UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Predictions From Eternal Inflation Permalink https://escholarship.org/uc/item/8761x1dq Author Leichenauer, Stefan Publication Date 2011 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Predictions From Eternal Inflation by Stefan Leichenauer 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 Associate Professor Yasunori Nomura Associate Professor Joshua Bloom Spring 2011 Predictions From Eternal Inflation Copyright 2011 by Stefan Leichenauer 1 Abstract Predictions From Eternal Inflation by Stefan Leichenauer Doctor of Philosophy in Physics University of California, Berkeley Professor Raphael Bousso, Chair We investigate the physics of eternal inflation, particularly the use of multiverse ideas to explain the observed values of the cosmological constant and the coincidences of cosmolog- ical timescales. We begin by reviewing eternal inflation, the multiverse, and the resulting measure problem. Then follows a detailed study of proposals to solve the measure problem, both analytical and numerical, including an analysis of their predictions for cosmological observables. A key outcome of this investigation is that the traditional anthropic calcu- lations, which take into account the necessity of galaxies and heavy elements to produce observers, are redundant in our framework. The cosmological coincidence problem, the seemingly coincidental equality of the timescales of observation and of vacuum domination, is solved for the first time without appeal to detailed anthropic assumptions: very general geometric considerations do the job automatically. We also estimate a 10% likelihood that evidence for eternal inflation will be found in upcoming measurements of the energy density of the universe. Encouraged by this success, we go on to construct a modified version of the light- cone time measure which has conceptual advantages but also reproduces the phenomenology of its predecessor. We complete our study of the measure problem by noting that for a wide class of proposed solutions, including the one developed here, there is an implicit assumption being made about a catastrophic end to the universe. Finally, as a by-product of this research program we find geometries which violate some of the accepted common knowledge on holographic entropy bounds. We point this out and conjecture a general result. i To my father, David Leichenauer, who would have distributed copies to everyone he knew. ii Contents List of Figures v List of Tables xii 1 Introduction 1 1.1 Background and Goals . 1 1.2 Outline . 3 2 A model for star formation 5 2.1 Introduction . 5 2.2 Model . 6 2.2.1 Goals . 6 2.2.2 Geometry and initial conditions . 7 2.2.3 Linear perturbation growth and halo formation . 9 2.2.4 Cooling and galaxy formation . 11 2.2.5 Star formation . 13 2.3 Results . 15 2.3.1 Our universe . 15 2.3.2 Varying single parameters . 18 2.3.3 Varying multiple parameters . 20 3 Applications of the star formation model 28 3.1 Introduction . 28 3.2 Making predictions in the landscape . 33 3.2.1 Two choices of measure . 33 3.2.2 Prior distribution and cosmological selection . 35 3.2.3 Three ways of modeling observers . 36 3.2.4 Summary . 38 3.3 Star formation and entropy production in the multiverse . 39 3.3.1 Star formation . 39 3.3.2 Entropy production . 41 3.4 Results . 43 3.4.1 Weighting by entropy production in the causal diamond . 46 3.4.2 Weighting by star formation + 5 Gyr in the causal diamond . 51 iii 3.4.3 Weighting by star formation + 10 Gyr in the causal diamond . 55 3.4.4 Weighting by entropy production in the causal patch . 59 3.4.5 Weighting by star formation + 5 Gyr in the causal patch . 63 3.4.6 Weighting by star formation + 10 Gyr in the causal patch . 67 3.5 Discussion . 71 3.5.1 Varying the cosmological constant only . 71 3.5.2 Varying spatial curvature only . 74 3.5.3 Varying both the cosmological constant and curvature . 77 3.5.4 Varying the density contrast only . 80 3.5.5 Varying the density contrast and the cosmological constant . 82 3.5.6 Varying the density contrast and spatial curvature . 83 3.5.7 Marginalizing . 84 4 The geometric origin of coincidences and heirarchies 86 4.1 Introduction . 86 4.2 Counting Observations . 89 4.3 Open FRW universes with cosmological constant . 91 4.4 The causal patch cut-off . 93 4.4.1 Positive cosmological constant . 93 4.4.2 Negative cosmological constant . 98 4.5 The apparent horizon cutoff . 101 4.5.1 Definition . 101 4.5.2 Positive cosmological constant . 103 4.5.3 Negative cosmological constant . 105 4.6 The fat geodesic cutoff . 108 4.6.1 Positive cosmological constant . 108 4.6.2 Negative cosmological constant . 110 5 Toward an improved definition of a measure 112 5.1 Introduction . 112 5.2 New light-cone time . 116 5.2.1 Probabilities from light-cone time . 116 5.2.2 A gauge choice for the conformal boundary . 117 5.2.3 The multiverse in Ricci gauge . 118 5.3 Properties of new light-cone time . 122 5.3.1 Rate equation . 122 5.3.2 Attractor solution . 125 5.3.3 Event counting . 125 5.4 The probability measure . 127 5.4.1 Probabilities from the new light-cone time cut-off . 127 5.4.2 Equivalence to the old light-cone time cut-off . 128 5.4.3 Equivalence to the new causal patch measure . 129 5.4.4 Equivalence to the old causal patch measure . 130 5.5 The general case . 131 5.5.1 Perturbative inhomogeneities . 131 iv 5.5.2 Singularities in the boundary metric . 133 5.5.3 Inhomogeneities . 135 5.6 Conformal factor for a homogeneous bubble universe . 136 6 An implicit assumption of geometric measures: the finiteness of time 141 6.1 Time will end . 141 6.2 The probability for time to end . 145 6.3 Objections . 147 6.3.1 Time cannot end in a late-time limit . 147 6.3.2 The end of time is an artifact . 148 6.4 The Guth-Vanchurin paradox . 150 6.5 Discussion . 151 6.5.1 Assumptions . 151 6.5.2 Observation . 153 6.5.3 Interpretation . 154 7 Saturating the holographic entropy bound 157 7.1 Introduction . 157 7.2 Saturating the holographic entropy bound in cosmology . 160 7.2.1 Light-sheets in open FRW universes . 161 7.2.2 Open universes with zero cosmological constant . 163 7.2.3 Open universes with positive cosmological constant . 165 7.2.4 Universes with negative cosmological constant . 167 7.2.5 Generalization to pressureless matter . 168 7.3 Maximizing observable entropy . 170 7.3.1 Entropy on the past light-cone at finite time . 170 7.3.2 Entropy in the causal patch . 172 7.4 Saturating the entropy bound in gravitational collapse . 174 7.4.1 A collapsing ball of dust . 174 7.4.2 Maximizing the entropy . 176 7.4.3 Example . 179 7.5 Saturating the entropy bound in a causal diamond . 179 7.5.1 Coleman-DeLuccia decay . 180 7.5.2 Shell of dust in AdS . 181 7.5.3 Slow feeding of a black hole . 183 3~4 7.6 A conjecture concerning the bound S ≲ A . 185 7.7 Saturating the generalized covariant entropy bound . 186 Bibliography 188 v List of Figures 2.1 The virial density is independent of curvature and has a simple scaling de- pendence on ρΛ. For positive ρΛ, ρvir asymptotes to 16ρΛ at late times. For negative ρΛ, the virial density formally goes to zero at t = πtΛ, which the latest possible time the universe can crunch. As.