Topics in Stringy Cosmology Naureen Goheer Thesis Presented for the Degree of Doctor of Philosophy in the Department of Mathematics and Applied Mathematics University of Cape Town December 2004 i Declaration The work presented in this thesis is partly based on collaborations with Marco Bruni (Insti- tute of Cosmology and Gravitation, Portsmouth University, UK), Alan Coley (Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada) Peter K. S. Dunsby, (Department of Mathematics and Applied Mathematics, University of Cape Town), Matthew Kleban (Institute for Advanced Study, Princeton, USA) and Leonard Susskind (Korea Institute for Advanced Study and Department of Physics, Stanford Uni- versity, USA). The list below identifies sections or paragraphs which are partially based on the listed publications: • Chapter 3 and Appendix A: N. Goheer and P. Dunsby, Phys. Rev. D66 (2002) 043527. N. Goheer and P. Dunsby, Phys.Rev. D67 (2003) 103513. • Chapter 6 and Appendix D: N. Goheer, M. Kleban, L. Susskind, JHEP 0307 (2003) 056. • Chapter 7 and Appendix C: N. Goheer, M. Kleban, L. Susskind, Phys. Rev. Lett. 92 (2004) 191601. • Chapter 4 and Appendix B: P. K. S. Dunsby, N. Goheer, M. Bruni and A. Coley, Phys. Rev. D69 (2004) 101303. N. Goheer, P. K. S. Dunsby, M. Bruni and A. Coley, Phys. Rev. D70 (2004) 123517. I hereby declare that this thesis has not been submitted, either in the same or different form, to this or any other university for a degree and that it represents my own work. Naureen Goheer ii Acknowledgments Hmm, I guess I should thank Scott for being there, and for constantly feeding me exciting ideas of your 17-dimensional super-dooper-symmetric universe. I sure am looking forward to reading your paper and finally being able to grasp all the details and implications - you changed my view of the world! I want to thank Peter, thank you so much for all your support, for giving me all the freedom I wanted and introducing me to things I thought I never wanted to know about! I have really had a great time under your supervision, I would have never hoped that getting (or trying to get) a PhD would be so much fun! Lenny and Matt, thank you so much for introducing me to a whole other world of physics and physicists. I hope you guys know how lucky I consider myself for having had the opportunity to work with you. I can’t begin to tell you how much I appreciate how you offered patient advice and friendship over the past years. On this note I also thank the Korean Institute for Advanced Studies for great hospitality, offering a wonderful environment for research and a large variety of interesting borderline peculiar pastries. Alan and Marco, it was great fun to write a couple of papers with you, everyone based on a different continent. Thank you guys! Many thanks also to Dalhousie University for inviting us – I must admit I had a great month in a surprisingly beautiful Nova Scotia. Jeff, dude! Thanks for telling me fun little stories about non-commutative lumps when I was just going to check whether you wanted to join my quest for a glass of wine or a Miller later on. Talking about Miller’s, Gavin, thanks for being there! Tyrone, I am really not so sure whether I will get a chance to write that book on how to climb 5.15, so I think we should play it safe and just dedicate this thesis instead of the hypothetical book to you. So here it is: I cordially thank you for all the long hours (and I mean hours!) that you patiently belayed me and never ever even in the most epic redpoint attempts decided to cut the rope. Herewith I would love to dedicate this thesis to you. I also must thank all the climbers who kept me sane during the past few years of increasing knowledge and confusion. I won’t mention your names since I won’t make you read this, except maybe the Chinese Three, Deon, Clinton and Jeremy... I further want to thank the Friedrich Ebert Stiftung and later UCT for financial support. Ich danke der FES insbesondere fuer ihre Flexibilitaet und meine kontinuierte Foerderung trotz meines unkonventionellen Studienweges. Danke! Last but definitely not least: my parents. Mama und Papa, ich hoffe ihr wisst wie schoen es ist zu wissen, dass ihr, was auch immer komme, nur eine (sei es auch 14-stellige) Telefonnummer entfernt seid. Papa, thanks for picking up the phone even if I occasionally miscalculated the time difference! Danke fuer eure Unterstuetzung, oder besser dafuer, dass ihr niemals an meinen recht ungewohnlichen Entscheidungen gezweifelt habt. Nicht viele Toechter kenned sich so gluecklich schaetzen, dass ihre Eltern sie kritiklos nach Afrika auswandern lassen. 1000 Dank fuer euer Vertrauen in mich und meinen Lebenswandel! iv v Abstract This thesis is composed of two distinct research topics. The first topic concerns Randall- Sundrum cosmological models, in particular their isotropization properties at early times. We first study the state space of inflationary braneworld models with exponential poten- tial and identify the past and future attractors within the homogeneous state space. Our main result is that the isotropic high energy BDL model is the generic past attractor in the state space of FLRW braneworld models. In the state space of Bianchi I models both the BDL model and the inflationary anisotropic Kasner model are past attractors. We then address the question of isotropization using a different approach by studying the evolution of general inhomogeneous and anisotropic perturbations around the spatially flat FLRW braneworld background for three different energy regimes. Here we find that for the high energy braneworld background, the physically relevant anisotropic, inhomogeneous pertur- bations decay when approaching the initial singularity. We confirm the standard results for the low energy limit: as in General Relativity, the anisotropy here grows in the past. We conclude that the initial singularity appears to be generically homogeneous and isotropic in the braneworld context. The second focus of this thesis is on classical cosmological spacetimes with horizons and their relation to non-classical concepts such as Horizon Complementarity and Holography. We first assume the de Sitter space version of black hole complementarity, which states that a single causal patch of de Sitter space is described as an isolated finite temperature cavity bounded by a horizon which allows no loss of information. After discussing how the de Sitter symmetries should be implemented, we prove that the finiteness of the horizon entropy stands in contradiction with the symmetry group of de Sitter space. We conclude that we must either give up the finiteness of de Sitter entropy or the exact symmetry of the classical space. We discuss the implications; in particular we argue that the lifetime of a de Sitter phase can not exceed the Poincar´e recurrence time. We find that the same arguments hold for 1+1 dimensional Rindler space. This means that compactifications of String Theory to maximally symmetric 1+1 dimensional Minkowski space are in conflict with Holography and expected to be unstable. vi Contents I Introduction 1 1 Why Stringy Cosmology? 3 1.1TheStandardModelofCosmology....................... 3 1.2ThesearchforaQuantumTheoryofGravity................. 5 1.3Stringcosmology................................. 8 1.4 The Randall-Sundrum cosmological models . ................. 10 1.5CosmologicalSpacetimeswithHorizons.................... 14 1.5.1 HorizonComplementarity........................ 14 1.5.2 TheHolographicPrinciple........................ 15 II Randall-Sundrum cosmology 17 2 Randall-Sundrum basics 19 2.1Introduction.................................... 19 2.2Geometricformulation.............................. 21 2.3CovariantlydecomposedKinematics...................... 24 2.3.1 1+3 Covariant Propagation and Constraint equations on the brane . 27 2.4ScalarFieldDynamicsontheBrane...................... 31 2.4.1 Braneworldinflation........................... 33 3 Dynamics of inflationary models with exponential potential 37 3.1DynamicalSystems................................ 38 3.1.1 Linearsystems.............................. 40 3.1.2 Localanalysisofnon-linearsystems.................. 41 3.2Thegoverningequations............................. 42 3.2.1 FLRWmodels.............................. 43 3.2.2 BianchiImodels............................. 44 3.3 AnalysisofFLRWmodelswithexponentialpotentials............ 46 vii viii CONTENTS 3.3.1 Sector A: ρ∗ ≥ 0and3R ≤ 0 ...................... 46 3.3.2 Sector B: ρ∗ ≥ 0and3R ≥ 0 ...................... 51 3.3.3 Sector C: ρ∗ ≤ 0and3R ≤ 0 ...................... 54 3.3.4 Sector D: ρ∗ ≤ 0and3R ≥ 0 ...................... 57 3.3.5 Thestructureofthestatespace..................... 59 3.4BianchiImodelswithanexponentialpotentials................ 71 3.4.1 Sector E: ρ∗ ≥ 0 ............................. 71 3.4.2 Sector F: ρ∗ ≤ 0 ............................. 73 3.4.3 ThefullBianchiIstatespace...................... 75 3.5DiscussionandConclusion............................ 76 4 Inhomogeneous Perturbations 79 4.1Introduction.................................... 79 4.2Cosmologicalperturbations........................... 80 4.2.1 Thesetup................................. 81 4.2.2 Dimensionlessfirstorderquantities................... 82 4.2.3 Dimensionless Linearized Propagation and Constraint Equations . 84 4.3HarmonicDecomposition............................ 86 4.4LowEnergyLimit:TheGRBackground.................... 88 4.5TheDarkRadiationEra............................. 92 4.6 High Energy Limit: The Fb Background...................