The Physics of Inflation A Course for Graduate Students in Particle Physics and Cosmology by Daniel Baumann Contents Preface 1 I THE QUANTUM ORIGIN OF LARGE-SCALE STRUCTURE 5 1 Classical Dynamics of Inflation7 1.1 Introduction . .7 1.2 The Horizon Problem . .8 1.2.1 FRW Spacetimes . .8 1.2.2 Causal Structure . .9 1.2.3 Shock in the CMB . 10 1.2.4 Quantum Gravity Hocus-Pocus? . 11 1.3 The Shrinking Hubble Sphere . 11 1.3.1 Solution of the Horizon Problem . 11 1.3.2 Solution of the Flatness Problem∗ ...................... 12 1.3.3 Conditions for Inflation . 13 1.4 The Physics of Inflation . 15 1.4.1 False Vacuum Inflation . 15 1.4.2 Slow-Roll Inflation . 15 1.4.3 Hybrid Inflation . 18 1.4.4 K-Inflation . 19 1.5 Outlook . 20 2 Quantum Fluctuations during Inflation 23 2.1 Motivation . 23 2.2 Classical Perturbations . 24 2.2.1 Comoving Gauge . 24 2.2.2 Constraint Equations . 25 2.2.3 Quadratic Action . 26 2.2.4 Mukhanov-Sasaki Equation . 27 2.2.5 Mode Expansion . 27 2.3 Quantum Origin of Cosmological Perturbations . 28 2.3.1 Canonical Quantization . 28 2.3.2 Non-Uniqueness of the Vacuum . 29 2.3.3 Choice of the Physical Vacuum . 30 2.3.4 Zero-Point Fluctuations in De Sitter . 32 2.4 Curvature Perturbations from Inflation . 33 2.4.1 Results for Quasi-De Sitter . 33 i ii Contents 2.4.2 Systematic Slow-Roll Expansion∗ ....................... 34 2.5 Gravitational Waves from Inflation . 36 2.6 The Lyth Bound . 37 3 Contact with Observations 39 3.1 Introduction . 39 3.2 Superhorizon (Non)-Evolution∗ ............................ 40 3.2.1 Weinberg's Proof . 40 3.2.2 Separate Universe Approach . 43 3.3 From Vacuum Fluctuations to CMB Anisotropies . 44 3.3.1 Statistics of Temperature Anisotropies . 44 3.3.2 Transfer Function and Projection Effects . 45 3.3.3 CMBSimple∗ .................................. 47 3.3.4 Coherent Phases and Superhorizon Fluctuations∗ .............. 52 3.3.5 CMB Polarization . 55 3.3.6 Non-Gaussianity . 58 3.4 From Vacuum Fluctuations to Large-Scale Structure . 59 3.4.1 Dark Matter Transfer Function . 59 3.4.2 Galaxy Biasing . 61 3.5 Future Prospects . 62 4 Reheating after Inflation 63 4.1 Introduction . 63 4.2 Elementary Theory of Reheating . 64 4.3 Parametric Resonance and Preheating . 66 4.3.1 QFT in a Time-Dependent Background . 66 4.3.2 Narrow Resonance . 67 4.3.3 Broad Resonance . 70 4.3.4 Termination of Preheating . 76 4.3.5 Gravitational Waves from Preheating . 77 4.4 Conclusions . 77 5 Primordial Non-Gaussianity 79 5.1 Why Non-Gaussianity? . 79 5.2 Gaussian and Non-Gaussian Statistics . 80 5.2.1 Statistics of CMB Anisotropies . 80 5.2.2 Sources of Non-Gaussianity . 81 5.2.3 Primordial Bispectrum . 81 5.2.4 Shapes of Non-Gaussianity . 83 5.3 Quantum Non-Gaussianities . 85 5.3.1 The in-in Formalism . 86 5.3.2 Single-Field Inflation . 90 5.4 Classical Non-Gaussianities . 99 5.4.1 The δN Formalism . 99 5.4.2 Inhomogeneous Reheating . 100 5.5 Future Prospects . 102 Contents iii II THE PHYSICS OF INFLATION 103 6 Effective Field Theory 105 6.1 Introduction . 105 6.2 Basic Principles of EFT . 106 6.2.1 Effective Actions . 106 6.2.2 Uses of EFTs . 108 6.2.3 Power Counting and Scaling . 109 6.2.4 Relevant, Irrelevant and Marginal . 110 6.2.5 Symmetries . 111 6.2.6 Quantum Corrections . 113 6.2.7 Matching and Running . 115 6.3 Seeing is Believing . 116 6.3.1 A Toy Model . 116 6.3.2 Tree-Level Matching . 116 6.3.3 Running . 118 6.3.4 One-Loop Matching . 122 6.3.5 Naturalness . 126 6.3.6 Summary . 128 6.4 The Standard Model as an Effective Theory∗ .................... 129 6.4.1 The Standard Model . 129 6.4.2 Accidental Symmetries . 131 6.4.3 Neutrino Masses . 132 6.4.4 Beyond the Standard Model Physics . 132 6.5 Conclusion . 132 7 Effective Field Theory and Inflation 133 7.1 UV Sensitivity . 133 7.2 Eta Problem . 134 7.3 Large-Field Inflation . 134 7.4 Non-Gaussianity . 135 8 Supersymmetry and Inflation 137 8.1 Introduction . 137 8.2 Facts about SUSY . 137 8.2.1 SUSY and Naturalness . 137 8.2.2 Superspace and Superfields . 139 8.2.3 Supersymmetric Lagrangians . 140 8.2.4 Miraculous Cancellations . 142 8.2.5 Non-Renormalization Theorem . 143 8.2.6 Supersymmetry Breaking . 144 8.2.7 Supergravity . 145 8.3 SUSY Inflation: Generalities . 146 8.3.1 Supergravity Eta Problem . ..