Physik-Department Max-Planck-Institut für Kernphysik Particle Physics and Dark Energy: Beyond Classical Dynamics Doctoral Thesis in Physics submitted by Mathias Garny 2008 Technische Universität München TECHNISCHE UNIVERSITÄT MÜNCHEN Max-Planck-Institut für Kernphysik Particle Physics and Dark Energy: Beyond Classical Dynamics Mathias Garny Vollständiger Abdruck der von der Fakultät für Physik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Lothar Oberauer Prüfer der Dissertation: 1. Prof. Dr. Manfred Lindner, Ruprecht-Karls-Universität Heidelberg 2. Univ.-Prof. Dr. Alejandro Ibarra Die Dissertation wurde am 24.09.2008 bei der Technischen Universität München eingereicht und durch die Fakultät für Physik am 24.10.2008 angenommen. Particle Physics and Dark Energy: Beyond Classical Dynamics Abstract In this work, quantum corrections to classical equations of motion are investigated for dynamical models of dark energy featuring a time-evolving quintessence scalar field. Employing effective quan- tum field theory, the robustness of tracker quintessence potentials against quantum corrections as well as their impact on cosmological observables are discussed. Furthermore, it is demonstrated that a rolling quintessence field can also play an important role for baryogenesis in the early universe. The macroscopic time-evolution of scalar quantum fields can be described from first principles within nonequilibrium quantum field theory based on Kadanoff-Baym equations derived from the 2PI ef- fective action. A framework for the nonperturbative renormalization of Kadanoff-Baym equations is provided. Renormalized Kadanoff-Baym equations are proposed and their finiteness is shown for a special case. Zusammenfassung In dieser Arbeit werden Quantenkorrekturen klassischer Bewegungsgleichungen in dynamischen Mo- dellen der Dunklen Energie untersucht, welche ein zeitabhängiges Quintessenz-Skalarfeld beinhalten. Im Rahmen effektiver Quantenfeldtheorie wird die Stabilität von Quintessenz-Potentialen bezüglich Quantenkorrekturen sowie deren Einfluß auf kosmologische Parameter diskutiert. Darüber hinaus wird gezeigt, daß ein zeitabhängiges Quintessenzfeld auch für die Baryogenese im frühen Univer- sum eine wichtige Rolle spielen kann. Die makroskopische Zeitentwicklung von skalaren Quanten- feldern kann basierend auf Grundprinzipien der Nichtgleichgewichtsquantenfeldtheorie mittels Ka- danoff-Baym Gleichungen beschrieben werden. Es wird ein Formalismus für die nichtperturbative Renormierung von Kadanoff-Baym Gleichungen entwickelt, renormierte Kadanoff-Baym Gleichun- gen vorgeschlagen, und deren Endlichkeit für einen Spezialfall nachgewiesen. Contents 1 Introduction 1 2 Dynamical Dark Energy5 2.1 Quintessence Cosmology................................6 2.2 Tracking Solutions...................................9 2.3 Interacting Quintessence................................ 12 3 Quantum Effective Action 15 3.1 1PI Effective Action.................................. 16 3.2 2PI Effective Action.................................. 19 3.3 nPI Effective Action.................................. 21 4 Quantum Corrections in Quintessence Models 23 4.1 Self-Interactions..................................... 24 4.2 Matter Couplings.................................... 44 4.3 Gravitational Coupling................................. 51 4.4 Summary........................................ 60 5 Leptonic Dark Energy and Baryogenesis 61 5.1 Quintessence and Baryogenesis............................. 61 5.2 Creation of a B−L-Asymmetry............................. 62 5.3 Stability......................................... 66 6 Quantum Nonequilibrium Dynamics and 2PI Renormalization 67 6.1 Kadanoff-Baym Equations from the 2PI Effective Action............... 68 6.2 Nonperturbative 2PI Renormalization at finite Temperature.............. 73 7 Renormalization Techniques for Schwinger-Keldysh Correlation Functions 79 7.1 Non-Gaussian Initial States............................... 79 7.2 Nonperturbative Thermal Initial Correlations..................... 86 7.3 Renormalized Kadanoff-Baym Equation for the Thermal Initial State......... 103 8 Renormalization of Kadanoff-Baym Equations 105 8.1 Kadanoff-Baym Equations and 2PI Counterterms................... 105 8.2 Renormalizable Kadanoff-Baym Equations from the 4PI Effective Action...... 106 8.3 Impact of 2PI Renormalization on Solutions of Kadanoff-Baym Equations...... 112 8.4 Summary........................................ 129 9 Conclusions 131 viii CONTENTS A Conventions 135 B Effective Action Techniques 137 B.1 Low-Energy Effective Action.............................. 137 B.2 Effective Action in Curved Background........................ 138 B.3 Renormalization Group Equations........................... 141 C Resummation Techniques and Perturbation Theory 145 C.1 Relation between 2PI and 1PI.............................. 145 C.2 Resummed Perturbation Theory............................ 146 D Quantum Fields in and out of Equilibrium 151 D.1 Thermal Quantum Field Theory............................ 151 D.2 Nonequilibrium Quantum Field Theory........................ 157 E Nonperturbative Renormalization Techniques 163 E.1 Renormalization of the 2PI Effective Action...................... 163 E.2 Renormalization of 2PI Kernels............................ 165 E.3 Two Loop Approximation............................... 167 E.4 Three Loop Approximation............................... 168 F Integrals on the Closed Real-Time Path 171 Acknowledgements 173 Bibliography 175 Chapter 1 Introduction According to the standard model of cosmology, the evolution of our universe experienced a rapidly inflating and highly correlated phase at its beginning. This phase ended in an explosive entropy production (reheating), during which all kinds of sufficiently light particles were produced and ther- malized, most of them highly relativistic. Reheating was followed by a controlled expansion during which the temperature decreased and more and more massive species became non-relativistic (radi- ation domination). Subsequently, pressure-less baryonic and cold dark matter became the dominant contribution to the total energy density, and underwent gravitational clustering (matter domination). However, in recent cosmic history, the expansion of the universe started to accelerate. This may be attributed to the so-called dark energy, which became more and more important and makes up over two third of the energy density of the universe today. All that is known about dark energy is based on its gravitational interaction. While the total energy density can be measured by observations of the anisotropy of the cosmic microwave background (CMB), the forms of energy which cluster gravitationally can be inferred from large-scale structure surveys together with appropriate models of structure formation. However, the clustered energy is much less than the total energy density, such that an additional, homogeneously distributed com- ponent is required. On top of that, such a dark energy component can precisely account for the accelerated expansion observed by measurements of the luminosity of distant supernovae [133]. This concordance of different observations makes the need for dark energy convincing and the question about its nature one of the most outstanding questions in astro-particle physics. The inclusion of a cosmological constant in Einstein’s equations of General Relativity provides a parameterization of dark energy which is compatible with cosmological observations [89]. The cos- mological constant can be viewed as a covariantly conserved contribution to the energy-momentum tensor which is invariant under general coordinate transformations. For any quantum field theory for which coordinate invariance is unbroken, this is precisely the property of the vacuum expectation value of the energy-momentum tensor. Therefore, the cosmological constant may be interpreted as the vacuum energy within quantum field theory [188]. However, since quantum field theory together with classical gravity determines the vacuum energy only up to a constant, it is impossible to predict the value of the cosmological constant. Furthermore, the naïve summation of zero-point energies of all momentum modes of a free quantum field leads to a divergent result. Once a cutoff between the TeV and the Planck scale is imposed, this amounts to a value which is between 60 and 120 orders of magnitude too large. This fact is known as the cosmological constant problem [178]. If the value inferred from cosmological observations is taken at face value, an enormous hierarchy between the vacuum energy density and the energy density of radiation and matter must have existed in the early universe (smallness problem). Subsequently, radiation and matter get diluted due to the cosmic ex- 2 1. Introduction pansion, and the cosmological constant becomes of comparable order of magnitude precisely in the present cosmological epoch (coincidence problem). These unsatisfactory features of the cosmological constant have motivated an extensive search for alternative explanations of dark energy. Apart from attempts to explain cosmic acceleration by modi- fications of the equations of General Relativity [74, 151], models of dynamical dark energy [65, 162] explore the possibility that the dark energy density might evolve with time and become diluted during cosmic expansion, similar to the radiation and matter components.