University of Southampton Faculty of Physical Sciences and Engineering Department of Physics and Astronomy Diffeomorphism-invariant averaging in quantum gravity and cosmology Anthony William Henry Preston Submitted for the degree of DOCTOR OF PHILOSOPHY arXiv:1612.06872v1 [gr-qc] 20 Dec 2016 September 2016 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF PHYSICAL SCIENCES AND ENGINEERING Department of Physics and Astronomy Doctor of Philosophy DIFFEOMORPHISM-INVARIANT AVERAGING IN QUANTUM GRAVITY AND COSMOLOGY Anthony William Henry Preston This thesis concerns research undertaken in two related topics concerning high-energy gravitational physics. The first is the construction of a manifestly diffeomorphism- invariant Exact Renormalization Group (ERG). This is a procedure that constructs effective theories of gravity by integrating out high-energy modes down to an ultraviolet cutoff scale without gauge-fixing. The manifest diffeomorphism invariance enables us to construct a fully background-independent formulation. This thesis will explore both the fixed-background and background-independent forms of the manifestly diffeomorphism- invariant ERG. The second topic is cosmological backreaction, which concerns the effect of averaging over high-frequency metric perturbations to the gravitational field equa- tions describing the universe at large scales. This has been much studied the context of the unmodified form of General Relativity, but has been much less studied in the con- text of higher-derivative effective theories obtained by integrating out the high-energy modes of some more fundamental (quantum) theory of gravity. The effective stress- energy tensor for backreaction can be used directly as a diffeomorphism-invariant ef- fective stress-energy tensor for gravitational waves without specifying the background metric. This thesis will construct the manifestly diffeomorphism-invariant ERG and compute the effective action at the classical level in two different schemes. We will then turn to cosmological backreaction in higher-derivative gravity, deriving the general form of the effective stress-energy tensor due to inhomogeneity for local diffeomorphism-invariant effective theories of gravity. This an exciting research direction, as it begins the con- struction of a quantum theory of gravity as well as investigating possible implications for cosmology. Contents 1 Introduction1 1.1 Physics at many scales of length . .1 1.2 Statistical mechanics and the Renormalization Group . .4 1.2.1 The Ising model . .4 1.2.2 Phase transitions . .5 1.2.3 Kadanoff blocking and RG transformations . .6 1.2.4 RG flow . .8 1.2.5 Statistical field theory . .9 1.2.6 Parisi-Sourlas supersymmetry . 11 1.3 Overview of General Relativity . 13 1.3.1 Conceptual background . 14 1.3.2 Geometry of curved spacetime . 15 1.3.3 Derivation of field equations from an action . 18 1.3.4 The stress-energy tensor . 20 1.3.5 Metric perturbations . 22 1.3.6 The diffeomorphism symmetry . 23 1.4 Observational status of modern cosmology . 24 1.4.1 Evidence for GR . 24 1.4.2 The big bang theory . 26 1.4.3 Homogeneity and isotropy . 28 1.4.4 The dark side of the universe . 32 1.4.5 Inflation . 36 2 Review of the manifestly gauge-invariant ERG 43 2.1 Kadanoff blocking continuous fields . 43 2.2 Classical ERG for massless scalar fields . 47 2.3 Gauge-invariant ERG for Yang-Mills theories . 50 v vi CONTENTS 2.4 Loopwise expansion . 55 2.5 Additional regularization . 56 3 Manifestly diffeomorphism-invariant ERG 59 3.1 Background-independent gravity flow equation . 59 3.2 Background-independent calculations . 64 3.2.1 Effective action in Weyl scheme . 69 3.2.2 Running away from the fixed point with dimensionful couplings . 73 3.2.3 Effective action in Einstein scheme . 75 3.3 Fixed-background formalism . 79 3.4 Ward identities . 81 3.4.1 Ward identities for an action . 82 3.4.2 Ward identities for the kernel . 85 3.4.3 Ward identity consistency in the flow equation . 88 3.5 Metric determinant expansion in the metric perturbation . 90 3.6 Functional derivatives of the covariantized kernel . 91 3.7 Transverse 2-point functions . 92 3.8 Classical 2-point functions at fixed-points . 95 3.8.1 Classical 2-point function in Weyl scheme . 95 3.8.2 Classical 2-point function in Einstein scheme . 96 3.9 Further regularization via supermanifolds . 97 3.10 Outlook . 99 4 Review of cosmological backreaction 101 4.1 Conceptual overview . 101 4.2 Notation . 103 4.3 Isaacson's shortwave approximation . 104 4.4 Burnett's weak-limit averaging formalism . 107 4.5 Green and Wald weak-limit framework . 110 5 Backreaction in modified gravity 115 5.1 Incorporating additional scales of length . 118 5.2 Zero tensor forms for modified gravity . 120 5.3 Stress-energy tensor in modified gravity . 121 5.4 Alternative derivation in scalar-tensor formulation . 122 5.5 Diffeomorphism invariance . 126 CONTENTS vii 5.6 Dark energy and new physics in the ultraviolet . 127 6 Generalized cosmological backreaction 133 6.1 General scaling properties . 133 6.2 Contributing action terms . 134 6.2.1 Riemann tensor expansions . 135 6.2.2 Covariant derivative expansions . 136 6.3 Zero tensors . 138 6.4 Stress-energy tensor . 139 6.5 Effective stress-energy tensor trace . 141 6.5.1 General consistency of the trace . 141 6.5.2 Evaluation of the generalized trace . 142 6.6 Diffeomorphism invariance . 143 6.7 Generalized backreaction discussion . 144 7 Summary 149 Bibliography 152 viii CONTENTS List of Figures 2.1.1 Example of a smooth infrared cutoff function in position representation 45 2.2.1 General flow equation for an action with a single scalar field at some n-point level . 48 2.3.1 Gauge-invariant n-point flow equation with a covariantized kernel . 54 ix Declaration of Authorship I, Anthony Preston, declare that this thesis, entitled “Diffeomorphism-invariant averag- ing in quantum gravity and cosmology", and the work presented in it, are my own and have been generated by me as the result of my own original research. I confirm that: 1. This work was done wholly or mainly while in candidature for a research degree at this University; 2. No part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution; 3. Where I have consulted the published work of others, this is always clearly at- tributed; 4. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work; 5. I have acknowledged all main sources of help; 6. Chapters1,2 and4 are reviews, written by me, describing the previously existing literature. Chapters3 and5 report on original research conducted in collaboration with my supervisor, Tim Morris [1,2]. Chapter6 reports on original research I conducted alone [3]. Chapter7 is a summary of the whole thesis. Signed: ........................ Date: .......................... Acknowledgements First of all, I would like to thank my supervisor, Tim Morris, for guiding me on this path and entertaining my wilder speculations while also teaching me how to formulate this research rigorously. I thank the University of Southampton for funding my PhD with a Mayflower Scholarship. I thank my parents and my brother for patiently supporting and encouraging me throughout my long education. During my time in Southampton, I have met many brilliant people who have made PhD life colourful and rewarding. In particular, I would like to thank my fellow PhD students in the Southampton High Energy Physics Group for exchange of ideas and friendship. Working clockwise through the weird and wonderful world of my office, the amazing 46/4007, I thank Thomas Neder, the Big Friendly German, for lively discussions on physics and the meaning of life; I thank Hamzaan Bridle for regularly surprising us with edible treats and unexplained machinery; I thank Marc Scott for his artistry in decorating the office and for lending a sympathetic ear throughout the highs and lows of the past four years; I thank the visiting students, David Garner, Diana Rojas and Rafael Delgado for the completely different ways they each enriched our office culture and mythology; I thank Andrew Meadowcroft for his surgical wit and unexpected pranks; I thank Tadeusz Janowski for outstanding contributions to 4007's physics knowledge, creative chaos and reliquary; I thank Alex Titterton for an endless stream of puns; I thank Jeff Choi for his helpfulness and stoicism; I thank Simon King for keeping us all on our toes; I thank Matt Spraggs for his entertainingly heated monologues; I thank Ronnie Rodgers for his remarkable enthusiasm for formal physics; I thank Maien Binjonaid for being the office’s wise elder statesman; I thank Azaria Coupe for injecting some optimism amid the madness; I thank Edwin Lizarazo for keeping the peace during the frequent office dramas; I thank Andrew Lawson for sound programming advice. Beyond the walls of 4007, I thank Jason Hammett for helping me to get settled into the office and maintaining a cheerful disposition amid the slings and arrows of outrageous leptoquarks. I thank the double-doctor Marc Thomas for inspirational conversations. I thank the Asymptotic Safety students, J¨urgenDietz and Zo¨eSlade, for interesting discussions and good humour in the face of my surreal tomfoolery. Many other people in Southampton and beyond have had a positive impact on my time here, and I thank you all for helping me along this adventure. Chapter 1 Introduction The layout of this introduction is as follows. Section 1.1 will give a broad discussion of how varying scales of length are studied in different physical contexts. Section 1.2 will discuss how problems with many degrees of freedom are tackled in statistical mechanics.