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Towards a Holographic Framework for Cosmology

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Towards a Holographic Framework for Cosmology TOWARDS A HOLOGRAPHIC FRAMEWORK FOR COSMOLOGY A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Xi Dong August 2012 © 2012 by Xi Dong. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/vd028jx9557 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Eva Silverstein, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Shamit Kachru I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Leonard Susskind Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii iv Abstract Studies on black hole physics have lead to the holographic principle, which states that a quantum gravitational system can be captured by a theory living in fewer dimen- sions. Given the observed accelerating expansion of our universe, it has been a major challenge to understand the realization of the holographic principle in cosmology. In this dissertation we review our progress in building such a framework. Starting from concrete AdS/CFT (anti de Sitter/conformal field theory) dual pairs, we obtain de Sitter (dS) and other general Friedman-Robertson-Walker (FRW) solutions by adding branes and other ingredients from string theory. In the de Sit- ter case, our brane construction gives a microscopic realization of the dS/dS cor- respondence. The degrees of freedom in the semi-holographic dual theory provide a parametric interpretation of the Gibbons-Harking entropy of the bulk de Sitter space. In the FRW case, we focus on a family of simple FRW solutions sourced by magnetic flavor branes. These solutions have a holographic dual interpretation which decouples from gravity at late time, opening up the possibility of a precise duality. Time-dependent effects play a crucial role in these dual theories. In particular, we find that time-dependent couplings in a quantum field theory can strongly affect long- distance physics, effectively shifting the infrared operator dimensions and generalizing known unitarity bounds. v Acknowledgements First and foremost I would like to thank my advisor, Eva Silverstein, for her constant support and guidance through the years. She has taught me how good theoretical physics is done, and her enthusiasm for physics always inspires me. Throughout my graduate career at Stanford University, I have been very fortunate to have fruitful collaborations with Ning Bao, Daniel Harlow, Sarah Harrison, Bart Horn, Shamit Kachru, Shunji Matsuura, Eva Silverstein, Gonzalo Torroba, Huajia Wang, and Alexander Westphal. Daniel Harlow has been a fantastic friend with whom I worked through many problems together. I have learned a tremendous amount of quantum field theory and string theory from Gonzalo Torroba. I have benefited greatly from interactions with other members of the Stanford Institute for Theoretical Physics and the SLAC Theory Group, especially Dionysis Anninos, Savas Dimopoulos, Daniel Green, Sean Hartnoll, Anson Hook, Jonathan Maltz, Michael Peskin, Michael Salem, Edgar Shaghoulian, Stephen Shenker, Douglas Stanford, Leonard Susskind, Vitaly Vanchurin, and Sho Yaida. I have also been very fortunate to spend a year at the Kavli Institute for Theoreti- cal Physics, UC Santa Barbara, as well as participating in many great programs there. I would like to thank especially Richard Eager, Thomas Faulkner, Zhengcheng Gu, Idse Heemskerk, Gary Horowitz, Nabil Iqbal, Vijay Kumar, Donald Marolf, David Morrison, Joseph Polchinski, and Matthew Roberts for many stimulating discussions. For this dissertation I would like to thank my reading committee members: Eva Silverstein, Leonard Susskind, and Shamit Kachru for their time, interest and in- sightful comments. I would also like to thank Andras Vasy for being the chair of my dissertation defense committee. vi Last and most, I would like to thank my family for all their love and support: for my parents who raised me with a love of science and supported me in all my endeavors, and for my wife Jing whose continuing encouragement and faithful support have made this dissertation possible. Thank you. Xi Dong Stanford University August 2012 vii Contents Abstract v Acknowledgements vi 1 Introduction 1 2 Micromanaging de Sitter holography 4 2.1 Introduction . 4 2.2 dS holography and microscopy . 5 2.3 General techniques . 11 2.3.1 The strategy for stabilization . 12 2.3.2 Stabilization procedure . 15 2.3.3 Effects from localized sources . 18 2.4 dS3 worked example . 20 2.4.1 Brane construction . 21 2.4.2 Stabilization mechanism . 26 2.4.3 D7-D7 stability analysis . 34 2.4.4 Stabilizing other moduli . 36 2.4.5 Localization of sources and the warp factor . 39 2.4.6 Entropy and brane construction . 42 2.4.7 Alternative examples . 43 2.5 Discussion and Future Directions . 45 viii 3 FRW solutions and holography from uplifted AdS/CFT 48 3.1 Introduction: keeping it real . 48 3.2 FRW solution sourced by magnetic flavor branes . 52 3.2.1 Magnetic flavor branes . 52 3.2.2 Solution and warping . 55 3.2.3 d − 1 Planck mass and its decoupling at late times . 61 3.2.4 Covariant entropy bound . 63 3.2.5 Basic relations among parameters . 65 3.3 Dynamics of particles and branes . 67 3.4 Degrees of freedom in FRW holography . 72 3.4.1 A microscopic count of degrees of freedom . 72 ~ 3.4.2 Deriving Ndof from the quasilocal stress tensor . 78 3.5 Correlation functions . 84 3.5.1 Massive Green's functions . 84 3.5.2 Massless Green's functions . 88 3.6 Future directions: Magnetic flavors and time-dependent QFT . 91 3.A Correlation functions in general CdL geometry . 93 3.A.1 Euclidean prescription . 93 3.A.2 Lorentzian prescription . 99 3.A.3 Our FRW spacetime . 103 3.A.4 Massive correlation functions . 106 4 Unitarity bounds and RG flows in time dependent QFT 108 4.1 Motivations . 108 4.2 Spacetime dependent double trace flows . 111 4.2.1 RG flow in the static limit . 112 4.2.2 Spacetime dependent case . 115 4.2.3 Long distance propagator . 118 4.2.4 Infrared physics and unitarity: two wrongs make a right . 121 4.3 Spacetime dependent RG flow for fermionic operators . 123 4.3.1 Results for the static theory . 124 ix 4.3.2 Infrared dynamics of the spacetime dependent theory . 127 4.4 Unitarity bounds in SUSY gauge theories and FRW holography . 128 4.4.1 N = 1 Supersymmetric QCD plus singlets . 129 4.4.2 Seiberg-Witten theory and flavor bounds . 134 4.4.3 Static theory with n > n∗ massive flavors . 138 4.5 Future directions . 139 4.A Dynamical couplings and unitarity . 140 4.B Gaussian theories with spacetime dependent masses . 141 4.B.1 Bosonic model . 142 4.B.2 Fermionic model . 144 4.C Expansion of the two-point functions . 145 4.C.1 Expansion around the nontrivial scale-invariant regime . 145 4.C.2 Expansion around the free fixed point . 147 4.D Higher-derivative toy model . 149 5 Simple exercises to flatten your potential 152 5.1 Motivation: realizing your potential . 152 5.1.1 Additional kinetic effects . 155 5.2 Warmup: review of axion monodromy inflation . 158 5.2.1 Flattening vs. moduli potential barriers . 161 5.3 Workout: axions pushing on heavy fields . 162 5.3.1 Bowflux: Sloshing of flux on fixed cycles . 163 5.3.2 Puffing on the kinetic term . 166 5.3.3 Weight lifting: pushing on moduli . 168 5.3.4 Circuit training: toward more generic UV complete examples . 169 5.4 Cooldown . 170 6 Analytic Coleman-De Luccia Geometries 172 6.1 Introduction . 172 6.2 General Properties of the Coleman-De Luccia Geometry . 173 6.2.1 Euclidean Preliminaries . 173 6.2.2 Lorentzian Continuation . 175 x 6.2.3 Constraints and a Definition . 177 6.2.4 Compact Coleman-De Luccia Geometries . 180 6.3 Examples . 181 6.3.1 De Sitter Domain Walls . 183 6.3.2 Decays from dS to dS . 184 6.3.3 Decays from dS to Minkowski Space . 187 6.3.4 Noncompact Examples . 189 6.A Null Energy Condition . 191 6.B A Bound on Parent dS Radius . 192 Bibliography 195 xi List of Tables 2.1 Ingredients for the brane construction of dS3 . 21 2.2 Numerical example for a dS3 solution . 32 xii List of Figures 2.1 Compact brane constructions for de Sitter solutions . 10 3.1 Contour C going above the double pole . 98 3.2 (a) Contour Ca surrounding simple poles; (b) Coutour Cb surrounding a double pole and simple poles below . 99 5.1 Combined data constraints on the tensor to scalar ratio r and the tilt ns together with the predictions for power-law potentials . 156 5.2 Effects of an inflationary flux on the three-term structure stabilized in a Minkowski minimum .
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