisely, the function ip joints the collection of holonomies
(in the arbitrary spin representation) into an SU{2) invariant complex num-
ber by contracting all the gauge indices with the intertwiners, the latter be-
ing invariant tensors localized at each node. The states (12.11) are called
cylindrical functionals since they have a one-dimensional support i.e. they
probe the SU{2) gauge connection on one-dimensional networks only. The
space of these functions is called Cyl.
Let us now promote the fundamental variables of the theory
(^[AJjPs^]) to quantum operators. Their action on the wave functions
(12.11) is given by
h f \A]^ T ^[A] = h t [A]9 T ^[A] (12.12a)
(^) *r,*[A] = i{(Ps[E]) a , *r,^L4]} , (12.12b)
where the relation (12.12) is defined by means of Eq. (12.9).
A key result in LQG is the construction of the kinematic scalar product
between two cylindrical functions. Indeed, the discreteness of area and
volume operators spectra, mainly based on the compactness of the SU{2)
group, can be obtained from it. The kinematic scalar product is defined as
(* r |* r ,) = i° if r^r'
l/n« e r^4c*/i.-)#'(^i,-) if r = r,
Modern Quantum Cosmology 505
where the integrals J hi are performed with the SU(2) Haar measure. Such
definition is based on the strong uniqueness theorem previously discussed.
The inner product vanishes if the graphs T and V do not coincide and it
is invariant under spatial diffeomorphisms, even if the states ^i and * 2
themselves are not. This happens because the coincidence between two
graphs is a diffeomorphism invariant notion. The information about the
position of the graphs, carried by the wave function, disappears in the
scalar product (12.13). The Hilbert space Jkin is the Cauchy completion
of the space of the cylindric functions Cyl with respect to the above inner
product.
Two remarks are appropriate:
i) The Hilbert space obtained is not-separable as it does not admit
a countable basis. The set of all spin networks is not numerable
and two non-coincident spin networks are orthogonal with respect
to the scalar product (12.13).
ii) States with negative norm are absent without imposing the con-
straints (12.2) and (12.4). This is in contrast with the usual gauge
theories where the negative norm states can be eliminated only
after imposing the constraints.
Let us now briefly analyze the area operator in LQG (the construction of
the volume operator is conceptually the same and it will not be discussed
here). The idea is to construct the areas as functions of basic variables
(holomomies and fluxes) and then promote them into operators. The area
element dA a of a surface S C E is expressed in terms of tetradic vectors
dA a = - e abc e b A e c = - e abc e b e^dx fi A dx 1 = e a01 E^dx P A dx 7 , (12.14)
where in the last step we have used the definition (2.133). The area of S is
then given by
As
-- J -<JdA a dA<
and the surface can be divided in N cells Si such that S = U/S 1 /.
then express the area as the limit of a sum, explicitly as
A s =YimA%, A^ = J2V(Ps,[E})a(PsAE}) a , (12-16)
where (P Sl [E]) a is the flux (12.8) of E% through the I-th cell. Such expres-
sion can be meaningfully quantized and the area operator can be written
Primordial Cosmology
A s = lim A%. (12.17)
It is well defined because the operator associated to the flux exists in the
Hilbert space Jki n - It is not difficult to show that, using the relations
(12.12) and (12.9), the action of the operator {Ps I ~) a (^T a on pj(h £ [A}) is
given by
(P^)a (P^) a Pj(hi) = i 2 (Kl) 2 (h il r a T a h i2 ) j = ( K1 ) 2 j(j + l) Pj (h e ).
(12.18)
In the first step we have assumed that the edge of the holonomy crosses Si
only once (I is decomposed into l\ and (V)- while the unitary irreducible
representation of SU(2) has been used in the second one 1 . The spin network
states are then eigenstates of the quantum area operator and we have
i s * = K7 ^ v / j p (i p + l)vI/, (12.19)
where p punctures in the 7-th cell are considered. The minimal accessi-
ble length appears to be of the Planck order and the spectrum is clearly
discrete. Notice that the spectrum depends on the Immirzi parameter 7.
12.1.2 Implementation of the constraints
Let us now discuss the implementation of the seven constraints of the theory
as operators. To deal with the constraints at a quantum level, one needs
to follow two steps: (i) to express them in terms of holonomies and fluxes,
and (ii) to investigate their properties.
The Gauss constraint (12.4) is the most simple one and it is already
implemented by the construction of the cylindrii al functionals (12.11), re-
quiring the SU(2) gauge invariance of the spin network states. This in-
variance is achieved by contracting the representation indices of a given
node in an SU(2) invariant manner. This way, the total (quantum) flux of
the gravitation,!! electric field vanishes. Hie Hilberf space of SU(2) gauge
invariant states is given by J"^ n C Jk in .
The diffeomorphism constraint T-i a = is more delicate to deal with,
because it cannot be treated in operatorial terms. The main difference
with respect to the WDW approach is that spin networks are supported on
one-dimensional space and not on all £. Diffeomorphism generators do not
exist as operators and diffeomorphism invariant states (with the exception
o the Casimir reads T a r a = —j(j + 1).
Modern Quantum Cosmology 507
of the empty spin network state * = 1) do not exist in Jki n - This feature
is due to the non-compactness of the gauge orbits of the diffeomorphism
group. This constraint is imposed implementing a group averaging method
in a way properly adapted to the scalar product (12.13). A key step arises
from solving the constraint on a larger space, i.e. the dual Cyl* of the
cylindric functions Cyl, such that
Cyl C J- kin C Cyl* . (12.20)
This is the so-called Gelfand triplet and the dual space is the space of the
distribution.
Cyl* can be characterized as follows. Given any operator O £ Cyl and
its adjoint O^ € Cyl, the action of its dual O* € Cyl* on any element
X £ Cyl* is given by
(0*X\V) = (X\0^), VtfeCyl. (12.21)
A diffeomorphism invariant state is then obtained by averaging *r € Cyl
over the diffeomorphism group as
r}(V T )= Yl </>*°*r, (12.22)
</>eDiff(s/r)
whore I ho diU'ooniorpliisins <> are in Diff(E) modulo those leaving invariant
the graph V. From the inner product (12.13), it turns out that only a finite
number of terms contributes to the kinematical scalar product between
a spin network state and a dill'ootnorpbisnis invariant one. This way, a
state ri(^>r) is well-defined making sense as a distribution living outside of
-^"kin- We denote the space of the difleouiorpbisuis invariant distributions
as Cyljjff. The scalar product between two diffeomorphism invariant states
is thus given by
(j?(*r')l»?(*r)>diff= <r/(ttr')l*r>. (12.23)
The diffeomorphism invariant Hilbert space Jdiff is obtained by completion
with respect to this norm, i.e. it is an averaged version of J-*kin-
Let us point out two remarks.
i) The new Hilbert space Jdiff is separable (differently from Jkm) and
it is a subspace of Cyl* .
ii) The elements of J^in related by a diffeomorphism arc mapped to
the same state of Cyl* .
508 Primordial Cosmology
The final challenge is to find a space annihilated by all the constraints and
defining a physical inner product which yields the final physical Hilbert
space J-'phys- This step is accomplished by quantizing the scalar constraint
(12.3) which encodes the dynamics of the theory. As mentioned in the
WDW formalism, the main problem of all (canonical) quantum gravity
approaches is to impose this constraint at a quantum level. Let us briefly
sketch its construction which is mainly due to Thiemann.
Let us firstly rewrite the function (12.3) in terms of basic variables
corresponding to well-defined quantum operators. This can be done using
classical identities to express the triads, the extrinsic curvature and the field
strength in terms of holonomies and well-defined operators like the volume.
In particular, introducing the integral trace of the extrinsic curvature of E
as
K = [ d 3 x v / ]h\K a , 3 h afs = [ d z xK a a E* , (12.24)
the smeared version of the (classical) scalar constraint turns out to be
H(N)= J d^xNe^iSabF^iA^V}
- 2 (! + T 2 )^M- {H E (1), V}}{A% {U E {1), V}}{A C 7 , V}
Here
v-f*«M-f*Jf
denotes the volume of £ and % (1) is the smeared Euclidean part of the
scalar constraint (12.3) as N = 1 defined as
H E (1) = f d 3 x^-e ab c F^ = f d 3 xe^6 ab F^{A b 7 ,V}. (12.27)
Such formulation of the scalar constraint, containing only the connection
and the volume, is the starting point for the quantization and, similarly to
ordinary QFT, this is achieved in three steps:
i) The constraint (12.25) has to be regularized (by a parameter e).
ii) The classical objects have to promoted to operators and the Poisson
brackets to commutators.
iii) The regularization has to be removed at the end of the computa-
tions. This last step is highly non-trivial and poses serious chal-
lenges.
Modern Quantum Cosmology 509
Given an infinitesimal loop P a p(e) with coordinate area e 2 , both cur-
vature F and Poisson brackets {A, V} can be regularized in terms of
holonomies and volumes. This procedure leads to the regularized quantum
super-Hamiltonian 7i{N, e) which is a sum over cells of volume e 3 (these
cells are centered on the nodes of the spin networks) and is given by
H(N,e) = Y,Nie af ^Tr{(h Pa ^ ) - /^ (e) ) fe-%, V"]
i
-2(1 + 1 2 )h- l [h ai K]h- p 1 [h^K]h- 1 [h 1 ,V]), (12.28)
where h a = h[ a . The essential piece in the expression above is the operator
(hp aB (e)-h- p l^ ( 12 - 29 )
which defines the action of the scalar constraint on a spin network state.
The quantum scalar constraint modifies the spin networks by creating a
blufinf/Ui !-'{< ) attached to a node.
The undcih i iapb I hit cb mgcd but two states supported on dif-
ferent networks are orthogonal by virtue of the scalar product (12.13). This
feature is reflected as soon as the regulator e has to be removed. In partic-
ular, for any cylindrical function \I/, the limit
]imH{N,e)V (12.30)
does not exist in Cyl. Usually one transfers the action of the scalar con-
straint to the dual space, i.e. adopting a weaker notion of limit. More
specifically, the limit e — > is defined by
{H'{N)X\9) = lim(X\ft(N,e)V) (12.31)
for all ^ G Cyl and X G Cyl^ iff . Such notion of convergence is mainly due
to the spatial diifoouiorphisui iuvariance. In fact, the limit e — > is just
a diffeomorphism and therefore, in the diifcouiorphism invariant Hilbert
space, it corresponds to a trivial operation. This can be considered as the
key point of LQG, i.e. a gauge theory is quantized in a diffeomorphism
invariant way.
Finally, we express three remarks on the quantum super-Hamiltonian
constraint :
i) it does not suffer from UV singularities;
ii) the new nodes created carry zero volume and are invisible to its
action;
iii) the modifications induced do not propagate over the whole graph
since they are localized in the neighborhood of a node.
510 Primordial Cosmology
12.1.3 Quantum constraints algebra
Space-time covariance is ensured at classical level by the closure of the
Dirac algebra (2.80), see Sec. 2.3. The most striking feature is its non-Lie
algebraic structure due to the field dependent structure constants in the
{ri(N),ri(N')} Poisson brackets. A crucial issue is to prove the closure of
this algebra at quantum level, i.e. the closure of the Dirac algebra generated
by the quantum operators corresponding to the constraints. This should
lead to the so-called quantum space-time covariance which can be regarded
as a fundamental requirement of any quantum theory of gravity.
The first two relations in Eqs. (2.80) do not pose any problem since they
can be reformulated in a finite manner and can be directly quantized. As we
have seen, the diffeomorphisni invariance is implemented by an averaging
procedure that makes the states invariant under finite diffeomorphisms.
The main challenge is thus to check the fate of the third relation in Eqs.
(2.80) at quantum level. The necessary use of the weaker limit (12.31) is
reflected on the quantum constraints algebra. In fact it turns out that the
quantum algebra closes since
([■H*(N),H*(N')}X\V)=0, (12.32)
for all ^ e Cyl and X £ Cyl^ iff . This relation is defined in Jdig and
ensures that the construction is anomaly-free. It is worth noting that the
closure of the algebra is required only after the imposition of the constraints,
i.e. it is partly on-shell. In this sense, it can be regarded as weaker than
the off-shell closure which manifestly addresses the quantum space-time
covariance. However, since the only states of interest are those invariant
under diffeomorphisms, such notion of closure works properly.
Let us conclude this paragraph by some considerations. LQG has consis-
tently improved, with respect to the geomerodynamics (WDW) approach,
the program of the canonical quantization of the Einstein GR and its main
results provide a rigorous kinematic construction of the quantum theory
and the derivation of a quantized space (often called as the clear realiza-
tion of the old Wheeler intuition about the foam of space) . LQG is however
not (yet) the final theory of quantum gravity and its main limit is the imple-
mentation of the dynamics. The Thiemann scalar quantum constraint is an
outstanding proposal which suffers of ambiguities mainly due to the strong
dependence on the regularization. Furthermore, no physical scalar product
has been constructed and no correct semiclassical limit is addressed. These
are two fundamental (related) open problems of LQG. Recently, it has been
to overcome these shortcomings by the Master Constraint Program or by
Modern Quantum Cosmology 511
the Spin Foam Models. The first approach is an implementation of the LQG
(canonical) formalism in which the Dirac algebra (2.80) is transformed in
a true Lie form. On the other hand, Spin Foam Models can be regarded
as an attempt to construct, a space-time version of spin networks, i.e. a
rigorous path integral formulation of LQG. Both approaches are matter of
current active research.
12.2 Loop Quantum Cosmology
A new quantum cosmology theory, motivated by LQG, has been recently
formulated and it is known as Loop Quantum Cosmology (LQC). LQC is
a minisuperspace model which is quantized according to the methods of
LQG. LQC however is not the cosmological sector of LQG since the in-
homogeneous fluctuations are switched off by hand ab initio rather than
being quantum-mechanically suppressed. Nonetheless, LQC is an impor-
tant arena to test the full theory leading to several relevant results such
(i) the absence of the classical Big Bah;:, singularity, replaced by a Big
Bounce, in the isotropic setting
(ii) a geometrical inflation as well as the suppression of the Mixmaster
chaotic behavior, in the homogeneous one.
From a technical point of view with respect to the WDW scheme, these
results are recovered here since the two quantum theories are (unitarily)
inequivalent. In fact, LQC can be considered as the implementation of
polymer quantum mechanics (see Sec. 11.2) to cosmological models written
in the Ashtekar-Barbero-Immirzi connection formalism.
This Section is devoted to analyze this approach in the isotropic frame-
work. The fate of the cosmological singularity and the differences with
respect to the WDW quantum cosmology are discussed in details. In the
next section, the theory will be implemented to the Bianchi IX dynamics.
12.2.1 Kinematics
Following what is done in the full theory, the first step in obtaining the
quantum cosmological scheme is to select the basic variables of the model.
As we have seen, the phase space of GR in its isotropic sector is two-
dimensional and the scalar factor a = a(l) is the only degree of freedom of
512 Primordial Cosmology
the system. Due to such symmetry of the manifold, the connection A a a and
the electric field E^ are respectively reduced to
A = c(t) (°e a r a ) , E = p(t) (Vh™°e a T a ^) . (12.33)
Here, a fiducial metric on the Cauchy surface S is fixed by the triad °e
and co-triad °e a as
k a b being the Killing-Cartan metric on symmetry group of the spatial sur-
face while the fiducial triad and co-triad satisfy the relation °e a (°e b ) = 5 b a .
The physical three- metric h a p is determined by the scale factor a(t) as
Kp = a 2 (t) h*J. (12.35)
The phase space of the model has coordinates (<■■{>), which are conjugate
variables satisfying
{c, p } = y- ^ 12 - 36 )
The connection formalism is related to the metric one via the relations
\p\=a\ c=i(^ + 7 a), (12.37)
where K = 0, ±1 is the usual curvature parameter of the FRW model.
The Gauss and the diffeomorphism constraints of the full theory are here
already satisfied by using Eq. (12.33). The scalar constraint rewrites as 2
A M?
n {c ^=--^\p\l—(c-T) 2 + T 2 J =0, (12.38)
where the spin connection T is given by T = K/2. In the flat case (K = 0),
the relation (12.38) reduces to the simpler form
w (c,p) = — L c 2yy^ = . ( 12 .39)
K 7
We stress that in general p e (— oo,+oo) and the classical singularity ap-
pears for p = 0. Differently from the WDW framework, it is not a boundary
of the configuration space. Changes in the sign of p are here allowed and
correspond to the changes of the orientation of the physical triad e" and
co-triad e°. These are related to °e" and °e a a via the equations
e a = ° e a sign(p)/"\/R> e a=° e a si g n (p)\/bi- (12.40)
2 Here we adopt the conventions adopted in li
Modern Quantum Cosmology 513
The construction of quantum kinematic of this model follows the lines
of LQG. As before, we need to construct SU(2) holonomies and electric
fluxes which can be meaningfully implemented as (quantum) operators.
The holonomy is formulated along the straight edges £ = \x °e" and is given
by
h a (c) = exp( M r aC ) = cos (^) I + 2r a sin (^) . (12.41)
Here I is the identity 2x2 matrix and fi € (— oo, +oo) is a real continuous
parameter along which the holonomies are computed. Due to isotropy, the
flux Ps(E) of the densitized triads across a surface S is proportional to the
momentum p itself
Ps(p) = A sP , (12.42)
where As is a factor determined by the fiducial metric. The elements of
the holonomies can be recovered from the functions
JV M (c) = e^/ 2 , (12.43)
which are almost periodic functions because /! is an arbitrary (non-integer)
real number. The cylindrical fund ions of this reduced model are thus given
by ^
*(c) = E^ e ^ c/2 ' ( 12 - 44 )
where £j e C. Such states are defined in the space of the symmetric cylin-
dric functions, denoted as Cyl s . The holonomy- flux algebra is generated
by e vc/2 an( j by p ^ e ^ ea j w j t ] 1 a hybrid representation between the
Heisenberg (p) and Weyl (e %ax ) ones.
The kinematical Hilbert space of LQC is obtained requiring that the
_/V M (c) form an orthonormal basis, i.e.
(N lt \N ltl )=6 lilli >, (12.45)
in analogy with the scalar product (12.13) of the full theory. Notice that on
the right-hand side there is a Kronecker-delta rather than the usual Dirac
distribution. From general theoretical considerations, the Hilbert space is
necessarily J-g = L 2 (M.b , <i/i) > where M.b is a compact Abelian group (the
so-called Bohr compactification of the real line) and d/i is an appropriate
measure on it. This is the space of the almost periodic functions and can
be characterized as the Cauchy completion of the space Cyl s with respect
Two important remarks are in order:
514 Primordial Cosmology
(i) assuming the scalar product as in Eq. (12.45), a new representation
of the Weyl algebra has been introduced. Such a representation
turns out to be inequivalent to the standard Schrodinger one since
the connection c itself does not exist as an operator in the Hilbert
space. This feature determines the failure of the Stone- Von Neu-
mann uniqueness theorem (see Sec. 11.2). New features are thus
allowed in the theory.
(ii) The Hilbert space Ts is not separable.
Let us now investigate the action of some fundamental operators in J-g.
The operators N^ and p act as multiplication and differentiation respec-
jfrtf = e^' 2 *, p ^ = -i1L^l. (12.47)
3 dc
Introducing the Dirac bra-ket notation as AT M (c) = (c|p), the action of p on
the eigenstates |/z) is
p| M > = ^fx\n) (12.48)
and the operator corresponding to the volume V = |p| 3 ' 2 has a spectrum
given by
The volume operator in LQC has a continuous spectrum. This feature is
in contrast with respect to LQG and can be attributed to the high degree
of symmetry. In fact, in LQG the spin networks are characterized by a
pair ('../) consist in; 1 , of a continuous edge £ and a discrete spin j. Due
to the symmetry, such pair now collapses to a single continuous label /i.
In the reduced theory, the spectrum is discrete in a weaker sense: all the
eigenvectors arc normalizable. Hence the Hilbert space can be expanded
as a direct sum, rather than as a direct integral, over the one-dimensional
eigenspaces of p.
In view of the analysis of the Big Bang singularity, we have to address
the role of the inverse scale factor fundamental operator. At a classical
level, the inverse scale factor sign(p)/y |p| diverges towards the singularity
(p — > 0). Let us express it in terms of holonomies and fluxes and then
proceed to the quantization. Let us note that the following classical identity
^ = -Tr (Vyfc, lh-\V^}) (12.50)
Modern Quantum Cosmology 515
holds, where the holonomy h a is evaluated along any given edge. Since of
the term hji^ 1 , the choice of the edge is not important, i.e. we do not
introduce a regulator and the expression (12.50) is exact. When dealing
with the scalar constraint, the situation will be different. The quantization
of the inverse scale factor (12.50) is now well-defined in Ts- The eigenvalues
are given by
— ) m-^ -"#)i*>.
where V u is the volume operator eigenvalue defined in Eq. (12.49), whose
main properties are of being bounded from above and to coincide with the
operator 1/ylW as ImI ^ 1- The upper bound is obtained for the value
fi = 1 and reads as
We recall that the classical Ricci scalar of curvature is given by R ~ I /a 2
and therefore, considering such bound, it assumes its maximum value at
R ~ 1/^p- According to the local characterization of a space-time singu-
larity (given by the divergence of the scalars built on the Riemann tensor),
the LQC model is singularity-free.
The physical picture emerging is intriguing. Although the volume op-
erator admits a continuous spectrum and a zero volume eigenstate (the
|/4 = 0) state), the inverse scalar factor is non-diverging at the classical sin-
gularity, but is bounded from above. This can indicate that, at a kinematic
level, the classical singularity is avoided in the quantum framework. The
semiclassical picture, i.e. the WDW behavior of the inverse scalar factor,
is recovered for |/*| 3> 1 and therefore far from the fully quantum regime.
Such behavior contrasts with the WDW formalism where the inverse scale
factor is unbounded from above. As we mentioned, the differences are due
to the non-standard Hilbert space adopted, or equivalently to the holon-
omy-flux algebra. As a matter of fact, differently from the WDW theory,
all eigenvectors of p are normalizable in LQC including the one with zero
eigenvalue. However, the boundedness of the inverse scale factor is by itself
(as a local criterion) neither necessary nor sufficient for the cosmological
singularity avoidance.
Let us investigate the quantum dynamics of the model.
516 Primordial Cosmology
12.2.2 Quantum dynamics and Big Bounce
To describe the quantum dynamics we have to impose the scalar constraint
at quantum level, so as to discuss the fate of the Big Bang singularity from
a dynamical point of view. In order to follow the lines of the full LQG
theory, the starting point will be the constraint (12.3) and not the reduced
one (12.38). In fact, the connection c itself is present in Eq. (12.38) rather
than the holonomies and therefore the operator %( c <p) is not well defined
at a quantum level, since c is not.
Let us investigate the flat model. Because of the spatial flatness, the
Euclidean (first) and Lorentzian (second) terms in the scalar constraint
(12.3) are proportional to each other. Mimicking the procedure followed in
the full theory (see the previous Section) , we rewrite the constraint in terms
of holonomies and fluxes. The term involving the triad can be rewritten
using a classical identity as
e abc ^^ = 4 £ ^M V* °e% Tr (h c { K \V}r a ) , (12.53)
where the holonomy is computed along an edge with length /j,q.
Let us consider the field strength part. As in the lattice gauge theo-
ries, the curvature F£g can be (classically) written in terms of a trace of
holonomies over a square loop D a {, (each side having length uq) with its
area shrinking to zero as
F* t c = -2 lim Tr (^^-) °e a a °e b p . (12.54)
Here h a b denotes the holonomy computed around the square D a b, explicitly
given by
h ab = hahbh' 1 ^ 1 - (12.55)
This way, the (regulated) scalar constraint (12.3), for the flat FRW cosmo-
logical model, rewrites as
U^v) = - 4Slg 3 n( 3 ) Ve afec Tr(fe^/if {(fef )- 1 ,F}) =0. (12.56)
K1 ^o abc
The dependence of the holonomies on /i is explicitly written and the lapse
function is fixed as N = 1. Differently from expression (12.50), the de-
pendence on u does not drop out and now plays the role of a regulator.
At a classical level, we can take the limit u — ¥ and verify that the
resulting expression coincides with the classical Hamiltonian (12.38), i.e.
Modern Quantum Cosmology 517
This limit does not exist at quantum level, essentially because the op-
erator c itself does not exist in the Hilbert space Ts. In the reduced theory
there is no way to remove the regulator. This feature is solved in compar-
ison with the full LQG theory. If we assume that the predictions of LQG
are true, then the regulator /j, can be shrunk until the minimal admissible
length given by the area operator spectrum. In this sense, the /xo — > limit
is physically meaningless and the failure of the limit to exist is a reminder
that an underlying quantum geometry exists. This way the field strength
operator (12.54) is non-local since /xo approaches a minimal non-zero value
related to the minimum of the area operator (12.19). Such a criterion is
the so-called minimal una gap argument.
However, how the reduced theory (LQC) can see a minimal length com-
ing out from LQG is not fully understood. In fact, LQC is not the cos-
mological sector of LQG, but a usual cosmological minisuperspace model
quantized through the LQG methods. A way for merging LQC in LQG will
be discussed in Sec. 12.4.
Let us now investigate the physical states. As in the full theory, the
physical states are those annihilated by all the constraints and live in some
larger space Cylg (the algebraic dual of Cyl s ), i.e. they do not need to
be normalizable. Promoting the expression (12.56) to a quantum operator
Hfj,o P , its action on the eigenstates |/i) is given by
*t' P V> = ^(V^ - V^ )(k, + 4 M0 ) - 2|/i> + lM-4/io)) • (12.57)
A generic state (the notation (ty\ is here adopted to the eventuality of
non-renormalizable states) can be expanded as
(*|=X>(/MXmI,
where </> represents a generic matter field included into the dynamics. Such
states satisfy the constraint equation
(tt | (#<£•">+■#*,) =0,
T~Lt representing the matter term appropriately regularized with hq. The
function ip(n, <f>) nas to satisfy the equation
-^T l(Vu+5u a - V n+3uo )i;(p + 4 Mo , <t>) - 2(V U+Ua - V^ )V(m, 4>)
7 Mo K
+(V3» " V»»)^(/' " ^o, ^)] = -^ V(/i, 4>) ■ (12-60)
518 Primordial Cosmology
This is nothing but a recurrence relation for the coefficients ?/>(/z, 0) which
ensures that (\P| is indeed a physical state. Even though \x is a continuous
variable, the quantum constraint (12.60) is an algebraic (difference) equa-
tion rather than a differential one. This is in evident contrast with respect
to the WDW theory.
We now discuss the implications of LQC on the fate of the classical
Big Bang singularity at a dynamical level. The singularity corresponds to
the state |/z = 0) and we have to analyze whether the quantum dynamics
breaks down at \i = 0. If this is not the case, the principle of quantum
hyperbolicitly (see Sec. 10.3) is satisfied and the classical singularity can be
considered as tamed by quantum effects. As we can see from Eq. (12.60),
starting at \i = —4Nfi we can compute all the coefficients tp(4:fio{ n — N),<p)
for n > 1. However, the coefficient tp(fi = 0,0) remains undetermined
because the generic coefficient vanishes if and only if n = N. The quantum
evolution seems to break down just at the classical singularity since it is
not possible to evolve the states beyond it, but this is anyhow not the case.
In fact, the coefficient ip(n = 0,0) is decoupled from the others thanks to
U Mo = V_ Mo and the condition H^ Q ip(fi = 0,0) = realizes. Therefore
the coefficients in (12.60) are such that one can unambiguously evolve the
states through the singularity even though ip(fj, = 0,0) is not determined.
We can thus conclude that the Big Bang singularity is solved in the LQC
framework.
We will now briefly discuss the outstanding recent results obtained by
Ashtekar and collaborators on the quantum Big Bounce. Such works have
significantly extended the previous disc i ion di playing in detail a clear
picture of the Universe evolution during the Planck era. The considered
model is the flat FRW Universe filled with a massless scalar field whose
energy density is
'-&■ (12 - 61)
For the analysis of this model in the WDW framework see Sec. 10.8. The
scalar constraint (10.128) rewrites in the connection formalism as
where, as usual, p^ is the momentum canonically conjugate to 0, i.e.
{<fr,P<p} = 1- As before, each trajectory can be specified in the (p, 0)-plane
since p<p is a constant of motion and they are given by Eq. (10.131) where
here \p\ = a 2 . The idea that a massless scalar field plays the role of a
Modern Quantum Cosmology 519
relational time (see Sec. 10.5) is implemented in LQC by demanding that
the whole Hilbert space is given by
J- tot = L 2 {RB,B(^)dfi) <g> L 2 (M, d(j>) , (12.63)
where B(/i) denotes the eigenvalue of the operator l/|p| 3 / 2 . The time <f> is
thus treated in the standard (Schrodinger) representation, while only the
effective degree of freedom \x is analyzed in the polymer representation (see
Sec. 11.2). In complete analogy to flic \\ L)\V framework, the quantum
dynamics of this model is summarized by the Klein-Gordon-type equation
(c^ + e L Qc)* = 0, (12.64)
where \I/ = \&(/z, (f>) is the wave function of the Universe and Olqc denotes
a difference operator which depends on fig. As before, it is possible to
construct wave packets peaked at late time and evolve them according to
(12.64) towards the singularity. Such analysis sheds light on the singular-
ity resolution in LQC. As a result, the semiclassical states remain sharply
peaked around modified classical trajectories (see the next subsection) by
which the classical Big Bang is replaced by a quantum Big Bounce. In
particular, the Universe experiences the bounce for a density of the Planck
order. At curvatures much smaller than the Planck one the states are
peaked on the classical trajectories (10.131). Loosely speaking, LQC leads
to a "quantum bridge" between the expanding and contracting classical
Uni
12.2.3 Effective classical dynamics
It is useful to obtain an effective Hamiltonian constraint which captures the
essential features of the discrete evolution equation of LQC.
As we have seen, the connection c itself cannot be implemented as an
operator in the Hilbert space essentially because of the form of the inner
product (12.45). From the polymer perspective, only its exponentiated
version can be defined as an operator. This way, let us consider the polymer
substitution formula (11.48) as
c ->■ sin(/i c)/^ • (12.65)
The scalar constraint (12.62) thus rewrites as
H cff = -^/H S in 2 (,oc) + ^ = 0, (12.66)
520 Primordial Cosmology
which summarizes the LQC quantum corrections to the standard Fried-
mann dynamics. The ordinary evolution is recovered as [i c <C 1. It is
worth noting that such expression can be analytically obtained by applying
the methods of geometric quantum mechanics directly to LQC. Thus, one
can obtain the effective Friedmann equation for the model. The equation
of motion for p is
V = -N- j— = V H sin(/i c) cos Mo c , (12.67)
3 oc 7/i
and the constraint (12.06) implies that
sin 2 (, 0C ) = ^^ = ^Hp. (12.68)
Combining Eqs. (12.67) and (12.68) the effective Friedmann equation of
LQC, in the synchronous reference frame (N = 1), is given by
ft = ^v!H' (mo)
denotes the critical energy density. As pio — >• 0, this quantity diverges and
the standard Friedmann dynamics is recovered. The modifications arising
from the LQC quantum effects in Eq. (12.69) are manifested in the form
of a p 2 term. Such factor is relevant in the high energy regime and, as p
reaches the critical value p c , the Hubble function vanishes and the Universe
experiences a bounce (or, more generally, a turn-around) in terms of the
scale factor. In the ordinary dynamics, the Hubble function cannot vanish
unless for the trivial case p = 0. For energy densities much smaller than
the critical one, the Friedmann dynamics appears.
The value of the critical density (12.70) is usually fixed by demanding a
relation with the minimum area operator eigenvalue of LQC The physical
area spanned by the elementary loop edge /io\/H is given by p^\p\ and this
way, from Eq. (12.19), the relation
l4\p\~%, (12.71)
is obtained and the critical energy density p c is of the Planck order since
Pp = 1/lp. It is worth noting that in this framework the regulator (1q has
been treated as a constant. However, for a consistent quantization it must
behave as p ~ V\/H- In order to include this varying parameter, one
Modern Quantum Cosmology 521
must change variables from the triad basis (fi) to the volume one, leading
to new basic variables to be quantized in the same way as above. In the
literature, this framework is known as "improved quantization" for which
we recommend the original works.
As a last point, let us consider the scalar field </> as the internal time for
the dynamics. As we have seen in Sec. 10.5, this fixes the gauge requiring
the lapse function to be
U3/2 J
N=^ = . (12.72)
2P0
2^75 '
The effective Friedmann Eq. (12.69) ii
i the (p, 4>) pk
\pd4>) 3\
«7 2 Mo Pi
3 \P\\
and, as p c -> oo, Eq. (10.130) is recov
ered. The sob
given by
p(4>) ~ e -VT (*-*») ^Ipl + e 2 V¥(<A-<Ao)^ ; (12 74)
where p| = 2j 2 p'j ) /ir 2 . The ordinary solutions (10.131) are recovered at
late times, i.e. for \<f>\ -» oo (see Fig. 12.1). The effective trajectory (12.74)
approximates quite well the loop quantum dynamics previously described
and clearly represents a bouncing solution (i.e. a singularity-free Universe)
over which the wave packets arc sharply peaked duri.u;>, the whole evolui i.oii.
The initial semiclassical state follows the classical trajectory (10.131) until
it reaches a purely quantum era where the effects of quantum geometry
become dominant. The resull ing dynamics is that of a quantum Big Bounce
replacing the classical Big Bang.
12.3 Mixmaster Universe in LQC
In this Section we describe the dynamics of the Mixmaster Universe in the
LQC framework. The discussion here presented covers the basic aspects,
referring for notation and details to the original literature.
The Bianchi IX evolution towards the singularity sees infinite sequences
of Kasner epochs characterized by a series of permutations as well as by
possible rotations of the expanding and contracting spatial directions (for
details, see Sec. 7.4.1). However, this infinite number of bounces within the
potential, at the ground of the stochastic properties, is a consequence of an
I'l-iiiioi-iiiid Cosmology
Figure 12.1 The effectiv
isotropic model. A Big B
LQC trajectory i
unce clearly replat
compared to the ordinary ones for an
ss the classical singularity at the Planck
unbounded growth of the spatial curvature. When the theory offers a cut-
off length and the curvature is bounded, the Bianchi IX model naturally
shows a finite number of oscillations and, in LQC, a quantum suppression
of the chaotic behavior takes place close to the singularity.
12.3.1 Loop quantum Bianchi IX
Bianchi IX model i:
Let us formulate the
The spatial metric
the connection formalism.
<B a = 5>2(a,
can be taken diagonal, leaving three degrees of freedom 3 only. The basic
variables for a homogeneous model are
A a a = c {a) {t)O a b w
*\t)O b a e\
(12.76)
where the indices in the brackets are not summed over. Here uj a are the
left-invariant 1-forms satisfying the Maurer-Cartan equation (7.28) and e a
are the vector fields dual to uj a (uj a (eb) =8%). These fields thus form an
3 Differently from Sec. 7.3, we denote the three scale factors a,b,c by 01,02,03-
Modern Quantum Cosmology 523
invariant basis. We remember (see Sec. 7.1) that an invariant basis {e }
is denned as Lie- invariant under the action of the Killing vectors field £ a ,
that is
L s e=[£a,e 6 ]=0, (12.77)
which in the case of Bianchi IX carry the 5*0(3) isometry. The isomorphism
between the Cauchy surfaces £ and the isometry group 5*0(3) ~ SU(2) ap-
pears and the 50(3)-matrix O b contains the pure gauge degrees of freedom.
The physical information of the model is provided by the gauge invariant
functions c a and p b which satisfy the Poisson brackets
{c a ,p b } = Kj8 b a . (12.78)
The connection variables are related to the metric formalism (scale factors
a c , spin connections T c and the extrinsic curvature K c = — a c /2) as
c a = T a - ~fK a , p a = \a b a c \sign(a a ) , (12.79)
r- 1 ( a » + a ° <\_ 1 (P C + P h p"p c \ mm
ra -2U + ^"^J"2U"V"WJ- (12 ' 80)
The classical dynamics is governed by the scalar constraint expressed as
2 r i i
Utx = - (r h r c - r a ) a a - -a a a b a c + cyclic = . (12.81)
k I 4 J
The potential term from (12.81) is given by
W{p) = 2 (p a p b (T a T b - r c ) + cyclic) (12.82)
which has infinite walls at small p a due to the divergence of the spin connec-
tion components. Thus the cosmological singularity in Bianchi IX appears
whenever a b = for some b.
The closed (K = 1) FRW model is recovered by setting a\ = a^ = &z =
a in the Bianchi IX phase space. In this case the triadic projection of the
Christoffel symbols ( 12 ,80) becomes the constant T = 1/2 and the isotropic
connection and momentum are respectively given by
c=\(l +1 d), \p\=a 2 . (12.83)
The loop quantization of the Bianchi IX model is performed straightfor-
wardly following the isotropic case. In fact, an orthonormal basis is given
by the p a -eigenstates
|/xi, /i 2 , Ms) = |/ii) ® |a*2> ® |A*3> (12-84)
524 Primordial Cosmology
and the Hilbert space is taken as the direct product of the isotropic ones.
It is separable and is a subspace of the kinematical non-separable Hilbert
space. The cylindrical functions are given by a superposition of the ba-
sis state (c|/x) ~ exp(vc/2). The basic quantum operators are the gauge
invariant triad operators p a (fluxes) and the holonomies
h a (c) = cos (^) + 20 b a n sin (^) . (12.85)
When implemented as operators in the Hilbert space, p a and h a act as dif-
ferentiation and multiplication, respectively. In particular, the eigenvalues
p a of the triad operator p a are given by
P a |/ii,^2,^3) = -y/ia|/il,/i2,/i3)- (12.86)
The volume operator is defined from p a as V = \/\p 1 p 2 p 3 \ and its action
on the eigenstates \fj,i, fi 2 , ^3) reads as
V\Hi,V2,V3) = [y) \/|Ml^2^3| \Hl,H2,H3) (12.87)
having a continuous spectrum also in the anisotropic case.
From these basic operators we can obtain, similarly to the isotropic
case, the inverse triad operator. Since it is diverging towards the classical
singularity, we are interested in its behavior at a quantum level. Concep-
tually, its construction in the anisotropic cosmological sector is the same as
in the isotropic one. The only difference resides in the computations and in
the appearance of some quantization ambiguities, as the half-integer values
of j and the continuous parameter I e (0, 1), although all the results are
independent of them. This behavior mimics the quantization ambiguities
present either in LQG or in the isotropic sector of LQC. While in the full
theory we find an ambiguous choice of the spin number j associated to a
given edge of the spin network, in the isotropic LQC such choice is reflected
on the Hamiltonian regulator fi . Nevertheless, this is but a parameter that
cannot be shrunk to zero, neither fixed in some way in the context of LQC
theory itself, but it comes out from extrapolating a prediction of another
theory (namely LQG). In the inverse scale factor term of isotropic LQC,
none of quantization ambiguity appears (see Sec. 12.2).
Let us construct the inverse triad operator. The technique is the same as
before: we express l/|p°| in terms of holonomies and volumes via a classical
identity and it can be meaningfully quantized as
b a | _1 |Mi ; M2,M3) = A jt if jt i(fj, a )\m,fj, 2 ,H3) , (12.88)
Modern Quantum Cosmology
/i,l(M-)= E fc l^a + 2fc|<
and A,-^ denotes a function of the quantization ambiguities j and I. The
values /j,;(/x) decrease for /i < 2j and the relation
|^p| / i o = 0) = 0, (12.90)
holds because of fj,i(0) = 0. Thus, the inverse triad operator annihilates
the stale corresponding to the classical singularity \fi a = 0).
The fundamental properties of the eigenvalues of the operator (12.88)
can be extracted from the asymptotic expansions of Fjj(n a ) = A^if^i{ix a ).
For large j one has /')./(//„) = Fi(v a ), with u a = /i a /2j, with no explicit
dependence on j and, in particular,
fl/l/„ ifl/„»l (Ma»j)
*iK) = {
J + iJ
hV Q «i (/i a «j).
The classical behavior of the inverse triad components is obtained for v a 3>
1, while the loop quantum modifications arise for v„ < 1. Rewriting the
spin connection in the triad representation, the potential (12.82) is given
by
W hl {v) = 2{k 1 j) 2 {v a v h (r a r 6 - r c ) + cyclic) , (12.92)
where
TaK) = \ [i/cSign(i/ )F,(^) + ^sign(i/ c )F,(i/ c ) - v b v c F?(v a )] . (12.93)
We are ready to quantize the scalar constraint of the model to extract
the dynamics. The loop quantization of the scalar constraint is however dif-
ferent in the homogeneous case. In fact, unlike the full theory, the Christof-
fel symbols are tensors in homogeneous space-times and cannot vanish as in
LQG. As far as they are chosen to be different from zero, it is not possible
to perform a diffeomorphism (maintaining the homogeneity) which makes
them vanish. Also the boloiiomics will depend also on the spin connections
and the quantum scalar operator leads to a partial difference equation, like
in the isotropic LQC. Such expression is anyway extremely complex and an
effective description turns out to be more suitable to extract the effects of
quantum geometry on the classical model.
526 Primordial Cosmology
12.3.2 Effective dynamics
We will discuss the Mixmaster Universe in the LQC framework, analyzing
its behavior at a semiclassical level, i.e. considering the modifications in-
duced to the dynamics by the loop quantization. To obtain an effective
Hamiltonian from the underlying quantum evolution, we will proceed in
two steps. At the first pace one develops a sort of continuum approxima-
tion, while at the second one a WKB expansion of the wave function is
performed, more explicitly
i) for a slow varying solution, the differences equation is specialized
for the continuum regime fi a ~ p a / 'ivy 3> 1, thus obtaining the
WDW-like equation
2 p a p b dWlP dpX» S - + cyclic ) + Wj ' 1 M vWVI *(p)
= - K \pVp 3 \ 3/2 P4>S(p), (12.94)
where <p denotes a generic matter field;
ii) the WKB limit of the wave function
is considered, i.e. T ~ e lA ' h . This approximation leads to the
Hamilton-Jacobi equation for the phase A to zeroth order in h.
The classical dynamics plus the quantum loop corrections are therefore
obtained. The key point for the classical analysis of the effective dynamics
is that the classical region a a ^> 1 (taking j > 1) can be separated into two
subregions. In fact, remembering the p-dependence in the WDW equation
(12.94) given by ^ a /2j ~ p a /JKj, we obtain the condition |[i a >l for
p a <C jivy, j > Ma > 1 (12.96)
P a > j ivy, fi a > j > 1. (12.97)
The second subregion (u a > j > 1) is the purely classical one, i.e. where
the Misner picture is still valid. From the expansion (12.91), the eigenval-
ues of the inverse triad operator correspond to the classical values. On the
other hand, the (classical) region where j 3> fi a 3> 1 is characterized by
loop quantum modifications since there the inverse triad operator eigen-
values have a power law dependence. The quantum modifications to the
Modern Quantum Cosmology 527
classical dynamics are controlled by the parameter j: if it is large enough,
one can move the quantum effects within the effective potential into the
semiclassical domain. However, the WKB limit (h — > 0) is strictly valid
only in the Misner region (/i a 3> j 3> 1), because in the first region a de-
pendence on (k7) _1 appears in the potential term. The validity of such
approximation in the first region (j 3> n a 3> 1) is a reasonable requirement
since the inverse triads vanish as p a — > 0.
A qualitative study of the modified classical evolution arises from ana-
lyzing the potential term, whose explicit expression is more complex than
the original one, making the dynamics more tricky. The volume variable is
regarded as a time variable (as in the ordinary approach) but in general the
dependence on it does not factorize out. Nonetheless, in the second region
the Misner potential is restored.
Considering the particular case /3_ = (namely the Taub Universe), we
can qualitatively study the effective potential of LQC. This case corresponds
to taking v 2 = v-^ = v and
v x = a = V 2 /{{2jfv) , (12.98)
and therefore T 2 = T 3 . The wall appears for v 3> 1 so that Fi(v) ~ v~ x ,
but a is not negligible, i.e. the relation Fi(cr) ~ a 2 holds. The potential
wall (12.92) becomes
Wj,i ~ ^F 2 a (3 - 2aFi(a)) , (12.99)
where V oc e 3a denotes the volume of the Universe.
For <7>1, Fi(cr) ~ er -1 and the classical wall e 4a-8 ^+ is restored. The
key difference with the standard case relies on the finite height of the wall.
As the volume decreases, the wall moves inwards and its height decreases
as well. In the subsequent evolution, the wall completely disappears as it
reaches its maximum for f3 + = —a and vanishes as e 12a oc V 4 towards the
classical singularity (a — > -co).
This peculiar behavior shows that the Mixmaster-like evolution breaks
at a given time and therefore the chaotic features of the model disappear. In
fact, when the volume is so small that the quantum modifications arise, the
point-Universe will never bounce against the potential wall and the Kasner
epochs will continue forever. It is worth noting that this behavior predicted
by the LQC framework produces (qualitatively) the same results as those
induced by a massless scalar field on the Universe dynamics: also in that
case, at a given time, the point-Universe performs the last bounce and then
it evolves freely (see Sec. 8.7.1).
528 Primordial Cosmology
12.4 Triangulated Loop Quantum Cosmology
As we have seen, LQC is the most remarkable application of LQG and
its results are explicitly grounded on the physical discreteness of quantum
geometry (see Sec. 12.1). However, LQC is still a minisuperspace theory in
which the symmetries are imposed at a classical level and the quantization
is performed by means of LQG techniques (essentially based on a singular
representation of the Weyl algebra). Thus LQC is not the cosmological
sector of LQG and the inhomogeneous fluctuations are switched off by hand
ab initio rather than being quantum-mechanically suppressed: there is not
a symmetry reduction of the quantum theory.
In this Section we will show how LQC can be merged into LQG and,
in particular, how the LQC dynamics naturally arises from a truncated
LQG model (namely with a finite number of degrees of freedom). This
model, based on a triangulation of a topological three-sphere, is known
as triangulated loop quantum cosmology and is based on a constrained
SU(2) lattice gauge theory describing the Bianchi IX Universe plus some
inhomogeneous degrees of freedom. We will first discuss the model and then
identify the isotropic as well as the homogeneous sectors of LQC, devoting
some space to the inclusion of inhomogeneity. In what follows, we will
3 the Immirzi parameter 7=1.
12.4.1 The triangulated model
The basic idea is to triangulate the spatial surfaces slicing the Bianchi IX
Universe which are (topologically) three-spheres S 3 . Let us consider an
oriented triangulation A„ of
S 3 ~ SO(3) ~ SU(2) (12.100)
formed by n tetrahedra t glued by their triangles 4 /. We consider a trian-
gulation in which the tetrahedra (and thus the triangles) are curved. This
kind of discretization is called a cellular complex decomposition as it differs
from the simplicial triangulation adopted for example in the Regge calcu-
lus, where the tetrahedra are flat. We associate a group and an algebra
element to each triangle given by
U f e SU(2) , E f = EJr a e su{2) , (12.101)
4 For a triangulation making use of n tetrahedra t there are In triangles ./'.
Modern Quantum Cosmology 529
respectively, where r a are generators of SU(2) denned in (12.6). The phase
space of this model is described by the Poisson brackets
{U f ,U fl } = 0, (12.102)
{Ef, Uf>} = 5ffT a U f , (12.103)
{E],E b f ,} = -5 ff >e abc E c f . (12.104)
The phase space is that of a canonical lattice SU{2) Yang-Mills theory being
the cotangent bundle of SU(2) 2n with its natural symplectic structure.
The dynamics is encoded in two sets of constraints: the Gauss constraint
(providing three constraints per tetrahedron) given by
G? = ^£? = 0, (12.105)
fet
where the sum is over the four faces of the tetrahedron, and the Hamiltonian
(scalar) constraint expressed as
Ut = Vf 1 ^ ^[UfU^Ef.Ef] =0, (12.106a)
V t = y/Tr[E f Ef,E f »] , (12.106b)
where the operator "Tr" denotes the trace in the su(2) Lie algebra, V t
is the volume of the tetrahedron t and UJ 1 = Uf-i. In what follow we
will consider fit = VtHt- The scalar constraint (12.106) provides the time
evolution of the system.
The interpretation of this model can be sketched considering the dual
triangulation A*, defined as follows: for each t £ A„ it is associated a
node n £ A* , while for each / £ A„ we associate a link I £ A*. Consider
now real Ashtekar-Barbero variables A° and E% on S 3 . Then, Uf is the
parallel transport of the connection along the link I and Ef is the flux of
the conjugate electric field across the triangle / (parallely transported to
the center of the tetrahedron).
The constraint in Eq. (12.106) is the discrete (triangulated) version of
the Euclidean part of the Ashtekar-Barbero Hamiltonian constraint 5 (12.3).
Firstly let us note that the Hamiltonian (12.3) has the correct continuum
limit as soon as considering the Gauss constraint. A holonomy U a along a
loop a can be approximated by
U a ~ exp J A ~ 1 - \a\ 2 F a + 0{\a\ 4 A 2 ) , (12.107)
5 In the Euclidean case, for 7=1, the constraint (12.3) reduces to the first part only.
In the Lorentzian case, this happens when considering the original complex Ashtekar
variables.
530 Primordial Cosmology
where F is the field strength of the connection A. By means of this holon-
omy expansion, Eq. (12.106) can be formally recast as
H t = V t Ht = Y, TT i E f E f] ~ M 2 Y, T ^ F ff E f' E A = ° • (12-108)
ff'et ff'et
The former term in the relation (12.108) vanishes because of the Gauss
constraint (12.105) and thus the second term undergoes the expected con-
tinuum limit
nVh = Tr[F afj E a E fi } = . (12.109)
Notice that this happens not only for small values of the length of the loop
|or|, i.e. for a fine triangulation, but also for a coarse triangulation (large
|a|) provided that \a\ 2 F is small. This model can be regarded as a lattice
approximation of the geometrodynamics of a closed Universe.
Let us now consider the simplest triangulation of S 3 , i.e. the one formed
by two tetrahedra glued together along all their faces. This model is called
dipole cosmology and is defined by the dual graph formed by two nodes
joined by four links as
This in turns specifies the cellular complex ( riangulation of the three-sphere.
The unconstrained phase space of the theory defined by this triangulation
has 24 dimensions and is coordinatized by (Uf,Ef). At each node there
are one Hamiltonian % = and three Gauss constraints G a = 0, but it is
easy to verify that the constraints of the two nodes are indeed the same,
giving a total of four constraints only, bringing the number of degrees of
freedom down to eight (or nine up to the dynamics generated by % = 0).
The dipole model is related to a homogeneous Universe with the topol-
ogy of a three-sphere, i.e. to Bianchi IX. As we have seen in Sec. 7.1, the
building blocks of homogeneous spaces are the left invariant 1-forms uj a
satisfying the Mauror-Cartan structure equation
dto a = )-C a bc uj h f\u c . (12.110)
If the symmetry group is SO (3), the Cauchy surfaces are S 3 and the struc-
ture constants are C\ c = e a bc . In order to translate this language in the
triangulated framework, let us consider the Plebanski 2-form of the con-
nection LO a
V a {uj) = \e a bc u b Au c . (12.111)
Modern Quantum Cosmology 531
Let u a j be the surface integral of this 2-form on the triangle / of the trian-
gulation. Using the Maurer-Cartan equation (12.110), we have
^=/ s °=U e "^ w =// ua =l
The flux of the Plebanski 2-form across a triangle is then equal to the line
integral of Lu a along the boundary of the triangle. Let us note two properties
of w?: firstly, for each tetrahedron t, the relation
£>? = W ^ = (12.113)
fet fet Jd f
holds. Secondly, because of the SU(2) symmetry, the vectors ujJ are pro-
portional to the normals of a regular tetrahedron in R 3 . This way, the angle
between two of them reads as cos#//' = w / w /< = 1/3. In conclusion, the
set of su(2) vectors ujJ forms a natural background fiducial structure for the
discrete theory, analogous to the w a fiducial connection in the continuous
theory.
Let us provide a physical meaning to the dipole cosmology model by
identifying its degrees of freedom with the ones of Bianchi IX plus inhomo-
geneous perturbations.
12.4.2 Isotropic sector: FRW
The phase space of the isotropic sector of Bianchi IX is coordinatized by the
connection c = c(t) and the conjugate variable p = p(t) (see Eq. (12.83)).
An embedding of the isotropic geometry in the phase space of the dipole
cosmology model is defined by
Uf = exp((c + 6)u a f T a ) , E f =pu a f T a , (12.114)
where S is a constant taking into account the curvature of the spatial ma-
nifold (see the discussion about the Christoffel symbols in Sec. 12.3) and
ujj has been defined in Eq. (12.112).
Let us discuss the dynamics of this model. The Gauss constraint
(12.105) is automatically satisfied because of (12.113), while the Hamil-
tonian constraint (12.106) becomes
H = — p 2 (cos(c-5)- 1) = 0. (12.115)
The value of 5 is fixed by requiring the matching with ordinary classical dy-
namics. In the case of small connection (|c| <C 1) we recover the constraint
H -> Hfrw oc -p 2 c(c - 1) . (12.116)
532 Primordial Cosmology
The appearance of the ordinary dynamics for small values of the connection
c is in agreement with the claim that a coarse triangulation well approxi-
mates the classical theory for a low-curvature space-time. The constraint
(12.115) can be seen as the effective constraint of the standard LQC. We
have thus obtained a "natural" effective Hamiltonian constraint by using
the discretized scalar constrained of the theory without need of a polymer-
ization of the classical theory. In the triangulated loop quantum cosmology
the polymerization is a direct consequence of the existence of the triangu-
lation.
The quantization of the isotropic sector of the triangulated model is
straightforward. The variable c multiplies the generator of a U{1) subgroup
of the compact group SU{2) 4 : c e [0,47r]. The kinematic Hilbert space is
that of the square integrable functions on a circle as
F iso = L 2 (S 1 ,dc/4n), (12.117)
and the eigenstates of p are labeled by an integer /i and read as (c|/i) =
e ifj,c/2_ rpj^ wave f unc tions ijj(c) can be decomposed in a Fourier series of
eigonstafes of p, labeled by an integer /j,, as
^(c) = E^e W2 - (12.118)
The fundamental operators on this representation are p and exp(ic/2),
whose action on generic states reads as
p\n) = (ji/2)\n) , ex P (ic/2) | M ) = \fx + 1) . (12.119)
This way, the quantum constraint operator corresponding to that in Eq.
(12.115) rewrites as a difference equation for the coefficients i \, = (c|//).
explicitly reading as
D+(ji) (c| M + 2) + £>°(a0 (c| h) + D-{p) (c\ y. - 2) = , (12.120)
where D (/j,) are some coefficients. Equation (12.120) has the structure
of the LQC difference equation (12.57). Notice that the discrete dynamics
is recovered in this manner without recurring to the minimal area gap
argument.
12.4.3 Anisotropic sector: Bianchi IX
The dynamics of a homogeneous anisotropic cosmological model is de-
scribed by three scale factors which identify three independent directions
(in the time evolution) of the Cauchy surfaces. In the connection formalism
Modern Quantum Cosmology 533
(see Section 12.3), relaxing the isotropy condition corresponds to consider
three different connections c a = c a {t) and momenta p a = p a (t). The trian-
gulated model can then be extended to an anisotropic setting by demanding
that the variables of the theory are given by
U f = exp ((c a + 5)u a f T a ) , E f = p a io a fTa . (12.121)
Let us notice a difference with respect to the usual LQC formulation of
an anisotropic model (see Sec. 12.3). In that case, the holonomies h a are
directional objects computed along the edges parallel to the three axes
individuated by the anisotropies, see Eq. (12.85). On the other hand, the
variables Uf in Eq. (12.121) are non-directional objects, because the four
faces of the triangulation do not have any special orientation with respect
to the three isotropy axes. The connection components are summed over
and they are thus independent on the a-direction. The Uf are in fact group
element of SU(2) that depends on the face /.
As in the isotropic case, the dynamics of this model is summarized in the
scalar constraint. The Gauss one does not carry out any informations since
it identically vanishes thanks to Eq. (12.113) thus leaving three degrees of
freedom. The Hamiltonian constraint is quite complex but, also in this case,
can be regarded as the effective one of the standard LQC. In particular,
the proper classical limit is recovered as soon as small connections \c a \ -C 1
are taken into account. For the complete analysis of this sector we refer to
the original papers.
12.4.4 Full dipole model
The dipole model has nine degrees of freedom and we have already identified
three of them which correspond to be Bianchi IX ones. The remaining six
are necessarily inhomogeneous, due to the generality of Bianchi IX, and can
be regarded as inhomogeneous perturbations to the Bianchi IX Universe.
This way, the inhomogeneous degrees of freedom are naturally captured by
a truncated LQG model.
As we have seen in Sec. 9.1, inhomogeneous perturbations to the Mix-
master Universe are expressed in terms of Wigner D-functions D J mm i(g)-
Let us translate this language to a first order formalism. The inhomogene-
ity implies that the basic 1-forms uj®(x 7 ) are replaced by time dependent
1-forms (i°(i, t) which, in turn, are decomposed on the homogeneous basis,
that is
w%{x) -> G>Z(x, t) = uj a a (x) + <f a a {x, t) . (12.122)
534 Primordial Cosmology
The functions <p' ( \(x.L) define the inhomogeneous perturbations to the
model. Similarly to Sec. 9.1, the projection of the perturbations on the
homogeneous basis oj°(x 7 ) and the subsequent expansion on the Wigner
D-functions is accomplished by
<p%(x,t) =tp ab (x,t)uj b a (x), (12.123)
^ 6 (M)=E E <Pfmm>(t)Di m ,(g(x)) , (12.124)
where g(x) are SU{2) group elements which coordinatize S 3 . Let us con-
sider the lowest nontrivial integer modes, i.e. the ones with spin and mag-
netic numbers are given by j = 1 and m! = 0, respectively. Assuming that
the dynamical inhomogeneous degrees of freedom tp ab mm , it) are diagonal in
the internal indices (a, 6), the lowest ones are given by
4™(t) = <5°Vm(*) m = -1,0,1, (12.125)
providing nine degrees of freedom.
This framework can be straightforwardly casted into the triangulation
formalism of the dipole model. The building blocks of the model are the
su(2) vectors uj'l which, by moans of the decomposition (12.122), rewrite as
Z a f = lj e a bcU b Aw c ~u a f + J e a bc co b Aip c =u] + ip% (12.126)
where we have neglected the second order terms, giving the inhomogeneous
discrete fiducial element. The dipole degrees of freedom are completely
specified in terms of the three connections c a (t) and the nine inhomo-
geneities ip"\l) as
Uf(c a , ip^) = exp (c a bj°jTi) exp (f^T a ) exp (5 uj^T a ) . (12.127)
Thus, all the 12 degrees of freedom of the dipole cosmology have been
univocally identified with geometrical quantities.
It is worth noting that the the Gauss constraint (12.105) does not iden-
tically vanish any longer but can be split into the homogeneous and the
inhomogeneous terms as
G " = E w / + E^/ = - ( 12 - 128 )
/ /
The first part is the constraint which appears within flic Bianchi IX frame-
work and identically vanishes because of the Stokes theorem. The second
Modern Quantum Cosmology 535
one gives three conditions on the inhomogeneous perturbations leading to
nine degrees of freedom.
Summarizing, we have analyzed a finite dimensional truncation of LQG.
In particular, a coarse triangulation of the physical space has been fixed
and we have considered a discretization and quantization of GR on this
triangulation. The model of a dipole SU{2) lattice theory, triangulating a
topological three-sphere by means of two tetrahedra, has been related to
a Bianchi IX cosmological model perturbed by six inhomogeneous degrees
of freedom. By means of this scheme, the LQC dynamics arises directly
from LQG without the need of heuristic arguments, like the minimal area
gap or polymerization and in this sense this framework represents a way
for merging LQC in LQG.
12.4.5 Quantization of the model
Let us finally perform the quantization of the triangulated model. We deal
with the generic triangulated model for which the dipole cosmology is a
particular case. The quantization procedure follows the methods developed
in lattice gauge theories.
A quant um representation of the observable algebra (12.102) is provided
in the auxiliary Hilbert space
J 7 aU x = L 2 [SU(2) 2n ,dU f }, (12.129)
where dUj is the Haar measure, i.e. the states have the form xj)(Uf). The
operators Uf are diagonal and the operators Ef are given by the left in-
variant vector fields on each SU(2) element.
Gauss invariance is achieved by SU{2) spin network states on the dual
graph A* . In the dual triangulation only four-valent vertices appear, which
intertwine spin-jy representations Dj f (U) associated to the holonomies
around the links If . The basis of these spin-network states is labeled by
\jf,H), (12.130)
where it denotes the intertwiner quantum number at the given node. The
spin network basis states are explicitly given by
VitJPt) = (Uf\jf,H) = ®/ DW(U f ) -® t Lt, (12.131)
in which the dot operator "•" indicates the contraction of the indices of the
matrices D^ f \U) with the indices of the intertwiners t t -
536 Primordial Cosmology
The quantum dynamics of the model is obtained following the Dirac
prescription (see Sec. 12.1). The Hamiltonian constraint (12.106) can be
directly implemented as a quantum operator rescaled by the volume V t .
The physical states are those satisfying the equation
n t V = Y] Tr[Uff,E f ,Ef]V = 0, (12.132)
ff'et
which corresponds to the early proposal by Rovelli and Smolin to perform
the quantization of the Hamiltonian constraint in LQG. On the other hand,
the Hamiltonian constraint can be defined a la Thiemann, as in Eq. (12.28),
rewriting the Hamiltonian constraint (12.106) in the form
Ht = Yl £ ff ' f "Tr[Uff,U r }{Uf«,V t }}=0 (12.133)
ff'f'et
and then defining the corresponding quantum operator by replacing the
Poisson brackets with the commutators. Here the operator associated to
the volume Vt turns out to be the standard LQG volume operator.
12.5 Snyder-Deformed Quantum Cosmology
This Section is devoted to present some results obtained in a recent ap-
proach to quantum cosmology. We discuss the implementation of a de-
formed Heisenberg algebra on the FRW cosmological models, having a
clear contact with non-commutative geometries. In particular, we quan-
tize a deformed symplectic structure of the minisuperspace and investigate
the dynamics of this quantum Universe, finally comparing it with LQC.
Since the algebra realizes the Snyder non-commutative space, we denote
such approach as the Snyder- deformed quantum cosmology.
Let us consider the modified symplectic geometry arising from the clas-
sical limit of the Snyder-deformed Heisenberg algebra (see Sec. 11.5). It
is then possible, considering s as an independent constant with respect to
h, to replace the quantum- mechanical commutator (11.74) via the Poisson
brackets
-i[q,p]=>{Q,P}= Vl-V, (12.134)
where, q refers to the non-commutative coordinate and s € R is the de-
formation parameter with the dimension of a squared length. The ordi-
nary algebra is recovered as s = 0. The relation (12.134) corresponds
exactly to the unique (up to a sign) possible realization of the Snyder non-
commutative space. In order to obtain the deformed Poisson brackets, one
Modern Quantum Cosmology 537
requires that they possess the same properties as the quantum mechanical
commutators, i.e. to be anti-symmetric, bilinear and to satisfy the Leibniz
rules as well as the Jacobi identity. This way, the Poisson brackets (for any
two-dimensional phase space function) are
\ dq dp dp dq J
The time evolution of the coordinate and momentum variables with respect
to a given deformed Hamiltonian H(q,p), are specified as
9 = {9,H} = ^ V / T^V, P = {p,n} = -^JT^. (12.136)
It is straightforward to implement such framework in a cosmological set-
ting. Let us consider the FRW model in the presence of a generic matter,
described by an energy density p = p(a) as in Sec. 10.8. The dynamics of
this model is encoded in the Hamiltonian constraint (10.127). The FRW
minisuperspace is then assumed to be Snyder-deformed and the commuta-
tor between the isotropic scale factor a and its conjugate momentum p a is
uniquely fixed by the relation
{a,Pa} = y/l-spl. (12.137)
Considering such approach, the classical equations of motion of the model
are modified as
a = N{a 7 H RW } = N \'T^ — y/l - sp>] , (12.138a)
12tt :
Pa = N{p a ,n WN ) (12.138b)
As in the standard case, the equation of motion for the Hubble function
(H = a/a) can be obtained solving the constraint (3.81) with respect to p a
and then considering Eq. (12.138a). Taking N = 1, it explicitly becomes
providing the deformed Friedmann equation which entails the modifications
arising from the Snyder-deformed Heisenberg algebra.
In order to make a comparison with respect to the LQC model described
in Sec. 12.2, it is interesting to consider the flat FRW Universe, i.e. with
K = 0. In this case, Eq. (12.139) reduces to
(~) 2 =?p(l-sign*f), Pc = 7T ^PP, (12.140)
538 Primordial Cosmology
where pp denotes the Planck energy density pp = l/lp. In the last step the
existence of a fundamental minimal length has been assumed, i.e. that the
scale factor (the energy density) has a minimum (maximum) at the Planck
The modifications arising from the deformed H.eiscuber;>, algebra in Eq.
(12.140) are manifested in the p 2 term. As soon as p reaches the critical
value p c (and s > 0), the Hubble function vanishes and the Universe expe-
riences a Big Bounce in the scale factor. For energy densities much smaller
than the critical one, the standard Lricdmaun dynamics is recovered. In
the same way, as the deformation parameter s vanishes, the correction term
disappears and the ordinary behavior of the Hubble function is obtained.
Two interesting features have to be stressed.
(i) The deformed Priedmann equation (12.140) in the s > case is
equivalent, at a phcnomenolojdcal level, to the one obtained for
the effective LQC dynamics (12.69).
(ii) The string inspired Randall-Sundrum brancworld scenario leads to
a modified Friedmann equation as in Eq. (12.140) with s < 0.
The opposite sign of the p 2 term, is the well-known key difference
between LQC and the Randall-Sundrum framework. In fact, the
former approach leads to a non-singular bouncing cosmology while
in the latter, because of the positive sign, a cannot vanish and
a cosmological bounce cannot take place. Of course, to obtain
a bounce, the correction term should be negative, i.e. make a
repulsive contribution.
Let us analyze the flat FRW model filled with a massless scalar field (j). As
usual (see Sec. 10.8), the energy density p<f, is given by p^ = p^/a 6 while
the phase space is four-dimensional with coordinates (a,p a ,(t>,p<p)- Since
P0 is a constant of motion, each classical trajectory can be described in the
(a, (/>)-plane. The scalar field <p ls considered as an internal clock (0=1)
as soon as the condition (10.129) for the lapse function holds. In this case,
the deformed Friedmann Eq. (12.139) rewrites as
_V%
B 2 a 2
= B \ l --52 J *\' ( 12J41 )
where B is defined in Eq. (10.130). The solution to Eq. (12.141) is giv
by
\s ( 1 \ 2
a(4>) = const x - _i ''" ' >
Modern Quantum Cosmology 539
This equation clearly predicts a Big Bounce if s > and from now on we
consider this case only.
As discussed in Sec. 10.8, the (effective) H ami] Ionian in the internal
time <fi description is expressed as
H c = B Pa a. (12.143)
Given any observable O, its evolution is governed by (10.133) taking into
account the deformed commutators as in (12.137). The equations of motion
-££). < 12 " 144 )
-Blpa
hold and the trajectories are in agreement with the deformed classical ones.
As before, to discuss the fate of the cosmological singularity at quantum
level, we have to analyze the evolution of a semiclassical initial state. Let us
remember that a semiclassical observable O requires an expectation value
close to the classical one with negligible fluctuations (AC) 2 , i.e. (AC) 2 -C
(C) 2 . The dynamics of the relative scale factor fluctuations is described by
the equation
d /(Aa) 2 \ 1 / a P%\ (a 2
(12.145)
As we have seen in Sec. 10.8, such quantity is conserved during the whole
evolution in the ordinary framework (s = 0) and thus the semiclassicity
of an initial state is there preserved. Such property is also valid in the
deformed scheme at late times \<f>\ — > oo, i.e. for large scale factor values
a 3> i/spip/lp. At the bouncing time, i.e. when the scale factor reaches its
Omin=^P*, (12-146)
Eq. (12.145) vanishes. Although the relative scale factor fluctuations are
in general not constant during the evolution, it is possible to show that the
difference in the asymptotic values
540 Primordial Cosmology
vanishes, since either the fluctuations (Aa) 2 (</>) either the mean value (a)(<j>)
are symmetric in time. This way, starting with a semiclassical state, for
example Gaussian, such that (Aa) 2 /(a) 2 < 1 at late times, this property
is satisfied on the other side of the bounce when the Universe approaches
large scales (a > y/sp^/l P ).
Summarizing, a bouncing cosmology is predicted by a Snyder-deformed
Friedmann dynamics and this model can be regarded as an attempt to
mimic the original LQC system by a simpler one.
Two remarks are however in order.
(i) LQC is based on a Weyl representation of the canonical commu-
tation relations which is inequivalent to the Schrodinger represen-
tation. On the other hand, the Snyder-deformed algebra cannot
be obtained by a canonical transformation of the ordinary Poisson
brackets of the system,
(ii) The p 2 term in the effective Friedmann Eq. (12.69) is not the only
correction from LQC unless the only matter source is a massless
scalar field. If it has mass or is self-interacting, there are infinitely
many other correction terms involving the pressure also. In the
deformed quantum cosmology, the structure of Eq. (12.140) is in-
dependent of the detailed matter content.
12.6 GUP and Polymer Quantum Cosmology: The Taub
Universe
In this Section we will compare the dynamics of the Taub Universe result-
ing from two different quantization techniques. The purpose is to quantize
a cosmological model by implementing in the formalism a minimal length
and to discuss the fate of the classical singularity. The model will be ana-
lyzed at classical and quantum level in both schemes. The two (quantum)
frameworks are the so-called generalized uncertainty principle (GUP) and
the polymer ones. For the first case, the model is quantized according to
the commutation relations associated to an extended formulation of the
Heisenberg algebra which reproduces the GUP (see Sec. 11.6). The poly-
mer quantum dynamics (see Sec. 11.2) of the Taub Universe will then be
considered. We stress that the polymer quantization is closely related to
the LQC techniques discussed in Sec. 12.2. We first analyze the classical
modified dynamics and then the quantum one.
Modern Quantum Cosmology 541
12.6.1 Deformed classical dynamics
As seen in Sec. 10.10.1, the Taub model can be interpreted as a massless
scalar relativistic particle (namely a photon) moving in the Lorentzian min-
isuperspace (r, x)-plane. The classical evolution corresponds to its light-
cone in the configuration space. More precisely, the incoming particle
(r < 0) bounces on the wall (x = x n = ln(l/2)) and falls into the clas-
sical cosmological singularity (r — > oo). Investigations on the modifications
of the dynamics within the GUP and polymer frameworks will show that
those two behaviors can be regarded as complementary.
Let us firstly discuss the GUP case. The GUP (classical) dynamics is
contained in the deformed phase space geometry arising from the classical
limit of (11.77). The fundamental minisuperspace Poisson brackets are thus
given by
{x,p} = l + sp 2 . (12.148)
Applying this scheme to the Hamiltonian (10.153), we obtain the equations
of motion for the model, i.e.
x(t) = (1 + sA 2 )t + const, p(r) = const = A, (12.149)
where x G [a; ,oo).
Let us consider at the classical level, the effects of the deformed Heisen-
berg algebra (12.148) on the Taub Universe. The angular coefficient in the
x{t) trajectory is given by (1 + sA 2 ) > 1 for s ^ 0. Thus the angle between
the two straight lines x(t), for r < and r > 0, decreases as s grows. The
trajectories of the particle (Universe) , before and after the bounce on the
potential wall at x = x$ = ln(l/2), are closer to each other than in the
canonical case (s = 0).
Let us discuss the polymer case. The polymer (classical) dynamics of
the model can be summarized as the substitution of Eq. (11.48) in the
Hamiltonian (10.153). This way, the equations of motion read as
% = {*, ^Idm} = cos( M oP), J = {P, nl DM } = , (12.150)
and are solved by
x(t) = cos(poP)T, p{t) = const = A. (12.151)
In the discretized (polymer) case, i.e. for /i 7^ 0, the one-parameter family
of trajectories flattens, as the angle between the incoming and the outgoing
trajectories is greater than tt/2, since p e {—tt/^ 0i t:/^ ). Since these
trajectories diverge rather than converge, we expect the polymer quantum
effects to be reduced in comparison to the classical case, as we will verify
below.
542 Primordial Cosmology
12.6.2 Deformed quantum dynamics
The quantum dynamics of the Taub Universe is here investigated according
to the GUP and polymer approaches above considered. Particular atten-
tion is paid to the wave-packet evolution and to the fate of the classical
cosmological singularity. In both frameworks, the variable r is regarded as
a time coordinate and therefore (r,p T ) are treated in the canonical way.
The deformed quantization (GUP or polymer) is then implemented only
on the submanifold describing the degrees of freedom of the Universe, i.e.
the phase space spanned by (x,p). Thus, we deal with a Schrodinger-like
equation as in the WDW case (see Sec. 10.10) given by
id T ^(r,p)=nl DM ^(r,p), (12.152)
where the operator Hj DM accounts for the modifications specific of the two
frameworks. As above, we have to square the eigenvalue problem in order
to correctly impose the boundary conditions: we will use the well-grounded
hypothesis that the eigenfunctions form be independent of the presence of
the square root, since its removal implies the square of the eigenvalues only.
Wave packets of the form (10.157) are then constructed for both models.
Some differences with respect to the ordinary scheme, as well as between the
two generalized approaches, however occur. In particular, the differences
are due to the distinct eigenfunctions and to the domain of definition of the
variables. Analyzing such evolutions, the GUP Taub Universe appears to
be probabilistically singularity- free. In the polymer case the cosmological
singularity is not tamed by the cut-off-scale effects.
Let us consider the model in the GUP approach. All the information
on the position is lost (see Sec. 11.6), so that the boundary conditions have
to be imposed on the quasiposition wave function (11.87), that is
</>(C = Co)=0. (12.153)
Here £o = (■0£™'|:ro|V'F li )' m agreement with the discussion in Sec. 11.6.
The form of the solution of Eq. (12.152) is the same as in the ordinary
framework (see Sec. 10.10), i.e.
MP,t) = Mp)e~ ikr , Mp) = 5{p 2 -k 2 ), (12.154)
where k is the momentum conjugate to r. The functions tp UJ (p) are however
modified with respect to the WDW case which in terms of the quasiposition
wave function (11.87) read a
MO = fc(1+ ^ 2)3/2 [ ex P (i^arctan(vGfc))
n(^fc))] ,
. N , (12.155)
V~s
Modern Quantum Cosmology 543
where A is a constant. In Eq. (12.155) the boundary conditions (12.153)
have already been imposed.
The deformation parameter s, i.e. a non-zero minimal uncertainty in
the anisotropy of the Universe, is responsible for the GUP effects on the
dynamics. The modifications induced by the deformed Heisenberg algebra
on the Universe dynamics are summarized in different s-regions. As soon
as s becomes more and more important, i.e. when we are at a scale such as
to appreciate the GUP effects, the evolution of the wave packets is different
from the canonical case. These effects are present when the product koy/s
becomes remarkable (namely as k 0y /s ~ 0(1)), k being the energy at
which the weighting function A(k) is peaked:
A(k) = k(l + sk 2 ) 3/2 e~ ik ~2^ } . (12.156)
For fixed fco and for growing s values, the wave packets begin to spread
and a constructive and destructive interference between the incoming and
outgoing waves arises. They escape from the classical trajectories and ap-
proach a stationary (independent of r) state close to the potential wall. A
probability peak "near" the potential wall thus appears (see Fig. 12.2).
Such behavior reflects what happens at the classical level. In fact, the
incoming and the outgoing trajectories shrink each other, so that a quantum
probability interference is a fortiori predicted. On the other hand, the
stationarity of the dynamics is a purely quantum GUP effect, and this
behavior cannot be inferred from a deformed classical analysis. From this
point of view, the classical singularity (r — ¥ oo) is strongly probabilistically
suppressed, because the probability to find the Universe is peaked just
around the potential wall. This way, the GUP-Taub Universe is singularity-
free.
Let us consider the polymer case. The quantum analysis of the model
is obtained by choosing a discretized x space and solving the corresponding
eigenvalue problem in the p polarization. Considering the time evolution
of the wave function \P, one obtains the following eigenvalue problem
(p 2 - k 2 )Mp) = — (1 - cos( M op)) - k 2 \ MP) ,
(12.157)
)se solution stands as
k 2 = k 2 M = ^(l- cos(/i p)) < *&« = A
Mo Po
(12.158a)
4>Ku (p) =AS(p- p kill0 ) + BS(p+ p kill0 )
(12.158b)
ipk,vL (x) = A [exp(ip k ^ x) - exp(ip k ^ ( 2x v ~ x ))] ■
(12.158c)
I'l-iiiioi-iiiid Cosmology
I*GUPI
2 £
-2 :
Figure 12.2 Wave packets |*(r,C)| of the Taub Universe
l the GUP framework a
1 (A'o =
■1).
Here Eq. (12.158b) provides the momentum wave function, with A and B
being two arbitrary integration constants, and Eq. (12.158c) is the coor-
dinate wave function, where an integration constant has been dropped by
imposing suitable boundary conditions. Moreover, the modified dispersion
relation
Pk 4 ,
.(.-£*)
(12.159)
has been obtained from (12.158a). Let us stress that k 2 is bounded from
above, as provided by Eq. (12.158a). However, its square root, considered
for its positive determination, accounts for the time evolution of the wave
function.
The next step is to construct suitable wave packets ^(x, r), weighted
with a Gaussian centered at kg , accounting for the previous discussion (note
that a maximum energy fc max is now predicted). At a fixed fcg, an interfer-
ence phenomenon between the wave and the wall appears and it becomes
more relevant as /j,q increases. Nevertheless, this interference cannot tame
the singularity (r — > oo), as it takes place in the "outer" region, in a way
complementary to the GUP approach (see Fig. 12.3). Then, the polymer-
:r 29, 2010 11:22
World Scientific Book - 9in x 6in
PrimordialCosmology |
Modern Quantum Cosmology
Taub Universe is then still a singular cosmological model.
m
1000
Figure 12.3 The spread polymer wave packet \<!f(x,
1/2 (/tin = 50, fco = 0.01, a = 0.125).
Summarizing, the Taub cosmological model offers a suitable s
to apply and compare different quantization techniques. It is possible to
single out a time variable, so that the anisotropy describes the real de-
gree of freedom of the Universe and therefore to investigate the fate of the
cosmological singularity without modifying the time variable.
The non-removability of the cosmological Mii-'jilariiv wilhin the polymer
framework could seem apparently in contradiction with other models (see
Sec. 12.2), but there are at least two fundamental differences:
(i) The variable r, which describes the isotropic expansion of the Uni-
verse, is not discretized but treated in the ordinary way. The only
anisotropy variable is discretized without modifying the volume
(time). In the FRW case, the scale factor of the Universe is di-
rectly quantized by the use of the polymer (loop) techniques, thus
the evolution itself of the wave packet of the Universe is deeply
modified,
(ii) In the Taub case the variable p, conjugated to the anisotropy, is a
546 Primordial Cosmology
constant of motion and, from the Schrodinger equation, it describes
also k, namely the energy of the system. According to the polymer
scheme (see Sec. 11.2), it is always possible to choose a scale ^ for
which the polymer effects are negligible during the whole evolution,
at classical level. On the other hand, the Hamiltonian constraint in
the FRW case does not allow for a constant solution of the variable
conjugate to the scale factor, and it is not possible to choose a scale
such that the polymer modifications are negligible throughout the
whole evolution.
Comparing the GUP to the polymer approach allows us to infer that
it is not always sufficient to "deform" the anisotropy variable to obtain
significant modifications on the Universe evolution. However, the poly-
mer paradigm is a Weyl representation of the commutation relations, while
the generalized commutation relations cannot be obtained by a canonical
transformation of the Poisson brackets.
12.7 Mixmaster Universe in the GUP Approach
As we have seen in the previous Section, the classical singularity of the Taub
cosmological model is tamed by GUP effects. Recalling that the Taub is a
special case of the Bianchi IX model, it is natural to investigate the GUP
effects on the Mixmaster dynamics. This Section addresses the Mixmaster
in the GUP framework, paying particular attention to the fate of its chaotic
behavior.
The ADM Hamiltonian of a homogeneous cosmological model (see Sec-
tion 8.2.4) is given by
-p a =nADM = (p 2 + +p 2 -+V) 1/2 , (12.160)
where the lapse function N = N(t) has been fixed by the time gauge d = 1
as in Eq. (8.41). The function %adm is a time-dependent Hamiltonian
from which it is possible to extract, for a given symplectic structure, all the
dynamical information.
Let us investigate the modifications to the classical dynamics induced by
the GUP algebra (see Sec. 11.6). This way, only the phase space spanned
by the anisotropy variables (and their conjugate momenta) is deformed.
The time variable a, i.e. the isotropic volume of the Universe, is treated
in the standard way. Considering the symplectic algebra (11.89), the time
Modern Quantum Cosmology 547
evolution of the anisotropics and momenta, with respect to the ADM Hamil-
tonian (12.160), is given by (i,j = ±)
ft = {A,Hadm} = T] [(! + S P 2 ) 6 *J + 2s PiPi\ Pj. (12.161a)
riADM
Pi = {ft, Wadm} = ~ 1 [(1 + sp 2 )8ij + 2sp iPj ] |^, (12.161b)
^ rtADM Cp.j
where the dot denotes differentiation with respect to a and p 2 = p 2 + + p 2 _ .
These are the deformed equations of motion for the homogeneous Universes,
while the ordinary ones are recovered for the s = case.
Let us discuss the Bianchi I model which corresponds to the case V =
and thus, from Eq. (12.160), it is described by a two-dimensional massless
scalar relativistic particle. The velocity of the particle (Universe) is modi-
fied by the deformed symplectic geometry and, from Eq. (12.161a), it reads
(12.162)
where /! = sp 2 . For s -J (/i « 1), the standard Kasner velocity p 2 = 1 is
recovered (see Sec. 8.2.2). The effect of a cut-off on the anisotropies then
implies that the point-Universe moves faster than the ordinary case. In
such deformed scheme the solution is still Kasucr-like, that is
$±=C ± (s), p±=0, (12.163)
but this behavior is modified by Eq. (12.162). In particular, the second
relation between the Kasner indices (see Eq. (7.51b)) pi,P2,P3 is deformed
Pi + p\ + p\ =
= l + 4/i + 6^ 2 ,
(12.164)
ile the first one p\ + p^ + Pz = 1 r
emains unchanged (s
eeEq.
(7.51a)).
Two considerations are in order.
(i) The GUP acts in an opposite way with respect to a massless scalar
field (or stiff-fluid, with pressure equal to density) in the standard
model (see Sec. 8.7.1). In that case the chaotic behavior of the
Mixmaster Universe is tamed. On the other hand, in the GUP
framework, all the terms on the right-hand side of Eq. (12.164) are
positive, thus the Universe cannot isotropize, i.e. it cannot reach
the stage such that the Kasner indices are all equal.
548 Primordial Cosmology
(ii) For every non-zero \i, two indices can be negative at the same time.
Thus, as the volume of the Universe contracts toward the classical
singularity, the distances can shrink along one direction and grow
along the other two. In the ordinary case the contraction is along
two directions.
The natural bridge between Bianchi I and the Mixmaster Universe is
represented by the Bianchi II model. Bianchi II is described by a potential
term V(a,(3±) oc e 4 "e _8 ^+ which can be directly recovered from the one of
Bianchi IX in the asymptotic region /3+ — > — oo. The BKL map of Bianchi
IX is obtained considering such a simplified model since it is, in the ordinary
framework, an integrable system. As we have seen in Sec. 7.4.2, the BKL
map is at the basis of the chaotic analysis of the Mixmaster Universe and
it is given by the reflection law of the particle (the point Universe) against
the potential walls.
A fundamental difference between the deformed and the ordinary frame-
works is that the ADM Hamiltonian 11 \y>\\ is no longer a constant of motion
near the classical singularity, since the wall velocity /3 wa ii is modified as
36//
(-4 + 22 1/3 S~ 1/3 + 2 2/3 S 1/3 ) , (12.165)
where
S = 2 + 81^/3 2 + 9yV/? 2 (4 + 81/i/3 2 ) . (12.166)
From the two velocity equations (12.162) and (12.165), it is possible to
understand the details of the bounce. In the standard case, the particle
(Universe) moves twice as fast as the receding potential wall, independently
of its momentum (namely, of its energy). In the GUP framework, the
particle velocity, as well as the velocity of the potential wall, depends on
the anisotropy momentum and on the deformation parameter s. Also in
this case the particle moves faster than the wall since the relation
Avail < $ (12.167)
is always verified (see Fig. 12.4). A bounce also takes place in the deformed
picture. Furthermore, in the asymptotic limit /j>1 the maximum angle
such that the bounce against the wall occur is given by
|6> max | = arccos ZZ2L j = - . (12.168)
q contrast to the ordinary case (/3 wa ii//3 = 1/2) where the maximum
e angle is given by |# max | = it/3. The particle bounce against the
Modern Quantum Cosmology 549
wall is thus improved in the sense that a maximum limit angle doesn't
appear anymore. The main difference with respect to the ordinary picture
is however that the deformed Bianchi II model is not analytically solvable.
No reflection map can be in general inferred since in the GUP picture it is
no longer possible to identify two constants of motion.
Figure 12.4 The potential v
the particle one $ in functio
/3wall//9 = 1/2 is recovered.
velocity /j wa n of the Bianchi IX model with respect t<
f fi = sp 2 . In the /i — > limit, the ordinary behavio
On the basis of the previous analysis, we get several features of the GUP
Mixmaster Universe. The potential term of Bianchi IX is given by (8.37b)
and its evolution is that of a two-dimensional particle bouncing an infinite
number of times against three walls which rise steeply toward the singular-
ity, with every single bounce described by the Bianchi II model. Between
two subsequent bounces the system is described by a Kasner evolution and
the permutations of the expanding-contracting directions is given by the
BKL map showing a chaotic behavior (see Sec. 7.4.1).
Two conclusions on the GUP-deformed Mixmaster Universe can be in-
ferred.
(i) When the ultra-deformed regime is reached (fi 3> 1), i.e. when the
550 Primordial Cosmology
point Universe has a momentum larger than the cut-off one, the
triangular closed domain appears to be stationary with respect to
the particle itself. The bounces of the particle are then increased by
the presence of a non-zero minimal uncertainty in the anisotropies.
(ii) In general, a BKL map (reflection law) cannot be obtained.
This arises analyzing the single bounce against any wall of the
equilateral-triangular domain, but the Bianchi II model is no longer
a solvable system in the deformed picture. The chaotic behavior of
the Bianchi IX model is not tamed by GUP effects, i.e. the GUP
Mixmastcr Viiiih i m in sUll a chaotic system.
In conclusion, it is interesting to point out the differences between this
model and the Mixmaster dynamics in LQC described in Sec. 12.3. In
the LQC scheme, the classical reflections of the point particle stop after
a finite amount of time and the Mixmaster chaos is suppressed. In that
framework, although the analysis is performed through the ADM reduction
of the dynamics similarly to the GUP case, all the three scale factors are
quantized using the LQG techniques. On the other hand, the time variable
(related to the volume of the Universe) is here treated in the standard
way and only the two physical degrees of freedom of the Universe (the
anisotropies) are deformed.
12.8 Guidelines to the Literature
The LQG theory presented in Sec. 12.1 is described in the books of Rov-
elli [398] and of Thiemann [438] and in the reviews [28,384,422,437]. Ped-
agogical expositions of the subject can be found in [120,341], while for a
critical point of view, see [364]. The first paper on the loop representa-
tion in GR is [400], while the spectrum of the area operator was firstly
obtained in [401]. The original construction of the Hamiltonian constraint
is in [436] . The original paper on the Wilson loop technique is [466] , while
a good textbook is that of Makeenko [334] .
LQC, discussed in Sec. 12.2, is reviewed in [23-25,94]. The absence of
singularity in LQC was shown in [91,92] and developed in [29,30]. The
effective equations of motion are described in [432] . For the improved dy-
namics in LQC see [31, 127]. For a comparison between LQC and WDW
approach, see [98].
The Bianchi IX model (Sec. 12.3) has be analyzed in LQC in [95-97].
For recent developments in homogeneous LQC, see for example, [33,34].
Modern Quantum Cosmology 551
The triangulated version of LQC analyzed in Sec. 12.4 has been pro-
posed in [49] developing the basic idea presented in [402].
The implementation of a Snyder-deformed Heisenberg algebra in quan-
tum cosmology as discussed in Sec. 12.5 has been developed in [46].
The GUP and polymer quantizations of the Taub Universe (Sec. 12.6)
are discussed in [54] and [48] respectively. The GUP quantum dynamics of
the FRW is studied in [53] . Different implementations of the GUP paradigm
in quantum cosmology can be found, for example, in [444-446].
The dynamics of the Mixmaster Universe in the GUP approach as pre-
sented in Sec. 12.7 is in [55]. For related studies, see [359].
This page is intentionally left blank
Bibliography
[1] Alberi, E., Bianchi, C. and Viviani, V. (1856). Le Opere dl Galileo Galilei
(Societa editrice fiorentina).
[2] Albrecht, A. and Steinhardt, P. J. (1982). Cosmology for grand unified the-
oric nil Hi i K mi hit i iii in 1 1 i i 1 i i PI i lie i i ' . ;
48, pp. 1220-1223, doi:10.1103/PhysRevLett.48.1220.
[3] Alexandrov, S. (2000). SO(4, C)-covariant Ashtekar-Barbero gravity and
the Immirzi parameter, Classical and Quantum Gravity 17, pp. 4255-4268,
doi:10.1088/0264- 9381/17/20/307, arXiv.gr-qc/0005085.
[4] Alexandrov, S. and Livine, E. R. (2003). SU(2) loop quantum gravity seen
from covariant theory, Physical Review D 67, 4, p. 044009, doi:10.1103/
PhysRevD. 67.044009, arXiv:gr-qc/0209105.
[5] Alpher, R. A. (1948). A neutron-capture theory of the formation and rel-
ative abundance of the elements, Physical Review 74, pp. 1577-1589, doi:
10.1103/PhysRev.74.1577.
[6] Alpher, R. A., Bethe, H. and Gamow, G. (1948). The origin of chemical
elements, Physical Review 73, pp. 803-804, doi:10.1103/PhysRev.73.803.
[7] Alpher, R. A. and Herman, R. (1948). Evolution of the universe, Nature
162, pp. 774-775, doi:10.1038/162774b0.
[8] Amati, D., Ciafaloni. M. and Veneziano, G. (1989). Can spacetime be
probed below the string size? Physics Letters B 216, pp. 41-47, doi:
10.1016/0370- 2693(89)91366-X.
[9] Amelino-Camelia, G. (2001). Testable scenario for relativity with minimum
length, Physics Letters B 510, pp. 255-263, doi:10.1016/S0370- 2693(01)
00506-8, arXiv:hep-th/0012238.
[10] Amelino-Camelia, G. (2002). Relativity in Spacetimes with Short-Distance
Structure Governed by an Observer-Independent (Planckian ; Lengl ii Scale,
International Journal of Modern Physics D 11, pp. 35-59, doi:10.1142/
S0218271802001330, arXiv:gr-qc/0012051.
[11] Amelino-Camelia, G. (2010). Doubly-Special Relativity: Facts, Myths and
Some Key Open Issues, Symmetry 2, pp. 230-271, arXiv: 1003.3942.
[12] Andersson, L., van Elst, H., Lim, W. C. and Uggla, C. (2005). Asymptotic
silence of generic cosmological singularities, Physical Review Letters. 94, p.
554 Primordial Cosmology
051101, arXiv.org/abs/gr-qc/0402051.
[13] Andriopoulos, K. and Leach, P. G. L. (2008). The mixmaster universe:
the final reckoning? Journal of Physics A 41, 15, p. 155201, doi:10.1088/
1751-8113/41/15/155201.
[14] Anosov, D. V. (1967). Geodesic Flows on Closed Riemann Manifolds with
Negative Curvature, Providence (American Mathematical Society).
[15] Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics
(Springer- Verlag, Berlin).
[16] Arnold, V. I. and Avez, A. (1968). Ergodic problems of classical mechanics
( Benjamin, New York).
[17] Arnowitt, R. and Deser, S. (1959). Quantum theory of gravitation: General
formulation and linearized theory, Physical Review 113, 2, pp. 745-750.
[18] Arnowitt, R., Deser, S. and Misner, C. W. (1959). Canonical variables for
general relativity, Physical Review 117, 6, pp. 1595-1602.
[19] Arnowitt, R., Deser, S. and Misner, C. W. (1962). The dynamics of gen-
eral relativity, in L. Witten (cd.). (invitation: an introduction to cur-
rent research, chap. 7 (Wiley, New York), pp. 227-265, arXiv.org/abs/gr-
qc/0405109.
[20] Artin, E. (1965). Ein mechanisches system mit quasi-ergodischen bahnen, in
S. Lang and J. T. Tate (eds.), The Collected Papers of Emil Artin (Addison-
Wesley, Reading, MA), pp. 499-501.
[21] Ashtekar, A. (1986). New variables for classical and quantum gravity, Phys-
ical Review Letters 57, pp. 2244-2247, doi:10.1103/PhysRevLett. 57.2244.
[22] Ashtekar, A. (1987). New Hamiltonian formulation of general relativity,
Physical Review D 36, pp. 1587-1602, doi:10.1103/PhysRevD.36.1587.
[23] Ashtekar, A. (2007). An introduction to loop quantum gravity through
cosmology, Nuovo Cimento B 122, pp. 135-155, doi:10.1393/ncb/
i2007-10351-5, arXiv:gr-qc/0702030.
[24] Ashtekar, A. (2009). Loop quantum cosmology: an overview, General Rel-
ativity and Gravitation 41, pp. 707-741, doi:10.1007/sl0714-009-0763-4,
arXiv:0812.0177.
[25] Ashtekar, A., Bojowald, M. and Lewandowski, J. (2003). Mathematical
structure of loop quantum cosmology, Advances in Theoretical anil Mathe-
matical Physics 7, pp. 233-268, arXiv.org/abs/gr-qc/0304074.
[26] Ashtekar, A., Fairhurst, S. and Willis, J. L. (2003). Quantum gravity,
shadow states and quantum mechanics, Classical and Quantum Gravity
20, pp. 1031-1061, arXiv:gr-qc/0207106.
[27] Ashtekar, A., Henderson, A. and Sloan, D. (2009). Hamiltonian general
relativity and the belinskii khalatnikov lifshitz conjecture, Classical and
Quantum Gravity 26, 5, p. 052001, doi:10. 1088/0264-9381/26/5/052001,
arXiv:0811.4160.
[28] Ashtekar, A. and Lewandowski, J. (2004). Background independent quan-
tum gravity: A status report, Classical and Quantum Gravity 21, pp. R53-
R152, arXiv.org/abs/gr-qc/0404018.
[29] Ashtekar, A., Pawlowski, T. and Singh, J. P. (2006a). Quantum nature of
the Big Bang, Physical Review Letters 96. 1 1. p. 1 11301, arXiv
qc/0602086.
[30] Ashtekar, A., Pawlowski, T. and Singh, P. (2006b). Quantum nature of the
bi;> bang: An analytical and numerical investigation. Physical lici'icir D
73, p. 124038, arXiv.org/abs/gr-qc/0604013.
[31] Ashtekar, A., Pawlowski, T. and Singh, P. (2006). Quantum nature of the
big bang: Improved dynamics, Physical Review D 74, 8, p. 084003. doi:
10.1103/PhysRevD.74.084003, arXiv:gr-qc/0607039.
[32] Ashtekar, A. and Stachel, J. (eds.) (1991). Conceptual problems of quantum
Til lii i P ii
[33] Ashtekar, A. and Wilson- Ewing, E. (2009a). Loop quantum cosmology of
Bianchi type i models, Physical Review D 79, 8, p. 083535, doi:10.1103/
PhysRevD. 79.083535, arXiv:0903.3397.
[34] Ashtekar, A. and Wilson- Ewing, E. (2009b). Loop quantum cosmology of
Bianchi type ii models, Physical Review D 80, 12, p. 123532, doi:10.1103/
PhysRevD.80.123532, arXiv:0910.1278.
[35] Banerjee, R., Kulkarni, S. and Samanta, S. (2006). Deformed symme-
try in Snyder space and relativistic particle dynamics, Journal of High
Energy Physics 5, p. 77, doi:10.1088/1126-6708/2006/05/077, arXiv:hep-
th/0602151.
[36] Bang, J. Y. and Berger, M. S. (2006). Quantum mechanics and the gen-
eralized uncertainty principle, Physical Review D 74, 12, p. 125012, doi:
10. 1103/PhysRevD. 74.125012, arXiv:gr-qc/0610056.
[37] Barbero G., J. F. (1995). Real Ashtekar variables for Lorentzian signa-
ture space times, Physical Review D D51, pp. 5507-5510, doi:10.1103/
PhysRevD.51.5507, arXiv:gr-qc/9410014.
[38] Bardeen, J. M. (1980). Gauge-invariant cosmological perturbations, Phys-
ical Review D 22, pp. 1882-1905, doi:10.1103/PhysRevD.22.1882.
[39] Barrow, J. D. (1981). Chaos in the einstein equations, Physical Review
Letters 46, 15, pp. 963-966.
[40] Barrow, J. D. (1982). Chaotic behavior in general relativity, Physics Reports
85, pp. 1-49.
[41] Barrow, J. D. (1987). Deflationary universes with quadratic lagrangians,
Physics Letters B 183, pp. 285-288, doi:10.1016/0370- 2693(87)90965-8.
[42] Barrow, J. D. (1988). String-driven inflationary and deflationary cos-
mological models, Nuclear Physics B 310, pp. 743-763, doi: 10. 1016/
0550-3213(88)90101-0.
[43] Barrow, J. D. and Stein-Schabes, J. (1985). Kaluza-Klein mixmaster uni-
verses, Physical Review D 32, pp. 1595-1596, doi:10.1103/PhysRevD.32.
1595.
[44] Barrow, J. D. and Tipler, F. J. (1979). Analysis of the generic singularity
studies by Belinskii, Khalatnikov, and Lifschitz. Physics Reports 56, pp.
371-402.
[45] Barvinsky, A. O. (1990). The general semiclassical solution of the Wheeler-
deWitt equations and the issue of unitarity in quantum cosmology, Physics
Letters B 241, 2, pp. 201 - 206, doi:DOI:10.1016/0370-2693(90)91278-J.
[46] Battisti, M. V. (2009). Cosmological bounce from a deformed heisenberg
556 Primordial Cosmology
algebra, Physical Review D 79, 8, p. 083506, doi:10.1103/PhysRevD.79.
083506.
[47] Battisti, M. V., Belvedere, R. and Montani, G. (2009). Semiclassical sup-
i 1 i i I i / j <
p. 69001, doi:10.1209/0295-5075/86/69001, arXiv:0905.3695.
[48] Battisti, M. V., Lecian, O. M. and Montani, G. (2008). Polymer quantum
dynamics of the Taub universe, Physical Review D 78, 10, p. 103514, doi:
10.1103/PhysRevD. 78.103514, arXiv:0806.0768.
[49] Battisti, M. V., Marciano, A. and Rovelli, C. (2010). Triangulated loop
1 ii i i ] L i i i i i i I i i i i i
tions, Physical Review D 81, 6, p. 064019, doi:10.1103/PhysRevD.81.
064019.
[50] Battisti, M. V. and Meljanac, S. (2009). Modification of Heisenberg uncer-
tainty relations in noncommutative Snyder space-time geometry, Physical
Review D 79, 6, p. 067505, doi:10.1103/PhysRevD. 79.067505.
[51] Battisti, M. V. and Meljanac, S. (2010). Scalar field theory on non-
commutative snyder space-time, Physical Review D 82, p. 024028, doi:
10.1103/PhysRevD.82.024028, arXiv:1003.2108.
[52] Battisti, M. V. and Montani, G. (2006). Evolut ion:)!', quantum dynamics of
a generic universe. Physics Letters B 637, pp. 203-209, arXiv.org/abs/gr-
qc/0604049.
[53] Battisti, M. V. and Montani, G. (2007a). The big-bang singularity in the
framework of a generalized uncertainty principle Physics Letters B 656.
pp. 96-101, arXiv.org/abs/gr-qc/0703025.
[54] Battisti, M. V. and Montani, G. (2007b). Quantum dynamics of the taub
universe in a generalized uncertainty principle framework, Physical Review
D 77, p. 023518, arXiv.org/abs/0707.2726.
[55] Battisti, M. V. and Montani, G. (2009). The Mixmaster Universe in a
generalized uncertaintj principle framework, Physics Letters B 681, pp.
179-184, doi:10.1016/j.physletb.2009.10.003, arXiv:0808.0831.
[56] Beaume, R., Manuceau, J., Pellet, A. and Sirugue, M. (1974). Translation
invariant states in quantum mechanics, Communications in Mathematical
Physics 38, p. 29.
[57] Belinskii, V. A. (2010). Cosmological singularity, in R. Ruffini &
G. Vereshchagin (ed.), American Lnstitute of Physics Conference Series,
American Lnstitute of Physics Conference Series, Vol. 1205, pp. 17-25, doi:
10.1063/1.3382327.
[58] Belinskii, V. A. and Khalatnikov, I. M. (1969). On the nature of the sin-
gularities in the general solutions of the gravitational equations, Soviet
Physics JETP 29, p. 911.
[59] Belinskii, V. A. and Khalatnikov, I. M. (1972). Effect of scalar and vector
fields on the natuie of tin cosmolo J i ilari 1 n I Eksperimental
noi i Teoreticheskoi Fiziki 63, 4, p. 1121.
[60] Belinskii, V. A. and Khalatnikov, I. M. (1973). Effect of scalar and vector
fields on the nature of the cosmological singularity, Soviet Physics JETP
36, p. 591.
[61] Belinskii, V. A. and Khalatnikov, I. M. (1975). Influence of viscosity on the
character of cosmological evolution, Soviet Journal of Experimental and
Theoretical Physics 42, p. 205.
[62] Belinskii, V. A. and Khalatnikov, I. M. (1977a). Viscosity effects in isotropic
cosmologies, Soviet Journal of Experimental and Theoretical Physics 45.
pp. 1-9.
[63] Belinskii, V. A. and Khalatnikov, I. M. (1977b). Viscosity effects in isotropic
cosii) ilogii Z i ' El pc.i /mental noi i Teoreticheskoi Fiziki 72, pp. 3-
17.
[64] Belinskii, V. A., Khalatnikov, I. M. and Lifshitz, E. M. (1970). Oscillatory
approach to a singular point in the relativistic cosmology, Advances in
Physics 19, pp. 525-573.
[65] Belinskii, V. A., Khalatnikov, I. M. and Lifshitz, E. M. (1982). A general
solution of the Einstein equations with a time singularity. Advances in
Physics 31, pp. 639-667.
[66] Belinskii, V. A., Lifshitz, E. M. and Khalatnikov, I. M. (1971). Re-
views of Topical Problems: Oscillatory Approach to the Singular Point
in Relativistic Cosmology, Soviet Physics Uspekhi 13, pp. 745-765, doi:
10.1070/PU1971v013n06ABEH004279.
[67] Belinskii, V. A., Nikomarov, E. S. and Khalatnikov, I. M. (1979). Inves-
tigation of the cosmological evolution of viscoelastic matter with causal
thermodynamics, Soviet Journal of Experimental and Theoretical Physics
50, p. 213.
[68] Benczik, S., Chang, L. N., Minic, D., Okamura, N., Rayyan, S. and
Takeuchi, T. (2002). Short distance versus long distance physics: The clas-
sical limit of the minimal length uncertainty relation, Physical Review D
66, 2, p. 026003, doi:10.1103/PhysRevD. 66. 026003, arXiv:hep-th/0204049.
[69] Benini, R., Kirillov, A. A. and Montani, G. (2005). Oscillatory regime in the
multidimensional homogeneous cosmological models induced by a vector
field, Classical and Quantum Gravity 22, pp. 1483-1491, arXiv.org/abs/gr-
qc/0505027.
[70] Benini, R. and Montani, G. (2004). Frame-independence of the inhomoge-
neous Mixmaster chaos via Misner-Chitre-like variables, Physical Review D
70, p. 103527, arXiv.org/abs/gr-qc/0411044.
[71] Benini, R. and Montani, G. (2007). Inhomogeneous quantum Mixmaster:
From classical toward quantum mechanics, Classical and Quantum Gravity
24, pp. 387-404, arXiv.org/abs/gr-qc/0612095.
[72] Bennett, C. L., Hill, R. S., Hinshaw, C, Larson, D., Smith, K. M., Dunkley,
J., Gold, B., Halpern, M., Jarosik, N., Kogut, A., Komatsu, E., Limon, M.,
Meyer, S. S., Nolta, M. R., Odegard, N., Page, L., Spergel, D. N., Tucker,
G. S., Weiland, J. L., Wollack, E. and Wright, E. L. (2010). Seven-year
Wilkinson Microwave Anisotropy Probe (WMAP) observations: Are there
Cosmic Microwave Background anomalies? arXiv:1001.4758.
[73] Berger, B. K. (1989). Quantum chaos in the mixmaster universe, Physical
Review D 39, pp. 2426-2429.
[74] Berger, B. K. (1990). Numerical study of initially expanding r
Primordial Cosmology
i. General Relativity and Gravitation 7, pp. 203-216.
[75] Berger, B. K. (1994). How to determine approximate mixmaster parameters
from numerical evolution of Einstein's equations, Physical Review D 49, pp.
1120-1123, arXiv.org/abs/gr-qc/9308016.
[76] Berger, B. K. (1999). Influence of scalar fields on the approach to a cosmo-
logical singularity, Physical Review D 61, 2, p. 023508, arXiv.org/abs/gr-
qc/9907083.
[77] Berger, B. K. (2002). Numerical approaches to spacetime singularities, Liv-
ing Reviews in Relativity 5, 1, arXiv.org/abs/gr-qc/0201056.
[78] Berger, B. K., Garfinkle, D., Isenberg, J., Moncrief, V. and Weaver,
M. (1998). The singularity in generic gravitational collapse is space-
like, local and oscillatory, Modern Physics Letters A 13, pp. 1565-1574,
arXiv.org/abs/gr-qc/9805063.
[79] Berger, B. K., Garfinkle, D. and Strasser, E. (1997). New algorithm for
Mixmaster dynamics, Classical and Quantum Gravity 14, pp. L29-L36,
arXiv.org/abs/gr-qc/9609072.
[80] Berger, B. K. and Moncrief, V. (1993). Numerical investigation of cosmo-
logical singularities, Physical Review D 48, pp. 4676-4687, doi:10.1103/
PhysRevD.48.4676, arXiv:gr-qc/9307032.
[81] Berger, B. K. and Vogeli, C. N. (1985). Path-integral quantum cosmology.
I. vacuum Bianchi type I, Physical Review D 32, 10, pp. 2477-2484, doi:
10.1103/PhysRevD.32.2477.
[82] Bernstein, J. (2004). Kinetic theory in the expanding universe (Cambridge
University Press).
[83] Bertone, G., Hooper, D. and Silk, J. (2005). Particle dark matter: Evidence,
candidates and constraints, Physics Reports 405, pp. 279-390, doi:10.1016/
j.physrep. 2004.08. 031, arXiv:hep-ph/0404175.
[84] Bertschinger, E. (1993). Gravitational clustering in cosmology, in J. Mis-
quich, G. Pelletier, & P. Schuck (ed.), Statistical Description of Transport
in Plasma, Astro- and Nuclear Physics, p. 193.
[85] Bertschinger, E. (1995). Cosmological dynamics, NASA STI/Recon Tech-
nical Report N 96, p. 22249, arXiv:astro-ph/9503125.
[86] Bianchi, L. (1898). On the three-dimensional spaces which admit a contin-
uous group of motions Memori( di Matematica e di Fisica della Societa
Italiana delle Scienze, Serie Terza XI, 267-352 (1898) 11, pp. 267-352.
[87] Bianchi, L. (1898). Sugli spazi a tre dimensioni che ammettono un gruppo
continuo di movimenti, Memoric di Ma.tcina.tica e di Fisica della Societa
Italiana delle Scienze, Serie Terza 11, pp. 267-352.
[88] Biesiada, M. (1995). Searching for invariani description of chaos in general
relativity, Classical and Quantum Grai'itg 12. p. 715.
[89] Bleher, S., Grebogi, C, Ott, E. and Brown, R. (1988). Fractal boundaries
for exit in hamiltonian dynamics, Physical Review A 38, 2, p. 930.
[90] Blyth, W. F. and C. J. Isham, C. J. (1975). Quantization of a friedmann
universe filled with a scalar field, Physical Review D 11, pp. 768-778.
[91] Bojowald, M. (2001). Absence of singularity in loop quantum cosmology,
Physical Review Letters 86, pp. 5227-5230, arXiv.org/abs/gr-qc/0102069.
Bojowald, M. (2001). Inverse scale factor in isotropic quantum geome-
try, Physical Review D 64, 8, p. 084018, doi:10.1103/PhysRevD. 64.084018,
arXiv:gr-qc/0105067.
Bojowald, M. (2007). Singularities and quantum gravity, in M. Novello
& S. E. Perez Bergliaffa (ed.), Xllth Brazilian School of Cosmololy and
Gravitation, American Institute of Physics Conference Series, Vol. 910, pp.
294-333, doi:10.1063/1.2752483.
[94] Bojowald, M. (2008). Loop Quantum Cosmology, Living Reviews in Rela-
tivity 11, 4, URL http://www.livingreviews.org/lrr-2008-4.
[95] Bojowald, M. and Date, G. (2004). Quantum suppression of the generic
chaotic behavior close to cosmological singularities, Physical Review Letters
92, p. 071302, arXiv.org/abs/gr-qc/0311003.
Bojowald, M., Date, G. and Mortuza Hossain, G. (2004). The Bianchi
IX model in loop quantum cosmology, Classical and Quantum Gravity
21, pp. 3541-3569, doi:10.1088/0264-9381/21/14/015, arXiv.org/abs/gr-
qc/0404039.
Bojowald. M.. Date, G. and Vandersloot, K. (2004). Homogeneous loop
quantum cosmology: The role of the spin connection, Classical and Quan-
tum Gravity 21, pp. 1253-1278, arXiv.org/abs/gr-qc/0311004.
Bojowald, M., Kiefer, C. and Vargas Moniz, P. (2010). Quantum cosmology
for the 21st century: A debate, arXiv:1005.2471.
Bond, J. R., Efstathiou, G. and Silk, J. (1980). Massive neutrinos and
the large-scale structure of the universe, Physical Review Letters 45, pp.
1980-1984, doi:10.1103/PhysRevLett.45.1980.2.
Bond, J. R. and Szalay, A. S. (1983). The collisionless damping of density
fluctuations in an expanding universe. The. Astro pit ysical Journal 274, pp.
443-468, doi:10. 1086/161460.
[101] Bondi, H. and Gold, T. (1948). The steady-state theory of the expanding
universe. Monthly Notices of the Royal Academy Society 108, p. 252.
[102] Brattelli, O. and Robinson, D. W. (1997). Operator Algebras and Quantum
i tistiral Mechanics, Vol. 1: C* - and IV*-Algebras. Symmetry Groups.
Decomposition ol ' fa I iprin < -Verlag, Berlin).
[103] Brau, F. and Buisseret, F. (2006). Minimal length uncertainty relation
and gravitational quantum well, Physical Review D 74, 3, p. 03000'-.'. doi:
10.1103/PhysRevD. 74.036002, arXiv:hep-th/0605183.
[104] Brown, J. D. and Kuchaf, K. V. (1995). Dust as a standard of space and
time in canonical quantum gravity, Physical Review D 51, pp. 5600-5629,
arXiv.org/abs/gr-qc/9409001.
[105] Burd, A. and Tavakol, R. (1993). Invariant lyapunov exponents and chaos
in cosmology, Physical Review D 47, 12, pp. 5336-5341.
[106] Burd, A. B., Buric, N. and Tavakol, R. (1991). Chaos, entropy and cosmol-
ogy, Classical and Quantum Gravity 8, pp. 123-133.
[107] Cabella, P. and Kamionkowski, M. (2004). Theory of Cosmic Microwave
Background polarization, arXiv:astro-ph/0403392.
Caldwell, R. R., Dave, R. and Steinhardt, P. J. (1998). Cosmological Im-
print of an Energy Component with General Equation of State, Physi-
Primordial Cosmology
cal Review Letters 80, pp. 1582-1585, doi:10.1103/PhysRevLett. 80.1582,
arXiv:astro-ph/9708069.
Capozziello, S. and Francaviglia, M. (2008). Extended theories of grav-
ity and their cosmological and astrophysical applications, General Rel-
ativity and Gravitation 40, pp. 357-420, doi:10.1007/sl0714-007-0551-y,
arXiv:0706.1146.
Capozziello, S. and Lambiase, G. (2000). Selection rules in minisuperspace
quantum cosmology, General Relativity and Gravitation 32, pp. 673-696,
doi: 10. 1023/A: 1001967102409, arXiv:gr-qc/9912083.
Carlevaro, N. and Montani, G. (2005). Bulk viscosity effects on the early
universe stability, Modern Physics Letters A 20, pp. 1729-1739, doi: 10.
1 142/S0217732305017998, arXiv:gr-qc/0506044.
Carlevaro, N. and Montani, G. (2008). Study of the quasi-isotropic so-
lution near the cosmological singularity in the presence of bulk viscos-
ity, International Journal of Modern Physics D 17, pp. 881-896, doi:
10.1142/S0218271808012553, arXiv:071 1.1952.
Carlip, S. (2001). Quantum gravity: A progress report, Reports on
Progress in Physics 64, p. 885, doi:10. 1088/0034-4885/64/8/301, arXiv:gr-
qc/0108040.
Catren, G. and Ferraro, R. (2001). Quantization of the Taub model with
extrinsic time, Physical Review D 63, 2, p. 023502, doi:10.1103/PhysRevD.
63.023502, arXiv:gr-qc/0006027.
Cavallaro, S.. Morchio. G. and Strocchi. F (1909). A generalization of the
Stone-von Neumann theorem to nonregular representations of the CCR-
algebra, Letters in Mathematical Physics 47, 4, pp. 307-320, doi:10.1023/
A:1007599222651, URL http : //dx . doi . org/10 . 1023/A : 1007599222651 .
Chandrasekhar, S. (1983). The Mathematical Theory of Black Holes (Ox-
ford University Press).
Chang, L. N., Minic, D., Okamura, N. and Takeuchi, T. (2002). Exact
solution of the harmonic oscillator in arbitrary dimensions with minimal
length uncertainty relations, Physical Review D 65, 12, p. 125027, doi:
10.1103/PhysRevD.65.125027.
Chernoff, D. F. and Barrow, J. D. (1983). Chaos in the mixmaster universe,
Physical Review Letters 50, 2, pp. 134-137.
Chitre, D. M. (1972). Investigation of vanishing of a horizon for Bianchi
type IX (The mirmastcr unii'crsc.j, Ph.D. thesis, University of Maryland,
technical Report No. 72-125.
Cianfrani, F., Lecian, O. M. and Montani, G. (2008). Fundamentals
and recent developments in noii-perturbative canonical quantum gravity.
arXiv:0805.2503.
Cianfrani, F. and Montani, G. (2009). Towards loop quantum gravity
without the time gauge, Physical Review Letters 102, 9, p. 091301, doi:
10.1103/PhysRevLett.l02.091301, arXiv:0811.1916.
Clowe, D., Bradac, M., Gonzalez, A. H., Markevitch, M., Randall, S. W.,
Jones, C. and Zaritsky, D. (2006). A direct empirical proof of the existence
of dark matter, The Astrophysical Journal Letters 648, pp. L109-L113,
doi:10.1086/508162, arXiv:astro-ph/0608407.
[123] Cole, S., Percival, W. J., Peacock, J. A., Norberg, P., Baugh, C. M., Frenk,
C. S., Baldry, I., Bland-Hawthorn, J., Bridges, T., Cannon, R., Colless,
M., Collins, C, Couch, W., Cross, N. J. C, Dalton, C, Eke, V. R., De
Propris, R., Driver, S. P., Efstathiou, C, Ellis, R. S., Glazebrook, K.,
Jackson, C, Jenkins, A., Lahav, O., Lewis, I., Lumsden, S., Maddox, S.,
Madgwick, D., Peterson, B. A., Sutherland, W. and Taylor, K. (2005). The
2<1F Galaxy Rrdsliift Survey: power-spectrum analysis of the final data set
and cosmological implications, Monthly Notices of the Royal Academy So-
ciety 362, pp. 505-534, doi:10.1111/j. 1365- 2966. 2005. 09318.x, arXiv:astro-
ph/0501174.
[124] Coleman, S. R. and Weinberg, E. J. (1973). Radiative corrections as the
origin of spontaneous symmetry breaking, Physical Review D 7, pp. 1888-
1910.
[125] Collins, C. B. and Ellis, G. F. R. (1979). Singularities in bianchi
cosmologies, Physics Reports 56, 2, pp. 65 - 65, doi:DOI:10.1016/
0370-1573(79)90065-6, URL http://www.sciencedirect.com/science/
article/B6TVP-46T4VVF-lC/2/3a87a0a761dd8a32798af70c6c2cl9aa.
[126] Contopoulos, B., G.and Grammaticos and Ramani, A. (1995). The last
remake of the Mixmaster universe model, Journal of Physics A 28, 18, p.
5313, URL http: //stacks. iop.org/0305-4470/28/i=18/a=020.
[127] Corichi, A. and Singh, P. (2008). Is loop quantization in cosmology unique?
Physical Review D 78, 2, p. 024034, doi:10.1103/PhysRevD. 78.024034,
arXiv:0805.0136.
[128] Corichi, A., Vukasinac, T. and Zapata, J. A. (2007a). Hamiltonian and
physical Hilbert space in polymer quantum mechanics, Classical and
Quantum Gravity 24, pp. 1495-1511, doi:10.1088/0264-9381/24/6/008,
arXiv:gr-qc/0610072.
[129] Corichi, A., Vukasinac, T. and Zapata, J. A. (2007b). Polymer quantum
mechanics and its continuum limit, Physical Review D 76, 4, p. 044016,
doi:10.1103/PhysRevD.76.044016, arXiv:0704.0007.
[130] Cornfeld, I. P., Fomin, S. V. and Sinai, I. A. G. (1982). Ergodic theory
(Springer, Berlin).
[131] Cornish, N. J. and Levin, J. J. (1997a). The mixmaster
chaotic farey tale, Physical Review D 55, pp. 7489-7510, arXiv
qc/9612066.
[132] Cornish, N. J. and Levin, J. J. (1997b). The mixmaster universe is chaotic,
Physical Review Letters 78, pp. 998-1001, arXiv.org/abs/gr-qc/9605029.
[133] Cotsakis, S. and Leach, P. G. L. (1994). Painleve analysis of the mixmaster
universe. Journal of Physics i Mathematical General 27, pp. 1625 Km I
[134] Csordas, A., Graham, R. and Szepfalusy, P. (1991). Level statistics of a
noncompact cosmological billiard, Physical Review A 44, 3, pp. 1491 1199.
[135] Damour, T. (2005). Poincare, relativity, billiards and symmetry, arXiv:hep-
th/0501168.
[136] Damour, T. and de Buyl, S. (2008). Describing general cosmological sin-
gularities in Iwasawa variables, Physical Review D 77, 4, p. 043520, doi:
562 Primordial Cosmology
10.1103/PhysRevD. 77.043520, arXiv:0710.5692.
[137] Damour, T., Henneaux, M. and Nicolai, H. (2003). Cosmological billiards,
Classical and Quantum Gravity 20, p. 020, arXiv.org/abs/hep-th/0212256.
[138] Damour, T. and Nicolai, H. (2008). Symmetries, singularities and the de-
emergence of space, International Journal of Modern Physics D 17, pp.
525-531, doi:10.1142/S0218271808012206, arXiv:0705.2643.
[139] de Bernardis, P., Ade, P. A. R., Bock, J. J., Bond, J. R., Borrill, J., Bosca-
leri, A., Coble, K., Contaldi, C. R., Crill, B. P., De Troia, G., Farese,
P., Ganga, K., Giacometti, M., Hivon, E., Hristov, V. V., Iacoangeli, A.,
Jaffe, A. H., Jones, W. C, Lange, A. E., Martinis, L.. Masi, S., Mason,
P., Mauskopf, P. D., Melchiorri, A., Montroy, T., Netterfield, C. B., Pas-
cale, E., Piacentini, F., Pogosyan, D., Polenta, G., Pongetti, F., Prunet,
S., Romeo, G., Ruhl, J. E. and Scaramuzzi, F. (2002). Multiple peaks in
the angular power spectrum of the Cosmic Microwave Background: Sig-
uificance < ii i < 1 consequences for cosmology. Tin Astrophi/sical .Journal 584.
pp. 559-566, doi: 10. 1086/324298, arXiv:astro-ph/0105296.
[140] de Bernardis, P., Ade, P. A. R., Bock, J. J., Bond, J. R., Borrill, J., Bosca-
leri, A., Coble, K., Crill, B. P., De Gasperis, G., Farese, P. C, Ferreira,
P. G., Ganga, K., Giacometti, M., Hivon, E., Hristov, V. V., Iacoangeli, A.,
Jaffe, A. H., Lange, A. E., Martinis. L., Masi, S., Mason, P. V., Mauskopf,
P. D., Melchiorri, A., Miglio, L., Montroy, T., Netterfield, C. B., Pascale,
E., Piacentini, F., Pogosyan, D., Prunet, S., Rao, S., Romeo, G., Ruhl,
J. E., Scaramuzzi, F., Sforna, D. and Vittorio, N. (2000). A flat universe
from high-resolution maps of the Cosmic Microwave Background radiation,
Nature 404, pp. 955-959, arXiv:astro-ph/0004404.
[141] de Moura, A. and Grebogi, C. (2001). Output functions and fractal dimen-
sions in dynamical systems, Physical Review Letters 86, 13, p. 2778.
[142] de Moura, A. and Grebogi, C. (2002). Countable and uncountable bound-
aries in chaotic scattering, Physical Review E 66, p. 046214.
[143] de Sitter, W. (1917). Einstein's theory of gravitation and its astronomical
consequences. Third paper, Monthly Notices of the Royal Academy Society
78, pp. 3-28.
[144] de Sitter, W. (1930). On the magnitudes, diameters and distances of the
extragalactic nebulae and their apparenl radial velocities (errata: 5 v, 230),
Bulletin of the Astronomical Institutes of the Netherlands 5, p. 157.
[145] Demaret, J., de Rop, Y. and Henneaux. M. (1988). Chaos in non-diagonal
spatially homogeneous cosmological models in spacetime dimensions < 10,
Physics Letters B 211, pp. 37-41, doi:10.1016/0370-2693(88)90803-9.
[146] Demaret, J., Hanquin, J. L., Henneaux, M., Spindel, P. and Taormina, A.
(1986). The fate of the mixmaster behaviour in vacuum inhomogeneous
Kaluza-Klein cosmological models, Physics Letters B 175, 2, pp. 129-132.
[147] Demaret, J., Henneaux, M. and Spindel, P. (1985). Non-oscillatory be-
haviour in vacuum kaluza-klein cosmologies, Physics Letters B 164, pp.
27-30.
[148] Desikan, K. (1997). Cosmological models with bulk viscosity in the presence
of particle creation. General Ilelafieili/ ami Graeilal ion 29 pp. 13'i i ['■'>.
[149] DeWitt, B. S. (1967a). Quantum theory of gravity I: The canonical theory,
Physical Review 160, pp. 1113-1148.
[150] DeWitt, B. S. (1967b). Quantum theory of gravity II: The manifestly co-
variant theory, Physical Review 162, pp. 1195-1239.
[151] DeWitt, B. S. (1967c). Quantum theory of gravity III: Applications of the
coyariaut theory. Physical Review 162, pp. 123!) L256.
[152] Dicke, R. H., Peebles, P. J. E., Roll, P. G. and Wilkinson, D. T. (1965).
Cosmic Black-Boch Radiation. Tin Astrophysical Journal 142, pp. 414-
419, doi:10.1086/148306.
[153] Dirac, P. (1964). Lecture on Quantum Mechanics (Belfer Graduate School
of Science, New York - Yeshiva University).
[154] Dittrich, B. (2006). Partial and complete observables for canonical general
relativity, Classical and Quantum Gravity 23, pp. 6155-6184, doi:10.1088/
0264-9381/23/22/006, arXiv:gr-qc/0507106.
[155] Dodelson, S. (2003). Modern Cosmology (Academic Press, Amsterdam).
[156] Doroshkevich, A. G. and Novikov, I. D. (1964). Mean density of radiation
in the metagalaxy and certain problems in relativistic cosmology, Soviet
Physics Doklady 9, p. 111.
[157] Eddington, A. S. (1930). On the instability of Einstein's spherical world,
Monthly Notices of the Royal Academy Society 90, pp. 668-678.
[158] Efstathiou, G., Moody, S., Peacock, J. A., Percival, W. J., Baugh, C,
Bland-Hawthorn, J., Bridges, T., Cannon, R., Cole, S., Colless, M., Collins,
C, Couch, W., Dalton, G., de Propris, R., Driver, S. P., Ellis, R. S., Frenk,
C. S.. Glazebrook, K., Jackson, C, Lahav, O., Lewis. I.. Lumsden, S., Mad-
dox, S., Norberg, P., Peterson, B. A., Sutherland, W. and Taylor, K. (2002).
Evidence for a non-zero A and a low matter density from a combined anal-
ysis of the 2dF Galaxy Redshift Survey and cosmic microwave background
misotropii \Ionthlt/ o < >j lit <yu\ \.i(iu mi >ri ij 330 |>] 1 _">
L35, doi:10.1046/j. 1365-8711. 2002. 05215.x, arXiv:astro-ph/0109152.
[159] Efstathiou, G. and Silk, J. (1983). The formation of galaxies, Fundamentals
of Cosmic Physics 9, pp. 1-138.
[160] Einstein, A. (1914a). Die formale Grundlage der allgemeinen Relativitats-
theorie, Sitzungsberichte der Koniglich Preufiischen Akademie der Wis-
senschaften (Berlin) , pp. 1030-1085.
[161] Einstein, A. (1914b). Physikalische Grundlagen einer Gravitationstheorie,
Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich 58, pp.
284-290.
[162] Einstein, A. (1914c). Prinzipielles zur verallgemeinerten Relativitatstheorie
ni ' i i ii n ill i ' / i i r > pp. 176-180.
[163] Einstein, A. (1915). Zur allgemeinen Relativitatstheorie, Sitzungsberichte
(lev i t/li<. i ,i i i ! i i i i i i I! haften (Berlin) , pp.
778-786.
[164] Einstein, A. (1916). Die Grundlage der allgemeinen Relativitatstheorie, An-
nalen der Physik 354, pp. 769-822, doi:10.1002/andp.l9163540702.
[165] Einstein, A. (1917). Kosmologische Betrachtungen zur allgemeinen Rela-
ii b ' ' / der
564 Primordial Cosmology
Wissenschaften (Berlin) , pp. 142-152.
[166] Einstein, A. and Grossmann, M. (1914). Kovarianzeigenschaften der Feld-
gleichmigoi] dor auf die verallgemeinerte Relativitatstheorie gcgrmidctcii
1 63, pp. 215—
225.
[167] Elgar0y, 0., Lahav, O., Percival, W. J., Peacock, J. A., Madgwick, D. S.,
Bridle, S. L., Baugh, C. M., Baldry, I. K., Bland-Hawthorn, J., Bridges,
T., Cannon, R., Cole, S., Colless, M., Collins, C, Couch, W., Dalton,
C, de Propris, R., Driver, S. P., Efstathiou, G. P., Ellis, R. S., Frenk,
C. S., Glazebrook, K., Jackson, C, Lewis, I., Lumsden, S., Maddox, S.,
Norberg, P., Peterson, B. A., Sutherland, W. and Taylor, K. (2002). New
upper limit on the total neutrino mass from the 2 Degree Field Galaxy
Redshift Survey, Physical Review Letters 89, 6, p. 061301, doi:10.1103/
PhysRevLett. 89. 061301, arXiv:astro-ph/0204152.
Ellis, G. F. R. and MacCallum, M. A. H. (1969). A class of homogeneous
cosmological models, Communications in Mathematical Physics 12, pp.
108-141, doi:10.1007/BF01645908.
169] Elskens, Y. and Henneaux, M. (1987). Ergodic theory of the Mixmaster
model in higher space-time dimensions, Nuclear Physics B 290, pp. Ill—
136.
170] Fang, L. Z. and Ruffini, R. (eds.) (1987). Quantum Cosmology, Advanced
Astrophysics and Cosmology, Vol. 3 (World Scientific).
171] Ferraz, K. and Francisco. G. (1992). Mixmaster numerical behavior and
generalizations, Physical Review D 45, 4, p. 1158.
172] Ferraz, K., Francisco, G. and Matsas, G. E. A. (1991). Chaotic and non-
chaotic behavior in the mixmaster dynamics, Physics Letters A 156, 7, pp.
407 - 409.
173] Fixsen, D. J. and Mather, J. C. (2002). The Spectral Results of the Far-
Infrared Absolute Spectrophotometer Instrument on COBE, The Astro-
physical Journal 581, pp. 817-822, doi: 10. 1086/344402.
174] Fredenhagen, K. and Reszewski, F. (2006). Polymer state approximation of
Schrodinger wavefunctions, Classical and Quantum Gravity 23, pp. 6577-
6584, doi:10.1088/0264-9381/23/22/028, arXiv:gr-qc/0606090.
175] Freidel, L. and Livine, E. R. (2006). 3d quantum gravity and effective
i 1 1 i itati [i ii ii field th / h I 9( '_ ,>
221301, doi:10.1103/PhysRevLett.96.221301, arXiv:hep-th/0512113.
176] Friedmann, A. (1922). Uber die Krummung des Raumes, Zeitschrift fur
Physik 10, pp. 377-386, doi:10.1007/BF01332580.
177] Friedmann, A. (1924). Uber die Moglichkeit einer Welt mit konstanter neg-
ativer Krummung des Raumes, Zeitschrift fur Physik 21, pp. 326-332,
doi:10.1007/BF01328280.
178] Frieman, J. A., Turner, M. S. and Huterer, D. (2008). Dark energy and the
accelerating universe, Annual Review of Astronomy & Astrophysics 46, pp.
385-432, doi:10.1146/annurev.astro.46.060407.145243, arXiv:0803.0982.
79] Fujio, K. and Futamase, T. (2009). Appearance of a classical Mixmaster
from the no-boundary quantum state, Physical Review D 80, 2,
p. 023504, doi:10.1103/PhysRevD.80.023504, arXiv:0906.2616.
Furusawa, T. (1986a). Quantum chaos of Mixmaster universe, Progress of
Tlicorctical Physics 75, pp. 59-67.
Furusawa, T. (1986b). Quantum chaos of Mixmaster universe. II, Progress
of Theoretical Physics 76, p. 67.
Furusawa, T. and Hosoya, A. (1985). Is anisotropic Kaluza-Klein model
of universe chaotic? Progress of Theoretical Physics 73, pp. 467-475, doi:
10.1143/PTP.73.467.
Galilei, G. (1632). Dialogo sopra i due massimi sistemi del mondo, Printed
by Giovan Battista Landini.
Galilei, G., Kepler, J., Foscarini, P. A. and Bernegger, M. (1663). Systema
cosmicvm (prostat voenale apud Thomam Dicas).
Galilei, G. and Manolessi, C. (1655). Opere di Galileo Galilei, Bologna.
Gambini, R. and Pullin. J. (2009). Free will, undecidability, and the prob-
lem of time in quantum gravity, arXiv:0903.1859.
Gamow, G. (1946). Expanding universe and the origin of elements, Physical
Review 70, pp. 572-573, doi:10.1103/PhysRev.70.572.2.
Gamow, G. (1948a). The evolution of the universe, Nature 162, pp. 680-
682, doi:10.1038/162680a0.
Gamow, G. (1948b). The origin of elements and the separation of galaxies,
PR 74, pp. 505-506, doi:10.1103/PhysRev.74.505.2.
Garay, L. J. (1995). Quantum gravity and minimum length, Inter-
national Journal of Modern Physics A 10, pp. 145-165, doi:10.1142/
S0217751X95000085, arXiv:gr-qc/9403008.
Garfinkle, D. (2004). Numerical simulations of generic singularities, Physi-
cal Review Letters 93, 16, p. 161101, doi:10.1103/PhysRevLett.93. 161101,
arXiv:gr-qc/0312117.
Gassoudi. P. (i(>.")3). Iiistitatio astronomica, juxla ' " '/ < tarn veterum
quam recentiorum: cui accesserunt Galili Gall * i It reus; et Jo-
hannis Kepleri Dioptrice (Londini).
Gerlach, U. H. (1969). Derivation of the ten Einstein field equations from
the semiclassical approximation to quantum geometrodynamics, Physical
Review 177, pp. 1929-1941, doi:10.1103/PhysRev. 177.1929.
Gibbons, G. W. and Hawking, S. W. (1977). Action integrals and partition
functions in quantum gravity, Physical Review D 15, pp. 2752-2756, doi:
10.1103/PhysRevD. 15.2752.
Gibbons, G. W., Hawking, S. W. and Perry, M. J. (1978). Path integrals
and the indefiniteness of the gravitational action, Nuclear Physics B 138,
pp. 141-150, doi:10.1016/0550-3213(78)90161-X.
Gibbons, G. W. & Hawking, S. W. (ed.) (1993). Euclidean Quantum Grav-
ity (World Scicnl ific . Singapore-).
Giulini, D. and Kiefer, C. (2007). The canonical approach to quantum grav-
ity: General ideas and geometrodynamics. in I.-O. Stamatescu & E. Seiler
(ed.), Lecture Notes in Physics, Berlin Springer Verlag, Vol. 721, p. 131.
Gold, B., Odegard, N., Weiland, J. L., Hill, R. S., Kogut, A., Bennett, C. L.,
Hinshaw, G., Chen, X., Dunkley, J., Halpern, M., Jarosik, N., Komatsu,
Primordial Cosmology
[201
202]
[203]
[21:
[2 1 2]
2 J.:! |
21 I J
[215]
E., Larson, D., Limon, M., Meyer, S. S., Nolta, M. R., Page, L., Smith,
K. M., Spergel, D. N., Tucker, G. S., Wollack, E. and Wright, E. L. (2010).
Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations:
Galactic foreground emission, arXiv:1001.4555.
Gotay, M. J. and Demaret, J. (1983). Quantum cosmologii :al singularities,
Physical Review D 28, pp. 2402-2413.
Gotay, M. J. and Demaret, J. (1997). A comment on singularities in quan-
tum cosmology, Nuclear Physics B Proceedings Supplements 57, pp. 227-
230, doi:10.1016/S0920-5632(97)00385-X, arXiv:gr-qc/9605025.
Graham, R. (1991). Supersymmetric Bianchi type IX cosmology, Physical
Review Letters 67, 11, pp. 1381-1383, doi:10.1103/PhysRevLett. 67.1381.
Graham, R. (1995). Chaos and quantum chaos in cosmological models,
Chaos Solitons and Fractals 5, p. 1103, arXiv.org/abs/gr-qc/9403030.
Graham, R., Hiibner, R., Szepfalusy, P. and Vattay, G. (1991). Level statis-
tics of a noncompact integrable billiard, Physical Review A 44, 11, pp.
7002-7015.
Grant. E. (1990). The foundation* oj modi in science in tilt Middle Ages:
Their religion* institutional, and intellectual contexts (Cambridge Univ
Grant. E. (2007). A history of natural philosophy: From the ancient world
to the nineteen',!! c.i ntari/ (( 'ambridgo Univ Press).
Grishchuk, L. P. and Rozhansky, L. V. (1990). Does the Hartle-Hawking
wavefunction predict the universe we live in? Physics Letters B 234, pp.
9-14, doi:10.1016/0370-2693(90)91992-K.
Grubisic, B. and Moncrief, V. (1993). Asymptotic behavior of the T 3 xR
Gowdy space-times, Physical Review D 47, pp. 2371-2382, doi:10.1103/
PhysRevD.47.2371, arXiv:gr-qc/9209006.
Grubisic, B. and Moncrief, V. (1994). Mixmaster spacetime, Geroch's
transformation, and constants of motion, Physical Review D 49, pp. 2792-
2800, arXiv.org/abs/gr-qc/9309007.
Guo, H., Huang, C, Tian, Y., Wu, H., Xu, Z. and Zhou, B. (2007). Snyder's
model-de Sitter special relativity duality and de Sitter gravity, Classical and
Quantum Gravity 24, pp. 4009-4035, doi:10. 1088/0264-9381/24/16/004,
arXiv:gr-qc/0703078.
Gurzadyan, V. G., Imponente, G. and Montani, G. (2002). Covariant
chaos in the mixmaster universe, in V. G. Gurzadyan, R. T. Jantzen and
R. Rufiini (eds.), The Ninth Marcel Grossmann Meeting, pp. 907-909.
Gurzadyan, V. G. and Ruffini, R. (eds.) (2000). The Chaotic Universe.
Guth, A. H. (1981). Inflationary universe: A possible solution to the horizon
and flatness problems, Physical Review D 23, pp. 347-356, doi:10.1103/
PhysRevD.23.347.
Guthrie, W. K. C. (1979). A History of Greek Philosophy: The Presocratic
tradition from Parmenides to Democritus (Cambridge Univ Press).
Haag, R. (1992). Local Quantum Physics (Springer, New York).
Halliwell, J. J. (1987). Correlations in the wave function of the Universe,
Physical Review D 36, pp. 3626-3640, doi:10.1103/PhysRevD. 36.3626.
[216] Halliwell, J. J. (1988). Derivation of the Wheeler-DeWitt equation from a
path integral for minisuperspace models, Physical Review D 38, pp. 2468-
2481, doi:10.1103/PhysRevD.38.2468.
[217] Halliwell, J. J. (1990). A bibliography of papers on quantum cosmology,
' a i I i ml ' 1 ' i i i ii n i "> i , _ '7 • .' I ' i i I 1.0.11 12
S0217751X90001148.
[218] Halliwell, J. J. (1991). Quantum cosmology, in J. C. Pati, S. Randjbar-
Daemi, E. Sezgin, & Q. Shafi (ed.), High Energy Physics and Cosmology,
p. 515.
[219] Halliwell, J. J. (1993). The interpretation of quantum cosmological models,
in R. J. Gleiser, C. N. Kozameh, & O. M. Moreschi (ed.), General Relativity
and Gravitation 1992, p. 63.
[220] Halliwell, J. J. (2002). The interpretation of quantum cosmology and the
problem of time, arXiv:gr-qc/0208018.
[221] Halliwell, J. J. and Hartle, J. B. (1990). Integration contours for the no
boundary wave function of the universe, Physical Review D 41, p. 1815,
doi:10.1103/PhysRevD.41.1815.
[222] Halliwell, J. J. and Louko, J. (1989a). Steepest-descent contours in the
path-integral approach to quantum cosmology. I. The de Sitter min-
isuperspace model, Physical Review D 39, pp. 2206-2215, doi:10.1103/
PhysRevD.39.2206.
[223] Halliwell, J. J. and Louko, J. (1989b). Steepest-descent contours in the
path-integral approach to quantum cosmology. II. Microsuperspace, Phys-
ical Review D 40, pp. 1868-1875, doi:10.1103/PhysRevD.40.1868.
[224] Halpern, P. (1986). Behavior of homogeneous five-dimensional space-times,
Physical Review D 33, pp. 354-362, doi:10.1103/PhysRevD.33.354.
[225] Hartle, J. B. and Hawking, S. W. (1983). Wave function of the u
Physical Review D 28, pp. 2960-2975.
[226] Hartle, J. B., Hawking, S. W. and Hertog, T. (2008). Classical r
of the no-boundary quantum state, Physical Review D 77, 12, p. 123537,
doi:10.1103/PhysRevD.77.123537, arXiv:0803.1663.
[227] Hartle, J. B., Hawking, S. W. and Hertog, T. (2008). The No-Boundary
Measure of the Universe. Physical Review Letters 100, p. 201301, doi:10.
1103/PhysRevLett. 100. 201301, arXiv:0711.4630.
[228] Hawking, S. and Ellis, G. F. R. (1973). The Large Scale Structure of Space-
Time (Cambridge University Press).
[229] Hawking, S. W. (1978). Quantum gravity and path integrals, Physical Re-
view D 18, pp. 1747-1753, doi:10.1103/PhysRevD. 18.1747.
[230] Hawking, S. W. (1982). Boundary conditions of the universe, in H. A.
Briik, G. V. Coyne and M. S. Longair (eds.), Astrophysical Cosmology.
Proceeding* oj the Simla W'cel, oil Cosmology unci Fundamental Physics
(Pontificia Academiae Scientarium) , pp. 563-574.
[231] Hawking, S. W. and Penrose, R. (1970). The singularities of gravitational
collapse and cosmology, in Proceedings of the Royal Society of London.
Series A, Mathematical and Physical Sciences (The Royal Society), pp.
529-548.
568 Primordial Cosmology
[232] Hawking, S. W. and Penrose, R. (1996). The Nature of Space and Time
(Princeton University Press).
[233] Hawkins, E., Maddox, S., Cole, S., Lahav, O., Madgwick, D. S., Nor-
berg, P., Peacock, J. A., Baldry, I. K., Baugh, C. M., Bland-Hawthorn,
J., Bridges, T., Cannon, R., Colless, M., Collins, C, Couch, W., Dalton,
C, De Propris, R., Driver, S. P., Efstathiou, C, Ellis, R. S., Frenk, C. S.,
Glazebrook, K., Jackson, C, Jones, B., Lewis, I., Lumsden, S., Percival,
W., Peterson, B. A., Sutherland, W. and Taylor, K. (2003). The 2dF Galaxy
Hcdshift Survey: correlation functions, peculiar velocities and the matter
den sit \ of the Universe, Monthly Not/res of tin Royal . \t nth mi) Soc.it ft/ 346.
pp. 78-96, doi:10.1046/j. 1365-2966. 2003. 07063.x, arXiv:astro-ph/02 12375.
[234] Heinzle, J. M. and Uggla, C. (2009a). A new proof of the Bianchi type
IX attractor theorem, Classical and Quantum Gravity 26, 7, p. 075015,
doi:10.1088/0264-9381/26/7/075015, arXiv:0901.0806.
[235] Heinzle, J. M. and Uggla, C. (2009b). Mixmaster: Fact and belief, Classi-
cal and Quantum Gravity 26, 7, p. 075016, doi:10. 1088/0264-9381/26/7/
075016, arXiv:0901.0776.
[236] Heinzle, J. M., Uggla, C. and Rohr, N. (2007). The cosmological billiard
attractor, Advances in Theoretical and Mathematical Physics 13, pp. 293-
407, arXiv:gr-qc/0702141.
[237] Henneaux, M. and Teitelboim, C. (1992). Quantization of Gauge Systems
(Princeton Univ. Press).
[238] Hobill, D., Burd, A. and Coley, A. (1994). Deterministic Chaos in General
R I it if (Plenum Press, New York).
[239] Hogg, D. W., Eisenstein, D. J., Blanton, M. R., Bahcall, N. A., Brinkmann,
J., Gunn, J. E. and Schneider, D. P. (2005). Cosmic Homogeneity Demon-
strated with Luminous Red Galaxies The Astrophysical Journal 624, pp.
54-58, doi:10. 1086/429084, arXiv:astro-ph/0411197.
[240] Hoist, S. (1996). Barbero's Hamiltonian derived from a generalized Hilbert-
Palatini action, Physical Review D 53, pp. 5966-5969, doi:10.1103/
PhysRevD.53.5966, arXiv:gr-qc/9511026.
[241] Horava, P. (1991). On a covariant Hamilton-Jacobi framework for the
Einstein-Maxwell theory, Classical and Quantum Gravity 8, pp. 2069-2084,
doi:10.1088/0264-9381/8/ll/016.
[242] Hoyle, F. (1948). A new model for the expanding universe, Monthly Notices
of the Royal Academy Society 108, p. 372.
[243] Hu, B. L. (1974). Separation of tensor equations in a homogeneous space
by group theoretical methods, Journal of Mathematical Physics 15, pp.
1748-1755, doi:10.1063/l. 1666537.
[244] Hu, B. L. (1975). Numerical examples from perturbation analysis of the
mixmaster universe, Physical Review D 12, pp. 1551-1562.
[245] Hu, B. L. and Regge, T. (1972). Perturbations on the mixmaster universe,
Physical Review Letters 29, pp. 1616-1620.
[246] Hu, W. and Dodelson, S. (2002). Cosmic Microwave Background
Ymsotropios. Annual Rcricic oj Astronomy !'■' Astrojih ysics 40 pp. 17.1
216, doi:10.1146/annurev.astro.40.060401.093926, arXiv:astro-ph/0110414.
[247] Hu, W. and Sugiyama, N. (1995a). Anisotropies in the cost
background: an analytic approach, The Astrophysical Journal AAA, pp.
489-506, doi:10.1086/175624, arXiv:astro-ph/9407093.
[248] Hu, W. and Sugiyama, N. (1995b). Toward understanding CMB
anisotropics and their implications. Physical Hcvicir D 51. pp. 2599 2030.
doi:10.1103/PhysRevD.51.2599, arXiv:astro-ph/9411008.
[249] Hu, W., Sugiyama, N. and Silk, J. (1997). The Physics of microwave
background anisotropies, Nature 386, pp. 37-43, doi:10.1038/386037a0,
arXiv:astro-ph/9604166.
[250] Hubble, E. (1929). A Relation between Distance and Radial Velocity among
Extra-Galactic Nebulae, Proceedings of the National Academy of Science
15, pp. 168-173, doi:10.1073/pnas.l5.3.168.
[251] Hubble, E. and Humason, M. L. (1931). The velocity-distance relation
among extra-galactic nebulae. The Astrophysical Journal 74, p. 43, doi:
10.1086/143323.
[252] Hubble, E. P. (1925). Cepheids in spiral nebulae, The Observatory 48, pp.
139-142.
[253] Hubble, E. P. (1926a). Extragalactic nebulae, The Astrophysical Journal
64, pp. 321-369, doi:10.1086/143018.
[254] Hubble, E. P. (1926b). A spiral nebula as a stellar system: Messier 33, The
Astrophysical Journal 63, pp. 236-274, doi:10.1086/142976.
[255] Immirzi, G. (1997a). Quantum gravity and Regge calculus, Nuclear Physics
B Proceedings Supplements 57, pp. 65-72, doi:10.1016/S0920-5632(97)
00354-X, arXiv:gr-qc/9701052.
[256] Immirzi, G. (1997b). Real and complex connections for canonical grav-
ity. Classical and Quantum Gravity 14, pp. L177-L181, doi:10.1088/
0264-9381/14/10/002, arXiv:gr-qc/9612030.
[257] Imponente. G. P. and Montani, G. (2001a). Covariant formulation of the
invariant measure for the mixmaster dynamics, International Journal of
Modern Physics D 11, pp. 1321-1330, arXiv.org/abs/gr-qc/0106028.
[258] Imponente, G. P. and Montani, G. (2001b). On the covariance of the mix-
master chaoticity, Physical Review D 63, p. 103501, arXiv.org/abs/astro-
ph/0102067.
[259] Imponente, G. P. and Montani, G. (2003). Quasi isotropic inflationary
solution, International Journal of Modern Physics D 12, pp. 1845-1858,
arXiv.org/abs/gr-qc/0307048.
[260] Imponente, G. P. and Montani, G. (2006). Classical and quantum behavior
of the generic cosmological solution, AIP Conference Proceedings 861, pp.
383-390, arXiv.org/abs/gr-qc/0607009.
[261] Isenberg, J. and Moncrief, V. (1990). Asymptotic behavior of the gravita-
tional field and the nature of singularities in gowdy spacetimes, Annals of
Physics 199, 1, pp. 84-122, arXiv.org/abs/gr-qc/9209006.
[262] Isham, C. J. (1992). Canonical quantum gravity and the problem of time,
arXiv:gr-qc/9210011.
[263] Isham, C. J. and Kuchaf, K. V. (1985). Representations of spacetime dif-
foomorphisms. II. canonical geometrodynamics, Annals of Physics 164, 2,
570 Primordial Cosmology
pp. 316 - 333, doi:DOI:10.1016/0003-4916(85)90019-3.
[264] Jarosik, N., Bennett, C. L., Dunkley, J., Gold, B., Greason, M. R., Halpern,
M., Hill, R. S., Hinshaw, G., Kogut, A., Komatsu, E., Larson, D., Limon,
M., Meyer, S. S., Nolta, M. R., Odegard, N., Page, L., Smith, K. M.,
Spergel, D. N., Tucker, G. S., Weiland, J. L., Wollack, E. and Wright, E. L.
(2010). Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) ob-
servations: Sky maps, systematic errors, and basic results, arXiv:1001.4744.
[265] Kamionkowski, M., Kosowsky, A. and Stebbins, A. (1997). Statistics of cos-
mic microwa.vo background polarization. Physical Hiviiw U 55. pp. 7368
7388, doi:10.1103/PhysRevD.55.7368, arXiv:astro-ph/9611125.
[266] Kasner, E. (1921). Einstein's Cosmological Equations, Science 54, pp. 304-
305, doi:10.1126/science.54.1396.304.
[267] Kasner, E. (1921). Geometrical theorems on einstein's cosmological equa-
tions, Advances in Physics 43, p. 217.
[268] Kato, M. (1990). Particle theories with minimum observable length and
open string theory, Physics Letters B 245, pp. 43-47, doi:10.1016/
0370-2693(90)90162-Y.
[269] Kempf, A. (1997). On quantum field theory with nonzero minimal uncer-
tainties in positions and momenta, Journal of Mathematical Physics 38,
pp. 1347-1372, doi:10.1063/1.531814, arXiv:hep-th/9602085.
[270] Kempf, A. and Mangano, G. (1997). Minimal length uncertainty rela-
tion and ultraviolet regularisation, Physical Review D 55, pp. 790!) 7920.
arXiv.org/abs/hep-th/9612084.
[271] Kempf, A., Mangano, G. and Mann, R. B. (1995). Hilbert space represen-
tation of the minimal length uncertainty relation, Physical Review D 52,
pp. 1108-1118, arXiv.org/abs/hep-th/9412167.
[272] Khalatnikov, I. M., Kamenshchik, A. Y. and Starobinsky, A. A. (2002).
Comment a.boui quasi-isotropic solution of eiusTehi equations near cos-
mological singularity, Classical and Quantum Gravity 19, p. 3845,
arXiv.org/abs/gr-qc/0204045.
[273] Khalatnikov, I. M. and Lifshitz, E. M. (1970). General cosmological solution
of the gravitational equations with a singularity in time. Phijs. Rev. Lett.
24, 2, pp. 76-79, doi:10.1103/PhysRevLett.24.76.
[274] Khalatnikov, I. M., Lifshitz, E. M., Khanin, K. M., Shchur, L. N. and
Sinai, Y. G. (1985). On the stochasticity in relativistic cosmology, Journal
of Statistical Physics 38, pp. 97-114, doi:10.1007/BF01017851.
[275] Kheyfets, A., Miller, W. A. and Vaulin, R. (2006). Quantum geometro-
dynamics of the Bianchi IX cosmological model, Classical and Quantum
Gravity 23, pp. 4333-4351, doi:10. 1088/0264-9381/23/13/003, arXiv:gr-
qc/0512040.
[276] Kiefer, C. (1991). On the meaning of path integrals in quantum cosmol-
ogy, Annals of Physics 207, 1, pp. 53-70, doi:DOI:10.1016/0003-4916(91)
90178-B.
[277] Kiefer, C. (1993). The semiclassical approximation to quantum gravity,
arXiv:gr-qc/9312015.
[278] Kiefer, C. (2000). Conceptual issues in quantum cosmology, Lecture Notes
in Physics 541, pp. 158-187, arXiv:gr-qc/9906100.
[279] Kiefer, C. (2007). Quantum Gravity (Oxford Univ. Press).
[280] Kiefer, C. (2009). Does time exist in quantum gravity? arXiv:0909.3767.
[281] Kiefer, C. (2009). Quantum geometrodynamics: Whence, whither?
General Relativity and Gravitation 41, pp. 877-901, doi:10.1007/
S10714-008-0750-1, arXiv:0812.0295.
[282] Kiefer, C. and Sandhoefer, B. (2008). Quantum cosmology,
arXiv:0804.0672.
[283] Kirillov, A. A. (1993). On the question of the characteristics of the spa-
tial distribution of metric inhomogeneities in a general solution to einstein
n 1 1 i i ii jeinitv of a < ill I i ill i ii '
76, p. 355.
[284] Kirillov, A. A. and Kochnev, A. A. (1987). Cellular structure of space in
the vicinity of a time singularity in the einstein equations, Pis ma Zhurnal
Eksperimental noi i Ti < i < >i F < pp 34. r -348.
[285] Kirillov, A. A. and Melnikov, V. N. (1995). Dynamics of inhomogeneities
of the metric in the vicinity of a singularity in multidimensional cosmology,
Physical Review D 52, 2, pp. 723-729, arXiv.org/abs/gr-qc/9408004.
[286] Kirillov, A. A. and Montani. G. (1997). Description of statistical properties
of the mixmaster universe, Physical Review D 56, pp. 6225-6229.
[287] Kirillov, A. A. and Montani, G. (2002). Quasi- isotropization of the inho-
niogeneous mi.x master universe induced by an inflationary process. Physical
Review D 66, p. 064010, arXiv.org/abs/gr-qc/0209054.
[288] Kodama, H. and Sasaki, M. (1984). Cosmological perturbation theory,
Progress of Theoretical PhysicsSupplement 78, p. 1, doi:10.1143/PTPS.
78.1.
[289] Kogut, J. B. (1983). The lattice gauge theory approach to quantum chro-
modynamics, Reviews of Modern Physics 55, pp. 775-836, doi:10.1103/
RevModPhys.55.775.
[290] Kolb, E. W. and Turner, M. S. (1994). The Early Universe (Westview
[291] Komatsu, E., Smith, K. M., Dunkley, J., Bennett, C. L., Gold, B., Hin-
shaw, G., Jarosik, N., Larson, D., Nolta, M. R., Page, L., Spergel, D. N.,
Halpern, M., Hill, R. S., Kogut, A., Limon, M., Meyer, S. S., Odegard,
N., Tucker, G. S., Weiland, J. L., Wollack, E. and Wright, E. L. (2010).
Seven- Year Wilkinson Microwave Anisotropy Probe (WMAP) observations:
Cosmological interpretation. arXi v:1001.4538.
[292] Konishi, K., Paffuti, G. and Provero, P. (1990). Minimum physical length
and the generalized uncertainty principle in string theory. Phi/sics Lcttt.r*
B 234, pp. 276-284, doi:10. 1016/0370- 2693(90)91927-4.
[293] Kontoleon, N. and Wiltshire, D. L. (1999). Operator ordering and consis-
tency of the wavefunction of the universe, Physical Review D 59, p. 063513,
doi:10.1103/PhysRevD.59.063513, arXiv:gr-qc/9807075.
[294] Kosowsky, A. (1996). Cosmic Microwave Background polarization, An-
nals of Physics 246, pp. 49-85, doi:10.1006/aphy.l996.0020, arXiv:astro-
ph/9501045.
Primordial Cosmology
[295] Kosowsky, A. (2002). The Cosmic Microwave Background, in S. Bonometto,
G. V. and U. Moschella (eds.), Modern Cosmology (IOP Publishing, Bristol
and Philadelphia), pp. 219-263.
[296] Kowalski, M., Rubin, D., Aldering, G., Agostinho, R. J., Amadon, A.,
Amanullah, R., Balland, C, Barbary, K., Blanc, G., Challis, P. J., Conley,
A., Connolly, N. V., Covarrubias, R., Dawson, K. S., Deustua, S. E., Ellis,
R., Fabbro, S., Fadeyev, V., Fan, X., Farris, B., Folatelli, C, Frye, B. L.,
Garavini, G., Gates, E. L., Germany, L., Goldhaber, G., Goldman, B.,
Goobar, A., Groom, D. E., Haissinski, J., Hardin, D., Hook, L, Kent, S.,
Kim, A. G., Knop, R. A., Lidman, C, Linder, E. V., Mendez, J., Meyers, J.,
Miller, G. J., Moniez, M., Mourao, A. M., Newberg, H., Nobili, S., Nugent,
P. E., Pain, R., Perdereau, O., Perlmutter, S., Phillips, M. M., Prasad, V.,
Ouimby. R., Regnault, N., Rich. ,1.. Rubenstein, E. P., Ruiz-Lapuente, P.,
Santos, F. D., Schaefer, B. E., Schommer, R. A., Smith, R. C, Soderberg,
A. M., Spadafora, A. L., Strolger, L., Strovink, M., SuntzefT, N. B., Suzuki,
N., Thomas, R. C, Walton, N. A., Wang, L., Wood-Vasey, W. M. and
Yun, J. L. (2008). Improved cosmological constraints from new, old, and
combined supernova data sets. The Astro physical Journal 686, pp. 749
778, doi:10.1086/589937, arXiv:0804.4142.
[297] Kuchaf, K. V. (1981). Canonical methods of quantization, in C. J. Isham,
R. Penrose and D. W. Sciama (eds.), Quantum Gravity II (Clarendon Press,
Oxford), p. 329.
[298] Kuchaf, K. V. (1993). Canonical quantum gravity, in R. J. Gleiser, C. N.
Kozameh and O. M. Moreschi (eds.), General Relativity and Gravitation,
p. 119.
[299] Kuchaf, K. V. and Torre, C. G. (1991). Gaussian reference fluid and inter-
pretation of quantum geometrodynamics, Physical Review D 43, p. 419.
[300] Landau, L. D. and Lifshitz, E. M. (1972). Fluid mechanics, 2nd edn.
(Butterworth-Heinemann) .
[301] Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields, 4th
edn. (Addison- Wesley. New York).
[302] Landford III, O. (1971). Selected topics in functional analysis, in C. DeWitt
and R. Stora (eds.), Proceedings of Les Houches Summer School "Statis-
tical Mechanics and Quantum Field Theory" (Gordon and Breach Science
Publishers, London), pp. 109-214.
[303] Lange, A. E., Ade, P. A., Bock, J. J., Bond, J. R., Borrill, J., Boscaleri,
A., Coble, K., Crill, B. P., de Bernardis, P., Farese, P., Ferreira, P., Ganga,
K., Giacometti, M., Hivon, E., Hristov, V. V., Iacoangeli, A., Jaffe, A. H.,
Martinis, L., Masi, S., Mauskopf, P. D., Melchiorri, A., Montroy, T., Net-
terfield, C. B., Pascale, E., Piacentini, F., Pogosyan, D., Prunet, S., Rao, S.,
Romeo, G., Ruhl, J. E., Scaramuzzi, F. and Sforna, D. (2001). Cosmologi-
cal parameters from the first results of Boomerang, Physical Review D 63,
4, p. 042001, doi:10.1103/PhysRevD.63.042001, arXiv:astro-ph/0005004.
[304] Larson, D., Dunkley, J., Hinshaw, G., Komatsu, E., Nolta, M. R., Ben-
nett, C. L., Gold, B., Halpern, M., Hill, R. S., Jarosik, N., Kogut, A.,
Limon, M., Meyer, S. S., Odegard, N., Page, L., Smith, K. M., Spergel,
D. N., Tucker, G. S., Weiland, J. L., Wollack, E. and Wright, E. L. (2010).
Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations:
Power spectra and WMAP-derived parameters, arXiv: 1001. 4635.
[305] Lemaitre, G. (1927). Un Univers homogene de masse constante et de
rayon croissant rendant compte de la vitesse radiale des nebuleuses extra-
galactiquos. Annates tic la Societe Scietifique de Bruxelles 47, pp. 49-59.
[306] Lemaitre, G. (1931a). Contributions to a British association discussion
on the evolution of the universe, Nature 128, pp. 704-706, doi:10.1038/
128704a0.
[307] Lemaitre, G. (1931b). Expansion of the universe, A homogeneous universe
of constant mass and increasing radius accounting for the radial velocity of
extra-galactic nebula, Monthly Notices of the Royal Academy Society 91.
pp. 483-490.
[308] Lemos, N. A. (1996). Singularities in a scalar field quantum cosmology,
Physical Review D 53, pp. 4275-4279, doi:10.1103/PhysRevD.53.4275,
arXiv:gr-qc/9509038.
[309] Lewis, A., Challinor, A. and Lasenby, A. (2000). Efficient computa-
tion of Cosmic Microwave Background anisotropies in closed Friedmann-
Robertson- Walker models, The Astrophysical Journal 538, pp. 473-476,
doi:10.1086/309179, arXiv:astro-ph/9911177.
[310] Lifschytz, G., Mathur, S. D. and Ortiz, M. (1996). A Note on the semiclassi-
cal approximation in quantuu i ' < i D E pp. 76 7 i
doi:10.1103/PhysRevD. 53.766, arXiv:gr-qc/9412040.
[311] Lifshitz, E. (1946). On the gravitational stability of the expanding universe,
Journal of Physics 10, p. 116.
[312] Lifshitz, E. M. and Khalatnikov, I. M. (1963). Investigations in rela-
tivistic cosmology, Advances in Physics 12, pp. 185-249, doi:10.1080/
00018736300101283.
[313] Lifshitz, E. M. and Khalatnikov, I. M. (1970). Oscillatory approach to sin-
gular point in the open cosmological model, Soviet Journal of Experimental
and Theoretical PhysicsLetters 11, pp. 123-125.
[314] Lifshitz, E. M., Khalatnikov, I. M., Sinai, Y. G., Khanin, K. M. and Shchur,
L. N. (1983). On the stochastic properties of relativistic cosmological mod-
els near the singularity, Pis ma ' i niid noi i Teoreticheskoi
Fiziki 38, p. 79.
[315] Lifshitz, E. M., Lifshitz, I. M. and Khalatnikov, I. M. (1970). Asymptotic
analysis of oscillatory mode of approach to a singularity in homogeneous
cosmological models, Zhurnal Eksperimental noi i Teoreticheskoi Fiziki 59,
pp. 322-336.
[316] Lifshitz, E. M., Lifshitz, I. M. and Khalatnikov, I. M. (1971). Asymptotic
analysis of oscillatory mode of approach to a singularity in homogeneous
cosmological models, Soviet Physics JETP 32, p. 173.
[317] Lifshitz, E. M. and Pitaevskii, L. P. (1981). Physical Kinetics (Butterworth-
Heinemann).
[318] Lin, X. and Wald, R. M. (1989). Proof of the closed-universe-recollapse
i I ] in i n i lal Biauchi type-] m I ; '
Primordial Cosmology
40, pp. 3280-3286, doi:10.1103/PhysRevD.40.3280.
[319] Lin, X. and Wald, R. M. (1990). Proof of the closed-universe recollapse
conjecture for general Bianchi type-IX cosmologies, Physical Review D 41,
pp. 2444-2448, doi:10.1103/PhysRevD.41.2444.
[320] Linde, A. D. (1982). A new inflationary universe scenario: A possible
solution of the horizon, flatness, homogeneity, isotropy and primordial
monopole problems, Physics Letters B 108, pp. 389-393, doi:10.1016/
0370-2693(82)91219-9.
[321] Linde, A. D. (1983). Chaotic inflation, Physics Letters B 129, pp. 177-181,
doi: 10. 1016/0370- 2693(83)90837-7.
[322] Linde, A. D. (1984). The inflationary universe, Reports on Progress in
Physics 47, pp. 925-986, doi:10.1088/0034-4885/47/8/002.
[323] Linde, A. D. (1990). The Physics of Elementary Particles and Inflationary
Cosmology (CRC Press).
[324] Linde, A. D. (2005). Particle physics and inflationary cosmology, Contem-
porary Concepts in Physiscs 5, pp. 1 362 ai , Xiv:l]op-tl)/0'}03203.
[325] Longair, M. (2006). The Cosmic Century: A History of Astrophysics and
Cosmology (Cambridge Univ Press).
[326] Lucchin, F. and Matarrese, S. (1985). Power-law inflation, Phys. Rev. D
32, 6, pp. 1316-1322, doi:10.1103/PhysRevD.32.1316.
[327] Lyth, D. H. and Liddle, A. R. (2009). The Primordial Density Perturbation
(Cambridge University Press).
[328] Lyth, D. H. and Riotto, A. (1999). Particle physics models of inflation and
the cosmological density perturbation, Physics Reports 314, pp. 1-146,
doi:10.1016/S0370-1573(98)00128-8, arXiv:hep-ph/9807278.
[329] Ma, C. and Bertschinger, E. (1995). Cosmological perturbation theory in
the synchronous and c< informal new Ionian gauges, The Astrophysical Jour-
nal 455, p. 7, doi:10.1086/176550, arXiv:astro-ph/9506072.
[330] Ma, Z.-Q. (2007). Group Theory for Physicists (World Scientific).
[331] Maartens, R. (1995). Dissipative cosmology, Classical and Quantum Grav-
ity 12, pp. 1455-1465, doi:10.1088/0264-9381/12/6/011.
[332] MacCallum, M. A. H. (1971). A class of homogeneous cosmological models
I i I l i ij [ i lia\ iour. ( I / 20
pp. 57-84, doi:10.1007/BF01646733.
[333] Magueijo, J. and Smolin, L. (2002). Lorentz invariance with an invari-
ant energy scale, Physical Review Letters 88, 19, p. 190403, doi:10.1103/
PhysRevLett. 88. 190403, arXiv:hep-th/0112090.
[334] Makeenko, Y. (2002). Methods of Contemporary Gauge Theory (Cambridge
University Press).
[335] Mandl, F. and Shaw, G. (1994). Quantum Field Theory, Revised Edition
(Wiley).
[336] Martinez, S. E. and Ryan, M. P., Jr. (1983). An Exact Solution for a Quan-
tum Taub Model, in C. Aragone (ed.), Relativity, Cosmology, Topological
Mass and Supergravity; SILARG IV, p. 262.
[337] Mather, J. C, Cheng, E. S., Cottingham, D. A., Eplee, R. E., Jr., Fixsen,
D. J., Hewagama, T., Isaacman, R. B., Jensen, K. A., Meyer, S. S., No-
erdlinger, P. D., Road. S. M.. Rosen. L. P.. Shafer. R. A., Wright, E. L.,
Bennett, C. L., Boggess, N. W., Hauser, M. G., Kelsall, T., Moseley, S. H.,
Jr., Silverberg, R. F., Smoot, G. F., Weiss, R. and Wilkinson, D. T.
(1994). Measurement of the cosmic microwave background spectrum by
the COBE FIRAS instrument, The Astrophysical Journal 420, pp. 439-
444, doi:10.1086/173574.
[338] Mather, J. C, Fixsen, D. J., Shafer, R. A., Mosier, C. and Wilkinson,
D. T. (1999). Calibrator design for the COBE Far-Infrared Absolute Spec-
trophotometer (FIRAS), The Astrophysical Journal 512, pp. 511-520, doi:
10.1086/306805, arXiv:astro-ph/9810373.
[339] Mauskopf, P. D., Ade, P. A. R., de Bernardis, P., Bock, J. J., Borrill,
J., Boscaleri, A., Crill, B. P., DeGasperis, G., De Troia, G., Farese, P.,
Ferreira, P. G., Ganga, K., Giacometti, M., Hanany, S., Hristov, V. V.,
Iacoangeli, A., Jaffe, A. H., Lange, A. E., Lee, A. T., Masi, S., Melchiorri,
A., Melchiorri, F., Miglio, L., Montroy, T., Netterfield, C. B., Pascale,
E., Piacentini, F., Richards, P. L., Romeo, G., Ruhl, J. E., Scannapieco,
E., Scaramuzzi, F., Stompor, R. and Vittorio, N. (2000). Measurement of a
peak in the Cosmic Microwave Background power spectrum from the North
American test flight of Boomerang, The Astrophysical Journal Letters 536.
pp. L59-L62, doi:10.1086/312743, arXiv:astro-ph/9911444.
[340] Mercuri, S. (2006). Fermions in the Ashtekar-Barbero connection formalism
for arbitrary values of the Iimimzi parameti t Physical Review D 73, 8, p.
084016, doi:10.1103/PhysRevD. 73.084016.
[341] Mercuri, S. (2009). Introduction to Loop Quantum Gravity, Proceedings of
Science ISFTG, p. 016, arXiv:1001.1330.
[342] Mercuri, S. and Montani, G. (2004). Revised canonical quantum gravity
via the frame fixing, International Journal of Modern Physics D 13, pp.
165-186, arXiv.org/abs/gr-qc/0310077.
[343] Milne, E. A. (1935). Relativity. Gravitation and World- Structure (Claren-
don Press, Oxford).
[344] Misner, C. W. (1969). Absolute Zero of Time, Physical Review 186, pp.
1328-1333, doi:10.1103/PhysRev.l86.1328.
[345] Misner, C. W. (1969a). Mixmaster universe, Physical Review Letters 22,
pp. 1071-1074.
[346] Misner, C. W. (1969b). Quantum cosmology I, Physical Review 186, pp.
1319-1327.
[347] Misner, C. W., Thorne, K. S. and Wheeler, J. A. (1973). Gravitation (Free-
man, W. H. and C, San Francisco).
[348] Montani, G. (1995). On the general behavior of the universe near the cosmo-
logical singularity, Classical and Quantum Gravity 12, 10, pp. 2505-2517.
[349] Montani, G. (1999). On the quasi- isotropic solution in the presence of ul-
trarelativistic matter and a scalar field, Classical and Quantum Gravity 1 6.
pp. 723-730.
[350] Montani, G. (2000). Quasi- isotropic solution containing an electromagnetic
and a real in ' >\ u fi< Id. / / ai Qm > vavity 17, pp.
2197-2204.
576 Primordial Cosmology
[351] Montani, G. (2001). Influence of particle creation on flat and negative
curved FLRW universes, Classical and Quantum Gravity 18, pp. 193-203,
doi:10.1088/0264- 9381/18/1/311, arXiv:gr-qc/0101113.
[352] Montani, G. (2001). Nonstationary corrections to the Mixmaster model
invariant measure, Nuovo Cimento B 116, p. 1375.
[353] Montani, G. (2002). Canonical quantization of gravity without "frozen
formalism", Nuclear Physics B 634, pp. 370-392, arXiv.org/abs/gr-
qc/0205032.
[354] Montani, G., Battisti, M. V., Benini, R. and Imponente, G. (2008).
Classical and quantum features of the Mixmaster singularity, Interna-
tional Journal of Modern Physics A 23, pp. 2353-2503, doi:10.1142/
S0217751X08040275, arXiv:0712.3008.
[355] Montani, G. and C'iaiifrani. F. (2008). General relativity as classical limit
of evolutionary quantum gravity, Classical and Quantum Gravity 25, p.
065007, doi:10.1088/0264-9381/25/6/065007, arXiv:0802.0942.
[356] Motter, A. E. (2003). Relativistic chaos is coordinate invariant, Physical
Review Letters 91, p. 231101, arXiv.org/abs/gr-qc/0305020.
[357] Mukhanov, V. (2005). Physical Foundations of Cosmology (Cambridge Uni-
versity Press), doi:10.2277/0521563984.
[358] Mukhanov, V. F., Feldman, H. A. and Brandenberger, R. H. (1992). Theory
of cosmological perturbations, Physics Reports 215, pp. 203-333, doi:10.
1016/0370- 1573(92)90044-Z.
[359] Myung, Y. S., Kim, Y.-W., Son, W.-S. and Park, Y.-J. (2010). Mixmaster
universe in the ; — '■> deformod Llofaya-LiMiitz gravity, Journal of High En-
ergy Physics 03, p. 085, doi:10.1007/JHEP03(2010)085, arXiv:1001.3921.
[360] Narlikar, J. V. and Padmanabhan, T. (1983). Quantum cosmology via
path integrals, Physics Reports 100, 3, pp. 151 - 200, doi:DOI:10.1016/
0370-1573(83)90098-4.
[361] Netterfield, C. B., Ade, P. A. R., Bock, J. J., Bond, J. R., Borrill, J., Bosca-
leri, A., Coble, K., Contaldi, C. R., Crill, B. P., de Bernardis, P., Farese,
P., Ganga, K., Giacometti, M., Hivon, E., Hristov, V. V., Iacoangeli, A.,
Jaffe, A. H., Jones, W. C, Lange, A. E., Martinis, L., Masi, S., Mason, P.,
Mauskopf, P. D., Melchiorri, A., Montroy, T., Pascale, E., Piacentini, F.,
Pogosyan, D., Pongetti, F., Prunet, S., Romeo, G., Ruhl, J. E. and Scara-
muzzi, F. (2002). A measurement by BOOMERANG of multiple peaks in
the angular power spectrum of the Cosmic Microwave Background, The
Astrophysical Journal 571, pp. 604-614, doi:10.1086/340118, arXiv:astro-
ph/0104460.
[362] Newton, I. (1726). Philosophiae Naturalis Principia Mathematica (Lon-
dini).
[363] Newton, I. and Halley, E. (1695). The true theory of the tides, extracted'
from hat admired treatise of Mr. Isaac Newton, intituled, Philosophiae
Naturalis Principia Mathematica; being a discourse presented with that
book to the Late King James, by Mr. Edmund Halley, Royal Society of
London Philosophical Transactions Series I 19, pp. 445-457.
[364] Nicolai, H., Peeters, K. and Zamaklar, M. (2005). Loop quantum grav-
ity: An outside view, Classical and Quantum Gravity 22, p. R193,
arXiv.org/abs/hep-th/0501114.
[365] Nojiri, S. and Odintsov, S. D. (2003). Modified gravity with negative and
positive powers of curvature: Unification of inflation and cosmic accel-
eration, Physical Review D 68, 12, p. 123512, doi:10.1103/PhysRevD.68.
123512.
[366] Nojiri, S. and Odintsov, S. D. (2008). Dark energy, inflation and dark mat-
ter from modified f(r) gravity, in Problems of Modern Theoretical Physics,
A Volume in honour of Prof. I. L. Buchbinder in the Occasion of His 60th
Birthday (TSPU Publishing, Tomsk), pp. 266-285, arXiv:0807.0685.
[367] Opik, E. (1922). An estimate of the distance of the Andromeda Nebula,
The Astrophysical Journal 55, pp. 406-410, doi:10. 1086/142680.
[368] O'raifeartaigh, L. and Straumann, N. (2000). Gauge theory: Historical
origins and some modern developments, Reviews of Modern Physics 72,
pp. 1-23, doi:10.1103/RevModPhys.72.1.
[369] Ott, E. (1993). Chaos in Dynamical Systems (Cambridge University Press,
Cambridge).
[370] Padmanabhan, T. (1993). Structure Formation in the Universe (Cambridge
University Press).
[371] Padmanabhan, T. and Chitre, S. M. (1987). Viscous universes, Physics
Letters A 120, pp. 433-436, doi:10. 1016/0375-9601(87)90104-6.
[372] Page, D. N. (1997). Space for both no-boundary and tunneling quantum
states of the universe. Physical Review D 56, pp. 2065-2072, doi:10.1103/
PhysRevD.56.2065, arXiv:gr-qc/9704017.
[373] Palm, E. (1957). On the zeros of Bessel functions of pure imaginary order,
Quarterly Journal of Mechanics and Applied Mathematics 10, 4, p. 500.
[374] Peacock, J. A. (1999). Cosmological Physics (Cambridge University Press).
[375] Peacock, J. A., Cole, S., Norberg, P., Baugh, C. M., Bland-Hawthorn, J.,
Bridges, T., Cannon, R. D., Colless, M., Collins, O, Couch, W., Dalton,
C, Deeley, K., De Propris, R., Driver, S. P., Efstathiou, C, Ellis, R. S.,
Frenk, C. S., Glazebrook, K., Jackson, O, Lahav, O., Lewis, I., Lumsden,
S., Maddox, S., Percival, W. J., Peterson, B. A., Price, I., Sutherland, W.
and Taylor, K. (2001). A measurement of the cosmological mass density
from clustering in the 2dF Galaxy Redshift Survey, Nature 410, pp. 169-
173, arXiv:astro-ph/0103143.
[376] Peebles, P. J. E. (1968). Recombination of the primeval plasma, The As-
trophysical Journal 153, p. 1, doi:10. 1086/149628.
[377] Peebles, P. J. E. (1980). The Large-Scale Structure of the Universe (Prince-
ton University Press).
[378] Peebles, P. J. E. (1993). Principles of Physical Cosmology (Princeton Uni-
versity Press).
[379] Peebles, P. J. E. and Ratra, B. (2003). The cosmological constant and
dark energy, Reviews of Modern Physics 75, 2, pp. 559-606, doi:10.1103/
RevModPhys.75.559.
[380] Peebles, P. J. E. and Yu, J. T. (1970). Primeval adiabatic perturbation
in an expanding universe, The Astrophysical Journal 162, p. 815, doi:10.
578 Primordial Cosmology
1086/150713.
[381] Penzias, A. A. and Wilson, R. W. (1965). A measurement of excess antenna
temperature at 4080 Mc/s. The Astrophysical Journal 142, pp. 419-421,
doi:10. 1086/148307.
[382] Percival, W. J., Baugh, C. M., Bland-Hawthorn, J., Bridges, T., Cannon,
R., Cole, S., Colless, M., Collins, C, Couch, W., Dalton, C, De Propris,
R., Driver, S. P., Efstathiou, C, Ellis, R. S., Frenk, C. S., Glazebrook, K.,
Jackson, C, Lahav, O., Lewis, I., Lumsden, S., Maddox, S., Moody, S.,
Norberg, P., Peacock, J. A., Peterson, B. A., Sutherland, W. and Taylor,
K. (2001). The 2dF Galaxy Redshift Survey: the power spectrum and the
matter ( -out cut of t lie FirivorM'. Monthly Not lot -■■ of tin Royal . [c.ailt my So: i-
ety 327, pp. 1297-1306, doi:10.1046/j. 1365-8711. 2001.04827.x, arXiv:astro-
ph/0105252.
[383] Percival, W. J., Sutherland, W., Peacock, J. A., Baugh, C. M., Bland-
Hawthorn, J., Bridges, T., Cannon, R., Cole, S., Colless, M., Collins, C,
Couch, W., Dalton, C, De Propris, R., Driver, S. P., Efstathiou, C, Ellis,
R. S., Frenk, C. S., Glazebrook, K., Jackson, C, Lahav, O., Lewis, I.,
Lumsden, S., Maddox, S., Moody, S., Norberg, P., Peterson, B. A. and
Taylor, K. (2002). Parameter constraints for flat cosmologies from cosmic
microwave background and 2dFGRS power spectra, Monthly Notices of the
Royal Academy Society 337, pp. 1068-1080, doi:10.1046/j. 1365-8711.2002.
06001.x, arXiv:astro-ph/0206256.
[384] Perez, A. (2004). Introduction to Loop Quantum Gravity and Spin Foams,
arXiv:gr-qc/0409061.
[385] Perez, A. and Rovelli, C. (2006). Physical effects of the Immirzi parameter
in Loop Quantum Gravity, Physical Review D 73, 4, p. 044013, doi:10.
1103/PhysRevD. 73. 044013, arXiv:gr-qc/0505081.
[386] Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A., Nugent, P., Cas-
tro, P. G., Deustua, S., Fabbro, S., Goobar, A., Groom, D. E., Hook, I. M.,
Kim, A. G., Kim, M. Y., Lee, J. C, Nunes, N. J., Pain, R., Pennypacker,
C. R., Quimby, R., Lidman, C, Ellis, R. S., Irwin, M., McMahon, R. G.,
Ruiz-Lapuente, P., Walton, N., Schaefer, B., Boyle, B. J., Filippenko, A. V.,
Matheson, T., Fruchter, A. S., Panagia, N., Newberg, H. J. M., Couch,
W. J. and The Supernova Cosmology Project (1999). Measurements of u>
an from 12 high lsh.il n ova ? I ' ' / 517.
pp. 565-586, doi:10.1086/307221, arXiv:astro-ph/9812133.
[387] Pinto-Neto, N. (2005). The Bohm interpretation of quantum cosmology,
Foundations of Physics 35, pp. 577-603, doi:10.1007/sl0701-004- 2012-8,
arXiv:gr-qc/0410117.
[388] Pokorski, S. (2000). Gauge Field Theories (Cambridge University Press).
[389] Puzio, R. (1994). On the square root of the Laplace-Beltrami operator as
a Hamiltonian, Classical and Quantum Gravity 11, 3, pp. 609-620.
[390] Riemann, B. (1868). Ueber die Hypothesen, welche der Geometrie zu
Grunde liegen. Ablwndltiiitjcii dcr Kit it it /lichen Gesellschaft der Wis-
sfiiscltaftt it iii GiUtiitt/cn .13. pp. 133-152.
[391] Riemann, B. (1873). On the hypotheses which lie at the bases of geometry,
Nature 8, pp. 14-17, 36-37.
[392] Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A., Diercks, A.,
Garnavich, P. M., Gilliland, R. L., Hogan, C. J., Jha, S., Kirshner, R. P.,
Leibundgut, B., Phillips, M. M., Reiss, D., Schmidt, B. P., Schommer,
R. A., Smith, R. C, Spyromilio, J., Stubbs, C, Suntzeff, N. B. and Tonry,
J. (1998). Observational evidence from supernovae for an accelerating uni-
verse and a cosmological constant, The Astronomical Journal 116, pp.
1009-1038, doi:10.1086/300499, arXiv:astro-ph/9805201.
[393] Ringstrom, H. (2000). Curvature blow up in Bianchi VIII and IX vacuum
i i i I 7 | I I i : 10. 1088/
0264-9381/17/4/301, arXiv:gr-qc/9911115.
[394] Ringstrom, H. (2001). The Bianchi IX attractor, Annates Henri Poincare
2, pp. 405-500, doi:10.1007/PL00001041, arXiv:gr-qc/0006035.
[395] Robertson, H. P. (1935). Kinematics and world-structure, The Astrophysi-
cal Journal 82, p. 284, doi:10. 1086/143681.
[396] Rovelli, C. (1991). Ashtekar formulation of general relativity and loop-
] mod] in I i i qua) i i i I in i ' i
Gravity 8, pp. 1613-1676, doi:10.1088/0264-9381/8/9/002.
[397] Rovelli, C. (1991). Time in quantum gravity: A hypothesis, Physical Review
D 43, p. 442.
[398] Rovelli, C. (2004). Quantum Gravity (Cambridge University Press).
[399] Rovelli, C. (2009). "Forget time", arXiv:0903.3832.
[400] Rovelli, C. and Smolin, L. (1990). Loop Space Representation of Quan-
tum General Relativity, Nuclear Physics B 331, p. 80, doi:10.1016/
0550-3213(90)90019-A.
[401] Rovelli, C. and Smolin, L. (1995). Discreteness of area and volume in
quantum gravity, Nuclear Physics B 442, pp. 593-619, doi:10.1016/
0550-3213(95)00150-Q, arXiv:gr-qc/9411005.
[402] Rovelli, C. and Vidotto, F. (2008). Stepping out of homogeneity in loop
quantum cosmology, Classical and Quantum Gravity 25, 22, p. 225024,
doi:10.1088/0264- 9381/25/22/225024, arXiv:0805.4585.
[403] Rubin, V. C. and Ford, W. K., Jr. (1970). Rotation of the Andromeda
Nebula from a spectroscopic survey of emission regions, The Astrophysical
Journal 159, p. 379, doi:10. 1086/150317.
[404] Rubin, V. C, Ford, W. K. J. and . Thonnard, N. (1980). Rotational prop-
erties of 21 SC galaxies with a large range of luminosities and radii, from
NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc/, The Astrophysical
Journal 238, pp. 471-487, doi:10.1086/158003.
[405] Rugh, S. E. (1994). Chaos in the Einstein equations-characterization and
importance? in D. Hobill, A. Burd and A. Coley (eds.), Deterministic
Chans in General Relativity (World Scientific, Singapore), p. 68.
[406] Ryan, M. P. and Shepley, L. C. (1975). Homogeneous Relativistic Cosmolo-
gies (Princeton University Press).
[407] Samuel, J. (2000a). Canonical gravity, diffeomorphisms and objective his-
tories, Classical and Quantum Gravity 17, pp. 4645 4654 doi:10.1088/
0264-9381/17/22/305, arXiv:gr-qc/0005094.
580 Primordial Cosmology
[408] Samuel, J. (2000b). Is Barbero's Hamiltonian formulation a gauge theory
of Lorentzian gravity? Classical and Quantum Gravity 17, pp. L141-L148,
doi: 10. 1088/0264- 9381/17/20/101, arXiv:gr-qc/0005095.
[409] Sanchez, A. G., Baugh, C. M., Percival, W. J., Peacock, J. A., Padilla,
N. D., Cole, S., Frenk, C. S. and Norberg, P. (2006). Cosmological pa-
rameters from cosmic microwave background measurements and the final
2dF Galaxy Redshift Survey power spectrum. Monthly Notices of the Royal
Academy Society 366, pp. 189-207, doi:10.1111/j. 1365- 2966.2005.09833.x,
arXiv:astro-ph/0507583.
[410] Schneider, J. and Neufeld, Z. (2002). Dynamics of leaking Hamiltonian
systems, Physical Review E 66, p. 066218.
[411] Seiberg, N. and Witten, E. (1999). String theory and noncommutative ge-
ometry, Journal of High Energy Physics 9, p. 32, doi:10. 1088/1126-6708/
1999/09/032, arXiv:hep-th/9908142.
[412] Seljak, IT., Makarov, A., Mandelbaum, R., Hirata, C. M., Padmanab-
han, N., McDonald, P., Blanton, M. R., Tegmark, M., Bahcall, N. A.
and Brinkmann, J. (2005a). SDSS galaxy bias from halo mass-bias rela-
tion and its cosmological implications, Physical Review D 71, 4, p. 043511,
doi:10.1103/PhysRevD.71.043511, arXiv:astro-ph/0406594.
[413] Seljak, IL, Makarov, A., McDonald, P., Anderson, S. F., Bahcall, N. A.,
Brinkmann, J., Buries, S., Cen, R., Doi, M., Gunn, J. E., Ivezic, Z., Kent,
S., Loveday, J., Lupton, R. H., Munn, J. A., Nichol, R. C, Ostriker, J. P.,
Schlegel, D. J., Schneider, D. P., Tegmark, M., Berk, D. E., Weinberg,
D. H. and York, D. G. (2005b). Cosmological parameter analysis including
SDSS Lya forest and galaxy bias: Constraints on the primordial spectrum
of fluctuations, neutrino mass, and dark energy, Physical Review D 71, 10,
p. 103515, doi:10.1103/PhysRevD. 71. 103515, arXiv:astro-ph/0407372.
[414] Seljak, U. and Zaldarriaga, M. (1996). A line-of-sight integration approach
to Cosmic Microwave Background Anisotropies, The Astrophysical Journal
469, p. 437, doi:10.1086/177793, arXiv:astro-ph/9603033.
[415] Senovilla, J. M. M. (2006). Singularity theorems in general relativity:
Achievements and open questions, arXiv:physics/0605007.
[416] Shestakova, T. P. and Simeone, C. (2004a). The problem of time and gauge
in vari alio .. I. Canonical quantization met lux Is Gravitation and < 'oxmologt/
10, pp. 161-176, arXiv:gr-qc/0409114.
[417] Shestakova, T. P. and Simeone, C. (2004b). The problem of time and gauge
invariance. II. Recent developments in the path integral approach, Gravi-
tation and Cosmology 10, pp. 257-268, arXiv:gr-qc/0409119.
[418] Silk, J. (1968). Cosmic black-body radiation and galaxy formation, The
Astrophysical Journal 151, p. 459, doi:10. 1086/149449.
[419] Silk, J. and Turner, M. S. (1987). Double inflation, Physical Review D 35,
pp. 419-428, doi:10.1103/PhysRevD.35.419.
[420] Slipher, V. M. (1915). Spectrographic observations of Nebulae, Popular
Astronomy 23, pp. 21-24.
[421] Smirnov, V. I. (ed.) (1992). Lyapunov, A. M. (Taylor & Francis).
[422] Smolin, L. (2004). An invitation to Loop Quantum Gravity, arXiv:hep-
th/0408048.
[423] Smoot, G. F., Bennett, C. L., Kogut, A., Wright, E. L., Aymon, J.,
Boggess, N. W., Cheng, E. S., de Amici, G., Gulkis, S., Hauser, M. G.,
Hinshaw, G., Jackson, P. D., Janssen, M., Kaita, E., Kelsall, T., Keegstra,
P., Lineweaver, C, Loewenstein, K., Lubin, P., Mather, J., Meyer, S. S.,
Moseley, S. H., Murdock, T., Rokke, L., Silverberg, R. F., Tenorio, L.,
Weiss, R. and Wilkinson, D. T. (1992). Structure in the COBE differential
microwave radiometer first-year maps. The Astroplit/siral Journal Letters
396, pp. L1-L5, doi:10.1086/186504.
[424] Snyder, H. S. (1947). Quantized space-time, Physical Review 71, 1, pp.
38-41, doi:10.1103/PhysRev.71.38.
[425] Sotiriou, T. P. and Faraoni, V. (2010). f{R) theories of gravity, Reviews of
Modern Physics 82, pp. 451-497, doi:10.1103/RevModPhys.82.451.
[426] Starobinsky, A. A. (1980). A new type of isotropic cosmological mod-
els without singularity, Physics Letters B 91, pp. 99-102, doi:10.1016/
0370-2693(80)90670-X.
[427] Stephani, H., Kramer, D., MacCallum, M. A. H., Hoenselaers, C. and
Herlt, E. (2003). Exact Solutions of Einstein's Field Equations (Cambridge
University Press, Cambridge, UK).
[428] Szabados, L. B. (2009). Quasi-local energy- momentum and angular mo-
mentum in general relativity, Living Reviews in Relativity 12, p. 4.
[429] Szydlowski, M. and Krawiec, A. (1993). Average rate of separation of tra-
jectories near the singularity in mixmaster models, Physical Review D 47,
12, p. 5323.
[430] Szydlowski, M. and Lapeta, A. (1990). Pseudo-riemannian manifold of mix-
master dynamical systems, Physics Letters A 148, 5, p. 239.
[431] Taub, A. H. (1951). Empty space-times admitting a three parameter group
of motions. Tl \ f Math i '< ! pp. 472-490.
[432] Taveras, V. (2008). Corrections to the Friedmann equations from loop quan-
tum gravity for a universe with a free scalar field, Physical Review D 78,
6, p. 064072, doi:10.1103/PhysRevD. 78.064072, arXiv:0807.3325.
[433] Tegmark, M., Eisenstein, D. J., Strauss, M. A., Weinberg, D. H., Blan-
ton, M. R., Frieman, J. A., Fukugita, M., Gunn, J. E., Hamilton, A. J. S.,
Knapp, G. R., Nichol, R. C, Ostriker, J. P., Padmanabhan, N., Percival,
W. J., Schlegel, D. J., Schneider, D. P., Scoccimarro, R., Seljak, U., Seo,
H., Swanson, M., Szalay, A. S., Vogeley, M. S., Yoo, J., Zehavi, L, Abaza-
jian, K., Anderson, S. F., Annis, J., Bahcall, N. A., Bassett, B., Berlind,
A., Brinkmann, J., Budavari, T., Castander, F., Connolly, A., Csabai, I.,
Doi, M., Finkbeiner, D. P., Gillespie, B., Glazebrook, K., Hennessy, G. S.,
Hogg, D. W., Ivezic, Z., Jain, B., Johnston, D., Kent, S., Lamb, D. Q., Lee,
B. C, Lin, H., Loveday, J., Lupton, R. H., Munn, J. A., Pan, K., Park, C,
Peoples, J., Pier, J. R., Pope, A., Richmond, M., Rockosi, C, Scranton,
R., Sheth, R. K., Stebbins, A., Stoughton, C, Szapudi, I., Tucker, D. L.,
vanden Berk, D. E., Yanny, B. and York, D. G. (2006). Cosmological con-
straints from the SDSS luminous red galaxies, Physical Review D 74, 12,
p. 123507, doi:10.1103/PhysRevD.74.123507, arXiv:astro-ph/0608632.
582 Primordial Cosmology
[434] Tegmark, M., Strauss, M. A., Blanton, M. R., Abazajian, K., Dodelson, S.,
Sandvik, H., Wang, X., Weinberg, D. H., Zehavi, I., Bahcall, N. A., Hoyle,
F., Schlegel, D., Scoccimarro, R., Vogeley, M. S., Berlind, A., Budavari, T.,
Connolly, A., Eisenstein, D. J., Finkbeiner, D., Frieman, J. A., Gunn, J. E.,
Hui, L., Jain, B., Johnston, D., Kent, S., Lin, H., Nakajima, R., Nichol,
R. C, Ostriker, J. P., Pope, A., Scranton, R., Seljak, U., Sheth, R. K.,
Stebbins, A., Szalay, A. S., Szapudi, I., Xu, Y., Annis, J., Brinkmann, J.,
Buries, S., Castander, F. J., Csabai, L, Loveday. J.. Doi. M.. Fukugita. M..
Gillespie, B., Hennessy, G., Hogg, D. W., Ivezic, Z., Knapp, G. R., Lamb,
D. Q., Lee, B. C, Lupton, R. H., McKay, T. A., Kunszt, P., Munn, J. A.,
O'Connell, L., Peoples, J., Pier, J. R., Richmond, M., Rockosi, C, Schnei-
der, D. P., Stoughton, C, Tucker, D. L., vanden Berk, D. E., Yanny, B.
and York, D. G. (2004). Cosmological parameters from SDSS and WMAP,
Physical Review D 69, 10, p. 103501, doi:10.1103/PhysRevD.69. 103501,
arXiv:astro-ph/0310723.
[435] Terras, A. (1985). Harmonic Analysis on Symmetric Spaces and Applica-
tions I (Springer- Verlag, New York).
[436] Thiemann, T. (1996). Anomaly-free formulation of non-perturbative, four-
dimensional Lorentzian quantum gravity, Physics Letters B 380, pp. 257-
264, doi:10.1016/0370- 2693(96)00532-1, arXiv:gr-qc/9606088.
[437] Thiemann, T. (2007a). Loop quantum gravity: An inside view, Lecture
Notes in Physics 721, p. 185, arXiv.org/abs/hep-th/0608210.
[438] Thiemann, T. (2007b). Modern Canonical Quantum General Relativity
(Cambridge University Press).
[439] Tomita, K. and Deruelle, N. (1994). Nonlinear behaviors of cosmological
inhomogeneities with a standard fluid and inflationary matter, Physical
Review D 50, 12, p. 7216.
[440] Torre, C. G. (1992). Is general relativity an "already parametrized" theory?
Physical Review D 46, 8, pp. R3231-R3234, doi:10.1103/PhysRevD.46.
R3231.
[441] Uggla, C, van Elst, H., Wainwright, J. and Ellis, G. F. R. (2003). The past
attractor in inhomogeneous cosmology, Physical Review D 68, p. 103502,
arXiv.org/abs/gr-qc/0304002 .
[442] Unruh, W. G. (1988). Time and Quantum Gravity, in M. A. Markov, V. A.
Berezin and V. P. Frolov (eds.), Quantum Gravity IV, p. 252.
[443] Unruh, W. G. and Wald, R. M. (1989). Time and the interpretation of
canonical quantum gravity, Physical Review D 40, pp. 2598-2614, doi:
10.1103/PhysRevD.40.2598.
[444] Vakili, B. (2008). Dilaton Cosmology, Noncommutativity and General-
ized Uncertainty Principle, Physical Review D 77, p. 044023, doi:10.1103/
PhysRevD. 77.044023, arXiv:0801.2438.
[445] Vakili, B., Khosravi, N. and Sepangi, H. R. (2007). Bianchi spacetimes
in noncommutative phase space, Classical and Quantum Gravity 24, pp.
931-949, doi:10.1088/0264-9381/24/4/013, arXiv.org/abs/gr-qc/0701075.
[446] Vakili, B. and Sepangi, H. R. (2007). Generalized uncertainty principle
in Bianchi type I quantum cosmology, Physics Letters B 651, pp. 79-83,
doi:10.1016/j.physletb.2007.06.015, arXiv.org/abs/0706.0273.
[447] Veneziano, G. (1986). A stringy nature needs just two constants, Euro-
physics Letters 2, 3, p. 199.
[448] Verde, L., Heavens, A. F., Percival, W. J., Matarrese, S., Baugh, C. M.,
Bland-Hawthorn, J., Bridges, T., Cannon, R., Cole, S., Colless, M., Collins,
C, Couch, W., Dalton, C, De Propris, R., Driver, S. R, Efstathiou,
C, Ellis, R. S., Frenk, C. S., Glazebrook, K., Jackson, C, Lahav, O.,
Lewis, I., Lumsdon. S., Maddox. S.. Madgwick, D., Norberg, P., Peacock,
J. A., Peterson, B. A., Sutherland, W. and Taylor, K. (2002). The 2dF
Galaxy Redshift Survey: the bias of galaxies and the density of the Uni-
verse, Monthly Notices of the Royal Academy Society 335. pp. 432 lit).
doi:10.1046/j. 1365-8711. 2002.05620.x, arXiv:astro-ph/0112161.
[449] Vilenkin, A. (1982). Creation of universes from nothing, Physics Letters B
117, pp. 25-28, doi:10.1016/0370-2693(82)90866-8.
[450] Vilenkin, A. (1988). Quantum cosmology and the initial state of the uni-
verse, Physical Review D 37, 4, pp. 888-897.
[451] Vilenkin, A. (1989). Interpretation of the wave function of the universe,
Physical Review D 39, 4, pp. 1116-1122.
[452] Vilenkin, A. (1997). Predictions from quantum cosmology, in N. Sanchez
and A. Zichichi (eds.), Proceedings of the International School of Astro-
physics "String (rrarity and Physics at the Plum', Energy Scale". Erice
1995, pp. 345-367, arXiv:gr-qc/9507018.
[453] Vittorio, N. and Silk, J. (1984). Fine-scale anisotropy of the cosmic mi-
crowave background in a universe dominated by cold dark matter, The
Astrophysical Journal Letters 285, pp. L39-L43, doi:10. 1086/184361.
[454] Wainwright, J. and Ellis, G. F. R. (1997). Dynamical Systems in Cosmology
(Cambridge University Press).
[455] Wainwright, J. and Hsu, L. (1989). A dynamical systems approach to
Biui 1 i ii I < > i 1 i 1 I i /
Gravity 6, pp. 1409-1431, doi:10.1088/0264-9381/6/10/011.
[456] Wald, R. M. (1984). General Relativity (University of Chicago Press,
Chicago) .
[457] Wald, R M 1.99 i). On ' inn 1 hi Tlicory in i i time and Black
Hole Thermodynamics (University of Chicago Press, Chicago, IL).
[458] Walker, A. G. (1937). On Milne's theory of world-structure, Proceedings of
the London Mathematical Society 42, 2, pp. 90-127.
[459] Weaver, M., Isenberg, J. and Berger, B. K. (1998). Mixmaster behavior in
n i ii i ii ii ! ical spacetimes, Physical Review Letters 80, pp.
2984-2987, arXiv.org/abs/gr-qc/9712055.
[460] Weiland, J. L., Odegard, N., Hill, R. S., Wollack, E., Hinshaw, G., Greason,
M. R., Jarosik, N., Page, L., Bennett, C. L., Dunkley, J., Gold, B., Halpern,
M., Kogut, A., Komatsu, E., Larson, D., Limon, M., Meyer, S. S., Nolta,
M. R., Smith, K. M., Spergel, D. N., Tucker, G. S. and Wright, E. L. (2010).
Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations:
Planets and celestial calibration sources, arXiv:1001.4731.
[461] Weinberg, S. (1971). Entropy generation and the survival of protogalaxies
Primordial Cosmology
in an expanding universe. The Astrophysical Journal 168, p. 175, doi:10.
1086/151073.
[462] Weinberg. S. (1072). Gnicitntion end Cosinoloi/ij: Principles and Applica-
tions of the General Theory of Relativity (Wiley).
[463] Weinberg, S. (2005). The Quantum Theory of Fields: Modem Applications,
Vol. 2 (Cambridge University Press).
[464] Weinberg, S. (2008). Cosmology (Oxford University Press).
[465] Wheeler, J. A. (1064). Geometrodynamics and the Issue of the Final State,
in C. DeWitt and B. DeWitt (eds.), Relativity, Groups and Topology (Gor-
don and Breach, New York), pp. 316-520.
[466] Wilson, K. G. (1074). Confinement of quarks, Physical Review D 10, 8, pp.
2445-2459, doi:10.1103/PhysRevD. 10.2445.
[467] Wiltshire, D. L. (1906). An introduction to quantum cosmology, in B. Rob-
son, N. Visvanathan and W. Woolcock (eds.), Cosmology: The Physics of
iin Universe. Proceedings of Hit Xih I'lii/sits Summer School (World Scien-
tific, Singapore), pp. 473-531.
[468] Wojtkowski, M. (1985). Invariant families of cones and Lyapunov expo-
nents, Ergodic Theory and Dynamical Systems 5, 1, pp. 145-161.
[469] Wright, E. L., Meyer, S. S., Bennett, C. L., Boggess, N. W., Cheng,
E. S., Hauser, M. G., Kogut, A., Lineweaver, C, Mather, J. C, Smoot,
G. F., Weiss, R., Gulkis, S., Hinshaw, G., Janssen, M., Kelsall, T., Lu-
bin, P. M., Moseley, S. H., Jr., Murdock, T. L., Shafer, R. A., Silverberg,
R. F. and Wilkinson, D. T. (1002). Interpretation of the Cosmic Microwave
Background Radiation anisotropy detected by the COBE differential mi-
crowave radiometer, The Astrophysical Journal Letters 396, pp. L13-L18,
doi:10.1086/186506.
[470] York, J. W. (1072). Role of conformal three-geometry in the dynamics of
gravitation, Physical Review Letters 28, 16, pp. 1082-1085, doi:10.1103/
PhysRevLett.28.1082.
[471] Zwicky, F. (1033). Die Rotverschiebung von extragalaktischen Nebeln, Hel-
vetica Physica Acta 6, pp. 110-127.
[472] Zwicky, F. (1937). On the masses of Nebulae and of clusters of Nebulae,
The Astrophysical Journal 86, p. 217, doi:10.1
Index
p-form, 390, 391
1-form, 59, 74, 75, 80, 81, 279, 280,
285, 320, 323, 329, 363, 364, 390,
453, 522, 530, 533
2-form, 59, 75, 78, 82, 481, 530, 531
2dF, 166, 192
Absolute magnitude, 176
Acceleration of the Universe, 40, 49,
113, 168, 175, 192, 233-239
Acoustic horizon, 186
Acoustic oscillations, 131, 166, 183,
184, 192
Adiabatic perturbations, 147,
149-151, 184, 188, 189, 229, 230
ADM reduction, 64, 71, 334, 339,
344, 348, 361, 413, 454, 550
Anisotropic stress, 59
Anisotropies, 241, 327, 330, 331, 333,
335, 340, 341, 357, 358, 380, 381,
404, 417, 437-441, 453, 454, 533,
543, 545-548, 550
Area operator, 505, 506, 517, 520, 550
Asymptotic limit, 124, 245, 266,
269-271, 322, 381, 382, 548
Attractor, 308, 317, 319, 320, 326
Auxiliary field, 238
Baryon asymmetry, 221
Baryon-to-photon ratio, 178
Bessel functions, 135, 464, 466-468
Bianchi I, 42, 279, 312, 313, 338, 451,
459, 547, 548
Bianchi identity, 53, 62, 245
Bianchi II, 315, 317, 338, 548-550
Bianchi IX, 42, 287, 298, 304, 317,
318, 320, 329, 338, 354, 361-364,
366, 398, 430, 453, 457, 470, 500,
511, 521-523, 528, 530-535, 546,
548-550
Bianchi VII, 279
Bianchi VIII, 304, 380
Big Bang, 34-37, 43, 45, 48, 85, 95,
106, 111, 113-115, 117, 123, 124,
166, 176, 177, 193-195, 205, 206,
242, 356, 357, 378, 421, 422, 426,
427, 448, 449, 451, 499, 511, 514,
516, 518, 519, 521
Big Bounce, 44-46, 85, 404, 427, 451,
491, 499, 511, 516, 518, 519, 521,
522, 538, 539
Big Crunch, 114, 448
Billiard, 327, 331, 338, 339, 341, 343,
344, 346, 360, 380, 386-388, 390,
391, 452, 465
BKL, 41, 42, 298, 316, 322, 323, 337,
344, 347-350, 355, 356, 360, 362,
367, 368, 375, 386, 388, 391, 392,
398, 399, 422
BKL map, 296-298, 300, 317, 326,
338, 349, 351, 361, 362, 372, 373,
375, 389, 391, 392, 397, 399,
548-550
BKL mechanism, 324, 373, 374, 381
Primordial Cosmology
Black body, 37, 165, 195
Boltzmann equation, 95, 96, 106-108,
127, 129, 142, 155-157, 159, 162,
186, 192, 199
BOOMERanG, 38, 49, 166, 187
Bose condensate, 227
Boundary conditions, 69, 70, 404,
409, 411, 419, 426, 428, 429,
441-443, 451, 455, 456, 459, 462,
464, 465, 470, 492, 542-544
Bubbles, 208, 209
Bulk viscosity, 96, 122-124, 126, 162,
241, 265, 266, 268, 272, 274, 275
CAMB, 187, 192
Canonical formulation, 64, 68, 78, 92,
384, 412
Cartan structure equation, first, 75
Cartan structure equation, second,
75, 77, 81
Causal horizon, 184, 195, 222, 223,
229, 267, 362
Causal regions, 105, 128, 195, 196,
208, 267, 375
Chaos, 5, 7, 43, 319, 320, 323,
325-328, 346, 348-351, 353,
359-361, 390, 399, 470, 550
Christoffel symbols, 52, 62, 74, 137,
138, 347, 366, 523, 525, 531
Clock, 23, 73, 413, 416, 417, 419, 421,
430-432, 450, 451, 469, 538
Co-Moving, 54, 55, 98, 100, 101, 104,
105, 109, 111, 112, 123, 125, 129,
147, 155-157, 229, 230, 242, 375,
414-416
COBE, 37, 38, 48, 177
Coherent oscillations, 187, 218, 220,
226, 227
Cold dark matter, 142, 147, 171, 172,
188, 190
Collision term, 107, 108, 157, 159
Commutators, 309, 405, 489, 491-494,
497, 500, 502, 508, 536, 537, 539
Conformal transformation, 237, 308
Conjugate momentum, 62, 66, 67, 71,
154, 155, 158, 334, 355, 357, 376,
379, 387, 414, 422, 423, 449, 537,
546
Continuity equation, 59, 109, 113,
121, 123, 124, 126, 127, 130, 237,
435, 438, 460, 461
Cosmic microwave background,
35-40, 43, 44, 47-49, 128, 165, 166,
168, 170, 176-183, 187-192, 195,
196, 221, 232, 357
Cosmic Microwave Background
Anisotropics, 37, 38, 49, 165, 166,
179, 180, 185-192
Cosmic scale factor, 98, 99, 104, 110,
126, 218, 221, 237, 258, 293, 298
Cosmic variance, 182
Cosmological constant, 31, 32, 40, 54,
110, 118, 119, 160, 161, 167, 176,
188-190, 210, 213, 214, 234, 235,
237, 241, 246, 257, 258, 328, 360,
438
Cosmological constant problem, 234,
235
Cosmological parameters, 38, 166,
187, 190
Cut-Off, 45, 46, 494, 522, 542, 547,
550
Damped oscillations, 215
Damping, 136, 142, 171, 172, 185,
186, 189, 191, 192, 206, 214,
216-218
Dark energy, 40, 165, 167, 168, 176,
177, 180, 188, 194, 233, 234,
237-239
Dark matter, 39, 40, 49, 151, 165,
167, 168, 170-172, 174, 188, 192
de Sitter solution, 32-34, 96, 118-119
Decay time, 219, 227
Deceleration parameter, 103, 113, 175
Decoupling, 91, 106, 168, 170, 171,
186, 227, 356
Deformed algebra, 500, 540
Density fluctuations, 44, 127, 129,
135, 141, 168, 171-173, 179, 196,
197, 229, 231
Diagonal, 112, 287, 289, 293, 310,
319, 320, 323-327, 331, 332, 359,
365, 369-371, 383, 522, 534, 535
Diffeomorphisms, 64, 65, 68-70, 76,
82, 83, 282, 348, 403, 405, 408, 411,
412, 422, 424, 427, 501, 503,
505-507, 509, 510, 512, 525
Dirac algebra, 68-70, 510, 511
Discrete spectrum, 45, 484
Effective number of degrees of
freedom, 111
Energy-Momentum tensor, 53-56, 58,
59, 63, 76, 92, 110, 123, 129, 136,
139, 142, 146, 152, 154, 157, 158,
160, 170, 234, 244, 268-270, 292,
312, 372, 415, 416, 480
Entropy, 26, 45, 111, 112, 125, 150,
195, 197, 198, 220, 221, 224, 225,
267, 303, 349
Equation of state, 54-57, 110, 113,
116, 119, 123, 126, 147, 154, 170,
205, 206, 233-235, 237, 242,
256-258, 265, 268, 274, 291, 312,
314, 315, 416
Evolutionary quantum gravity, 415
Expansion of the Universe, 30-37, 40,
44, 47, 99-107, 109, 113, 114, 117,
122, 124, 125, 129, 130, 133, 134,
175, 215, 219, 222, 233, 235, 333,
363, 438, 453
Expansion rate, 101, 104, 105, 123,
198, 199, 267
Expansion scalar, 87, 90, 91, 311
Expansion tensor, 87
Expansion, inflationary, 193, 205-211,
213-215, 221, 229, 239, 263, 422
Extrinsic curvature, 65, 66, 74, 81,
264, 411, 443, 508, 523
False vacuum, 44, 207-212, 228
Fell theorem, 475, 478-481, 498
FIRAS, 37, 48, 177, 178
Flatness paradox, 195, 196, 213, 221,
223, 257
Fractal, 328, 350-353, 360
Free streaming, 171
Friedmann equation, 96, 112, 113,
115, 123, 126, 135, 176, 197, 199,
217, 237, 520, 537, 538
Friedmann Universe, 40, 104, 105,
116, 195, 197
Galaxy, 7, 20, 30-34, 37, 39, 47-49,
95, 96, 100-102, 109, 128, 165-175,
179, 181, 191, 192, 229, 232
Galaxy clusters, 168
Galaxy recession, 100
Galaxy surveys, 166, 192
Gauge invariance, 506
Gauge mode, 137, 144, 149
Gauge tensor, 61, 204
Gauss constraint, 62, 63, 82, 84, 312,
315, 355, 428, 501, 504, 506,
529-531, 534
Generalized uncertainty principle,
495, 540
Geodesic, 41, 52, 53, 55, 65, 73,
86-91, 99, 101, 106, 155, 156, 331,
342-348, 350, 359, 380, 382, 387,
416, 425, 452, 464, 490
Geodesic deviation, 52, 89, 347, 350
Geometrodynamics, 28, 29, 54, 92,
96, 206, 208, 236, 238, 405, 469,
499, 501, 502, 530
GNS construction, 476, 478, 479, 482,
498
Goldstone boson, 201, 203, 204
Good reheating, 220, 227
Graph, 4, 15, 39, 485, 486, 500,
503-505, 507, 509, 530, 535
Gravitational instability. 37, 128, 129,
136, 165, 167, 169, 170, 229
Hamilton- Jacobi, 70, 92, 357, 358,
377, 415, 434, 435, 438, 441
Harmonic oscillator, 439, 440, 447
Higgs field, 204, 208, 211, 212, 227
Hilbert space, 407, 409, 418, 420, 421,
426, 431, 439, 450, 474, 476-486,
495, 499, 500, 502-509, 513, 514,
517, 519, 524, 532, 535
Hodge operator, 77
Primordial Cosmology
Holonomy, 481, 499, 502-504, 506,
508, 509, 513-516, 524, 525, 529,
530, 533, 535
Hoist action, 51, 78, 80, 81, 84, 92
Homogeneous space, 18, 55, 282-284,
286, 308, 320, 322, 325, 328, 362,
363, 372, 430, 525, 530
Homogeneous Universe, 128, 279,
286, 288, 320, 356, 530, 547
Horizon paradox, 105, 129, 195-197,
221-224
Hot dark matter, 171, 191
Hubble constant, 36, 101, 113, 188,
230
Hubble diagram, 40, 165, 176
Hubble function, 113, 216, 223, 226,
311, 520, 537, 538
Hubble law, 32, 33, 48, 95, 96,
100-103, 175
Hubble length, 97, 103-105, 107, 116,
117, 119, 128, 137, 154, 195, 197,
221, 222, 267, 311
Hubble parameter, 101, 123, 198, 232,
312
Hubble radius, 104, 215, 221, 222,
224, 231, 236
Hydrodynamical equations, 266, 270,
Immirzi parameter, 45, 78, 84, 501,
506, 528
Inhomogeneous scale, 228, 375, 382,
383
Instability, 41, 42, 85, 114, 217, 344,
346, 347, 357, 361, 382
Integrated Sachs- Wolfe effect, 184
Invariant measure, 328, 344, 345, 359,
382, 383, 462
Invariant operator, 362, 365, 366
Island, 33, 373, 374
Isocurvature perturbations, 149-151
Isometry, 28, 97, 279-284, 287, 310,
368, 523
Isotropic metric, 252
Isotropic Universe, 43, 45-47, 95-98,
106, 112, 113, 119-121, 124, 127,
128, 160, 162, 194, 205, 241, 246,
252, 256, 257, 267, 269, 272, 274,
357, 404, 432, 447, 522, 533
Isotropization, 356-358, 404, 437, 438
Jacobi metric, 331, 341, 359, 381
Jeans length, 131, 132, 135, 136, 169,
170, 183
Jeans mass, 134
Jeans mechanism, 162, 167, 169, 183,
194
Kasner epoch, 42, 279, 294, 296-301,
304, 306, 318, 321, 323, 324, 327,
337, 347, 348, 356, 362, 367, 371,
372, 388-390, 521, 527
Kasner era, 279, 280, 297, 298, 300,
348
Kasner indices, 290, 291, 296, 300,
323, 392, 398, 547
Kasner Universe, 470
Killing vectors, 280, 282, 283, 364,
418, 445, 512, 523
Klein-Gordon, 56-58, 418, 431, 434,
445, 519
Lapse function, 64, 65, 67, 71, 72,
120, 121, 334, 341, 343, 345, 355,
358, 375, 380, 386, 403, 410, 414,
423, 429, 431, 448, 516, 521, 538,
546
Last scattering surface, 178-180, 184,
186, 188, 190
Lattice, 481, 485, 486, 516, 528-530,
535
Left invariant, 530, 535
Legendre transformation, 66, 67, 121,
327, 330, 384, 453
Liouville measure, 359
Loop quantum cosmology, 45, 85,
427, 511-521, 528, 532
Loop quantum gravity, 45, 51, 77,
487, 500-511
LTB model, 159
Luminosity distance, 102, 103,
175-177
Lyapunov exponent, 346-350, 359,
Matter dominated Universe, 114, 132,
136, 180, 197
Maurer-Cartan equation, 287, 522,
531
Maxwell field, 58, 59
Minimal length, 473, 487, 489, 490,
494, 498, 517, 538, 540
Minisupermetric, 423
Mixmaster, 42, 43, 257, 258, 292, 317,
319, 320, 326-328, 330, 331, 335,
339, 341, 344, 346-351, 353, 354,
356, 357, 359-362, 367, 371,
373-375, 380, 382, 388, 399, 404,
438, 451, 457, 459, 460, 462, 463,
467, 470, 499, 511, 521, 526, 527,
546-551
Negative curvature, 47, 120, 343, 344,
382
Neutrinos, 112, 118, 142, 171, 188,
191
No-boundary, 404, 411, 442-444, 440,
447, 470
Notion of Universe, 30
Off-diagonal, 242, 293, 369, 371, 372,
387
Operator ordering, 464
Ordering, 21, 314, 373, 391, 394, 396,
397, 406, 409, 424, 431, 462, 463,
489, 502
Particle creation, 124-126
Particle decays, 107
Particle horizon, 104, 105, 221, 223
Path integral, 403-405, 410, 411, 424,
428-430, 443, 470, 489, 511
Perfect fluid, 54-56, 59, 89, 124, 139,
142, 143, 149, 151, 152, 154, 160,
185, 205, 206, 237, 244, 256, 258,
291, 292, 310, 312, 314, 325, 367,
373, 416
Perturbations, 41, 58, 96, 101, 123,
124, 127-143, 145-154, 156-159,
162, 163, 168, 170-172, 174, 179,
180, 183, 184, 186-188, 191, 192,
194, 196, 229-233, 235, 242, 245,
257, 258, 262-264, 271, 272, 274,
275, 294, 296-299, 325, 357,
362-366, 531, 533-535
Plane wave, 132-134, 445, 496
Point-Universe, 378
Polymer, 473, 474, 478, 480-484, 486,
498, 500, 511, 519, 535, 540-546,
551
Poor reheating, 220, 227
Power spectrum, 38, 39, 149, 151,
165, 172-174, 180, 181, 183, 188,
192, 230, 231, 262
Problem of time, 407, 413, 419, 425,
437, 469
Radiation-dominated Universe, 111,
149, 241
Raychaudhuri equation, 74, 88, 91,
310
Recombination, 128, 178-180,
183-186, 188, 189, 192, 194-196,
221, 222
Redshift, 32-34, 46-49, 98-103,
175-176
Regularization, 489, 508, 510
Reheating, 215-221, 226-227
Reionization, 180, 186, 188, 190
Relativistic species, 188, 215, 216,
218, 220, 227
Ricci coefficients, 76, 308, 369
Ricci scalar, 66, 71, 76, 99, 161, 236,
310, 311, 324, 411, 422, 515
Ricci tensor, 53, 74, 76, 99, 137, 138,
244, 245, 249, 260, 264, 265, 288,
289, 292, 293, 310, 321, 368-370,
374, 388
Riemann tensor, 52, 53, 91, 260, 425,
515
Sachs- Wolfe effect, 184
Scalar field, 40, 44, 45, 56-58, 141,
151, 193, 200-203, 205-220,
Primordial Cosmology
224-231, 233-235, 237, 238, 241,
246-248, 250-254, 256-259, 261,
262, 264, 274, 328, 353, 354, 357,
358, 360, 399, 404, 417, 418,
430-432, 443, 445, 447, 449, 450,
470, 518, 521, 527, 538, 540, 547
SDSS, 166, 192
Self-dual, 78, 79, 84, 92
Semiclassical, 45, 46, 201, 404, 418,
425, 433-438, 441, 444, 449, 459,
510, 515, 519, 521, 526, 527, 539,
540
Shear, 87, 268, 269, 310-312
Shear viscosity, 122, 154, 185, 265,
268, 269
Shift vector, 64, 65, 67, 71, 72, 120,
380, 403, 410, 423
Silk damping, 185
Slicing, 83, 405, 409, 414-416, 528
Slow-Rolling, 209, 210, 212, 213, 215,
221, 224-226, 228-230, 241, 248,
252, 257, 258, 263, 446
Snyder space-time, 491-494, 498, 536,
537, 540, 551
SO(3), 80, 81, 84, 287, 363, 378, 453,
523, 528, 530
Sound horizon, 184-186, 189
Spatial gradients, 71, 112, 205, 250,
256, 260, 262, 279, 361, 374, 377,
380, 381, 399
Spectral index, 188, 189, 232
Spin connection, 75-81, 84, 512, 523,
525
Spin network, 504-507, 509, 511, 514,
524, 535
Splitting, 13, 64, 65, 70, 80, 378, 425
Standard ruler, 103, 186
Stationary distribution, 280, 302, 303
Stochasticity, 42, 339, 344
Stone-von Neumann theorem, 477
String theory, 44, 487, 489-491, 497,
498
Structure formation, 96, 131, 136,
141, 163, 170-172, 228-230, 232,
257, 263
Structure formation, bottom-up, 172
Structure formation, top-down, 172
Super-metric, 67
Super-momentum, 67, 68, 70, 72, 331,
376, 378, 379, 403, 414, 415, 427
Supernovae la, 40, 102, 175
Superspace, 45, 46, 350, 403-405,
408, 410, 414, 417, 422-425,
427-430, 433, 435-437, 443, 445,
447, 450, 469, 470, 473, 481, 497,
499, 500, 511, 528, 536, 541
Symmetry breaking, 200, 207, 239,
257, 357
Symmetry group, 60, 61, 97, 283, 284,
286, 287, 320, 363, 453, 465, 493,
501, 512, 530
Synchronous reference frame, 51, 55,
73, 92, 96, 98, 100, 112, 119, 126,
129, 137, 144, 145, 159, 162, 205,
237, 242, 243, 253, 258, 259, 274,
293, 300, 305, 350, 382, 388, 416,
520
Taub Universe, 344, 404, 453, 455,
500, 527, 540-545, 551
Tetrad, 51, 74-76, 80, 84, 92, 283,
286, 288, 309, 321, 346, 347, 379,
505
Time-gauge, 80, 84, 341
Topological defects, 187, 209
Topological term, 77, 79
Triad, 81, 279, 284, 287, 288, 290,
325, 368, 503, 508, 512, 513, 516,
521, 523-527
Triangulation, 528-535
True vacuum, 44, 58, 207, 208, 211,
213, 215, 216, 226, 228
Tunneling, 208-210, 239, 404, 442,
445-447, 470
Two-point correlation function, 165,
172-174, 181
Ultrarelativistic energy density, 241,
251
Unitarily equivalent, 439, 478, 482
Universe dumpiness, 127
Universe volume, 45, 333, 335, 404
Vector potential, 63, 328
Viscosity, 54, 122, 139, 152, 154, 267,
268, 271, 272, 274
Volume operator, 504, 505, 514, 515,
524, 536
Wave function of the Universe, 407,
410, 424-426, 429, 433, 435, 437,
438, 442, 443, 450, 470, 519
Wave packet, 426, 450, 456, 457, 519,
521, 542-545
Weyl algebra, 477-479, 481, 483, 514,
528
WKB approximation, 415, 425, 432,
526
WMAP, 38, 49, 166, 179, 187,
190-192, 232
Yang-Mills, 45, 54, 60-63, 71, 77, 78,
81, 82, 92, 410, 501, 529
Zero-point energy, 234
PRIMORDIAL
COSMOLOGY
World Scientific
www.worldsclentilic.com
I'JFJ II' III I V-'P' III <
II III llli
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11