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Cosmological Singularity Resolution : Classical and Quantum Approaches

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Cosmological Singularity Resolution : Classical and Quantum Approaches Cosmological Singularity Resolution Classical and Quantum Approaches DISSERTATION zur Erlangung des akademischen Grades DOCTOR RERUM NATURALIUM (Dr. rer. nat.) im Fach Physik Spezialisierung: Theoretische Physik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Humboldt-Universit¨atzu Berlin von Sebastian F. Bramberger Pr¨asidentin der Humboldt-Universit¨atzu Berlin: Prof. Dr.-Ing. Dr. Sabine Kunst Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at: Prof. Dr. Elmar Kulke Tag der Disputation am 19.12.2019 Gutachter: Prof. Dr. Hermann Nicolai Prof. Dr. Claus Kiefer Dr. Olaf Hohm Abstract In the face of ever more precise experiments, the standard model of cosmology has proven to be tremendously robust over the past decades. Inflation or ekpyrosis provide a basis for solving some of its remaining conceptual issues - they are a beautiful and natural simplifi- cation to our understanding of the universe's early history; yet they leave many questions unanswered and raise new problems. For example, inflationary theories fail to be predictive as long as eternal inflation is not better understood. At the same time, ekpyrotic theories struggle to explain the transition from a contracting to an expanding phase - the so-called bounce. Both of them lack any understanding or description of the origin of everything and contain cosmological singularities. Here, we provide concrete steps towards shedding a light on these mysteries. The overarching theme that guides most chapters in this thesis is how to deal with cosmologi- cal singularities and whether they can be resolved without invoking extraordinary physics. In the first part, we construct classically non-singular bounces in the most general closed, homo- geneous but anisotropic space-time. In special cases we find analytic solutions to Einstein's equations which, in addition, describe inhomogeneities and electro-magnetic fields. Looking at the general case, we find bounces numerically and show that they leave the universe in a state well-suited for inflation to commence. In the second part we analyze the effect of intro- ducing quantum mechanics semi-classically to cosmology. Our methods, which are based on Feynman's sum over histories framework, reveal novel and interesting properties of the early universe. We scrutinize both processes responsible for eternal inflation: false vacuum decay and slow-roll inflation. In the first case, we are able to show that instabilities may occur dur- ing false vacuum decay independent of the scale at which the decay happens. In the second case, we provide a new framework which can be used to describe quantum effects during an inflationary phase and goes beyond the usual treatment of Quantum Field Theory in curved space-time. We calculate the dominant contributions to transition amplitudes during slow- roll and eternal inflation as well as their properties. Finally, we show that quantum effects are helpful in resolving cosmological singularities. We demonstrate that anisotropies do not hinder the universe's creation from nothing. Furthermore, we construct numerical solutions in which the universe tunnels to a different state before reaching a singularity. With that, we resolve for the first time cosmological singularities without the use of extravagant physics. i Zusammenfassung Das Standardmodell der Kosmologie stellte sich in den letzten Jahrzehnten, trotz immer genauerer experimenteller Tests, als sehr robust heraus. Dar¨uber hinaus schaffen ekpyro- tische und inflation¨areTheorien eine Grundlage um viele konzeptuelle Probleme des fr¨uhen Universums zu l¨osen.Dennoch bleiben viele Fragen unbeantwortet. So ist es in inflation¨aren Theorien schwierig pr¨aziseVorhersagen zu treffen so lange die ewige Inflation nicht besser verstanden wird. Auf der anderen Seite haben ekpyrotische Theorien Schwierigkeiten den Ubergang¨ zwischen kontrahierenden und expandierenden Phasen - den so-genannten kosmis- chen R¨uckprall - zu erkl¨aren.Zudem beschreibt keine der beiden Theorien den Ursprung von Allem und beinhalten kosmologische Singularit¨aten. Hier stellen wir Denkans¨atzebereit um diese Unklarheiten n¨aherzu beleuchten. Im ersten Teil der Arbeit konstruieren wir klassische, singularit¨atenfreie R¨uckpr¨allein der generellsten geschlossenen, homogenen aber anisotropischen, Raumzeit. In speziellen F¨allen finden wir analytische L¨osungender Einsteingleichungen die zus¨atzlich sogar Inhomogenit¨aten und elektromagnetische Felder beschreiben. Im Allgemeinen finden wir r¨uckprallende L¨osungen numerisch und zeigen, dass sie das Universum in einen Zustand lassen, der f¨ureine subse- quente Inflationsphase gut geeignet ist. In dem l¨angeren,zweiten Teil besch¨aftigenwir uns mit den Konsequenzen auf die Kos- mologie, die eine konsistente, semiklassische Quantisierung mit sich bringt. Unsere Methoden, die auf Feynmans Summe ¨uber Pfade basiert, offenbart neue und interessante Ph¨anomene des fr¨uhenUniversums. Im Speziellen analysieren wir beide Prozesse, die ewige Inflation verur- sachen: Der Zerfall des falschen Vakuums und langsam-rollende Inflation. Im ersten Fall zeigen wir, dass Instabilit¨atenw¨ahrenddes Zerfalls auftreten k¨onnenunabh¨angigvon der Energie des Zerfalls. Im zweiten Fall stellen wir einen neues Konzept vor mit dem Quanten- effekte w¨ahrendder inflation¨arenPhase beschrieben werden k¨onnenund das ¨uber die ¨ubliche Beschreibung in der Quantenfeldtheorie in gekr¨ummter Raumzeit hinausgeht. Wir berech- nen die dominanten Beitr¨agezu Amplituden die typische und ewige inflation¨areProzesse beschreiben sowie deren Eigenschaften. Schlussendlich zeigen wir wie Quanteneffekte f¨urdie Aufl¨osungkosmologischer Singularit¨atenhilfreich sind. Wir zeigen explizit, dass Anisotropien kein Problem f¨urden Ursprung des Universums durch ein Tunneln aus dem Nichts darstellen. Zudem konstruieren wir numerische L¨osungen,in denen das Universum vor dem Erreichen einer Singularit¨atin einen anderen Zustand tunnelt. Damit l¨osenwir zum aller ersten Mal kosmologische Singularit¨atenohne den Einsatz von extravaganter Physik auf. ii Contents Publications vii 1 Introduction 1 2 The Standard Model of Cosmology 4 2.1 The Lagrangian description of General Relativity . 4 2.2 The Big Bang model . 5 2.3 The PLANCK measurements . 8 2.4 The Cosmic Microwave Background . 8 2.5 Problems of the Big Bang model . 11 2.5.1 The singularity problem . 12 2.5.2 The flatness problem . 12 2.5.3 The horizon problem . 13 2.5.4 The topological defects puzzle . 13 2.5.5 The classicality puzzle . 13 3 Beyond the Standard Model of Cosmology 14 3.1 Inflation . 14 3.1.1 De Sitter space . 15 3.1.2 Slow-roll inflation . 15 3.1.3 The end of inflation? . 17 3.1.4 Eternal inflation . 18 3.2 Ekpyrosis . 19 3.3 Scaling solutions . 20 3.4 Problems of inflation and ekpyrosis . 21 3.4.1 Inflation . 21 3.4.2 Ekpyrosis . 22 3.5 Beyond FLRW: symmetries in cosmology . 23 3.5.1 Rotations and Translations . 23 3.5.2 All Cosmological Models . 24 3.5.3 The Bianchi Classification . 24 3.5.4 Bianchi IX . 26 3.6 Beyond General Relativity . 29 iii 4 Classically Bouncing Cosmologies 29 4.1 Anisotropic bounces . 32 4.1.1 Adding an electromagnetic field . 32 4.2 Inhomogeneous and anisotropic bounces . 34 4.2.1 Adding an electromagnetic field . 34 4.3 A black hole - bounce correspondence . 36 4.4 Examples . 39 4.5 Bounces in the presence of a cosmological constant . 43 4.5.1 Time symmetric bounces . 43 4.5.2 Time asymmetric bounces . 49 4.5.3 Axial Bianchi IX: Comparing to the exact solution . 53 4.6 Bounces in the presence of a scalar field . 57 4.7 Discussion . 61 5 Quantum Tunnelling 65 5.1 The Simplest Case: 1D Quantum Mechanics . 65 5.2 Tunneling via complex time paths . 67 5.3 Examples . 72 5.3.1 Inverted harmonic oscillator . 73 5.3.2 Inverted Higgs potential . 76 5.3.3 Potential barrier with singularities . 78 5.4 Discussion . 81 5.5 False Vacuum Decay . 84 5.5.1 Coleman DeLuccia Instantons . 85 5.6 The negative mode problem . 86 5.7 Negative mode problem for a polynomial potential . 89 5.7.1 Numerical example of negative Q far from Planck scale . 89 5.7.2 Negative Q in the thin wall approximation . 91 5.7.3 Existence of Coleman - De Luccia solutions . 93 5.7.4 Comparison with numerics . 93 5.8 Negative mode problem for Higgs-like potentials . 95 5.9 Discussion . 96 iv 6 Quantum Cosmology 98 6.1 The Hamiltonian Formulation of General Relativity . 98 6.2 Quantization . 99 6.3 Minisuperspace . 100 6.3.1 Canonical Quantization . 101 6.3.2 Path Integral Quantization . 102 6.4 Boundary Conditions . 102 6.5 Classicality . 104 7 Quantum Singularity Resolution 105 7.1 The Anisotropic Minisuperspace Model . 105 7.2 The Anisotropic No-Boundary Proposal . 107 7.2.1 No-Boundary Conditions . 107 7.2.2 Classicality . 110 7.2.3 Existence and Basic Features of Anisotropic Instantons . 111 7.2.4 Scaling of the classicality conditions . 117 7.3 Quantum Transitions of the Universe . 123 7.3.1 Quantum Transitions: from Inflation to Inflation . 125 7.3.2 Quantum Transitions: from Ekpyrosis to Inflation . 131 7.4 Discussion . 136 8 Lorentzian Quantum Cosmology 139 8.1 Picard-Lefschetz Theory . 140 8.2 Exactly Soluble Scalar Field Minisuperspace Models . 142 8.2.1 The Simplest Case: Pure Gravity . 142 8.2.2 Gravity and a Scalar Field . 144 8.3 Homogeneous Transitions During Inflation . 146 8.3.1 Inflation - Rolling Down the Potential . 149 8.3.2 Jumping Up the Potential . 156 8.3.3 Avoiding Off-Shell Singularities . 169 8.4 Discussion . 170 9 Conclusion 172 v A The Variational Principle 175 A.1 Dirichlet Conditions . 175 A.2 Neumann Conditions . 178 A.3 Robin Conditions . 179 B Cosmological Perturbation Theory 179 C Kantowski-Sachs bounces 181 D Quantum Bounces 183 D.1 Contours of Integration . 183 D.2 Perturbative Results . 185 D.2.1 Large scalar field . ..
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