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UNIVERSITE¶ DE GENEVE FACULTE¶ DES SCIENCES D¶epartement de physique th¶eorique Professeur R. DURRER Cosmology of Brane Universes and Brane Gases THESE pr¶esent¶ee a la Facult¶e des sciences de l'Universit¶e de Geneve pour l'obtention du grade de Docteur es sciences, mention physique par Timon Georg BOEHM de B^ale (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 ¯eld 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 speci¯c 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 ood 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). i . ii Figure 1: iii . iv Remerciements Je tiens d'abord a remercier ma directrice de these, Ruth Durrer, pour m'avoir enseign¶e les sujets fascinants de la relativit¶e g¶en¶erale et cosmologie, mais ¶egalement pour son soutien et ses conseils, ainsi que sa compr¶ehension 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'Universit¶e de Geneve: Cyril Cartier, Stefano Fo®a, 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 th¶eorique a Paris-Orsay, ou j'ai pro¯t¶e de nombreuses discussions avec Emilien¶ Dudas et Jihad Mourad. Et ¯nalement 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 corrig¶e les parties anglaises et fran»caises de cette these. Examinateurs Le jury de cette these se compose de Prof. Pierre Bin¶etruy, Universit¶e Paris-XI, Orsay Cedex, France. ² Prof. Robert Brandenberger, Brown University, Providence, USA. ² Prof. Ruth Durrer, Universit¶e de Geneve, Suisse. ² Prof. Michele Maggiore, Universit¶e de Geneve, Suisse. ² Je tiens a les remercier d'avoir accept¶e de faire partie du jury de ma these. v . vi Meinen Eltern gewidmet vii . viii Contents Abstract i Remerciements v R¶esum¶e 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-Lema^³tre 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-Lema^³tre space-times . 25 1.3 The gravitational ¯eld equations . 25 1.3.1 Cosmological equations . 26 1.3.2 Energy `conservation' . 27 1.3.3 Past and future of a Friedmann-Lema^³tre 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 ix 2.4.1 Spaces with positive, negative, and zero Ricci curvature . 47 2.4.2 The gravitational ¯eld equations . 48 2.5 Toroidal compacti¯cation . 50 2.5.1 Kaluza-Klein states . 50 2.5.2 Dimensional reduction of the action . 52 2.5.3 Ho·rava-Witten compacti¯cation . 53 2.6 The modi¯cations 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 e®ective Planck mass . 57 2.7.2 The idea of `large' extra-dimensions . 58 3 Branes 61 3.1 Extended objects . 62 3.2 Di®erential geometrical preliminaries . 62 3.2.1 Embedding of branes . 63 3.2.2 The ¯rst 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 ¯eld . 95 x 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 ¿ . 104 5.3.3 Comments on an analysis in conformal time ´ . 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 ¯ve-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 anisotropies 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 uctuations in the CMB . 157 8.3.1 Induced vector perturbations on the brane . 157 8.3.2 Sources of CMB anisotropies . 158 8.3.3 Temperature uctuations as a function of the perturbation variables . 159 xi 8.3.4 The perturbed geodesic equation .
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