inflation and quantum geometrodynamics in scalar-tensor theories DISSERTATION zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg vorgelegt von Matthijs van der Wild 2019 Dekan Prof. Dr. Gregor Herten Referent Prof. Dr. Jochum van der Bij Koreferent Prof. Dr. Harald Ita Tag der mündlichen Prüfung 23.10.2019 colophon The work described in this document was funded in part by the dfg, under grant number grk 2044, and in part by Freiburg University, under the Abschlussstipendium. This document was typeset using ConTEXt. The body text is set in the extended Baskervald font at 10 points. The mathematical text is set in the TEX Gyre Pagello Math font at 10 points. The diagrams were made using METAPOST. The text on the next page quotes Terry Pratchett’s The Wyrd Sisters. This book was written using 100% recycled words. ABSTRACT The work described in this thesis is an investigation of scalar-tensor theories. Scalar- tensor theories generalise Einstein’s theory of relativity by replacing the universal coupling constant of gravity, the Planck mass, with a dynamical scalar field. One motivation to consider scalar-tensor theories can be found within cosmology itself. Observations indicate that, during the very early universe, space underwent a pe- riod of accelerated expansion. This phase of expansion is called inflation. The most common method to implement inflation introduces a new scalar field to generate the required vacuum energy. It is not yet clear what the fundamental nature of this scalar field should be, and general relativity does not provide an answer to this question. Scalar-tensor theories, on the other hand, naturally explain the existence of such a field. During inflation it can be expected that tiny quantum gravitational corrections can be found in the imprint of the cosmic microwave background radiation. These cor- rections can be studied within the formalism of quantum geometrodynamics, which is a canonical, non-perturbative theory of quantum gravity. An important object in this formalism is the Wheeler-DeWitt equation, which governs the dynamics of the quantised degrees of freedom. Although the Wheeler-DeWitt equation describes a full theory of quantum gravity, it has a well-defined classical limit, and quantum grav- itational corrections can be systematically derived from a semiclassical expansion. The thesis starts with a review of the formalism of general relativity. It introduces a general class of scalar-tensor theories, and presents its properties. It reviews the formalism of inflation, and ends with the presentation of the formalism of inflation within the context of a general scalar-tensor theory. Furthermore, it presents the consequences of a general scalar-tensor theory in the context of semiclassical quantum gravity. The basis for this investigation is the canonical approach to quantum gravity, called quantum geometrodynamics. This is a canonical and non-perturbative formulation of quantum gravity, in which the Hamiltonian of the theory of general relativity is quantised in an analogous way as in non-relativistic quantum mechanics. The formalism is generalised to a general class of scalar-tensor theories. It is demonstrated how quantum geometrodynamics can be used to estimate the form and magnitude of the first quantum gravitational correc- tions. Then, the thesis considers the formalism of quantum geometrodynamics within an inflationary model. The corrected Schrödinger equation that arises in the semiclassi- cal expansion of the Wheeler-DeWitt equation is related to inflationary observables. The form of these corrections is derived and the magnitude is estimated. It is found that the quantum gravitational corrections in the inflationary power spectrum are too small to be detected by current experiments, although they are in principle observ- able. The corrections are found to have a clear observational signature. v CONTENTS Abstract v Publications xi 1 Introduction 13 1.1 The theoretical description of gravity 13 1.2 Contents of this thesis 15 1.3 Conventions 16 2 Dynamics of spacetime 17 2.1 Covariant formulation of gravity 17 2.2 Foliation of spacetime 18 2.3 Intrinsic hypersurface geometry 19 2.4 Extrinsic hypersurface geometry 20 2.4.1 Decomposition of curvature 23 2.4.2 Decomposition of the d’Alembert operator 25 2.5 adm line element 25 2.6 adm action and boundary terms 26 2.7 Symmetry reduction 28 3 Scalar-tensor theories 29 3.1 Scalar-tensor theories of gravitation 30 3.2 Dynamics of scalar-tensor theories 31 3.3 Conformal transformations 32 3.4 Field parametrisations of scalar-tensor theories 33 3.5 Examples of scalar-tensor theories 34 3.5.1 Higgs inflation 34 3.5.2 푓 (푅)-theory as a scalar-tensor theory 35 3.6 Equivalence of frames 37 4 Cosmology 39 4.1 The cosmological principle 39 4.2 The flrw universe 41 4.2.1 Time reparametrisations 42 4.3 De Sitter space 43 4.4 Evolution of the universe 44 4.4.1 Epochs of the universe 45 4.5 Inflation 46 4.5.1 The cosmic microwave background radiation 46 4.5.2 The flatness and horizon problems 47 4.5.3 Scalar-field inflation 49 4.5.4 The slow-roll approximation 50 4.5.5 Frame invariance of cosmological observables 52 4.6 Cosmological perturbations 53 vii 4.6.1 Gauge dependence of the perturbations 53 4.6.2 Gauge invariant perturbations 54 4.6.3 Quantised scalar perturbations 56 4.6.4 Quantised tensor perturbations 58 4.7 Inflationary observables 59 4.7.1 Scalar power spectrum 59 4.7.2 Tensor power spectrum 61 4.7.3 Empirical values 62 5 Quantum geometrodynamics 63 5.1 Hamiltonian formulation 64 5.2 Quantisation of constrained systems 67 5.3 The semiclassical limit of quantum geometrodynamics 69 5.3.1 The semiclassical expansion 70 5.4 Unitarity in quantum geometrodynamics 72 5.5 Example: minimally coupled theories 74 5.5.1 Expansion of the heavy sector 76 5.5.2 Expansion of the fast sector 77 5.5.3 Unitarity in the semiclassical expansion 78 6 Quantum geometrodynamics in scalar-tensor theories 81 6.1 Scalar-tensor theory and foliation of spacetime 82 6.2 Canonical formulation and Hamiltonian constraint 84 6.3 Quantum theory and Wheeler-DeWitt equation 86 6.4 Weighting and transition between frames 89 6.4.1 Born-Oppenheimer approximation 89 6.4.2 Transition to the Einstein frame 91 6.4.3 Weighting of the Hamiltonian 93 6.5 Semiclassical approximation 95 6.5.1 Representation of the corrections 99 6.6 Conclusion 104 7 Quantum gravitational corrections in inflation 107 7.1 Scalar-tensor theories of inflation 108 7.1.1 Field equations and energy-momentum tensor 108 7.1.2 Spacetime foliation 108 7.1.3 Cosmological background evolution 109 7.1.4 Inflationary background dynamics in the slow-roll approximation 111 7.1.5 Cosmological perturbations 112 7.2 Quantum Geometrodynamics 114 7.2.1 Hamiltonian formalism 114 7.2.2 Quantum Geometrodynamics and Wheeler-DeWitt equation 114 7.3 Semiclassical expansion of the Wheeler-DeWitt equation 115 7.3.1 Implementation of the semiclassical expansion 115 7.3.2 Hierarchy of background equations 118 7.3.3 Hierarchy of perturbation equations 119 viii 7.4 Cosmological power spectra in the Schrödinger picture 120 7.4.1 Power spectra without quantum gravitational corrections 121 7.4.2 Power spectra including quantum gravitational corrections 125 7.5 Observational signatures of the corrections 128 7.6 Conclusion 132 8 Discussion and outlook 135 A Geometry of configuration space 137 A.1 Christoffel symbols 137 A.2 Metric determinant 138 A.3 Riemann tensor configuration space 139 A.4 Ricci tensor configuration space 140 A.5 Ricci scalar configuration space 140 B Quantum gravitational corrections in the slow-roll approximation 143 C Correspondence between quantisation pictures 149 Acknowledgements 151 Bibliography 153 Index 163 ix PUBLICATIONS This work is based in part on the following works: •[1] C. F. Steinwachs and M. L. van der Wild, Quantum gravitational corrections from the Wheeler-DeWitt equation for scalar-tensor theories, Class. Quant. Grav. 35(13), 135010, 2018, eprint arXiv:1712.08543 [gr-qc]. •[2] C. F. Steinwachs and M. L. van der Wild, Quantum gravitational corrections to the inflationary power spectrum for scalar-tensor theories, accepted for publication in Class. Quant. Grav., eprint arXiv:1904.12861 [gr-qc]. xi 12 INTRODUCTION 1.1 THETHEORETICALDESCRIPTIONOFGRAVITY In recent years experiments, such as those performed with the Planck satellite, have given a new wealth of data that can be used to piece together the structure of 1 the cosmos. This has resulted in a theoretical description in the form of the Lambda cold dark matter (Λcdm) model of cosmology which, ultimately, is based on Einstein’s theory of general relativity. As a description of gravity, the theory of general relativity is highly successful. It accurately describes gravitational time dilatation and redshift, planetary motion in the solar system, and the bending of light as it moves through space. General relativity has withstood the test of time up to the present, as it provides an accurate description of recent observations. Perhaps the most prominent examples are the experimental observation of gravitational waves and the shadow of the supermassive black hole in the centre of the galaxy M87. With all these data in favour of Einstein’s theory one cannot help but wonder whether there is a deeper, more fundamental theory of which general relativity is a low-energy limit. These considerations are partially based on observations that cannot be explained by general relativity alone. The observation of galaxies at the edge of the observable universe found that the universe is to a large extent the same in every direction.