UNIVERSITE DE GENEVE FACULTE DES SCIENCES
Departement de physique theorique Professeur R. DURRER
Cosmology of Brane Universes
and Brane Gases
THESE
presentee a la Faculte des sciences de l'Universite de Geneve
pour l'obtention du grade de
Docteur es sciences, mention physique
par
Timon Georg BOEHM
de
Bale (BS)
These N° 3481
GENEVE
Atelier de reproduction de la Section de physique
2004
Abstract
The standard big bang model gives a fairly good description of the cosmological
evolution of our universe from shortly after the big bang to the present. The
existence of an initial singularity, however, might be viewed as unsatisfactory in a
comprehensive model of the universe. Moreover, if this singularity indeed exists,
we are lacking initial conditions which tell us in what state the universe emerged
from the big bang.
The advent of string theory as a promising candidate for a theory of quantum
gravity opened up new possibilities to understand our universe. The hope is that
string theory can resolve the initial singularity problem and, in addition, provide
initial conditions.
String theory makes a number of predictions such as extra-dimensions, the
existence of p-branes (fundamental objects with p spatial dimensions) as well
as several new particles. Consequently, over the past few years, a new field of
research emerged, which investigates how these predictions manifest themselves
in a cosmological context. In particular, the idea that our universe is a 3-brane
embedded in a higher-dimensional space received a lot of attention.
In this thesis we investigate the dynamical and perturbative behavior of string
theory inspired cosmological models. After an introduction to extra-dimensions
and p-branes, we identify our universe with a 3-brane embedded in a 5-dimensional
bulk space-time. We then study the cosmology and the evolution of perturbations
due to the motion of this brane through the higher-dimensional space-time. Se-
condly, we point out the dynamical instabilities of the Randall-Sundrum model.
And thirdly, vector perturbations on the brane induced by perturbations in the
bulk are calculated, and the resulting CMB power spectrum is estimated. In
all cases, dynamical instabilities are encountered, which suggests that existing
attempts to realize cosmology on branes are at least questionable.
In the last part of this thesis, we study string and brane gas cosmology. In this
scenario, the role of strings and branes is to drive the background dynamics. A
string theory specific symmetry between large and small scales (T-duality) is used
to avoid the initial big bang singularity. We show that the evolution of an initially
small and compact nine-dimensional space-time leads to three large dimensions,
which then become our visible universe. Brane gas cosmology seems to us very
promising to bring together string theory and cosmology.
This work is only part of a tremendously growing flood of literature, but we
hope that it is nevertheless a piece of mosaic in the work of art which is science.
We hope that the interplay between string theory and cosmology enables us to
advance to new spheres and dimensions (see Fig. 1).
Remerciements
Je tiens d'abord a remercier ma directrice de these, Ruth Durrer, pour m'avoir
enseigne les sujets fascinants de la relativite generale et cosmologie, mais egalement
pour son soutien et ses conseils, ainsi que sa comprehension et sa patience.
J'ai eu le plaisir de collaborer avec Robert Brandenberger, Carsten van de
Bruck, Christophe Ringeval et Daniele Steer.
Je remercie les membres du groupe de cosmologie de l'Universite de Geneve:
Cyril Cartier, Stefano Foffa, Yasmin Friedmann, Kerstin Kunze, Simone Lelli,
Michele Maggiore, Alain Riazuelo, Anna Rissone, Marti Ruiz-Altaba, Marcus
Ruser, Riccardo Sturani, Filippo Vernizzi, Peter Wittwer et en particulier Christo-
phe Ringeval et Roberto Trotta ainsi que les cosmologistes musicaux Thierry
Baertschiger et Sam Leach.
Pendant cette these j'ai eu la chance de passer un mois au Laboratoire de
physique theorique a Paris-Orsay, ou j'ai profite de nombreuses discussions avec
Emilien Dudas et Jihad Mourad.
Et fmalement je suis reconnaissant aux mains magiques de Cyril Cartier et An-
dreas Malaspinas pour leur soutien 'informatique et latex' ainsi qu'a Sam Leach
et a Christophe Ringeval pour avoir corrige les parties anglaises et franchises de
cette these.
Examinateurs
Le jury de cette these se compose de
• Prof. Pierre Binetruy, Universite Paris- XI, Orsay Cedex, France.
• Prof. Robert Brandenberger, Brown University, Providence, USA.
• Prof. Ruth Durrer, Universite de Geneve, Suisse.
• Prof. Michele Maggiore, Universite de Geneve, Suisse.
Je tiens a les remercier d'avoir accepte de faire partie du jury de ma these.
Meinen Eltern gewidmet
Contents
Abstract i
Remerciements v
Resume 1
Introduction 13
I EXTRA-DIMENSIONS AND BRANES 19
1 The cosmological standard model 21
1 . 1 Isotropy and homogeneity of the observable universe 22
1.2 Friedmann-Lemaitre space-times 23
1.2.1 Isotropic manifolds 23
1.2.2 Riemannian spaces of constant curvature 24
1.2.3 The metric of Friedmann-Lemaitre space-times 25
1.3 The gravitational field equations 25
1.3.1 Cosmological equations 26
1.3.2 Energy 'conservation' 27
1.3.3 Past and future of a Friedmann-Lemaitre universe 27
1.3.4 Cosmological solutions 29
1.3.5 A remark on conformal time 30
1.4 The cosmic microwave background 31
1.5 Appendix 32
2 Extra-dimensions 35
2.1 Motivation 36
2.2 Supergravity 37
2.3 String theory 39
2.3.1 Introductory remarks 39
2.3.2 10- and 26-dimensional space-times 41
2.3.3 Mass spectrum of a closed string on a compact direction . . 44
2.4 Geometrical remarks on compact spaces 46
2.4.1 Spaces with positive, negative, and zero Ricci curvature . . 47
2.4.2 The gravitational field equations 48
2.5 Toroidal compactification 50
2.5.1 Kaluza-Klein states 50
2.5.2 Dimensional reduction of the action 52
2.5.3 Hofava-Witten compactification 53
2.6 The modifications of Newton's law 54
2.6.1 Newton's law in d non compact spatial dimensions 54
2.6.2 Newton's law with n compact extra-dimensions 55
2.7 The hierarchy problem 57
2.7.1 Fundamental and effective Planck mass 57
2.7.2 The idea of 'large' extra-dimensions 58
Branes 61
3.1 Extended objects 62
3.2 Differential geometrical preliminaries 62
3.2.1 Embedding of branes 63
3.2.2 The first fundamental form 64
3.2.3 The second fundamental form 65
3.2.4 First and second fundamental form for one co-dimension . . 66
3.2.5 The equations of Gauss, Codazzi and Mainardi 68
3.2.6 Junction conditions 70
3.3 Branes from supergravity 72
3.3.1 The black p-brane geometry 72
3.3.2 Anti-de Sitter space-time 74
3.4 Branes from string theory 76
3.4.1 T-duality for open strings 76
3.4.2 Some properties of D-branes 77
II BRANE COSMOLOGY 81
4 Cosmology on a probe brane 83
4.1 Introduction 84
4.2 Probe brane dynamics 84
4.3 Friedmann equation 86
4.4 Discussion 88
5 Perturbations on a moving D3-brane and mirage cosmology (ar-
ticle) 89
5.1 Introduction 91
5.2 Unperturbed dynamics of the D3-brane 94
5.2.1 Background metric and brane scale factor 94
5.2.2 Brane action and bulk 4-form field 95
5.2.3 Brane dynamics and Friedmann equation 99
5.3 Perturbed equations of motion 101
5.3.1 The second order action 101
5.3.2 Evolution of perturbations in brane time r 104
5.3.3 Comments on an analysis in conformal time r\ 110
5.4 Bardeen potentials 112
5.5 Conclusions 115
6 Cosmology on a back-reacting brane 117
6.1 Introduction 118
6.2 The Randall-Sundrum model 119
6.2.1 Warped geometry solution 119
6.2.2 Scales and the hierarchy problem 120
6.2.3 Non compact extra-dimension 121
6.3 Brane cosmological equations 122
6.3.1 The five-dimensional Einstein equations 122
6.3.2 Junction conditions and the Friedmann equation of brane
cosmology 124
6.3.3 Solution of the Friedmann equation and recovering standard
cosmology 125
6.4 Einstein's equations on the brane world 127
7 Dynamical instabilities of the Randall-Sundrum model (article) 131
7.1 Introduction 133
7.2 Equations of motion 135
7.2.1 General case 135
7.2.2 Special case: The Randall-Sundrum model 138
7.3 A dynamical brane 139
7.4 Gauge Invariant Perturbation equations 144
7.4.1 Perturbations of the Randall-Sundrum model 144
7.4.2 Gauge invariant perturbation equations 145
7.5 Results and Conclusions 148
8 On CMB anisotropics in a brane universe 151
8.1 Introduction 152
8.2 Bulk vector perturbations in 4 + 1 dimensions 153
8.2.1 Background variables 153
8.2.2 Perturbed metric and gauge invariant variables 153
8.2.3 Perturbed Einstein equations 155
8.3 Temperature fluctuations in the CMB 157
8.3.1 Induced vector perturbations on the brane 157
8.3.2 Sources of CMB anisotropics 158
8.3.3 Temperature fluctuations as a function of the perturbation
variables 159
8.3.4 The perturbed geodesic equation 161
8.4 Observation of the temperature fluctuations 163
8.5 The Ci's of vector perturbations 165
CMB anisotropics from vector perturbations in the bulk (article) 169
9.1 Introduction 171
9.2 Background 173
9.2.1 Embedding and motion of the brane 173
9.2.2 Extrinsic curvature and unperturbed junction conditions . . 175
9.3 Gauge invariant perturbation equations in the bulk 178
9.3.1 Bulk perturbation variables 178
9.3.2 Bulk perturbation equations and solutions 179
9.4 The induced perturbations on the brane 183
9.4.1 Brane perturbation variables 183
9.4.2 Perturbed induced metric and extrinsic curvature 184
9.4.3 Perturbed junction conditions and solutions 185
9.5 CMB anisotropics 186
9.6 Conclusion 191
9.7 Appendix 193
9.7.1 Comparison with the Randall-Sundrum model 193
9.7.2 CMB angular power spectrum 195
III BRANE GAS COSMOLOGY 203
10 The cosmology of string gases 205
10.1 Avoiding the initial singularity 206
10.1.1 T-duality 206
10.1.2 String thermodynamics 207
10.2 Why is space three-dimensional? 209
10.3 Equation of state of a brane gas 212
11 On T-duality in brane gas cosmology (article) 217
11.1 Introduction 219
11.2 Review of brane gas cosmology 221
11.3 Energy, momentum, and pressure of p-branes 223
11.4 Winding states and T-duality 226
11.5 Mass spectra and T-duality 227
11.5.1 Masses of p-branes with single winding 227
11.5.2 Multiple windings 229
11.6 Cosmological implications and discussion 230
11.7 Appendix 231
Conclusions 233
Resume
Introduction
La theorie des cordes est une theorie fondamentale, qui unifie la gravitation
et les interactions de jauge d'une maniere consistante et renormalisable. Etant
une theorie de la gravite quantique, on pense, qu'elle a joue un role important tot
dans l'histoire de l'univers, et qu'elle est necessaire a la comprehension de celui-ci.
D'autre part, due a la recolte d'une grande quantite de donnees astrophysiques,
la cosmologie est devenue une science de precision. On espere pouvoir tester
pour la premiere fois les predictions de la theorie des cordes dans les processus
cosmologiques.
Pendant les annees passees l'interaction entre la theorie des cordes et la cos-
mologie est devenue un domaine de recherche important, qui a apporte de nou-
velles connaissances aux deux sujets. L'objectif de cette these est d'etudier, com-
ment se manifestent certaines predictions de la theorie des cordes dans un cadre
cosmologique. Cette these comporte quatre articles, qui ont ete elabores en col-
laboration avec d'autres chercheurs, ainsi que des chaptires d'introduction et de
revue.
Le modele standard de la cosmologie
Le modele standard de la cosmologie s'appuie sur trois piliers: l'isotropie
de l'expansion cosmique, l'isotropie du fond de rayonnement diffus ainsi que la
synthese des elements legers. La geometrie d'un univers isotrope autour de chaque
point (et done homogene) est donnee par la metrique de Friedmann-Lemaitre,
dsj = g^dx^dx" = -At 2 + a 2 (r) \ ^ . + r 2 (d9 2 + sin 2 0d</> 2 )l . (1)
\1-Kr 1 J
ou t est le temps cosmique, a(r) le facteur d'echelle, et /C la courbure des surfaces
a r constant. La dynamique du champ gravitationnel g^ v est determinee par les
equations d 'Einstein,
G^ + A i g^ = 8TrG i T^, (2)
ou le tenseur d'Einstein Q^ v decrit la geometrie courbee de l'espace-temps, et le
tenseur d'energie-impulsion T^ v son contenu materiel. En supposant que celui-
ci est un fluide parfait, on a T^ v = di&g(—p,P,P,P). La quantite A 4 est la
constante cosmologique quadridimensionnelle. Avec la metrique (1) les equations
d'Einstein donnent
+ ^ = ^ P+ T' (3)
Le point designe une derivee par rapport a r, et H = a/a est le parametre de
Hubble. De l'identite de Bianchi, V V Q^ = 0, et ainsi V ' V T^ V = 0, on deduit une
loi de 'conservation' d'energie,
p + 3H (p + P) = 0. (5)
En supposant une equation d'etat P = ujp, l'integral de l'equation (5) donne
fa A 3(l+o;)
P = Pi {{) • (6)
Les trois equations (3). (4). (5). dont seulement deux sont independantes, sont
les equations de base pour un univers Friedmann-Lemaitre. Dans l'histoire de
l'univers, une phase dominee par radiation, u = 1/3, a ete suivie d'une phase
dominee par matiere, uj = 0. Dans ces cas le comportement du facteur d'echelle
est
air) oc r 1 ' 2 pour u = 1/3,
(7)
air) oc r 2 ' 3 pour to = 0,
selon les equations (3) et (6). Les equations (7) decrivent un univers en expansion.
L'instant r = dans le passe, ou les sections spatiales etaient d'epaisseur nulle,
correspond a une singularite initiale, le big bang. Pour resoudre des problemes
cosmologiques, comme celui de l'horizon et de la platitude, on peut supposer
une phase inflationaire apres le big bang. Apres une duree tres courte, celle-ci
passe a la phase de radiation. A present, les observations indiquent, que nous
nous trouvons dans une phase d'expansion acceleree, dans laquelle l'energie est
dominee par la constante cosmologique A 4 ,
a(r) ex e ^ X ^ T . (8)
Dimensions supplementaires
Du point de vue cosmologique, la singularite initiale est une des raisons les plus
importantes de chercher des nouvelles theories fondamentales, qui permettent une
description de revolution cosmologique sans singularites. A present le candidat
le plus prometteur est la theorie des cordes, qui est nee de la physique des partic-
ules, lorsqu' on cherchait une theorie pour l'interaction forte. Dans la theorie des
cordes, les objets fondamentaux ne sont plus des particules ponctuelles, mais des
cordes unidimensionnelles, dont les oscillations font naitre un spectre de masse.
En particulier, ce spectre contient une particule a masse nulle et spin deux, le
graviton. A basses energies Paction de la theorie des cordes se reduit a Paction
de la relativite generale. Ainsi, la theorie des cordes contient la gravitation. Ce
fait n'est pas evident du tout et represente un argument fort en faveur de cette
theorie. La plupart des consequences phenomenologiques cependant se manifes-
tent seulement a des tres hautes energies ou a des tres petites echelles de longueur
et ne sont pas (encore?) verifiees experiment alement.
Ann d'etre consistante, la theorie des cordes doit etre supersymmetrique et se
derouler dans un espace-temps a dix dimensions. Elle predit ainsi l'existence de
six dimensions spatiales supplementaires. D'habitude on suppose, que celles-ci
sont compactes et petites, arm d'expliquer, pourquoi elles n'etaient pas observees
jusqu'a present. Remarquons ici que l'idee des dimensions supplementaires a ete
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introduite deja a partir de 1914 par Nordstrom, Kaluza et Klein dans un essai
d'unifier la gravitation et Felectromagnetisme.
Dans le chapitre sur les dimensions supplement aires nous expliquons en detail
comment on trouve le nombre de dix pour les dimensions de l'espace-temps par
un argument d'invariance de Lorentz. Pour passer a un espace-temps a dimension
plus basse, on compactifie un nombre de dimensions spatiales. Nous discutons des
apects geometriques d'espaces compacts ainsi que la compactification de Hofava
et Witten. Celle-ci mene a un espace-temps effectivement cinq-dimensionnel, qui
est utilise frequemment dans la cosmologie branaire.
Les experiences de la physique des particules testent les interactions de jauge
jusqu'a des echelles de 1/200 GeV -1 ~ 10~ 15 mm. Done on sait, que des particules
comme le photon doivent etre liees dans dans notre universe quadridimensionnel.
Comme nous allons le voir, cette observation peut etre expiquee par la theorie des
cordes. Au contraire, la gravitation est sensible au nombre total des dimensions
spatiales. Par consequent on s'attend a ce que la loi de Newton soit changee. En
effet, en presence de n dimensions supplement aires compactes la loi de Newton
est
" = ^ M
a des distances r petites devant la largeur L des dimensions supplementaires. La
forme habituelle F ~ r~ 2 est confirmee experiment alement seulement au-dessus
de 20 /im. Des mesures plus precises de la loi de Newton pourraient ainsi reveler
l'existence des dimensions supplementaires dans une experience de laboratoire
quadridimensionnelle .
Dans l'equation (9) la quantite G D est la constante de Newton fondamentale,
qui donne la 'vraie' grandeur de la gravitation dans un espace-temps a D di-
mensions. La constante de Newton dans notre univers est une quantite derivee,
G 4 oc^. (10)
Cette relation permet une solution au probleme de hierarchie entre les forces de
gravitation et de jauge. En supposant que G D est du meme ordre de grandeur
que la constante de couplage electro-faible, la hierarchie est enlevee de la theorie
fondamentale. Dans la theorie effective quadridimensionnelle la gravitation ap-
parait beaucoup plus faible que les interactions de jauge, parce que elle est diluee
(voir le facteur 1/L n ) dans les dimensions supplementaires.
Branes
Dans la cosmologie quadridimensionnelle on connait des objets etendus comme
des cordes cosmiques et des murs de domaine. En generalisant cette observation,
un espace-temps D-dimensionnel peut contenir des sous-varietees Lorentziennes
de dimension p + 1 < D. Ces objets sont nommes p-branes, ou p designe le
nombre leurs dimensions spatiales. Dans les theories de supergravite on trouve
des p-branes comme solutions solitoniques, et dans la theorie des cordes comme
des hypersurfaces contenant les degres de liberte de jauge.
Dans la cosmologie branaire notre univers est identifie avec une 3-brane, et la
matiere et les champs de jauge sont restreints sur cette brane. Un observateur sur
la brane ne peut percevoir les dimensions supplementaires que par la gravitation.
Dans le chapitre sur les branes nous introduisons des elements de la geometrie
differentielle, qui seront utiles pour la description geometrique des p-branes. Par
exemple, la courbure d'une p-brane par rapport a l'espace-temps D-dimensionnel
est decrite par le tenseur de courbure extrinseque. Dans la cosmologie branaire one
considere souvent le cas, ou il n'y a effectivement qu'une dimension supplementaire.
Dans ce cas la on peut lier le tenseur de courbure extrinseque K AB au contenu
materiel de la brane S AB par les conditions de raccordement de Israel,
K> B ~ K< B = 4 (s AB - ± q AB S^j . (11)
Le signes > et < denotent la valeur de K AB sur les deux cotes de la brane, q AB
est la premiere forme fondamentale et S est la trace de S AB . Cette relation nous
permettera de trouver des equations cosmologiques (en analogie avec les equations
de Friedmann) sur la brane, qui represente notre univers.
Une brane peut agir comme source d'un champ gravitationnel ainsi que d'un
champ de jauge. Nous discutons une geometrie statique cree par une collection
de N 3-branes en analogie avec la solution de Reissner-Nordstrom pour un trou
noir d'une masse M est d'une charge Q oc N. Cette geometrie nous sert comme
background pour des applications cosmologiques. Dans une certaine limite elle
se reduit a un espace-temps anti-de Sitter a cinq dimensions plus une partie
spherique. La metrique de 1'AdSs s'ecrit
r 2 T 2
ds 2 = — {-At 2 + Sij dxM^') + ^dr 2 , (12)
oil r est la coordonnee de la dimension supplementaire. La constante L est le
rayon de courbure de 1'AdSs . Nous allons voir une premiere application de la
metrique (12) pour la cosmologie branaire dans le chapitre suivant.
Cosmologie d'une brane test
Dans la suite nous placons la 3-brane, qui represente notre univers, dans le
sous-espace Minkowski (avec les coordonnees (t, x x ,x 2 ,x 3 )) de la metrique (12).
Dans l'espace-temps courbe la brane bouge le long de la direction radiale, ce qui
mene a une expansion homogene et isotrope de la brane. Le facteur d'echelle a
est proportionnel a la position radiale r de la brane. Cette idee se generalise a
d'autres backgrounds provenant de la theorie de supergravite ou des cordes.
Ici l'expansion est due seulement au mouvement de la brane et non pas a
son contenu materiel. Ce pour ca, qu'on appelle ce scenario la 'cosmologie mi-
rage' [90]. On travaille dans une approximation, ou la back-reaction de la brane
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sur la geometrie environnante peut etre negligee. Done les equations du mouve-
ment peuvent etre trouvees a partir d'une action du type Nambu-Goto. Dans le
cas p = 3 celle-ci s'ecrit 1
3 /dVe-°V^-T 3 fd 4 ad 4
S D3 = -T 3 / dVe-*^"^ / d 4 f rC 4 . (13)
Pour la metrique (12) par exemple, on voit, que l'energie totale de la brane
est conservee selon le theoreme de Noether. A l'aide de cette observation, on
trouve une equation differentielle, qui decrit le mouvement radial de la brane. En
utilisant la relation a ex r, celle-ci se transforme ensuite en une equation de type
Friedmann.
Perturbations on a moving D3-brane and mirage cosmology (article)
Ce chapitre correspond a l'article 'Perturbations on a moving D3-brane and
mirage cosmology', dans lequel nous etudions revolution des perturbations cos-
mologiques sur une 3-brane en mouvement dans un espace-temps anti-de Sitter-
Schwarzschild a cinq dimensions. Tout d'abord on trouve que le mouvement de
la brane non-perturbee mene a une expansion homogene et isotrope donne par
une equation de type Friedmann,
oil E = E — q^fi, et r H est le rayon de Schwarzschild. Le parametre q est egal
a +1 pour une brane dont la masse est egale a sa charge (appelee une brane
BPS). Le terme en a~ 8 , dominant a des temps tots, correspond a un fluide avec
l'equation d'etat oj = 5/3, tandis que le terme en a~ 4 represente de la radiation
dite 'noire', qui ne correspond pas a de la radiation physique. Si la supersymmetrie
sur la brane est brisee, q ^ ±1, l'equation (14) a egalement une contribution
q 2 — 1, qui joue le role d'une constante cosmologique. Tous ces termes sont dus
seulement au mouvement de la brane. La solution de l'equation (14) dans le cas
supersymmetrique, q = ±1, est
a ( T )^ T l/4 + r l/2^ (15)
Concernant les perturbations, nous ne voulons etudier egalement que les effets
due au mouvement de la brane. En negligeant la back- reaction, les seules pertur-
bations possibles sont celles par rapport au plongement non-perturbe. Celles-ci
peuvent etre decrites par un champ scalaire (j>. Nous derivons une equation du
mouvement pour <f>, oil la geometrie du bulk 2 intervient comme masse effective.
1 T3 est la tension de la brane, a^ les coordonnees intrinseques, $ le dilaton a dix din
a determinante de la metrique induite et C4 un champ de jauge Ramond-Hauiond.
2 En general le terme 'bulk' designe l'ensemble des dimensions de l'espace-temps.
Pour une brane BPS, parametrisee par une energie conservee E = 0, on trouve
pour les modes superhorizons
k = A k a 4 + B k a- 3 , (16)
ou les constantes A k and B k sont determinees par les conditions initiales. Done les
modes superhorizons croissent comme a 4 (a~ 3 ), quand la brane est en expansion
(contraction). Pour les modes subhorizons on trouve
e ikrj e -ikrj
(f> k =A k —- + B k —— 7 (17)
oil i] est le temps conforme sur la brane. Les modes subhorizons sont stables,
lorsque la brane est en expansion, mais ils croissent sur une brane en contraction.
En particulier, la brane devient instable, quand elle s'approche du trou noir de la
geometrie AdSs-Schwarzschild (r \ 0, a \ 0), parce que tous les modes croissent.
Dans l'article nous discutons egalement les cas E > et des branes avec q ^ 1.
Si on identifie alors la 3-brane avec notre univers, les perturbations <fi k in-
duisent des perturbations cosmologiques scalaires. On peut demonter, que les
perturbations du plongement (f> sont lies directement aux potentiels de Bardeen
par
(18)
Nous pensons, que ces perturbations sont importantes, egalement si on inclut de la
matiere sur la brane. Remarquons aussi, que cette approche peut etre generalisee
aux cas, que la co-dimension de la brane est plus grande que un.
Cosmologie sur une brane avec back-reaction
Un probleme de la cosmologie mirage est certainement, qu'on a neglige la
back-reaction de la brane sur la geometrie du bulk. S'il n'y a qu'une dimension
supplementaire (si le nombre des co-dimensions d'une brane est un), on peut tenir
compte de la back- reaction grace aux conditions de raccordement (11). Pour
cette raison la cosmologie branaire utilise souvent un espace-temps effectivement
cinq-dimensionnel, selon une compactification de l'espace-temps dix-dimensionnel
proposee par Hofava et Witten. Dans ce scenario la brane est fixee par rapport a
la dimension supplementaire, mais le bulk est dependent du temps (en contraste
avec la cosmologie mirage 3 ).
A l'aide des conditions de raccordement, Binetruy et al. [18] ont trouve une
equation d'evolution sur la brane,
g2 = f^ + 3> 2 -$ + ^ (^
3 On peut montrer, que les deux points de vue sont lies par une transformation de coordonnees et
done equivalents.
8 RE
ou Kg est lie a la constante de Newton cinq-dimensionelle, et p B est la densite
d'energie correspondante a une constante cosmologique dans le bulk. La densite
d'energie sur la brane (d'un fluide parfait quelconque) intervient comme p 2 , con-
trairement a ce qu'on trouve dans l'equation de Friedmann standard (H 2 ex p).
Le terme en a~ 4 correspond a radiation noire et peut etre identifie avec le terme
a~ 4 , qu'on a trouve deja dans l'equation (14). En supposant, que la constante
cosmologique dans le bulk est compensee par un terme sur la brane, on peut
montrer, que les solutions de l'equation (19) sont de la forme
o(r)~(r + r 2 ) 1/9 , (20)
ou q = 3(1 + lo). Done a grands temps le deuxieme terme domine et on retrouve
le comportement standard pour radiation et matiere.
Dans ce chapitre nous discutons egalement les equations d 'Einstein dans un
univers branaire [142],
Gpv = -A 4 fiv + 87rG47>„ + k\tx^ - E^. (21)
Ici Q^ v est le tenseur d'Einstein construit a partir de la metrique induite g^ u ,
et t^ v est le tenseur d'energie-impulsion sur la brane. A 4 est la constante cos-
mologique effective. Le terme tt^ u est quadratique dans t^ v (menant au p 2 dans
l'equation (19)), et E^ v est une projection du tenseur de Weyl dans le bulk. Ce
dernier terme represente des ondes gravitationelles a cinq dimensions.
Dynamical instabilities of the Randall-Sundrum model (article)
Randall et Sundrum ont propose un modele branaire statique, ou la metrique
sur la brane est multipliee par un facteur exponentiellement decroissant le long
de la cinquieme dimension. Ceci permet d'avoir une dimension supplement aire
non-compacte [127] ainsi que de resoudre le probleme de la hierarchie [128]. Leur
modele se base sur un fine-tuning entre la constante cosmologique (negative) A
dans le bulk et la tension V de la brane.
Dans l'article 'Dynamical instabilities of the Randall-Sundrum model' nous
etudions une generalisation dynamique de ce modele. Dans une premiere partie
nous essayons de realiser ce fine-tuning d'une fagon dynamique. Nous montrons,
que l'energie potentielle d'un champ scalaire sur la brane ne peut pas annuler A.
Ce resultat est generalise pour toute matiere satisfaisant la condition d'energie
faible.
Dans la deuxieme partie nous derivons les equations d'Einstein pour une 3-
brane dans un bulk a cinq dimensions. En perturbant ces equations nous pouvons
etudier la stabilite du modele RS. Dans ce but le fine-tuning est perturbe par une
constante O,
V = v / -12A(l + 0). (22)
Nous trouvons des solutions analytiques aux equations de perturbation a cinq
dimensions dont celle pour le facteur d'echelle sur la brane s'ecrit,
a 2 (T) ~ l + 2Qr + 4a 2 nT 2 . (23)
Done une instabilite quadratique dans le temps cosmique r apparait. En plus, il
y a un mode Q, qui represente une instabilite lineaire en r, meme si la condition
de fine-tuning n'est pas perturbee du tout. Comme toutes les equations sont
invariantes de jauge, ce mode n'est pas simplement du au choix des coordonnees.
On peut montrer, que Q correspond a la vitesse de la brane, si on relache la
condition, que celle-ci est fixee.
Les anisotropies dans le rayonnement du fond diffus pour des univers
branaires
La prochaine demarche est d'etudier les consequences observables des univers
branaires. Le rayonnement du fond diffus (CMB) s'est avere comme moyen puis-
sant pour tester des modeles cosmologiques, et done on espere d'obtenir des con-
traintes sur les modeles branaires a partir de leurs predictions sur le CMB. Dans ce
but la theorie de perturbations cosmologiques a cinq dimensions a ete developpee
pendant les dernieres annees [129].
Dans ce chapitre et dans l'article suivant nous nous interessons aux pertur-
bations vectorielles induites dans un univers branaire par des ondes gravitation-
nelles dans le bulk. Comme nous allons voir, leur comportement est radicalement
different de celui dans la cosmologie standard.
Nous considerons un bulk avec la metrique AdSs (voir l'equation (12)), et
nous derivons les equations d'Einstein perturbees. Une 3-brane, qui represente
notre univers, est placee ensuite dans la geometrie perturbee. Semblablement a
la cosmologie mirage, cette brane est en mouvement, mais cette fois-ci on tient
compte de la back-reaction par les conditions de raccordement (11). Celles-ci nous
permettent en meme temps de trouver les perturbations vectorielles induites sur
la brane.
Une fois que les perturbations sur la brane sont connues, on peut proceder en
ulilisant la theorie standard du CMB. Dans ce cadre nous etablissons le lien entre
les perturbations vectorielles et les fluctuations de temperature dans le CMB ainsi
que le spectre de puissance associe.
CMB anisotropies from vector perturbations in the bulk (article)
Dans l'article 'CMB anisotropies from vector perturbations in the bulk' nous
estimons les anisotopies vectorielles dans un univers branaire. Nous resolvons les
equations d'Einstein mentionnees ci-dessus dans le cas le plus general et pour des
conditions initiales quelconques. Les perturbations vectorielles sur la brane sont
obtenues par les conditions de raccordement. Certaines des solutions montrent
une croissance exponentielle dans le temps conforme sur la brane, contrairement
aux modes vectoriels dans la cosmologie standard, qui decroissent comme a~ 2
quelques soient les conditions initiales. Les modes croissants sont d'energie finie
et parfaitement normalisables et posent done un probleme severe pour les univers
branaires.
10 RE
Le fait, que ces modes sont normalisables est du a la structure particuliere
du bulk. Comme dans la plupart des modeles branaires, on a impose, que le
bulk soit symmetrique sous des reflexions, qui laissent la position radiale de la
brane fixe (symmetric Z-i). Pour voir l'essence de la physique derriere ces modes,
considerons l'example suivant: les solutions de l'equation de Klein-Gordon pour
une masse negative au carre sont de la forme exp[±fc(r ± t)\. En particulier, les
solutions exp[— k(r — t)] avec r > sont initialement petites, mais croissent dans
le temps, si on ne pose pas l'amplitude initiale a zero. La situation pour les modes
vectoriels est analogue. Dans notre cas la frequence spatiale Q des perturbations
dans le bulk joue le role de la masse au carre. Si elle est negative, il y a des
solutions exponentiellement croissantes. En plus de ces modes exponentiels, ils
existent des modes, qui croissent seulement comme une loi de puissance du facteur
d'echelle, mais qui menent neanmois a des effects importants dans le CMB.
Nous estimons les anisotropics causees par les modes exponentiellement crois-
sants en calculant analytiquement le spectre de puissance Ct dans certaines ap-
proximations. Le fait, que les fluctuation de temperature aujourd'hui sont de
l'ordre de 10~ 10 contraint l'amplitude primordiale des modes vectorielles d'etre
enormement petite:
A Q (Q) < e _1 ° 3 , pour O/a ~ 10 -26 mm -1 , (24)
et
A (Q) < e~ 10 , pour Q/a ~ 1mm -1 , (25)
ou o est le facteur d'echelle aujoud'hui. Dans l'equations ci-dessus on a demande
un spectre invariant d'echelle (autour de £ ~ 10), et on a fixe le rayon de courbure
de AdSs a la valeur L = 10~ 3 mm.
Comme les modes vectoriels dans le bulk sont d'energie finie, ils peuvent etre
excites par divers processus, par exemple de l'inflation dans le bulk. S'il n'existe
pas de mecanisme, qui interdit leur production, les universe branaires anti-de
Sitter ne peuvent pas reproduire une cosmologie homogene et isotrope.
Cosmologie des gas de cordes
Dans les derniers deux chapitres nous presentons une idee, qui utilise une
symmetric intrinseque de la theorie des cordes pour eviter la singularite initiale.
Ce scenario s'appelle la 'cosmologie des gas de cordes' ou simplement la 'cosmolo-
gie des cordes'. II y a un nombre de differences fondamentales par rapport aux
univers branaires, qu'on a discute precedemment.
Supposons que l'espace possede la topologie d'un tore neuf-dimensionnel, sur
lequel se propagent des cordes fondamentales. Les etats d'une corde fermee sont
des etats oscillatoires, des etats d'impulsion (correspondant au mouvement du
centre de masse de la corde) et des etats d'enroulement. Ces derniers sont pos-
sibles a cause de la topologie toroidale. Chaque etat excite contribue a la masse
d'une corde selon
M 2 = (^-) +(^V) + oscillateurs, (26)
ou R est le rayon d'une dimension compacte du tore, et n et u sont des nombres
d'excitation. On observe, que la formule (26) est invariante sous la transformation
R^%, n^co, uj^n, (27)
qui peut etre appliquee a chaque direction R du tore. D'apres cette symmetric,
appelee la dualite T, les spectres de masse sont les memes sur le tore original avec
des rayons R et le tore 'dual' avec des rayon a'/R. Dans la theorie des cordes on
pense, que tout processus doit satisfaire a cette symmetric En particulier, on peut
decrire revolution de l'univers en termes de R ou en termes de a'/R sans difference
pour les resultats physiques. Ainsi on peut eviter de tomber sur un singularity
lorque R \ en considerant la theorie duale ou a'/R / oo. Remarquons que
l'equation de Friedmann n'est pas symmetrique sous la transformation a — > \/a
(ou a oc R), et la singularite initiale est souvent inevitable.
Pour un gas de cordes on a pu montrer [29] que la temperature satisfait a
T(fl) = r0Q. (28)
La temperature reste done toujours finie, tout en evitant la singularite initiale.
Ce scenario propose par Brandenberger et Vafa offre egalement une expli-
quation elegante, pourquoi nous vivons dans un espace trois-dimensionnel. Sup-
posons, que initialement toutes les dimensions du tore etaient compactes et petites
(de l'ordre de l'echelle des cordes). L'etat initial est done un gas chaud et dense de
cordes, et comme condition initiale on demande, que toutes les directions soient
en expansion isotrope. Cependant les modes d'enroulement empechent, que le
tore s'aggrandisse. Seulement dans un sous-espace tridimensionnel les modes
d'enroulement peuvent s'annihiler, car la probabilite, qu'ils s'y intersectent est
non-nulle. Ce sous-espace tridimensionnel peut done devenir grand et constitue
fmalement notre univers observable. Les autres six dimensions du tore restent
petites, telles qu'elles soient invisibles aujourd'hui.
On T-duality in brane gas cosmology
Cette idee a ete generalisee pour le cas ou la matiere sur le tore contient
aussi des p-branes. Dans une geometrie toroidale une p-brane peut avoir des
modes d'enroulement en analogie avec une corde fondamentale. Dans Particle
'On T-duality in brane gas cosmology' nous etablissons une formule analogue
a l'equation (26) pour les differentes p-branes, et nous montrons, que sous la
transformation (27) une p-brane devient une (9-p)-brane dont la masse est
M* Q _ p = M p . (29)
Done de nouveau on trouve les memes degres de liberte dans la theorie originale
et duale, condition necessaire pour que les deux soient equivalentes. Pour prouver
12 RE
l'absence d'une singularite initiale, il faudrait encore montrer, que la relation (28)
est valable egalement pour un gas de p-branes.
Remarquons que dans la cosmologie des gas de branes nous ne vivons pas sur
une brane particuliere. Le role des branes est seulement de regler la dynamique de
l'espace-temps. En effet, en generalisant l'argument donne ci-dessus, nous avons
montre, que le nombre de dimensions, qui deviennent 'grandes' est egalement
Irois.
Dans les modeles des univers branaires, qu'on a etudie dans cette these, on a
toujours trouve des instabilites dynamiques. Ceci indique, qu'il est tres difficile
de construire une cosmologie valable avec ce genre de modeles. D' autre part,
le scenario de la cosmologie des gas de branes offre une alternative interessante,
meme s'il y a encore beaucoup de questions ouvertes. En comparant les deux, il
semble finalement, que la cosmologie des gas de branes soit la plus prometteuse
pour unifier la theorie des cordes et la cosmologie.
Introduction
14 INTRODUCTION
Superstring theory is a fundamental theory, which unifies gravity and gauge in-
teractions in a consistent and renormalizable way. The fundamental constituents
are no longer point-like particles, but 1-dimensional 'strings', whose oscillations
give rise to a spectrum of particles. In particular, this spectrum contains a mass-
less spin-2 state, which is identified with the graviton. It was also shown that the
low energy action of string theory reduces to the Einstein-Hilbert action of gen-
eral relativity. Therefore, string theory includes gravity. At the same time, string
theory gauge groups such as E$ contain SU(3) x SU(2) x U(l) as subgroups, and
hence the standard model of particle physics can be accommodated.
For consistency requirements, such as Lorentz invariance and anomaly cancel-
lation, superstring theory requires the number of space-time dimensions to be 10.
It therefore predicts the existence of six spatial extra-dimensions. Certainly, this
is a logical possibility, which however has not been empirically verified until now.
In fact, these extra-dimensions could be rolled up to small circles, such that they
are visible only at very high energies.
String theory also predicts the existence of (p+l)-dimensional hypersurfaces
to which standard model fields are confined. An observer on such a 'p-brane'
would be able to notice the presence of extra-dimensions only by gravitational
interactions, because gravity is the only fundamental force propagating in the
whole 10-dimensional space-time. Aside from the Einstein-Hilbert term, the low
energy action of string theory contains also a number of new fields that are not
present in the standard model, for instance the dilaton and the axion, which may
play an important role in cosmology.
The idea of extra-dimensions has originally been introduced by Nordstrom,
Kaluza, and Klein in order to unify gravity and electromagnetism. Many decades
later, the development of supergravity theories led to a revival of this idea. In
the framework of supergravity, the presence of seven additional dimensions is
required, and p-branes arise naturally as classical solitonic solutions.
The significance of string theory for cosmology is that it can possibly resolve
the initial singularity (big bang) problem and, moreover, provide initial condi-
tions. Therefore, it is important to investigate how string theory predictions such
as extra-dimensions and branes manifest themselves in a cosmological context.
This is the principal aim of this thesis.
String theory is not supposed to modify the cosmological evolution from nu-
cleosynthesis onward, where the physics are quite well understood and agree with
observations. But as a theory of quantum gravity, string theory is expected to
play an important role near the Planck scale. Stringy physics could have left an
imprint in the early universe and is probably necessary to understand it. On the
other hand, one hopes to learn more about string theory through cosmological
observations.
The main body of this thesis consists of four articles corresponding to chap-
ters 5, 7, 9, and 11. The published versions have been retained unchanged, apart
from some adaptations of the notation for overall consistency. The purpose of
the remaining chapters is to embed the research carried out in the literature that
already exists, as well as to provide a short introduction to each topic. Thesis est
omnis divisa in partes tres 4 .
Part I: Extra-dimensions and branes
In part I we briefly review the standard cosmology, particularly emphasiz-
ing issues which are of direct relevance for the present work. To start with,
we point out the reason why string theory and supergravity require D = 10 or
D = 11 space-time dimensions. Then, we work out the compactification of extra-
dimensions in general, as well as on a torus and on an orbifold in particular. If
extra-dimensions indeed exist, Newton's law would get modified: in the presence
of n compact extra-dimensions, it would be a \/ r 2+n law at scales much smaller
that the compactification radius. While the 4-dimensionality of gauge interactions
has been tested down to 1/200 GeV -1 ~ 10~ 15 mm, Newton's law is experimen-
tally confirmed only above L ~ 20 /zm, thus leaving room for new physics below
that scale. This is an example where the scenario of extra-dimensions, and indi-
rectly also sting theory, can be tested even in the laboratory.
Recently, it has been argued that relatively large compact extra-dimensions
(i.e. with L ~ /nm) can solve the hierarchy problem: the effective 4-dimensional
Newton constant is given by G4 ex G D /L n , where G D is the fundamental gravita-
tional constant, which can be of the order of the electroweak scale so that, in the
fundamental theory, the gap between the gravitational and the electroweak scale
disappears.
Part I is kept rather general and often goes somewhat beyond cosmology in
order to put our work into a broader context.
Part II: Brane cosmology
Part II is devoted to brane cosmology. Superstring theory and M theory
suggest that our observable universe could be a (3+l)-dimensional hypersurface,
a 3-brane, embedded in a 10 or 1 1-dimensional space-time. This idea has recently
received a great deal of interest. Such brane worlds have also been studied earlier
in the context of topological defects, before branes were discovered to have a
string theory realization.
A natural link between string theory and cosmology can be made within a
framework called the mirage cosmology [90]. In this approach, our universe is
identified with a probe 3-brane moving in a higher-dimensional space-time, which
is given by a supergravity solution. If the bulk metric has certain symmetry
properties, the unperturbed brane motion leads to a homogeneous and isotropic
expansion or contration with a scale factor a(r) on the brane. In the article 'Per-
turbations on a moving D3-brane and mirage cosmology' in Chap. 5, we study
the evolution of perturbations on such a moving brane. Deviations from the un-
perturbed embedding give rise to perturbations around the Friedmann-Lemaitre
n de bello gallico, liber prim
16 INTRODUCTION
solution, and those 'wiggles' can be directly related to the gauge invariant Bardeen
potentials. We show that on an expanding brane superhorizon modes grow as a 4 ,
while subhorizon modes are stable. On a contracting brane, both super- and sub-
horizon modes are growing. These perturbations evolve as a consequence of the
brane motion only and are not sourced by matter. However, they are expected to
be important if matter is also included. Given the probe nature of the brane, our
method has many similarities with the study of topological defects. For example,
the dynamics and perturbations are derived from the Dirac-Born-Infeld action,
which is a generalization of the Nambu-Goto action. Therefore, it is not difficult
to apply this method even when the number of co-dimensions is greater than one.
On the other hand, we cannot include the back-reaction of the brane onto the
bulk geometry, and this is a major shortcoming of the mirage cosmology.
In Chap. 6 and the following, we therefore investigate other brane world mo-
dels, which accommodate the back-reaction via junction conditions linking the
(real) energy content of the brane to the geometry of the bulk. This approach
in turn is limited to the case of one co-dimension because, when there are more
than one extra dimensions, the junction conditions do not apply anymore. There-
fore, much work has focused on the case in which our universe is a 3- brane in a
5-dimensional bulk. This scenario is motivated by Hofava-Witten compactifica-
tion, where the brane is located at an orbifold fixed point. The bulk is in general
time-dependent, and via the junction conditions this leads to a cosmological evo-
lution on the brane. Binetruy et a\. derived a Friedmann-like equation for the
brane world and showed, that the standard evolution can be recovered at late
times [18]. A more general approach is to derive equations for the Einstein tensor
on a 3-brane, which has been carried out by the authors of Ref. [142].
Randall and Sundrum proposed a bulk geometry in which the metric on the
3-brane is multiplied by an exponentially decreasing 'warp' factor, such that trans-
verse lengths become small at short distances along the fifth dimension [127]. This
allows for a non compact extra-dimension without coming into conflict with obser-
vational facts. A related model was proposed to solve the hierarchy problem [128].
However, both models rely on a fine-tuning between the brane tension and the
cosmological constant in the bulk. In the article 'Dynamical instabilities of the
Randall-Sundrum model' in Chap. 7, we construct a dynamical generalization of
the RS model, and show that, in a cosmological context, small deviations from
fine-tuning lead to runaway solutions. We also formulate a no-go theorem show-
ing that the fine-tuning cannot be obtained by a dynamical mechanism involving
a scalar field or a fluid on the brane.
The next step is to derive observational consequences of brane world models.
The cosmic microwave background (CMB) anisotropics represent some of the
most important cosmological observations. Measurements of the temperature
fluctuations in the CMB provide us with a window on the early universe, and are
therefore also suited to confirm or rule out various brane world models. To that
end, a lot of work has been invested recently to derive gauge invariant perturbation
theory in brane worlds with one co-dimension [129].
17
In the article 'CMB anisotropics from vector perturbations in the bulk' in
Chap. 9, we take our universe to be a 3-brane moving in a 5-dimensional anti-de
Sitter (AdSs ) bulk. The setup is similar to the one in the mirage cosmology,
but this time the back- reaction is taken into account. We consider vector per-
turbations in the bulk, which are modes of 5-dimensional gravity waves, and we
find analytically the most general solution of the perturbed Einstein equations
for arbitrary initial conditions. Via the junction conditions, these bulk perturba-
tions induce vector perturbations on the brane, and we find exponentially growing
modes, which are nonetheless perfectly normalizable. This differs radically from
the usual behavior in the standard cosmology, where vector modes are always
decaying. We estimate the effect on the angular power spectrum and discuss new
severe constraints for brane worlds.
Part III: Brane gas cosmology
In the last part, we present a scenario called brane gas cosmology. It is also
motivated by superstring theory, but differs in many aspects from brane world
models. In particular, we are not thought to live on a brane, but rather in the
bulk. The topology of the background space-time is that of a nine-torus, and the
matter source consists of a gas of strings and p-branes.
Initially, all nine spatial dimensions are small and compact and, by a dynamical
decompactification mechanism involving the winding modes of the strings and
branes, three dimensions grow large.
The main motivation which led to the development of brane gas cosmology is
the initial singularity problem in the standard cosmology. In their original work,
Brandenberger and Vafa considered a gas of strings and showed that the T-duality
symmetry of string theory can be used to give a singularity-free description of the
cosmological evolution [29]. T-duality is a symmetry between large and small
scales, and it allows to describe the 'region' near the big bang in terms of low
curvature scales. With this symmetry it can be shown that for a string gas the
temperature remained always finite in the past.
This idea was later extended to include various p-branes in addition to strings.
In the article 'On T-duality in brane gas cosmology' in Chap. 11, we establish
the action of T-duality on the states making up the brane gas, and show that
the mass spectrum is indeed invariant. Based on this, we claim that the initial
singularity can be avoided, also in the case that the matter source consists of a
gas of branes.
Notations and conventions
Unless explicitly mentioned, we shall use the following notations and conven-
tions throughout this thesis:
• D denotes the total number of space-time dimensions, while d is the total
number of purely spatial dimensions, and n the number of extra-dimensions.
Thus D=l + d=l + 3 + n.
18 INTRODUCTION
• M, N label coordinates of the D-dimensional space-time, /x, v of a Lorentzian
submanifold (e.g. a brane), i,j of a Riemannian (mostly Euclidean) subma-
nifold, and /, J of a Riemannian submanifold which represents the space of
extra-dimensions.
• It is often convenient to split the coordinates of the Z?-dimensional space-
time as (x M ) = (t,x,x') = (t,x l ,x'), where i = 1, ■ ■ ■ ,p and / = p+1, ■ ■ ■ ,d.
In Chaps. 10 and 11 we write (x M ) = (t, x n ) where n = 1, ■ ■ ■ , d. If the focus
is on a single extra-dimension x d , we shall write (x M ) = {x tl ,x d ). In the
case D — 5 we set x d = y or x d = r.
• (a^) = (r, a 1 ) = (r, a 1 , ■ ■ ■ , a p ) denote internal coordinates on a submani-
fold. On a brane, a = r corresponds to cosmic time. In some places, we
use conformal time rj instead.
• Boldface vectors are always 3- vectors, e.g. the direction of observation of
CMB photons is indicated by n.
• The metric of the D-dimensional space-time is G MN , and the induced or
internal metric on a Lorentzian submanifold is g^ v . The two are related by
a pull-back or push-forward. For clarity, we sometimes use a hat to stress
that a quantity is associated with g^ rather than with G MN . For example,
R^vpa is the induced (or internal) Riemann tensor on a brane.
• The metric signature is — h • • ■ +. The .D-dimensional Minkowski metric is
(r ?UJV )=diag(-l,+l,--- ,+1).
• We use = for definitions, ~ for approximately equal, ~ for a rough cor-
respondence, and oc for a proportionality.
• M D denotes the D-dimensional (fundamental) Planck mass, and M4 the
effective Planck mass in our 4-dimensional universe.
• The D-dimensional cosmological constant is denoted by A D , and the cosmo-
logical constant in our 4-dimensional universe by A4. As an exception we
use A instead of A5 for the 5-dimensional cosmological constant due to its
frequent appearance.
We work in units h = c = k B = 1, such that there is only one dimension, energy,
which is usually measured in GeV. Then,
[energy] = [mass] = [temperature] = [length] -1 = [time] -1 .
Part I
EXTRA-DIMENSIONS AND
BRANES
Chapter 1
The cosmological standard
model
22 CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
1.1 Isotropy and homogeneity of the observable uni-
verse
The cosmological standard model relies on three main observations:
1. Isotropic expansion of the universe. In 1929 Hubble discovered that
the spectra of most galaxies are redshifted and interpreted this as a Doppler
shift 1 resulting from their motion away from us (z = v/c). Furthermore, he
observed that the escape velocity of galaxies is proportional to their distance,
v — Hd, where H is Hubble's constant. The value of H today is 70 s ^° c .
2. Isotropy of the cosmic microwave background (CMB) radiation.
In 1965 Penzias and Wilson discovered a uniform background of 'cosmic
photons' corresponding to black body radiation of 3K. In fact, in 1948
Gamov had already predicted the existence of this radiation as left over
after the combination of electrons and protons into hydrogen during an
earlier hotter phase of the universe. Since that moment the CMB photons
have travelled freely through the universe and have cooled down to their
present temperature of 2.725 K due to the cosmic expansion.
3. Abundance of light elements. The fact that high energies are needed to
synthesize elements gives another hint that the early universe must have
been hot. (Formation in stars alone would not yield the correct abun-
dances.) Nucleosynthesis calculations in the early universe predict the mea-
sured abundances of hydrogen, helium, lithium, and deuterium. Nucleosyn-
thesis also gives indirect evidence that the already the early universe must
have been very isotropic.
These three pieces of evidence tell us that our universe has emerged from a very
hot and dense state called the big bang. In general relativity, the big bang is an ini-
tial singularity. During the subsequent expansion, the universe cooled down such
that successively more and more structures (nuclei, atoms, molecules) could form.
Small gravitational instabilities gave rise to the large scale structure observed to-
day (solar system, galaxies, galaxy clusters). On scales above roughly 100 Mpc,
the distribution of matter becomes isotropic 2 . Assuming that we do not inhabit a
preferred position in space, we think of the whole universe as being homogeneous
on large scales 3 . The dynamics of the expansion can then be described by highly
symmetric solutions of Einstein's equations. They are called Friedmann-Lemaitre
space-times (sometimes including also Robertson and Walker) after the Russian
mathematician Alexander Friedmann, who in 1922 derived his cosmological equa-
tions, and the Belgian priest Georges Lemaitre, who is regarded as the father of
1 In the framework of general relativity the red-shift is not interpreted as a Doppler shift, but as
the expansion of space-time itself.
2 This is still small compared to the Hubble radius today, c/H Q = SOOO/i^ 1 Mpc.
3 Nowadays, there are heated debates whether matter has a fractal distribution.
1.2. FRIEDMANN-LEMAITRE SPACE-TIMES 23
the big bang theory. We introduce this geometry in the next section (following
the treatment of Ref. [149]).
1.2 Friedmann-Lemaitre space-times
1.2.1 Isotropic manifolds
We start with the definitions of Riemannian and Lorentzian metrics and mani-
folds.
Definition 1. A Riemannian metric on a differentiable manifold M. is a
covariant tensor field g of order two with the following properties:
(i) g(X,Y) = g(Y,X) VX,Y E X(M), where X(M) is the set of all infinitely
many times differentiable vector fields on A4 .
(ii) g is non degenerated at each point of M, i.e. if g{X,Y) = VI" =>■ X = 0.
(Hi) The signature of g is (+, ■ ■ ■ ,+)■
Definition 2. A pseudo-Riemannian metric satisfies (i) and (ii) of the
above definition, but the signature of g is ( — , • • • ,—,+,••• ,+)•
The special case (—, +, ■ ■ ■ , +) is called a Lorentzian metric.
Definition 3. The pair (A4,g) is called a Riemannian, pseudo-Riemannian
or Lorentzian manifold according to the type of g defined above.
Isotropy is defined in the following way:
Definition 4. A 4- dimensional Lorentzian manifold (A4,g) is isotropic with re-
spect to the time-like velocity field v, g(v,v) = —1, if at each point q
{T q <p | up E lso q (M), (T q tp)v = v}? S0 3 (v). (1.1)
Here, T q (p is the map between the two tangent spaces at the points q and ip(q)
i.e. T q ip : T q M — > T V ^M, and Iso q (.A/f) is the group of local isometries of M
leaving the point q invariant. SOs(v) is the group of linear transformations in
T q A4 leaving v invariant and inducing special orthogonal transformations in the
space orthogonal to v. Somewhat loosely speaking, definition (1.1) states that a
space-time is isotropic if the length and angle preserving maps with v invariant
contain the group of rotations. One can then show [148] that locally M. can be
foliated into a one-parameter family of spatial hypersurfaces E T (with parameter
r) which are orthogonal to v. The integral curves of v are geodesies of (A4,g),
i.e. V v v = 0, and the geodesic distance between two hypersurfaces is constant
independent of q E S T - This allows to identify r with a 'cosmic time'. Further-
more, one can prove that (S T ,7 T ) (where 7 T is the induced metric on E T ) is a
space of constant curvature. Then the map </> : S T — > £ T < induced by the flux <j>
along the integral curves of v satisfies 4>*^ T ' = const -7 T , where </>* is the pull-back
associated with <f). This means that, in comoving coordinates, the metric tensors
on all hypersurfaces E T are equal up to a 'scale factor'.
24 CHAPTER 1. THE CQSMOLOGICAL STANDARD MODEL
One may therefore decompose the metric tensor g on M. as 4
5 =-dT 2 + a 2 (r)7, (1.2)
where dr is a 1-form obtained by applying the exterior derivative d on the co-
ordinate function t, and 7 is the metric of a 3-dimensional Riemannian space
of constant curvature. The scale factor a depends only on the hypersurface and
thus on r. A space-time (A4,g) with g given by Eq. (1.2) is called a Friedmann-
Lemaitre space-time. Notice that a manifold which is isotropic at each point q is
also homogeneous 5 .
We now investigate in more detail the Riemannian manifold (£ T ,7 T ).
1.2.2 Riemannian spaces of constant curvature
In the following, we consider (£ T ,7 T ) for some fixed value of r and omit the
subscript. Let V denote an affme connection on £, and X,Y,Z,W vector fields
in X{Ti) C T q Ti. The curvature is the map
R : *(£) x *(£) x *(£) — *(£)
R(X,Y)Z = V X (V Y Z) -V Y (V X Z)-V [X:Y] Z.
The last term vanishes in a coordinate basis. The Riemann tensor is defined via
R(X,Y,Z,W) =g(X,R(Z,W)Y), (1.4)
which is the scalar product of the two vectors X and R(Z, W)Y. Now let E c T q T,
denote an arbitrary plane in the tangent space T q T, and X, Y two orthonormal
vectors spanning E. For each plane E, the sectional curvature is defined by
K q (E) = R(X,Y,X,Y). (1.5)
Note, that this expression is independent of the basis of E.
Definition 5. If, for all points q € £ and for all planes E c T q Y,, the sectional
curvature IC q (E) is equal to a constant K,, then £ is called a space of con-
stant curvature. In the following, we consider K. to be normalized, such that
K, = +1,0, —1 for positive, negative and zero curvature.
Spaces of constant curvature are frequently used also in brane cosmology, and
we shall discuss them in more detail in Sec. 2.4. In the next paragraph, we give
several coordinate expressions for £.
4 Often g is also denoted by As 2 .
5 A manifold (A4,g) is called homogenous if locally g = — dr 2 + 7;.jda:Ma: J , in a coordinate basis
(t, x'). The requirement of homogeneity is weaker than that of isotropy around each point.
1.3. THE GRAVITATIONAL FIELD EQUATIONS
1.2.3 The metric of Friedmann-Lemaitre space-times
Let us introduce coordinates x 1 , x 2 , x 2 on S. A basis of the co-tangent space T*M.
is then given by the 1-forms (dx° = dr,dx 1 ,dx 2 ,dx 3 ), and the metric tensor on
M. can be developed as
In a space of constant curvature one can always choose x x ,x 2 ,x 3 , such that the
metric (1.2) takes the form
g = _dr 2 + a 2 (r) * [(dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 ] , (1.7)
(1 + JCg 2 /4)
where g 2 = (x 1 ) 2 + (a; 2 ) 2 + (a; 3 ) 2 . Note that, unlike S, the manifold M. is in
general not a space of constant curvature. On going to polar coordinates one has
(dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 = dg 2 + g 2 (d8 2 + sin 2 0d</> 2 ), (1.8)
and by defining a new radial coordinate
one can rewrite Eq. (1.7) in the form
dr 2
Finally, we can substitute
sinx K, = +1,
X /C = 0, (1.11)
sinh% K — — 1,
to get the form
5 =-dr 2 + a 2 (r) [d X 2 + £ 2 (x)(d# 2 + sin 2 Od(f) 2 )] , (1.12)
where
£ 2 (x) = <! x 2 £ = 0, (1-13)
(1.10)
{sin 2 x /C = +1
x 2 2 £ = o,
sinh x K, — —1
1.3 The gravitational field equations
In general relativity the metric <7 M „ is a dynamical variable describing the gravi-
tational field. Its dynamics are governed by Einstein's equations
R»v - \ 9^R + A4<7^ = G»v + A 4 ff^ = k\T^ (1.14)
26 CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
which relate the energy-momentum tensor T^ v to the curvature of space-time
encoded in the Einstein tensor Qy, v . The latter is a combination of the Ricci
tensor R^ and the Riemann scalar R. The strength of the coupling is given by
the constant k\, which is related to the Planck mass M 4 and the Newton constant
G 4 by
«2 = ^2=87rG 4 . (1.15)
This is the only free parameter in general relativity. Finally, the quantity A 4 is
the cosmological constant.
1.3.1 Cosmological equations
From the gravitational field equations (1.14) one can derive cosmological equa-
tions. This is most readily done in an orthonormal basis
Then the components of the Einstein tensor, constructed from the Friedmann
metric (1.7), read
Qm =
(1.17)
where the dot denotes a derivative with respect to cosmic time r.
On scales where the universe is isotropic and homogeneous (> 100 Mpc), mat-
ter can be regarded as a continuous medium, and we assume this to be a perfect
fluid. In the basis (1.16) the energy-momentum tensor then takes the form
T m = Pl Tij^PSij. (1.18)
With these expressions, the 00 component of (1.14) leads to the first Friedmann
equation
a 2 K 8ttG 4 A 4 , „,
a z cr 3 3
giving the expansion rate d of the universe as a function of its energy content,
the spatial curvature and the cosmological constant. It is a first order differential
equation, because the 00 Einstein equation is a constraint.
One defines the Hubble parameter by
H=- 7 (1.20)
in analogy to the Hubble constant, which is the ratio v/d. In general, H depends
on time.
1.3. THE GRAVITATIONAL FIELD EQUATIONS
From the 11 component of Einstein's equations, and by using Eq. (1.19), one
obtains the second Friedmann equation
l = ~ip^) + Y- (L21)
Notice that Eqs. (1.19) and (1.21) are relations between measurable quantities
and must therefore be independent of the basis or of the metric signature.
1.3.2 Energy 'conservation'
The Einstein tensor satisfies geometrical identities
VvQ*" = 0, (1.22)
which are called contracted Bianchi identities. Via Einstein's equations and for
A 4 — 0, one has
V V T^ = 0. (1.23)
This is, however, not an energy conservation law, as it is not of the form dJTv- v =
0. Indeed, this non conservation is due to the fact that matter exchanges energy
with the gravitational field. But for the gravitational field there is no energy-
momentum tensor: at any point q G M., it can be transformed away, as locally
g^il) = ^nv and T fJ, l/ \(q) = 0, and without field there is no energy nor mo-
mentum. Notice also that in special relativity the conservation laws for energy
and momentum are based on the invariance of a closed system under translations
in space and in time. On a curved manifold, however, such translations are in
general not symmetries anymore, and therefore there exists no general energy-
conservation law in general relativity 6
For the Friedmann-Lemaitre metric (1.7), the v = component of Eq. (1.23)
yields
p+3H(p + P) = (1.24)
giving the local rate of change of the energy density in a Friedmann-Lemaitre
space-time. This equation is nevertheless called 'the energy conservation law'.
The three equations (1.19), (1.21), and (1-24) are the basic equations gov-
erning the dynamics of a Friedmann-Lemaitre universe. Only two of them are
independent. For example, by using the 'conservation' law (1.24), the second
order equation (1.21) can be integrated to obtain the first order equation (1.19).
1.3.3 Past and future of a Friedmann-Lemaitre universe
Without solving explicitly the Friedmann equations, we can already make some
qualitative statements about the dynamics of a Friedmann-Lemaitre universe.
Consider Eq. (1.21) with A4 = 0. As long as p+3P is positive, a must be negative.
6 If however a Killing field K exists, i.e. a field satisfying C K g = 0, one can construct conserved
quantities T^ V K V , provided that Eq. (1.23) holds.
28 CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
Since the scale factor a is positive, and d is positive (due to the observed Hubble
expansion), a(r) is a concave function of r. Therefore, a must have been zero at
some finite time in the past. This singularity of the Friedmann-Lemaitre universe
corresponds to the big bang. From equation (1.19) one can show, using similar
arguments, that when A4 = the future behavior is entirely determined by the
curvature of the spatial sections. For /C = — 1 and /C = the universe expands
eternally, while for K. = +1 the expansion stops and turns into a contraction. The
final fate of the universe is then a 'big crunch' singularity where again a = 0.
Alternatively, the qualitative behavior can be obtained by specifying the en-
ergy density p with respect to a critical energy density p c . For A 4 = the
Friedmann equation (1.19) yields
The spatial curvature K. is positive or negative, according to whether p is greater
or less than the critical density
*-^i- (126 »
Therefore, the universe eternally expands for p < p c and collapses for p > p c .
The value of p c today is
p c = 1.88 -l(r 29 ^-^, H = 100/i — — , /i = 0.70, (1.27)
cm J s • Mpc
where Hq denotes the Hubble parameter today.
It is useful to measure energy densities in terms of the critical energy density
by introducing the dimensionless parameter
n x = —, (1.28)
where X labels a particular contribution or particle species. Here, X will refer
to the curvature /C or to the cosmological constant A4, and it will be omitted for
the usual matter density p. To write to the Friedmann equation in terms of O x ,
we divide equation (1.19) by H 2 ,
K. 8irG4 4
1 = -^ + ^^+^^- (1-29)
where we now have included A4 and attributed an energy density pa 4 to it, such
that finally
1 = Ok; + + Q A4 . (1.30)
1.3. THE GRAVITATIONAL FIELD EQUATIONS 29
This form of the Friedmann equation is particularly useful for cosmological pa-
rameter estimation.
Let us remark here, that the qualitative behavior of cosmological models is
often determined by rather general requirements on the energy-momentum tensor
T M „, without need to specify a particular matter model. Consider, for example,
matter satisfying
>0, (1.31)
( r .-i*. r ).
where v is an arbitrary unit time-like 4-vector and T = T^ v g^ u is the trace of
the energy-momentum tensor. For a perfect fluid this corresponds to p + 3P > 0.
The condition (1.31) is called strong energy condition. Via Einstein's equations
(for A 4 = 0),
V = «2 (^ - ^ r ) > (i.32)
the strong energy condition is equivalent to
R^v v > «=> Ric(v, v) > 0. (1.33)
Now, if the Ricci tensor of a manifold (Ai,g) satisfies this condition, one can prove
that, a finite time back in the past, M. is geodesically incomplete, and hence there
must have been an initial singularity (for a textbook treatment of these so-called
singularity theorems see e.g. [160]).
It is however likely that Einstein's equations do not hold near the singularity,
and then the strong energy condition (1.31) does not translate into the condi-
tion (1.33) for the singularity theorems 7
Let us mention that there exists also a 'weak energy condition', T^v^v" > 0,
which means that the energy density of matter as measured by an observer with
4- velocity v 11 has to be greater or equal than zero. For a perfect fluid, this amounts
to p > 0. Finally, the 'dominant energy condition' requires that —T^ v v v be a
future directed time-like or null vector. Physically, this quantity corresponds to
the energy-momentum 4-current density of matter as seen by an observer with
velocity v. For a perfect fluid, this reduces to p > \P\.
1.3.4 Cosmological solutions
The cosmological equations (1.19), (1.21), and (1.24) can be easily integrated,
when the pressure is related to the energy density by an equation of state P = top.
First, the energy 'conservation' law (1-24) gives
P=Pi{-) , (1-34)
7 In fact, one of the main motivations to introduc
initial singularity. We are presenting a possibility tc
'On T-duality in brane gas cosmology' in Chap. 11.
30 CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
where a* and pi are determined by the initial conditions. In the following we take
them to be the values today, a j = ao • In the matter dominated era one has u> =
and p ~ a~ 3 , whereas in the radiation dominated era lo = 1/3 and p ~ a -4 .
The extra factor a -1 is due to the stretching of wavelengths during the cosmic
expansion.
Let us assume that /C = and A4 = 0, and integrate the Friedmann equa-
tion (1.19) by inserting the solution (1.34) for p. This leads to the relation
(1.35)
with Hq = 87I " 3 4 po. Eq. (1.35) is solved by
where r* is some initial time, and a; = a(rj). For w = 0, a ~ r 2 / 3 , whereas for
a; = 1/3, a ~ r 1 / 2 . It will be interesting to compare these solutions with those
obtained in brane cosmology in paragraph 6.3.3.
1.3.5 A remark on conformal time
Sometimes it is convenient to work with a different time coordinate, namely con-
formal time 77, defined via
9=-dT 2 + a 2 (r) 1
^a 2 (r 1 )(-d V 2 + 1 ).
The Hubble parameter in conformal time is
H=- = Ha, (1.38)
where the prime denotes a derivative with respect to 77. The Friedmann equa-
tions (1.19) and (1.21) take the form
(1-39)
^ = -^fV(, + 3P ) + ^i, (1.40)
and the energy 'conservation' law is
p' + 3H(p + P) = 0. (1.41)
The relation corresponding to Eq. (1.35) is
n^nU^) 1+3 \ (i.42)
1.4. THE COSMIC MICROWAVE BACKGROUND
and the solution to that equation is
-(l + MU (r,- m )\ . (1.43)
Here, r\i is some initial time and ai = a(rji). For u> = 0, one has a ~ r\ ' , and for
uj = 1/3, one has a ~ 77.
1.4 The cosmic microwave background
At the beginning of this chapter we mentioned that small gravitational fluctua-
tions in the early universe gave rise to the formation of large scale structures.
Similarly, small perturbations in the matter density at the time when the universe
was 300000 years old, result in temperature fluctuations in the cosmic microwave
background (CMB) radiation today. Since those fluctuations are of the order of
10~ 5 , the universe must have been very isotropic at the age of 300000 years.
In this section, we give a brief description of the origin of the CMB. Historically,
its existence was predicted by Gamov in 1946 and was confirmed by Penzias and
Wilson in 1965, providing strong support for the idea of a hot and dense beginning
of the universe.
In a hot and dense initial phase of the universe, the photons were tightly
coupled to baryons and leptons, for example by Thompson scattering 8 and other
collision processes. Thereby, the baryons play the role of the 'walls of a cavity'
with temperature T. Consequently, the spectral distribution of the photons is
that of black body radiation with temperature T.
When T has dropped to about 3000 K (corresponding to an age of 300000
years) the electrons can combine 9 with protons to form hydrogen atoms, and
the free electron fraction suddenly drops below 10~ 4 . Consequently, the photon-
electron interaction rate, T ~ n e a T , (n e is the free electron density, and a T is
the Thompson cross section) becomes small compared to the expansion scale,
T <C H, and the photons propagate freely through the universe. Conventionally,
one defines a 'last scattering surface' by the condition n e = 1/2 (rather than some
constant time parameter).
During the subsequent cosmic expansion, the photons maintain their black
body spectrum, but their temperature decreases according to T ~ a -1 from
3000 K down to 2.73 K. The spectrum observed today is the most perfect black
body spectrum ever measured. Since its maximum is at micrometer wavelengths,
the ensemble of photons is called cosmic microwave background. Because those
photons have hardly interacted since their moment of emission, the CMB provides
us with an image of the universe when it was 300000 years old,
misleading: the c
8 the scattering between photons and non relativistic electrons
9 Almost universally, the term 'recombine' is used, which is hows
32 CHAPTER 1. THE COSMOLOGICAL STANDARD MODEL
If the universe was completely isotropic at last scattering, photons incident
from different direction in the sky would have exactly the same temperature.
However, this is not quite the case: various experiments have measured small
temperature fluctuations in the CMB, called CMB anisotropics. The provide us
with rich information on cosmological parameters. We establish the link between
matter perturbations and CMB anisotropics in Chap. 8, and apply this formalism
to anti-de Sitter brane worlds in Chap. 9.
1.5 Appendix
In this appendix, we fix our conventions (following [150] and [111]), in particu-
lar those for the Riemann and Ricci tensors, as they are often a source of sign
confusion. Let (Ai,g) be a 4-dimensional Lorentzian manifold and let
x-.U CM — > F C R 4
(1.44
peUt — > x^(p) E V, with q\ — >0
be a map from an open neighborhood U of q 6 M. to an open neighborhood V of
zero in M 4 . The set (x^) = (x^x^x^x 3 ) are the coordinates of the point q. A
basis of the tangent space T R 4 is given by the derivations
rn ^ ( d d d d \ n An
and a basis of the co-tangent space T*A4 is given by the 1-forms
(dx") = (dx°,dx 1 ,dx 2 ,dx 3 ). (1.46)
Since T q M. is isomorphic to ToM 4 , the vectors (1-45) are also a basis of T q M., and
analogously for the co-tangent space.
Let V denote an afhne connection on M. . The link with the covariant derivative
Va in the direction d$ is
(VX)(dx 7 ,a 5 ) = (dx 7 ,V 5 X), (1.47)
where X e T q Ai and ( ) denotes the contraction. Notice that VX is a 1-times co-
variant and 1-time contravariant tensor, whereas V 7 X is a vector. The connection
1-forms cu a /3 and the Christoffel symbols T a ps are defined by
V 5 dp = Lo a p{d 5 )d a = T a p5 d a , (1.48)
thus measuring the 'twist' of the basis vectors as they are transported on the
curved manifold M. . The Christoffel 10 symbols are the coefficients of the connec-
tion 1-forms
u a p = T^dx 7 , (1.49)
tairs (upstairs), they are called Christoffel symbols of the first (second)
1.5. APPENDIX 33
because the basis (1-45) and (1-46) are dual to each other: (dx 1 ,ds) = dx 7 (9,5) =
dgx 1 = <5J. Furthermore, in the coordinate basis (1.45), the Christoffel symbols
can be expressed in terms of metric derivatives as
H9A 7 ,/3-ff/3 7 ,A), (1-50)
where the subscript ,7 is an abbreviation for the partial derivative <9 7 . Notice,
that the Christoffel symbols are not components of some tensor.
We have already defined the curvature in Eq. (1.3) as well as the Riemann
tensor in Eq. (1.4). The components of the Riemann tensor are
R aPl 5 = g{d a ,R{d 1 ,d & )d p ). (1.51)
Alternatively, this can be written as a contraction
R a p l5 = (&x a ,R(d 1 ,d 5 )dp) = (&x a ,(V 1 V 5 d p -V 5 V 7 ^ ))
J££^ =^v^, (1.52)
= r a /35, 7 + r A ,35r Q A 7 - (7 «-► 6),
where in the first line, we have used the fact that in a coordinate basis the
commutator [<9 7 ,<9,5] vanishes. The Riemann tensor has the following symmetries
(the square brackets denote antisymmetrization) :
Ra/3-yS = —R(3a~f& Antisymmetry in the first two indices,
Ra/3-yS = —Raps-/ Antisymmetry in the last two indices,
R a [p~f5] ~ Vanishing of antisymmetric parts, ^ ' '
Ra/3[-y5;e] — Bianchi identity.
In the last equation, the semicolon denotes V e . Eqs. (1.53) are the complete set
of symmetries of the Riemann tensor. From these, one can deduce the additional
symmetries
Ra/3-yS = R-ySap Symmetry under pair exchange, , .
^[a/3 7 5] =0.
The Ricci tensor is defined as the contraction of the Riemann tensor on the first
and third (or second and fourth) indices,
R/35 = 9 al R a l3 7 5 = R a (3a5, (1-55)
and is symmetric in its indices. From Eq. (1.52), the Ricci tensor in terms of the
Christoffel symbols reads
R/38 = r a ^ a + r (3 S r a \ a - T a p a ^ 5 - r /3Q r Q A5 . (1.56)
The extension of these definitions to a D-dimensional space-time is straightfor-
ward. We shall need some of these expressions in Chap. 8, when we calculate the
perturbed Einstein equations in an AdSs background.
Chapter 2
Extra-dimensions
36 CHAPTER 2. EXTRA-DIMENSIONS
2.1 Motivation
The idea that a higher-dimensional approach may help to understand (3+1)-
dimensional phenomena appeared several times in the history of physics. In the
twenties Theodor Kaluza [87] and Oskar Klein [94] proposed a 5-dimensional the-
ory to unify gravity and electromagnetism. They observed that the 4-dimensional
gravitational and electromagnetic fields can be understood as components of the
five-dimensional metric tensor. Unfortunately, it turned out that the Kaluza-
Klein mechanism is not a phenomenologically working theory, but it still serves
as a prototype for many higher-dimensional models (see Sec. 2.5).
Until today, there is no experimental evidence for the existence of extra-
dimensions. Nevertheless, theories with extra-dimensions were and are develo-
ped, because a higher-dimensional approach is often useful to understand 'un-
connected' phenomena from a more fundamental and unified perspective. For
instance, the existence of extra-dimensions would opened up new possibilities to
connect particle physics and cosmology, as well as to answer questions on renor-
malizability, the unification of forces, and the big bang singularity. Let us take a
closer look at this, proceeding cronologically
In the seventies an 1 1-dimensional theory called supergravity was constructed.
Until then, all attempts to unify general relativity and quantum field theory
turned out to be non renormalizable. The hope in supergravity was that the di-
vergence problems of quantum gravity theories could be solved if supersymmetry
is added. The peculiar dimensionality 'eleven' arises, as it is the largest possi-
ble number of space-time dimension (and thus incorporates the largest possible
symmetry group) in which a consistent field theory can be constructed. However,
these hopes were dashed when it became clear that local supersymmetry is not
enough to overcome the divergence problems. Nevertheless, supergravity remains
important since it was discovered, later on, to be the low energy limit of string
theory. As the supergravity equations of motion are needed in the article 'Pertur-
bations on a moving D3-brane and mirage cosmology' in Chap. 5, we introduce
them in Sec. 2.2.
Nowadays the most promising theory of quantum gravity is string theory. Its
fundamental constituents are no longer point particles, but 1-dimensional objects
called strings. They are characterized by a tension, and their excitations give rise
to states representing various massless and massive particles, notably the graviton.
SupcTsymmetric string theories require the number of space-time dimensions to
be ten. The success of string theory in unifying gravity and gauge interactions
without divergences is the main reason today to take extra-dimensions seriously.
Some basic ideas of string theory are presented in Sec. 2.3. In particular, it is
demonstrated how string theory predicts the number of space-time dimensions,
and what the spectrum of a closed string looks like. The latter is relevant for the
article 'On T-duality in brane gas cosmology' in Sec. 11.
In all these theories there is a number n of spatial extra-dimensions in addition
to the observed three. One can think of this as attaching an n-dimensional space
2.2. SUPERGRAVITY 37
at each point of our 4-dimensional space-time. The total number of space-time
dimensions is then D=l + d=l + 3 + n. In the previous examples: D = 5
(Kaluza-Klein) , D = 11 (supergravity) , and D = 10 (superstring theory).
Why have these extra-dimensions not been observed (yet)? A simple expla-
nation is that they could be curled up to small circles of the order of the Planck
length, and that therefore they are 'visible' only at energies near the Planck mass
~ 10 19 GeV. There exist also scenarios in which this 'compactifiaction scale' is
around 1 TeV, such that it could be accessible in future collider experiments.
A rather intuitive explanation of this is the following: think of a thin and
lengthy object, for example a pencil. If it is looked at from very far, the pencil
appears as a 1-dimensional object, because the eye cannot discern the thickness.
But looking from a small distance, the second rolled up dimension of the pencil
becomes visible. In physics, 'looking close' means 'looking at high energies', and
therefore compact extra-dimensions can only be seen at high-energies.
We make some general remarks about compact spaces in Sec. 2.4, and present
two specific types of compactifications, namely toroidal compactification and
Hofava-Witten compactification, in section 2.5 and in paragraph 2.5.3.
The presence of extra-dimensions would modify Newton's law of gravitational
attraction. Instead of the F ~ 1/r 2 form, one finds that the force depends on
the number n of extra-dimensions according toF~ l/r 2+n . This is encouraging,
for it allows to test the scenario of extra-dimensions in the laboratory without
resorting to high energy experiments. A derivation of the modified Newton's law
is given in Sec. 2.6.
Finally, in Sec. 2.7 we discuss an idea to solve the hierarchy problem with
'large' extra dimensions.
2.2 Supergravity
In general relativity the (local) symmetry group is the Poincare group. The
corresponding set of generators can be extended to include supersymmetry trans-
formations. One then obtains a classical field theory called supergravity. In order
to make the symmetry group as large as possible one tries to construct a the-
ory with the highest possible number of space-time dimensions. This turns out
to be eleven, since for D > 11 the spectrum contains massless particles of spin
greater than two, and hence is inconsistent already on the classical level. There-
fore, from a theoretical point of view, 1 1-dimensional supergravity is the 'most
general' starting point for a unifying theory of gravity and gauge interactions.
Instead of the original 1 1-dimensional action of supergravity we shall write
down its form after a dimensional reduction to D = 10. We postpone explaining
how to do this until Sec. 2.5, and simply give the result here. The 10-dimensional
reduced supergravity action is particularly useful because supergravity is the low
energy limit of type IIB super string theory. Obtaining one action from the other
by a simple dimensional reduction is possible as the supersymmetry algebras of
CHAPTER 2. EXTRA-DIMENSIONS
the two theories are the same.
The low energy action of type IIB superstring theory is 1 [125]
4(V„$)(V"$)---ff AEC ^ BC
23 ' 7 (2-1)
where k\ is related to the 10-dimensional gravitational constant. The indices
are pulled up and down by the 10-dimensional metric G MN . The field <& is the
dilaton 2 , and the 3-form field strength H ABC derives from an antisymmetric tensor
field B MN via the exterior derivative H 3 = dB 2 - The fields in the first line are
multiplied by a factor e~ 2 * which is the inverse squared of the string coupling
constant g s . The power —2 corresponds to the coupling in a tree-level diagram
in string theory. Furthermore, there is a 5-form field strength F ABCDE associated
with a 4-form potential C MNRS via F 5 = dC^. The reason why this term is
interesting is that the 4-form naturally couples to a 3-brane 3 .
We are interested only in certain features of the low energy action that we
will use for cosmology, and for simplicity we have omitted a 1-form Fi, a 3-form
F 3 as well as fermionic and Chern-Simons terms (gauge invariant terms where
wedge products of the p-form potentials appear rather that their derivatives).
Notice that the high degree of supersymmetry (16 or 32 generators) completely
determines the low energy action (2.1) 4 .
This action is written in the so-called string frame: compared to the usual
Einstein-Hilbert action or the action for a scalar field, there is a unusual factor
e~ 2 * and a wrong sign of the scalar field kinetic term. This can be changed by
going to a physical frame called Einstein frame (labelled by an e) by a conformal
transformation
G e MN = n 2 G MN = e^G„. (2.2)
Here D = 10, thus in the action (2.1) we have to replace G MN by e + ^/ 2 G e MN
which yields, for example
H ABC H ABC = H e ABC H* BC e-^ /2 , (2.3)
where the sub- or superscript e simply means that now the indices are pulled up
and down with the metric in the Einstein frame. Under the conformal transfor-
1 In the article in Chap. 5 we use the more compact notation J d 10 x^/— G-^F AB cdeF abcde =
J F A *F for the second integral, where *F is the Hodge dual of F.
1 Physically, the dilaton measures the size of the eleventh compactified dimension.
3 It is well known that a 1-form, e.g. the electromagnetic potential A^, couples to the charge of
a point particle which is 0-brane. In analogy, a 2-form potential couples to a 1-brane or a string, a
3-form potential couples to a 2-brane and so on.
4 In string theory the action Eq. (2.1) corresponds to the low energy action of type IIB, and the
fields in the first line, G MN , H ABC , and $, are associated with the Neuveu-Schwarz (NS-NS) sector,
whereas Cmnrs with the Ramond-Ramond (RR-RR) sector.
2.3. STRING THEORY 39
mation (2.2) the Riemann scalar transforms as (see Ref. [160])
R = n 2 [R e + 2(D - l)Gf w V m Vjv Infi - (D - 2){D - l)G A e IN (V M Infi)(V w In ft)]
= e-* /2 Lr £ - -Gf N V M V N $- -Gf"(V JV/ $)(V JV $) ,
(2.4)
where i? and R e are the Riemann scalars constructed from the metrics G MN and
G e MN respectively. The action written in the Einstein frame is
-(V A/ $)(V-$) - e-*H e ABC H^ c
(2.5)
where the exponential is not present anymore, and the dilaton kinetic term comes
with the right sign and prefactor. We have used the fact that V M V M 3> = which
is the dilaton equation of motion if the source is a 5- form (see below) .
This action is of great importance for brane cosmology, for it admits solutions
which are extended objects, so-called p-branes (see Chap. 3). Cosmologists often
tend to use simplified effective theories in space-times with other dimensionalities
than ten, mostly four, five or six, and to keep only fields which they can make
use of. Therefore let us generalize the action to a D-dimensional space-time, the
field strength to a q-form, and omit the field H3. We shall also suppress the sub-
or superscripts e as now all quantities are in the Einstein frame
s ° = 4 / d °^ ( R - ^"*>< v "*> - k$r* F " ■■■<•** -*) ■
(2.6)
which leads to the equations of motion
R MN = ^(V a/ $)( Vn $)
2(g-l)! V 2 q q(D-2)
= V M ( e a ^F MA *- A *),
2q\ Al A *
(2.7)
where a q — (5 - q)/2.
2.3 String theory
2.3.1 Introductory remarks
Despite its great success, the SU(3) x SU{2) x U{\) standard model, together with
general relativity, is certainly not a complete description of nature. Firstly, the
40 CHAPTER 2. EXTRA-DIMENSIONS
standard model depends on roughly twenty free parameters and is therefore too
arbitrary. Secondly, attempts to unify quantum field theory with general relativity
following the usual perturbative methods have failed, because the theory turned
out to be non renormalizable. This indicates that at 'high energies' new elements
come into play And thirdly, the singularities of general relativity survive even in
this unified theory.
A number of ways have been suggested to solve these problems. One idea
is extra-dimensions, first suggested to unify electromagnetism and gravity. An-
other idea is to include supersymmetry, which resulted in supergravity theories.
However, none of these concepts have led to a theory that solves all of the prob-
lems mentioned above. The first plausible and promising candidate for a working
theory of quantum gravity is string theory.
From quantum field theory we are used to point-like particles that are excita-
tions of some field and which interact locally by exchange of gauge particles. In
string theory the fundamental constituents are strings, i.e. 1-dimensional objects.
Like a chord, a string can oscillate, and there is a spectrum of masses or energies
associated with the different oscillatory states. To a low energy observer an oscil-
lating string looks like a point-like particle with a rest mass equal to the energy of
the oscillatory state. In particular, a single string can give rise to different types
of particles depending on its state of oscillation.
When building a consistent quantum theory based on strings one finds that:
a) such a theory includes gravity: the spectrum contains a massless spin-2 state,
the graviton, which arises as an excitation of a closed string. Moreover, the
Einstein- Hilbert action of general relativity turns out to be part of the low energy
action of string theory. This is the 'string theory miracle', b) String theory is
a consistent and divergence-free theory of quantum gravity. In particular, the
ultraviolet divergence, arising from short distances, is absent. Roughly speaking
this is due to the fact that extended objects such as strings 'smear out' the
interactions over space-time, and this softens the UV divergence. The technical
argument is that, when calculating string amplitudes, the UV divergent region is
not in the domain of integration, c) String theory gauge groups, e.g. E$, contain
the standard model groups, d) A consistent string theory must be supersymmetric
(called superstring theory), because bosonic string theory has a tachyon. Unlike in
quantum field theory, fermions are now required for consistency, e) String theory
has only one free parameter, a' = 1/2ttt f , where r F is the string tension 5 . This
sets the string length scale t s = a' 1 / 2 and the string mass scale m s = a' -1 / 2 , f)
Superstring theory predicts the number of space-time dimensions to be D = 10.
It requires the existence of extra-dimensions for consistency. Points a) and f)
are particularly interesting for cosmology. If string theory really is the correct
description of nature, one must ask the question how cosmology looks like within
this framework.
There exist five types of superstring string theories, called type I, IIA, IIB,
is a fundamental sting, in contrast to D-strings (1-branes) thai
'Smic strings.
2.3. STRING THEORY 41
heterotic 50(32), and heterotic EgxEg. Type I is a theory of open and closed un-
oriented superstrings, whereas type II contains only closed oriented superstrings.
Here, I and II is the number of supersymmetry generators, and A and B indicate
that the left- and right-moving oscillators transform under separate space-time
supersymmetries that have opposite (A) or equal (B) chirality. These five string
theories can be viewed being different corners in the moduli space of a single the-
ory, called M theory 6 . A particular compactification of M theory due to Hofava
and Witten is highly relevant for brane world models. We shall discuss it in
paragraph 2.5.3.
It would be beyond the scope of this thesis (and the knowledge of its author) to
enter into more details here. In the following sections, however, we shall discuss in
some detail the string theory prediction of the space-time dimensionality, as well
as the mass spectrum of a closed string, since these issues are of direct relevance
for the articles in this thesis.
2.3.2 10- and 26-dimensional space-times
In general relativity, space-time is a given fundamental quantity, whose dynamics
are described by Einstein's equations. The number of space-time dimensions is set
to four by hand. In string theory space-time is a derived concept: the fundamental
quantity is the world-sheet W, swept out by a moving string, and the dynamics
are described by the Polyakov action
S= —7 / dTdav / ^jg^(d^X M )(d„X N )G MN . (2.8)
Here, r and a are internal coordinates on the world-sheet with — oo < t <
oo, < (J < s, where s is the length of a string, and g^ v (with /i = r, a) is
the internal metric. Diffeomorphism invariance on W requires that X' m (t' , a') =
X M (r, a). Therefore, from the world-sheet perspective, the set {A M } corresponds
to D massless scalar fields covariantly coupled by the metric G MN (setting M —
0, • • • , D — 1). On W everything can be described in terms of a 2-dimensional
conformal field theory. On the other hand, one may want to embed the world-
sheet into a £>-dimensional target space-time M. D . Then, X m {t, a) play the role
of embedding functions. A priori we do not know, what properties A4 D should
have for the theory to be consistent, so for the moment we must leave the value
of D open. Now here is the crucial point: specifying the 'couplings' of the D
scalar fields to each other is equivalent to specifying G MN . Therefore, space-time
is encoded in a 2-dimensional conformal field theory, and string theory is able
to predict its dimensionality. It is D = 26 for bosonic strings, and D = 10 for
super strings. Since there is no intuitive explanation of this result, and since it is
crucial for brane cosmology, we would like to present a short calculation following
6 Much about M theory is yet obscure, and also concernin
'M' stands for matrix, membrane, mother, mystery, magic,
42 CHAPTER 2. EXTRA-DIMENSIONS
Ref. [124] to assert this prediction. For simplicity, the demonstration is made for
open bosonic strings.
We use light cone coordinates 7 x M = (x + , x~ , x l ) where x ± = (x° ± x x )/\/2,
i = 2, • • • ,D — 1, and the canonical momenta are denoted p M = (p + ,p~ ,p z ).
The aim is to determine D. The position of a point on the string is given by
x hI = X m (t 7 (j) 7 and the transverse embedding functions X % satisfy the wave
equation 8
d 2 X l = c 2 d 2 a X\ (2.9)
with velocity c = s/2-Ka'p + . We impose Neumann boundary conditions by re-
quiring that the derivative tangential to the string vanishes ('no momentum is
flowing off the string'): d a X l = at a = 0, s. Then the general solution to
Eq. (2.9) is 9
X\t, <j) = x l + 4^ + *(2a') 1/2 Y. ^-^ ncTla cos ( — ) , (2.10)
where the sum runs over all n from minus infinity to plus infinity, except for
n = 0. In fact, the n = term corresponds to the center of mass momentum of
the string 10
P\r) = — [ dad T X\ T ,a), (2.11)
whereas the first term is center of mass position
daX*(r,a). (2.12)
Each oscillator mode n has an amplitude a l n which, upon quantization, becomes
an operator satisfying the bosonic commutation relations
[ajn.a*] = m6 ij 5 m - n (2.13)
for each i,j,n,m separately. The requirement that X % be real leads to {a l n )^ =
a l _ n . The Hamiltonian is [124]
H =2^E P^ + ^|E a -n< + <oL n J ■ (2.14)
Note that creation and annihilation operators are not yet normal ordered. By
using the commutation relations (2.13) for m = n and summing over the D — 2
JJv asm" light cone coordinates, the diffeomorphism and Weyl redundancy of the Polyakov action
is eliminated.
8 After a particular gauge fixing of the induced metric on the world-sheet.
9 This solution extends to X + and X~ , but for our purpose only the transverse coordinates are
relevant.
10 The exact definition of p+ is made via the Lagrangian of the string, see Ref. [124]. Here, we
content ourselves with the observation that p+ corresponds to the energy of the string.
2.3. STRING THEORY
transverse coordinates, one finds
^a i n a i _ n =^ j a i _ n a i n + n(D-2). (2.15)
i=2 i=2
Upon inserting this into Eq. (2.14) one encounters an infinite sum over all n which
can be regularized to give
f>=-7V ( 2 - 16 )
Thus the Hamiltonian becomes
1 d-i
- 2 P + 5 p¥ + %h ( ? 5 a ~ n< + 2 ~^
The mass squared of an oscillatory state is now
D-l
9 + v- • • 1 ( 2-D\
M> = 2 P +H-Y^ P y=-[N + —y
(2.18)
where we have replaced the sum over the oscillators by the level number N. Let
us look at the first excited state with N = 1
a'-xlO), M2 = y~^- ( 2 -19)
Since i runs form 2 to D — 1, these are D — 2 internal states. This is exactly the
amount of states that a massless spin 1 particle has in a D-dimensional space-
time. This comes as follows: for a massless particle the momentum can be written
as p M = (E, E, 0, ■ ■ ■ ,0), hence the invariance group is the little group SO(D — 2).
Now the spin 1 representation of SO(D — 2) is [D — 2) -dimensional, giving rise to
D — 2 internal states. Therefore the state (2.19) must be massless which requires
D = 26. Summarizing, by simply demanding Lorentz invariance, we have found
that the bosonic string can only live in a target space-time that has 1 time and
25 spatial dimensions. A similar argument, including fermionic fields, shows that
for superstrings the number of space-time dimensions must he D — 10.
These predictions, which appear to be wrong at first sight, are usually rec-
onciled with the observed D = 4 by compactifying the superfluous dimensions
(see Sec. 2.5). But here is the problem: the effective 4-dimensional theory, and
consequently the physical constants in our universe, crucially depend on the de-
tails of the compactification process. Therefore, string theory looses virtually
all of is predictive power. It is not clear how to overcome this deficiency. In
Chap. 10 we discuss a dynamical 'decompactification' mechanism which possibly
helps avoiding the problem.
44 CHAPTER 2. EXTRA-DIMENSIONS
2.3.3 Mass spectrum of a closed string on a compact direction
In preparation for part III of this thesis, we derive the mass spectrum of a closed
string put on a compact direction. We shall encounter momentum, winding, and
oscillatory modes which are interesting for string and brane gas cosmology.
A closed string solution of the wave equation (2.9) has to satisfy X m (t, a+s) =
X M (t, cr), in addition to the Neumann boundary condition. The mode expansion
X M (r,a)
-2nn(iCT + i<r)/s , ^n_ -2* n{ icr-i<,)/ a
(2.20)
We have replaced the index i by M, as all D space-time coordinates solve Eq. (2.9).
It is easy to see that Eq. (2.20) is the correct mode expansion for the closed
string by comparing it with the form (3.68) of the open string solution 11 . In
both cases the first term of the oscillatory part corresponds to a left moving wave
and the second to a right moving wave. On a closed string, those modes travel
independently in opposite senses. Therefore, the two amplitudes a™ and a™ are
different, whereas for the open string they are tied together by the Neumann
boundary condition. Notice also that there is a factor 2tt instead of n in the
exponent to satisfy the periodicity condition mentioned above.
One may rewrite Eq. (2.20) in terms of complex coordinates z = e ao+la and
z = e a °- ia , where a = icr becomes real after an analytic continuation to a
Minkowskian world-sheet. In the momentum term, r can be expressed as r =
\n(\z\ 2 )/(2ic). The length of the closed string is set to be s = 2ir. Then the
velocity defined under Eq. 2.9 becomes c = l/(a'p + ). With these substitutions
Eq. (2.20) reads
X M (z,z) = X M (z)+X M (z)
I / /\ 1/2
(2.21)
5"Pr Hz
r £^
We have split the center of mass momentum p M into left and right moving mo-
menta p" and pfl , in order to write the mode expansion as a sum of two in-
dependent parts: the 'left-movers' are described by a holomorphic function, the
'right 'movers' by an anti-holomorphic function.
Let us focus for a moment on the zero modes. From Eq. (2.21) one finds,
substituting back r and a,
X m (t, a) = x M + j {p™ +P™)ct+^ (j# - O a. (2.22)
"which is just a rewriting of the open string mode expansion given in Eq. (2.10)
2.3. STRING THEORY 45
In a non compact space-time the embedding functions are single valued and, as
one runs around the closed string, X m (t, a + 2ir) = X M (r,a). Thus p h * = pi' ,
and Eq. (2.22) is just a trivial rewriting of Eq. (2.20). Suppose now that one
direction, x d say, is a circle of radius R. Then
X d (T,a + 2ir) = X d (T,a) + 27rRij, (2.23)
where 2ttR is the periodicity, and 10 € Z. This leads to the constraint
^ {p d L - p d R ) = loR. (2.24)
On the other hand, the momentum flowing around a compact direction must be
quantized according to 12
rf+P«=^ + ^=^ n L ,n R ,neZ. (2.25)
From Eqs. (2.24) and (2.25) one finds
H n ujR rI n u)R
R a' R a'
Momentum along a compact direction looks like mass in the remaining D — 1
dimensions (see Eq. (2.47) later on). Intuitively, the momenta in Eq. (2.26) give
therefore rise to an 'averaged' mass
«-4[KMtfhS^(?)' (**)
More carefully, one requires that the Virasoro generators 13 L and L vanish
la'
Q - L ^\\W + A
2 2 , L ' ' (2.28)
0-Lo=^y[K) 2 +P 2 ]+iV-l,
where p 2 denotes the space-time momentum in the D — l non compact dimensions.
N and iV are the total level number of left- and right-movers, and —1 is a constant
from normal ordering the expressions. The D — 1-dimensional mass is given by
M 2 = —p 2 . Inserting expressions (2.26) into (2.28) and solving for M 2 yields
m2 =(t;Y +(^r) +^(N + N-2). (2.29)
\RJ \ a 'J a'
12 This can be seen as follows: the general translation operator is exp(ix d p d ). Under a full transla-
tion around the compact direction, i.e. x d = 2nR, a state must be invariant. Thus exp(i27r Rp d ) = 1,
and hence p d = n/R, where n £ Z.
13 A profound explication would be beyond the scope of this thesis. Here, we just remark that the
Virasoro generators are coefficients of the Fourier expansion of the energy-momentum tensor on the
world-sheet of a string. The requirement that Lq = and Lq = for left- and right-movers is a
physical state or on-shell com lit ion.
46 CHAPTER 2. EXTRA-DIMENSIONS
Eq. (2.29) is the main result of this section. It shows how momentum-, winding-,
and oscillatory modes contribute the mass of a closed string. Notice that the
contribution of the oscillators is set by the string tension which is proportional to
1/a' . The number w is a winding number, counting how many times the string
winds around the compact direction. It is positive or negative according to the
orientation of the string. Two states with opposite winding ±u> may annihilate,
leaving a state with w = 0. Strings with non zero winding number are topological
solitons. The existence of winding states is an intrinsically stringy phenomenon.
Furthermore, it can be shown that the combination L ~~ L generates trans-
lations along the string. Hence, one requires additionally that Lq — L = to
ensures that the physics remain invariant. Subtracting the two equations in (2.28)
leads to the 'level matching condition'
nu + N-N = 0. (2.30)
If there are equally many left and right moving oscillators, either the center of
mass momentum along the string or the winding number have to be zero. A
possibility consistent with the level matching condition (2.30) is to set n = w =
N = N = 0. Then the mass formula (2.29) gives
M 2 = -, (2.31)
a'
which is precisely the mass of the tachyon of bosonic string theory. In superstring
theory this mode is not present anymore.
We have presented the derivation of the closed string spectrum in some detail,
because the result (2.29) will be crucial in string and brane gas cosmology.
2.4 Geometrical remarks on compact spaces
In the previous sections we have introduced and motivated the idea of extra-
dimensions. Naive observation, however, suggests that we live in a universe with
three spatial dimensions. How can the string theoretical prediction be reconciled
with this observational fact? A simple explanation for why extra-dimensions
(assuming that they exist) are invisible in our low energy world is the following:
suppose that all n extra-dimensions are not 'large' and 'straight', but curled
up to small circles of radius R. Then, they are visible only at energies above
~ 1/R. For instance, if R is the Planck length, it would take 10 19 GeV to excite
states associated with the extra-dimensions (see paragraph 2.5.1 for the precise
meaning of this). At energies lower that that, our world looks 3-dimensional.
Some properties of such 'Kaluza-Klein states' are described in Sec. 2.5, and more
quantitative statements on the size of extra-dimensions are made in Sec. 2.7.
The aim of the present section is to become familiar with some geometrical
properties of compact and non compact spaces. With a few 'reasonable' assump-
tions it is possible to exclude certain topologies for our universe as well as for the
2.4. GEOMETRICAL REMARKS ON COMPACT SPACES 47
extra-dimensional space right from the start. We closely follow the treatment of
Ref. [147].
2.4.1 Spaces with positive, negative, and zero Ricci curvature
Consider a gravity theory on a Z?-dimensional manifold M. D with the metric G
and the action
S D = —j / dVR(G) + matter, (2.32)
where dV is the invariant volume form on M. D . By choosing local coordinates
x M = (x°, ■ ■ ■ , x D_1 ), one can write dV = d D x\J — det(G). The constant k 2 d is
related to the D-dimensional Newton constant, and R(G) is the Riemann scalar
constructed from the metric G.
We are looking for solutions which are locally of the form M. D = M.\ y. M. 2
and G = n*gi + ft 2 g2 under the map 7Tj = A4\ x M. 2 — > Mi- In the following we
take (Mi,gi) to be a Lorentzian manifold and (A^2i52) a Riemannian manifold.
Our universe is identified with M.\ (when dim(.Mi) = 4), or with a subspace
of M.\. The manifold M.2 contains the remaining extra-dimensions. Both are
assumed to be Einstein spaces, i.e.
Ric( 9l ) = C l9l , Ric{g 2 ) = C 2 g 2 . (2.33)
In component language, the Riemann tensor of a n-dimensional Einstein space
reads
Rabcu = -r- — zR(g AC g BD - g A o9Bc), (2.34)
n(n-l) yyAcyBD y AD » BC )i v >
and the Ricci tensor is
R AB - 9 mn Rmanb = -g^ B R- (2.35)
n
The normalization in Eq. (2.34) is chosen such that the contraction g AB R AB indeed
yields the Riemann scalar R. Finally, the Einstein tensor is
G AB = R A B-\g A BR= 2 -^g AB R- (2.36)
Eqs. (2.34)-(2.36) are valid for M\ or M 2 with n = dim(Mi).
The simplest Lorentzian manifolds (^1,51) satisfying the relations (2.33) are
those with maximal symmetry, namely de Sitter (C\ > 0), Minkowski [C\ = 0),
and anti-de Sitter {C\ < 0). If M.\ is embedded in a £)-dimensional flat space
with D = dim(A^i) + 1, then the isometry groups of de Sitter and anti-de Sitter
space are SO(l,D - 1) and SO{2 1 D - 2), respectively (see e.g. Eq. (3.66)). To
proceed we make the 'reasonable' assumption that C\ < in order to satisfy the
strong energy condition (1.33) on .Mi.
For (.M2, 52) we could for example take a sphere (C 2 > 0) or a torus (C 2 = 0).
When M. 2 has positive Ricci curvature (C 2 > 0), one can prove the following-
theorem:
48 CHAPTER 2. EXTRA-DIMENSIONS
Theorem 1. (Myers): Let (Mi, 92) be an n- dimensional connected complete Rie-
mannian manifold with positive definite Ricci curvature Ric > (n — l)fco- Then
(i) The diameter of M. 2 is at most n/y/ko.
(ii) M.2 is compact.
The first point is relevant, for example, in the proposal of 'large' extra-dimensions
in Sec. 2.7: the size of the extra-dimensions cannot be chosen independently of
their geometry. The second point tells us, that the extra-dimensions are com-
pact if C2 > (if M2 is simply connected). As mentioned at the beginning, the
assumption of compact extra-dimensions is commonly made as a way to render
them 'invisible'. For a proof of this theorem, see e.g. Ref. [143].
When M.2 has negative Ricci curvature, then the following theorem applies (see
Ref. [95]):
Theorem 2. (Bochner): Let (Mi^Qi) be a compact Riemannian manifold with
negative definite Ricci curvature. Then, there exist no non trivial Killing fields.
In other words, if X is a Killing field, then X = 0.
And finally, if M2 has zero Ricci curvature (see Ref. [95]):
Theorem 3. If (Mi,9i) * s a compact Riemannian manifold with vanishing Ricci
tensor field, then every infinitesimal isometry of M2 is a parallel vector field. In
other words, any Killing field X is parallel transported, VX = (where V denotes
the connection on Mi)-
From this, one derives the
Corollary 1. (Lichnerowicz, Ref. [95]): If a connected compact homogeneous
Riemannian manifold (Mi-, 92) has zero Ricci tensor field, then M2 is a Eu-
clidean torus.
We shall often refer to this case later on, for instance, when we discuss toroidal
compactification and Kaluza-Klein theories. In the brane gas model, presented
in part III of this thesis, all spatial dimensions are assumed to have the topology
of a torus. Unfortunately, this geometry does not lead to phenomenologically in-
teresting particle physics models, since the isometry group of the manifold is the
gauge group, which in the case of a torus turns out to be abelian. More realistic
compactifications involve Calabi-Yau spaces.
2.4.2 The gravitational field equations
In the previous paragraph, we considered the case C\ < 0, C2 > as potentially
interesting for brane world models. Our universe would be contained in Mi and
the space of the remaining extra-dimensions is M2- Let us now take a look at
2.4. GEOMETRICAL REMARKS ON COMPACT SPACES 49
the corresponding field equations. With the assumptions M. D = M.\ x M. 2 and
G = 7r*c/i + ir 2 g 2 , the connection forms and the curvature forms separate, and
hence the Ricci tensor oi M. D splits into
Ric(G) = 7r*ffic( 5l ) + ir*Ric(g 2 ). (2.37)
Notice, that this is not the case for the Einstein tensor. In a pure gravity theory,
i.e. with no cosmological constant, the field equation is
Ric(G) = 0, (2.38)
which forces C\ = C 2 = 0. C\ = is desirable, as it represents a Minkowski
vacuum, but C 2 = leads to uninteresting gauge groups. We therefore include a
cosmological constant A D into the Einstein equation,
G(G)+A D G = 0. (2.39)
The Ricci tensor now reads
and the Riemann scalar is
on
R(G) = TJ3^ A - ( 2 - 41 )
The difference between C\ and C 2 is set by A D . Particularly interesting for brane
world cosmology is the case C\ < 0, C 2 > 0, where M. \ is a 5-dimensional anti-de
Sitter space-time and A4 2 a 5-sphere. The space-time M = A4i x M 2 is then
AdSsxS 5 . Eq. (2.41) is valid also in the subspace Mi, such that the cosmological
constant of AdSs is given by
A=^fl( fll ) (2.42)
with n = 5. Since the Ricci curvature of AdSs is negative, also the 5-dimensional
cosmological constant is negative 14 . We shall derive the AdSsxS 5 geometry in
Sec. 3.3, and show that our universe could be a 3- brane embedded in AdSs .
We end this section with the following observation. Identify our 4-dimensional
universe with M\. For the extra-dimensions to be compact, chose again C 2 > 0,
and to have them of Planckian size, C 2 ~ AfJ, according to Myers' theorem.
Here, M4 denotes the 4-dimensional Planck mass. One 'naturally' expects that
C\ ~ C 2 , such that the cosmological constant in Aii is of the order M%, and thus
120 orders of magnitude too large.
14 In Chap. 6 v,
by a coordinate t
50 CHAPTER 2. EXTRA-DIMENSIONS
2.5 Toroidal compact ificat ion
As a particular example we discuss toroidal compactification. It serves to illus-
trate some features which are common also to other compactifications, and it will
lead us to the Kaluza-Klein unification of gauge theories and gravity. Consider
a D-dimensional manifold with a metric, (Ai D ,G), which can be decomposed
locally into
M D = M d xS\ x M = (x^,x d ). (2.43)
Here, S 1 is a circle with radius R, and x d a periodic coordinate, i.e.
x d = x d + 2ttR. (2.44)
This is an example of toroidal compactification. In general, further or even all
spatial dimensions can be periodic.
2.5.1 Kaluza-Klein states
To see the effect of periodicity, consider a massless scalar field <f>(x M ) in a D-
dimensional Minkowski space-time. The x d dependence can be expanded into a
Fourier series
<K* M ) = 0(^, x d ) = Y, M^> mxd/R - (2.45)
nez
The D-dimensional wave equation for a free scalar field, d N d N (p{x M ) = 0, yields
(dpd" - ^r) </>„(*") = 0, (2-46)
which is an equation of motion for the Fourier coefficients </>„. The </>„'s are
therefore d-dimensional scalar fields with masses
_ t.2
(2.47)
Thus, the d-dimensional mass or energy spectrum is an infinite tower of equally
spaced states, where the spacing is given by the size of the compactified dimension,
see Fig. 2.1. The lowest state is massless, and it takes an energy \/R to excite the
first massive state. At energies much lower than this, only the massless ground
state is relevant, and the physics are effectively d-dimensional. Hence, for any
field in the theory, one can integrate out its dependence on the periodic direction
to obtain a low energy effective description.
Now, consider a curved space-time where again the x d direction is compact.
The D-dimensional metric can be Fourier expanded according to
G MN (x M ) = G MN (x»,x d ) = Y, G n MN (xne inxd/R . (2.48)
Assuming that the metric excitations are very massive, we can restrict ourselves to
the zero mode, G^^x''). The issue now is to explain the Kaluza-Klein unification
2.5. TOROIDAL COMPACTIFICATION
Figure 2.1: Infinite tower of Kaluza-Klein mass states.
of the electromagnetic and gravitational fields. To that end, parameterize the zero
mode (omitting the superscript) as
ds 2 = G MN dx M dx N = G^dx^dx" + (f>(dx d + A^dx^) 2 . (2.49)
The fields G^, A^, and (f> = Gdd depend only on the non compact coordinates
x^. In a more suggestive way, we write the decomposition (2.49) in matrix form
From the ci-dimensional point of view, the Z)-dimensional metric separates into
g^ v = G^u + (f)A tl A l ,, Afj,, and (f>, which transform as a tensor, vector, and scalar
under the irreducible representations of some symmetry group in M. d . The ten-
sor g^ v is the metric in the subspace A4 d , and the vector A^ represents the
electromagnetic potential. They are both components of a higher-dimensional,
purely gravitational field. In the original formulation of Kaluza and Klein, the
space-time M. D was taken to be 5-dimensional, in order to unify the observed 4-
dimensional gravitational and electromagnetic fields. Clearly, the idea also holds
if d is arbitrary.
Furthermore, note that Eq. (2.49) is the most general form of the metric
invariant under reparametrizations
x d ^x' d = x d + X(x^), (2.51)
if simultaneously
52 CHAPTER 2. EXTRA-DIMENSIONS
This is easily seen by applying the transformation laws:
dx' d + A'^dx" = dx d + dX + A^dx" - (<9 M A)da^ = dx d + A^dx". (2.53)
Notice that the scalar A does not depend on x d , and thus dA = (d^ t X)dx 11 . Hence,
remarkably, the d-dimensional gauge transformations (2.52) arise from higher-
dimensional coordinate transformations. The conserved quantity associated with
invariance under (2.51) is the momentum along S l .
Unfortunately, the theory of Kaluza and Klein does not work in practice,
because the field acts as a Brans-Dicke scalar, which modifies 4-dimensional
gravity in a non acceptable way. Nevertheless, their idea shows the virtues (and
dangers) of a higher-dimensional approach, and it is the prototype of other uni-
fying theories.
Nowadays, string theory is the most promising candidate of such a theory.
We have seen in Sec. 2.3, that it predicts a 10-dimensional space-time. To make
contact with the observed world, the first step is to take the low energy limit, in
which case the action of type IIB string theory is given by Eq. (2.1). The second
step is to compactify six spatial dimensions. In the next paragraph, we show how
this affects the low energy action.
2.5.2 Dimensional reduction of the action
Consider the action (2.1) and, for simplicity, keep only the Riemann scalar and
the dilaton kinetic term. One then obtains the 'dilaton-gravity action'
> = — 7T / d D x v^i
2«| J
-Ge- i *(i? + 4(V A/ $)(V M $)). (2.54)
Commonly, this action is used for D = 10, D = 5, and D — 4, passing from one
to the other by a dimensional reduction. To see how this works in detail, assume
that the x d -direction is compact (where D = d+ 1). The dependence on x d can
then be Fourier transformed according to Eq. (2.45), and only the zero modes are
kept, for instance, <& = ^(a^), and the index on the derivative M is replaced by
/i.
The Ricci scalar for the metric (2.49) can be decomposed as
R=R d - 2e- 6 (V A1 V A1 e 6 ) - - e 2b F^F^, (2.55)
where we have defined e 2b = Gdd, F^v = d^A w — d„A^, and R is constructed
from the full D-dimensional metric G MN , whereas Rd from G ^ v (but not from
9iiv = G^ lv +e 2b A^A v ). Inserting this into the action (2.54) yields
Sd= 2^[ dxd [ ddx V-^ e&e ~ 2 * { Rd ~ 2e- fe (V A1 V^e 6 )
+ 4(V AJ $)(V M $) - -e 3b F ltv F flv \
(2.56)
2.5. TOROIDAL COMPACTIFICATION 53
where Gd denotes the determinant of G^ v . The integral over x d simply gives 2ttR,
and the factor e b , coming from the volume element of the compact direction, can
be absorbed by defining a (i-dimensional dilaton $d(x M ) = $(x^) — b(x fi )/2. This
,= ^/d^V^e-^(^-
(V At 6)(V /J 6)
-4(V M $ d )(V M $ d ) - -e^F^F^Y
(2.57;
We have assumed that the equation of motion for the modulus field b is V^V^fr =
0, such that the corresponding term from the derivative is zero. The dimensional
reduction leads to kinetic terms for the fields 6(i''),$ ( j(i f '),i fl (x''). In Sec. 2.2,
we have seen that the wrong sign and prefactor of the dilaton kinetic term is due
to the fact that the action is written in the string frame instead of the physical
Einstein frame. On the other hand, the dimensional reduction itself does not lead
to potential terms, and so the fields 6(2^), ^(x^), A fJi (x fJ ') are massless.
In addition to toroidal compactifications, there exist also compactifications on
Calabi-Yau manifolds. These manifolds have a more complex structure and lead
to phenomenologically more realistic particle physics theories.
2.5.3 Hofava-Witten compactification
For brane world cosmologies, a compactification scheme due to Hofava and Wit-
ten [77] is particularly interesting, and we shall briefly discuss it here. In this
scenario, one starts from 1 1-dimensional M theory, which contains the five 10-
dimensional superstring theories in different corners of its moduli space. Assume
that the eleventh dimension is curled up to a circle S 1 , and that this circle has a
mirror symmetry Z2. The coset space S 1 /Z2 is called an orbifold 15 . If 8 denotes
the coordinate on S 1 , there are two fixed points at 9 = and 9 = vr, at which the
coset space becomes singular. This can be avoided by placing a 10-dimensional
hypersurface at each singular point. Since those hypersurfaces with nine spatial
and one time direction have particular properties, e.g. a tension, they correspond
to 9-branes 16 . As a condition for stability, the sum of the two tensions must
be zero. In other words, there is a positive and a negative tension brane. In a
cosmological framework, the sign of the tension is important, for it determines
whether gravity on a brane is attractive or repulsive. Moreover, the positivity of
the tension is a condition for local stability.
So far, space-time has the structure M 10 x S 1 . Now, on each 9-brane, one can
compactify six spatial dimensions to end up with two parallel 3-branes residing
at the orbifold fixed points. The resulting space-time is effectively 5-dimensional,
and the S 1 plays the role of the fifth dimension.
15 In general, an orbifold is the coset space M./H, where H is a group of discrete symmetries of
the manifold M.
16 The meaning and properties of p-branes are introduced in Chap. 3
54 CHAPTER 2. EXTRA-DIMENSIONS
This setup is very common in brane cosmology, where one of the two 3-branes
is identified with our universe. Particularly interesting is that, before compacti-
fication, each of the 9-branes carries an E$ gauge group of the heterotic string.
After compactification, the Eg is broken to SU(3) x SU(2) x U(l), and hence the
standard model of particle physics can be accommodated on a brane.
2.6 The modifications of Newton's law
Gravity propagates in all dimensions, because it is the dynamics of space-time
itself. In particular, it is sensitive to the presence of extra-dimensions. This is
not the case for the electromagnetic, the strong, and the weak forces. In string
theory, the graviton corresponds to an excitation of a closed string, and as such
it is free to propagate everywhere. Gauge particles, instead, correspond to open
string ends, which are confined to lower-dimensional hypersurfaces (see Sec. 3.4).
In this section, we study the modifications of Newton's law in the presence
of n extra-dimensions. In principle, this allows to detect extra-dimensions in a
laboratory experiment.
2.6.1 Newton's law in d non compact spatial dimensions
First, let us generalize Newton's law to a number d = 3 + n non compact spatial
dimensions. Given a mass M, we want to find the force F exerted on a test mass
\i a distance r away. In a ci-dimensional space, the source M is enclosed by a
(d — 1) = (2 + n)-dimensional sphere S 2+n . Gauss' theorem is
/ F ■ dS = / (V • F) dV,
(2.58)
where B 3+n is a 3 + n-dimensional ball, whose surface is S 2+n . Since F is per-
pendicular onto every surface element dS, the left hand side gives
F ( dS = Fr 2+n \S 2+n \,
(2.5'))
where the surface area of the (2 + n)-sphere is given by |S' 2+ "| = ra3+n)/2) • ^°
evaluate the right hand side of Eq. (2.58), we use the gravitational field equation 17
V-g=\S 2+n \G D p, (2.60)
where G D denotes Newton's constant in D = d + 1-dimensional space-time. The
normalization is chosen to cancel the surface area on the left hand side. Recall
that for n = 0, this factor gives Air. Substituting F = /ig, we get
p I (V -g)dV = fi\S 2+n \G D I pdV = n\S 2+n \G D M. (2.61)
17 The second field equation would be V A g = 0, i.e. the gravitational field has no curl.
2.6. THE MODIFICATIONS OF NEWTON'S LAW 55
Equating Eqs. (2.59) and (2.61) gives
F = G D ^. (2.62)
This is Newton's law if there are n non compact extra-dimensions. The factor
\/r n reflects the dilution of field lines into the additional space. Notice also
the appearance of the D-dimensional instead of the 4-dimensional gravitational
constant.
2.6.2 Newton's law with n compact extra-dimensions
Now assume that there are n compact extra-dimensions, each of size L, in addition
to the usual three non compact directions. This situation can be reproduced by
a setup with 3 + n non compact dimensions and mirror masses M placed at the
points kL, k € Z, along each of the extra-dimensions. Again, we want to find the
force F on a test mass // at a distance r from the origin. If r <C L, the influence of
the mirror masses is negligible, and the result is the same as for a single source M
and d = 3 + n non compact spatial dimensions, derived in the previous paragraph:
F = G D -^ for r<^L. (2.63)
In particular, this is true if r is a distance in our universe. This allows us to test,
in principle, whether there exist extra-dimensions by measuring the force between
two test bodies in the laboratory. There is no need to access the extra-dimensions
as the lines of force do it for us.
At distances r>L, a test mass feels the influence of the mirror masses, but
it cannot discern their discrete spacing. Let us take r to be a distance in our
universe, and discuss first the case n = 1. Then the mirror masses look like
line of uniform mass density M/L. For symmetry reasons, we enclose it with a
cylinder of length L c and suitable end caps whose contribution will vanish in the
limit L c — » oo, see Fig. 2.2. Again, we use Gauss' law (2.58), where on the left
hand side the integration domain is now the surface of the cylinder dC = M. x S 2 .
Then
F f dS = FL c (4irr 2 ). (2.64)
JdC
For the right hand side of (2.58) one uses again the field equation (2.60) (with
n = 1) as all dimensions are non compact
H\S 2+X \G D f pdV = f i\S 2+1 \G D (lJ£\ . (2.65)
Solving for F one finds
4fL r 2 ' y ' '
The artificial cut-off L c has dropped out. It is now straightforward to generalize
CHAPTER 2. EXTRA-DIMENSIONS
Figure 2.2: A uniform mass density (represented by the solid line) along the
extra-dimension y is enclosed by a cylinder of length L c with two hemispheres as
endcaps. Here, r is a distance 'along' our 3-dimensional universe.
this result to n extra-dimensions by replacing the mass density by M/L n , the
integration domain by dC = M. x • • • x K x S 2 , and the surface area of the
sphere in the field equations by |S ,2+n |. The integral over p gives then L™(M/L n ).
Summarizing,
F JS^,_M r>>L
4ttL" r 2
In terms of the original description without mirror masses, we have found New-
ton's law for the gravitational attraction between two masses [x and M at a
distance r in our 3-dimensional universe, but in the presence of n compact extra-
dimensions. It is still a 1/r 2 law, independently of n, as the curled up directions
cannot be seen from distances large compared to L. Intuitively, since the extra-
dimensional are compact, there is no room for the gravitational field lines to
spread into them.
As a by-product of the result (2.67), one can read off the value of the 4-
dimensional Newton constant (measured in gravitational strength experiments)
in terms of the fundamental D-dimensional Newton constant.
\S^\G D
Gi = ^L^- (2 ' 68)
The main result of this section is that in the presence of n extra-dimensions,
Newton's law is F ~ i / r 2 +™ a t distances much smaller than the size of the extra-
dimensions. Surprisingly, the 1/r 2 form is experimentally confirmed only above
2.7. THE HIERARCHY PROBLEM 57
r ~ 20 /mi [42]. From the experimenter's point of view, this is due to the weakness
of gravity compared to the Coulomb force. It is extremely difficult to eliminate
disturbing influences such as small electric charges on the probes to obtain a clear
signal. The lower limit on the validity of Newton's law is at the same time the
upper limit on the size of L. Extra-dimensions are thus not excluded as long as
their size is smaller than 20 //m.
Whereas gravity is sensitive to extra-dimensions, gauge interactions are not.
This is because gauge particles are confined to our universe (see Sec. 3.4). For that
reason, extra-dimensions have not been detected in collider experiments, which
test gauge interactions down to 1/(200 GeV).
2.7 The hierarchy problem
The Einstein- Hilbert action of general relativity is
S 4 = Y ^-jd 4 x^^R(g)^^jd 4 xV^R(g). (2.69)
Here, we are interested only in the prefactor which has dimensions of a mass
squared in order to make the action dimensionless. This mass is called the Planck
mass 18 and is defined via the only free parameter in general relativity, Newton's
constant
M 4 = 1 = 2.4 x 10 18 GeV. (2.70)
This relation defines the 'energy scale of gravity'. For gauge interactions, instead,
an important scale is the 'electroweak scale' defined as the energy at which the
running coupling constants a EM and a w are of the same size, and the electro-
magnetic and the weak force are unified to a SU(2) x U(l) gauge theory. The
numerical value of the electroweak scale is roughly 1 TeV.
Why do these two scales differ by a factor 10 15 ? Another way of formulating
this 'hierarchy problem' is: why is gravity so much weaker than the the other
fundamental forces? If one attempts to unify gravity and gauge interactions,
their coupling constants should be at least of the same order, otherwise one has
to specify a mechanism which explains this huge difference. In the following, we
are discussing an idea to solve the hierarchy problem.
2.7.1 Fundamental and effective Planck mass
In Sec. 2.6 we have seen that the 4-dimensional Newton constant is related to
the D-dimensional gravitational constant by Eq. (2.68). Here, we derive this
relation in an alternative way, and in terms of mass scales rather than gravitational
iuced Planck mass in which numerical factors of order
58 CHAPTER 2. EXTRA-DIMENSIONS
constants. The action of a D-dimensional theory (supergravity, low energy string
theory) is
s,=4/d^ ( « + ...)^/d»*
-G(R+---), (2.71)
where we have defined a D-dimensional Planck mass M D . The power on M D
is D — 2 = 2 + nin order to make the action dimensionless. Note that, as we are
interested in physical masses, we have placed ourselves in the Einstein frame.
Assuming that the x d direction is a circle, and carrying out a dimensional
reduction as described in Sec. 2.5, one finds
S D = ^— J dx d ^/G7 d J d D - 1 x^J^G~ d (R d + ■
(2.72)
Therefore, the (D — l)-dimensional Planck mass is given by 19
M o-i 3 = Mn~ 2 V, (2-73)
where V the volume of the compactified dimension. To arrive at an effective
theory in four space-time dimensions, this procedure is repeated for the remain-
ing n — 1 extra-dimensions, and the overall coefficient is identified with the 4-
dimensional Planck mass
Ml = M°- 2 V n = Ml +n V n , (2.74)
where V n denotes the volume of the n-dimensional compact space. Note that this
formula holds not only for toroidal, but all kinds of compactifications in which
space-time has a product structure M. D = M.\ x M.2, where our universe is a
(sub)space of M.\ and the space of extra-dimensions is M.2-
2.7.2 The idea of 'large' extra-dimensions
A solution to the hierarchy problem based on the relation (2.74) was suggested
by the authors of Refs. [12, 13]. Their idea is that the 4-dimensional Planck mass
is only a derived scale, whereas the relevant scale is the Z)-dimensional Planck
mass which could be around 1 TeV.
Let us write V n = L n where L is the size of an extra-dimension. Setting
M D ~ ITeV, Eq. (2.74) gives
L ~ 10( 3 °/™)- 17 cm. (2.75)
ein frame the factor \/Gdd cannot be absorbed in a redefinition of the
2.7. THE HIERARCHY PROBLEM 59
For n = 1, L ~ 10 13 cm, which is excluded because in this case the modified
Newton law (2.63) would apply up to solar-system distances. Obviously this is
not the case. For n = 2, however, L ~ 1 mm which is only marginally excluded by
experiments (see the end of Sec. 2.6). Therefore, a scenario with two 'large' extra-
dimensions could possibly solve the hierarchy problem. To satisfy simultaneously
other experimental constraints, coming for example from the production of mass-
less higher-dimensional gravitons at the TeV scale, at least three extra-dimensions
are actually needed. One also has to keep in mind that this result relies on the as-
sumptions that the fundamental Planck mass really is around 1 TeV, and that the
space-time does have a product structure. In Chap. 6 we present an alternative
attempt to solve the hierarchy problem due to Randall and Sundrum.
Chapter 3
Branes
62 CHAPTER 3. Bl
3.1 Extended objects
In general relativity the elementary objects are point-like particles moving on a
four-dimensional manifold with a metric and a certain symmetry group. Those
particles have to be test particles in the sense that their masses are negligible,
otherwise the metric along the trajectory becomes singular. In quantum field
theory, the elementary objects are approximated by point-like particles, although
their wave function is spread in space.
The first example of classical elementary extended objects were topological
defects formed during phase transition, e.g. cosmic strings and domain walls. In
a 4-dimensional space-time domains walls are the biggest (apart from space-filling)
objects that can exist. A D-dimensional space-time, however, can be populated
by extended objects, having up to p = D — 1 spatial dimensions. A general
definition of such p-branes is [37]: a p-brane is a dynamical system defined in
terms of fields with support confined to a {p+ 1) -dimensional world-sheet surface
in a background space-time manifold of dimension D > p + 1. The category p-
brane includes particles (zero-branes), strings (1-branes), membranes (2-branes),
and so on, up to space filling branes (p + 1 = D), which are continuous media,
and where the confinement condition is redundant. The latter are not of primary
interest for us. In this context, our 4-dimensional universe is conceived as a
3-brane, to which gauge fields and fermionic fields are confined. String theory
provides a natural explanation for why matter can be trapped on a hypersurface.
In addition, there exist also more phenomenological approaches [133].
We begin this chapter with some geometrical preliminaries, which are essential
to describe p-branes. Section 3.3 explaines how p-branes emerge as solitonic
solutions of supergravity, and how anti-de Sitter space arises in this context. In
Sec. 3.4 we introduce Dp-branes from a string-theoretical point of view.
3.2 Differential geometrical preliminaries
This section serves as an introduction to a few geometrical concepts which are
useful in brane cosmology. We try to give a self-consistent overview, mainly
following Refs. [37] and [150]. The theory is fully covariant and applies to classical
relativistic branes. These include supergravity p-branes, string theory D-branes
in the low energy limit, as well as topological defects and brane gases. It also
covers a wide range of physical systems such as metal plates in an electromagnetic
field or soap bubbles. In all cases, it is assumed that the branes are infinitely thin,
which is a good approximation if their curvature radius is much larger than their
actual thickness.
In the first part, we define the first and second fundamental form for an ar-
bitrary number of co-dimensions. Afterwards, we restrict ourselves to one co-
dimension and discuss the equations of Gauss, Codazzi, and Mainardi, as well
as the junction conditions and their relevance for brane cosmology. At the same
3.2. DIFFERENTIAL GEOMETRICAL PRELIMINARIES 63
time, this section is a good place to fix our notations and conventions and a few
delicate signs.
3.2.1 Embedding of branes
Consider a D-dimensional Lorentzian manifold M. D with a non degenerate metric
G. Locally, it is always possible to choose coordinates x M , M = 0, ■ ■ ■ , D — 1.
The corresponding metric components are denoted G MN . On A4° imagine a num-
ber of intersecting or non intersecting hypersurfaces Mi, i = 1, ■ ■ ■ , n, which are
sul (manifolds of M. D . The number of co-dimensions of a submanifold is equal to
dim(.M D ) — dim(A/i). In cosmological applications those submanifolds or hyper-
surfaces are world-sheets of branes, and therefore the number of co-dimensions is
equal to the number of spatial extra-dimensions.
For practical purposes, it is useful to introduce locally a system of internal
coordinates <r M on Mi. An immersion 1 of Mi in M. D is given by the mapping
X M :Ni-^M D , N
a^ i — > x
with, for example, x° = a ,--- ,x p = a p ,x p+1 — 0,x d = 0, if Mi is a (p+1)-
dimensional submanifold. Conversely, if x M is given, it is always possible to
choose locally internal coordinates <t m on Mi according to the prescription above,
since there exist p + 1 coordinate diffeomorphisms a — > cr(a), which leave the
Lagrangian invariant. Globally, this need not always be the case, for example if
the branes are intersecting, or in a cluster of soap bubbles.
At each point q of AA one can define the tangent space T q Ni- A basis of T q Mi
is given by the p + 1 vectors -^ , ■ ■ ■ , -Jj^ . An arbitrary vector in T q Mi has then
components
Knowing e" one can explicitly construct the internal 2 metric g^ on the brane
Mi via the pull-back
9^ = G MN e^e N v . (3.3)
Suppose now that the action of the compound system of branes is the sum 3
S = Si -\ h S n , where
Si= [d p+1 aV=g£i (3.4)
is the action of a single brane. The Lagrangian density Li depends only on fields
living on the i-th brane, which, however, may be induced by external fields. The
a locally injective mapping.
2 We shall use 'internal', 'intrinsic', and 'induced' metric as synonyms.
3 This is for example the case in the Randall-Sundrum model with n = 2. Notice also that for
his consideration we do not take into account possible bulk terms.
64 CHAPTER 3. B)
conserved Noether currents and the surface energy-momentum tensor of the i-th
brane are
t^ = -2J^+£ % g^, (3.6)
for each canonical momentum p^. The symbol S denotes a variational (Eulerian)
derivative. There are other internal quantities, for example the Riemann tensor
constructed from g^ Vl if the action includes curvature terms. For specific compu-
tational purposes, it may be convenient to work with g^ Vl j^^ and t^ v . However,
if there are more or intersecting branes, the choice of internal coordinates is awk-
ward. In this case, it is more advantageous to work with D-dimensional tensors
of the background. One defines the D-dimensional energy momentum tensor by
the push-forward
T MN =t*"e%eZ. (3.7)
This tensor is defined only at the location of the brane. In this way, all internal
tensorial quantities can be lifted to become tensors of the full space-time M. D .
The calculations are then carried out with those background tensors. Further
advantages of this approach are the absence of delta functions, which otherwise
appear at the position of the branes, as well as conceptual simplicity. However in
general, which view is more favorable to adopt, depends on the specific problem.
3.2.2 The first fundamental form
To construct T MN according to Eq. (3.7), one must nevertheless specify internal
coordinates a^ and an embedding to obtain the basis vectors ejjf . So in the end,
using D-dimensional quantities would not be more economic or simple. There
is an alternative way to find T MN , or in fact any Z)-dimensional tensor, without
falling back on the use of internal coordinates. Assume that Mi is a strictly
time-like or space-like hypersurface. Then, at any point q of Mi, it is possible to
decompose the D-dimensional metric tensor into a part tangential and orthogonal
to the surface element as
G MN = q MN + L MN . (3.8)
The tangential part q MN is called first fundamental form. It is a D-dimensional
tensor, but only defined on the hypersurface Mi. The action of the mixed tensor
q M N is to project any D-dimensional tensor at a point q E Mi onto the tangent
space T q Mi- Similarly, _L m jv is an orthogonal projector. They satisfy
1 M a<1 A n = ( 1 M n; J- M a-L% = ±- U N , q M A ± A N =0. (3.9)
For example, one can define a .D-dimensional covariant derivative, that differen-
tiates along the hypersurface, by
g° v V = "V*. (3.10)
3.2. DIFFERENTIAL GEOMETRICAL PRELIMINARIES 65
Notice that " V M is not the covariant derivative associated with the internal metric
g^, but only a tangential projection of the D-dimensional connection. We shall
make extensive use of this in the next paragraph.
With the aid of q MN , the Z?-dimensional tensor T MN in Eq. (3.7) can be
obtained without a detour on the induced metric simply by varying the Lagrangian
with respect to the bulk metric
r ™ = _ 2 JA_ + £ .^™ (3 . n)
oLj mn
In the second term, we have written q MN instead of G MN , since T MN is equal
to its own tangential part 4 . Calculations involving the energy- momentum tensor
on the brane can now be carried out using this equivalent background quantity.
Recall that the link with the original internal quantity is still given by Eq. (3.7).
An alternative definition of the first fundamental form is
q MN =g^e K 'e N v . (3.12)
This is seen by writing Eq. (3.8) as
QMN = q MN + J_MN = gpOfMfg + _L M ™, (3.13)
i.e. the full metric on A4 D can be constructed by lifting the internal metric to a
D-dimensional tensor, namely the metric along the submanifold Mi, and adding
the orthogonal complement. Here, also the difference between q M N and e™ be-
comes evident: e™ promotes (pushes- forward) a tensor on Mi to a D-dimensional
tensor and vice versa. In contrast, q M N simply projects a D-dimensional tensors
to its own tangential part. A useful relation for calculations (see for example
paragraph 3.2.5) is
qMN = G AB q M A q N B , (3.14)
which simply states that the projection of the D-dimensional metric yields the
(also D-dimensional) metric along Mi. In particular, q MN can be used to raise
and lower indices of Z3-dimensional tangential tensors.
3.2.3 The second fundamental form
At any point q of the p-dimensional submanifold Mi, one can define the second
fundamental form
K MN A = -q B N q c M V c q A B = -q B N ll V M q A B , (3.15)
which is tangential to Mi in its first two indices and orthogonal in the last. The
second fundamental form is the generalization of the extrinsic curvature tensor to
an arbitrary number of co-dimensions. From the Weingarten equation, it follows
mental form are pulled up and down with G MN , as for any D-
66 CHAPTER 3. B)
that K[ MN] A = 0, where the square bracket denotes antisymmetrization. By
contracting over the tangential indices, one obtains the extrinsic curvature vector
K A = K M M A , (3.16)
which is purely orthogonal onto A/i, i.e. ^K B = q B A K A is identically zero. The
specification of K A provides an equation of motion for the hypersurface. When
there are no external forces, it takes the simple form K A = 0. If the hypersurface
is subject to an external force F A , e.g. due to some gauge field in the bulk, the
equation of motion is
T MN K MN A = ± F A . (3.17)
The surface energy-momentum tensor T MN is provided by Eq. (3.7) or (3.11), and
X F A = L A B F B is the orthogonal projection of the external force. For instance,
for a Nambu-Goto string (see Ref. [158]), T M N = —p8 M N , and form T M N K M N A =
X F A one finds
K M M A = K A = -- ± F A . (3.18)
P
In addition, there exists a formalism to treat perturbations on a hypersurface in
terms of the first and second fundamental forms, see e.g. Ref. [16]. We shall make
use of this in the case of one co-dimension in the article in Chap. 5.
3.2.4 First and second fundamental form for one co-dimension
So far, everything we have said is true for a hypersurface with an arbitrary number
of co-dimensions. In brane cosmology, motivated by Hofawa-Witten compactifi-
cation (see paragraph 2.5.3), one identifies our universe with a 3-brane embedded
in a 5-dimensional bulk. We therefore make the identification, that this 3-brane
corresponds to A/i., and that the bulk is .M 5 . Some models also require the pres-
ence of a second brane A/2 which is aligned paraded to A/i. In the following,
however, we focus on a single brane and drop the subscript 1 to avoid cluttering
the notation.
The unit normal n M ± T q J\f at the point q E J\f is defined by the orthogonality
and normalization conditions
G MN n M e™ — 0. a = 0, ■ ■ ■ ,p,
1 (3.19)
G MN n n = ±1.
The plus sign applies for a space-like unit normal (time- like brane) , and the minus
sign for time-like unit normal (space- like brane). Causality requires that the
trajectory of a brane in the bulk is time- like, and therefore we shall always assume
the unit normal to be space-like, if not otherwise indicated 5 . The orthogonal
metric tensor can now be simply expressed in terms of the unit normal
± MN =n M n N , (3.20)
5 In the well-known 3 + 1 splitting in general relativity, the unit normal is time-like. Therefore.
care must be taken, when comparing our expressions with those in general relativity textbooks.
3.2. DIFFERENTIAL GEOMETRICAL PRELIMINARIES 67
such that full space-time metric can be decomposed as
G MN = q MN + n M n N . (3.21)
Also the second fundamental form can now be defined in terms of the unit normal
K MN = -q B N q c M V c n B = -q B N l] V M n B . (3.22)
Note that V c n B is in general not in the tangent space T q N, therefore it needs to
be projected onto T q J\f by the first fundamental form. The indices M and N are
'tangential', whereas B and C stand for arbitrary directions. K MN is symmetric
in M and N. In Gaussian normal coordinates 6 one has n c V c n B = 0, since the
unit normal is tangential to a geodesic of A4 5 , and Eq. (3.22) reduces to
K MN = -V M n N . (3.23)
The second fundamental form is linked to the usual extrinsic curvature tensor
k, n/ . which measures 'extrinsically' the curvature of a hypersurface via the tilt of
the unit normal as it is transported along the hypersurface. The relation is
K MN = fc^ejfe?. (3.24)
Like in Eq. (3.7), K MN is defined only on the hypersurface A/". According to the
specific problem, it is more convenient to work with k^ v or K MN . A definition of
k^ v consistent with (3.22) is
^ = -< e " V « n '" ( 3 - 25 )
Indeed,
K MN — k^e^e" = F"e^e" = -(e fJ ') B (e u ) c V c n B e"e^ J
= -{e») B el{e v ) c el , V c n B . (3.26)
Alternatively, k^ v can be defined via the Lie derivative of the five-dimensional
metric in the direction of the unit normal
V = --e™e%£ n G MN , (3.27)
because by definition of the Lie derivative 7
C n G MN = n A G MN _ A + G AN n A , M + G MA n A , N
= n A G MN , A + (n N:M - G AN:M n A ) + (n M>N - G MA , N n A )
= n NiM + n MiN -2n A T ANM
= V«% + V N n M ,
6 Gaussian normal coordinates are defined in the following way: from each point p in the neigh-
borhood of Af, a geodesic in A4 5 is dropped onto Af, such that it intersects Af orthogonally at a
point 7. \> is then uniquely characterized by indicating the coordinates of q and a proper distance
along the geodesic.
7 We adopt the common, but untidy, notation C n G MN , for what is acutally (C n G) MN . Similarly,
one writes V M n w instead of (V Ju n) Jv .
68 CHAPTER 3. Bl
where we have used the product rule and replaced metric derivatives by Christoffel
symbols. Since the Lie derivative is symmetric, the definition (3.27) is indeed
equivalent to fc M „ = —e"e"V M n N given in Eq. (3.25).
A third equivalent definition is the following [150]: the extrinsic curvature is a
symmetric bilinear form K (., .) acting on the tangent space T q J\f, which satisfies
V x Y = V x Y + K(X,Y)n. (3.29)
Here, X, Y 6 T q N and nLT q J\T. The symbol V denotes the connection with
respect to the metric G in M. 5 , and V the connection for the internal metric g
on Af, given by Eq. (3.3). By taking the scalar product with n on both sides,
(V X Y, n) = (V X Y, n) +K(X, Y) (n, n), (3.30)
K(X,Y) = (V x Y,n). (3.31)
We now use the Ricci identity XG(Y, Z) = (V X Y, Z) + (Y, V X Z). The left hand
side is zero, for V is a metric connection, i.e. VG = 0. Setting Z = n, one finds
K(X,Y) = -(Y,V x n), (3.32)
which is equivalent to Eq. (3.23). Indeed, to write this equation in component
form, we choose a basis {eo, e±, e-2,, e^, n} of T q A4 5 , and set X = e M , Y = e N (both
in the tangential space). Then
K MN = K(e M ,e N ) = -(e N ,X7 eM n) = -(e N , (V M n L )e L )
\ N) \ N-, e M ) K N,\ I LI
= -(V M n L )(e N ,e L ) = -V M n N .
3.2.5 The equations of Gauss, Codazzi and Mainardi
The Gauss equation (theorema egregium) provides a link between the intrinsic
Riemann tensor on a submanifold with co-dimension one with the Riemann tensor
of the embedding manifold. In brane cosmology, the Gauss equation is useful to
find the Riemann, Ricci, and Einstein tensors on a brane from those in the bulk.
For concreteness, we consider as before a time- like 4-dimensional submanifold
Af, embedded in a 5-dimensional Lorentzian manifold Ai 5 . The Gauss theorem
(theorema egregium) is [150]:
Theorem 4. (Gauss)
(R(X, Y)Z, W) = (R(X, Y)Z, W) + K{X, Z)K{Y, W) - K(Y, Z)K(X, W),
(3.34)
where (R(X,Y)Z,W) denotes the Riemann tensor of A4 5 , constructed from the
metric G, and R(X,Y)Z,W) is the Riemann tensor of J\f , constructed from g
given by the pull-back (3.3). All vector fields are tangential to the submanifold,
3.2. DIFFERENTIAL GEOMETRICAL PRELIMINARIES 69
i.e. X, Y,Z,W e T q AT.
The components of the Gauss equation are obtained by setting X = e c ,Y =
e D ,Z = e B ,W = e A . Hence
(3.35)
(R(e c ,e D )e B ,e A ) = (R(e C7 e D )e B ,e A ) + K(e c , e B )K(e D ,e A
-K(e D: e B )K(e c ,e A )
or, equivalently,
R MNRS q M A q N B q R c q s D = R ABC o + K CB K DA - K DB K CA . (3.36)
Notice that we have used bulk indices M = 0, • • ■ ,4 according to the formalism
introduced before. Since R ABCD is an internal brane quantity, it is clear that
its indices actually run only from to 3. The bulk Riemann tensor is evaluated
at the position of the brane and tangentially projected 8 by q M A q N B q R c q s D . On
the other hand, the extrinsic curvature is by definition located on the brane and
tangentially projected onto it.
A complementary equation is that of Codazzi and Mainardi [150]:
Theorem 5.
(R(X, Y)Z, n) = {V X K){Y, Z) - (V Y K)(X, Z), (3.37)
where X,Y,Z 6 T q M and n±T q Af.
In component language, with n = n A e A , it reads
n A (R(e c ,e D )e B ,e A ) = V C K DB - V D K CB (3.38)
or, equivalently,
n A R ANRS q N B q R c q s D = V C K DB - V D K CB . (3.39)
Here, only the last three indices of R ANRS are projected onto the brane. Upon
contraction with G BD , a term q N B q s D G BD = q NS appears on the left hand side,
which is the 5-dimensional metric along J\f, and therefore can be used to contract
over the second and fourth index of the Riemann tensor. One easily finds
R AB n A q B c = V c K-V D K c D . (3.40)
We have used the notation K = K D ° for the trace.
The equations of Gauss, Codazzi, and Mainardi are useful to split the 5-
dimensional Einstein tensor,
Qab=Rab-\g ab R, (3.41)
8 Alternatively, we could have written I B AL:ri , for the left hand side.
70 CHAPTER 3. Bl
into 'mixed', 'orthogonal', and 'parallel' components with respect to the time-
like hypersurface M (see Ref.[19]). The mixed component corresponds directly to
Eq. (3.40), as the second term in Eq. (3.41) does not contribute upon contraction
with q B c . Thus
Q AB n A q B c = V c K -V D K C D . (3.42)
Similarly, the orthogonal component is
(3.43)
where R denotes the Riemann scalar constructed from the internal metric g^.
Finally, the parallel component is
Q AB q A c q B o = Qcd - KK co + n E V E K CD + V c a D + 2n^ c K 0)E a B - a c a D
+ (-K 2 + -K AB K AB - n A V A K - V B a B \q CD ,
(3.44)
where Q CD is the Einstein tensor constructed from the internal metric, a B =
n c "V c n B is the 'acceleration field', and the round brackets denote symmetriza-
tion with weight one half. Notice also the difference between the two covariant
derivatives that appear in the above formula: V associated with the metric G MN
and V associated with g^.
3.2.6 Junction conditions
So far, we have considered H to be an empty hypersurface. In a cosmological
context, however, it corresponds to a brane with a certain energy density due
to matter and radiation. The energy content has an effect on the bulk space-
time, and therefore the geometry on both sides must be matched in a proper way.
The problem is somewhat subtle, because the smoothness of the bulk metric is
influenced not only by the presence of the brane, but also by the particular choice
of coordinates to describe the bulk 9 . An example is the treatment of thin shells in
ordinary 4-dimensional general relativity. In Ref. [82] a formalism was developed,
making no reference to coordinate systems, in order to study boundary surfaces or
shock waves, where the density jumps, as well as surface layers, where it becomes
infinite. Earlier work on the subject has been done by Refs. [99], [141], and [45].
The same formalism is equally suited for brane cosmology.
We want to know about the local geometry due to some matter distribution.
Therefore we start with the 5-dimensional Einstein equations written in the form
R AB = «s \T AB - - G ab t\ , (3.45)
9 In Newtonian theory, the solution of a matching problem is straightforward: there is no problem
in setting up a well-defined coordinate system, and one simply has to impose continuity and jump
conditions for the Newtonian potential and its first derivative across the hypersurface.
3.2. DIFFERENTIAL GEOMETRICAL PRELIMINARIES 71
where T = T A A denotes the trace of the 5-dimensional energy-momentum tensor.
The unusual factor 1/3 arises because G AB G AB = 5 rather than 4. Assume that
the brane has a finite thickness e, and let Af < and A/" > denote the two boundaries
of A/". A Gaussian normal coordinate y is defined such that y = on Af < and
y = £ on M > . In Gaussian normal coordinates, the 5-dimensional Ricci tensor,
projected onto the brane, can be written as [82]
R CD q c A q° B = —^- + Rab ~ KK AB + 2K A C K CB . (3.46)
dy
This form of the Ricci tensor is inserted into the Einstein equations (3.45), and
both sides are integrated from H < to A/" > ,
aK ~ ■ ■ hi*.-*.,,
1„ , „ „\ ( 3 ' 47 >
Notice that, on the left hand side, all tensors are by definition tangential, whereas
on the right hand side the bulk energy- momentum tensor and the bulk metric have
to be projected onto each hypersurface in the integration domain. It is reasonable
to expect that the second term on the left hand side is bounded between M < and
Af > , and therefore the integral vanishes in the limit e — > 0.
We define the surface energy-momentum tensor of Af by
S AB = lim /
dyT CD q c A q D B , (3.48)
where S = S AB q AB denotes the trace. As a first junction condition, we assume
that the first fundamental form is continuous across A/", i.e.
?A*=?AS- ( 3 - 49 )
In the limit of an infinitely thin brane, Eq. (3.47) then becomes
Kab - K ab = 4 (s AB ~ \ QabS) . (3.50)
In the literature, this is known as Israel's junction condition 10 . The relations (3.49)
and (3.50) guarantee the consistent matching of the bulk geometry across the
brane. In brane cosmology, motivated by the Hofava-Witten compactificaton,
one often works with a ^-symmetric bulk, i.e. with a symmetry under y —> —y.
Then, K< B = -K> B = -K AB , and Eq. (3.50) becomes 11
K AB = ^(s AB - 1 -q AB s) 1 (3.51)
10 Often they are writtei
KJj (SW - Ism" 5 )- where S^
11 The choice of the 'right' i
the discussion in the article '
„ = diag(-p, P,P,P),
5ide of the brane as rej
On CMB anisotropics
rial brane
for the whole bi
r perturbation ii
nely k> v - k< u
ilk can be subtle,
i the bulk'.
See
72 CHAPTER 3. Bl
i.e. the extrinsic curvature of the brane is completely determined by its matter
content.
The power of this approach lies in the fact that it makes no reference to a
particular coordinate system, and that no knowledge is needed about the global
structure of the manifold A4 5 . The basic formalism is valid for time- like (as
assumed here) and space- like hypersurfaces, but not for surfaces that are null,
since then the concept of extrinsic curvature breaks down.
In this section, we have introduced various tensors to describe the geometry of
hypersurfaces, both from the full space-time point of view (q AB ,K AB ), and from
an intrinsic point of view (g^, k^). We have shown how they are related to each
other, and how they can be used for practical calculations. Now, our aim is to
motivate the existence of such hypersurfaces by introducing p-branes, which are
physical objects arising in supergravity and string theory.
3.3 Branes from supergravity
3.3.1 The black p-brane geometry
Originally, p-branes were found as classical solutions to supergravity. Since 11-
dimensional supergravity, compactified on a circle, is equivalent to the low energy
limit of 10-dimensional type IIB string theory, we can start from the equations
of motion (2.7). We are looking for solutions that correspond to a p-dimensional
source of charge Q p under the 4- form potential C4 . The metric is required to have
p-dimensional Euclidean symmetry and, for simplicity, spherical symmetry in the
remaining 10 — p dimensions. The spherical part, including time, has Lorentzian
signature. In the string frame, the solution for the metric is [2]
ds 2 Q = -J J ^dt 2 + Vf-( P )Y, dxldxl + l TTT d P 2 + P 2 f-(py dn2 s- P > ( 3 - 52 )
Vf-(P) i= i J+^Pi
where p is the radial coordinate of the spherical part, x 1 are the spatial coordinates
along the source, and dfig_ p is the volume element of an 8— p sphere. Furthermore,
we have defined
f±{p) = 1 -{-j) ' ^ = -2-7^' ^ = 2-7-7 (3 ' 53)
The solution for the dilaton field is
e- 2 * = g- 2 f-{p)- P -^. (3.54)
Since f-(p) = 1 in the limit p — » 00, g s represents the asymptotic value of the
string coupling constant. For p = 3, the dilaton is constant.
Eq. (3.52) is called black p-brane solution, in analogy with the charged black-
hole or Reissner-Nordstrom geometry. In particular, it has an inner and outer
3.3. BRA GRAVITY 73
horizon, r_ and r + . When r + > r_, the singularity is always shielded by the
horizon, and the p-brane can be regarded as a black-hole. In analogy with the
Reissner-Nordstrom case, this corresponds to the mass (density) of the source
being larger than its charge (density), M p > Q p .
In a string theoretical context, the source is formed by a stack of N Dp-
branes on top of each other. The charge of a Dp-brane is equal to its tension t p ,
given in Eq. (3.73), such that Q p = Nr p . When r + < r_, there is a time-like
naked singularity, which corresponds to the unphysical regime M p < Q p in the
Reissner-Nordstrom case. Finally, if r_ = r + , the lower bound on the mass is
saturated, M p = Q p . Such a solution is referred to as extremal p-brane, and it
corresponds to a ground state. For p ^ 3, the horizon and the singularity coincide,
and the supergravity description for an extremal p-brane breaks down close to
the horizion. Then, one has has to resort to the full string theory description.
For p = 3, however, the surface p = r_ = r + corresponds to a regular horizon,
and it is this case that is particularly interesting for cosmology, if our universe is
identified with a 3-brane moving in the background of a black 3-brane solution.
To discuss extremal and non extremal solutions with r + > r_ , we introduce a
parameter r H measuring the departure from extremality,
r 7 H - p = r 7 + - p - r 7 _- p , (3.55)
and a new radial coordinate r,
r 7- P = pi-v _ r r_-P : ( 356 )
which is suited to describe the region outside the inner horizon. Note that the
p-brane source is located at p = 0, whereas r = denotes the location of the
inner horizon. We also define
H p (r) = l+(^—) 7 ~\ Fp (r) = l-(^) 7 ~ P , (3.57)
where H p is a harmonic function. Then
'--£■ h = F t (3 - 58 '
Inserting Eqs. (3.58) into the solution (3.52), and using that r 7-P-\- r ^ p — H p r 7 -p
and dp = ^g^dr, the black p-brane geometry becomes
As 2 w = H p 1/2 (-F p (r)dt 2 + ^dxMxM + H^ 2 (^— + r 2 drt 2 a _ p J . (3.59)
The metric (3.59) is completely equivalent to the original metric (3.52) in the
region outside the inner horizon. In the extremal limit, r H = and F p = 1, and
the Euclidean symmetry is enhanced to Poincare symmetry. Eq. (3.59) describes
74 CHAPTER 3. Bl
the geometry induced by a p-dimensional source of mass M p and charge Q p .
These two parameters can be expressed in terms of r_ and r + , and are therefore
contained in the function H p (r) and F p (r). For our purpose, they are not relevant,
as we will be merely interested in a particular limit of the metric (3.59), namely an
anti-de Sitter-Schwarzschild space-time (AdSs-S) which we are going to derive in
the next paragraph. Nevertheless, we find it instructive to see how the parameters
of AdSs-S are linked to the fundamental supergravity solution.
Notice that p-branes are solitons of supergravity. When making the analogy
with topological defects, a 0-brane corresponds to a monopole, a 1-branes to a
cosmic string, and a 2-brane to a domain wall. The peculiar case p = —1 is an
instanton. On the other hand, there is a close correspondence to string theory D-
branes. We will comment on this and on the validity of the supergravity solution
in Sec. 3.4.
3.3.2 Anti-de Sitter space-time
The curvature of the geometry (3.59) is set by the horizon r_, which in turn is
determined by the mass and the charge of the p-brane source. It is interesting
to consider the black p-brane solution in the limit r<r_. Because r = corre-
sponds to the horizon itself, this limit means being near the horizon. Therefore,
the limit r <C r_ is commonly called the near horizon limit. Let us define the
variable L = r_ for future convenience, and set p = 3. This choice will be justified
below. Essentially, the near horizon limit then corresponds to the approximation
H 3 (r) = l+(^) 4 ^) 4 . (3.60)
The metric (3.59) then takes the form
= -f(r)dt 2 + 5 (r)^dxMx l + ^— + L 2 dQ 2 .
+ L 2 <mi
(3.61)
Thus, the 10-dimensional space-time splits into A4 5 x S 5 , where A4 5 is a manifold
with anti-de Sitter-Schwarzschild (AdSs-S) geometry and S 5 is a 5-sphere. A4 5 is
a non compact space-time, labelled by coordinates t,x 1 ,x 2 ,x 3 ,r, and contains a
3-dimensional Euclidean subspace. The volume element of the 5-sphere is given
by
dQ 2 , = d$l + sin 2 ((9i) {d6>! + sin 2 (6> 2 ) [d<9 2 + sin 2 ((9 3 ) (d(9| + sin 2 (6» 4 )d6»|)] } .
(3.62)
Moreover, we have defined metric functions f(r) and g{r) for later convenience
(see. Chap. 5). Here, the parameter r H , which classified before whether the
solution is extremal (r H = 0) or non extremal (r H > 0) plays the role of a
3.3. BRA GRAVITY 75
horizon. In fact, without knowing the supergravity origin, one could interpret the
geometry (3.61) as arising from a black-hole at r = 0.
Let us now consider the extremal limit r H = of the metric (3.61). Then,
M. b becomes pure 5-dimensional anti-de Sitter space-time (AdSs ), and together
with the spherical part, this is called the AdSsxS 5 geometry. The metric of the
AdSs part is
r 2 T 2
ds 5 = 72 ( _di2 + MxMx 3 ) + ~^ dr2 - ( 3 - 63 )
In summary, we have shown that AdSs arises from the black 3-brane super-
gravity solution by taking both the near horizon limit and the extremal limit. The
inner horizon r_ has become the curvature radius L of AdSs , since by definition
L — r_. Of course, anti-de Sitter space-time exists also on its own right as a
maximally symmetric solution of Einstein's equations with a negative cosmologi-
cal constant (see Sec. 2.4).
In brane world cosmology, AdSs is particularly important due to the general-
ized Birkhoff theorem [26]. Most brane world models consider an AdSs background,
and our universe is identified with a 3-brane moving in it. The observed homo-
geneity and isotropy can be accommodated by choosing an embedding in which
the brane is parallel to the Minkowskian part of the metric (3.63). This also
explains why we have chosen p = 3 at the beginning.
We end this section by giving some alternative forms of the AdSs metric (3.63)
which are useful in various calculations. First, by defining a new radial coordinate
z = L 2 /r, the metric (3.63) can be cast into the form
ds 2 = — (-di 2 + 6 ij dx i dx j + dz 2 ) . (3.64)
Alternatively, one may work with a radial coordinate g = — L In (r/L), such
that r 2 /L 2 = eT 2e l L . In these coordinates the metric (3.63) reads
ds 2 = e- 2e/L (-dt 2 + Sydx*^) + dg 2 , (3.65)
which is the form used in the so-called Randall- Sundrum model: a Minkowski
metric on a 3-brane is multiplied by an exponentially decreasing 'warp' factor.
Finally, AdSs space-time can be understood as a hyperboloid embedded in a
6-dimensional flat space with coordinates (xo, £1,22, £3, £4, 25) an d a metric
dsg = — dx 2 — dx 2 + da; 2 + dx\ + dx 2 + dx 2 (3.66)
subject to the constraint
-x% -xl + xj + xl + xl + x 2 4 = -const 2 . (3.67)
76 CHAPTER 3. Bl
3.4 Branes from string theory
In string theory there exist objects called Dp-branes, which are hypersurfaces
representing the geometrical locus where open strings end. The letter 'D' refers to
Dirichlet boundary condition, and 'p' stands for the number of spatial dimensions
as we are familiar with from the previous section. Their existence was first claimed
in Ref. [126]. In this section we follow however the more pedagogic treatment
of [124], [125], and [83].
3.4.1 T-duality for open strings
To see how D-branes come about, recall the mode expansion for the open string,
Eq. (2.10). An open string supports oscillatory modes travelling from its left to its
right end, and vice versa. To make this evident, one rewrites the solution (2.10)
™=*"+£*+< m e?
:*)/s
v 2y
(3.68)
The first term in the oscillatory modes is due to left-, the second due to right-
movers. By the Neumann boundary condition they are not independent, therefore
there is just one amplitude a™ for both. Note that we have replaced the index i
in Eq. (2.10) by M = 0, • • • , D — 1 because, after all, the expansion is valid for
all directions. We proceed now similarly as in paragraph 2.3.3: by an analytic
continuation to a Minkowskian world-sheet, one defines a real coordinate cto =
icr and a complex variable z = e cro+lf7 . The time t can then be expressed as
t = ln(|z| 2 )/(2ic). This time we chose the length of the (open) string to be s = tt.
Hence the velocity defined below Eq. (2.9) becomes c = l/(2a'p + ). In terms of
the complex variables z and its conjugate z, Eq. (3.68) reads
X M (z,z) = X M (z) + X M (z)
/ /\ 1/2 M
.,.,,, , . / a \ V"^ a r,
(3.69)
-ia P hx{z) + i\-\ l^—z
This splitting is analogous to Eq. (2.21) and attributes a holomorphic function
to the left-movers and an anti-holomorphic function to the right-movers. We
have added a coordinate x' M whose meaning will soon become obvious. Assume
that one direction, x d say, is a circle with radius R. One can reparameterize the
3.4. BRANES FROM STRING THEORY
compact direction in terms of a so-called T-dual coordinate x' d defined by
X"(z, z) = X*{z)-X*(z)
< + *®. + <*o""E£
(3.70)
Here, x' d is the center of mass in terms of the T-dual coordinate.
Along a compact direction the momentum is quantized according to p d =
n/R, n E Z. However, there is no time dependence of the zero mode and, in
addition, since the oscillator terms vanish at a = 0, n, the endpoints of the open
string do not move in the x' d direction. From Eq. (3.70) one sees that the T-dual
coordinate respects Dirichlet boundary conditions d T X' d = at a = 0, n, whereas
in the other non compact directions, one still has Neumann boundary conditions
d a X M = 0, M 7^ d. If D = 10 say, the open string ends are still free to move
along eight spatial dimensions thereby defining a 8-dimensional hypersurface, a
D8-brane. Note that translational invariance in x' d direction is broken.
The distance between the open string endpoints in the direction x' d is
X' d (a = tt) - X' d (a = 0)= 2a' (^) n = 2ttR'uj, (3.71)
where R' = a'/R is the radius of the circle in the T-dual coordinate, and uj = n
now denotes a winding number. The interpretation of Eq. (3.71) is that for uj =
an open string begins and ends on the same D-brane without winding around
the compact direction, while for uj ^ 0, the string winds around w times before
coming back (see Fig. 3.1).
The existence of winding modes is an intrinsically stringy phenomenon and
will be of great importance in string gas and brane gas cosmology (part III of
this thesis). We end this paragraph by stressing the crucial result that, under
a T-duality transformation, lengths are inverted and momentum modes becomes
winding mode and vice versa
R^^, ti^uj. (3.72)
3.4.2 Some properties of D-branes
So far we have T-dualized only in one direction. One may also compactify other
directions and repeat the procedure described above for the corresponding coordi-
nates. Thereby, one successively generates a D7-brane, a D6-brane and so on. A
Dl-brane is also called D-string to distinguish it from a fundamental string. (As
a difference, for instance, D-strings are much heavier than fundamental strings at
small string coupling.)
CHAPTER 3. Bit
Figure 3.1: The horizontal axis corresponds to the T-dual coordinate x' d . An
open string with winding number zero starts and ends on the same Dp-brane,
whereas an open string with single winding can be represented by identifying the
hyperplanes on the left and on the right. Note that those are not Dp-branes; they
just symbolize the periodic identification.
Let us take a closer look at the action of T-duality. Along the brane, the open
string ends are subject to the Neumann boundary conditions. Upon T-dualizing
in a tangential direction, an additional Dirichlet boundary condition arises, and
a Dp-brane is turned into a D(p-l)-brane. Similarly, a T-duality in an orthogonal
direction turns a Dp-brane into a D(p+l)-brane. T-dualizing twice in the same
direction yields back the original brane. The effect of T-duality on various winding
states is discussed in detail in Sec. 11.4.
For the theory to be consistent, one finds that the dimensionalities of D-branes
must not be arbitrary. In type IIA string theory the allowed branes are those with
even p, hence DO, D2, D4, D6, D8-branes, while in type IIB the brane must have
odd p, hence there are Dl, D3, D5, D7, D8-branes.
As a next issue, we remark that the open string ends can be endowed with
degrees of freedom whose associated wave functions are called Chan-Paton factors.
Imagine a number N of D-branes on top of each other, while the ends of a string
connect some pair in the stack. The end-states are labelled \i) and \j), i =
1, ■ ■ ■ ,N, indicating the branes on which the string begins and ends. Those states
play the role of gauge charges. In a regime where g 8 N <C 1, perturbative field
theory is valid, and the dynamics on the stack of branes are described by a U(N)
gauge theory [2]. This is interesting if the stack is identified with our universe:
gauge particles are confined to the brane world, and therefore forces mediated
by them are not sensitive to the existence of extra-dimensions. In contrast, the
3.4. BRA -iTRING THEORY 79
graviton is an excitation of a closed string, which is not bound to any brane,
and hence gravity feels all space dimensions. This observation could reconcile the
postulate of 'large' extra-dimensions (see Sec. 2.7) with the fact that there is no
evidence for extra-dimensions from collider experiments probing electromagnetic,
weak and strong forces down to 1/(200 GeV).
A Dp-brane is not just a hyperplane, but also a physical object with localized
energy and certain conserved quantum numbers: it is a soliton in string theory.
In particular, any Dp-brane has a (positive) tension
which can be obtained from the amplitude of closed string exchange between two
parallel Dp-branes. Since closed strings give rise to the graviton, this interaction is
attractive. The tension (3.73) is proportional to the inverse of the string coupling-
constant whatever the value of p. (This is a non perturbative effect.)
At the same time, a Dp-brane carries a charge p p , which couples to a Ramond-
Ramond (p+l)-form field as discussed in Sec. 2.2. Conversely, each Dp-brane is
the source of a gauge field in the bulk. Since Dp-branes have the same sign of
charge, the interaction is repulsive. The sum of the two forces is zero if the Dp-
branes are parallel to each other, and if they are BPS states, i.e. their tension
(mass) is equal to their charge, r p = p p . In this case, there is no net force
between the branes, and the system is stable. We investigate BPS and non BPS
configurations in paragraph 5.2.2 by means of an effective potential, and discuss
how instabilities of the latter translate into cosmological perturbations.
It is believed that Dp-branes and p-branes are different descriptions of the
same object. The p-branes are classical solutions of super-gravity, whereas the
Dp-branes arise from a theory taking quantum effects into account. They are
thus the full string theoretical version of p-branes. This interpretation fits well
in the picture, that supergravity is the low energy limit of string theory [2]. The
supergravity description is valid, if the curvature of the p-brane geometry, set by
the horizon r_, is small compared to the curvature defined by the string length
£ a = a' 1 / 2 , i.e. r_ ^> £ s , such that stringy corrections are negligible.
We also remark that the classical parameters r_ , r + can be expressed in terms
of string theory quantities such as g s and N, and that the comparison of p-branes
and Dp-branes has led to the discovery of the anti-de Sitter/conformal field theory
(AdS/CFT) correspondence. For more fundamental issues concerning Dp-branes
we refer the reader to [125].
Finally, we remark that there is an action governing the dynamics of Dp-
branes: this so-called the Dirac-Born-Infeld (DBI) action is the string theory
generalization of the Nambu-Goto action for topological defects. We introduce it
in the next chapter.
After these preliminaries on extra-dimensions and branes, we now turn to main
subject of this thesis which is brane cosmology. Henceforth, we shall use p-branes
and Dp-branes in a cosmological context.
Part II
BRANE COSMOLOGY
Chapter 4
Cosmology on a probe brane
84 CHAPTER 4. COSMOLOGY ON A PROBE BRANE
4.1 Introduction
In chapters 2 and 3 we introduced the idea of extra-dimensions and extended
objects, generalizing the physics of point particles in 4-dimensional space-time to
that of p-branes in D-dimensional space-time. We also mentioned the idea that
in this context our universe may be a 3-brane embedded in a higher-dimensional
space-time. In this chapter, we are going to present a simple model which makes
a natural link between string theory and cosmology.
Let us consider a 10-dimensional space-time, whose geometry is given by a
low energy solution of string theory. We take this to be a 5-dimensional anti-de
Sitter-Schwarzschild geometry multiplied by a 5-sphere (AdSs-SxS 5 ), as derived
in Sec. 3.3. This geometry describes the gravitational field generated by a stack
of 3-branes in the near horizon limit. Suppose now that our universe is another
3-brane moving along a geodesic in this curved background. Consequently, some
dynamics are generated on the brane, even though the background itself is static.
In fact, the motion through the higher-dimensional space-time induces a cosmo-
logical evolution on the brane, mimicking the effect of matter that usually drives
the expansion. In this scenario, there is no real matter on the brane, and in
allusion to the illusion, this scenario is called the 'mirage' cosmology.
We thus consider a 10-dimensional non-compact space-time (A4 10 ,G) with
coordinates x M = (t, x 1 , x 2 ,x 2 , r, 9 1 , 9 2 , 9 s , 9 4 , 9 5 ) and a metric
ds 2 1Q = G MN dx M dx N = -f(r)dt 2 + g(r)S ij dx i dx j + h(r)dr 2 + s(r)dO^. (4.1)
For h = 1//, s = L 2 and where / and g are the functions given in Eq. (3.61),
this is the AdSs-SxS 5 metric, or in the extremal limit AdSsxS 5 . For the time
being, we just assume that the metric coefficients f,g,h,s are some arbitrary
functions of the radial coordinate r, in order to accommodate also other string
theory backgrounds. With this ansatz, there is a Euclidean symmetry along the
3-brane and a spherical symmetry in the other spatial directions. We shall write
out the volume element of the 5-sphere as dfl 2 = s I .,d9'd9 J where /, J = 6, ■ ■ ■ ,10
(see also Eq. (3.62)). The generalization for arbitrary p is straightforward: in this
case, there is a p-dimensional Euclidean subspace, and the spherically symmetric
part is represented by a 8 — p sphere.
We investigate the geodesic motion of a 3-brane (to be identified with our
universe) in the background (4.1). This brane is assumed to be 'light', such that
the back-reaction onto the ambient geometry can be neglected. In analogy with a
test particle, we shall often refer to it as a test or probe brane. Throughout this
chapter we follow closely Ref. [90], in which this idea of mirage cosmology was
first developed.
4.2 Probe brane dynamics
Our aim is to derive an equation of motion for the brane which, in the end, will
translate into a Friedmann-like equation on the brane. A simple way of doing this,
4.2. PROBE BRANE DYNAMICS 85
is to start from the Dirac-Born-Infeld (DBI) action which governs the dynamics
of a Dp-brane, see Ref. [90]. The action we consider is (with p = 3)
Sds = -T 3 / d 4 ae-*^-T 3 / d 4 aC 4 . (4.2)
The first term is a special case of the full DBI action (see Eq. (5.1)), where we
have neglected the antisymmetric tensor field B^ v and the gauge field F^ v . Here,
T 3 is the brane tension, <7 P are the coordinates on the brane worldsheet, <f> is
the 10-dimensional dilaton, and g is the determinant of the induced metric. The
second term is a Wess-Zumino term describing the coupling of the 3-brane to the
Rami itid-Ramond 4-form field in the bulk. The quantity d^aC^ under the integral
represents the pull-back of this 4-form onto the brane.
To calculate the induced metric on the 3-brane, we need to specify an embed-
ding x M = X M {(j^). The requirements of isotropy and homogeneity on the brane
are met by choosing
X° = t, X 1 = x\ X 4 = R(t), 1 = e'(t). (4.3)
Then, the induced metric is
ds 2 A = g^dx^dx" = (-/(#) + h{R)R 2 + s{R)s IJ Q'Q J ^ dt 2 + g(R)S ij dx i dx j ,
(4.4)
the dot denoting derivative with respect to t. From the DBI action, one reads off
the Lagrangian density
£ = -e-*v /3 5-C 4
/ ; (4-5)
= -\JA{R) - B{R)R 2 - D(R)Q 2 - C(R),
where we have cast the determinant of the metric g, and the exponential prefactor
e~* into the functions A(R) , B (R) , D "(R) , to be evaluated at the location of the
brane r = R(t), where
A(R) ee e- 2 *f(R)g(R)\
B(R)^e- 2<s> g(Rfh(R), (4.6)
Z^itO^e- 2 *^) 3 ^).
Furthermore, we have assumed that C(R) = C± also depends only on r = R(t).
The kinetic energy associated with the rotation of the brane is 2 = s /J 0'0 J '.
From the form of the Lagrangian it is clear that there is a conserved energy as well
as a conserved angular momentum around the 5-sphere, according to Noether's
theorem.
The canonical momenta are
ac _ BR
OR ^ A - BR 2 - DQ 2 ' (47)
p _ dc _ p SlJ e J
d® 1 \/A - BR 2 - DQ 2 '
86 CHAPTER 4. COSMOLOGY ON A PROBE BRANE
The Hamiltonian for the probe brane is found by the Legendre transformation
H = PR + P i e 1 -C, (4.8)
and the conserved energy, E = H 7 is
E = E(R,Q) = A +C. (4.9)
\JA - BR 2 - DQ 2
The conserved norm squared of the angular momentum is
J 2 = J 2 (R,G) = s IJ P I P J = ^ — , (4.10)
A - BR 2 - DQ 2
where s IJ is the inverse of s IJy and where we have used that s IJ s IK s JL Q K Q L = G 2 .
Eqs. (4.9) and (4.10) can now be inverted to yield R and as functions of E and
J. First, from Eq. (4.10) one finds
I A _ RE>2^ T 2
(4.11)
D(D + J 2 )
Inserting this expression into Eq. (4.9) and solving for R 2 , one obtains the equa-
tions of motion for the probe brane
*>-^N 4_£±£)1 ( 4. 12)
B [ (E-C) 2 D J '
Here, E and J are considered as parameters of the brane trajectory, and A, B, C, D
are quantities describing the background geometry as well as the 4-form field in
the bulk. Since they depend just on the radial coordinate r = R(t), the problem
is analogous to that of a test particle in a central potential. Eq. (4.12) gives the
radial velocity of the brane as a function of bulk time t. In principle, it can be
integrated to give the radial trajectory r = R(t) of the brane through the bulk.
4.3 Friedmann equation
The induced metric on the brane (4.4) implicitly depends on time. The function
g(R(t)), multiplying the spatial part of the metric, plays the role of the scale
factor, and hence the motion will induce an expansion or contraction on the
brane. Let us write
d s 2 = (-f(R) + h(R)R 2 + s(R)e 2 ) d£ 2 + g(R)5 ij dx i dx j
V / (4.14)
= -dr 2 + a 2 (r) ( 5 ii dxMx i ,
4.3. FRIK STATION
where cosmic time r on the brane is defined by
(S) =-f( R ) + HR)R 2 + s(R)Q 2
{E-cy
The second equality follows by substitution with Eqs. (4.12) and (4.13). By
calculating the derivative of a 2 (r) = g(R(t(r))) with respect to r, one obtains the
relation
where a T is the abbreviation for ^. Using Eqs. (4.15) and (4.12), one gets finally
r(£-C) 2 S e 2 *-/( 5 3 S + J 2 e 2 *)l
&-m i
fg 3 hs J / 417 s
8ttG a
= ~ ^
In the second line, we have reinterpreted the cosmological evolution in terms of a
mirage energy density p e g, which is driving the expansion. However, /9 e g does not
correspond to any real matter on the brane. If there is no motion, g' = dg/dR = 0,
then also p e g = 0.
This Friedmann equation is valid for any background of the form (4.1). In the
special case of AdSs-SxS 5 , the metric is given by (3.61), and the solution 1 for
the 4- form field is C = C4 = — jj + ^fi- Then the Friedmann equation (4.17)
becomes
(*:)"»[(**+ £H_ £4 + #4 + !i£-y, («8>
\ a J L 2 [\ L 4 J a 4 L 2 a 6 a s L 6 a 10 J
where we have absorbed the constant part of C in a redefinition of the energy:
E = E — ^fj. Since g(R) = R 2 /L 2 = a 2 , a cosmological expansion is induced if
the brane moves towards bigger r and vice versa.
Each term in Eq. (4.18) can be associated with a different equation of state
P e ff = uPeS, namely
a 4 3' a e ' a» 3' a 1 " 3" { '
The 1/a 4 term dominates at late times or at big r, and reproduces the behavior
of radiation. This term is always greater or equal than zero because E > 0.
Further remarks on this point are made in paragraph 5.2.3. The early evolution
is characterized by exotic matter with w = 5/3 and u = 7/3.
A deficiency is the absence of a term 1/a 3 , mimicking the evolution due to
dust matter. On the other hand, one has to keep in mind, that Eq. (4.18) is valid
iFor a derivation see [93] and Sec. 5.2.
88 CHAPTER 4. COSMOLOGY ON A PROBE BRANE
only for an AdSs-SxS 5 background. It is quite possible that other string theory
backgrounds could lead to a more realistic Friedmann equation.
In the next chapter, we derive the Friedmann equation on a probe 3-brane in
a 5-dimensional AdSs-S bulk (see Eq. 5.29). Not surprisingly, it turns out that
this is simply the above equation (4.18) with J = 0: by omitting the S 5 part, we
have considered the brane to be located at a fixed point on the sphere, and so its
angular momentum is zero. On the other hand, when there are more than one
extra-dimension, there is greater freedom in the brane motion, and this leads to
the additional terms in the Friedmann equation (4.18) with respect to the one in
five dimensions.
4.4 Discussion
A general interesting feature of higher-dimensional theories is that the initial sin-
gularity problem can be naturally resolved. In Eq. (4.18), a(r) = appears as a
singularity to an observer on the 3-brane, whereas from the 10-dimensional per-
spective, r = is a regular point corresponding to the horizon of the underlying-
black 3-brane solution. In this example, the initial singularity on the brane is
an artefact because the embedding breaks down. More generally speaking, one
expects that also general relativity is only an effective description of a more funda-
mental higher-dimensional theory, which should always be regular. Singularities
such as the big bang then correspond to points, where the effective description
breaks down.
An advantage of mirage cosmology is certainly its relative simplicity, and the
fact that the formalism can be applied to arbitrary p-branes in a D-dimensional
space-time. A major shortcoming, however, seems to be the lack of back-reactions
or self-gravity We are going to comment on this point at the end of Sec. 6.3.
In Chap. 6 and the following, we investigate other brane world models, which
accommodate the back-reaction via junction conditions linking the real energy
content of the brane to the geometry of the bulk. The disadvantage there is that
the approach is limited to the case of one co-dimensison.
It not clear whether the exotic matter contributions (4.19) are compatible
with experiments. Nucleosynthesis constraints suggest that the expansion of the
universe is driven by real matter, and that it is not due to motion only. On
the other hand, real matter can be included in this picture, for instance via the
gauge field F^ v in the DBI action (see Ref. [90]). The most general situation
is obviously a combination of both. It is worthwhile investigating the degree to
which the mirage matter contribution is important.
In the subsequent article, we are studying the evolution of perturbations on a
probe brane moving through an AdSs-S space-time.
Chapter 5
Perturbations on a moving
D3-brane and mirage
cosmology (article)
90 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
This chapter consists of the article 'Perturbations on a moving D3-brane and
mirage cosmology', published in Phys.Rev.D66 (2002), see Ref. [22].
It is also available under http://lanl.arXiv.org/abs/hep-th/0206147.
5.1. INTRODUCTION
Perturbations on a moving D3-brane and mirage
cosmology
Timon Boehm
Departement de Physique Theorique, Universite de Geneve, 24 quai E. Ansermet, CH-1211 Ger
J Switzerland.
D.A. Steer
)ire de Physique Theorique, Bat 210, Universite Paris XI, Orsay Cedex, Fra
Federation de Recherche APC, Universite Paris VII, France.
We study the evolution of perturbations on a moving probe D3-brane coupled
to a 4-form field in an AdSs-Schwarzschild bulk. The unperturbed dynamics
are parameterized by a conserved energy E and lead to a Friedmann-Robertson-
Walker 'mirage' cosmology on the brane with a scale factor a(r). The fluctuations
about the unperturbed worldsheet are then described by a scalar field <f>(r,x).
We derive an equation of motion for </>, and find that in certain regimes of
a the effective mass squared is negative. On an expanding BPS brane with
E = superhorizon modes grow as a 4 while subhorizon modes are stable. When
the brane contracts, all modes grow. We also briefly discuss the case when
E > 0, BPS anti-branes as well as non BPS branes. Finally, the perturbed brane
embedding gives rise to scalar perturbations in the FRW universe. We show
that tj> is proportional to the gauge invariant Bardeen potentials on the brane.
PACS number: 98.80.Cq, 04.25. Nx
5.1 Introduction
The idea that our universe may be a 3-brane embedded in a higher dimensional
space-time is strongly motivated by string and M theory, and it has recently
received a great deal of attention. Much work has focused on the case in which the
universe 3-brane is of co-dimension one [136, 127, 128] and the resulting cosmology
(see e.g. [19, 43, 44]) and cosmological perturbation theory (e.g. [156, 115, 100,
129, 33, 96, 97]) have been studied in depth. When there is more than one extra
dimension the Israel junction conditions, which are central to the 5-dimensional
studies, do not apply and other approaches must be used [35, 90, 6]. In the
'mirage' cosmology approach [90, 92] the bulk is taken to be a given supergravity
solution, and our universe is a test D3-brane which moves in this background
space-time so that its back-reaction onto the bulk is neglected. If the bulk metric
has certain symmetry properties, the unperturbed brane motion leads to FRW
cosmology with a scale factor a(r) on the brane [90, 144]. Our aim in this paper
92 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
is to study the evolution of perturbations on such a moving brane. Given the
probe nature of the brane, this question has many similarities with the study
of the dynamics and perturbations of topological defects such as cosmic strings
[76, 64, 72, 16].
Though we derive the perturbation equations in a more general case, we con-
sider in the end a bulk with AdSs-Schwarzschild xS 5 geometry which is the near
horizon limit of the 10-dimensional black D3-brane solution. In this limit (using
the AdS-CFT correspondence) black-hole thermodynamics can be studied via the
probe D3-brane dynamics [36, 139]. As discussed in paragraph 5.2.1, we make the
assumption that the D3-brane has no dynamics around the S 5 so that the bulk
geometry is effectively AdSs-Schwarzschild. Because of the generalized Birkhoff
theorem [26], this 5-dimensional geometry plays an important role in work on
co-dimension one brane cosmology. Hence links can be made between the unper-
turbed probe brane FRW cosmology discussed here and exact brane cosmology
based on the junction conditions [144]. Similarly, the perturbation theory we
study here is just one limit of the full, self-interacting and non Z 2 -symmetric
brane perturbation theory which has been studied elsewhere [129]. Comments
will be made in the conclusions regarding generalizations of this work to the full
10-dimensional case.
Regarding the universe brane, the zeroth order (or background) solution is
taken to be an infinitely straight brane whose motion is now constrained to be
along the single extra dimension labelled by coordinate r. The brane motion is
parameterized by a conserved positive energy E [90]. In AdSs-Schwarzschild geo-
metry and to an observer on the brane, the motion appears to be FRW expansion
or contraction with a scale factor given by a oc r. Both the perturbed and un-
perturbed brane dynamics will be obtained from the Dirac-Born-Infeld action for
type IIB superstring theory (see e.g. [14]),
S D 3 = -T 3 f dV-^-det^ + 2ira'F liV + B„ v ) - p 3 f dV C 4 . (5.1)
Here a 11 (/x = 0, 1,2, 3) are coordinates on the brane world-sheet, T 3 is the brane
tension, and in the second (Wess-Zumino) term p 3 is the brane charge under a
Ramond-Ramond 4-form field living in the bulk. We will write
Ps = qT 3 , (5.2)
so that q = (— )1 for BPS (anti-)branes. In Eq. (5.1) g^ is the induced metric
and Fpv the field strength tensor of gauge fields on the brane. The quantities B^ v
and C4 are the pull-backs of the Neveu-Schwarz (NS) 2-form, and the Ramond-
Ramond (RR) 4-form field in the bulk 1 . In the background we consider, the dila-
ton is a constant and we set it to zero. In general the brane will not move slowly,
and hence the square root in the DBI part of (5.1) may not be expanded: we will
,ain quantity is the pull-back of a bulk tensor, we use the widehat
e denote the induced metric by g^y rather than <? M „
5.1. INTRODUCTION 93
consider the full non linear action. Finally, notice that since the 4-dimensional
Riemann scalar does not appear in (5.1) (and it is not inherited from the back-
ground in this probe brane approach) there is no brane self-gravity. Hence the
'mirage' cosmology we discuss here is solely sourced by the brane motion, and it
leads to effects which are not present in 4-dimensional Einstein gravity. The lack
of brane self-gravity is a serious limitation. However, in certain cases it may be
included, for instance by compactifying the background space-time as discussed
in [30] (see also [35]). Generally this leads to bi- metric theories. Even in that
case, the mirage cosmology scale factor a(r) which we discuss below plays an
important role and hence we believe it is of interest to study perturbations in this
'probe brane' approach.
Deviations from the infinitely straight moving brane give rise to perturbations
around the FRW solution. Are these 'wiggles' stretched away by the expansion,
or on the contrary do they grow, leading to instabilities? To answer this ques-
tion, we exploit the similarity with topological defects and make use of the work
developed in that context by Garriga and Vilenkin [64], Guven [72], and Battye
and Carter [16]. The perturbation dynamics are studied through a scalar field
(j>(a) whose equation of motion is derived from the action (5.1). We find that, for
an observer comoving with the brane, </> has a tachyonic mass in certain ranges of
r which depend on the conserved energy E characterizing the unperturbed brane
dynamics. We discuss the evolution of the modes <fik for different E and show that
in many cases the brane is unstable. In particular, both sub and superhorizon
modes grow for a brane falling into the black-hole. It remains an open question
to see if brane self-gravity, neglected in this approach, can stabilize the system.
Finally, we also relate cj) to the standard 4-dimensional gauge invariant scalar
Bardeen potentials <& and 1 3/ on the brane. We find that <f> oc VP oc (p (no derivatives
of <p enter into the Bardeen potentials).
The work presented here has some overlap with that of Carter et ai. [41] who
also studied perturbations on moving charged branes in the limit of negligible
self-gravity Their emphasis was on trying to mimic gravity on the brane, and in
addition they included matter on the brane. Here we consider the simplest case in
which there is no matter on the brane, namely F^ = in Eq. (5.1). Our focus is
on studying the evolution of perturbations solely due to motion of the brane: we
expect the contribution of these perturbations to be important also when matter
is included. Moreover, we hope that this study may more generally be of interest
for the dynamics and perturbations of moving D-branes in non BPS backgrounds.
The outline of the paper is as follows. In section 5.2 we link our 5-dimensional
metric to the 10-dimensional black D3-brane solution and specify the unperturbed
embedding of the probe brane. To determine its dynamics from the action (5.1)
the bulk 4-form RR field must be specified. We discuss the normalization of
this field. At the end of the section we summarize the motion of the probe
brane by means of an effective potential. Comments are made regarding the
Friedmann equation for an observer on the brane. In section 5.3 we consider small
deviations from the background brane trajectory and investigate their evolution.
94 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
The equation of motion for (f> is derived, and we solve it in various regimes,
commenting on the resulting instabilities. In section 5.4 we link </> to the scalar
Bardeen potentials on the brane. Finally, in section 5.5 we summarize our results.
5.2 Unperturbed dynamics of the D3-brane
In this section we discuss the background metric, briefly review the unperturbed
D3-brane dynamics, and comment on the cosmology as seen by an observer on the
brane. The reader is referred to [90, 122] for a more detailed analysis on which
part of this section is based.
5.2.1 Background metric and brane scale factor
For the reasons mentioned in the introduction, we focus mainly on an AdSs-
SxS 5 bulk space-time. This is closely linked to the 10-dimensional black 3-brane
supergravity solution [78, 2, 93] which describes N coincident D3-branes carrying
RR charge Q = NT 3 and which is given by
ds 2 = J? 3 ~ 1/2 (-Fdt 2 + dx ■ dx) + H^ 2 (-^r + r 2 dQ 2 J , (5.3)
where the coordinates (t,x) are parallel to the N D3-branes, dH 2 , is the line
element on a 5-sphere, and
jy 3 (r) = l + ^-, F=l- r -^. (5.4)
The quantity L is the AdSs curvature radius and the horizon r H vanishes when
the ADM mass equals Q. The link between the metric parameters L, r H and the
string parameters N, T 3 is given e.g. in [93] . The corresponding bulk RR field
may also be found in [93].
The near horizon limit of the metric (5.3) is AdS5-SxS 5 space-time [2]. Our
universe is taken to be a D3-brane moving in this background. We make the
following two assumptions: first, the universe brane is a 'probe' so that its back-
reaction on the bulk geometry is neglected. This may be justified if N 3> 1.
Second, the probe is assumed to have no dynamics around S 5 so that it is con-
strained to move only along the radial direction r. This is a consistent solution
of the unperturbed dynamics since the brane has a conserved angular momentum
about the S 5 , and this may be set to zero [90, 144]. In section 5.3 we assume that
is also true for the perturbed dynamics. Thus in the remainder of this paper we
consider an AdSs-S bulk space-time with metric
ds 2 = -f(r)dt 2 + g(r)dx ■ dx + h(r)dr 2
= G MN dx M dx N ,
5.2. UNPERTURBED DYNAMICS OF THE D3-BRANE 95
where (for r > r H )
/«-£(»-£)• «)-£ w-wy (5 - 6)
In the limit r H — > this becomes pure AdSs.
More generally, by symmetry, a stack of non rotating D3-branes generates a
metric of the form ds\ = ds 2 + k(r)dQ 2 , where ds 2 is given in Eq. (5.5) [145].
In this case, since the metric coefficients are independent of the angular coordi-
nates (0 1 , ■ • • ,9 5 ), the unperturbed brane dynamics are always characterized by
a conserved angular momentum around the S 5 [90] . As a result of the second as-
sumption above, we are thus effectively led to consider metrics of the form (5.5).
Hence, for the derivation of both the unperturbed and perturbed equations of
motion, we keep f,g,h arbitrary and consider the specific form (5.6) only at the
cud.
The embedding of the probe D3-brane is given by x M = X M (x^). (We have
used reparametrization invariance to choose the intrinsic worldsheet coordinates
a ii = x fi -j p or fa e unperturbed trajectory we consider an infinitely straight brane
parallel to the x^ hyperplane but free to move along the r direction:
X» = x*, X i = R(t). (5.7)
Later, in section 5.3, we will consider a perturbed brane for which X 4 = R(t) +
5R(t,x).
The induced metric on the brane is given by
so that the line element on the unperturbed brane worldsheet is
ds 2 4 = g^dx^dx" = -(/(.R) - h(R)R 2 )dt 2 + g(R)dx ■ dx = -dr 2 + a 2 (r)dx ■ dx.
(5.9)
An observer on the brane therefore sees a homogeneous and isotropic universe in
which the time r and the scale factor a(r) are given by
-H>
(f - hte)dt, a(r) = VgjRijT)- (5-10)
The properties of the resulting Friedmann equation depend on f(R),g(R),h(R)
(i.e. the bulk geometry) as well as R (the brane dynamics) as discussed in [90, 144]
and summarized briefly below.
5.2.2 Brane action and bulk 4-form field
In AdSs-S geometry, B MN vanishes, and we do not consider the gauge field F^
on the brane. (For a detailed discussion of the unperturbed brane dynamics with
96 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
and without F^, which essentially corresponds to radiation on the brane, see
[90. 144]. Non zero B MN has been discussed in [163].) Thus the brane action
(5.1) reduces to
S m = -T 3 jd 4 xV^-p3Jd 4 xC 4 , (5.11)
, , , - dX M dX N dX s dX R . nN
, = det(^), C^C^ — — — — , (5.12)
and C MNSR are components of the bulk RR 4-form field. The first term in
Eq. (5.11) is just the Nambu-Goto action.
In the gauge (5.7), g and C 4 depend on t only through R. Thus rather than
varying (5.11) with respect to X M and then integrating the equations of motion,
it is more straightforward to obtain the equations of motion from the Lagrangian
-\/fg 3 -g 3 hE?-C, (5.13)
where C = C(R) = ^~C 4 = qC 4 . Since £ does not explicitly depend on time, the
brane dynamics are parameterized by a (positive) conserved energy E = ^-R — C
from which
v-ft-A)- (5 - 14)
Transforming to brane time r defined in equation (5.10) yields
where the subscript denotes a derivative with respect to r.
In order to analyze the brane dynamics in AdSs-S space-time, where /, g and
h are given in Eq. (5.6), one must finally specify C(R) or equivalently the 4-form
potential C MNSR . To that end 2 recall that the 5-dimensional bulk action is
S 5 = ^[d 5 xV=G(R-2A)-^[F 5 A*F 5 , (5.16)
where A is the bulk cosmological constant 3 and F 5 = dC 4 is the 5-form field
2 For the 10-dimensional AdSs-SxS 5 geometry the solution for the 4-form field is given, for exam-
ple, in [93]. For completeness, we rederive the result starting directly from the 5-dimensional metric
(5.6).
3 Note that there is no cosmological constant in the fundamental 10-dimensional supergravity
action 2.1. Here, A appears because we consider only the AdSs part. It represents the negative
cosmological constant of AdSs which, together with the positive Ricci curvature form the 5-sphere,
adds up to zero. Thus A is needed for consistency of the equations of motion.
5.2. UNPERTURBED DYNAMICS OF THE D3-BRANE 97
strength associated with the 4-form C4. The resulting equations of motion are
iW = \KG MN + ^ (f mbcde F n — - ^F ABCDE F—G MN ) ,
(5.17)
d*F 5 = i 7 l=((^ + 3^ + ^)F 012 3 4 -2^ 12 3 4 )d, = 0, (5.18)
where the prime denotes a derivative with respect to r. In AdSs-S, R MN =
— -p-G MJV and Eq. (5.18) gives
^ (^01234 " Fo 1234 ) = => F 01234 = C-^, (5.19)
where c is a dimensionless constant (see for example [40]). (Note that this solution
satisfies dF$ = since the only non zero derivative is c?4-Fbi234 which vanishes on
antisymmetrizing. ) Integration gives
C i23 = v^+w, (5.20)
where v — c/4 and w are again dimensionless constants. Hence the function C{r)
appearing in Eq. (5.13) is
C(r) = qCoi23 = qv T jj + qw. (5.21)
In 10 dimensions the constant c (and hence v) is fixed by the condition J *F =
Q , and w may be determined by imposing (before taking the near horizon limit
and hence with metric (5.3)) that the 4-form potential should die off at infinity
[93]. This second argument is not applicable here. Instead, we fix v and w in
the following way: consider the motion of the unperturbed brane seen by a bulk
observer with time coordinate t. One can define an effective potential V* s through
^R 2 + Kg = E (5.22)
so that on using equation (5.14),
V^E, qi R)=E- l -{^)\
(see Fig. 5.2) where
and C = C{R) is given in Eq. (5.21). We now use the fact that there is no net
force between static BPS objects of like charge, and hence in this case the effective
98 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
potential should be identically zero. Here, such a configuration is characterized
by r H = 0,q = 1,E = 0: imposing that V* s = for all R, forces v — ±1 and,
in this limit, w = 0. Second, we normalize the potential such that V* s (E,q =
1, R — > oo) = for arbitrary values of the energy E and r H . This leads to
v=-l, „ = +£. (5.24)
In particular for E = 0, the brane has zero kinetic energy at infinity. Even in this
case the potential is not flat, unless r H = 0, as can be see in Fig. 5.1. According
to this normalization
C(r) = -?p + ?^<0 (5.25)
as in the 10-dimensional case [93]. Notice that, since the combination appearing
in the equation of motion for R is E — C, the constant w only acts to shift the
energy. For later purposes, we define the shifted energy E by
E = E-qw = E-q^. (5.26)
Figure 5.1: V* s (E,q,R) for E = 0, q = 1, L = 4 and different values of r H . For
R — > oo the potential goes to zero according to our normalization. When r H = 0,
the potential is exactly flat.
Finally, we comment that substitution of Eq. (5.19) into Eq. (5.17) determines
the bulk cosmological 4 constant to be given by L 2 A = — 6 — c 2 /4 = —10.
4 Equivalently we could have started from the 10-dimensional SUGRA action, used the 10-
dimensional solution for F5 (which is identical to (5.25)) and then integrated out over the 5-sphere.
After definition of the 5-dimensional Newton constant in terms of the 10-dimensional one, the above
cosmological constant term is indeed obtained, coming from the 5-sphere Ricci scalar.
5.2. UNPERTURBED DYNAMICS OF THE D3-BRANE 99
5.2.3 Brane dynamics and Friedmann equation
We now make some comments regarding the unperturbed motion of the 3-brane
through the bulk, R(t), as seen by an observer on the brane. This will be useful
in section 5.3 when discussing perturbations. Recall that since a(r) = R(t)/L
(see Eq. (5.10)), an 'outgoing' brane leads to cosmological expansion. Contraction
occurs when the brane moves inward. For the observer on the brane, one may
define an effective potential by
\r t 2 + Kff = E, (5.27)
whence, from Eq. (5.15),
V^{E iq ,R) = E+ l -(^ U^J S -(E-C) 2 . (5.28)
Consider a BPS brane q = +1 (see Fig. 5.3). As noted above, for r H = E =
one has V^ s = so that the potential is flat. For r H =t 0, VJ S contains a term
ex — i?~ 6 , and the probe brane accelerates toward the horizon, which is reached
in finite (r-)time. On the other hand, for a bulk observer with time t, it takes
infinite time to reach the horizon where V^* ff = E (see Fig. 5.2).
From Eqs. (5.15) and (5.21) it is straightforward to derive a Friedmann-like
equation for the brane scale factor a(r) [90, 144]:
a 2
E +-(2qE+ r -f-\+{q 2 -l) . (5.29)
The term in 1/a 8 (a 'dark fluid' with equation of state P = 5/3p) dominates at
early times. The second term in 1/a 4 is a 'dark radiation' term. As discussed in
[144], the part proportional to r H corresponds to the familiar dark radiation term
in conventional ^-symmetric (junction condition) brane cosmology, where it is
associated with the projected bulk Weyl tensor. When E is non zero, Z-i symmetry
is broken 5 [144] and this leads to a further dark radiation term [40, 98]. The
last term in Eq. (5.29) defines an effective 4-dimensional cosmological constant
4^ — tM^ 2 ~~ 1) wn i cn vanishes if the (anti) brane is BPS (i.e. q = ±1). All these
terms have previously been found in both mirage cosmology and conventional
brane cosmology [144, 40].
Notice that the dark radiation term above has a coefficient
V^2qE+ r ^ [ = 2qE- r ^(q 2 -l), (5.30)
which is positive for q = +1 (since E > 0). However, for BPS anti-branes q = — 1,
the coefficient (5.30) is negative unless E = 0. Thus when E ^ and q = — 1
5 When making the link between mirage cosmology and the junction condition approach, E <x
M_ — M + where M± are the black-hole masses on each side of the brane [144].
CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
Figure 5.2: V* s (E,q,R) for E = 2, r H = 1, L = 4. For a BPS-brane (q = 1),
K*ff — > as it! — > oo according to our normalization. This should be contrasted
with a non BPS brane e.g. with q = 1.2. Note that V* s (E,q,R = r H ) = E. Any
inwardly moving (contracting) brane takes an infinite amount of i-time to reach
the horizon.
there is a regime of R for which H is negative. In Fig. 5.3 this is represented by
the forbidden region where the potential exceeds the total energy E. At V e ^ = E
the Hubble parameter is zero and an initially expanding brane starts contracting.
On the contrary, we do not obtain bouncing solutions in our setup, regardless
of the values of q and E. Bouncing and oscillatory universes are discussed in
e.g. [112, 84, 31].
The Friedmann equation (5.29) can be solved exactly. In the BPS case, A 4 — 0,
the solution is
«W 4 = af + g(r - n) 2 ± i(r - n)(E 2 + fiaf) 1 / 2 , (5.31)
where a; is the value of the scale factor at the initial time Tj, and the ± determines
whether the brane is moving radially inward or outward. In the next section when
we solve the perturbation equations, it will be sufficient to consider regimes in
which only one of the terms in Eq. (5.29) dominates. These will be given in
section 5.3.
One might wonder whether it is possible to obtain a term oc 1/a 3 (dust) in
the Friedmann equation, and also one corresponding to physical radiation on the
brane (rather than dark radiation). Physical radiation comes from taking F^ v /
in (5.1) [90], and a 'dark' dust term has been obtained in the non BPS background
studied in [30]. Finally, a curvature term 1/a 2 has been obtained in [164].
5.3. PERTURBED EQUATIONS OF MOTION
Figure 5.3: V^ S (E, q, R) for the same parameters as in Fig. 5.2. A BPS brane has
zero kinetic energy at infinity corresponding to a vanishing cosmological constant
on the brane. Otherwise, the cosmological constant is oc q 2 — 1. A BPS anti-brane
is allowed to move only in a restricted range of R: after having reached a maximal
scale factor, the universe starts contracting. Any inwardly moving brane falls into
the black-hole in a finite r.
5.3 Perturbed equations of motion
In this section we consider perturbations of the brane position about the zeroth
order solution R(t) given in Eq. (5.14). Once again we work with the metric (5.5),
specializing to AdSs-S geometry only at the end. The perturbed brane embedding
X 4 — R(t) + 8R(t, x) leads to perturbations 5g^ v of the induced metric on the
brane and these are discussed in section 5.4. Note that these perturbations about
the fiat homogenous and isotropic solution are not sourced by matter on the
brane, and their evolution will depend on the unperturbed brane dynamics and
hence on E. We now derive an equation for the evolution of the perturbed brane
to see if there are instabilities in the system.
5.3.1 The second order action
Since we consider a co-dimension one brane, the fluctuations about the unper-
turbed moving brane can be described by a single scalar field 0(x M ) living on
the unperturbed brane world sheet [72]. To describe the dynamics of 0(x p )
(which is defined below), we use the covariant formalism developed in [72] to
study perturbed Nambu-Goto walls. (For other applications, see also [64, 34].)
102 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
The perturbed brane embedding is given by
X M (t,x) = X M (t) + 0(t,x)n M (t) 7 (5.32)
where X M {t) is the unperturbed embedding, and physical perturbations are only
those transverse to the brane (see also section 5.4). The unit space-like normal
to the unperturbed brane, n M {t) = n M (X(t)), is defined through
G MJV n M ^— = 0, G MN n M n N = 1, (5.33)
so that
ti m ^(rJ ^^o,0,0,J ^-^)- (5.34)
Thus for a 5-dimensional observer comoving with the brane, <f> (which has dimen-
sions of length) is the measured deviation from the background solution of the
previous section [64]. For an observer living on the brane, the perturbations in
the FRW metric generated by </> are discussed in section 5.4 in terms of the gauge
invariant scalar Bardeen potentials.
An equation of motion for <f> can be obtained by substituting (5.32) into the
action (5.11) and expanding to second order in </>. The terms linear in <j) give the
background (unperturbed) equations of motion studied in the previous section;
now we are interested in the terms quadratic in <f> which give the linearized equa-
tions of motion. A similar analysis was carried out by Garriga and Vilenkin [64]
for Nambu-Goto cosmic domain walls in Minkowski space and was generalized by
Guven [72] for arbitrary backgrounds. For the action (5.11), the quadratic term
is [34]
S^ = ~\ |d 4 aV=^ [(V^XV'V) - {k%k\ + R MN n M n N ) </> 2 ] .
(5.35)
Here V is the covariant derivative with respect to the induced metric g^ y , and
the extrinsic curvature tensor k^ v on the brane is given by
^ = -(V N n A/ )^^, (5.36)
ox^ ox"
where V is the covariant derivative with respect to the 5-dimensional metric G M N .
Finally, R MN is the Ricci tensor of the metric G MN . Apart from </>, all the terms in
(5.35) are unperturbed quantities. Note that there is no contribution to S^i from
the Wess-Zumino term of the action (5.11): all terms quadratic in </> cancel since
C0123 is the only non zero component of the 4-form field. However, C does enter
into the term linear in (j> and hence into the background equations of motion, as
analyzed in the previous sections.
Variation of the action (5.35) with respect to <j) leads to the equation of motion
V^V^ + (fc^fc% + R MN n M n N ) <p = (5.37)
5.3. PERTURBED EQUATIONS OF MOTION
or equivalently
V^V^ - m 2 (p = 0, (5.38)
where
m 2 = -(k fi u k v li + R MN n M n N ). (5.39)
To determine the extrinsic curvature contribution to (5.39), it is simpler to cal-
culate first the 5-dimensional extrinsic tensor or second fundamental form defined
by Eq. (3.22)
K MN = -q B N q c M V c n B , (5.40)
where q B N acts as projection tensor onto the brane. It is the mixed tensor asso-
ciated with the first fundamental form
q MN = G MN - n M n N . (5.41)
We then use the fact that
k^ v k v ^ = K M N K N M . (5.42)
On defining T by
T.m-f-**-,>''
\dtj J (E-C) 21
the non zero components of K M N are
K\ - -^fW(A-!L# + ™# + lLy (5.44)
(5.45)
h R r
T"
a "' - -Mi) \i=**=#« <" 6 >
K\ = ^-K\ (5.47)
so that
^w"»*"-MK7) +3 te + (^) ■ (5 - 48 »
The Ricci term is
3 I
Th\f g g \gj g h\
(5.49)
104 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
Collecting these results gives
h'J C
f9 3 h \f g g \gj g h gE-C
A (—)
3 \E-CJ
ft / \f) f g fh\
(5.50)
In the remainder of this section we try to obtain approximate solutions for <f>
from equation (5.38). Some aspects of this calculation are clearer in brane time
t, and others in conformal time 1} (where r\ = f dr/a^r)). Of course the results
are independent of the coordinate system. For these reasons we have decided to
present both approaches, beginning with brane time.
5.3.2 Evolution of perturbations in brane time r
On using the definition of brane time r in equation (5.10), the kinetic term in
(5.37) is given by
V^V M = -0 TT - 3tf0 T + ^ (<t>xi x i + 4>x*& + 0x3.3) . (5.51)
(In conformal time the factor of a~ 2 multiplying the spatial derivatives disappears,
see below.) We now change variables to ip = a 3 ^ 2 4> so that Eq. (5.38) becomes
<Ptt ~ -o (Vxi«* + VxW + V&x*) + M 2 (t)<p = 0, (5.52)
where
M 2 (r) = m 2 -
3 \g'
g 4 \g J ' g
3 (E-C) 2 \ lfg' g" 13 (g'V 1 g' h' g' C
' 4 fgtfi | 2 / g g 4 \g J 2g h gE-C
3 \E-CJ
.lIoII- flL\ 2 J'9^_f' h ' q 5 "_ 3 f 9 'Y_ 39 ' h ']
"47 / \1) + 77 7^ + 3 7 iW 27X
(5.53)
5.3. PERTURBED EQUATIONS OF MOTION
This expression is valid for any /, g, and h. We now specialize to AdSs-S geometry
in which case
33 E 2 3 1/-, r|\ 25. 2 .1
4 o'
(5.54)
^r ? "y"
Notice that there are regimes of a in which M < such as, for instance, for
small a where the a~ 8 term dominates, and furthermore that the location of these
regimes depends on the energy E of the brane. We also see that since M 2 ~ H 2 ,
instabilities will occur for modes with a wavelength greater than H _1 . Figure 5.4
shows the typical shape of M 2 as a function of a for fixed energy and different
q. In the following, we discuss only cases with q 2 > 1 as then the 4-dimensional
cosmological constant is positive.
Figure 5.4: The dimensionless quantity M 2 L 2 (on the vertical axis) is plotted as
a function of a for E = 1,L = l,r H = 1. Here, the effective mass squared is
positive in a certain range only for the BPS-brane. Note that the negative M 2 L 2
region is not hidden behind the horizon.
Analysis of equation (5.52) is simpler in Fourier space where
y? fc (r) = J d 3 x p(-r, x)Q~ its , (5.55)
and k is a comoving wave number related to the physical wave number k p by
k = ak p . Thus (5.52) becomes
- 2 (k 2 - k 2 (r)) y k = 0, (5.56)
106 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
where the time dependent critical wave number k 2 (r) is given by
k 2 (r) = -M 2 (r)a 2 . (5.57)
One might suppose that for M 2 > all modes are stable. However, due to
the r-dependence of k c this is not necessarily true (as we shall see in equation
(5.71)).
Our aim now is to determine the a-dependence of ipk- We proceed in the
following way: notice first that the Friedmann equation (5.29) and the expression
for M 2 (t) in (5.54) both contain terms in a~ 8 , a~ 4 , and a . We will focus on
a regime in which one of these terms dominates. Then the Friedmann equation
can be solved for a(r) which, on substitution into (5.54), gives M 2 (t). A final
substitution of M 2 (t) into the perturbation equation (5.56) for ip^ enables this
equation to be solved in each regime. We consider the following cases: i) q = +1,
ii) q = — 1 and Hi) q 2 > 1.
BPS brane: q = +1
For a BPS brane, the Friedmann equation (5.29) and effective mass M 2 (t) are
given by
U
E 2
(5.58)
The E dependence of these equations slightly complicates the analysis of these
equations, and hence we begin with the simplest case in which E = 0.
Case 1 E = 0:
When E = (the static limit in which the probe has zero kinetic energy at
infinity (see Fig. 5.1)) only the term proportional to a~ 8 survives in (5.58) and
(5.59). Recall that when r H vanishes the potential VJ S is fiat. Furthermore, since
E oc r\ = 0, it follows from (5.59) that M 2 {t) = in this limit: as expected, a
BPS probe brane with zero energy in AdSs has no dynamics and is completely
stable.
When r H ^ 0, M 2 (t) < Vr, and the solution of (5.58) is
a(r) A = a1±-^-(T-n). (5.60)
Here at > a H = r H /L is the initial position of the brane at r = n, and the
choice of sign determines whether the brane is moving radially inward ( — ) or
outward (+): this is a question of initial conditions. Let R^ = \/\Ha\ denote the
5.3. PER' : CATIONS OF MOTION 107
(comoving) Hubble radius. Then it follows from (5.59) and the definition of k\ in
(5.57) that
l~|fc c ( T )|~|jy | = -^-, (5.61)
where we neglect numerical factors of order 1. Thus the critical wavelength is
A c ~ Rh- (Notice that Rh is minimal at a H and increases with a.)
For superhorizon modes A ^> Rh or \k\ >C \k c \, and in this limit the perturba-
tion equation (5.56) becomes
Vk,TT " ^f^' = 0- (5-62)
On inserting solution (5.60) into fc 2 one obtains
^k=^ = A k a 4 + B k a- 3 , (5.63)
(where the constants A k and B k are determined by the initial conditions). Hence
if the brane moves radially outward the superhorizon modes grow as a 4 oc r. If
the brane is contracting they grow as a~ 3 . In the near extremal limit, r H <C L
or a H <C 1, the amplitude of these superhorizon modes can become very large,
suggesting that they are unstable. Of course our linear analysis will break down
when (j> becomes too large.
Consider now subhorizon modes A <C Rh or \k\ ^> \k c \. Then (5.56) is just
ip kTT + (fc 2 /a 2 ) ipii = 0. However, in this case it is much easier to solve the
equation in conformal time rj where the factor of a~ 2 is no longer present. We
anticipate the result from paragraph 5.3.3: it is
ikr] p-ifei?
4> k = A k + B k . (5.64)
a a
For an outgoing brane a increases and subhorizon modes are stable. For an ingoing
brane a decreases, and the amplitude of the perturbation becomes very large in
the near extremal limit. (Note that, as the brane expands, superhorizon modes
eventually become subhorizon, and similarly, on a contracting brane, subhorizon
modes become superhorizon.)
To conclude, when r H ^ 0, E = 0, and the brane expands, superhorizon modes
are unstable while subhorizon modes are stable. For a contracting brane, and in
the near extremal limit, both super and subhorizon modes are unstable.
Case 2 E / 0:
When the energy of the brane is non zero the situation is more complicated.
Notice first from (5.59) that M 2 (t) has one zero at a = a c given by
Oc=^- (5-65)
108 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
Hence M 2 {t) is negative when a < a c and positive for a > a c (see Fig. 5.5).
However, since a c is E dependent, there may be ranges of E for which the negative
mass region is hidden within the black-hole horizon. Indeed, we find
a c < a H ^=^> E_ <E < E + , (5.66)
Et = g(13±4V3). (5.67)
The situation is shown schematically in Fig. 5.5.
Figure 5.5: The curve represents a c , the zero of M 2 (r), as a function of the energy
E as given in Eq. 5.65. Below the curve the effective mass squared is negative,
above it is positive. For E < E- and E > E + the M 2 {t) already becomes
negative outside the horizon, whereas for energies within the interval E-, E + the
M 2 (t) < region is hidden within the horizon. The parameters chosen are q = 1,
r H — 1) and L — 1.
Now consider H 2 given in Eq. (5.58). The two terms are of equal magnitude
when a = a c = (E 2 /2E) 1 / 4: ~ a c . Thus when aCfl c (and hence in the regions in
which M 2 < in Fig. 5.5), the dominant term in H 2 is the one proportional to
a~ 8 . The system is therefore analogous to the one considered above when E = 0,
and for superhorizon modes the solution is given in (5.63): for an outgoing brane
4>k ~ a 4 . When E ~ E + or E ~ E_, these regimes extend down to the black-
hole horizon: thus in the near extremal limit the contracting brane will again be
unstable since 4>k ~ a -3 -
When a 3> a c (and hence in the regimes in which M 2 > in Fig. 5.5), the
5.3. PER' : . NATIONS OF MOTION 109
dominant term in H 2 is ex a~ 4 so that
a(r) 2 = o?±2^|— (r-Ti) (5.68)
and
kl{r) = -M\r)a 2 = -\-^- 2 . (5.69)
On superhorizon scales the mode equation is
At first sight one might expect the solution to this equation to be stable since
M 2 > 0. However, surprisingly, it is not. (Indeed, below we will see that in
conformal time the effective mass is actually negative in this regime.) A change
of variables to u = a 2 shows that the solution of (5.70) is
ip k = A k o?l 2 + B k a x ' 2 (5.71)
which grows as r 3 / 4 ^ 1 / 4 respectively. Finally,
</> k = A k + B k a- 1 . (5.72)
For E within the band E_ ~ E ~ E + , the solution (5.72) for the modes is valid
for all a so that superhorizon modes grow as a -1 as the brane approaches the
black-hole horizon.
When E ~ E + or E ~ E_ these solutions are valid for a 3> a c . Thus for
an expanding brane <p k tends to a constant value. For a contracting brane, the
term a a -1 could become important, although for small enough a the relevant
regime is that considered above in which case the solution is given by (5.63) and
the superhorizon modes grow as a~ 3 .
For subhorizon modes, the solution is still as given in (5.64).
BPS anti-branes: q = — 1
Now the Friedmann equation (5.29) and effective mass M 2 (t) become
(5.73)
-V> - 4 [5* + ||]
so that M 2 is always negative, independently of E. Note that H 2 > for a < a c
where a c = (E 2 /2E) 1 ^. However, since E = E -\- a%/2 for anti-branes, it follows
that a c > a H for all E (i.e. there are no energy bands to consider in the case of
anti-branes). When a ^C a c , H 2 oc M 2 oc a~ 8 and once again this is analogous
to the case studied above for E = 0: superhorizon modes grow as a 4 , and in the
near extremal limit the subhorizon modes on an ingoing brane are unstable.
110 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
Non BPS branes: q ^ ±1
Here we shall only briefly discuss the case q 2 > 1 for large a. Now, independently
of E, there is a cosmological constant dominated regime (see Eq. (5.29)). There
the solution for the scale factor is
o(r) = a(r i )e ± v / ^W3(--^) where ^ = ^7^- (5-75)
In this regime, however, M 2 is negative with
25
-12 A4
(5.76)
and R h = l/\Ha\ = (A^r^V 1 .
For subhorizon modes (A <C Rh) the solution for ifk is again given by (5.64).
For superhorizon modes, and considering an outgoing brane, there is an exponen-
tially growing unstable mode
<j) k = A A; eV / ^W3( r - r -) = AfeCL ( 5 7? )
Hence, this non BPS brane is unstable for large a. It is not clear to us why the
acceleration due to the positive cosmological constant does not rather stretch the
perturbations away.
5.3.3 Comments on an analysis in conformal time 77
It is instructive to carry out a similar analysis in conformal time rj rather than
cosmic time r, and we comment briefly on it here. In conformal time and trans-
formed to Fourier space, Eq. (5.38) becomes
<t>k m + 2H(t>k,r, + (k 2 + a 2 m 2 )<f> k = 0, (5.78)
where H = aH. The friction term can be eliminated by a change of variables to
ijj = a<f>, and the above equation becomes
TPk, vv + (k 2 -k 2 c (r)))^k = 0, (5.79)
k 2 M = -M 2 (t 1 )
5.3. PERTURBED EQUATIONS OF MOTION
and
M 2 (f]) = a 2 m 2 — a m /a
(E-C) 2 \lf>g> g" fg'V 1 g> h> b g> C
f9 2 h [2fg g \g J 2gh 2gE-C ( 5 .S
2h[f 2\f) + 2fg 2fh + g 2 g h
Specializing to A0IS5-S yields
^ 2 W = -^te + 6(g 2 -l)a 2 |. (5.81)
Notice that in conformal time and for \q\ > 1, M 2 (77) is always negative inde-
pendently of E. From this, one can immediately see the instability for small k in
Eq. (5.71), even though M 2 (t) can be positive in that case. It is clear that the
results on brane (in)stability must be independent of whether or not the analysis
is carried out in r/ or r time. We will see that this is indeed the case: the reason
is that not only the sign of the effective mass squared but also its functional de-
pendence on time determine the stability properties. We now summarize briefly
some of the aspects that differ between the r and r\ analysis.
Consider the simplest case: q = +1 and E = 0. The solution of the (conformal
time) Friedmann equation is a 3 = a 3 ± 3a? H (ri — rji)/2L 7 and k c (rj) ~ \H\ = 1/Rh-
For superhorizon modes |fc| <C \k c \ Eq. (5.79) reduces to V ; a.-. w — 'vO^V-'fc = 0.
Given a{rj) and hence k c {a{rf)) it is straightforward to find the solution which is,
as expected, exactly that given in (5.63). For subhorizon modes, |fe| ^> \k c \, the
solution was given in (5.64).
Consider now q = +1,E > 0. Recall that in the r time analysis both M 2 (t)
and H 2 contained terms in a~ 4 and a~ 8 and, in particular, there was a regime
in which M 2 (t) was positive and proportional to a~ 4 ex H 2 . In 77 time, however,
Tl oc a~ 6 + a~ 2 with Ai 2 is always negative and oc — a~ 6 . Thus while the a <C a c
regime reduces to that discussed above for E = 0, the a ^> a c regime is a little
less clear. There 7i 2 ~ a(rj)~ 2 , but A4 2 ~ — a(r/) -6 . Thus
a( v ) = a i ±^-(r,-r h ) (5.82)
112 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
and
10E 2
k c (r,) 2 = -M(r)) 2 = j^r. (5.83)
Now |fe c (?7)| ~ \H\ 3 L 2 = L 2 j 'R\, and so one can no longer identify the criti-
cal wavelength with the Hubble radius. For |fc| <C \k c \ Eq. (5.79) reduces to
d 2 ^fe/da 2 — (5E 2 /E)(iljk/a 6 ) = 0. The solution is expressed in terms of Bessel
functions which, however, show exactly the same behavior as (5.72): namely
4>k = ipk/o, tends to a constant as a — > oo. The other limit a — > is not relevant
as the above equation is only valid for a^> a c .
We do not discuss further the case of q = —1 and q =/= 1 since the results
obtained in this approach are exactly as discussed in paragraphs 5.3.2 and 5.3.2.
5.4 Bardeen potentials
So far we have discussed the evolution of <f> which is the magnitude of the brane
perturbation as seen by a 5-dimensional observer comoving with the brane. For an
observer living on the brane, the perturbed brane embedding gives rise to pertur-
bations about the FRW geometry. Recall (see Eq. (5.9)) that for the unperturbed
brane
ds\ = g^ lv Ax tl dx v
= -(f(R) - h(R)R 2 )dt 2 + g(R)dx ■ dx (5.84)
= -n 2 (t)dt 2 + a 2 (t)dx • dx.
The scale factors n 2 (t) and a 2 (i) pick up their time dependence through R(t),
for instance a 2 (t) = g(R(t)). In this section we calculate Sg^ resulting from the
perturbed embedding (5.32) and relate it to the Bardeen potentials.
Initially, rather than using the covariant form (5.32), let us write more gene-
rally
X°(t,x) = t + (°(t,x), (5.85)
X i (t,x) = x t + C i (t,x), (5.86)
X i (t,x) = R(t) + e(t,x). (5.87)
Below we will see that the perturbations ( z do not enter into the two scalar
Bardeen potentials which correspond to the two degrees of freedom £° and e.
This is expected since perturbations parallel to the brane are not physical and
can be removed by a coordinate transformation [47]. Then only right at the end
will we set (°/n° = e/n 5 = cp. We will find that the two Bardeen potentials are
proportional to each other and to <fi.
By definition, the perturbed induced metric is given by
9^ = SV + Sg^
d d (5.88)
= G MN (X + 6X) — (X M + 5X M ) — (X N + 5X N ). V '
OXu_ ox v
5.4. BARDEEN POTENTIALS 113
Evaluating 8g^ v to first order for the perturbed embedding (5.85)-(5.87) and the
general bulk metric (5.5), one obtains
5goo = £(-/' + h'R 2 )+2(-C°f + ehR) 1 (5.89)
Sg 0i = -(diOf + Cg+id^hR, (5.90)
5 9l0 = eg'Sij + idiQ + d^g. (5.91)
Note the terms proportional to e come from the Taylor expansion of G MN {X + 8X)
in (5.88) to first order.
In the usual way, the perturbed line element on the brane is written as
d§ 2 4 = -n 2 (l + 2A)dt 2 - 2anB i dtdx i + a 2 (% + h ij )dx i dx j , (5.92)
where n(t) and a(t) are defined in (5.84), and as usual vectors are decomposed
into a scalar part and a divergenceless vector component e.g.
Bi = d t B + B { (5.93)
with d % B{ = 0. We will use a similar decomposition for Q defined in (5.86) as
well as the usual one for tensor perturbations. Thus from (5.89)-(5.91) we have
(5.94)
where we have used standard notation defined e.g. in [129]. By considering coordi-
nate transformations on the brane and doing standard 4-dimensional perturbation
theory one can define the usual two Bardeen potentials, as well as the brane vec-
tor and tensor metric perturbations. For the first Bardeen potential we find, after
some algebra,
A=^[^(f'-h'R 2 ) + (C°f-
-ehR)
B=±-{ C °f-ta 2 -ehR},
Bi = - a -L
°-m>
E = C,
Ei = C«,
Eij = 0,
$ = -<7+-
■G)
- E )
(5.95)
Notice that all terms containing Q in B and E have cancelled as expected since
a J n z R L
114 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
they are not physical degrees of freedom. Similarly,
$ = A - -d t (aB + —&)
1 l" <^ i (5 ' 96)
' n 2 R
[<«H ['*-'(£)]
The important point to notice in this second case is not only the absence of £ l ,
but that all derivatives of the perturbations (° and e (which appear in A) have
also cancelled. Hence we will find that the Bardeen potentials are proportional to
<j> only and not to any of its derivatives. Finally, the gauge invariant vector and
tensor perturbations are identically zero.
We now set
e = n 4 </>, C° = n°<f>, (5-97)
(where n 4 is the radial component of the unit normal n M to the brane) in order to
make contact with the covariant formalism of section 5.3. Then the combination
that appears in both * and <fr is
C°i?- e =-(^n 4 ^, (5.98)
where n 4 is the radial component of the normal to the unperturbed brane. Thus
which, on going to AdSs-S and using the expression for R 2 in (5.14), yields
tf = 3<5> + 4qlj-j. (5.101)
Even though there are no anisotropic stresses, the Bardeen potentials here are not
equal. This is because of the absence of Einstein's equations in mirage cosmology 6 .
We see that for superhorizon modes on an expanding brane (for which, from
section 5.3, (f>k oc a 4 ), we also have $& oc a 4 . Similarly, 3>fc also grows rapidly for
a brane falling into the black-hole horizon.
To obtain a true (i.e. gauge invariant) measure of the deviation from the FRW
metric, it is useful to look at the ratio of the components of the perturbed Weyl
tensor and the background Riemann tensor which in the FRW case is roughly
given by {krj) 2 \$k + ^fc|, see [52]. For $& oc a 4 this ratio grows, because a ~ r/ 1 ^ 3
when n 2 ~ a~ 6 .
6 In standard 4-dimensional cosmological perturbation theory, Einstein's equations impose that
the (scalar) anisotropic stress is proportional to <& — $.
5.5. CONCLUSIONS 115
5.5 Conclusions
In this paper we have studied the evolution of perturbations on a moving D3-brane
coupled to a bulk 4-form field, focusing mainly on an AdSs-Schwarzschild bulk.
For an observer on the unperturbed brane, this motion leads to FRW expansion
or contraction with a scale factor oocr. We assumed that there is no matter on
the brane and ignored the back-reaction of the brane onto the bulk. Instead, we
aimed to investigate the growth of perturbations due only to motion, and also
to study the stability of moving D3-branes. For such a probe brane, the only
possible perturbations are those of the brane embedding. The fluctuations about
the straight brane world sheet are described by a scalar field (ft which is the proper
amplitude of a 'wiggle' seen by an observer comoving with the unperturbed brane.
Following the work of [64, 72, 34] we derived an equation of motion for </>, and
investigated whether small fluctuations are stretched away by the expansion, or,
on the other hand, whether they grow on a contracting brane. The equation for
(f> is characterized by an effective mass squared and we noted that if this mass is
positive, the system is not necessarily stable: indeed, in section 5.3 we discussed a
regime in which the effective mass squared is positive in brane time but negative in
conformal time, and therefore the perturbations grow. Another important factor
in the evolution of <j> is the time dependence of that mass.
In section 5.3 we found that on an expanding BPS brane with total energy E =
0, super horizon modes grow as a 4 , whereas subhorizon modes decay and hence
are stable. For a contracting brane, on the contrary, both super and subhorizon
modes grow as a~ 3 and a -1 respectively. These fluctuations become large in the
near extremal limit, n H « 1. We therefore concluded that the brane becomes
unstable (i.e. the wiggles grow) as it falls into the black-hole. We also discussed
the case E > for BPS branes and BPS anti-branes. Non BPS branes were found
to be unstable at late times when a positive cosmological constant dominates.
We have discussed the evolution of the fluctuations 4> as measured by a five-
dimensional observer moving with the unperturbed brane. However, for an ob-
server at rest in the bulk, the magnitude of the perturbation is given by a Lorentz
contraction factor times the proper perturbation 4>. (For a flat bulk space-time
this was pointed out in [64].) Hence, if perturbations grow for the 'comoving'
observer, they do not necessarily grow for an observer at rest in the bulk.
Finally, the fluctuations around the unperturbed world-sheet generate pertur-
bations in the FRW universe. In section 5.4 we discussed these perturbations
from the point of view of a 4-dimensional observer living on the perturbed brane.
We calculated the Bardeen potentials <I> and * which were both found to be
proportional to (j>. Furthermore, we saw that the ratio 'Weyl to Riemann' which,
expressed in terms of $ and \I/, gives a gauge invariant measure for the 'deviation'
from the FRW metric, also grows.
A limitation of this work is that the back-reaction of the brane onto the bulk
was neglected. One may wonder whether inclusion of the back-reaction could
stabilize <f>. To answer that question, recall that the setup we have analyzed here
116 CHAPTER 5. PERTURBATIONS ON A MOVING D3-BRANE
corresponds, in the junction condition approach, to one in which Z^ symmetry
across the brane is broken. Then the brane is at the interface of two AdSs-S
space-times, and its total energy is related to the difference of the respective
black-hole masses: E ex M+ — M_. Perturbation theory in such a non Z-i-
symmetric self-interacting case has been set up in [129], though it is technically
quite complicated. However, in the future we hope to try to use that formalism
to include the back-reaction of the brane onto the bulk.
It would be interesting to extend this analysis to branes with n codimensions:
in this case one has to consider n scalar fields - one for each normal to the brane.
Formalisms to treat this problem have been developed in [16, 73]. In that case the
equations of motion for the scalar fields are coupled, and it becomes a complicated
task to diagonalize the system.
Finally, it would also be interesting to consider non zero F^ v , and hence the
effect of perturbations in the radiation on the brane.
Acknowledgments
We thank Ph. Brax, E. Dudas, S. Foffa, M. Maggiore, J. Mourad, M. Parry,
A. Riazuelo, K. S telle, and R. Trotta for numerous useful discussions and encour-
agement. We especially thank R. Durrer for her comments on the manuscript.
T.B. thanks LPT Orsay for hospitality.
Chapter 6
Cosmology on a back-reacting
brane
118 CHAPTER 6. COSMOLOGY ON A BACK-REACTING BRANE
6.1 Introduction
In the previous chapters 4 and 5, we identified our universe with a probe 3-brane
moving in a background geometry without including the back-reaction. However,
depending on the coupling constant of the theory, a brane can be 'heavy', in
which case the assumption that the back-reaction can be neglected is not valid
anymore 1 . To take the back-reaction into account, one has to resort to Einstein's
equations. There are two possible ways to do that, which are however limited to
the case that the co-dimension is one. In the following, we therefore consider a
3-brane in a 5-dimensional bulk, and assume that the bulk dynamics are governed
by the 5-dimensional Einstein equations. The first method is to use the junction
conditions (3.50), such that the dynamics on the brane includes effects from the
back- reaction. The second method is to calculate the induced 4-dimensional Ein-
stein equations by the relations of Gauss, Codazzi, and Mainardi (3.35) and (3.38).
We present the two approaches in Sec. 6.3 and 6.4, respectively.
In mirage cosmology, we have used a static background motivated by super-
gravity, and the dynamics were generated by letting the brane move. Here, we
shall use a set-up in which the 3-brane is at rest at the price of having a time-
dependent bulk. It has been shown by the authors of Ref. [142], that the two view-
points are equivalent, in that the AdSs metric is related to the time-dependent
metric by a coordinate transformation. This reconciles the supergravity com-
pactification AdSsxS 5 with the Hofawa-Witten compactification.
We start this chapter by giving a short introduction to the model of Randall
and Sundrum. In Sec. 2.7 we presented an idea to solve the hierarchy problem
with 'large' extra-dimensions, in which the 4-dimensional Planck mass is related
to the fundamental scale of gravity by (see Eq. (2.74))
Mj = Ml +n V n , (6.1)
where V n is the volume of some compact space, for instance an n-sphere S n .
For this idea to work, it is crucial that the higher-dimensional space-time has a
product structure M. D = M 4 x S n , i.e. that the metric is factorizable. Other-
wise, it would be impossible to carry out the dimensional reduction (2.72). The
model of large extra-dimensions has been criticized by Randall and Sundrum, for
it introduces a new hierarchy between the electroweak scale M EW (TeV) and the
compactification scale M c , which is the energy associated with the first excited
Kaluza-Klein state. While it considerably relieves the hierarchy problem, the
Randall-Sundrum model suffers from a fine-tuning problem which, in a cosmolo-
gical context, is very severe, as we point out in the article 'Dynamical instabilities
of the Randall-Sundrum model' in Chap. 7. Nevertheless, the Randall-Sundrum
model is a prototype in brane cosmology, and so it is worth mentioning it.
Explicit mass formulae for p-branes are given in Eqs. (11.18)-(11.22) and Eqs. (11.23)-(11.27).
6.2. THE RANDALL-SUNDRUM MODEL 119
6.2 The Randall- Sundrum model
The Randall-Sundrum (RS) model is inspired from Hofava-Witten compactifica-
tion (paragraph 2.5.3), in which space-time is 5-dimensional and two 3-branes are
located at orbifold fixed points. One of the branes is our 'visible' universe and
supports the standard model matter fields, the other brane will be referred to
as 'hidden' brane. Mass scales on the two branes are related to each other by
an exponential factor arising from the particular form of the background metric.
Thus, by an exponential suppression, small scales on one brane can be generated
from large scales on the other. In particular, this mechanism offers a solution to
the hierarchy problem in which all scales are derived from a single scale.
6.2.1 Warped geometry solution
In next two paragraphs, we briefly discuss the first model of Randall and Sundrum,
Ref. [128]. Let us consider a 5-dimensional space-time M 4 x S 1 /Z2, parameterized
by coordinates (x M ,</>), where </> lies in the interval [— 7r,7r] with the identification
4> <-> —(/)■ Two 3-branes parallel to M. 4 reside at the orbifold fixed points <f) = 0, n.
They are the boundaries of the RS space-time, which is the half space {0 < <
tt}.
We choose an embedding, such that the induced metrics on the branes are,
respectively
flj^V) = G M „(^ = 0). 9™(xn = G^(x»^ = it), (6.2)
where G^ denote the components of the 5-dimensional metric along M 4 . The
action of the RS model is 2 ' 3
s 5 = s gravity + s v[s + s hid ,
Mr 3 f „ r I
Cavity = ^jd 4 xj d<Pr c V^G(R - A),
r * ~- ( 6 - 3 )
Svis = yd 4 x A /=^-(£ vis -Kis),
S hid = Jd 4 xV^m^(Uid - V hld ).
Here, A is a 5-dimensional cosmological constant, £ vis is the Lagrangian density of
the standard model fields, and V^ s is the tension of the brane. The corresponding
Einstein equations can be found from Eqs. (7.49)-(7.52) in the article in Chap. 7
by setting all time-dependent terms to zero. Here, we are looking for a static
solution respecting 4-dimensional Poincare invariance,
ds 2 = e 2i?.(0) v ^ dx ^ dx ^ + r 2 d0 2 , (6.4)
2 For consistency with the notation in this thesis as well as with that of most other authors, we have
normalized R and A differently than in the original RS paper: A RS = (M|/2)r c A, M 3 RS = (M|/4)r c .
3 In the article in Chap. 7, we work in units where M|/2 = 1/2k§ = 1.
120 CHAPTER 6. COSMOLOGY ON A BACK-REACTING BRANE
where r c denotes the compactification radius. In contrast to the large extra-
dimensions scenario, this metric is not factorizable: there is a so-called 'warp
factor' multiplying the Minkowski metric along the brane. This is the crucial
idea of the RS model. Inserting the ansatz (6.4) into the equations of motion,
one finds 4
R^) = -rM\[^§- (6.5)
For later purpose we define a parameter a = —y^ which represents a mass
scale set by the cosmological constant in the bulk. This will become important in
the next paragraph. From Eq. (6.5), clearly A must be negative, which reflects the
fact that the ansatz (6.4) together with Eq. (6.6) is a parametrization of AdSs
(see also (2.41)). There is a second equation of motion to satisfy, containing
Dirac delta functions from the boundaries. This requires a fine-tuning between
the brane tension and the cosmological constant in the bulk,
Vhid = -VUb = ^yV=l2A. (6.6)
The first equality arises from the requirement that the sum of the tensions on an
orbifold must be zero. It turns out, that the tension of the brane we are living on
is negative, which is rather unphysical. This will be changed in the second model
ofRS.
Notice again, that the RS model is purely static. In Chap. 7, we construct
a cosmological, i.e. dynamical version of it and perform a 5-dimensional pertur-
bation analysis. We find, that the model is unstable, as soon as the fine-tuning
relation (6.6) is perturbed. Intuitively, this is clear, as a 'de-tuning' corresponds to
turning on a 4-dimensional cosmological constant on the brane (see also Sec 6.4).
6.2.2 Scales and the hierarchy problem
Let us write the warped geometry solution as
ds 2 5 = G^dx^dx" + r 2 c d<f> 2 = e - 2 ^IH ( Via , + hp V ) da^dz" + r 2 d0 2 , (6.7)
where h fJiU (x fJ ') parameterizes small fluctuations around the Minkowski background
and corresponds to the massless zero mode in the Kaluza-Klein expansion 5 . By
comparison with Eq. (6.2), one finds
9^ = %v + K„, g™ = e- 2r ^ Q l 5 hw (6 . 8 )
Therefore, any two mass scales on the branes are inversely related by
M vis = e +r eW |a|j kf Md_ (g g)
"See also Eq. (7.27) with the identification y = r c cf>.
5 We are not considering perturbations in r c , but simply assume that the radius of the e
s fixed to the vacuum expectation value of some modulus field.
6.2. THE RANI) )EL 121
This is readily understood, for instance, by transforming the action of a canoni-
cally normalized scalar field from the visible to the hidden brane or vice versa.
Eq. (6.9) is the key to the solution of the hierarchy problem: assume that the
5-dimensional Planck mass, M5, is a mass parameter on the hidden brane, giving
the 'true strength' of gravity. Set A/5 ~ M EW ~ 1 TeV, to put gravity and gauge
interactions on an equal footing. If one now requires that the corresponding mass
parameter on the visible brane is 10 18 GeV = 10 15 TeV, one obtains
10 15 TeV = e +r ^ H TeV. (6.10)
In order not to create just another hierarchy, we have to set the mass parameter
a also to the TeV scale. Consequently, r c ~ 10/TeV, such that the mass scale
associated with the compactification radius, l/r c is also roughly at the TeV scale.
Thus, in this scenario, there is no large hierarchy between the scales: M 5 ~
Af EW — \ a \ — l/ r c — 1 TeV, and the huge value of the Planck mass in our
visible universe is explained by an exponential factor. Conversely, one can view
Af 4 ~ 10 18 GeV as the fundamental scale, and the TeV as a derived scale, which is
done so in the original formulation of RS . Regarding the solution of the hierarchy
problem, the RS proposal is clearly an improvement compared to the scenario of
large extra-dimensions. There, the relation between the 'effective' Planck mass
and the 'fundamental' scale is M\ = Mj?r™ (with n = 1 here). Using the same
values for A/4 and AI 5 as above, one obtains l/r c = 10~ 30 /" TeV, i.e. a huge
hierarchy between the compactification scale and the TeV.
As an experimental signature of the RS model, one expects new excitations
with TeV energies, since l/r c ~ 1 TeV is the energy of the first Kaluza-Klein
state. Such energies should be accessible to future collider experiments.
6.2.3 Non compact extra-dimension
Due to the warp factor, distances parallel to the brane become rapidly very small
as one moves away from the location of the brane into the fifth dimension. Effec-
tively, this dimension becomes 'invisible' for an observer on the brane. This hints
at the possibility that the extra-dimension may even be infinite, still leading to an
acceptable 4-dimensional phenomenology. Subsequently, RS proposed a second
model [127], which emerges from the first one by taking the limit r c — > 00. In
the orbifold set up, this is accomplished by pushing one brane to infinity, thereby
decompactifying the S 1 . The remaining brane, with positive tension, is now iden-
tified with our universe. For this second model, the key observation is, that the
curved background (6.7) can support a single normalizable bound state of the
higher-dimensional graviton. Since this bound state is localized at the position
of the brane, it plays the role of the 4-dimensional graviton, and the usual 1/r 2
Newton law is recoverd. In addition, there is a continuum of KK states, because
the extra-dimension is infinite. Even though there is no gap between the con-
tinuum and the ground state, 4-dimensional gravity is very well approximated,
because the continuum states contribute only weakly. In fact, in Ref. [127], it is
122 CHAPTER 6. COSMOLOGY ON A BACK-REACTING BRANE
shown that n non compact extra-dimensions are compatible with the 1/r 2 form of
Newton's law. This takes away some observational pressure on KK type models,
discussed in Sec. 2.7. Unfortunately, it seems difficult to embed the phenomeno-
logical approach of RS in string theory.
Because the RS solution is static, it cannot be a viable model for brane cos-
mology. The next step is therefore to look for time-dependent solutions of the
5-dimensional Einstein's equations.
6.3 Brane cosmological equations
We shall start from an action similar to the one of the RS model. This time, we
are looking for time-dependent solutions of the 5-dimensional Einstein equations.
A Friedmann equation governing the expansion rate of the brane is then found
using Israel's junction conditions. Together with the usual energy conservation
law, this equation completely describe the cosmological evolution on the brane.
By construction, this approach fully takes into account the back-reaction due to
the presence of the brane.
6.3.1 The five- dimensional Einstein equations
In this section we follow the approach of Binetruy et al. [18]. These authors
considered a 5-dimensional space-time (A4 5 ,G) with an action
Sr>= 2~2 / dW^-R- fd 5 x^GC m
(6.11)
where 1/2k| = Mf/2, and £ m is the Lagrangian density describing matter fields
in the bulk and on the brane. The gravitational action is obtained from the
dilaton-gravity action (2.54) for D = 5 by setting the dilaton to zero (see para-
graph 2.5.2). The corresponding Einstein's equations are
Gun = Rmn ~ \ G MN R = k 2 5 T mn , (6.12)
and we assume, that they completely describe the dynamics of the bulk.
To take into account the isotropy and homogeneity of our universe, we require
A4 5 to contain a maximally symmetric 3-dimensional subspace (_M 3 ,7). This
leads to the ansatz
ds 2 = G MN dx M dx N
= -n 2 (t,y)dt 2 + a 2 (t,y)j ij dx i dx j +b 2 (t,y)dy 2 ,
where y is the coordinate of the extra-dimension. Here, we do not specify whether
it is compact or not. Furthermore, if our universe is taken to be the hypersurface
{y = 0}, the usual scale factor is simply given by a(t,y = 0). Like in the RS
6.3. BRANE COSMOLOGICAL EQUATIONS
model this metric is not factorizable. In some places we use the alternative (but
completely equivalent) parametrization
dsl = -e 2N ^dt 2 + e 2R ^>7^dxW + e 2B ^dy 2 . (6.14)
The RS metric (6.4) is recovered by setting N(y) = R(y),B = l,7ij = <%, and
we shall mostly adopt this form in Chap. 7.
The components of the Einstein tensor constructed with the metric (6.14) are
I Goo = R 2 + RB + e 2 ( N - B \-R" - 2R' 2 + R'B') + K,S N ~ R \
Qoi = 0,
Qij = e 2{R - N) ~Hj{-2R -B-3R 2 - B 2 + 2NR + NB - 2RB)
+ e 2( - R - B) -y i:j (N" + 2R" + N' 2 + 3P' 2 + 2N'R' - N'B' - 2R'B') - /C 7ij ,
- ^04 = -R' - RR' + N'R + R'B,
1 g 4A = e 2 ( B - N \-R - 2R 2 + NR) + N'R' + R' 2 - /Ce 2 ^- R) ,
(6.15)
where a dot denotes the derivative with respect to t, and a prime that with respect
to y. The constant /C is the curvature of the subspace .M 3 with K, = +1,0,-1.
Later, in (7.12)-(7.17), we give Einstein's equations for a more general setup
including the dilaton and a scalar field on the brane and potential terms for both
of them.
The 5-dimensional energy-momentum tensor is decomposed into a bulk part
T B and a part describing matter fields on the brane 6 ,
T MN = (T B ) MN + S^S v n ^t^, (6.16)
vl N = diag(-p B , P B ,P B ,P B ,P T ),
■'*„ = diag(-p, P,P,P).
(6.17)
In this ansatz, one supposes that T B is independent of y, and that (T B )o4 = 0,
i.e. there is no energy flow along the fifth dimension.
Equating the components (6.15) of the Einstein tensor with their correspond-
ing matter parts (6.17), one obtains a coupled system of non linear partial differen-
tial equations of second order. Remarkably, it is possible to find a first integral 7 ,
provided that —p B = P B = P T , i.e. if the energy-momentum tensor in the bulk is
that of a cosmological constant A = K 2 p B . In the notation n = e N ,a = e R , b = e B
6 The factor 1/6 is included to assure that / dyVG^^^-roo = p with G44
7 See Ref. [18] for a more detailed disc
124 CHAPTER 6. COSMOLOGY ON A BACK-REACTING BRANE
the integrated Einstein equations read
/ a V 4 (a'\ 2 JC C ,„.,„,
— = 1TP*+[t) -^ + -1' 6 ' 18
\naj 6 \baj a 2 a 4
where C is an integration constant. This equation is valid for all values of y.
Notice also that the presence of the brane has not yet been taken into account.
If the bulk is stabilized, i.e. b = 0, it possible to integrate Eq. (6.18) to find a
global solution for a(t,y) and n(t,y). Here, we are only interested in evaluating
Eq. (6.18) at the location of the brane, y = 0, in order to find an evolution
equation for the scale factor ao = a (t, y = 0). To do so, we need Israel's junctions
conditions.
6.3.2 Junction conditions and the Friedmann equation of brane
cosmology
In Eq. (6.18) the value of a'(t,y = 0) is unknown, and therefore we cannot just
evaluated it at y = to find a Friedmann-like equation on the brane. However,
we can assume that the bulk is ^-symmetric. Then, the metric (and thus a) is
continuous at y = 0, and the scale factor satisfies a{—y) = a{y) and a'{—y) =
—a'(y). One defines the jump of a' across y = by
[a'] = lim (a'(t,y)- a' \t,-y)), (6.19)
and hence in a .^-symmetric setup
a'(t,y = 0) = ^[a'}. (6.20)
Israel's junction conditions state that this jump is non zero if there is a hypersur-
face with some energy content at y = 0. Here, this hypersurface is our universe
brane, which affects the bulk geometry due to its matter and radiation content.
For the general form of the junction conditions, we refer to paragraph 3.2.6
and in particular to Eqs. (3.50) and (3.51). Here, we undertake a more 'pedes-
trian' approach: because of the jump in a', the second derivative a" , and thus
Einstein equations (6.15), contain Dirac delta functions. To treat the regular and
distributional parts separately, one writes
a" = a'; e& + 5{y)[a'l (6.21)
and similarly for n or, in the alternative notation, for the exponents R and
N. Equating the distributional parts in Eqs. (6.15) with those in the energy-
momentum tensor (6.16) leads to the conditions
3 <.„&„ 3 ^ (622)
6.3. BRANE COSMOLOGICAL EQUATIONS 125
These equations can be understood as the integral of the Einstein equations across
the brane. The derivative a' is proportional to the extrinsic curvature. Equiva-
lently, we could have obtained the Eqs. (6.22) via the general expression (3.51).
Let us now go back to Eq. (6.18). With a suitable change of the time coordinate
t, for the fixed value y = 0, one can always obtain tiq = 1. The new time
coordinate then corresponds to cosmic time r, and the dots will henceforth denote
derivatives with respect to r. Using Eq. (6.20), [a'] = 2a' , together with the first
equation in (6.22), yields
tf 2 -(^V = fp s + |p 2 -^ + ^. (6-23)
\Oo/ 6 36 ag Oq
This is a Friedmann-like equation for the expansion rate H of a 3-brane embedded
in 5-dimensional bulk. Remarkably, the brane energy density p enters quadrati-
cally, in contrast to the usual Friedmann equation H 2 ~ p. The cosmological
evolution also depends on a bulk cosmological constant A = K 2 p B and a term
C/clq, where C is determined by initial conditions in the bulk. This term corre-
sponds to radiation, though it is not physical radiation. The curvature term /C/oq
is as in standard cosmology.
Eq. (6.23) is valid for any equation of state, P = ujp, on the brane, since the
only assumption we have made on t^ v is that it represent a perfect fluid. The
above equation is sufficient to study the cosmological evolution of a brane universe.
There is no need to know the geometry outside the brane nor the evolution of the
extra-dimension b(t,y), provided that p B and C are fixed.
From the Bianchi identities V M Q MN = one can derive an energy conservation
equation. If (T B )o4 = it takes the usual form
p + 3H(p + P) = 0. (6.24)
If (Tb)o4 7^ there is an additional term describing the energy flow from the
brane into the bulk and vice versa.
6.3.3 Solution of the Friedmann equation and recovering stan-
dard cosmology
The p 2 term in Eq. (6.23) leads to a cosmological evolution which is different from
the standard one. In particular, in the early universe, the p 2 term is important.
Strong constraints on this non standard behavior arise from the expansion rate
during Nucleosynthesis. It has been shown in [19] that a pure H 2 ~ p 2 law does
not satisfy Nucleosynthesis bounds. A possible solution, suggested in [18], is to
decompose the energy density into a part due to real matter, p, and a part due to
the brane tension, p T , by replacing p — > p+ p T in Eq. (6.23). One then obtains
h 2 = i PB + ipi + i PT p +ip 2 -- 2 +- 4 - (6.25)
126 CHAPTER 6. COSMOLOGY ON A BACK-REACTING BRANE
At late times, p <C p T so that the p 2 contribution is negligible. Furthermore, the
negative bulk energy density can be cancelled by the term in p\ by imposing that
?'-+lM= o - < 6 - 26 >
which is nothing else but the RS fine-tuning condition (6.6). The fine-tuning (6.26)
is the brane world version of the cosmological constant problem, and perturbations
of this relation will be investigated in Chap. 7. Finally, we make the identification
Thus, Newton's constant G4 is determined by K5 and the brane tension. The
tension needs to be positive, otherwise gravity on the brane would be repulsive.
This causes some trouble for the first model of RS.
With these assumptions equation (6.25) takes the standard form
H 2 = —^p--^ + - i , (6.28)
of the 4-dimensional Friedmann equation (apart form the last term, which can be
neglected in the matter dominated era).
It is interesting to compare Eq. (6.28) to the corresponding equation in mirage
cosmology, Eq. (4.18) 8 . There, we have also found a term 1/dg, mimicking the
behavior of radiation, which was due to the conserved energy E of the moving
brane as well as to the black hole horizon r H . Therefore, the parameter C can
be related to those quantities. The terms 1/ag, l/a®, l/aj found in mirage cos-
mology are absent here. In contrast, the p term allows to take into account real
matter and real radiation on the brane, which source the cosmological expansion.
For these reasons, and since it is not clear how to reproduce a 1/ajj term in mirage
cosmology, it seems necessary to include the back-reaction. In the co-dimension 1
case, a detailed discussion on the the link between the mirage cosmology approach
and the junction condition approach has been made in Ref. [144].
We conclude this section by giving a particular solution of the brane Fried-
mann equation (6.25). Consider the case where the fine-tuning condition (6.26) is
satisfied and set /C = C = 0. Furthermore, assume an equation of state P = cup,
such that the energy conservation law on the brane, Eq. (6.24), can be integrated.
Thus,
/ X 3(1+0,)
P = Pi( — ) , (6.29)
where pi and a* are constants determined by the initial conditions (see also
Eq. (1.34)). Substituting this expression into Eq. (6.25), one finds after inte-
gration,
a (r)=a^4 Pl (^4p T r 2 + q -r^ (6.30)
8 In Chaps. 4 and 5 we have denoted the scale factor on the brane by a rather than ag.
6.4. EINSTEIN'S EQUATIONS ON THE BRANE WORLD 127
with q = 3(1 + uS). Thus, the early universe evolution is characterized by an
a ~ T X / q law, which in the case of radiation [u> = 1/3, q = 4) gives a ~ t 1 / 4 .
At late times, the first term in Eq. (6.30) dominates, and the standard evolution
a ~ T 2 l q is recovered. Notice that this crucially relies on the assumption of the
fine-tuning (6.26).
Another interesting possibility arises, if the relation (6.26) is not quite zero,
but slightly positive. Then, this term has the effect of a cosmological constant on
the brane, which could drive late time acceleration.
6.4 Einstein's equations on the brane world
In the previous section we have shown how to derive an evolution equation on the
brane from the 5-dimensional Einstein equations (6.12). Alternatively, one can
derive equations for the 4-dimensional Einstein tensor on the brane by using the
relations of Gauss, Codazzi and Mainardi (see paragraph 3.2.5). Those induced
Einstein equations can be written mostly in terms of internal brane quantities.
Since the fundamental gravity, however, is 5-dimensional, one expects corrections
to the usual 4-dimensional Einstein equations. In this section we closely follow
Ref. [142].
Consider a 3- brane (AA 4 ,g) embedded in a 5-dimensional space-time (M. 5 ,G).
Locally the 5-dimensional metric can be decomposed according to Eq. (3.21)
G MN = q MN + n M n N , (6.31)
where q MN is the first fundamental form and n M the unit normal one-form to the
brane. We only consider the simplest case where the brane is fixed at y = 0,
and so the tangent vectors are e 1 ^ = 8" L . Since the brane is not moving, the
unit normal vector n M has a single non zero component in the y-direction. The
relation (3.12) between the first fundamental form and the induced metric is
q MN = g^5™ 5™. (6.32)
The derivation of the induced Einstein equations is somewhat technical, and so
we refer the reader to Ref. [142]. Essentially, it is performed using the differential
geometry formalism presented in Sec. 3.2. Here, we just mention the important
steps. According to the Gauss equation (3.35), the internal Riemann tensor on the
brane can be written in terms of the 5-dimensional Riemann tensor and the second
fundamental form K MN . By contraction one obtains the 4-dimensional Einstein
tensor Q^ u as a function of (R MNRS , R MN , R, K 2 IN ). The 5-dimensional Riemann
tensor R M nrs is then decomposed into (C MNRS ,R MN , R), where C denotes the
Weyl tensor. Furthermore, the Ricci tensor R MN and the Riemann scalar R can
be eliminated in favor of T MN using the 5-dimensional Einstein equations (6.12).
One obtains, schematically,
Guv ~ (T MN ,C MNRS ,K 2 MN ). (6.33)
128 CHAPTER 6. COSMOLOGY ON A BACK-REACTING BRANE
So far, the recipe for the derivation is completely general. To proceed, we choose
Gaussian normal coordinates 9 based on the brane,
ds 2 — q iLV dx >1 dx v + dy 2 = g^^dx^dx" + dy 2 , (6.34)
and write the 5-dimensional energy-momentum tensor as
T MN = -p B G MN + S^5 u N d(y)S IM/ , (6.35)
where p B denotes the energy density associated with a cosmological constant in
the bulk, p B = A./k 2 . The surface energy- momentum tensor,
Sp V = -p T g^ + t^, (6.36)
contains the energy density p T associated with the brane tension and t^ v , the
usual 4-dimensional energy-momentum tensor of fluid matter. Notice that this
corresponds to the splitting p — > p + p T in the previous section. Eqs. (6.35) and
(6.36) are now inserted into Eq. (6.33).
Finally we assume that the bulk is ^-symmetric. Then, one can use Israel's
junction conditions (3.51) to express K MN in Eq. (6.33) in terms of S^. The final
form for the Einstein equations in a 3-brane world reads
Gpv = %v - ^ 9nvR = -A 4 fiv + SirGiT^ + k£ti> - E^, (6.37)
where
° 4 ~ 4M pT > (6.38)
*"" = ~\ T ^ + lV^ + Wra/,7^ - ^r 2 , r = t%,
E^u = C A BRS n A n a q'^ql.
The quantity A 4 is the effective 4-dimensional cosmological constant on the brane,
which is a combination of the negative bulk energy density p B and the energy
density corresponding to the brane tension, p T . The case A 4 = corresponds
to the RS fine-tuning (6.6) in a cosmological context. As in the section before,
it is found that Newton's constant G 4 is given by k 5 and p T . In the above
Einstein equations, the matter source is represented by r^ v and 7r M „. The latter
is quadratic in T /JbV , which traces back to the Gauss equation (3.35): there, the
extrinsic curvature tensor enters quadratically, which via the junction conditions
translates into terms S 2 U and finally t 2 v . As a consequence for cosmology, the
unusual p 2 term shows up in the Friedmann equation (6.23). Furthermore, there
is a term E^ V1 which is a projection of the 5-dimensional Weyl tensor onto the
6.4. EINSTEIN'S EQUATIONS ON THE BRANE WORLD 129
brane, and which cannot be determined from internal brane quantities alone.
The Weyl tensor represents 5-dimensional gravitational waves propagating in the
bulk, and E^ v corresponds to their energy-momentum deposited on the brane.
In a cosmological setting, this contribution shows up as C/clq in the Friedmann
equation 10 .
In the Einstein equations, E^ must be evaluated either at y = +0 or y = — 0,
rather than on the brane itself. Because of that, the equations on the brane are
not a closed system. For certain issues, for example in perturbation theory, one
has to resort to the full 5-dimensional dynamics.
Both unusual terms, 7r Ml/ and E^ u , play a role in the early (brane) universe,
but can be neglected today. Hence, in a low energy limit, the usual Einstein
equations are recovered.
Chapter 7
Dynamical instabilities of the
Randall- Sundrum model
(article)
132 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
This chapter consists of the article 'Dynamical instabilities of the Randall-
Sundrum model', published in Phys.Rev.D64. (2001), see Ref. [21].
It is also available under http://lanl.arXiv.org/abs/hep-th/0102144.
In this article, we work in units 2k\ = 1. The fine-tuning condition (6.26) then
becomes Eq. (7.28), where we have expressed it in terms of the 5-dimensional
cosmological constant A rather than p B . In this chapter, denotes the dilaton,
as we need the variable $ for a Bardeen-like potential.
7.1. INTRODUCTION
Dynamical instabilities of the Randall- Sundr um
model
Timon Boehm and Ruth Durrer
Departement de Physique Theorique, Universite de Geneve, 24 quai E. Ansermet, CH-1211
4 Switzerland.
Carsten van de Bruck
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
Wilberforce Road, Cambridge CB3 OWA, UK.
We derive dynamical equations to describe a single 3-brane containing fluid
matter and a scalar field coupling to the dilaton and the gravitational field in a
five-dimensional bulk. First, we show that a scalar field or an arbitrary fluid on
the brane cannot evolve to cancel the cosmological constant in the bulk. Then we
show that the Randall— Sundrum model is unstable under small deviations from
the fine— tuning between the brane tension and the bulk cosmological constant
and even under homogeneous gravitational perturbations. Implications for brane
world cosmologies are discussed.
PACS number: 98.80.Cq
7.1 Introduction
Until now, string theories are the most promising fundamental quantum theories
at hand which include gravity. Open strings carry gauge charges and end on so-
called Dp-branes, (p+l)-dimensional hypersurfaces of the full space-time. Corre-
spondingly, gauge fields may propagate only on the (p+l)-dimensional brane, and
only modes associated with closed strings, such as the graviton, the dilaton and
the axion, live in the full space-time [125]. Superstring theories and especially M
theory suggest that the observable universe is a (3+ 1) -dimensional hypersurface,
a 3-brane, in a 10 or 1 1-dimensional space-time. This fundamental space-time
could be a product of a four-dimensional Lorentz manifold with an n-dimensional
compact space of volume V n (n being the number of extra-dimensions). Then,
the relation between the (4 + n)-dimensional fundamental Planck mass M± +n and
the effective four-dimensional Planck mass M4 = ^/lffinG^) ~ 2.4 x 10 18 GeV
is
Mi = M 4 2 +"V„- (7-1)
If some of the extra dimensions are much larger than the fundamental Planck
scale, M i+n is much smaller than M 4 and may even be close to the electroweak
134 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
scale, thereby relieving the long-standing hierarchy problem [12, 9]. For example,
if one allows for two 'large' extra dimensions of the order of 1 mm, one obtains
a fundamental Planck mass of 1 TeV. However, a new hierarchy between the
electroweak scale and the mass scale associated with the compactification volume,
V n ~ is introduced.
Clearly, this idea is very interesting from the point of view of bringing together
fundamental theoretical high energy physics and experiments, which have been
diverging more and more since the advent of string theory. While the four-
dimensionality of gauge interactions has been tested down to scales of about
1/200 GeV -1 ~ 10~ 15 mm, Newton's law is experimentally confirmed only above
1 mm. Therefore, 'large' extra dimensions are not excluded and should be tested
in the near future by refined microgravity experiments [103, 79]. The fundamental
string scale might in principle be accessible to the CERN Large Hadron Collider
(LHC) [110, 10, 69, 134].
In the past, it was commonly assumed that the fundamental space-time is
factorizable, and that the extra-dimensional space is compact. Recently, Randall
and Sundrum [127] proposed a five-dimensional model, in which the metric on
the 3-brane is multiplied by an exponentially decreasing 'warp' factor such that
transverse lengths become small already at short distances along the fifth dimen-
sion. This idea allows for a non compact extra-dimension without getting into
conflict with observational facts. In this scenario the brane is embedded in an
anti-de Sitter space, and a fine-tuning relation
A = -^V 2 , (7.2)
between the brane tension V and the negative cosmological constant in the bulk
A has to be satisfied. Here, k 5 is related to the five-dimensional Newton constant
by Kg = 6tt 2 G^ = A/g~ 3 . Randall and Sundrum also proposed a model with
two branes of opposite tension which provides an elegant way to relieve both
hierarchy problems mentioned above [128]. However, also this model requires the
fine-tuning (7.2). Here, we will only consider the case of a single brane.
The main unattractive feature of the Randall-Sundrum (RS) model is the
fine-tuning condition (7.2). Both from the particle physics and the cosmological
point of view this relation between two a priori independent quantities appears
unlikely. One would like to put it on a physical basis, such as a fundamental
principle, or explain it due to some dynamical process.
The purpose of this paper is to point out the cosmological problems associated
with the fine-tuning condition (7.2). The outline of the paper is as follows: In
Sec. 7.2, we derive dynamical equations describing the gravitational field and the
dilaton in the bulk coupling to fluid matter and a scalar field on the brane. These
equations allow for a dynamical generalization of the RS model, which is a special
static solution of our equations with vanishing dilaton. Our equations also provide
a starting point for further studies of various issues in cosmology, for example
inflation. In Sec. 7.3 we discuss a cosmological version of the RS model and show
7.2. EQUATIONS OF MOTION 135
that the fine-tuning condition (7.2) cannot be stabilized by an arbitrary scalar
field or fluid on the brane. In Sec. 7.4 we discuss linear perturbations of the static
RS model and derive gauge invariant perturbation equations from our general
setup. We prove that the full RS space-time is unstable against homogeneous
processes on the brane such as cosmological phase transitions: The solutions run
quadratically fast away from the static RS space-time. This instability reminds
that of the static homogeneous and isotropic Einstein universe [59]. In linear
perturbation theory we also find a mode which represents an instability linear in
time. In the last section we present our results and the conclusions.
7.2 Equations of motion
In this section we derive the equations of motion. For generality and for future
work we have included the dilaton, although it does not play a role in the present
discussion of RS stability. Works on dilaton gravity and the brane world have
also been done by the authors of Refs. [109] and [108].
7.2.1 General case
We consider a five-dimensional space-time with metric G MN parameterized by
coordinates (x M ) = (x^,y), where M = 0,1,2,3,4 and n = 0,1,2,3, with a 3-
brane fixed at y = 0. We use units in which 2/tg = 1. In the string frame the
action is
S 5 = J d 5 xV^G e - 2<t ' (R + 4(V M (p)(V N cp)G MN - A(cp))
U ^-"- s (73)
— / d XyJ—ge '
which describes the coupling of the dilaton </> to gravity, as well as to a scalar
field ip with potential V(y>) and to a fluid with Lagrangian £ on the brane. The
graviton, the dilaton and the 'bulk potential' A.(<j>) live in the five-dimensional
bulk space-time, whereas the fluid and the scalar field are confined to the brane.
The induced four-dimensional metric is 1
g^ = S»;5ZG MN (y = 0). (7.4)
The action in the Einstein-frame is obtained by the conformal transformation
G MN ^e~T^G MN (7.5)
with D = 5. We find
S^ = Jd 5 x^G^R-^(V M ^(V N ^)G MN -eW 3 ^A(^
- J ^x^g Q e-( 2 / 3 ^(V^)(V^)<r + e^* (V(<p) + C)
1 Where confusion could arise, we use a hat to denote four-dimensional quantities.
(7.G)
136 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
where G MN now denotes the metric tensor in the Einstein-frame, and R and V are
constructed from this metric. The equation of motion for the dilaton is obtained
by varying this action with respect to </>. We find
6 6 Ocp
(7.7)
Similarly, the equation for <p is
e-tW^.elW^^o. (7.8)
ay
Finally, the 5-dimensional Einstein tensor is
Q MN = \ f(V M ^)(V N 0) - I Gmjv (V0) 2 ) - l G MN e^^m
<%)« ( - e-P/3)0i ((V^)(V^) - 1 ff/lv (Vp) 2 ) (7 .9)
where r M „ is the energy-momentum tensor of the fluid on the brane. As we are
interested in cosmological solutions, we require the 3-brane to be homogeneous
and isotropic and make the ansatz
ds 2 = - e 2N ^dt 2 + e 2R ^dx 2 + e 2B ^dy 2 , (7.10)
where the ordinary spatial dimensions are assumed to be flat. Note that this
metric is not factorizable as the scale factor on the brane e R ^ t,v ^ and the lapse
function e^'*' 9 ' depend on time as well as on the fifth dimension. The factor
e B(t,y) j s a modulus field. The energy-momentum tensor of a homogeneous and
isotropic fluid, representing matter in the universe, is
(T"„(t)) = dmg(-p(t),P(t),P(t),P(t)), (7.11)
and for the dilaton and the brane scalar field we shall assume (f> = (p(t, y),ip = tp{t).
Finally, the Lagrangian density of the fluid, £, is given by its free energy density
F (seeRef. [153]).
With these assumptions the equations of motion take the form below. An
overdot and a prime refer to the derivatives with respect to t and y, and quantities
on the brane carry a subscript zero, for example N = N(t, y = 0).
7.2. EQUATIONS OF MOTION
The equation of motion for the dilation is
^ e- 2N {ij) - j>N + 3J>R + 4>B) - ? e- 2B ((/)" + <j>'N' + 3<f>'R' - <//B')
+ leW^m+eW^ d ^ ( 7 .12)
+S(y)e- B Q e^ 2 / 3 ^ 2 V + \ e^{V{ V ) + F)\ = 0.
The equation of motion for the scalar field on the brane is
e- 2N ° {(p - <pN + 3^R ) + e^ 3 ^ ^^ = . (7.13)
The 00-component of the 5-dimensional Einstein equations is
3e- 27V (ii 2 + RB- 2 -jA + 3e- 2B (-R" - 2R' 2 + R'B' - ^'A
_ 1 e (4/3)0 A(0) _ 5{y)e -B Q e -(2/3) ^-2^2 + 1 e ( 2 /3)^ (V( ^ + ,)) = (J .
(7.14)
The 11-component is
e~ 2N (-2R -B-3R 2 - B 2 + 2NR + NB- 2RB - -cp 2 ]
+e~ 2B ( N" + 2R" + N' 2 + 3R' 2 + 2N'R' - N'B' - 2R'B' + V 2 )
+ 1 e (4/3)0 A((/)) _ 5{y)e -B n e -(2/3)0 e -2^2 + 1 e (2/3)0 ( p _ y {(f))) \ = Q
(7.15)
The 04-component is
The 44-component is
3e~ 27V (-R - 2R 2 + NR- ^ 2 J + 3e- 2B (n'R 1 + R' 2 - \
+ I e (4/3)0 A(0) = O .
Finally, for the fluid we assume an equation of state of the form
P = P(p). (7.18)
(7.17)
In order to have a well defined geometry, the metric has to be continuous
across y = 0. However, first derivatives with respect to y may not be continuous
138 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
at y = 0, and second derivatives may contain delta functions. Such distributional
parts can be treated separately by writing
/" = /£ g + %)[/'], (7.19)
where
[/'] = lim i(/'(y) - /'(-y)) (7.20)
is the jump of /' across y = 0, and / r " g is the part which is regular at y = 0.
By matching the delta functions from the second derivatives of (j>, N and R with
those in equations (7.12), (7.14) and (7.15), one obtains the junction conditions
[</>'] = l e -(V^o e Bo-2N ^2 + l e (2/3)0o e B o(y((/3)+jF) ^ (?21)
[JV'] = ^ e -(2/3)0o e Bo-2%^2 + 1 e (2/3)<Ao e B 0(3p + ^ _ y {(p)) ^ (? 22)
[#] = __L e -(2/3)0„ e Bo-27V o ^2 _ 1 e (2/3)*o e B 0(7( ^ + p) . (7.33)
Equations (7.22) and (7.23) are equivalent to Israel's junction conditions [82].
Our equations agree with those found by other authors in special cases, see
e.g. Refs. [19] and [85].
In the remainder of this paper, we assume Z 2 symmetry. Furthermore, we
neglect the dilaton and consider A(<p) to be a constant.
7.2.2 Special case: The Randall Sundrum model
The RS model is a special static solution of the equations derived in the previous
section with N(y) = R(y),B = 0, when A is taken to be a pure cosmological
constant, and V represents a constant brane tension. All other fields are set to
zero. The RS metric is
dsj = e 2aM (-dt 2 + dx 2 ) + dy 2 . (7.24)
Our equations of motion then reduce to
6i?' 2 = -|, (7.25)
3J2" = -*(i/)y. (7.26)
Equation (7.25) can now be solved by
R{y) = -^\y\^a\y\, (7.27)
which respects Z 2 symmetry and leads to an exponentially decreasing 'warp fac-
tor'. To satisfy simultaneously equation (7.26), one must fine-tune the brane
tension and the (negative) bulk cosmological constant
A+^ = 0. (7.28)
7.3. A DYNAMICAL BRANE 139
This is the RS solution. A priori, A and V are independent constants, and there is
no reason that such a relation should hold. However, in a realistic time-dependent
cosmological model this relation must be satisfied in order to recover the usual
Friedmann equation for a fluid with /)« F see Ref. [43, 44, 18]. In the next
section we study whether Eq. (7.28) can be obtained by some dynamics on the
brane.
7.3 A dynamical brane
We first consider a dynamical scalar field on the brane. The fine-tuning condition
(7.28) corresponds to the requirement that the negative bulk cosmological con-
stant A can be cancelled by the brane tension V which we try to identify with the
potential energy of the scalar field p. If, starting with some initial conditions on ip
and tp, the evolution of the system would stabilize at A+ j- = 0, the cancellation
could be accomplished dynamically. If this would be the case for a 'large class' of
initial conditions, the RS solution (7.27) would be an attractor of the system.
We start from Eqs. (7.13)-(7.17) for the case of a vanishing dilaton. Taking
the 'mean value' of the 44 equation across y = 0, inserting the junction conditions
(7.22), (7.23) and taking into account Z2 symmetry, one obtains (see Ref. [19])
R +2Rl = -^p b ( Pb + 3P b ) + ^, (7.29)
where p b = p + p v and P b = P + P v are the total energy density and the total
pressure on the brane due to the fluid and the scalar field. In this section the
overdot denotes the derivative with respect to the time coordinate r given by
dr = e N °^dt. Using the energy conservation equation on the brane,
p b = -3R (p b + P b ), (7.30)
one can eliminate the pressure and integrate Eq. (7.29) to obtain a 'Friedmann'
equation for the expansion of the brane (see Refs. [43, 44, 18])
where ao(t) = e R (. t ^y= ) denotes the scale factor on the brane, H = ao/ a o = Rq,
and C is an integration constant. If the dilaton vanishes, Eq. (7.13) becomes the
ordinary equation of motion for a scalar field
dV
<p + 3Hcp+ — = (7.32)
dip
with an energy density and pressure
P v = \<? + V{<p), (7.33)
P v = \<? ~ V(<p). (7.34)
140 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
We now assume that the energy density of the scalar field dominates any other
component on the brane, that is p v ^> p. This may be the case in the early
universe. Later in this section we will see that this assumption does not affect
our result. In the same sense we neglect the radiation term, so that Eq. (7.31)
reduces to
b =+\/>H a+ t£)- (7 - 35)
The positive sign corresponds to an expanding brane. The question of whether the
system evolves towards A + j^ = is now translated into the question of whether
the Hubble parameter vanishes at some time T\. From Eqs. (7.32) and (7.35)
together with Eq. (7.33) one finds
H = -^P^\ (7.36)
which is always negative. (The case y>( T i) = simultaneously with H{t\) =
will be treated separately.) Starting with an expanding universe, H > 0, this
implies that H is indeed decreasing and H = may well be obtained within finite
or infinite time depending on the details of the potential V(tp). However, at t\
the scale factor has reached a maximum (ao( T i) = o-o{ti)H{t\) < 0) and, after a
momentary cancellation of A with p v 2 , H changes sign and the brane begins to
contract with
" = l/M A+ 3r)- (7 - 37)
In order for H to stop evolving at T\ when the RS condition A + p v 2 /12 = is
satisfied, we need £-^H(ti) = for all n > 0, which implies £-^p v = and also
^ry(Ti) = for all n > 1. Therefore, the scalar field has to be constant with
value (pi = </?(ti) and V(</?i) = \l — 12A. But this is only possible if V(tpi) is a
minimum of the potential, and we have to put tp into this minimum with zero
initial velocity from the start. This of course corresponds to the trivial static
fine-tuned RS solution.
We conclude that the fine-tuning condition (7.28) cannot be obtained by such
a mechanism. Note that our arguments have been entirely general and we have
thus shown that the fine-tuning problem cannot be resolved by an arbitrary brane
scalar field.
To illustrate the dynamics, we consider the potential V{ip) = ^m 2 ip 2 . Eq. (7.35)
then takes the form
It is convenient to use dimensionless variables x, y, z and 77 related to ip, tp, H
7.3. A DYNAMICAL BRANE 141
and r by
/24T 1
tp=\—x, (f = V24my, H = mz, r = —r\. (7.39)
V m m
Equations (7.32) and (7.38) are equivalent to a two-dimensional dynamical system
in the phase space (x,y) with
x' = y, y' = -x - 3zy, (7.40)
with the constraint equation
z 2 = -K + (x 2 + y 2 ) 2 . (7.41)
The prime denotes the derivative with respect to the 'time parameter' r\ and
K = — 12 x m 2 A. In Fig. 7.1 two typical trajectories found by numerical solution of
the system (7.40)-(7.41) are shown in the phase space (x,y). For large initial y,
the damping term first dominates and lowers y until the potential term becomes
comparable. Then, the system evolves towards the minimum of the potential until
the curve hits the circle x 2 + y 2 = \[~K (after finite time), where the damping
term changes sign, and the trajectories move away nearly in the y direction. In
Fig. 7.2 the corresponding evolution of the Hubble parameter z is shown.
In ordinary four-dimensional cosmology there exists a 'no-go theorem' due
to Weinberg [162], which states that the cosmological constant cannot be can-
celled by a scalar field. The argument is based on symmetries of the Lagrangian.
In brane cosmology it is known [142] that for a 3-brane, embedded in a five-
dimensional space-time, Einstein equations on the brane are the same as the
usual four-dimensional Einstein equations, apart from two additional terms: a
term -n^ which is quadratic in the matter energy-momentum tensor t m „ on the
brane 2 ,
and a term E^ V1 which is the projection of the five-dimensional Weyl-tensor,
C A bcdi onto the brane
E»v = C A BCD n A n c q fl B q„ D , (7.43)
where n A is the normal vector to the brane, q MN is the first fundamental form,
and t is the trace of the energy momentum tensor. Being constructed purely
from t^ v1 the quantity tt^ does not introduce additional dynamical degrees of
freedom. It just contributes the term p v 2 to the 'Friedmann' equation (7.31).
It is also clear that E^ v , which is traceless, cannot cancel the cosmological con-
stant on the brane. However, since the effective Einstein equations on the brane
cannot be derived from a Lagrangian, and since E^ contains additional infor-
mation from the bulk, it is not evident that Weinberg's theorem holds in our case.
CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
Figure 7.1: Two trajectories in the phase space (x,y) which represent typical
solutions of the system (7.40)-(7.41) for K = 1. The trajectory on the left (dotted,
red), starting with an initial condition x in = — 3,y in = 2, winds towards the circle
x 2 + y 2 = VK , which corresponds to the condition z — H = 0. After reaching
the circle, the solution moves away showing that it is not an attractor. The
trajectory on the right (solid) with initial conditions x in = 2,y in = 2 shows a
similar behavior. It takes much longer to pass the region around the kinks than
to trace out the remaining parts of the trajectories.
More generally, our 'no-go' result also holds for any matter obeying an equa-
tion of state P = up when uj > — 1. This can be seen in a similar way: initially
the Hubble parameter is
H = + \jh{ A + £)- < 7 - 44 >
Using the energy conservation equation
p=-3H(l + io)p, (7.45)
one finds
H = -±(l + UJ )p 2 , (7.46)
and hence H < as long as the condition u > — 1 (or P > —p) is satisfied. To
relate our finding to previous results [43, 44, 18, 107], let us note that Eqs. (7.44)
7.3. A DYNAMICAL BRANE
Figure 7.2: The time evolution of the dimensionless Hubble parameter z for the
two trajectories shown in Fig. 1.
and (7.46) imply the following condition for inflation on the brane
For a brane energy density given by the brane RS tension V = \/— 12A and
an additional component indicated by a subscript j, so that p = V + pf and
P - -V + P f this gives
^ = -[V(l + 3u f ) + Pf (2 + 3w/)]^ > , (7.48)
which coincides with Eq. (8) of Ref [107]. If the RS term dominates, V 3> pj we
obtain the usual strong energy condition for inflation, 1 + 3ujf < 0, but if V <§C pj
the condition is stronger, namely 2 + 3ujf < 0.
As in the case of the scalar field, the brane starts to contract as soon as H =
is reached. We have thus shown that a relation such as Eq. (7.28) cannot be
realized in a cosmological setting with matter satisfying ui > — 1.
After this section, in which we adopted the viewpoint of the brane, we now
come back to the full five-dimensional space-time to investigate the stability of
the RS model.
144 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
7.4 Gauge Invariant Perturbation equations
We formally prove that the five-dimensional RS space-time is unstable under small
homogeneous perturbations of the brane tension.
7.4.1 Perturbations of the Randall Sundrum model
The equations of motion derived in Sec. 7.2 provide with N(t,y),R(t,y), and
B(t, y) a dynamical generalization of the RS model. We consider A and V to
be constant and set the dilaton, the scalar field on the brane and the energy-
momentum tensor of the fluid to zero. Equations (7.14)-(7.17) now reduce to
00 : 3e- 2W (i? 2 + RB) + 3e- 2B (-R" - 2R' 2 + R'B')
1 1
(7.49)
N (-2R -B-3R 2 -B 2 + 2NR + NB - 2RB)
B (N" + 2R" + N' 2 + 3R' 2 + 2N'R' - N'B' - 2R'B')
(7.50)
04 : R! + RR' - N'R - R' B = , (7.51)
44 : 3e~ 2N (-R - 2R 2 + NR) + 3e- 2B (N'R' + R' 2 ) + ^A = (7.52)
The RS solution (7.27) is a static solution of these equations, provided that condi-
tion (7.28) holds. We now derive linear perturbation equations from Eqs. (7.49)-
(7.52) which describe the time evolution of small deviations from RS. To this goal
we set
N(t,y) = a\y\+n(t,y), (7.53)
R(t,y) = a\y\+r(t,y), (7.54)
B(t,y) = b(t,y), (7.55)
where a — —y^- and n(t,y),r(t,y),b(t,y) are small at t = 0. The perturbed
metric is
dS 2 5 = G MN dx M dx N
= _ e Mv\+2n dt 2 + e 2aM + 2, da? 2 + ^6^2 ' ' ^
We consider an energy-momentum tensor deviating from RS only by a slight
mismatch of the brane tension
T MN = -KG MN - 8(y)8^8 u N e- h Vg^, (7.57)
7.4. GAUGE INVARIANT PERTURBATION EQUATIONS 145
with
V = v / -12A(l + n), (7.58)
where |fi| <C 1 parameterizes the perturbation of the brane tension, G MN is the
perturbed metric (7.56) and g^ is its projection onto the brane. Clearly, if already
this restricted set of perturbation variables contains an instability, the RS solution
is unstable under homogeneous and isotropic perturbations. Inserting this ansatz
into equations (7.49)- (7.52) and keeping only first order terms, we find
r" - Aa 2 b + 6(y)a{Ar' - b') - S(y)2a(b + O) = , (7.59)
e -**\ y \ (2? + b)- n" - 2r" + 12a 2 b - 9(y)a(iri + 8r' - 3b')
+S(y)6a(b + n) = , (7.60)
f'-6(y)ab=0, (7.61)
e -2a|y|f + 4a 2 b _ 6l(y) a ( n ' + 3r') = , (7.62)
where
The junction conditions are
[/] = [ri] = 2a(b Q + 0) . (7.64)
Since we want to consider ^-symmetric perturbations, we require the functions
n, r and b to be symmetric in y. In order to make coordinate-independent state-
ments, we rewrite these equations in a gauge invariant way.
7.4.2 Gauge invariant perturbation equations
Under an infinitesimal coordinate transformation induced by the vector field
X = T(t,y)d t + L(t,y)d y , (7.65)
the metric perturbations 5G MN (corresponding to the first order terms in the
metric (7.56)) transform according to
8G MN -> SG MN + C X G MN , (7.66)
where LxG MN is the Lie derivative of the static background metric (7.24). One
obtains the following transformation laws for the variables n, r and b:
n^n + 9(y)aL + f, (7.67)
r^r + 6(y)aL, (7.68)
b^b+L'. (7.69)
Since we require the 04 component of the metric to vanish, it must remain zero
under the coordinate transformation. This implies
146 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
From Eq. (7.69), together with Zi symmetry, one finds that L' must be continuous
and symmetric in y. Therefore L must be continuously differentiable and odd in
y, which implies L(t, y = 0) = 0. Hence, the perturbation r restricted to the brane
ro is gauge invariant. Note that L(t, y = 0) = also follows from Eq. (7.68) and
L 6 C 1 . Hence the gauge invariance of ro is not a consequence of Z^ symmetry,
but is also preserved for non ^-symmetric perturbations. By computing the Lie
derivative of the background energy-momentum tensor from Eq. (7.57) one finds
that the perturbation of the brane tension H is gauge invariant. Condition (7.70)
and the symmetry property of L' ensure that there is no energy flow onto or off
the brane. With the following set of gauge invariant quantities
$ = r' - 0(y)ab , (7.71)
* = ri - 9(y)ab - 9(y)-e- 2aM f , (7.72)
a
r o = r(t,y = 0) 7 (7.73)
O, (7.74)
we can rewrite the perturbation equations (7.59)-(7.62) in terms of these variables
$' + %)4a$ - 5{y)2an = , (7.75)
*' + 2$' + 0(y)4a(tt + 2$) + S(y) ( -r - 6aO J = , (7.76)
6 = 0, (7.77)
^ + 3$ = 0. (7.78)
The junction conditions are
[*] = 2aO , [*] = 2aO - -r . (7.79)
The solutions of equations (7.75) with (7.77) and (7.78) are given by
1 T
(7.80)
Z2 symmetry requires $ to be odd in y and thus $0 = 0. Inserting Eq. (7.78) in
(7.76) one obtains
r = 4a 2 (7.81)
and after integration
r (t) = 2a 2 nt 2 + Qt, (7.82)
where Q is a small but arbitrary integration constant determined by the initial
conditions. (An additive constant to ro can be absorbed in a redefinition of the
spatial coordinates on the brane.) The scale factor on the brane is
e 2r (t) _ 1 + 2 ro (t) = 1 + Aa 2 VLt 2 + 2Qt. (7.83)
7.4. GAUGE INVARIANT PERTURBATION EQUATIONS 147
We have thus found a dynamical instability, which is quadratic in time, when the
brane tension and the bulk cosmological constant are not fine-tuned. Our state-
ment is valid in every coordinate system as r$ is gauge invariant. In addition,
more surprisingly, in linear perturbation theory there is no constraint on Q, and
it cannot be gauged away. This linear instability remains even for SI = 0, that is
if the brane tension is not perturbed at all.
Let us finally discuss our solutions in two particular gauges. As a first gauge
condition we set r' = 0, which fixes V = — 6{y)-r' . The integration constant
on L is determined by the condition L(t, y = 0) = 0. (Note that r' contains a
function and therefore V is continuous.) For all values of y we have
r(t) = 2a 2 nt 2 + Qt. (7.84)
Since b(y) = -e(y)^(y),
b(y) = -ne- iaM . (7.85)
From the definition of \P it follows that
ri = -3$ + 6(y)ab + 9(y)- e~ 2aM r , (7.86)
a
which can be integrated to give
n{t, y) = Qe~ 4QM - 20e~ 2a|!/l + N{t) . (7.87)
These n,r, and b solve Eqs. (7.59)-(7.62). The integration constant Af(t) can be
absorbed in the gauge transformation T. Together with the choice of L', this fixes
the gauge, and the solutions are therefore unique up to an additive purely time
dependent function to T.
Another possible gauge is b = 0. Then, from r' = <f>,
r(t,y) = -^e- 4a ^+TZ(t), (7-88)
K(t) = r (t) + ^ = 2a 2 nt 2 + Qt + ^ (7.89)
n(t, y) = -fie" 4a|! ' 1 - 20e~ 2a|y| + Af(t) . (7.90)
Again, the integration constant Af(t) can be gauged away by choosing an appro-
priate T, and the solutions are uniquely determined by the gauge fixing.
Inserting these solutions in the perturbed metric
dS 2 = _ e 2a\y\+2n di 2 + ^1,1+2^2 + ^6,2 {7M)
148 CHAPTER 7. DYNAMICAL INSTABILITIES OF THE RS-MODEL
we find that the full RS space-time, not only the brane, is unstable against ho-
mogeneous perturbations of the brane tension.
We must require the initial perturbations to be small, that is at some initial
time t = 0, the deviation from RS has to be small for all values of y. In the case
of a compact space-time \y\ < j/ max , this just requires |fi| <C e~ 4aymax (remember
that a is a negative constant). For a noncompact space-time — oo < y < oo, we
have to require O = 0. In other words, for V ^ \l — 12A there exists no solution
which is 'close' to RS in the sense of L 2 or sup y at any given initial time.
Finally, we present a geometrical interpretation of the gauge invariant quanti-
ties $ and \&. Since the five-dimensional Weyl tensor of the RS solution vanishes,
the perturbed Weyl tensor is gauge invariant according to the Steward- Walker
lemma [146]. The 0404 component of the Weyl tensor of the perturbed metric
(7.56) is up to first order
0,404 = \ e 2a M(n" - r" + e(y)a(n' - r')) + l -(f - b), (7.92)
which can be expressed in terms of gauge invariant quantities
C 404 = - \ e a ^ U a ^{<5> -*))' + %)-ro. (7.93)
All other nonvanishing Weyl components are multiples of C0404:
C0101 = C0202 = C0303 = C1212 = C1313 = C2323 = — o e a C0404 (7.94)
and
C1414 = C2424 = C3434 = -C0404 • (7.95)
In first order the projected Weyl tensor (defined in Ref. [142]) is En = E22 =
E33 = ^Eoo = 2a 2 with E^^ = 0. The Weyl-tensor completely vanishes for
O = 0.
7.5 Results and Conclusions
In this paper we have addressed two main questions: First, we investigated
whether the RS fine-tuning condition can be obtained dynamically by some mat-
ter component on the brane. As a concrete example, we studied a scalar field on
the brane and found that a bulk cosmological constant cannot be cancelled by
the potential of the scalar field in a nontrivial way. This result can be generalized
for any matter with an equation of state u> > — 1.
Second, we studied the stability of the RS model in five dimensions. We have
found that the RS solution is unstable under homogeneous and isotropic, but
time dependent perturbations. For a small deviation of the fine-tuning condition
parameterized by O 7^ 0, this instability was expected. It is very reminiscent of
7.5. RESULTS AND CONCLUSIONS 149
the instability of the static Einstein universe, where the fluid energy density and
the cosmological constant have to satisfy a delicate balance in order to keep the
universe static. But even if tt = 0, there exists a mode (Q ^ 0), which represents
an instability in first order perturbation theory. It is interesting to note, that, this
mode can be absorbed into a motion of the brane, where Q represents the velocity
of the brane. In a cosmological context, our result means that a possible change
in the brane tension, e.g. during a phase transition, or also quantum corrections
to the bulk energy density (see Ref. [61]) could give rise to instabilities of the full
five-dimensional space-time. For related work see Refs. [91], [86], [121].
Even if one would consider a dynamical scalar field in the bulk (which does
not couple to brane fields), which settles into a vacuum state such that its energy
density is constant along the fifth dimension, one would not be able to solve the
cosmological constant problem without falling back to some fine-tuning mecha-
nism. From our results we can conclude that in order to have a chance to solve
the RS fine tuning problem dynamically, we have to consider fully dynamical bulk
fields. This can in principle be done with the system of equations, which we have
presented in Sec. 7.2, and which also applies to the effective five-dimensional
low-energy theory suggested by heterotic M theory [104].
Acknowledgments
We wish to thank Arthur Hebecker, Kerstin Kunze, Marius Mantoiu Daniele
Steer, Toby Wiseman and Peter Wittwer for useful discussions and comments.
C.v.d.B thanks Geneva university for hospitality. This work was supported by
the Swiss National Science Foundation. C.v.d.B was supported by the Deutsche
Forschungsgemeinschaft (DFG) .
Chapter 8
On CMB anisotropies in a
brane universe
152 CHAPTER 8. ON CMB ANISQTRQPIES IN A BRANE UNIVERSE
8.1 Introduction
A great deal of our knowledge in cosmology comes from measurements of the
cosmic microwave background (CMB), in particular its anisotropics. Recent ex-
periments have provided us with precision data which can be used to constrain
the space of cosmological parameters, and thus to confirm or reject cosmological
models. From a theoretical point of view, this is possible because perturbation
theory in a (3+l)-dimensional universe is very well established.
Of course, one would also like to test brane world models with the CMB, in
particular, whether the idea of extra-dimensions is viable at all. Brane world
models should leave distinct imprints in the CMB, such that they can hopefully
be strongly constrained in the future. Unfortunately, perturbation theory with
extra-dimensions is far more complicated than in the standard (3+l)-dimensional
case. We have seen in Chap. 6, that the induced Einstein equations (6.37) on
the brane are not a closed system. Therefore, we cannot simply perturb those
equations, since we miss information on perturbations from the bulk. Instead,
we have to perturb the 5-dimensional Einstein equations, which contain the full
dynamics of gravity. The brane is then taken into account via Israel's junction
conditions.
Recently, a number of works on 5-dimensional perturbation theory have ap-
peared (see [129] and references therein). In these treatments the brane is fixed at
some value of the extra-dimensional coordinate, and the bulk is dynamical. When
one tries to actually solve the corresponding perturbation equations, it turns out
that it is much simpler to consider a static bulk (AdSs ) and a moving brane.
Technically, this is due to the fact that if already the background equations are
dynamical, the perturbation equations become extremely cumbersome. In this
context, one should note that Israel's junction conditions do also apply if the
brane is moving.
In this chapter and in the article 'CMB anisotropics from vector perturbations
in the bulk' in Chap. 9, we focus on vector perturbations, leaving scalar and tensor
modes for future studies. The vector perturbations in the bulk induce vector
perturbations on the brane. In the standard cosmology, vector perturbations
are known to decay for any initial conditions. We shall see in Chap. 9 that
this is completely different for brane worlds, and therefore the study of vector
perturbations is very interesting.
The second reason to consider vector perturbations is that massless vector
modes in the bulk always remain massless, because they are protected by a gauge
symmetry. Therefore, they should be observable at any (in particular low energy)
scale. And finally it is fair to say that vector modes are simpler to treat than
scalars, since they involve only three instead of seven differential equations.
Compared to the mirage cosmology approach, some progress has been made in
that, perturbing Einstein's equations, the back-reaction is now taken into account.
Unfortunately, a direct comparison between the two approaches is not possible,
as the perturbation on the probe brane was a scalar describing the fluctuations
8.2. BULK VECTOR PERTURBATIONS IN 4 + 1 DIMENSIONS 153
of its embedding.
The outline of this chapter is as follows: in Sec. 8.2 we perturb an AdSs bulk
and derive the perturbed vector Einstein equations. In the remaining sections,
we place ourselves on the brane and recall, how temperature fluctuations in the
CMB arise as a result of perturbed photon trajectories (Sec. 8.3), the definition
and some properties of the power spectrum (Sec. 8.4), and finally, how the Cg's
are actually calculated from the metric and matter perturbations (Sec. 8.5). The
treatment in the last three sections is the same as in 4-dimensional standard
cosmology.
8.2 Bulk vector perturbations in 4 + 1 dimensions
8.2.1 Background variables
As mentioned in the introduction, the background is taken to be 5-dimensional
anti-de Sitter space-time (AdSs ). We begin by recalling the relevant background
tensors. For the AdSs metric, we shall use the parametrization given in Eq. (3.63),
dsj = G MN dx M dx N
r 2 T 2 (8-1)
= — (-di 2 + (%dxMx J ') + — dr 2 ,
where r is the coordinate of the extra-dimension, and L is the curvature radius
of the AdSs . The Christoffel symbols are calculated using the formula (1.50),
r^ 4 = ^, rV = -^v. r4 44 = A (8.2)
where \i = 0, 1,2,3, and the other components are zero. Since AdSs is a space-
time of constant negative curvature, we can use the expression for the Ricci tensor
and the Riemann scalar from Sec. 2.4,
-§.
Throughout this and the next chapter, we assume that the bulk is empty, i.e. that
there is no other form of energy than a (negative) cosmological constant A. Then,
the 5-dimensional Einstein equations are
= G MN + KG MN = R MN - X - G MN R + KG MN = ^G MN + AG MN , (8.4)
thus forcing A = — 6/L 2 .
8.2.2 Perturbed metric and gauge invariant variables
The splitting of a general 5-dimensional perturbation into scalar, vector, and
tensor parts is performed with respect to a 3-dimensional subspace M. 3 , because
154
CHAPTER 8. ON CMB ANISOTROPIES IN A BRANE UNIVERSE
then the results are directly comparable to the (3+l)-dimensional cosmology. The
scalar, vector, and tensor modes are irreducible components under 50(3) x E 3 ,
which is the group of isometries of the unperturbed space time. Here, S0(3)
and E3 are the groups of spatial rotations and translations, respectively, and
the terms 'scalar', 'vector', and 'tensor' correspond to the spin-O, spin-1, and
spin-2 representations. The advantage of this decomposition is that, in linear
perturbation theory, scalar, vector, and tensor modes evolve independently. In
the following, for the reasons mentioned in the introduction, we restrict ourselves
to vector modes.
In a 5-dimensional space-time there are three vector variables, Bi,Ci, and Ei
needed to parameterize a general vector perturbation,
dSJ = G MN dx M dx N = (G MN + SG MN )dx M dx N
= -^ [-dt 2 + 2B i dtdx i + (Sij + ViEj + VjEJdx'dx*] + 2C i dx i dr + ^-dr 2 ,
r (8.5)
where Vj denotes the connection in A4 3 . Since Bi,d, and Ei are divergenceless 1
3-vectors, each of them has only two independent components, and we can set
i= 1, 2 for a mode with wave vector k = (0, 0, k). For a more detailed treatment,
as well as for the counting of the degrees of freedom, we refer to [129].
Alternatively, we can write Eq. (8.5) in matrix form,
{SG M
( B 1 B 2 B 3 \
1 -Hil H\2 -"13 ~^lG\
2 B\2 Hq,2 H23 72" C2
3 -^13 H 2 3 H 33 72" C3
V gci £c 2 £c 3 /
(8.6)
with H i:j = ViEj + VjEt.
In order to make coordinate independent statements, the perturbed metric is
written in terms of gauge invariant variables. Under an infinitesimal coordinate
transformation, induced by the divergenceless vector e M ,
x M -> x M +e M , e M = (0,£i,0), (8.7)
with e % = S^Sj, the metric perturbations transform as
SG MN -> SG MN + C £ G MN , (8.8)
where C e G MN is the Lie derivative of the unperturbed metric in the direction e' u .
1 The divergence term yields a scalar.
8.2. BULK VECTOR PERTURBATIONS IN 4 + 1 DIMENSIONS 155
In terms of the variables Bi,d, and Ei, the transformation law (8.8) reads
L 2
B i ^B i +—d t e l1
C i ^C i + d r s i --£ i , (8.9)
Since there are three divergenceless vectors and one gauge freedom (8.7), there
exist two divergenceless gauge invariant vectors which are
E t = Bi - d t E i7 (8.10)
5* = Q - ^d r Ei. (8.11)
We lose no generality, but considerably simplify the calculations, by setting Ei —
0. Then the gauge invariant variables simply reduce to Ej = Bi and Sj = Cj. We
shall work with this gauge fixing in the remainder of this section.
8.2.3 Perturbed Einstein equations
The perturbations of the Christoffel symbols are
ST° Q0 = 8T 0i =
«•-*(£-*)
^ fc 04 = - s' fc + -js,
^r° 44 = o
sr* il = --, { E k:l - Si ,,
«r 4 00 = o
<jr A:
, = - (S M - E i)fc )
<5r\
,- = ^
5r fc 4 ,
^ 2 f-/ 1- \
<JT 4 o«
ST%
- o r ,2 ( H *J + S A*)
(8.12)
The perturbations of the Ricci tensor are calculated by using its expression in
terms of Christoffel symbols (see Eq. (1.56) for the definition),
SR MN = SR A MAN =8T A MN A + ST A BA T B MN + T A BA 8T B MN
A ' A f (8.13)
ST A MA N - 5T A BN T B MA - T A BN ST B MA .
CHAPTER 8. ON CMB ANISOTROPIES IN A BRANE UNIVERSE
This results in
SR 00 = 0,
«fc = £(£* + 3^ -£=?"
-AEi-
-«£«
SRoi = 0,
«««- 5(2^, + **)+»£<*
J + %
.) - A,.
1 ( l2 ~ l \~ 8 ^
-«).
5R U = 0,
with the spatial Laplacian
(8.14)
5 U*v
U* 2 / + \dx 3 )
(8.15)
The Riemann scalar is pure scalar and therefore for vector perturbations one
has
SR = 0. (8.16)
The perturbations of Einstein's equations are
(8.17)
Using that the unperturbed Riemann scalar and the 5-dimensional cosmological
constant are given by R = —20/L 2 and A = —6/L 2 according to Eqs. (8.3) and
(8.4), one obtains the following perturbation equations,
L 2 •• L 2
-^E z - -^-AE* - £ = 0,
r z r 2
which are the 0i,ij, and i4 components of Einstein's equations. So far, Sj =
Sj(i, x, r), and Sj = Sj(i, x, r). In terms of Fourier components Si = Sj(t, k, r),
and Si = Ej(£,k, r), the second equation in (8.18) reads
^ + 3^3,-^ = 0, (8.19)
which is a constraint equation. Its derivative with respect to t can be inserted into
the Oi equation to eliminate the mixed derivative E' t . Similarly, the r derivative is
8.3. TEMPERATURE FLUCTUATIONS IN THE CMB 157
used in i4 to eliminate S^. Then, the first and third equation in (8.18) decouple,
where £j and Sj are now understood as Fourier modes.
In this derivation we have assumed that the bulk contains no other form of
energy than a cosmological constant A which is not perturbed. Therefore, any
bulk matter perturbations are absent in our equations. The perturbed vector
Einstein equations (8.20) can be solved analytically for arbitrary initial conditions.
The solutions are given in Eqs. (9.45) and (9.46) in the article 'CMB anisotropics
from vector perturbations in the bulk' in Chap. 9.
8.3 Temperature fluctuations in the CMB
8.3.1 Induced vector perturbations on the brane
Our previous derivation of the vector perturbations in the bulk is independent of
the presence of a brane. We now consider a 3-brane embedded in the subspace
M 3 , which is moving along a geodesic of AdSs . This setup is the same as in
mirage cosmology in chapters 4 and 5, but now the back- reaction will be taken
into account: the perturbation equations on the brane are derived via the Israel
junction conditions.
From mirage cosmology, we know that the motion of the brane induces a cos-
mological expansion, where the scale factor is proportional to the radial position
of the brane. With a particular embedding, the brane has the properties of a
Friedmann-Lemaitre universe. For a detailed description of the embedding and
the background dynamics we refer to Sec. 9.2.
The vector perturbations in the bulk induce vector perturbations on the brane 2 .
Like in the standard cosmology, they are described by two divergenceless variables
bi and e,,
d5b = 9^Ay^dy v = (fiV + Sg^dy^dy"
= -dr 2 + 2ab i drdy i + a 2 (r)(*y + v>j + V j e i )dy i dy j ,
where y M = (r, y l ) denote internal coordinates on the brane 3 , and t stands for
cosmic time. Since the unperturbed spatial subspace M 3 is flat, its metric is Sij.
The perturbations bi and a are related to Bi,Ei, and d by the pull-back of the
metric (8.5) onto the brane (see also Eq. (9.65)). For the detailed treatment we
2 Since the decomposition into scalars, vectors, and tensors was done with respect to M a , vectors
in the bulk correspond to vectors on the brane, and no further scalars arise.
3 We use y 1 * instead of the usual a^ , because ctj will be used to denote the gauge invariant vector
perturbation on the brane.
158 CHAPTER 8. ON CMB ANISQTRQPIES IN A BRANE UN J
refer to Sec. 9.4; here we just mention that the gauge invariant metric perturbation
is
ai = bi- a&i, (8.22)
and the gauge invariant velocity perturbation,
# i= Vi + aet. (8.23)
The dot denotes a derivative with respect to r, and Vi is the parametrization of
the perturbed 4- velocity of a perfect fluid on the brane, defined in Eq. (9.57). Via
the perturbed Israel junction conditions one finds (see also Eq. (9.72) and (9.73))
<7i = \Zl + L 2 H 2 Zi + LHSi, (8.24)
a l + ^= Vl + L2R2 (a 2 5 r S, - dt-Zi) . (8.25)
2LHa V '
The gauge invariant quantity <t, + i?j is known as vorticity. Given the solutions
for T,i and Hj, these two quantities are the input for the calculation of the vector
CMB anisotropics in a brane universe. Once cr 2 : and i?j are known in terms of
brane coordinates (r, y l ), one can forget about their 5-dimensional origin and
apply the standard theory of CMB anisotropics in 3+1 dimensions. This is what
we are going to do in the remainder of this chapter.
8.3.2 Sources of CMB anisotropies
Photons incident from different directions in the sky have nearly the same tem-
perature of 2.73 K. This almost uniform background radiation is the strongest
indication that our universe was isotropic already at the time of decoupling of
photons and baryons. Today, various experiments are able to measure the tem-
perature fluctuations in the CMB with great accuracy. The major effects and
sources giving rise to those CMB anisotropies are
1. Density fluctuations of photons at the time r\ E of emission from the last
scattering surface. A denser region emits hotter photons according to the
Stefan-Bolzmann law p 1 ~ T 4 .
2. Fluctuations in the gravitational potential. Photons emitted in a potential
well get red-shifted while they climb out of it. Thus, today they are slightly
colder. This is called the Sachs— Wolfe effect.
3. Since the moment of decoupling, the photons have travelled through the
perturbed geometry (8.21). All the gains and losses in energy along a tra-
jectory from ry E to today, rj R , have to be summed up. This is called the
integrated Sachs— Wolfe effect.
4. A Doppler shift, because the matter that last interacted with the CMB
photons was in relative motion to an comoving observer today.
8.3. TEMPERATURE FLUCTUATIONS IN THE CMB 159
Further sources of CMB anisotropics can be found e.g. in Ref. [154]. The first two
effects are associated with perturbations of scalar quantities such as the photon
density contrast. Therefore, we do not take them into account here. The last
two effects involve also vector perturbations, and so they are important for our
calculation of the CMB anisotropics in brane worlds later on. One has to keep
in mind that in our treatment the vector perturbations are created in the bulk.
When projected onto the brane, they manifest themselves in the same way as the
sources listed above. We neglect any vector perturbation that have their origin
on the brane itself.
The power spectrum that we are going to calculate in the article in Chap. 9
looks very different from the standard one, as physics in the extra-dimension leave
a distinct imprint in the CMB. We can use this to strongly constrain at least a
certain class of brane world models with the CMB.
In the following two paragraphs, we investigate in detail the temperature
fluctuations associated with the integrated Sachs- Wolfe effect.
8.3.3 Temperature fluctuations as a function of the perturba-
tion variables
Consider a 4-dimensional Lorentzian manifold .M 4 with internal coordinates y M =
(ij,y), r? denoting conformal time, and a perturbed metric g^ lv = rj^ + h^ v . We
want to investigate the trajectory of a photon through Ai 4 , starting at emission
on the last scattering surface and ending in a telescope today. In parametric form,
this trajectory can be written as y p (A) = (77(A), y(A)), where A denotes an affine
parameter. The perturbed 4-momentum of the photon is
P = ^-. (8.26)
dA
The unperturbed 4-momentum is parameterized as
k* = (u, -vn), (8.27)
where n = (n 1 ,n 2 ,n 3 ) is the direction of observation. Since fc p is a light-like
4- vector (rj^k^k^ = 0), n must be normalized: n 2 = 1. The perturbed photon
momentum is parameterized as
k" = k^ + Sk^ = (1/, -i/n) + (Sk°,6k). (8.28)
We shall see later on, that Sk° has a vector component arising from the metric
perturbation a. We also need to specify the 4- velocity of an observer comoving
with the baryon fluid,
u" = u" + (fa" = (-,0J + (0, — ) , (8.29)
Uf, = TV + 5u^ = (-a, 0) + (0, a( Vi + 6,-)) . (8.30)
160 CHAPTER 8. ON CMB ANISQTRQPIES IN A BRANE UNIVERSE
The scale factor a shows up in these expressions, since we have used conformal
time 4 . Furthermore, we define T R = T(r/ R ,n) to be the temperature of photons
observed today in the direction n, and T E = T(i] E ,x E ) to be the temperature at
the emission point (r/ E ,x E ) on the last scattering surface. The ratio of the two
temperatures is given by
£ = &k. (8.31)
From Eqs. (8.28) and (8.30), one finds
%»&? = -av(l + n i (vi + b i ) + -5k a \ . (8.32)
The combination Vi + bt is equal to ~&i + Uj according to Eqs. (8.22) and (8.23)
and therefore gauge invariant. The denominator in Eq. (8.31) is Taylor expanded
in the perturbed quantities such that Eq. (8.31) becomes, up to first order,
1 E a E v E \ [E v \
(8.33)
The T ~ l/a form of the unperturbed temperature curve is recovered by 5 setting
v ~ l/a 2 . Since the last scattering surface is not a surface of constant time, but a
surface of constant free electron density, we make the following expansions around
the mean time of emission, fj E ,
Ve —Vb + 8rj B ,
a E = a(rj E ) = a(fj B ) + a'(fj E )Sr] E = a(fj E ) (1 + H(fj s )) Si] E ,
T E =T(r ]E ,x E ) = T(f] E )+T'(f] E ,x E )8r lE =T(f] E )+5T E (T] E ,x E ),
T R = T( Vr ,ii) = T( Vr ) + T'(r ]R ,x R )5r lR = T{t] r ) + 5T R (ri R ,n).
Here, the prime denotes a derivative with respect to conformal time rj. The
temperature T(f} E ) = T E is a spatial average over the last scattering surface in
the sense of an ensemble average, whereas T(r) R ) = T R represents an average over
all directions of observations in a unique measurement. Under the assumption of
ergodicity, the two ways of averaging are equivalent. From (8.34) one derives
T R f R ( 8T R ST E \ /n n N
t e = t e { 1 + t;-t;)- (8 - 35)
The left hand side of Eq. (8.33) is now replaced by Eq. (8.35). Isolating 8T R /T R
in the resulting expression and using
g(y E ) _
a R
T R
' T~e
4 Ab
a physical qu
^responding to
antity. (Ik
the affint
i four-velocity is defir
. dr;
? parametrization —
icd as
(8.36)
dy^ drj dy 11 1
dr] dr drj a
8.3. TEMPERATURE FLUCTUATIONS IN THE CMB 161
leads to
S ^ = n\# l + a l )\ n E +±6k°\ + S ^+H(f lE )S VE . (8.37)
The left hand side depends only on the direction of observation n and the time
of observation r] R , the right hand side depends on the point of emission and the
geometry along which the photon travels to us. The term 8T E /T E in Eq. (8.37)
has to be evaluated at (t] E7 x e ). Using the definition of 5T E in Eqs. (8.34) and
the relation (8.36) one finds
^(r? E ,x E ) = ^(ij^x,) -n(fj B )8 Vs , (8.38)
such that in Eq. (8.37) the term in TL is cancelled. The remaining term can be
written as
fW-) = ^ (8.39)
T E 4 p 1
according to the Stephan-Boltzmann law p 1 ~ T 4 . This term is a scalar and
therefore we do not consider it in our treatment. With these steps Eq. (8.37)
becomes
S ^=n i ($ l + a i )\ R E +-5k°\ . (8.40)
8.3.4 The perturbed geodesic equation
We are left with the calculation of the term 5k° /v. When a photon travels through
the perturbed geometry on Ai 4 , its 4- momentum is continuously deflected. In-
stantaneously, this is measured by the quantity Sk^, which is found by solving
the perturbed geodesic equation. The calculation can be considerably simplified
by noting that two conformally equivalent metrics, ds 2 = a 2 ds 2 , have the same
light-like geodesies. We therefore need only consider the perturbed metric,
ds 2 = (v^ + h^dy^dy" = -drj 2 + 2b i dr]dy i + (<% + V;^- + V j e i )dy i dy j . (8.41)
The perturbed 'normalized' 4-momentum of the photon is then
fr=K, (8.42)
dA
where A denotes an affine parameter, and the hat indicates that a quantity is
associated with the metric (8.41). We shall use the parameterization
!" = £'* + Sh? = (1, k l ) + (flSVJfe*), (8.43)
with the light-like normalization (k ) 2 = 1.
The perturbed geodesic equation reads
^+r% /3 fc4 /3 = 0. (8.44)
Sk"
162 CHAPTER 8. ON CMB ANISQTRQPIES IN A BRANE UNIVERSE
Since the background is time-independent, the unperturbed Christoffel symbols
vanish, and the first order perturbation of the geodesic equation is simply
isk" + 5f %k a hP = 0, (8.45)
dA
with
STa/3 = \lf V (hva,p + Kp,a ~ h a p, v ). (8.46)
Using that h va ,pk^ = jrh ua and that, from the unperturbed geodesic equation,
^~- — 0, one can integrate Eq. (8.45) from the moment of emission to the moment
of reception. This results in
-V^Kak^ + \^ v f d\ha0,uk a k^. (8.47)
Without loss of generality, we may now choose hij = V^ + V ' jti = 0. Then
hoi = bi = <Ji- Furthermore, /iqo = as we are considering only vectors. Upon
inserting this into Eq. (8.47), one finds for the zero component
Sk Q \ E = aM" - I dXidrjdi)^. (8.48)
Comparing with Eq. (8.27), we make the identifications fc M = (1, — n l ) and A = r\.
Then Sk° = 8k° /v. Inserting the result (8.48) into (8.40), one finds the final
result
H^= ?M|*+ I d V (d 71 ai)n l . (8.49)
The first term on the right hand side is the perturbation in the Doppler shift
due to the relative motion of the observer and the emitter. The second term is
the integrated Sachs- Wolfe (ISW) effect: the changes of the geometry perturba-
tion along the trajectory, d v ai, are projected onto the 'line of sight' n\ and the
resulting corrections are summed up.
Sometimes it is useful to write the result (8.49) in an alternative form. To
that end the total derivative along the trajectory is written as
" a ' <h '' - > ' (8.50)
Upon partial integration, Eq. (8.49) becomes
-= 1 = n i (0 i + <T i )\l+ d?7 (djaiWtf. (8.51)
This form makes the vorticity appear explicitly.
Eqs. (8.49) and (8.51) are the final results for the temperature fluctuations in
the CMB in the direction n due to vector perturbations in the standard universe
or in a brane world. Notice, that ST R /T R is written entirely in terms of gauge
invariant variables, as it must be the case for a measurable quantity.
8.4. OBSERVATION OF THE TEMPERATURE FLUCTUATIONS 163
8.4 Observation of the temperature fluctuations
In practice, measurements of the temperature fluctuations in the CMB are ex-
pressed in terms of the angular power spectrum Ci, where i is the multipole
number. In this section, we explore the link between 5T R /T R and Ce, following
Refs. [154], [152], and [53]. Our mathematical conventions are those of Ref . [70].
From the angular power spectrum a variety of cosmological parameters can be
extracted.
The temperature fluctuations can be expanded into spherical harmonics Y™,
which form a complete set on the 2-sphere:
^ ee ^(n,^,x R ) = £ Jl a im (r, R ,ic a )Yr(n), (8.52)
R l=0m=-l
where the coefficients ae m are given by
o/mfe,x B ) = y a dfi^(n ))7j „x Jt )y/ n (n)*. (8.53)
Here, T R = T = 2.728 ± 0.002 K denotes the mean temperature of the CMB. In
the following, we are omitting the arguments ry R and x R , which indicate the time
of observation and the position of the observer.
The degree to which the temperature of photons incident from — n and — n' is
correlated, is measured by the two point function
(f(n)f>')). (8.54)
The symbol () denotes the hypothetical average over an ensemble of universes,
i.e. the sum over of all possible different realizations of the CMB for constant
cos 9 = n • n'. From the definition (8.52) one finds
^(n)^(n')^ = E E lflim*i'm>) Y e m (n)Y e m (n'y, (8.55)
where,
(a, m a^)=|do|dO'^(n)^(n')^r(n)*r/(n')
(8.56)
The coefficients ae m originate from a stochastic process, e.g. quantum fluctuations
or phase transitions, that generates perturbations in the early universe. Assuming
that this process is statistically homogeneous and isotropic, one can write
(ag m ae> m ') = 5u>5 mrn >Ci, (8.57)
164 CHAPTER 8. ON CMB ANISOTRQPIES IN A BRANE UNIVERSE
where the quantity Cg does not depend on x H (homogeneity) or m (isotropy).
The CVs are the CMB power spectrum. Using Eq. (8.57), we obtain
/^(n)^(n')\=^C^yr(n)^ m (n')* = ^E( 2£+1 ) C ^( n - n ')'
= 5i_A ,>,.(„.„')
(8.58)
where we have used the addition theorem for spherical harmoincs, and the Pg's
are Legendre's polynomials. In a spatially flat universe, the index t is related to
the angle 9 between two points on the sky in the directions n and n' by 6
Thus the quantity (2£ + l)Cg appearing in Eq. (8.58) is the amplitude of the
temperature correlation on the angular scale 9. The CMB power spectrum has its
highest peak at I ~ 220, which means that the strongest temperature correlation
is between points at an angular distance 9 = ^^- ~ 0.8°.
The relation (8.58) can be inverted to give Cg as a function of the temperature
correlator. Multiplying both sides with Pg>{^) (where fi = nn'), integrating over
the interval [—1,1], and using the orthogonality relation J_ 1 dfj,Pg([j,) Pf (/J,) =
2^p[^U'j one finds
'f-A
C,-2xJ An(— (n) — (n')^P,(/l). (8.60)
In this 'hypothetical' treatment ST/T,ag m , and Cg are random variables in
an ensemble of universes. Since in practice we cannot reproduce our universe
many times, one makes the ergodic hypothesis, that the ensemble average can be
replaced by a spatial average. For a given multipole I, there are 2^+1 independent
numbers ag m over which this average can be taken. One thus defines an estimator
for the observed CMB power spectrum by
' x ' "'-■' (8.61)
The a°^ are assumed to be gaussian random variables following a % distribution
with 2£ + 1 degrees of freedom. Then the expectation value of Cf 03 is equal to
the hypothetical value Cg, and the variance is
eg -V2ZTT ( }
6 This conversion is similar to that between real space and Fourier space quantities, e.g. A = 2ir/k.
8.5. THE C t 'S OF VECTOR PERTURBATIONS 165
For large £, the estimator defined in (8.61) is good, whereas for small £ (large
scales) the variance is a fundamental limitation on the precision of the measure-
ments. This is called 'cosmic variance' and is due to the fact that we can observe
only a single realization of the CMB and this only from one given position.
Since in linear perturbation theory scalar, vector, and tensor perturbations
are independent, also the corresponding a^ m 's and C/s are independent
(af m aj/ m /) = (af m aj, m ,) = (a£ m aj, m ,) = 0. (8.63)
8.5 The CV's of vector perturbations
With the definitions from the previous section, we can now calculate the angular
power spectrum Cg for vector perturbations §i and 0{. The basic idea is to insert
expression (8.49) for 6T/T into the correlator {a^ m a^/ m <} given in Eq. (8.56), and
carry out the integrals over the sphere. In detail, the procedure is as follows.
First, ST/T is expanded into vector Fourier modes,
( <S.(i I !
which allows us to treat each k independently. For a fixed k, one defines a set of
orthogonal basis vectors (k,e + ,e x ) as shown in Fig. 8.1 by
e A -k = 0, e x -e x ' = S X \>, A,A' = +, X. (8.65)
Since a photon is transversely polarized, the amplitudes ^(77, k) and 0^(77, k) can
be decomposed into
£«(»/, k) = Y, $x(v,k)a x (k)e x (k),
*;fo,k)= Y. vx(v,k)a x (k)e x (k),
where a\(k) are random amplitudes for each mode k, and e x are the components
of e A .
We start our calculation from Eq. (8.49),
— (n) = -n l 7?i(r/ E ,x B ) + / di] (d v ai)n l .
(8.67)
The term i}i(r] R ,x R ), which is a dipole corresponding to the observers (our) mo-
tion, has been omitted since it can be set to zero by a redefinition of the coordi-
CHAPTER 8. ON CMB ANISOTROPIES IN A BRANE UNIVERSE
Figure 8.1: The direction of observation, n, and the set of orthogonal basis vectors
(k,e+,e x ) over which the amplitudes ^(77, k) and <5^(?7,k) are decomposed.
nates. On inserting the above Fourier decompositions one finds for the tempera-
ture correlator,
A ' A ' (8.68)
+ / dr] n l e$ a' x e tk '* > x {k —> k',n —> n', A — > A'}* (a\(k)a\>(k')} .
Assuming that the process generating the perturbations was isotropic, the corre-
lator of the a\(k) satisfies
(a x (k)a x ,(k')) = S 3 (k-k')6 xx ,. (8.69)
Next, the temperature correlator (8.68) has to be integrated over the 2-sphere.
Therefore, it is useful to write n l e\ and e lk ' x in terms of angular variables. With
respect to the basis (k,e + ,e x ), one has n = (sin 9 cos <f>, sin 9 sin (f>, cos 9), where
we use 9 for the angle between k and n (rather than between n and n' as before).
Setting e+ = (l,0,0),e x = (0,1,0), and k = (0,0, k) one finds
n l e i — 5 X sin 9 sin </> + 5 + sin 9 cos (f>
1 1 _ (8.70)
= 8 X sin 9— (e l< ^ — e l *j + S + sin 9- (e l< ^ + e J *) .
From the definition ^— = fc p = (1,— n l ) in the previous section, we have x =
8.5. THE C/'S OF VECTOR PERTURBATIONS
n (Vn ~ v)> an< i hence
k • x = k • n(r] R — 77) = /c(r/ R — ??) cos #.
(8.71)
Eqs. (8.69), (8.70), and (8.71) are inserted into Eq. (8.68), and the fc'-integral and
the sum over A' drop out.
When the temperature correlator is integrated over the sphere (see Eq. (8.56)),
the quantity
lf m (r],k)= I dOy f m (n)* sin6»e ±i V fe( ' 7R -' 7)cos0 (8.72)
appears. The plane wave ^{vr-^cosO - ls decomposed into spherical Bessel func-
tions and Legendre polynomials,
e ik( VR -n) cose = J^ i l \2e' + i)je[k(ri H - i])]Pe' (cos 9)
''"" (8-73)
= ]T i e 'W + VMKVx ~ r,)]J-^-Y$(n),
l'=0 S M +1
such that T lm becomes
lf m = ^ i e ' V^(2l' + l)je< [k(vn ~ V)} f dH Y e m (n)*Y?,(n) sin ee ±i<t> . (8.74)
To solve the integral on the right, one uses the recursion relations for spherical
harmonics to write
The remaining integral turns out to be simply the orthogonality relation
(' dnY e m (n)*Yl ±1 (n) = 6 t , e ±i6 m ,i. (8.76)
Putting everything together, one finds
which can be simplified using the relation
-MX)
(8.78)
if = ±6 m ±1 i l+1 J^(U + 1) J 1(1 + 1) 3 J^h — p. ( 8 j9)
k(r] R - 77)
168 CHAPTER 8. ON CMB ANISOTRQPIES IN A BRANE UNIVERSE
After this side-calculation we now resume the calculation of the the power
spectrum Cg. The temperature correlator (8.68) is inserted into Eq. (8.56) for
£ = £' and m = m' . Since all information about the stochastic correlations are
contained in a\(k), and because of the assumption (8.69), we can replace (a( m a.( m )
by |a^ m | 2 . Then Eq. (8.56) becomes
+ f dr t [-ia' x (l+ m - J-J + a' + (l+ m +J- m )] | 2 .
(8.80)
We have carried out the remaining sum over A to remove 8* , S x coming from
Eq. (8.70). The amplitudes i? x , $+, <3"' x , a' + were defined in Eq. (8.66). Notice also
that the curly bracket {k — > k', n — ► n', A — ► A'}* in Eq. (8.68) has combined with
the first one to give a norm squared, and the factor 1/4 is due to Eq. (8.70).
From Eq. (8.80), one finds Cg by averaging over the 2£+ 1 possible values of m
according to Eq. (8.61). Thereby, the sum over m cancels with (5 TO) ±i in Eq. (8.77)
and one obtains the final expression
c e = -eu+i) f dkk 2 \A e (k)\ 2 ,
i" Jo
ma,) =~{h+ iK) y - - ^ )] + r d^ {o> + + ^ y - - f ,
V2 \ ' k( VR -r) E ) J R yj2 K + ' k(rj R - rj)
where i}\ = $\(r] E ,k), a' x = <j\(ri,k), and the prime denotes a derivative with
respect to rj.
We give an alternative form for A^(fc) which might be useful in some cases. It
is obtained by partial integration and using
(£ - i) J Jrl _ {£ + 2 ) J J±l = j' l+1 { x ) + j' e _^ x ). (8.82)
A t (k) = — =((#+ + *+) + i(# x
\ Je[k{r] R -7] E )}
^V+^o+J^^x^oxjj k{rJR _
+ 2ZT1 / dr] 72 ^ + + ^ x) ^ j ' £ + l[k ^ R ~ v)] + 3t-AKnR -v)}}-
(8.83)
Here, the first term involves the vorticity i}\ + a\, and the amplitude &\ appears
without derivative.
Chapter 9
CMB anisotropies from
vector perturbations in the
bulk (article)
170 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
This chapter consists of the article 'CMB anisotropics from vector perturba-
tions in the bulk', see Ref. [132].
It is also available under http://lanl.arXiv.org/abs/hep-th/0307100.
In this article, the internal coordinates on the brane are denoted by y^ instead
of cr p , in order to avoid confusion with the gauge invariant vector perturbation
on the brane, o^.
9.1. INTRODUCTION
CMB anisotropies from vector perturbations in the
bulk
Christophe Ringeval, Timon Boehm, and Ruth Durrer
Departement de Physique Theorique, Universite de Geneve, 24 quai E. Ansermet, CH-1211 Geneva
4 Switzerland.
The vector perturbations induced on the brane by gravitational waves propa-
gating in the bulk are studied in a cosmological framework. Cosmic expansion
arises from the brane motion in a non compact ^-symmetric five-dimensional
anti-de Sitter space-time. By solving the vector perturbation equations in the
bulk, for generic initial conditions, we find that they give rise to growing modes
on the brane in the Friedmann-Lemaitre era. Among these modes, we exhibit
a class of normalizable perturbations, which are exponentially growing with re-
spect to conformal time on the brane. The presence of these modes is strongly
constrained by the current observations of the cosmic microwave background
(CMB). We estimate the anisotropies they induce in the CMB, and derive quan-
titative constraints on the allowed amplitude of their primordial spectrum.
PACS number: 04.50.+h, ll.10.Kk, 98.80.Cq
9.1 Introduction
The idea that our universe may have more than three spatial dimensions has been
originally introduced by Nordstrom [119], Kaluza [87] and Klein [94]. The fact
that superstring theory, the most promising candidate for a theory of quantum
gravity, is consistent only in ten space-time dimensions (11 dimensions for M
theory) has led to a revival of these ideas [124, 125, 77]. It has also been found that
string theories naturally predict lower dimensional "branes" to which fermions and
gauge particles are confined, while gravitons (and the dilaton) propagate in the
bulk [8, 126, 105]. Such brane worlds have been studied in a phenomenological
way already before the discovery that they are actually realized in string theory [3,
136].
Recently it has been emphasized that relatively large extra-dimensions (with
typical length L ~ /xm) can solve the hierarchy problem: the effective four-
dimensional and the fundamental D-dimensional Newton constant are related by
Gi oc G D /L n . Thus G4 can become very small if the fundamental Planck mass
is of the order of the electroweak scale. Here n denotes the number of extra-
dimensions [12, 13, 9, 128]. It has also been shown that extra-dimensions may
even be infinite if the geometry contains a so-called "warp factor" [127].
The size of the extra-dimensions is constrained by the requirement of recover-
ing usual four-dimensional Einstein gravity on the brane, at least on scales tested
172 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
by experiments [102, 155, 7]. Models with either a small Planck mass in the
bulk [12, 13, 9], or with non compact warped extra-dimensions [128, 127], have
been shown to lead to an acceptable cosmological phenomenology on the brane [19,
44, 43, 142, 60, 107, 135], with or without Z 2 symmetry in the bulk [40, 17, 41].
Explicit cosmological scenarii leading to a nearly Friedmann-Lemaitre universe at
late times can be realized on a 3-brane at rest in a dynamical bulk [18, 159] or,
alternatively, on a brane moving in an anti-de Sitter bulk [98, 81]. It has been
shown that both approaches are actually equivalent [116].
One can also describe brane worlds as topological defects in the bulk [23,
71, 11, 118, 66]. This is equivalent to the geometrical approach in the gravity
sector [133], while it admits an explicit mechanism to confine matter and gauge
fields on the brane [133, 130, 15, 56, 49, 55, 48, 50, 120, 67, 4, 117]. Depending
on the underlying theory, the stability studies of these defects have shown that
dynamical instabilities may appear on the brane when there are more than one
non compact extra-dimensions [65, 88, 123], whereas this is not the case for a five-
dimensional bulk [68], provided that a fine-tuning between the model parameters
is fixed [21].
The next step is now to derive observational consequences of brane world
cosmological models, e.g. the anisotropics of the cosmic microwave background
(CMB). To that end, a lot of work has recently been invested to derive gauge in-
variant perturbation theory in brane worlds with one co-dimension [138, 100, 156,
32, 33]. Again, the perturbation equations can be derived when the brane is at
rest [129], or when it is moving in a perturbed anti-de Sitter space-time [116, 114,
113, 115, 47, 46]. Whatever the approach chosen, the perturbation equations are
quite cumbersome and it is difficult to extract interesting physical consequences
analytically. Also the numerical treatment is much harder than in usual four-
dimensional perturbation theory, since it involves partial differential equations.
Nevertheless, it is useful to derive some simple physical consequences of per-
turbation theory for brane worlds before performing intensive numerical studies.
This has been done for tensor perturbations on the brane in a very phenomenolog-
cal way in Ref. [101] or on a more fundamental level in Ref. [62]. Tensor modes
n the bulk which induce scalar perturbations on the brane have been studied
n Ref. [54] and it was found that they lead to important constraints for brane
worlds.
In this article we consider a brane world in a five-dimensional bulk where
cosmology is induced by the motion of a 3-brane in AdSs . The bulk perturbation
equations are considered without bulk sources and describe gravity waves in the
bulk. The present work concentrates on the part of these gravity waves which
results in vector perturbations on the brane.
For the sake of clarity, we first recall how cosmology on the brane can be
obtained via the junction conditions, particularly emphasizing how Z 2 symme-
try is implemented [19, 44, 43, 142, 60, 107, 135]. After rederiving the bulk
perturbation equations for the vector components in terms of gauge invariant
variables [116, 114, 113, 115, 47], we analytically find the most general solutions
9.2. BACKGROUND 173
for arbitrary initial conditions. The time evolution of the induced vector pertur-
bations on the brane is then derived using the perturbed junction conditions. The
main result of the paper is that vector perturbations in the bulk generically give
rise to vector perturbations on the brane which grow either as a power law or
even exponentially with respect to conformal time. This behavior essentially dif-
fers from the usual decay of vector modes in standard four-dimensional cosmology,
and may lead to observable effects of extra-dimensions in the CMB.
The outline of the paper is the following: in the next section, the cosmological
brane world model obtained by the moving brane in an anti-de Sitter bulk is
briefly recalled. In Sec. 9.3 we set up the vector perturbation equations and solve
them in the bulk. In Sec. 9.4 the induced perturbations on the brane are derived
and compared to those in four-dimensional cosmology, while Sec. 9.5 deals with
the consequences of these new results on CMB anisotropics. The resulting new
constraints for viable brane worlds are discussed in the conclusion.
9.2 Background
As mentioned in the introduction, our universe is considered to be a 3- brane
embedded in five-dimensional anti-de Sitter space-time
r 2 T 2
(9.1)
The capital Latin indices A,B run from to 4 and the fiat spatial indexes i,j
from 1 to 3. Anti-de Sitter space-time is a solution of Einstein's equations with a
negative cosmological constant A
G AB +AG AB =0, (9.2)
provided that the curvature radius L satisfies
L 2 = ~l (9.3)
Another coordinate system for anti-de Sitter space can be defined by the co-
ordinates transformation r 2 /L 2 = eT 2e l L . Then the metric takes the form
ds 2 5 = G AB dx A dx B = e~ 2e/L (-dt 2 + <%dxMx j ) + dg 2 , (9.4)
which is often used in brane world models.
9.2.1 Embedding and motion of the brane
The position of the brane in the AdSs bulk is given by
x M = X M (y»), (9.5)
174 CHAPTER 9. CMB ANISOTROPICS FROM VECTOR PERTURBATIONS
where X M are embedding functions depending on the internal brane coordinates
y^ (fi = 0, ■ ■ • ,3). Using the reparametrization invariance on the brane, we choose
x % = X % = y l . The other embedding functions are written
X° = t h (r), X 4 = r h (r), (9.6)
where r = y° denotes cosmic time on the brane. Since we want to describe a
homogeneous and isotropic brane, X° as well as X 4 are required to be independent
of the spatial coordinates y l . The four tangent vectors to the brane are given by
<d M = d ^d M , (9.7)
M dyK
and the unit space-like normal 1-form n M is defined (up to a sign) by the ortho-
gonality and normalization conditions
n M e™ = 0, G AB n A n B = 1. (9.8)
Adopting the sign convention that n points in the direction in which the brane is
moving (growing r\> for an expanding universe), one finds using
e°=i b , 4 = n,, e) = 5), (9.9)
the components of the normal
Uq = — r'b, n 4 = tb, rti = 0. (9.10)
The other components are vanishing, and the dot denotes differentiation with
respect to the brane time r.
This embedding ensures that the induced metric on the brane describes a
spatially flat homogeneous and isotropic universe,
dsl = g^dy^dy" = -dr 2 + a 2 (r)<%djW, (9.11)
where a(r) is the usual scale factor, and g^ v is the pull-back of the bulk metric
onto the brane
9^ = G AB e*e B v , (9.12)
(see e.g. [38, 131]). The first fundamental form q AB is now defined by
i.e. the push- forward of the inverse of the induced metric tensor [39, 38]. One
can also define an orthogonal projector onto the brane which can be expressed in
terms of the normal 1-form
± AB =n A n B =G AB -q AB , (9.14)
in the case of only one codimension.
9.2. BACKGROUND 175
Upon inserting the equations (9.1), (9.10) and (9.13) into the above equation,
one finds a parametric form for the brane trajectory [98, 81, 115, 47]
r h (r) = a(r)L,
1 / (9-15)
a
where H = a/a denotes the Hubble parameter on the brane. Alternatively, this
result can be obtained by comparing expression (9.12) with the Friedmann met-
ric (9.11).
Therefore, the unperturbed motion induces a cosmological expansion on the
3-brane if r^ is growing with t^,.
9.2.2 Extrinsic curvature and unperturbed junction conditions
The cosmological evolution on the brane is found by the Lanczos-Sen-Darmois-
Israel junction conditions 1 . They relate the jump of the extrinsic curvature across
the brane to its surface energy-momentum content [99, 141, 45, 82]. The extrinsic
curvature tensor projected on the brane can be expressed in terms of the tangent
and normal vectors as
' l " ' (9.16)
Here V denotes the covariant derivative with respect to the bulk metric, and C n
is the five-dimensional Lie-derivative in the direction of the unit normal on the
brane. With the sign choice in Eq. (9.16), the junction conditions read [111]
k>„ ~ k% = 4 [s^ - \s 9va \ = njS^, (9.17)
where S^ is the energy momentum tensor on the brane with trace S, and
^6^5 = ^, (9.18)
where M 5 and G5 are the five-dimensional (fundamental) Planck mass and New-
ton constant, respectively. The superscripts ">" and "<" stand for the bulk sides
with r > rb and r < ?v As already noticed, the brane normal vector n M points
into the direction of increasing r [see Eq. (9.10)]. Eq. (9.17) is usually referred to
as second junction condition. The first junction condition simply states that the
first fundamental form (9.13) is continuous across the brane.
In general, there is a force acting on the brane which is due to its curvature
in the higher dimensional geometry. It is given by the contraction of the brane
energy momentum tensor with the average of the extrinsic curvature on both sides
of the brane [17]
■S" 1 " (*£„ + k< v ) = 2/. (9.19)
1 In the following, they will be simply referred to as "junction conditions'.
176 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
This force /, normal to the brane, is exerted by the asymmetry of the bulk with
respect to the brane [38, 17]. In this paper, we consider only the case in which the
bulk is ^-symmetric across the brane, hence / = 0. In this case the motion of
the brane is caused by the stress energy tensor of the brane itself which is exactly
the cosmological situation we have in mind.
From Eqs. (9.10), (9.11), (9.15) and (9.16), noting that the extrinsic curvature
can be expressed purely in terms of the internal brane coordinates [47, 115], one
has
*W = ~ [°ab {e*d v n B + e A v d^n B ) + e*e B n a G ABtC ] • (9-20)
A short computation shows that the non vanishing components of the extrinsic
curvature tensor are
1 + L 2 H 2 + £ 2 H
iVH-^-H- (9.21)
k tj = -^-y/i + UHHij.
It is clear, that the extrinsic curvature evaluated at some brane position r^ does
not jump if the presence of the brane does not modify anti-de Sitter space. Like
in the Randall-Sundrum (RS) model [127], in order to accommodate cosmology,
the bulk space-time structure is modified by gluing the mirror symmetric of anti-
de Sitter space on one side of the brane onto the other [116]. There are two
possibilities: one can keep the "r > r^" side and replace the "r < r^" side to get
*>„ = fc/iio k % = -*>> ( 9 - 22 )
where fc M „ is given by Eq. (9.21). Conversely, keeping the r < r\> side leads to
k^u = -ku.u, k< v = fc M „. (9.23)
Note that both cases verify the force equation (9.19). From the time and space
components of the junction conditions (9.17) one obtains, respectively
± VT±TJP = _i_ 4(p + pT) (925)
Here the brane stress tensor is assumed to be that of a cosmological fluid plus a
pure tension p T , i.e.
S^ = {P + P )u^u v + Pg^-p T g^ 1 (9.26)
p and P being the usual energy density and pressure on the brane, and u^ the
comoving four-velocity The "±" signs in Eqs. (9.24) and (9.25) are obtained by
keeping, respectively, the r > r^, or r < rt,, side of the bulk. In order to allow
9.2. BACKGROUND 177
for a positive total brane energy density, p + p T , we have to keep the r < r^ side
and glue it symmetrically on the r > r^ one 2 . In the trivial static (H = 0) case
this construction reproduces the Randall-Sundrum II [127] solution with warp
factor exp(—\g\/L), for — oo < g < oo if we choose r\> = L = constant. In our
coordinates, we just have < r < r^ on either side of the brane, and the bulk is
now described by two copies of the "bulk behind the brane" . Even if r only takes
values inside a finite interval, and even though the volume of the extra dimension,
is finite, the bulk is semi-compact and its spectrum of perturbation modes has no
gap (like in the RS model).
From Eqs. (9.24) and (9.25), one can check that energy conservation on the
brane is verified
p + 3H (P + p) = 0. (9.28)
Solving Eq. (9.25) for the Hubble parameter yields
ff2 =#'( i+ £) + !^-i- < 9 - 29 »
At "low energies" , \p/p T \ <C 1, the usual Friedmann equation is recovered provided
the fine-tuning condition
**-h (9 - 30)
is satisfied. The four-dimensional Newton constant is then given by
(9.31)
Thus a positive tension is required to get a positive effective four-dimensional
Newton constant. Note also that low energy means r 2 ~ H~ 2 ^> L 2 . In the
Friedmann-Lemaitre era, the solution of Eq. (9.29) reads
„•)/■>
CU2)
= "o(H
for a cosmological equation of state P = wp with constant w. The parameters
H and a refer, respectively, to the Hubble parameter and the scale factor today.
For the matter era we have w — 0, and during the radiation era w = 1/3.
2 Note that we obtain the same result as in Ref. [47]: a positive brane tension for an expanding
universe is obtained by keeping the anti-de Sitter side which is "behind the expanding brane with
respect to its motion" .
178 CHAPTER 9. CMB ANISQTRQPIES FROM VECTOR PERTURBATIONS
9.3 Gauge invariant perturbation equations in the
bulk
A general perturbation in the bulk can be decomposed into "3-scalar" , "3- vector"
and "3-tensor" parts which are irreducible components under the group of isome-
tries (of the unperturbed space time) SO(3) x E 3 , the group of three dimensional
rotations and translations. In this paper we restrict ourselves to 3- vector pertur-
bations 3 and consider an "empty bulk", i.e. the case where there are no sources
in the bulk except a negative cosmological constant. With respect to the bulk,
and its four spatial dimensions, only bulk gravity waves are therefore considered
since they are the only modes present when the energy momentum tensor is not
perturbed. It is well known (see e.g. Ref. [129]) that gravity waves in 4 + 1 di-
mensions have five degrees of freedom which can be decomposed with respect to
their spin in 3 + 1 dimensions into a spin 2 field, the ordinary graviton, a spin
1 field, often called the gravi-photon and into a spin field, the gravi-scalar. In
this work we study the evolution of the gravi-photon in the background described
in the previous section.
After setting up our notations, we find the gauge invariant vector perturbation
variables in the bulk and write down the perturbed Einstein equations. We derive
analytic solutions for all vector modes in the bulk.
9.3.1 Bulk perturbation variables
Considering only vector perturbations in the bulk, the five dimensional perturbed
metric can be parameterized as
(9.33)
+ ^-dr 2 + 2B~dtdx i + 2C i dx i dr,
where V; denotes the connection in the three dimensional subspace of constant t
and constant r. Assuming this space to be flat one has V^ = di. The quantities
E l , B\ and C l are divergenceless vectors i.e. diE 1 = diB 1 = diC 1 = 0.
As long as we want to solve for the vector perturbations in the bulk only, the
presence of the brane is not yet relevant. Later it will appear as a boundary con-
dition for the bulk perturbations via the junction conditions as will be discussed
in paragraph 9.4.3.
Under a linearized vector type coordinate transformation in the bulk, x M — »
x M + e M , with e M = (0, £j,0), the perturbation variables defined above transform
3 The prefix "3-" will be dropped in what follows, and the term "vector" will be always applied
here for spin 1 with respect to the surfaces of o
9.3. GAUGE INVARIANT PERTURBATION EQUATIONS IN THE BULK 179
as
E i -> E i + ^2 £ U
Bi ^ Bi+ ^ dteij (9.34)
C i ^C i + d r e i --e i .
As expected for three divergenceless vector variables and one divergenceless
vector type gauge transformation, there remain four degrees of freedom which are
described by the two gauge invariant vectors
Y>i = Bi- d t Ei, (9.35)
Z z = Ci- j^d r E % . (9.36)
Note that in the gauge Ei = these gauge-invariant variables simply become B t
and Ci respectively.
9.3.2 Bulk perturbation equations and solutions
A somewhat cumbersome derivation of the Einstein tensor from the metric (9.33)
to first order in the perturbations leads to the following vector perturbation equa-
tions,
T. /r-3 \
(9.37)
^^S + 5^a r S-L 2 a t 2 E + L 2 AS = 0, (9.38)
£<5 h ) - 5<5 h ) - M5 h ) + l Ht* e ) = °- < M9 »
where A denotes the spatial Laplacian, i.e.
A = 6 ij didj, (9.40)
and the spatial index on S and H has been omitted. One can check that these
equations are consistent, e.g. with the master function approach of Ref. [113].
A complete set of solutions for these equations can easily be found by Fourier
transforming with respect to x l , and making the separation ansatz:
E(t,r,k) = S T (t,k)S R (r,k), (9.41)
S(t,r,k) = S T (i,k)3 a (r,k). (9.42)
The most general solution is then a linear combination of such elementary modes.
180 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
Eq. (9.38) splits into two ordinary differential equations for S T and S R ,
,<9 r E E , , ~
r s R
d 2 ^
(9.43)
(9.44)
where k is the spatial wave number, and ±fi the separation constant having the
dimension of an inverse length squared. The frequency Q represents the rate of
change of £ R at r ~ L, while the rate of change of E T is \/\Q 2 T k 2 \. From the
four-dimensional point of view, — O 2 can also be interpreted as the mass m 2 of
the mode so that ±0 2 = — m 2 . The signs in Eqs. (9.43) and (9.44) come from
the choice O 2 > 0. Eq. (9.43) is a Bessel differential equation of order two for
the "— " sign and a modified Bessel equation of order two for the "+" sign [1],
while Eq. (9.44) exhibits oscillatory or exponential behavior in bulk time. From
Eq. (9.39), similar equations are derived for S T (i, k) and S R (r, k). This time, the
radial function is given by Bessel functions of order one. The constraint equation
(9.37) ensures that the separation constant ±Q 2 is the same for both vectors and
it also determines their relative amplitude. The general solution of Eqs. (9.37) to
(9.39) is a superposition of modes f2,k which are given by
o K
fiJHr\ p±ty /n^¥
^! Y (— \ P ±W^ 2 + fc 2
(9.45)
(')..!.(,)
Here K p and I are the modified Bessel functions of order p while J p and Y p are the
ordinary ones. The ± signs in Eqs. (9.45) and (9.46) correspond to the two linearly
independent solutions of Eq. (9.44), whereas the sign of the separation constant
determines the kind of Bessel functions: +fi 2 (or m 2 < 0) for the modified Bessel
function K and I; — O 2 (or m 2 > 0) for the ordinary Bessel functions J and Y. In
9.3. GAUGE INVARIANT PERTURBATION EQUATIONS IN THE BULK 181
general, each of these modes 4 can be multiplied by a proportionality coefficient
which depends on the wave vector k and ft. Eq. (9.37) ensures that this coefficient
is the same for S and H. Furthermore, notice that if Q 2 > k 2 the K- and I
modes can have an exponentially growing behavior, whereas for Q 2 < k 2 one sets
\/i! 2 — k 2 = i\/\fl 2 — k 2 \ such that the modes become oscillatory. The J- and
Y-modes are always oscillating.
For a given perturbation mode to be physically acceptable one has to require
that, at some initial time £; n , the perturbations are small for all values < r <
r b(tin), compared to the background. To check that, we use the limiting forms
of the Bessel functions [1]. For large arguments, the ordinary Bessel functions
behave as
J p (x) - J — cos(x--p-
Y p (*) x :
while the modified Bessel functions grow or decrease exponentially
(9.47;
v (9.48)
I (x) ~ -;=e x .
Therefore, in Eqs. (9.45) and (9.46) all modes, except for the K-mode, diverge as
r — > 0. Hence the only regular modes are
s = A(k, n)^K 2 (^n\ e ±f v / o 2 -fc 2 ; (9 49)
where the amplitude A(k, ft) is determined by the initial conditions and carries
an implicit spatial index. For small wave numbers k 2 < O 2 the growing solution
rapidly dominates, whereas for large wave numbers k 2 > H 2 both solutions are
comparable and oscillating in time. It is easy to see that the K-mode is also
normalizable in the sense that
£ VW\\Z\ 2 dr oc £ I [K 2 (^V)] 2 dr <
£ v1*| 2 dr oc £ I [ Kl (^)] 2 dr <
(9.51)
Note that also the J-modes and Y-modes are normalizable. One might view
this integrability condition as a requirement to insure finiteness of the energy of
4 In the following, they will be labeled by the kind of Bessel function they involve, e.g. "K-mode" ,
182 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
these modes. This suggests that the J- and Y-modes could also be excited by
some physical process. Indeed, from Eq. (9.4), their divergence for r — > can be
recast in terms of the g coordinate, with g — > oo. Expressed in terms of g, the
integrability condition (9.51) ensures that the J- and Y-modes are well defined
in the Dirac sense, and thus that a superposition of them may represent physical
perturbations 5 [151].
Let us briefly discuss also the zero-mode Q = 0. The solution of Eqs. (9.37),
(9.38) and (9.39) are then
(9.52)
(9.53)
These solutions (which can also be obtained from Eqs. (9.45) and (9.46) in the
limit H — > 0) diverge for r — > but the A-mode is normalizable in the sense that
the integrals defined in Eq. (9.51) converge.
Clearly the most intriguing solutions are the K-modes, especially for values
of the separation constant verifying H 2 > k 2 . Then, if present, these fluctuations
soon dominate the others in the bulk. The fact that m 2 = — O 2 < in this case
implies that the K-modes are tachyonic modes, and it is thus not surprising that
they may generate instabilities.
Before we go on, let us just note, that all these solutions are also valid solutions
of the bulk vector perturbation equations in the RS model. In their original
work [127], Randall and Sundrum have obtained very similar equations (we used
somewhat different variables). However, they considered only the solutions with
m 2 = +H 2 > and therefore did not find the growing K-modes. As we shall see
in the next section, this choice is justified when one considers boundary conditions
which do not allow for any anisotropic stresses on the brane. This is indeed well
motivated as far as cosmology is not concerned. A more detailed discussion of
the relevance of these modes for the RS model is given in appendix 9.7.1.
In our cosmological framework however, if there is no physical argument which
forbids these modes, they have to be taken seriously since they represent solutions
of the perturbations equations which are small at very early times and grow
exponentially with respect to bulk time. Note that this instability is linked to the
particular bulk structure considered here where the brane lies at one boundary
of the space-time. In the full AdSs , < r < oo, the K-modes are clearly not
normalizable since K p (L 2 Q/r) diverges for r — > oo.
At last, one may hope that the K-modes are never generated. However, during
any bulk inflationary phase which leads to the production of 4 + 1 dimensional
gravity waves, as we shall see now, if the anisotropic stresses on the brane do
not vanish identically the K-modes are perfectly admissible solutions. For a
given inflationary model, it should be also possible to calculate the spectrum of
fluctuations, |A(k,0)| 2 .
5 This is of course not the case for the I-modes.
9.4. THE INDUCED PERTURBATIONS ON THE BRANE 183
At this stage, the perturbation modes have been only derived in the bulk. In
the next section, we shall determine the induced perturbations on the brane using
the perturbed junction conditions.
9.4 The induced perturbations on the brane
9.4.1 Brane perturbation variables
Since we are interested in vector perturbations on the brane induced by those in
the bulk, we parameterize the perturbed induced metric as
d5b = 9iJ,v&y^&y u
= -At 2 + 2ab i drdy i (9.54)
+ a 2 (Sij + Vtej + Vj-ei) dyMy j ,
where e l and b l are divergence free vectors. The junction conditions which relate
the bulk perturbation variables to the perturbations of the brane can be written
in terms of gauge invariant variables. Under an infinitesimal transformation y^ — >
yM _|_ £A» ; w here £ M = (0, a 2 ^), we have
e% -> e; + &,
(9.55)
bi->bi + a€i.
Here the dot is the derivative with respect to the brane time r and £j is a di-
vergence free vector field. Hence the gauge invariant vector perturbation is [52]
(Ti = bi- oeV (9.56)
This variable fully describes the vector metric perturbations on the brane.
The brane energy momentum tensor S^ u given in Eq. (9.26) has also to be
perturbed. As we shall see, the junction conditions (together with Z 2 symmetry)
do in general require a perturbed energy momentum tensor on the brane. Since
we only consider vector perturbations 8p = SP = 0. However, the perturbed
four- velocity of the perfect fluid does contain a vector part u M = u^ + 8u^ , with
Su" = [ v * | , (9.57)
and where v l is divergenceless. Under j/ M — > j/ M + £^ ,
Vi^Vi-aiu (9.58)
where Vi = 8ijV l . A gauge invariant perturbed velocity can therefore be defined
as
$i = v i + oe i . (9.59)
184 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
In addition, the anisotropic stresses contain a vector component denoted 7Tj. Since
the corresponding background quantity vanishes, this variable is gauge invariant
according to the Stewart-Walker lemma [146].
In summary, there are three gauge invariant brane perturbation variables. We
shall use the combinations
di = bi — adi,
d i = v i + ae i , (9.60)
To apply the junction conditions we need to determine the perturbations of
the reduced energy momentum tensor defined in Eq. (9.17). In terms of our gauge
invariant quantities they read
SS TT = 0, (9.61)
SSn = -a(p+ 2 -p- l -p T ^j a.-aiP + p) ^, (9.62)
SSij = a 2 P (duTj + djiTi) . (9.63)
9.4.2 Perturbed induced metric and extrinsic curvature
We now express the perturbed induced metric, and the perturbed extrinsic cur-
vature in terms of the bulk perturbation variables [47] . In principle there are two
contributions to the brane perturbations: perturbations of the bulk geometry as
well as perturbations of the brane position. A bulk perturbed quantity has then to
be evaluated at the perturbed brane position [see Eq. (9.6)]. Using reparametriza-
tion invariance on the brane [47], the perturbed embedding can be described in
terms of a single variable T,
X M =X M +Tn M , (9.64)
where all quantities are functions of the brane coordinates y^ . Since T is a scalar
perturbation it does not play a role in our treatment, and we can consider only
the perturbations 8G AB due to the perturbed bulk geometry evaluated at the
unperturbed brane position. The induced metric perturbation is then given by
Sg^u = g^u - g^u = e*e*6G AB . (9.65)
From Eqs. (9.33), (9.35), (9.36) and (9.65) one finds in the gauge E t =
Sg T r = 0,
8g Ti = aVl + £ 2 # 2 £; + aLHZi, (9.66)
Sg tJ = SG tJ = 0.
The time component vanishes as it is a pure scalar, and the purely spatial compo-
nents can be set to zero without loss of generality by gauge fixing (Ei = e^ = 0).
9.4. THE INDUCED PERTURBATIONS ON THE BRANE 185
In the same way, perturbing Eq. (9.20), and making use of Eqs. (9.7), (9.8)
in order to derive the perturbed normal vector, leads to (again we use the gauge
Sk TT = 0, (9.67)
5k Ti = -d t Zi - -a 2 d r Hi - aHy/l + L 2 H 2 E
+ -a^l + L 2 H 2 (diEj + djZi) , (9.69)
where 8k ^ v = k^ v — k^, and all bulk quantities have to be evaluated at the brane
position. In the derivation we have also used that on the brane <9 M = e^d A .
9.4.3 Perturbed junction conditions and solutions
The first junction condition requires the first fundamental form q AB to be conti-
nuous across the brane. Therefore, the components of the induced metric (9.54)
are given by the explicit expressions (9.66). This leads to the following relations
e-i = E h
i (9-70)
b t = Vl + mPBi + LHCi,
where the bulk quantities have to be evaluated at the brane position (£t>, ?~b)- For
(Ti = hi — aei we use
aet = a (t h d t Ei + r h d r Ei)
= \/l + L 2 H 2 d t Ei + a 2 LHd r Ei.
Together with Eqs. (9.35) and (9.36) this gives
Vi = y/l + L 2 H 2 Zi + LHZi. (9.72)
The equations corresponding to the second junction condition are obtained by
perturbing Eq. (9.17) (using Z 2 symmetry, k> v = —k< v = —k^ y ) and inserting
the expressions (9.68), (9.69) for the perturbed extrinsic curvature tensor, with
Eqs. (9.62), (9.63) for the perturbed energy-momentum tensor on the brane. After
some algebra one obtains for the Oi and the ij components, respectively
. 2Lg a(<T i + i ) = a 2 drXi-d t E i , (9.73)
K 2 5 aPTTi = -LHZi - \/l + L 2 H 2 Ei, (9.74)
186 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
where we have used the unperturbed junction conditions, Eqs. (9.24) and (9.25),
and the fact that on the brane d^ = e^d A .
In the RS model one has H = H = and the requirement that the anisotropic
stresses vanish identically. We show in appendix 9.7.1 that the well-known results
of Refs. [127] and [151] are recovered in this limit.
Hence, if by some mechanism, like e.g. bulk inflation, gravity waves are pro-
duced in the bulk, their vector parts Ej and Hj will induce vector metric perturba-
tions &i on the brane according to Eq. (9.72). The vorticity ai + fii and anisotropic
stresses 7Tj on the brane define boundary conditions for the bulk variables accord-
ing to Eqs. (9.73) and (9.74). In general, the time evolution of 7T; may be given
by some additional matter equation, like e.g. the Boltzmann equation or some
dissipation equation which usually depends also on the metric perturbations. It
is interesting to note that for generic initial conditions in the bulk, the amplitude
of the K-mode does not vanish, which means that the anisotropic stresses on the
brane may grow exponentially. At late time (for LH <C 1) Eq. (9.74) reduces to
K^aPTTi = — Hj. A generally covariant equation of motion for m must be com-
patible with this behavior since it is a simple consequence of the 5-dimensional
Einstein equations for a certain choice of initial conditions.
In the following we do not want to specify a particular mechanism which ge-
nerates Ej and Hj, and just assume they have been produced with some spectrum
given by A(k,fl). In usual 4-dimensional cosmology it is well-known that vector
perturbations decay. Therefore, in ordinary 4-dimensional inflationary scenarios
they are not considered. Only if they are continuously re-generated like, e.g. in
models with topological defects (see e.g. Ref. [51]), vector modes affect CMB
anisotropics. Here the situation is different since the modes considered are either
exponentially growing or oscillating with respect to bulk time. Therefore, we
expect the behavior of vector perturbations to be very different from the usual
4-dimensional results even in the absence of K-modes.
In the following, we assume that the boundary and initial conditions are such
the 7Ti(ti n ) ^ 0. They therefore allow for K-mode contributions. Clearly, if this
happens it leads to exponential growth of at and 7Tj. However, before concluding
about the viability of these modes, one has to check if they have observable
consequences on the brane. Indeed, anisotropic stresses are often very small
(e.g. of second order only) and one may therefore hope that the initial amplitudes
of the K-modes are also very small and do not lead to destructives effects, at
least on time scales equal to the age of the universe. By estimating the induced
CMB anisotropics, we show in the next section that this is not the case.
9.5 CMB anisotropies
To calculate the CMB anisotropies from the vector perturbations induced by bulk
gravity waves, the relevant quantities are a and ■& + a given in terms of the bulk
variables by Eqs. (9.72) and (9.73). Inserting the solutions (9.49) and (9.50) for
9.5. CMB ANISOTROPIES
the K-mode into (9.72) and (9.73) yields
a(t hl k) = A(k,n) | VTTl^p-kJ — )
V O 2 a 2 ^ a )\
3 ±t b ^0 2 - k 2
(9.75)
(9.76)
( CT + .)(t b ,k) = ^(k,0)4^±^^4K 1 f^e^^^,
where again we have omitted the spatial index i on tx, •& and A. Similar equations
can be obtained for the J- and Y-modes by replacing, in Eqs. (9.75) and (9.76),
the modified Bessel functions by the ordinary ones, plus the transformations:
-k 2 -> k 2 and ± -> ±i.
These equations are still written in bulk time t\, which is related to the con-
formal time rj on the brane by
dt h = \/l + L 2 H 2 dr]. (9.77)
Therefore, at sufficiently late time L 2 H 2 <C 1 such that dt^ ~ di]. Note that L
is the size of the extra-dimension which must be smaller than micrometers while
H^ 1 is the Hubble scale which is larger than 10 5 light years at times later than
recombination which are of interest for CMB anisotropics.
As a result, the growing or oscillating behavior in bulk time carries over to con-
formal time. Moreover there are additional time dependent terms in Eqs. (9.75)
and (9.76) with respect to Eqs. (9.49) and (9.50) due to the motion of the brane.
As can be seen from Eqs. (9.75) and (9.76), the modes evolve quite differently for
different values of their physical bulk wave number fi/a. In the limit fi/a <C \/L
and for fl 2 > k 2 , the growing K-modes behave like
A k 2 / ( 9 - 78 )
CT + ^(W2^ '
where use has been made of L 2 H 2 <C 1, and of the limiting forms of Bessel
function for small arguments [1]
K ^ x Kzol T{p) {l) P - (9J9)
In the same way, from Eq. (9.48), the K-modes verifying CI/ a ^> 1/L reduce to
(9.80)
188 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
They are exponentially damped compared to the former [see Eq. (9.78)]. As a
result, the main contribution of the K-mode vector perturbations comes from
the modes with a physical wave number fi/a smaller than the energy scale 1/L
associated with the extra-dimension. As the universe expands, a mode with fixed
value Q remains relatively small as long as the exponents in Eq. (9.80) satisfy
n
L - vVtt 2 -k 2 ~-[pL- rVn 2 - k 2 j > 0. (9.81)
When this inequality is violated, for k <C O this is soon after r ~ L, the mode
starts growing exponentially. The time r ~ L also corresponds to the initial time
at which the evolution of the universe starts to become Friedmannian.
In the same way, one can derive the behavior of the J- and Y-modes on the
brane for physical bulk wave numbers greater or smaller than the size of the extra-
dimension. This time, the exponentially growing terms are replaced by oscillatory
ones, and the ordinary Bessel functions are approximated by (see Eq. (9.47) and
Ret [1])
J .w,:.f5Ti)(I) P ' , (
1 /2\» (9 ' 82)
From Eqs. (9.47) and (9.82), the equivalents of Eqs. (9.75) and (9.76) for J-
and Y-modes can be shown to oscillate always. From Eq. (9.32), their amplitude
is found to decay like a~ 3 / 2 in the short wavelength limit fi/a 3> l/L. In the
long wavelength limit fi/a <C 1/L, the amplitude of the Y-mode stays constant
whereas the J-mode decreases as a -4 .
The vorticity is also found to oscillate in conformal time. This time, the
amplitude of the long wavelength Y-modes always grows as a 3w+1 while for the
J-modes it behaves like a 3 ™ -1 . Finally, in the short wavelength limit, both Y and
J vorticity modes grow like a iw+1 / 2 .
Whatever the kind of physical vector perturbation modes excited in the bulk,
we have shown that there always exist bulk wave numbers H that give rise to
growing vector perturbations on the brane. Although the J- and Y-modes ge-
nerate vorticity growing like a power law of the scale factor, they can be, in a
first approximation, neglected compared to the K-modes which grows like an ex-
ponential of the conformal time. We therefore now concentrate on the K-modes
and derive constraints on their initial amplitude A(k, fi) by estimating the CMB
anisotropics they induce.
In order to determine the temperature fluctuations in the CMB due to vector
perturbations on the brane, we have to calculate how a photon emitted on the last
scattering surface travels through the perturbed geometry (9.54). A receiver to-
day (with conformal time rj R ) therefore measures different microwave background
temperatures T(r] R ,n l ) for incident photons coming from different directions n\
With the conventions and formulae in the paragraphs 8.3.3 and 8.3.4 the vector-
9.5. CMB ANISOTROPIES
type temperature fluctuations read
-r
- ,, ,. , , n nWdA,
^ ^ 7 <^' (9.83)
where A denotes the amne parameter along the photon trajectory and the prime
is a derivative with respect to conformal time ??. The "R" and "E" index refer
to the time of photon reception (today) and emission at recombination. For the
second equality we have used
dci , -d<Ti . ,
— - = o' i -n>— -, 9.84
dA dxi v '
where — n l is the direction of the photon momentum. We have also neglected the
contribution from the upper boundary, "R" , in the first term since it simply gives
rise to a dipole term. The first term in Eq. (9.83) is a Doppler shift, and the
second is known as integrated Sachs- Wolfe effect. To determine the angular CMB
power spectrum Cg, we apply the total angular momentum formalism developed
by Hu and White [80]. According to this, a vector perturbation v is decomposed
as
v = e + v + +e x v x , (9.85)
where e + ' x are defined so that (e + , e x , and k = k/fc) form a righthanded or-
thonormal system. Using this decomposition for #; and ctj, one obtains the an-
gular CMB perturbation spectrum Cg via
C e =-£(£+ 1) f k 2 (\A e (k)\ 2 )dk, (9.86)
A ( (k) = -#+fa
kri n — kri„
/ ° 4.// , n li(kri n — kri) , v '
In Eq. (9.87) we have assumed that the process which generates the fluctuations
has no preferred handedness so that (|o" + | 2 ) = (|cr x | 2 ) as well as (|$ + | 2 ) = (|i? x | 2 ).
Omitting the + and x superscripts, we can take into account the other mode
simply by a factor 2. Furthermore, we have redefined i] = r\ R .
As shown in the previous section, the main contribution of the K-modes comes
from those having long wavelengths a/fl 3> L, and k < O. In the following, only
these modes will be considered. Since they are growing exponentially in r], the
integrated Sachs- Wolfe contribution will dominate and we concentrate on it in
what follows. A more rigorous justification is given in appendix 9.7.2. Inserting
190 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
the limiting form (9.78) for a in Eq. (9.87) gives
A e (k) ~ 2A o n n k n e™oV
(9.88)
where a simple power law ansatz has been chosen for the primordial amplitude
^/(|A(k,ft)| 2 ) = A (n)n 2 L 2 k n . (9.89)
A dimensionless wave number k, and conformal time ro, have also been introduced
as
k=-, zu = r)Q, (9.90)
in order to measure their physical counterparts in units of the bulk wavelengths.
The condition k < O now becomes k < 1. The integration over r\ in the inte-
grated Sachs- Wolfe term is transformed into an integration over the dimensionless
variable x defined by
x = k(7 lo -r 1 ) = k(w -w), (9.91)
the subscript "0" refers to the present time. Note that x E = k (i] — r] E ) ~ kr\ .
By observing the CMB, one may naturally expect that the perturbations with
physical wavelength greater than the horizon size today have almost no effect. In
terms of our parameters, this means that the main contribution in the Cg comes
from the modes verifying O/a > H , hence vj > 1.
In appendix 9.7.2, we derive a crude approximation for the angular power
spectrum induced by the exponentially growing K-modes, in a range a little more
constrained than the one previously motivated, namely
f max ff„ < — < ^— , (9.92)
% 1 + z E
where z E is the redshift at photon emission which is taken to coincide with re-
combination, z E ~ 10 3 . In order to simplify the calculation, we do not want
the transition between the damped K-modes (ft /a > L~ l ) and the exponentially
growing ones (O/a < L^ 1 ) to occur between the last scattering surface and today.
This requirement leads to the upper limit of Eq. (9.92). Moreover, in order to de-
rive the Cg, we perform an expansion with respect to a parameter l/ru assumed
small, and £ max refers to the multipole at which this approximation breaks down.
The lower limit in Eq. (9.92) comes from this approximation. Using the values
L =s 10- 3 mm, H- 1 ~ 10 29 mm, f max ~ 10 3 , and z E ~ 10 3 , one finds
(9.93)
9.6. CONCLUSION 191
The corresponding allowed range for the parameter w Q becomes [see Eq. (9.116)]
10 3 < w < 10 29 . (9.94)
Clearly the detailed peak structure on the CMB anisotropy spectrum would
have been different if we had taken into account the oscillatory parts (k > O) of
the K-modes, as well as the Y- and J-modes, but here we are only interested
in estimating an order of magnitude bound. As detailed in appendix 9.7.2, for a
scale invariant initial spectrum, i.e. n = —3/2, we obtain
From current observations of the CMB anisotropics, the left hand side of this
expression is about 10~ 10 , and for I ~ 10, one gets
e -K-51nK,)]
A(V)< Wj ■ (9.96)
From Eq. (9.116) and (9.94), one finds that the primordial amplitude of these
modes must satisfy
A (n) < e- w \ for O/a ~ lO^mm" 1 (9.97)
and, more dramatically,
A (O) < e- lf)29 , for O/o ~ lmm" 1 (9.98)
for the short wavelength modes. As expected, the perturbations with wavelength
closer to the horizon today (smaller values of f2) are less constrained than smaller
wavelengths [see Eq. (9.97)]. Moreover, one may expect that the bound (9.98)
is no longer valid for H/a > L _1 /(l + z E ) since the modes in Eq. (9.80) start
to contribute. However, the present results concern more than 20 orders of mag-
nitude for the physical bulk wave numbers n/a , and show that the exhibited
modes are actually very dangerous for the brane world model we are interested
in.
It seems that the only way to avoid these constraints is to find a physical
mechanism forbidding any excitation of these modes.
9.6 Conclusion
In this paper we have shown that vector perturbations in the bulk generically lead
to growing vector perturbations on the brane in the Friedmann-Lemaitre era. This
behavior radically differs from the usual one in four-dimensional cosmology, where
vector modes decay like a~ 2 whatever the initial conditions.
Among the growing modes, we have identified so called K-modes which are
perfectly normalizable and lead to exponentially growing vector perturbations on
192 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
the brane with respect to conformal time. By means of a rough estimate of the
CMB anisotropies induced by these perturbations, we have shown that they are
severely incompatible with a homogeneous and isotropic universe; they light up
a fire in the microwave sky, unless their primordial amplitude is extremely small.
No particular mechanism for the generation of these modes has been specified.
However, one expects that bulk inflation leads to gravitational waves in the bulk
which do generically contain them. Even if they are not generated directly, they
should be induced in the bulk by second order effects. Usually, these effects are
too small to have any physical consequences, but here they would largely suffice
due to the exponential growth of the K-modes [see Eqs. (9.97) and (9.98)]. This
second order induction seems very difficult to prevent in the models discussed
here.
It is interesting to note that this result is also linked to the presence of a
non compact extra-dimension which allows a continuum of bulk frequencies Q.
A closer examination of Eq. (9.44) shows that the mode Q = 0, admits only J-
and Y-mode behaviors. In a compact space, provided the first quantized value
of O is sufficiently large, one could expect the exponentially growing K-modes
to be never excited by low energy physical processes. Another more speculative
way to dispose of them could be to consider their causal structure: as we have
noticed before, the modes with separation constant +Q 2 are tachyons of mass
— H 2 from the four-dimensional point of view. From the five-dimensional point of
view, these are not "propagating modes", but "brane-modes" which decay into
the fifth dimension with penetration depth d = L 2 il.
In a more basic theory, which goes beyond our classical relativistic approach,
these modes may thus not be allowed at all.
Finally, we want to retain that even if the K-modes can be eliminated in
some way, the growing behavior of the Y- and J-modes remains. Although their
power law growth is not as critical as the exponential growth of the K-modes,
they should have significant effects on the CMB anisotropies. Indeed, they lead
to amplified oscillating vector perturbations which are entirely absent in four-
dimensional cosmology.
We therefore conclude that, if no physical mechanism forbids the generation
of the discussed vector modes with time dependence oc exp(ryv0 2 — k 2 ). anti-
de Sitter infmitly thin brane worlds, with non compact extra-dimension, cannot
reasonably lead to a homogeneous and isotropic expanding universe.
Acknowledgments
It is a pleasure to thank Robert Brandenberger, Cedric Deffayet, Roy Maartens,
Filippo Vernizzi, David Wands and Peter Wittwer for helpful discussions. This
work is supported by the Swiss National Science Foundation. We also acknowledge
technical and moral support by Martin Zimmermann in the last phase of the
project.
9.7. APPENDIX 193
9.7 Appendix
9.7.1 Comparison with the Randall- Sundrum model
As already mentioned in the text, if the brane is at rest (H = 0) at r^ = L, our
model reduces to the RS II model. One may ask therefore, quite naturally, why
has our dangerous K-mode never been discussed in the context of RS II? In this
appendix we address this question.
First of all, the bulk solutions S and S for vector perturbations of AdSs with
a brane, remain valid. The solutions with m 2 = — O 2 < have, however not
been discussed in the literature so far. Also, when constructing the Green's
function [63, 127, 151], these solutions have not been considered. As we shall see
now, for most problems that is most probably very reasonable.
In the RS II model one considers perturbations which do not require anisotropic
stresses on the brane, iti = 0. Eq. (9.74) then reduces to
S(r = L,t,fi,fc) = , (9.99)
such that S has to vanish on the brane. We insert this into a general solution of
the form
~(r,t,n,k) = ±t^i + ky^^ e ±lt ^ n2 + k2
(9.100)
for to 2 = n 2 > 0,
3(r,t,fi,fc) = ±^l-k 2 /n 2 ^ I e ±t ^ n2 - k2
H^) + M,(£2)], (MO.,
for m 2 = -ST < .
The boundary condition (9.99) then implies
D = ~C^j^-, for m 2 = -S! 2 <0. (9.103)
Eq. (9.102) is exactly the relation which has also been found in Ref. [127], while
Eq. (9.103) is new. However, if the solution is not allowed to grow exponentially
when approaching the Cauchy horizon r — > 0, one has to require D = 0, which
implies C = since K 1 has no zeros. With this physically sensible condition
(see Ref. [151]), we can discard these solutions. Nevertheless, in cases where the
I-modes can be regularized, e.g. by compactification, presence of a second brane,
194 CHAPTER 9. CMB ANISOTROPICS FROM VECTOR PERTURBATIONS
the most general Green's function would include them. It is interesting to note
that the calculation of the static potential of two masses M\ and M^ at distance x
generated by the exchange of the zero-mode and the two continua of Kaluza-Klein
modes with positive and with imaginary masses, simply leads to
V{x) ~ G^^ ( 1 + / mL 2 e- mx dm - f mL 2 e- imx dm)
' V Jo Jo J (9.104)
The short distance modification hence deviates by a factor of 2 from the result
of Ref. [127], if we include the tachyonic modes. One has to be aware of the fact
that, like so often, the result is sensitive to the choice of the Green's functions.
Anyway, small initial perturbations of the RS solution which allow for small
anisotropic stresses, so that the condition (9.103) does not need to be imposed,
will in general contain a small K-mode which grows exponentially and renders
the cosmological model unstable. It seems to us that this possibility has been
overlooked in the literature so far.
We give a simple example which sketches the presence of this instability. We
consider a 1 + 1 dimensional Minkowski space-time, with orbifold-like spatial sec-
tions which can be identified with two copies of y > 0. The "brane" is represented
by the point y = and the "bulk" by the two copies of y > 0. For an initially
small perturbation /(j/, t) in the bulk, which satisfies a hyperbolic wave equa-
tion, we want to analyze whether an instability can build up. We are looking for
solutions of
d 2 f-d 2 y f = 7 (9.105)
with small initial data, say f(t = 0, y) <C 1 and d t f(t = 0,j/) C 1 for all y >
0. By separation of variables one can find a complete set of solutions, / =
f±(k)exp[±ik(y ± t)]. For a sufficiently small value of f±(k) these solutions
satisfy the initial conditions. These solutions oscillate in time; they have constant
amplitude. However, there are other solutions, / = g±(k)exp[±k(y ± t)]. Since
the initial data has to be small, the solutions oc exp(+ky) are not allowed. But
the solutions f = g- exp[— k{y — t)\ have perfectly small initial data and they
represent an exponential instability. If we fix the boundary conditions, setting
f(t, y = 0) = 0, or dtf(t, y = 0) = 0, this instability disappears, but if /(£, y = 0)
is free, even a very small initial value /(0, 0) ^C 1 can induce an exponential
instability. Clearly, this leads also to an exponential growth of the boundary
value f(t,y = 0).
If we give f(0,y) — Aexp(-ky) and dtf(0,y) = kAexp(-ky) as initial condi-
tions, the function f(t,y) = Aexp[k(t — y)\ solves the equation and generates the
growing exponential. If we would require, as an additional boundary condition
that, e.g. the solutions at y = remain at least bounded, this mode would not
be allowed and we would have to expand the initial data in terms of the oscilla-
tory modes. However, it seems to us acausal to pose conditions of what is going
9.7. APPENDIX 195
to happen "on the brane" in the future. But mathematically, without any such
"acausal" boundary conditions, the initial value problem is not well posed. This
example is a simple analog of our instability. As long as anisotropic stresses va-
nish identically, only the J- and Y-modes are relevant. However, if the brane has
arbitrarily small but non vanishing anisotropic stresses on which we do not want
to impose any constraints for their future behavior, an exponential instability can
build up. This is a rather unnatural behavior which may cast doubts on the RS
realisation of brane worlds in the context of cosmological perturbation theory.
9.7.2 CMB angular power spectrum
We first present a crude and then a more sophisticated approximation for the
Cg power spectrum from the exponentially growing K-modes. As we shall see,
at moderate values of t ~ 10-50, both lead to roughly the same bounds for the
amplitudes which are also presented in the text.
Crude approximation
Here we start from Eq. (9.88). In the integral
f * iMe-x^^dx, (9.106)
Jo x
we replace ji by its asymptotic expansion for small £,
This is a good approximation if either x E ~ kr/o < £/2 or (£/2)(l/k 2 — l) 1 / 2 >
1. Since k 2 < H 2 , the first condition is always satisfied if the first of the two
inequalities in (9.92) are fulfilled. The integral of x then gives
nA 2 Q 2n k 2n ( 1
<|A,(fc)| 2 > * ° 22 ^3 [ ¥ ~ l ) e*-.Vi-". (9.108)
Integrating over k, we must take into account that our approximation is only valid
for k < fc max = (O 2 — rjQ 2 ) 1 / 2 . Since we integrate a positive quantity we certainly
obtain a lower bound by integrating it only until A: max . To simplify the integral
we also make the transformation of variables y = y 1 — k 2 . With this and upon
inserting our result (9.108) in Eq. (9.86), we obtain
«.)=(§)'
ec, > ^Aiii'^'j' (i - j,y+<-'/y-=v».»di,.
1 > 1 on the entire range of integration. Henc<
(9.109)
For t > 2, y 3 ~ 2e > 1 on the entire range of integration. Hence we have
(9.110)
196 CHAPTER 9. CMB ANISOTRQPIES FROM VECTOR PERTURBATIONS
This integral can be expressed in terms of modified Struve functions [1]. In the
interesting range, w Q ^> 1, we have
[\l ~ y 2 )^ 1 ^^ c T{U + n l+V 2) e*°o. (9.111)
Jo Aw Q '
Inserting this result in Eq. (9.110) we finally obtain
t N
+£+1/2
J 2 2 *+ 1 \w 0/
V2^Vle- e / jr
2 2l+1 Wo,
A 2 n 2r
(9.112)
■4V' ;
where we have used Stirling's formula for T(£ + n + 1/2) and set n = —3/2 after
the ~ sign.
In the next paragraph we use a somewhat more sophisticated method which
allows us to calculate also the Doppler contribution to the C/s. For the ISW
effect this method gives
ec ^f*i^{^T A '^ ( " 13)
for n = —3/2. Until t ~ 15 the two approximations are in reasonable agreement
and lead to the same prohibitive bounds for A (£l). For t > 15, Eq. (9.113)
becomes more stringent.
Sophisticated approximation
In Eq. (9.88) we have only considered the dominant contribution coming from the
integrated Sachs- Wolfe effect. The general expression is obtained by inserting the
solutions (9.78) for a and ■d in Eq. (9.87),
d.r.
To derive the first term we have used Eq. (9.32) in the matter era. The parameter
P=6(l + z E )(^) (9.115)
reflects the change in behavior of the modes, redshifted by z E to the emission
time, which are either outside or inside the horizon today. It is important to note
that the parameter H a /Q completely determines the effect of the bulk vector
perturbations on the CMB, together with the primordial amplitude A . Indeed,
9.7. APPENDIX 197
solving Eq. (9.29) in terms of conformal time, and using Eqs. (9.28) and (9.32),
yields r) ~ 2/(a H ) in the Friedmann-Lemaitre era. Thus
Q/o n 1 n/a n
<^ 2 T7f' "■"IT^T (9 - 116)
We now replace the spherical Bessel functions j e in the integrated Sachs- Wolfe
term (ISW) using the relation [1]
Je{x) = V^ +i/2(x) ' (9 ' 117)
In the ISW term the upper integration limit can be taken to be infinity as the
contribution from x E to infinity can be neglected provided x\/l/k 2 — 1 > 1. This
restriction is equivalent to k 2 < 1 — l/t*7 which is verified for almost all values of
k up to one, given that w varies in the assumed range (9.94). We remind that
for the exponentially growing K-mode k < Q, and hence < k < 1. This allows
for the exact solution [70]
(9.118)
where F is the Gauss hypergeometric function. In regard to the subsequent
integration over k we approximate F as follows. For small values of k, F is nearly
constant with value 1, at k = 0. As k — > 1 the slope of F diverges and it cannot
be Taylor expanded anymore. However, by means of the linear transformation
formulas [1], F can be written as a combination of hypergeometric functions
depending on 1 — k 2
„(tt ,. 3„n r H) r Q) „at .i. „\
\2'2 ' 2' J _(l 3\-{t 1\ V2'2
ifrfr
(9.119)
These in turn can be expanded around 1 — k 2 = 0, and gives
F ~ 2 e+1/2 (l-eVl-k 2 ) . (9.120)
These two approximations intersect at k — \/\ — \/t 2 . In this way, we can
evaluate the mean value of F by integrating the two parts over the interval [0, 1].
Thus, the hypergeometric function is replaced by
>M-l/2
(9.121)
198 CHAPTER 9. CMB ANISQTROPIES FROM VECTOR PERTURBATIONS
Furthermore, the Gamma functions in (9.118) can be approximated using Stir-
ling's formula [1]
T(£) 1
r(£ + 3/2) e 3 / 2 '
Putting everything together and squaring Eq. (9.114) we obtain
(9.122)
\ M k)\ 2 = 2^P«o-| (i - 1) 2 ^'^ a +1/2 IH^o - -J]) 2
r— —\ 2 ^-
v-
-J f + 1/2 [fc(w„-C7 H )] —
- 1} (l-fc 2 ) ]
36£ 7 J ■
The Cn 's are then found by integrating over all k- modes
3 Ci 2 \^t(i
H(k)\ 2 dk
(9.124)
= 4A 2 o e(e + i)o 2n+3 (cf ] + cf ) + cj 3) ) ,
where the C] correspond to the three terms in Eq. (9.123). In the following we
keep only the zero order terms in w /w E . From Eqs. (9.123), (9.124) one finds
CP = A A 2 ™" 1 ( l ~ ^) e 2ro E vCT rj r kw \] 2 dk.
w 3j o \ k 2 J e+1/2
(9.125)
First, notice that if the argument is larger or smaller than the index, the Bessel
functions are well approximated by their asymptotic expansions (9.47) and (9.82),
respectively. Therefore, we split the fc-integral into two integrals over the intervals
[0, kf\ and [ki, 1], in each of which the Bessel function is replaced by its limiting
forms. The transition value kg is given by ki ~ £/zv . In the integral from kg
to 1, the sin 2 (fcro ) is then replaced by its mean value 1/2 which is justified if
the multiplying function varies much slower in k than the sine. To carry out the
integration we make the substitution y 2 = 1 — k 2 , and in order to simplify the
notation we define the integral
r(a,M) = /
„(1 _ y 2 ) "e aw «»dy. (9.126)
9.7. APPENDIX
In this way we can write Eq. (9.125) in the form
2
~\
I / :r N 2f +! r
Cf = —. fl(0, y t , n - 3/2) - 2^1(0, y t , n - 1/2) + 2-1(0, w , n + 1/2)1
r 2 (^ + 3/2) *
(9.127)
Since ye = \J 1 — kj is very close to one, and the integrand is continuous in the
interval [0, 1], integrals of the form l(yi, 1, v) can be well approximated by the
mean value
-(-) . (9.128)
For the integrals of the type 1(0, ye, v) we distinguish between three cases:
Case a: v > — 1. This case corresponds to a spectral index n > 1/2 in the first
integral in Eq. (9.125). We write l(0,ye,v) = 1(0,1, u) -l(y t ,l,v). The
solution of the latter is given by Eq. (9.128), whereas the former can be
solved in terms of modified Bessel and Struve functions [70]
x(0 ' *' v) = 2(^Ti) + ^Y w ^'^ vv{y + 1} [U/.( 2w b) + k +3/2 (2^J] .
(9.129)
Since our derivation assumes zu E > £, the large argument limit applies and
we have
e 2ro E
I^ +3/2 (2tt7 E ) + L, +3/a (2tS7 B ) ~ , (9.130)
independently of the index v.
Case b: i/ = — 1. Since the above expressions, Eq. (9.129), diverge for v = — 1,
we approximate the integral by
1(0, y e , v) s e 2ro E ^ y(l - y 2 )-^ = -e 2ro E m (J-\ . (9.131)
We have checked that the numerical solution of 1(0, ye, v) agrees well with
the approximation, provided yg_ is close to 1.
Case c: v < — 1. We use the same simplification as in Eq. (9.131), and now the
integral yields
l(0, yi , v )^e 2 ^ J^ y(l-y 2 y&y--^—^[ — ) . (<>.]:L>)
\2(*+l)
2(v + 1) \vu
200 CHAPTER 9. CMB ANISOTROPIES FROM VECTOR PERTURBATIONS
For the particular value n = —3/2, Eq. (9.127) contains terms 1(0, y£,— 3) and
X(0, ye, —2) which can be evaluated according to (9.132), and a term 1(0, yi, —1)
for which we use (9.131). The remaining three integrals over the interval [yt,l\
are evaluated by (9.128). The result is
r-n e^o/^E r f 2 ( t 2 \ 2 ( P \ e 2/3£ ( I 2 \ 2 1
133)
The parameter (3 is a constant of order unity, within in our approximation it is
(3= l-ln2~0.3.
The second term cf ) in Eq. (9.123) reads
(9.134)
where only the zero order terms in w e Jvj has been kept. Using the limiting
forms for the Bessel function for arguments smaller and larger than the transition
value kg, yields
1 ~ ZPI 2 wl l2 T(t+Zl2) h
1+1/2
Ak
(9.135)
/i3F(i-£)*.(fa.-!«)
For consistency with the derivation of C\ , we have assumed that the main
contribution comes from the first integral, while the second one is small due to
the oscillating integrand. Since kg ^C 1, we can use again the mean value formula
to evaluate the first integral, and by the Stirling formula for T(£+3/2), Eq. (9.135)
becomes
df~ •" fAV^Vd-JM (9136)
Since I < vj q , the spectrum is damped at large £, while the other terms can lead
to the appearance of a bump, depending on the value of w and n.
The last terms C) reads
tf'-js.jf.-^*"^!-*").*-
(9.137)
9.7. APPENDIX 201
Splitting this expression in two terms over 1 — k 2 , and using the substitution
y 2 = 1 — k 2 yields
-l(o,l,n + £+^)\, (9.138)
where X is given by Eq. (9.126) with w E — > w Q . As before, these two integrals
can be expressed in terms of modified Bessel and Struve functions [70]. From
Eq. (9.129), taking their limiting forms at large argument, and expanding the T
function by means of the Stirling formula gives
r <*)„ 1 . /^^ fM B+W/2 f9 139)
£ ~ 72e^(n + £ + 3/2)(n + £+l/2) \j 2 £^/ 2 36 \wj ' { '
Clearly, C\ dominates over the others since it involves exp(2ro ) while C\
and C\ appear only with fractional power of this factor, namely exp(o7 ) and
exp(w /z E ). This is due to the fact that we are concerned with incessantly grow-
ing perturbations leading to the predominance of the integrated Sachs- Wolfe ef-
fect.
Inserting Eqs. (9.133), (9.136) and (9.139) for the particular value n = -3/2
into Eq. (9.124) gives the final CMB angular power spectrum
~~2tt
+ 1)_ 2.Je w o/*B [ I 2 ( I 2 \ 2 f £\ #1* ( I 2 \ 2 1
e ro o (J_ V- 1 m f _ _P_\ /^e 2ro o e- £ f ±V^ \
12^/2^3/2 y w J e [ l 24zJ + \l2 P/2 36 \ W J J"
(9.140)
Part III
BRANE GAS COSMOLOGY
Chapter 10
The cosmology of string gases
206 CHAPTER 10. THE COSMOLOGY OF STRING GASES
In the last part of this thesis, we are going to address two main questions: first,
how can one avoid the initial singularity in the standard cosmology, and second,
why does our universe have apparently three spatial dimensions? As different as
they are, both questions can possibly be resolved in a single framework called
'string gas' or 'brane gas' cosmology. This proposal is quite different than the
brane world models we have encountered so far.
Roughly speaking, brane world models fall in two categories: models in which
one or two parallel 3-branes are at orbifold fixed points of a compact or non
compact extra-dimension, and models in which a 3-brane is moving through an
anti-de Sitter space-time. Both (in the end equivalent) scenarii are based on a
compactification of 1 1-dimensional M theory down to a five-dimensional space-
time (see paragraph 2.5.3).
In contrast, in brane gas cosmology all nine spatial dimensions predicted by
superstring theory are compact from the start. The matter source is a variety of
p-branes and strings, and the term 'gas' is used because they are homogeneously
distributed in space and have no particular orientation with respect to each other.
In contrast to brane world models, we do not live on one of these branes, but
simply somewhere in the bulk.
In the next two sections, 10.1 and 10.2, we closely follow the original work of
Brandenberger and Vafa [29], where only fundamental strings were considered.
An extension including p-branes was made in Ref. [5]. The aim of the article
'On T-duality in branes gas cosmology' in the next chapter is to show that the
arguments for a non singular cosmological evolution given in Ref. [29] carry over
when branes are also included.
10.1 Avoiding the initial singularity
10.1.1 T-duality
Crucial in string and brane gas cosmology is the concept of T-duality introduced
in Sec. 3.4. Let us consider closed fundamental strings moving in a compact space
which, for simplicity, is taken to be a 9-dimensional torus. The strings form a gas
in the sense mentioned above. The possible string states are oscillatory modes,
corresponding to the vibration of the string, momentum modes, corresponding to
the center of mass motion, and winding modes, describing the winding of a string
around a compact direction. Each excited state contributes to the energy of a
string which is (for a derivation see Sec. 3.4)
-(f)
- oscillators. (10-1)
Here, R denotes the radius of a compact direction and plays the role of a dimen-
sionful scale factor. The square root of the constant a' is the fundamental string
length, and the integers n and u are the excitation numbers of the momentum
10.1. AVOIDING THE INITIAL SINGULARITY 207
modes and the winding modes. The latter is counting how many times a closed
string winds around a compact direction. Expression (10.1) is invariant under the
transformation
i?^^-, n^co, uj^n, (10.2)
i.e. under the exchange of large and small lengths, as well as of the simultaneous
exchange of momentum and winding modes. This symmetry is called T-duality,
and there are strong arguments that it holds not only in this particular example,
but that it is an intrinsic and fundamental symmetry of the whole string theory.
All physical processes should be equivalent, whether they are looked at with the
'i?-eye' or the 'l/it'-eye'. This is possible because the notion of distance is a
derived concept. It can be defined either as the Fourier transform of momentum
states
\x)= J2 eiX ' P \P) ' with \x) = \x + 2wR), (10.3)
v
or as the Fourier transform of winding states
\x') = ^ e ix ''" \u>) , with \x') = \x' + 2tt(x'/R). (10.4)
Note that in i?-space the winding modes are not localized. In the 1/R space,
however, they are localized, since they correspond to momentum states in R-
space. Neither of the two definitions (10.3) and (10.4) is more fundamental. For
example, there is no experiment which could decide whether we live in a universe
which is 10 10 or 10~ 10 lightyears wide.
Effectively, there exists a minimal length R = a' 1 / 2 , in the sense that the
range R < a' 1 / 2 can be equally well described using the i?'-picture where R' >
a' 1 / 2 . Ultimately, this is due to the extended nature of strings, and it is the key
to formulate a singularity-free theory. We shall see below how this affects the
behavior of the temperature as R — > 0.
The Friedmann equation in the standard 4-dimensional cosmology, a/a ~
1/a 4 or 1/a 3 , where a ex R, is not invariant under T-duality. This is a hint
that Einstein's equations break down at least at the self-dual radius a' 1 / 2 , and
that the standard cosmology is valid only for R ~^> a' 1 / 2 . A reason why the
usual Friedmann equations lacks T-duality symmetry is that usually space-time
is considered to be non compact, and therefore the winding modes are absent. To
restore T-duality symmetry in a cosmological equation, one would have to include
a graviton associated with the winding modes.
10.1.2 String thermodynamics
As one approaches the initial big bang singularity (going backwards in time) the
spatial sections shrink to zero size, and the temperature rises up to infinity. The
208 CHAPTER 10. THE COSMOLOGY OF STRING GASES
singularity theorems (see paragraph 1.3.3) state that this is unavoidable for mat-
ter satisfying the strong energy condition 1 (under the assumption that Einstein's
equations are valid) . Surprisingly, in string thermodynamics this temperature sin-
gularity does not show up. To see this, consider the canonical 2 partition function
for a single string
dEp(E)e- E/T , (10.5)
■r
where
p(E) = e wl/2£ (10.6)
is the number density (degeneracy) of string states of energy E (see Ref. [124]).
Note that, due to the exponential growth, the degeneracy becomes rapidly enor-
mous. The integral in Eq. (10.5) is finite provided that the temperature T is
smaller than
T " = tAu^ ( 10 - 7 )
47ra' 1 /2
which is called Hagedorn temperature 3 . This suggests that there exists a maxi-
mum attainable temperature: for T > T H , the partition function diverges, and
therefore the mean energy
E mean = -T 2 -^lnZ (10.8)
would be infinite. This is in contrast to the partition function for point particles.
How does the temperature of the string gas behave as a function of the (di-
mensionful) scale factor R as we go to smaller and smaller radii R? To answer this
question we make two assumptions: firstly, that the back-reaction of the string
gas on the space-time geometry can be neglected. This is justified if the string
coupling constant g s is much smaller than one. And secondly, that the evolu-
tion of the universe is adiabatic, i.e. the entropy remains constant. This second
assumption allows us to find T(R) without knowing the particular dynamics of
gravity. We can already guess the qualitative shape of the T(R) curve: at large
R, the T ~ R~ x behavior in the standard cosmology should be recovered. As one
goes to smaller radii, the curve will flatten out at a temperature close to T H , and
at even smaller radii the temperature drops again in order to satisfy the T-duality
symmetry
T(R) = T 0Q . (10.9)
x For a perfect fluid this is p + 3P > 0.
2 We remind the reader that a canonical ensemble is a very large number of copies of the original
system that all have the same temperature. This is realized for example by bringing them into
thermal contact with a thermostat of temperature T. However, the system is still exchanging heat
with the thermostat, and therefore its energy is fluctuating. The probability of finding a state with
energy E is encoded in the exponential factor in Eq. (10.5). Notice that T is an external parameter,
whereas E is an inner variable of the system.
3 The original work of Hagedorn on thermodynamics of strong interactions can be found under
Ref. [74]
10.2. WHY IS SPACE THREE-D L? 209
The maximum temperature is reached at the self-dual radius R = a 1 / 2 . A careful
computation shows that this corresponds not quite to the Hagedorn temperature,
but to T = T H — c/S 2 , where c is some constant, and S is the entropy of the string
gas. The precise shape of the curve has been calculated numerically in Ref. [29] 4 .
A physical interpretation is the following: at large R the winding modes are
irrelevant, and all energy is stored in momentum modes (see Eq. (10.1)). Since
these correspond to a center of mass motion, one expects the same behavior of
temperature as in a gas of pointlike particles, namely an increase in T as R
decreases. On the other hand, as R — > 0, it becomes energetically favorable
to excite winding modes instead of momentum modes, and an energy transfer
to the latter takes place. When R is very close to zero, the spacing between
winding states (given by R) becomes zero, such that they are in the ground state
corresponding to zero temperature. In conclusion, the temperature must be zero
both for R — > and R — > oo, with a maximum in between.
From the point of view of T-duality, the absence of a temperature singularity
can be understood by noticing that for an observer with the R'-eye, the collapse
R — > means an expansion R' — > oo, and so it is clear that the temperature has
to drop to zero. These arguments show that the initial singularity can be avoided
in a cosmological scenario with strings as a matter source.
The authors of Ref. [29] also pointed out that close to the Hagedorn tempera-
ture the energy fluctuations become large, and one therefore has to resort to the
microcanonical ensemble 5 . In particular, they showed that at sufficiently large
energies the specific heat becomes negative, unless all spatial directions are com-
pact. That is an important reason why in string gas cosmology, and later also in
brane gas cosmology, only compact spaces are considered.
10.2 Why is space three-dimensional?
We have seen that in string gas cosmology all nine spatial dimensions are compact
from the beginning. In brane world models instead, one starts with a non compact
space and then compactifies a number of spatial dimensions. A serious shortcom-
ing of this procedure is that the effective 4-dimensional physics will depend on
the geometry of the compact space. For example, we have seen that Newton's
constant depends on the volume of the extra-dimensions (see Eq. (2.68)). More
generally, each compactification involves a large number of so-called modulus
fields describing the shape and size of the compact space. In the simplest case of
toroidal compactification, such a modulus field is e.g. the radius R. Unfortunately,
it is to a large extent arbitrary on which particular space (a torus, a Calabi-Yau
manifold or others) the compactification must be carried out. Certain restric-
tions come from the requirement of obtaining the standard model gauge groups
4 Strictly speaking, all this applies only for type II superstrings. '
heterotic strings, but it can be argued that they lead to a similar curve T(R)
5 In the microcanonical ensemble all systems have an energy contained in a
which is by definition not fluctuating as in the canonical ensemble.
210 CHAPTER 10. THE COSMOLOGY OF STRING GASES
or a certain number of supersymmetries. Still, the parameter space is huge, and
therefore string theory looses virtually all of its predictive power. This is one of
the main problems today in connecting string theory to low energy physics. Note
that string theory itself has only one free parameter, namely a' .
Therefore, to avoid the moduli problem and to have a well defined thermody-
namical description, string gas cosmology postulates that all nine spatial dimen-
sions are compact from the start. The initial conditions are that all directions are
small and of equal (string scale) size 6 , and that the strings form a hot and dense
gas.
Given these 'natural' initial conditions one would like to explain why finally
three spatial dimensions have become much larger that the others. Such a dy-
namical 'decompactification' mechanism has been proposed in Ref. [29] for the
case that the compact manifold is a 9-torus. Let us assume that initially all nine
directions are of string scale size and start to expand isotropically. The winding-
modes wrapping around all 1-cycles would like to slow down the expansion since
their energy is proportional to the radius R of a cycle. However, their number
decreases as the size of the torus increases in order to minimize the energy. Hence
the expansion can continue. This is true provided that the winding modes are in
thermal equilibrium with other modes, string loops, and radiation. As soon as
they fall out of thermal equilibrium, a large number of winding modes remain,
and this will stop the expansion. The process that maintains thermal equilibrium
is of the form
uj + Q <-> unwound states, (10.10)
where ui stands for a string with opposite winding number, and unwound states
are, for instance, string loops that have split off or radiation. The annihila-
tion (10.10) between a winding and an anti-winding state happens by the inter-
commutation process shown in Fig. 10.1. Here we assume that, as the cycles grow
larger, the strings behave as classical objects with a thickness given by the string
length.
The crucial point is now that, in order for the process (10.10) to happen, the
two world-sheets of the strings have to intersect. In a 9-dimensional space, strings
will generically miss each other, so the winding modes fall out of equilibrium
and stop the expansion of the torus. In a 3-dimensional subspace, however, the
winding modes will generically meet, because their world-sheets fill all of space-
time: (1 + 1) + (1 + 1) = 3+1. Thus, in a 3-dimensional subspace, thermal
equilibrium can be maintained due to the process (10.10), and the expansion can
go on, while the other dimensions stay small. This is called the Brandenberger-
Vafa mechanism to dynamically explain, why we see three large spatial dimensions
today. Notice that, according to this scenario, the three large dimensions are still
compact. This is not in contradiction with observations as long as their radius is
much larger than the Hubble radius.
6 This is the third crucial assumption, together with the smallness of the string coupling conM ant .
and the adiabaticity of the evolution.
10.2. WHY IS SPACE THREE-DIMENSIONAL?
Figure 10.1: A toroidal topology can be represented as a cube, where opposite
faces are identified. The left panel shows two closed strings winding around
different compact directions. These strings will generically intersect each other if
they are propagating in 3-dimensional space. After the intersection, the strings
regroup as displayed in the right panel. Effectively, this process corresponds to
an unwinding from the compact direction.
These rather qualitative arguments are supported by numerical simulations [137]
and by the equations of motion of string cosmology which govern the evolution
of the background: in Ref. [153] it was demonstrated that the winding modes, if
not annihilated, indeed stop the expansion because of their negative pressure.
There is a number of points that remain to be clarified. First, if the ex-
pansion rate R/R of the torus if too slow, all processes will equilibrate, and in
particular the winding modes do not fall out of equilibrium anymore. One should
therefore quantitatively compare the time scales of the two processes. Second,
what happens if there is a slight asymmetry between the number of winding and
anti-winding modes? (Recall the asymmetry between matter and anti-matter in
the early universe, which is crucial for our existence.) Third, one should check
whether the classical treatment of strings is justified. Forth, it is not clear why
the number of large spatial dimensions is not less than three: after all, it is more
efficient to maintain thermal equilibrium in one ore two dimensions. Fifth, the
scenario does not take into account long distance interactions. According to the
classical dimension counting argument, point particles would always fail to meet
each other and to interact, unless they live on a line. And finally, in this discus-
sion we have not yet taken into account p-branes, which are important objects in
string theory. The role of p-branes within the BV mechanism has been studied
in Ref. [5], and the question of avoiding the initial singularity in a theory with
p-branes will be investigated in the article in the next chapter.
Despite these shortcomings, a dynamical decompactification mechanism seems
more 'natural' than starting from an non compact space-time and compactifying
by hand six dimensions. In particular, those scenarios do not explain why one
CHAPTER 10. THE COSMOLOGY OF STRING GASES
ends up with three large dimensions.
The dynamical decompactification process takes place at very early times and
high energies, and so experimental confirmation is lacking. Perhaps, it is at
least possible to verify whether we live in a compact universe or not by cosmic
crystallography, see e.g. Ref. [106].
10.3 Equation of state of a brane gas
We now calculate the equation of state of a gas consisting of p-branes in a D-
dimensional space-time. This section gives the detailed version of the calculation
presented in the article 'On T-duality in brane gas cosmology' in the next Chapter.
The equation of state is important for cosmological issues such as brane gas
inflation.
We start our calculation from the D-dimensional energy-momentum tensor
describing a single p-brane whose location is encoded in the embedding functions
X*(a),
T"»(x-
-/
r p / d'+W V " X K (a))V^ 9 ^X^X-
(10.11)
For simplicity, the ensemble of internal coordinates <t p is abbreviated as a. Here,
t p = T p /g 8 is the physical brane tension, g^ is the induced metric on the brane,
and g = det(g fll/ ). Since we are interested in the energy density and the pressure
of a gas of arbitrarily oriented p-branes, we do not fix a particular embedding and
leave the functions X K unspecified. Instead, we use the freedom to reparameterize
the p + 1 intrinsic world-sheet coordinates a^ to choose a gauge in which goo =
— \/—g, ffoi = 0, ■ • • , go P = 0. The induced metric is then
and its inverse is given by
5ii
V o
( -l/^/=g
g 11
\
gi P
(10.12)
(10.13)
V g pl ■■■ g-
Notice that the spatial pxp submatrices are also inverses to each other. Further-
more, we make a gauge choice for the time coordinate a
X°(a) = (j°.
(10.14)
10.3. EQUATION OF STATE OF A BRANE GAS
Then the energy density of a p-brane is
T 00 (x K ) = -t p f d p+1 aS {D \x K -X K (a))V^gg' LV X^X l/
= -T P f<r*-WHx*-x*w)y/=g (10 - 15)
x [ 9 00 X X° + s 11 !}!" + <? 12 X° X° + ■■■],
but because of the gauge choice (10.14), all terms with spatial derivatives are
zero. Inserting the expression for g 00 one obtains
T 0( V") = ~r P f dP +1 aS {D \x K - X K (a))^g(-l/V^9)
■' (10.16)
= +r p / d p+1 aS {D \x K - X K (a)).
The delta function is split according to
6 {D \x K - X K (a)) = S(t - a°)6 {d) (x n - X n (a°, a 1 )), (10.17)
where a^ = (a°,a l ),x K = (t,x n ), d = D — 1. Finally, one integrates out time to
obtain the energy density of a p-brane in d spatial dimensions:
p p = T°°{t,x n ) = +t p f d p aS {d) (x n - X n (£,CT*)).
(10.18)
Next, we calculate the (spatial) trace of the energy- momentum tensor, noting
that the background is Minkowski with signature ( — ,+,■■■ ,+),
T™ = Tl + ■ ■ ■ + Tf = T 11 + h T dd
= -t p / d p+1 a5 {D) (x K - X K (<j))y/=g~ (10.19)
x [<rx^X% + ■■■ + g^X%X d v ] .
Collecting the metric coefficients yields
T™{x K ) = -Tp f d p+1 aS {D \x K - X K (a))yf=g~
x [g™{Xlxl + --- + X d ) X d t )
+ g ll {X\x\ + ■■■ + X* X X$) + ■■■
+ g pp (X l pX]p+--- + X d p X d p)
+ 2g l2 (X\x] 2 + ■■■ + Xf t X%) + ■■■ ^ 10 - 2 °)
+ 2g lp (X\x]p + ■■■ + X%X d p )
+ 2g 2 \X%X% + ■■■ + X%X%) + ■■■
+ 2g lp {X) 2 X)p + ■■■ + X d 2 X d p ) + ■■■
+ 2gV- 1 *(XJ p _ l Xf p + ■■■ + X%_ X X%)] .
214 CHAPTER 10. THE COSMOLOGY OF STRING GASES
The idea is now to re-express the derivatives of the embedding functions in terms
of g M „. For instance
912 = ~X\X\ + X\x\ + ■■■ + X\x\
_ i i d d ( 10 - 21 )
where in the second line we have used the fact that the first term vanishes ac-
cording to gauge (10.14). With X m = (X 1 , ■ ■ ■ ,X d ) one finds
T™(x K ) = -t p / d p+ W D) (x A " - X K (a)) v / ^
x [g 00 X m X m + g n 9ll + ■■■+ gPPg pp
+ 2g i2 gi2 + ... + 2g i Pgip (10-22)
+ 2g 23 g 23 + ■■■+ 2g 2p g 2p + ■■■
+ 2g p - 1 < p g p _ ltP \ .
The dot denotes a derivative with respect to a . Next, one uses the fact that the
spatial submatrices g^ and gij are also inverses
1 = 9 ll gu + 9 12 §2i + ■■■ + g lp g P i
= g 11 gn + g 12 gi2 + — Vg lp 9i P ,
such that T™ can be simply written
T£(x K ) = -r p J <F +l a5^\x K - X K {a))^-g ^-l/^^)X m X m +p] .
(10.24)
To eliminate the remaining y/—g, one uses the gauge choice for goo :
-V=9 = 9oo = -X° Q X° + X] X] + ■■■ + X%X% = -1 + X m X m . (10.25)
Substituting this in Eq. (10.24) and integrating out time, the final result for the
(spatial) trace of the energy-momentum tensor is
T™(t, x n ) = +t p J d p aS {d) (x n - X n (t, a*)) Up + l)X m X m - p\ . (10.26)
The quantity X m X m is the squared velocity of a point on the brane parameterized
by (t, a 1 ). To take into account all points and all embeddings, one has to average
over the spatial coordinates a 1 . This leads to a 'brane gas averaged' velocity
v 2 (t) = (X m X m ) which depends on time only, and hence can be taken out of the
integral:
(T™(i,x))= [(p + l)v 2 - p] t p f d p a8 (d \x n -X n (t,a i )). (10.27)
10.3. EQUATION OF STATE OF A BRANE GAS
Comparing Eq. (10.27) with the expression for the energy density (10.18), one
finds the equation of state
where V p is the pressure of the p-brane gas. This pressure is within the range
n 1
(10.29)
where the lower bound is assumed in the non relativistic limit, v — > 0, and the
upper bound in the relativistic limit, v — > 1.
To illustrate this formula, take for example a domain wall in three spatial
dimensions (p = 2,d = 3). Then the equation of state (10.28) gives
V 2 = \v 2 - !] p 2 , (10.30)
which in the limit v -^ gives V2 = — 3P2 as expected, and for v — > 1 gives
V 2 = \P2-
It is amusing to note that the case p = d in the non relativistic limit leads to
an equation of state V = —p. Hence a cosmological constant can be interpreted
as a space-filling brane.
Chapter 11
On T-duality in brane gas
cosmology (article)
218 CHAPTER 11. ON T-DUALITY IN BRANE GAS COSMOLOGY
This chapter consists of the article 'On T-duality in brane gas cosmology',
published in JCAP 06 (2003), see Ref. [20].
It is also available under http://lanl.arXiv.org/abs/hep-th/0208188.
Like in the previous chapter, the pressure of a p-brane gas will be denoted by
V p , because the letter P is here used for the p-brane momentum.
11.1. INTRODUCTION
On T-duality in brane gas cosmology
Timon Boehm
Jniversite de Geneve, 24 quai E. Ansermet, CH-1211 Genev
4 Switzerland.
Robert Brandenberger
Institut d'Astrophysique de Paris, 98bis Blvd. Arago, F-75014 Paris, France,
Physics Department, Brown University, Providence, RI 02912, USA.
In the context of homogeneous and isotropic super-string cosmology, the T-
duality symmetry of string theory has been used to argue that for a background
space-time described by dilaton gravity with strings as matter sources, the cos-
mological evolution of the universe will be non singular. In this letter we discuss
how T-duality extends to brane gas cosmology, an approximation in which the
background space-time is again described by dilaton gravity with a gas of branes
as a matter source. We conclude that the arguments for non singular cosmolo-
gical evolution remain valid.
PACS number: 98.80.Cq, ll.25.-w
11.1 Introduction
In [29] it was suggested that due to a new string theory-specific symmetry called
T-duality, string theory has the potential to resolve the initial singularity problem
of standard big bang cosmology, a singularity which also plagues scalar field-driven
inflationary cosmology [25, 24].
The framework of [29] was based on an approximation in which the mathema-
tical background space-time is described by the equations of dilaton gravity (see
[153, 157]), with the matter source consisting of a gas of strings. The background
spatial sections were assumed to be toroidal such as to admit one-cycles. Thus,
the degrees of freedom of the string gas consist of winding modes in addition to
the momentum modes and the oscillatory modes. Then, both momentum and
winding numbers take on discrete values, and the energy spectrum of the theory
is invariant under inversion of the radii of the torus, i.e. R — > a'/R, where a' 1 / 2 is
the string length l s . The mass of a state with momentum and winding numbers
n and uj, respectively, in a compact space of radius R, is the same as that of a
state with momentum and winding numbers to and n, respectively, in the space of
radius a'/R. This symmetry was used to argue [29] that as the radii of the torus
decrease to very low values, no physical singularities will occur. Firstly, under
220 CHAPTER 11. ON T-DUALITY IN BRANE GAS COSMOLOGY
the assumption of thermal equilibrium, the temperature of a string gas at radius
R will be equal to the temperature of the string gas at radius a' /R. Secondly,
any process computed for strings on a space with radius R, is identical to a dual
process computed for strings on a space with radius a' /R. Therefore there ex-
ists a 'minimal' radius a' 1 ' 2 in the sense that physics on length scales below this
radius can equally well be described by physics on length scales larger than that.
Since the work of [29] our knowledge of string theory has evolved in important
ways. In particular, it has been realized [126] that string theory must contain
degrees of freedom other than the perturbative string degrees of freedom used
in [29] . These new degrees of freedom are Dp-branes of various dimensionalities
(depending on which string theory one is considering). Since the T-duality sym-
metry was used in an essential way (see e.g. [125]) to arrive at the existence of
Dp-branes (p-branes for short in the following), it is clear that T-duality symme-
try will extend to a cosmological scenario including p-branes. However, since a
T-duality transformation changes the dimensionality of branes, it is useful to ex-
plicitly verify that the arguments of [29] for a non singular cosmological evolution
carry over when the gas of perturbative string modes is generalized to a gas of
branes. A model for superstring cosmology in which the background space-time
is described (as in [153, 157]) by dilaton gravity, and the matter source is a gas
of branes, has recently been studied under the name of "brane gas cosmology"
[5, 27] (see also [57, 58] for extensions to backgrounds which are not toroidal, and
[161] for an extension to an anisotropic background).
In this letter, we establish the explicit action of T-duality in the context of
brane gas cosmology on a toroidal background. For a solution of the background
geometry appropriate for cosmological considerations in which the radii of the
torus are decreasing from large to small values as we go back in time, we must
consider T-dualizing in all spatial dimensions. We demonstrate that the mass
spectrum of branes remains invariant under this action. Thus, if the background
dynamics are adiabatic, then the temperature of the brane gas will be invariant
under the change R — > a'/R, i.e.
<$
t{r) = t [r)> (1L1)
thus demonstrating that superstring cosmology can avoid the temperature sin-
gularity problem of standard and inflationary cosmology. In the appendix we
make further remarks on why we expect brane gas cosmology to be non singu-
lar. Another crucial assumption is that the string coupling constant g s is small
(compared to one) such that back-reactions of the string and brane gas on the
curvature of space-time can be ignored. Similarly, our results can be used to show
that the arguments for the existence of a minimal physical length given in [29]
extend to brane gas cosmology.
The outline of this letter is as follows. In the following section we give a
brief review of brane gas cosmology. In section 11.3 we derive the energy, the
momentum, and the pressure for p-branes. Next, we review the action of T-duality
11.2. REVIEW OF BRANE GAS COSMOLOGY 221
on winding states of p-branes. The main section of this letter is section 11.5 in
which we show that the mass spectrum of a p-brane gas of superstring theory is
invariant under T-duality. Section 11.6 contains a discussion of some implications
of the result, and conclusions.
11.2 Review of brane gas cosmology
As already mentioned in the introduction, the framework of brane gas cosmology
consists of a homogeneous and isotropic background of dilaton gravity coupled to
a gas of p-branes as a matter source. We are living in the bulk 1 .
The initial conditions in the early universe are 'conservative' and 'democratic';
conservative in the sense that they are close to the initial conditions assumed to
hold in standard big bang cosmology (i.e. a hot dense gas of matter), democratic
in the sense that all 9 spatial dimensions of critical string theory are considered
on an equal basis 2 . Thus, matter is taken to be a gas of p-branes of all allowed
values of p in thermal equilibrium. In particular, all modes of the branes are
excited, including the winding modes.
The background space-time is taken to be 1 x T 9 where T 9 denotes a nine-
torus. The key feature of T 9 which is used in the analysis is the fact that it admits
one-cycles which makes it possible for closed strings to have conserved winding-
numbers 3 . It is also assumed that the initial radius in each toroidal direction is
the same, and comparable to the self-dual radius a' 1 / 2 . Initially, all directions
are expanding isotropically with R > a' 1 / 2 (the extension to anisotropic initial
conditions has recently been considered in [161]).
As shown in this letter, as a consequence of T-duality symmetry, brane gas cos-
mology provides a background evolution without cosmological singularities. The
scenario also provides a possible dynamical explanation for why only three spatial
dimensions can become large. Winding modes (and thus T-duality) play a cru-
cial role in the argument. Let us first focus on the winding modes of fundamental
strings [29].
The winding and anti-winding modes u and u) of the strings are initially in
thermal equilibrium with the other states in the string gas. Thermal equilibrium
1 In this sense, brane gas cosmology is completely different in ideology than brane world scenarios
in which it is assumed (in general without any dynamical explanation) that we live on a specific
brane embedded in a warped bulk space-time. From the point of view of heterotic M theory [77],
our considerations should be viewed as applying to the 10-dimensional orbifold space-time on which
2 For consistency, critical superstring theories need a 10 dimensional target space-time which is
in apparent contradiction with the observed four. Usually it is assumed that six dimensions are
compactified from the outset due to some unknown physics. However, following the usual approach
in cosmology it seems more natural that initially all nine spatial dimensions were compact and small,
and that three of them have grown large by a dynamical decompactification process. A corresponding
scenario was originally proposed in [29] .
3 Recently, the scenario of [29, 5] was generalized [57, 58] to spatial backgrounds such as Calabi-Yau
manifolds which admit 2-cycles but no 1-cycles.
222 CHAPTER 11. ON T-DUALITY IN BRANE GAS COSMOLOGY
is maintained by the process
uj + uj <=^ loops, radiation. (H-2)
When strings cross each other, they can intercommute such that a winding and
an anti-winding mode annihilate, producing fundamental string loops or radia-
tion without winding number. This process is analogous to infinite cosmic strings
intersecting and producing cosmic string loops and radiation during their inter-
action (see e.g. [158, 28] for reviews of cosmic string dynamics).
As the spatial sections continue to expand, matter degrees of freedom will
gradually fall out of equilibrium. In the context of string gas cosmology with
R > a' 1 / 2 , the winding strings are the heaviest objects and will hence fall out of
equilibrium first. Since the energy of a winding mode is proportional to R, New-
tonian intuition would imply that the presence of winding modes would prevent
further expansion. This is contrary to what would be obtained by using the Ein-
stein equations. However, the equations of dilaton gravity yield a similar result to
what is obtained from Newtonian intuition [153]: the presence of winding modes
(with negative pressure) acts as a confining potential for the scale factor.
As long as winding modes are in thermal equilibrium, the total energy can
be minimized by transferring it to momentum or oscillatory modes (of the fun-
damental string). Thereby, the number of winding modes decreases, and the
expansion can go on. However, if the winding modes fall out of equilibrium, such
that there is a large number of them left, the expansion is slowed down and even-
tually stopped. If we now try to make d of the original 9 spatial dimensions much
larger than the string scale, then an obstruction is encountered if d > 3: in this
case the probability for crossing and therefore for equilibrating according to the
process (11.2) is zero. On the other hand, in a three-dimensional subspace of the
nine-torus, two strings will generically meet. Therefore, the winding modes can
annihilate, thermal equilibrium can be maintained, and, since the decay modes of
the winding strings have positive pressure, the expansion can go on 4 . As a result,
there is no topological obstruction to three dimensions of the torus growing large
while the other six are staying small of size R ~ a' 1 / 2 . Large compact dimensions
are not in contradiction with observations if their radius is bigger than the Hubble
radius today.
It is not hard to include p-branes into the above scenario [5]. Now the initial
state is a hot, dense gas of all branes allowed in a particular theory. In particular,
brane winding modes are excited, in addition to modes corresponding to fluctua-
tions of the brane. Since the winding modes play the most important role in the
dynamical decompactification mechanism of [29, 5], we will focus our attention
on these modes. The analogous classical counting argument as given above for
strings yields the result that p-brane winding modes can interact in at most 2p+ 1
4 Since quantum mechanically, the thickness of the strings is given by the string length [89], it
is important for the brane gas scenario that the initial size of the spatial sections was string scale.
Otherwise, it would always be the total dimensionality of space which would be relevant in the
classical counting argument of [29], and there could be no expansion in any direction.
11.3. ENERGY, MOMENTUM, AND PRESSURE OF P-BRANES 223
spatial dimensions. Since for weak string coupling and for spatial sizes larger than
the self-dual radius the mass of a p-brane (with a fixed winding number in all of
its p spatial dimensions) increases as p increases, p-branes will fall out of equilib-
rium earlier the larger p is. Thus, for instance in a scenario with 2-branes, these
will fall out of equilibrium before the fundamental strings and allow five spatial
dimensions to start to grow [5]. Within these five spatial dimensions, the funda-
mental string winding modes will then allow only a three-dimensional subspace to
become macroscopic. Thus there is no topological obstruction to the dynamical
generation of a hierarchy of internal dimensions.
11.3 Energy, momentum, and pressure of p-branes
This section is devoted to the derivation of physical quantities describing the brane
gas which determine the cosmological evolution of the background space-time.
Starting from the Nambu-Goto action, we obtain expressions for the energy
and the momentum of a p-brane in D = d + 1 dimensional space-time. We show
that there is no momentum flowing along the p tangential directions. From the
energy-momentum tensor one can also define a pressure, and hence an equation
of state, for the whole brane gas. Even though some of the results in this section
are already known, we find it useful to give a self-consistent overview.
Let <t = {a°,a l ),i = !,••• ,p, denote intrinsic coordinates on the world-
sheet of a p-brane. Its position (or embedding) in £)-dimensional space time
is described by x M = X M (a), where M = 0, ■ ■ ■ ,d, and the induced metric is
9 iiv = VmnX^X^,, where /x, v = 0, ■ ■ ■ ,p.
In the string frame, the Nambu-Goto action is
S P = -T p /"d p+ Ve-*V=^, (11.3)
where T p denotes the tension of a p-brane, g = det(g fJi ^), and <fr is the dilaton of
the compactified theory. For our adiabatic considerations, we assume that it is
constant (taking its asymptotic value), and absorb it into a physical tension
T =e -*T = 5. = — (1141
p p g s (2it)p ^a'CP+D/ 2 ' { '
where in the final step we have used the explicit expression for T p (see e.g. [125]).
Note that for any p-brane t p goes like l/g s , and that hence, in the weak string
coupling regime which we are considering, the branes are heavy. The action (11.3)
can be written as an integral over D-dimensional space-time
S p = fd D x (-t p fd p+1 aS^\x K -X K {a))^g\ . (11.5)
As the integration domain is a torus, both integrals are finite.
224 CHAPTER 11. ON T-DUALITY IN BRANE GAS COSMOLOGY
Varying the action (11.5) with respect to the background metric and comparing
with the usual definition of the space-time energy-momentum tensor, one obtains 5
T MN (x K ) = -t p J d p+1 aS {D] (x K -rfff))^?"^,"' C 11 - 6 )
The Nambu-Goto action (11.3) is invariant under p + 1 reparametrizations
a — > a(cr), and we can use this freedom to choose
9oo = -v /3 5, 9o t = 0. (11.7)
Notice that for the spatial sub-matrix with indices i,j,k = 1, ■ ■ ■ ,p, det(gij) =
— \f—g and g lk gkj = Sj. By choosing the gauge (11.7), we do not specify a
particular embedding. This will be convenient later when treating a brane gas,
where the branes have arbitrary orientations. Furthermore, it is consistent to set
X° = a .
To calculate the energy E p of a p-brane, one observes that in the gauge (11.7)
=» T°°(x k ) = t p fd p+1 a6^ D \x K -X K (a)). (11.8)
Writing x K = (t, x n ),n — 1, • • • , d, and splitting the delta-function into a product,
the integral over a can be carried out. The energy density of a p-brane in d spatial
dimensions is then
p p =T 00 (t,x n )=T p I d p aS {d \x n -X n (t,a 1 )), (11.9)
and its total energy is
E p = I d d x Pp = t p f d p a = t p Vol p . (11.10)
The volume of a p-brane in its rest frame, Vol p , is finite as the brane is wrapped
around a torus. Eq. (11.10) provides a formula for the lowest mass state, M p = E p ,
which will be used in section 11.5. As expected intuitively, the minimal mass is
equal to the tension times the volume of a brane.
To calculate the space-time momentum P™ of a p-brane, one first evaluates
T 0n( x K-) = Tp j A p+1 a5 (D \x K -X K {a))X 1 } ) . (11.11)
ir gets multiplied with
11.3. ENERGY, MOMENTUM, AND PRESSURE OF P-BRANES 225
Proceeding similarly as before, the total momentum of a p-brane is found to be
Pp=T P ld p aX n (t,a i ), (11.12)
where the dot denotes the derivative with respect to time t. The gauge condi-
tions (11-7) can be written as = goi = X m X m ^, where the sum over m =
1, ■ ■ ■ ,d is the ordinary Euclidean scalar product. This is equivalent to saying
that the (spatial) velocity vector X is perpendicular onto each of the tangential
vectors X™. Therefore, only the transverse momentum is observable 6 . Assuming
that the brane is a pointlike classical object with respect to the transverse direc-
tions, this momentum is not quantized despite of the compactness of space. In
particular, the question whether there might exist a T-duality correspondence be-
tween transverse momentum modes and winding modes does not arise. Moreover,
we neglect the possibility of open strings travelling on the brane which would in
fact lead to a non zero tangential momentum 7 . Hence, in what follows, we focus
on the zero modes of p-branes.
Finally, the pressure V p of a p-brane is given by averaging over the trace T™.
First, notice that
= g 00 X™X mfi + g ll g lx + ■■■+ g™g pp
+2g l2 g 21 + ■ ■ ■ + 2g lp g pl
+2g 23 g 32 + ■■■ + 2g 2p g p2 + ■■■
+2g v ~ l,p g p , p -i
= --Lx™X m , +p. (11.13)
V~9
In the first step we have used the fact that the products of the embedding functions
can be expressed in terms of the induced metric, for instance gu = X™X m _\, and
in the second step that g %k gkj = $j- Inserting this into Eq. (11.6), eliminating the
remaining yf—g by using —y/—g = goo = — 1 + X r ^X rrit0 , an d integrating out the
a dependence, one finds
T%(t, x n ) =t p J <Fa5 {d \x n - X n (t, <r*))[(p + l)X m X m - p) . (11.14)
The quantity X m X m is the squared velocity of a point on a brane parameterized
by (t 7 a l ). We define the mean squared velocity of the branes in the gas by
averaging over all a 1 , i.e. v 2 (t) = ( X m X m ). In the averaged trace, (T 1 ™), the
velocity term can be taken out of the integral. Comparing with Eq. (11.9), one
6 Note the analogy with topological defects in field theory, where also only the transverse momen-
tum of the defects - here taken to be straight - is observable.
7 A relation between brane winding modes and open string momentum modes, quantized as
n m /R m , was shown in [140].
226 CHAPTER 11. ON T-DUALITY IN BRANE GAS COSMOLOGY
obtains the equation of state of a p-brane gas
In the relativistic limit, v — > 1, the branes behave like ordinary relativistic par-
ticles: V p = \p p , whereas in the non relativistic limit, v — > 0, V p = —\p v - For
domain walls this result was obtained in [165].
The pressure V p and the energy density p p are the source terms in the Einstein
equations for the brane gas [5, 153].
11.4 Winding states and T-duality
We briefly review some of the properties of T-duality that are needed subse-
quently.
Consider a nine-torus T 9 with radii (Ri, ■ ■ ■ ,Rq). Under a T-duality trans-
formation in n-direction
R n -»• R' n = ^- , (11.16)
and all other radii stay invariant. T-duality also acts on the dilaton (which is
constant in our case), and hence on the string coupling constant, as
/ «' 1/2
g s ^g' s = ^-gs. (11.17)
Note, however, that the fundamental string length l s = a' 1 / 2 is an invariant. The
transformation law (11.17) follows from the requirement that the gravitational
constant in the effective theory remains invariant under T-duality. In general, T-
duality changes also the background geometry. However, a Minkowski background
(as we are using here) is invariant.
For a p-brane on T 9 , a particular winding state is described by a vector uj =
(u>i, • • • , ujg). There are 9!/[p!(9 — p)\] such vectors corresponding to all possible
winding configurations. For illustration take a 2-brane on a three-torus: it can
wrap around the (12), (13), (23) directions, and hence there are 3!/2! = 3 vectors
u = (vi,u; 2 ,Q),u; = (u)i,0,u; 3 ),u) = (0,w 2 ,w 3 ).
Whereas T-duality preserves the nature of a fundamental string, it turns a p-
brane into a different object. To see this consider a brane with p single windings
to = (1, ■ ■ ■ , 1, 0, ■ ■ ■ ,0) which represents a p-dimensional hypersurface on which
open strings end. Along the brane the open string ends are subject to Neumann
boundary conditions. These become Dirichlet boundary conditions on the T-dual
coordinate R' n (if n denotes a tangential direction), i.e. for each string endpoint
R' n is fixed. Thus a T-duality in a tangential direction turns a p-brane into a
(p-l)-brane. Similarly, a T-duality in an orthogonal direction turns it into a
(p+l)-brane (see e.g. [125] for more details).
11.5. MASS SPECTRA AND T-DUALITY 227
Next, consider a T-duality transformation in a direction in which the p-brane
has multiple windings ui n > 1. One obtains a number w„ of (p-l)-branes which are
equally spaced along this direction. As an example take a 1-brane with winding
lo\ = 2 on a circle with radius R\. This configuration is equivalent to a 1-brane
with single winding on a circle with radius 2R\. T-dualizing in 1-direction gives a
single 0-brane on a circle of radius a' /2R\ which is equivalent to two 0-branes on
a circle of radius a' / R\ (see e.g. [75]). Since applying a T-duality transformation
twice in the same direction yields the original state (up to a sign in the Ramond-
Ramond fields), also the inverse is true: a number ui n of (p-l)-branes correspond
to a single p-brane with winding uj n .
So far we have discussed T-duality transformations in a single direction. For
applications to isotropic brane gas cosmology we need to consider T-dualizing in
all nine spatial directions. Given a gas of branes, £>, on a nine-torus with radii
(Ri, • • • , Rg) consisting of a large number of branes of all types admitted by a
particular string theory, we want to find the corresponding gas B* on the dual
torus T* with radii (R[, • • ■ , R' 9 ). To that end one performs a T-duality transfor-
mation in each of the nine spatial directions. From what we have discussed so far,
it is now easy to see that a p-brane in a winding state uj = (ui\, • ■ • , oj p , 0, • ■ • ,0)
is mapped into a number ui\ ■ ■ ■ ui p of (9-p)-branes, each of which is in a state
uj* = (0, • • ■ , 0, 1, . . . , 1). The (9-p)-brane wraps in the (9-p) directions orthogo-
nal to the original p-brane. It is clear that the above considerations hold for any
winding configuration.
After these preparatory steps, we now turn to the main part of this letter.
11.5 Mass spectra and T-duality
11.5.1 Masses of p-branes with single winding
In this section we show that each mass state in a brane gas B has a corresponding
state with equal mass in the brane gas B* . Based on type IIA superstring theory
we take B to consist of 0, 2, 4, 6 and 8-branes. Then, by the discussion in the pre-
ceding section, the brane gas B* contains 9, 7, 5, 3, 1-branes which are the states
of type IIB as we have carried out an odd number (nine) of T-duality transfor-
mations. Notice that this follows from the T-duality symmetry for fundamental
strings, not from T-duality arguments applied to the above brane gases which
we actually want to show. Our demonstration is done by carrying out explicitly
nine T-duality transformations on a mass state in B, and showing that there is a
corresponding and equal mass state in B* . In this sense the two brane gases are
T-dual.
Suppose that the branes in B are wrapped around some of the cycles of a nine-
torus with radii (R\, ■ ■ ■ , Rg). Then, the volume Vol p of a p-brane in Eq. (11.10)
is simply the product of the p circumferences, and the minimal masses M p = E p
CHAPTER 11. ON T-DUALITY IN BRANE GAS COSMOLOGY
(in the string frame) are
M o = ^T72> ( 1L18 )
g s a n/z
M 2 = (2tt) 2 R 9 R 8 t 2 = -^^ , (11.19)
(11-20)
g 8 a D ' z
M 6 = (2TrfR 9 ---R i r e = Rg ' "^ , (11.21)
g s a ' l z
M 8 = (27rfR 9 ---R 2 r s = R9 "; R 2 \ (11.22)
g s a y / z
(see also [125]) where in the second step we have used expression (11.4) for the
tension of a p-brane. For notational convenience we have fixed a particular wind-
ing configuration. The argument is generalized for arbitrary winding configura-
tions and winding numbers at the end of this section. If, as we have assumed,
R n > a' 1 / 2 , then the heaviest object in the theory is the 8-brane.
The Nambu-Goto action (11.3) is invariant under T-duality. Hence, all for-
mulae derived from it (energy, mass and pressure) are valid in both the original
brane gas B and in the dual brane gas B* . Thus, the mass spectrum of the B*
brane gas is
M; = (2nfR' 9 .--R' 1 r; = ^ 7J §, (11.23)
M 7 * = (2ir) 7 R' 7 ■ ■ ■ R' x t* 7 = R ' 7 '" R ' 1 , (11.24)
g*a a/z
M* = (2ir) 5 R' ■ ■ ■ RWi = Rb ' \' Rl , (11.25)
g * a 6/2
M* = (2^) 3 i«^3* = 4^| , (11-26)
g*a 4 / 2
Mi* = 2-kR'tI = — . (11.27)
Since R n < a' 1 / 2 , the heaviest brane of the dual gas B* is now the 1-brane. The
coupling constant in B* is given by
a' 9 / 2
9*s = Rg ... Ri 9s- (H-28)
Note that if the radii (R\, ■ ■ ■ ,R 9 ) of the initial nine-torus are bigger than the
self-dual radius a' 1 / 2 , then g* < g s , and thus the assumption of a small string
coupling constant is safe.
Given the two mass spectra, one can easily verify that each state in the brane
gas B has a corresponding state with equal mass in the dual brane gas B*:
M;_ p = M p . (11.29)
11.5. MASS SPECTRA AND T-DUALITY 229
This establishes explicitly that the T-duality of the string gas used in [29] extends
to the brane gas cosmology of [5].
As an explicit example, consider a 2-brane wrapped around the 8 and 9 direc-
tions. Its mass is (11.19)
M 2 = — ?-^ . (11.30)
g s a d ' 2
If we replace the string coupling constant g s by the dual string coupling constant
g* via (11.28), and the radii R% and Rg by the dual radii R 8 and R g via (11.16),
one obtains
Af 2 = R \", g R J = m; . (n.31)
g*a *l z
In the above example, we have specified a particular winding configuration for
simplicity, but clearly the argument holds as well in the general case, where a
p-brane wraps around some directions n\ ■ ■ -n p :
M p = {2,YR ni ---R np T p =^^. (11.32)
Via the same steps as in the above example, it follows that
R'rrn ~'Kn„
M* = (2TT) 9 - p R' mi ---R' J
•m 9 _ p '9-p ~
= M p , (11.33)
where {mi, • • • ,mg_ p } ^ {m, ■ ■ ■ ,n p }.
11.5.2 Multiple windings
Consider now a p-brane with multiple windings 10 = (wi, ■ ■ • ,uj p ,0, • ■ • ,0). Its
mass is
M p (u)=u 1 --'U P M P . (11.34)
In the B* brane gas this corresponds to a number wi ■ ■ ■ u> p of (9-p)-branes each
with winding uj* = (0, ■ ■ ■ , 0, 1, ■ ■ ■ ,1) and mass Mg_ . Since Mg_ p = M p , the
total mass of this 'multi-brane' state is equal to the mass of the original brane,
namely
(wi • • • uj p )M;_ v = M p {u) , (11.35)
which establishes the correspondence of B and B* in the case of multiple windings.
One should also add fundamental strings to the brane gas B. Since their mass
squared is
*={£)' +{¥)'' <"- 36 >
is its clear that every fundamental string state in B has a corresponding state in
B* when n n ^ u n .
230 CHAPTER 11. ON T-DUALITY IN BRANE GAS COSMOLOGY
11.6 Cosmological implications and discussion
We have demonstrated explicitly how T-duality acts on a brane gas in a toroidal
cosmological background, and have in particular shown that the mass spectrum of
the theory is invariant under T-duality. Thus, the arguments of [29] which led to
the conclusion that cosmological singularities can be avoided in string cosmology
extend to brane gas cosmology.
Whereas T-duality does not change the nature of fundamental strings, but
simply interchanges winding and momentum numbers, it changes the nature of
branes: after T-dualizing in all d spatial dimensions, a p-brane becomes a (d-p)-
brane which, however, was shown to have the same mass as the 'original' brane.
In [5] it was shown that the dynamical decompactification mechanism pro-
posed in [29] remains valid if, in addition to fundamental strings, the degrees
of freedom of type IIA superstring theory are enclosed. We briefly comment on
the decompactification mechanism in the presence of a type IIB brane gas on a
nine-torus. As before, we assume a hot, dense initial state where, in particular,
the brane winding modes are excited and in thermal equilibrium with the other
degrees of freedom. All directions of the torus are roughly of string scale size, / s ,
and start to expand isotropically For the 9,7, and 5-brane winding modes there
is no dimensional obstruction to continuously meet and to remain in equilibrium,
thereby transferring their energy to less costly momentum or oscillatory modes:
these degrees of freedom do not constrain the number of expanding dimensions.
However, the 3-branes allow only seven dimensions to grow further, and out of
these, three dimensions can become large when 1-brane and string winding modes
have disappeared. As far as the 'intercommutation' and equilibration process is
concerned, the 1-branes and the strings play the same role, but since the winding
modes of the former are heavier (—^7 2> ^7 at weak coupling), they disappear
We have focused our attention on how T-duality acts on brane winding modes.
However, since in a hot and dense initial state we expect all degrees of freedom of
a brane to be excited, we should also include transverse fluctuations (oscillatory
modes) in our considerations concerning T-duality. To our knowledge, the quan-
tization of such modes is, however, not yet understood, and we leave this point
for future studies.
Another interesting issue is to investigate how the present picture of brane
gas cosmology gets modified when gauge fields on the branes are included. These
correspond to U{N) Chan-Paton factors at the open string ends. In this case, a
T-duality in a transverse direction yields a number N of parallel (p-l)-branes at
different positions [124].
11.7. APPENDIX 231
Acknowledgments
We would like to thank S. Alexander, L. Alvarez-Gaume, J. Fernando-Barbon, S.
Foffa, S. Lelli, J. Mourad, R. Myers, Y. Oz, F. Quevedo, A. Rissone, M. Rozali
and M. Vasquez-Mozo for useful discussions. R.B. wishes to thank the CERN
Theory Division and the Institut d'Astrophysique de Paris for their hospitality
and support during the time the work on this project was done. He also ac-
knowledges partial support from the US Department of Energy under Contract
DE-FG02-91ER40688, TASK A.
11.7 Appendix
In this appendix we would like to give some further arguments for why we be-
lieve brane gas cosmology to be non singular. First, let us recall the results of
[29]. In the case of a string gas, where the strings are freely propagating on a
9-torus, it was shown that the cosmological evolution is free of singularities. The
background space-time has to be compact, otherwise the thermodynamical de-
scription of strings is not sound, in particular the specific heat becomes negative
at large energies. An important assumption in the derivation was that the evo-
lution of the universe is adiabatic, i.e. the entropy of the string gas is constant.
Making use of this assumption, one can find the temperature as a function of
the scale factor, T(R), without referring to the dynamics of gravity or Einstein's
equations. Using a microcanonical approach, it was shown that there exists a
maximum temperature, called Hagedorn temperature T H , and hence there is no
temperature singularity in string gas cosmology. The curve T(R) is invariant un-
der a T-duality transformation, T(R) = T(a'/R). Another crucial assumption
is that the string coupling constant, g s , is small enough such that the thermody-
namical computations for free strings are applicable, and that the back-reaction
of the string condensate on the background geometry can be neglected. In lack of
knowledge about brane thermodynamics we simply postulate that the statements
above extend to brane gases.
We conclude by making a comment on our work in the light of the well-known
singularity theorems in general relativity. These theorems make assumptions
about the geometry of space-time such as R^v^v 1 " > for all timelike vectors v 1 *
(for a textbook treatment see e.g. [160]). By Einstein's equations this is equiva-
lent to the strong energy condition for matter. However, we do not trust Einstein's
equations in the very early universe as they receive corrections which are higher
order in a' as well as g s , and also they lack invariance under T-duality transfor-
mations R — > a'/R. Therefore we cannot invoke the energy momentum tensor
given in Eqs. (11. 9), (11. 11), (11.14) to decide whether the universe described by
our model is geodesically complete or not.
Conclusions
234 CONCLUSIONS
In this thesis, we have investigated different topics at the interface between
string theory and cosmology. In the first part, we identified our universe with a 3-
brane embedded in a 5-dimensional bulk. Starting with a rather simple approach
to perturbations on moving branes, we have considered a dynamical generalization
of the Randall-Sundrum model, and finally estimated the CMB power spectrum
for anti-de Sitter brane worlds. In all of these cases, dynamical instabilities were
encountered, which suggests that existing attempts to realize cosmology on branes
are at least questionable.
In the second part, we considered an alternative model, in which we do not live
on a particular brane. Instead, the role of strings and branes is only to 'regulate'
the background dynamics. Though this approach is less well developed, it seems
to us a more promising avenue to bring together string theory and cosmology. In
particular, progress has been made toward the resolution of the initial singularity
problem.
Common to both approaches is that a higher-dimensional view allows to dis-
cover new relations between seemingly unconnected phenomena and to under-
stand them from a unified perspective. Below, we summarize the conclusions of
each article.
Perturbations on a moving D3-brane and mirage cosmology
We have studied the evolution of perturbations on a moving 3-brane coupled
to a 4-form field in an AdSs-Schwarzschild bulk. The unperturbed dynamics lead
to homogeneous and isotropic expansion on the brane, where the scale factor ci(t)
is related to the position of the brane via a oc r. The fluctuations we consider
are due only to this motion, and can be described in terms of a scalar field (f>.
For a BPS brane 8 , parameterized by a conserved energy E = 0, we find that
superhorizon modes evolve as
cp k = A k a 4 + B k a~ 3 ,
where the constants A k and B k are determined by the initial conditions. Hence,
if the brane is expanding, superhorizon modes grow as a 4 . When the brane is
contracting, superhorizon modes grow as a~ 3 . Of course, at some point our linear
analysis will break down. For subhorizon modes we find
Q ikr] Q-ikri
</>fc = A k + B k .
a a
For an expanding brane subhorizon modes are stable whereas for a contracting
brane they grow large. In particular, the brane becomes unstable as it falls into
the AdSs-Schwarzschild black- hole (r \ 0,a \ 0). We have also discussed the
cases E > 0, BPS anti-branes (i.e. with opposite Ramond-Ramond charge) as
well as non BPS branes.
8 A BPS Dp-brane is characterized in that its charge is equal to its tension, see Eq. 5.2. In the
low energy description, it corresponds to an extremal p-brane.
235
The perturbations <f> gives rise to scalar perturbations in the Friedmann-
Lemaitre (brane) universe. For a 4-dimensional observer living on the perturbed
brane, (j> is related to the gauge invariant Bardeen potentials in the following way:
^ = 3$ + 4g(^j ,
where E = E — qjfj, r H is the horizon, L is the curvature radius of the AdSs-
Schwarzschild geometry, and q is a parameter taking the value 1 for BPS branes,
and —1 for BPS anti-branes. Our approach is limited to the case in which the
back-reaction of the brane onto the bulk can be neglected.
We have also derived a Friedmann equation on the 3-brane and found terms in
1/a 8 , 1/a 4 as well as a cosmological constant if the brane is non BPS. Thus, pure
brane motion in an AdSs-Schwarzschild background cannot mimick the effect of
dust matter. It is not excluded though that in other backgrounds a term 1/a 3
can be obtained.
Dynamical instabilities of the Randall-Sundrum model
The Randall-Sundrum model relies on a fine-tuning between the brane tension
V and the (negative) cosmological constant A in the bulk. We have investigated
whether this fine-tuning could be achieved by a dynamical mechanism. As a
concrete example, we studied a scalar field on the brane, and we found that the
bulk cosmological constant cannot be cancelled by the potential energy of the
scalar field. More generally, this result also holds for any matter on the brane
with an equation of state w > — 1.
Furthermore, we have derived Einstein's equations for a 3-brane, which con-
tains fluid matter and a scalar field, and which is coupled to the dilaton and to
the gravitational field in a 5-dimensional bulk. They allow for a dynamical ge-
neralization of the RS model and ultimately to investigate its stability. To that
end, the fine-tuning 9 is perturbed by a constant fl,
V = v /r l2A(l + 0).
We have found analytic solutions of the 5-dimensional perturbation equations,
which for the scale factor on the brane yield
a 2 (r) ~ l + 2QT + 4a 2 0r 2 ,
showing that a runs away from the static RS solution as r 2 (where r is cosmic
time on the brane). In a cosmological context, the change in brane tension, Q,
may be due to a phase transition, for example.
236 CONCLUSIONS
More surprisingly, there exists also a mode Q, which represents an instability,
even if the fine-tuning is not perturbed at all. Within linear perturbation theory,
there is no constraint on Q, and it cannot simply be gauged away, because all
our statements are gauge invariant. This mode can be understood as the velocity
of the brane, thus indicating that it is not consistent to keep the brane fixed at
y = while perturbing the metric as in Eq. (7.56).
CMB anisotropics from vector perturbations in the bulk
We considered again the evolution of perturbations on a 3-brane moving in
an AdSs bulk 10 . This time the back-reaction was taken into account by imposing
Z 2 symmetry with respect to the brane position and by using Israel's junction
conditions.
We have found the most general analytic solutions for vector perturbations in
the bulk, which generically lead to growing vector perturbations on the brane.
This differs radically from the usual behavior in 4-dimensional cosmology, where
vector modes decay like a~ 2 whatever the initial conditions. We have identified
vector modes on the brane, which are normalizable and exponentially growing in
conformal time. Ultimately, the existence of these modes is due to Z 2 symmetry in
the bulk, and they are normalizable only because the bulk is finite (see Eq. (9.51)).
We comment on this issues as well as on the role of these modes in the RS model
in the appendix of our article.
Regarding the vector modes, the spatial frequency of the bulk perturbations,
H, plays the role of an effective mass, and SI 2 might be positive or negative 11 .
The latter case represents a bulk tachyonic mode, which is another way to see the
instability. In addition, there exist modes which grow as a power law, but they
still lead to significant effects in the CMB.
An estimate of the CMB anisotropics for the exponentially growing modes
stringently constrains their primordial amplitude:
A a (£l) < e~ 10 , for O/a ~ 10~ 26 mm _1 ,
A Q (Q) < e~ w ' , for n/a Q ~ lmm -1 ,
where a is the scale factor today. We have required a scale invariant spectrum
around £ ~ 10, and we have set the anti-de Sitter curvature radius L to 10~ 3 mm.
Vector modes are part of gravitational waves in the bulk, which are generated
for instance in bulk inflationary models. They are also likely to be produced
by second-order effects in the bulk. If such processes are not forbidden by some
physical mechanism, then anti-de Sitter brane worlds cannot reasonably lead to
a homogeneous and isotropic expanding universe.
10 For the calculation it turns out to be much simpler to consider a moving brane in a static
background, instead of fixing the brane in a dynamical bulk.
11 Because the Greek alphabet is finite, unfortunately Q happened to label the perturbation of the
RS fine-tuning as well as the spatial frequency of the bulk vector modes.
237
Since we have considered only vector perturbations here, we cannot compare
our results to those found in mirage cosmology. There, the back-reaction was
neglected, and so the only possible perturbations are those in the embedding, thus
a single scalar. Certainly, it would be interesting to solve the scalar perturbations
in the above framework, also to decide how important the inclusion of the back-
reaction is.
On T-duality in brane gas cosmology
In this alternative approach, we considered a gas of strings and p-branes in a
10-dimensional background space-time, where the spatial sections have the topo-
logy of a 9-torus with radii R n , n = 1, ■ ■ ■ ,9. The possible values of p are either
1, 3, 5, 7, 9 or 0, 2, 4, 6, 8 according to the type of string theory one is considering.
The principal aim of brane gas cosmology is to avoid the initial singularity. To
that end, we have derived a formula for the mass of a p-brane with single or
multiple windings around the cycles of the torus, as well as the equation of state
for the brane gas. Consider this brane gas on a 'dual' torus, which is obtained by
applying the transformation
onto all nine spatial dimensions. This so-called T-duality transformation turns a
p-brane with mass M p into a (9— p)-brane with mass Mg_ p . We have explicitly
shown that
M 9 % = M p .
Thus, there are the same degrees of freedom on the original and the dual torus.
For a gas of strings, invariance of the mass spectrum under a T-duality trans-
formation has been used to show that the dynamical evolution of the torus can
be described in a singularity- free way [29]. Given that this invariance holds also
for branes, we argued that the initial singularity is avoided in the framework of
brane gas cosmology. It remains to be shown that the temperature of the brane
gas on the dual torus is the same, T* = T.
We also addressed the dynamical decompactification mechanism proposed
in [29], now including p-branes. In a cosmological scenario, the initial state is
thought to be a hot and dense gas of strings and branes. All cycles of the torus
are roughly of string scale size and start to expand isotropically. We demonstrate
that the winding modes of strings and p-branes allow only a 3-dimensional sub-
space to become large. In this scenario, we do not live on a particular brane but
in the bulk. The extra-dimensions are invisible today because they remain of
string scale size.
Certainly, another attractive feature of brane gas cosmology is that the prob-
lems associated with compactification are avoided.
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