FRAME COVARIANCE AND FINE TUNING IN INFLATIONARY COSMOLOGY A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2019 By Sotirios Karamitsos School of Physics and Astronomy Contents Abstract 8 Declaration 9 Copyright Statement 10 Acknowledgements 11 1 Introduction 13 1.1 Frames in Cosmology: A Historical Overview . 13 1.2 Modern Cosmology: Frames and Fine Tuning . 15 1.3 Outline . 17 2 Standard Cosmology and the Inflationary Paradigm 20 2.1 General Relativity . 20 2.2 The Hot Big Bang Model . 25 2.2.1 The Expanding Universe . 26 2.2.2 The Friedmann Equations . 29 2.2.3 Horizons and Distances in Cosmology . 33 2.3 Problems in Standard Cosmology . 34 2.3.1 The Flatness Problem . 35 2.3.2 The Horizon Problem . 36 2 2.4 An Accelerating Universe . 37 2.5 Inflation: More Questions Than Answers? . 40 2.5.1 The Frame Problem . 41 2.5.2 Fine Tuning and Initial Conditions . 45 3 Classical Frame Covariance 48 3.1 Conformal and Weyl Transformations . 48 3.2 Conformal Transformations and Unit Changes . 51 3.3 Frames in Multifield Scalar-Tensor Theories . 55 3.4 Dynamics of Multifield Inflation . 63 4 Quantum Perturbations in Field Space 70 4.1 Gauge Invariant Perturbations . 71 4.2 The Field Space in Multifield Inflation . 74 4.3 Frame-Covariant Observable Quantities . 78 4.3.1 The Potential Slow-Roll Hierarchy . 81 4.3.2 Isocurvature Effects in Two-Field Models . 83 5 Fine Tuning in Inflation 88 5.1 Initial Conditions Fine Tuning . 88 5.2 Parameter Fine Tuning . 92 5.3 Trajectory Fine Tuning . 94 6 Models of Inflation 97 6.1 Single-Field Models . 97 6.1.1 Induced Gravity Inflation . 98 6.1.2 Higgs Inflation . 102 6.1.3 F (R) Models . 104 3 6.2 Multifield Models . 107 6.2.1 Minimal Two-Field Inflation . 107 6.2.2 Non-minimal Two-Field Inflation . 114 6.2.3 F ('; R) Models . 120 7 Beyond the Tree Level 122 7.1 The Conventional Effective Action . 123 7.2 The Vilkovisky Effective Action . 127 7.3 The De Witt Effective Ection . 130 7.4 The Conformally Covariant Vilkovisky{De Witt Formalism . 132 8 Conclusions 137 A Frame-Covariant Power Spectra 140 Bibliography 147 4 List of Tables 2.1 Evolution of energy density and scalar factor for different eras of the Universe. 32 2.2 Density parameters for different components of the Universe [53, 55]. The total radiation density is the sum of the photon density and the neutrino density. 33 3.1 Conformal weights and scaling dimensions of various frame- covariant quantities. 67 6.1 Observable inflationary quantities for the minimal two-field model at N = 60. Note that the running of the tensor spectral index αT is not quoted in [54], as no tensor modes were measured by PLANCK. It is derived from the consistency relation (4.42) with transfer angle Θ = 0, and serves as a constraint on a pos- sible future measurement of αT , in the slow-roll approximation. The parameter βiso is constrained by assuming different non- decaying isocurvature modes: (i) the cold dark matter density isocurvature mode (CDI), (ii) the neutrino density mode (NDI), and (iii) the neutrino velocity mode (NVI). 113 6.2 Observable inflationary quantities for the non-minimal model at N = 60. The limits on these quantities from 2015 PLANCK data [54] are the same as in Table 6.1. 118 5 List of Figures 1.1 Illustration of quantisation in different frames. It is not im- JF EF mediately obvious that Γ1−loop = Γ1−loop. Figure reproduced from [14]. 16 2.1 Schematic representation of the metric expansion of the Uni- verse. As time passes, the \density" uniformly decreases such that points recede from each other at a speed proportional to their distance (Hubble's law). 28 5.1 Probability that a randomly selected value for the α-attractor potential corresponds to an initially inflationary trajectory in the slow-roll approximation. 92 5.2 Trajectory flow between two isochrone surfaces N(') = N1 and N(') = N2 in field space. 96 6.1 End-of-inflation curve for the minimal two-field model (6.42) 2 2 with m =(λMP ) = 1. 109 6.2 Field space trajectories and isochrone curves for the minimal two-field model (6.42). 109 6.3 Sensitivity parameter Q∗ for the minimal model (6.42) at N = 60 to boundary conditions given by '0. The dashed line corre- sponds to Q∗ = 1. 110 6 6.4 Power spectrum normalisation for the minimal two-field model (6.42) −12 −6 with λ = 10 and m=MP = 10 at N = 60 for different boundary conditions in terms of '0 and the corresponding hori- zon crossing values '∗. Solid lines correspond to the theoretical predictions while the horizontal line corresponds to the observed obs power spectrum PR given in (6.15). 111 6.5 Predictions for the inflationary quantities r; nR; αR; αT , fNL and βiso in the minimal model (6.42) for boundary condition given by '0=MP = 0:496. 112 6.6 Evolution of ! andη ¯ss along the inflationary trajectory with '0 = 0:495 for the minimal two-field model (6.42). 113 6.7 End-of-inflation curve for the non-minimal model (6.47)with −6 −12 m = 5:6 10 MP , λ = 10 , and ξ = 0:01. 115 × 6.8 Field space trajectories and isochrone curves for the non-minimal two-field model (6.47). 115 6.9 Power spectrum normalisation for the non-minimal two-field −6 −12 model (6.47) with m = 5:6 10 MP ; λ = 10 , and ξ = 0:01 × for different boundary conditions in terms of '0 and the corre- sponding horizon crossing values '∗. Solid lines correspond to the theoretical predictions while the horizontal dashed lines cor- respond to the allowed band for the observed power spectrum obs PR given in (6.15). 116 6.10 Predictions for the inflationary quantities r; nR; αR; αT , fNL, and βiso in the non-minimal two field model (6.47) for boundary conditions admissible under normalisation of PR to the observed obs power spectrum PR . 117 6.11 Sensitivity parameter Q∗ for the non-minimal two-field model (6.47) to boundary conditions given by '0. The dashed line corre- sponds to Q∗ = 1. 118 6.12 Evolution of ! andη ¯ss along the observationally viable infla- tionary trajectories for the non-minimal two-field model (6.47). 119 7 Abstract This thesis presents the development of a fully covariant approach to scalar- tensor theories of gravity in the context of inflation, as well as a covariant treatment of trajectory fine tuning in multifield models. Our main result is the introduction of frame covariance as a way to confront the frame problem in inflation. We treat the choice of a gravitational frame in which a theory is presented as a particular instance of gauge fixing. We take frame covariance beyond the tree level by virtue of the Vilkovisky{De Witt formalism, which was originally developed with the aim of removing the gauge and reparametri- sation ambiguities from the path integral. Adopting an analogous approach, we incorporate conformal covariance to the Vilkovisky{De Witt formalism, demonstrating that the choice of a conformal frame is not physically impor- tant. This makes it possible to define a unique action even in the presence of matter couplings. We therefore show that even if the matter picture of the Universe may appear different in conformally related models, the underlying theory is independent of its frame representation. We further examine the relation between parameter fine tuning and initial condition fine tuning. Even though conceptually distinct, they both adversely affect the robustness of us- ing established particle physics models to drive inflation. As a way to remedy this, we note that the presence of additional scalar degrees of freedom can \rescue" particular models that are ruled out observationally by shifting the burden of fine tuning from the parameters to the choice of slow-roll trajectory. We refer to this uniquely multifield phenomenon as \trajectory fine tuning", and we propose a method to quantify the sensitivity of multifield models to it. We illustrate by presenting examples of both single-field and multifield models of inflation, as well as F (R) and F ('; R) theories. 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