General Properties of Landscapes: Vacuum Structure, Dynamics and Statistics by Claire Elizabeth Zukowski A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Raphael Bousso, Chair Professor Lawrence J. Hall Professor David J. Aldous Summer 2015 General Properties of Landscapes: Vacuum Structure, Dynamics and Statistics Copyright 2015 by Claire Elizabeth Zukowski 1 Abstract General Properties of Landscapes: Vacuum Structure, Dynamics and Statistics by Claire Elizabeth Zukowski Doctor of Philosophy in Physics University of California, Berkeley Professor Raphael Bousso, Chair Even the simplest extra-dimensional theory, when compactified, can lead to a vast and complex landscape. To make progress, it is useful to focus on generic features of landscapes and compactifications. In this work we will explore universal features and consequences of (i) vacuum structure, (ii) dynamics resulting from symmetry breaking, and (iii) statistical predictions for low-energy parameters and observations. First, we focus on deriving general properties of the vacuum structure of a theory independent of the details of the geometry. We refine the procedure for performing compactifications by proposing a general gauge- invariant method to obtain the full set of Kaluza-Klein towers of fields for any internal geometry. Next, we study dynamics in a toy model for flux compactifications. We show that the model exhibits symmetry-breaking instabilities for the geometry to develop lumps, and suggest that similar dynamical effects may occur generically in other landscapes. The questions of the observed arrow of time as well as the observed value of the neutrino mass lead us to consider statistics within a landscape, and we verify that our observations are in fact typical given the correct vacuum structure and (in the case of the arrow of time) initial conditions. Finally, we address the question of subregion duality in AdS/CFT, arguing for a criterion for a bulk region to be reconstructable from a given boundary subregion by local operators. While of less direct relevance to cosmological space-times, this work provides an improved understanding of the UV/IR correspondence, a principle that underlies the construction of many holographically-inspired measures used to make statistical predictions in landscapes. i Contents Contents i 1 Introduction 1 2 Kaluza-Klein Towers on General Manifolds 5 2.1 Introduction . 5 2.2 Scalar . 7 2.3 Vector . 8 2.4 p-form . 10 2.5 Graviton . 14 2.6 Flux Compactifications . 26 2.7 Discussion . 42 3 Flux Compactifications Grow Lumps 44 3.1 Introduction . 44 3.2 The Symmetric Solution . 47 3.3 The Lumpy Solutions . 54 3.4 Ellipsoidal Solutions and the ` = 2 Instability . 58 3.5 Higher-` Solutions . 62 3.6 The Effective Potential for Lumpiness . 64 3.7 Discussion . 66 4 Multi-Vacuum Initial Conditions and the Arrow of Time 68 4.1 Introduction . 68 4.2 Conventions and Approximations . 70 4.3 Two Toy Models . 74 4.4 A Large Landscape . 78 4.5 Hartle-Hawking Initial Conditions . 80 4.6 Dominant Eigenvector Initial Conditions . 83 5 Anthropic Origin of the Neutrino Mass from Cooling Failure 89 5.1 Introduction . 89 ii 5.2 Predictions in a Large Landscape . 98 5.3 Calculation of d =d log mν ............................ 102 P 6 Null Geodesics, Local CFT Operators and AdS/CFT for Subregions 110 6.1 Introduction . 110 6.2 The Reconstruction Map . 114 6.3 A Simple, General Criterion for Continuous Classical Reconstruction: Cap- turing Null Geodesics . 125 6.4 Arguments for an AdS-Rindler Subregion Duality . 130 6.5 Discussion . 133 A Hodge Eigenvalue Decomposition for Forms 134 B Killing Vectors on Closed Einstein Manifolds 141 C Conformal Killing Vectors on Closed Riemannian Manifolds 144 D Hodge Eigenvalue Decomposition for Symmetric Tensors 148 E Perturbation theory 151 F Cosmological Constant and the Causal Patch 156 G Structure Formation with Neutrinos 159 H Late-Time Extrapolation of Numerical Results 162 I Cooling and Galaxy Formation 165 Bibliography 167 iii Acknowledgments I would especially like to thank my Ph.D. advisor, Raphael Bousso, whose support and guidance have truly made this work possible, and whose curiosity and discernment have inspired me to become a better physicist myself. I would also like to especially thank my undergraduate advisor Janna Levin, an excellent role model, physicist and collaborator; Kurt Hinterbichler for incessant support and for devoting many hours to answering physics questions; and my parents for setting a great example and encouraging me to focus on my studies. This work would also not have been possible without many additional collaborators who contributed to the papers represented in individual chapters: Alex Dahlen, Ben Freivo- gel, Dan Mainemer Katz, Stefan Leichenauer, and Vladimir Rosenhaus. Finally, I would like to thank several particularly influential professors through the years, including Allan Blaer, Ori Ganor, Lawrence Hall and Robert Lipshitz, as well as some of the additional peers and younger role models who helped create a lively and exciting academic atmosphere at various stages of my education, including Atanas Atanasov, Camille Avestruz, Adam R. Brown, Yael Degany, Ethan Dyer, Netta Engelhardt, Zachary Fisher, Kevin Grosvenor, Daniel Harlow, Nathan Haouzi, Marilena Loverde, David J. E. Marsh, Emily Maunder, Mudassir Moosa, Hechen Ren, Michael P. Salem, Kevin Schaeffer, Shamil Shakirov, and Ziqi Yan. This work was supported by an NSF Graduate Fellowship, by the Berkeley Center for Theoretical Physics, by the National Science Foundation (award numbers 0855653, 0756174, 1214644 and 1316783), by fqxi grants RFP-1004 and RFP3-1323, and by the U.S. Department of En- ergy under Contract DE-AC02-05CH11231. It draws on research that was partly completed at both Kavli IPMU and the Perimeter Institute. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. 1 Chapter 1 Introduction Our own universe may be part of a larger landscape consisting of many different vacua, each of which realizes different possibilities for low-energy physics. Indeed such a scenario follows naturally from any compactified theory of extra dimensions such as string theory. Different low-energy solutions correspond to different compactified geometries, and from the four- dimensional viewpoint these solutions are connected dynamically through the deformations of moduli governing the size and shape of the extra dimensions. In the case of string theory, where the internal space is a complicated Calabi-Yau manifold with typically hundreds of cycles for flux to wrap in different ways, the resulting low-energy landscape is famously believed to be extremely large, on the order of 10500 vacua [1, 2]. To make progress within such an incredibly complex framework, it is useful to ask questions that rely only on generic properties of a theory rather than intricate details of its solutions. Consequently, this thesis will aim to explore generic properties of landscapes, especially but not necessarily restricted to landscapes resulting from compactification. One important (and in the case of large landscapes, generic) property of landscapes such as our own that contain de Sitter vacua is eternal inflation [3, 4]. We have evidence via the observation of the cosmic microwave background (CMB) and the distribution of galaxies in the sky that our own universe underwent ordinary inflation, a period of accelerated expansion driven by the vacuum energy of a scalar field [5], and we also know that we live close to the time of matter-Λ equality, after which it will again become vacuum energy dominated [6, 7]. Within any inflating vacuum there will be pocket regions that exit inflation, for instance by tunneling to a different vacuum in the landscape, and other regions that continue inflating. If the Hubble expansion rate of the false vacuum is large enough there will forever be regions that continue inflating, in other words inflation will be eternal. In a large landscape such as the string landscape, it is reasonable to consider that at least one vacuum will have a Hubble expansion rate large enough compared to its rate to decay out, thus allowing eternal inflation to occur; it should be emphasized that it is only necessary for this to be satisfied by a single vacuum. Once an eternally inflating phase is entered, infinite volume is produced and not only do all possibilities for different low-energy physics occur, but each occurs infinitely many times. CHAPTER 1. INTRODUCTION 2 One can ask about the relative probability of different low-energy observations, but naively the answer is nonsensical: divided by . To make progress one must first regulate these infinities using a measure.1 In modern cosmology1 much dissatisfaction has resulted from this realization, since it is true that the the choice of measure is highly non-unique; this is the so-called \measure problem." One may hope that a correct and complete understanding of quantum gravity will come with a prescription for regulating these infinities, but at a point in time where a non-perturbative formulation of string theory remains elusive, this may seem like a far-off optimism. However, all is not lost because in the meantime it is possible to make progress phe- nomenologically. Many naive choices of measure can be immediately ruled out for obvious reasons, because they exhibit a youngness paradox [8, 9] or predict that typical observers are Boltzmann brains, freak observers that fluctuate locally from equilibrium and, with over- whelming probability, see only an empty universe [10, 11]. The set of measures is further reduced by dualities that have been discovered between measures that are defined locally, for instance by restricting to the neighborhood of a geodesic, and ones that are defined via some global time cutoff [12, 13].