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Cosmological Fluctuations in Standard and Conformal Gravity University of Connecticut OpenCommons@UConn Doctoral Dissertations University of Connecticut Graduate School 6-5-2020 Cosmological Fluctuations in Standard and Conformal Gravity Matthew Phelps University of Connecticut - Storrs, [email protected] Follow this and additional works at: https://opencommons.uconn.edu/dissertations Recommended Citation Phelps, Matthew, "Cosmological Fluctuations in Standard and Conformal Gravity" (2020). Doctoral Dissertations. 2532. https://opencommons.uconn.edu/dissertations/2532 Cosmological Fluctuations in Standard and Conformal Gravity Matthew Phelps, Ph.D. University of Connecticut, 2020 In the theory of cosmological perturbations, methods of simplifying the equations of motion, eliminating non-physical gauge modes, and constructing the solutions broadly fall into one of two approaches. In the first approach, one first fixes the gauge freedom by imposing coordinate constraints and then solves the fluctuation equations directly in terms of the metric perturbation. In the context of confor- mal gravity, we implement this method by constructing a gauge condition that is invariant under conformal transformations, allowing the full set of exact solutions to be obtained within any background geometry that is conformal to flat (thus covering all cosmologically relevant geometries). With this construction, we show that in a radiation era Robertson-Walker cosmology, conformal gravity fluctua- tions grow as t4. The other approach, standard within modern cosmology, is to decompose the metric perturbation into a basis of scalars, vectors, and tensors defined according to their transformation behavior under three-dimensional rota- Matthew Phelps { University of Connecticut, 2020 tions. With this basis automatically facilitating gauge invariance, it is asserted that the scalar, vector, and tensor sectors decouple within the fluctuation equa- tions themselves, `ala the SVT decomposition theorem. Here we investigate the status of the decomposition theorem by solving the fluctuation equations exactly within standard cosmologies as applied to both Einstein gravity and conformal gravity, finding that in general the various SVT gauge-invariant combinations only separate at a higher-derivative level. To achieve separation at the level of the fluctuation equations themselves one must additionally impose spatial boundary conditions, with an exact form depending on the given background geometry. To match the underlying transformation group of GR and thus provide a manifestly covariant formalism, we introduce an alternate scalar, vector, tensor basis with components defined according to general four-dimensional coordinate transforma- tions. In this basis, the fluctuation equations greatly simplify, and while one can again decouple them into separate gauge-invariant sectors at the higher-derivative level, in general we find that even with boundary conditions we do not obtain a decomposition theorem in which the fluctuations separate at the level of the fluc- tuation equations themselves. Cosmological Fluctuations in Standard and Conformal Gravity Matthew Phelps B.S., University of Colorado Colorado Springs, 2014 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Connecticut 2020 Copyright by Matthew Phelps 2020 i APPROVAL PAGE Doctor of Philosophy Dissertation Cosmological Fluctuations in Standard and Conformal Gravity Presented by Matthew Phelps, B.S. Major Advisor Philip Mannheim Associate Advisor Alex Kovner Associate Advisor Vasili Kharschenko University of Connecticut 2020 ii To Aaron M. Phelps and Renee S. Phelps, lost but never forgotten iii ACKNOWLEDGEMENTS To Philip Mannheim - Working under you will always be one of the great honors of my life. Your brilliance is outshined only by your love and driven curiosity of the field, which have served as a continual source of inspiration throughout. To my family, your continual encouragement has meant a great deal to me. To Ray Retherford - Not of my blood but forever my brother true and through. You were there every time I needed you. To Michaela Poppick - Your unwavering support is nothing short of admirable and I am indebted to you for leading me to become a better person. To H. Perry Hatchfield, for helping me not lose touch of the most important aspects of myself amongst such a transformational time. Your taste in music may only be rivaled by the gods themselves. To Candost Akkaya, for injecting high doses of humor into an otherwise mechanical environment, and for so many parallel journeys we have shared together. To Jason Hancock - Thank you for your faith in my abilities and treating me like an equal. Working with you was an unexpectedly great pleasure. To Lukasz Kuna - Our foray into the world of cryptocurrency could never be forgotten. To my past advisors, Robert Camley and Karen Livesey, for mastering the balance of challenging my mind while building my intellectual confidence. To all other faculty and friends, I thank you for helping me through this entire journey. iv TABLE OF CONTENTS 1. Introduction :::::::::::::::::::::::::::::::: 1 1.1 Overview................................6 2. Formalism ::::::::::::::::::::::::::::::::: 9 2.1 Einstein Gravity........................... 10 2.1.1 Fluctuations Around Flat in the Harmonic Gauge..... 14 2.2 Conformal Gravity.......................... 14 2.2.1 Conformal Invariance..................... 17 2.2.2 Fluctuations Around Flat in the Transverse Gauge.... 23 2.2.3 On the Energy Momentum Tensor.............. 25 2.3 Cosmological Geometries....................... 30 3. Scalar, Vector, Tensor (SVT) Decomposition ::::::::::: 33 3.1 SVT3................................. 35 3.1.1 SVT3 in Terms of hµν in a Conformal Flat Background.. 36 3.1.2 SVT3 in Terms of the Traceless kµν in a Conformal Flat Background.......................... 38 3.1.3 Gauge Structure and Asymptotic Behavior......... 39 3.2 SVTD................................. 48 3.2.1 Gauge Structure (D = 4)................... 53 3.3 Relating SVT3 to SVT4....................... 55 v 3.4 Decomposition Theorem and Boundary Conditions........ 58 3.4.1 SVT3............................. 58 3.4.2 SVT4............................. 64 4. Construction and Solution of SVT Fluctuation Equations ::: 66 4.1 SVT3................................. 71 4.1.1 dS4 ............................... 71 4.1.2 Robertson Walker k = 0 Radiation Era........... 76 4.1.3 General Roberston Walker.................. 86 4.1.4 Robertson Walker k = −1.................. 105 4.1.5 δWµν Conformal to Flat................... 124 4.2 SVT4................................. 130 4.2.1 Minkowski........................... 131 4.2.2 dS4 ............................... 134 4.2.3 General Robertson Walker.................. 157 4.2.4 δWµν Conformal to Flat................... 160 4.3 Gauge Invariants, SVT Mixing, and the Decomposition Theorem. 162 4.3.1 SVT3 Fluctuations Around an Anti de Sitter Background. 163 4.3.2 SVT3 Fluctuations Around a General Conformal to Flat Background.......................... 165 4.3.3 Imposition of SVT Gauge Conditions............ 170 vi 5. Construction and Imposition of Gauge Conditions :::::::: 172 5.1 The Conformal Gauge and General Solutions in Conformal Gravity 174 5.1.1 The Conformal Gauge.................... 174 5.1.2 δWµν in an Arbitrary Background.............. 178 5.1.3 δWµν in a Conformal to Flat Minkowski Background... 185 5.1.4 Calculation Summary..................... 188 5.2 Imposing Specific Gauges within δGµν ............... 189 5.3 Robertson Walker Radiation Era Conformal Gravity Solution... 193 6. Conclusions :::::::::::::::::::::::::::::::: 200 Bibliography ::::::::::::::::::::::::::::::::: 204 A. SVT by Projection ::::::::::::::::::::::::::: 209 A.1 The 3 + 1 Decomposition....................... 209 A.2 Vector Fields............................. 210 A.3 Transverse and Longitudinal Projection Operators for Flat Space- time Tensor Fields.......................... 211 A.4 Transverse-Traceless Projection Operators for Flat Spacetime Ten- sor Fields............................... 213 A.5 Transverse and Longitudinal Projection Operators for Curved Space- time Tensor Fields.......................... 215 vii A.6 D-dimensional SVTD Transverse-Traceless Projection Operators for Curved Spacetime Tensor Fields................. 216 B. Conformal to Flat Cosmological Geometries :::::::::::: 219 B.1 Robertson-Walker k = 0....................... 219 B.2 Robertson-Walker k > 0....................... 219 B.3 Robertson-Walker k < 0....................... 220 B.4 dS4 and AdS4 Background Solutions................ 221 B.5 dS4 and AdS4 Background Solutions - Radiation Era....... 222 C. Computational Methods :::::::::::::::::::::::: 223 C.1 GitHub Repository.......................... 223 C.2 Mathematica and xAct ........................ 223 C.3 Detailed Calculation Example.................... 225 C.3.1 Conformal Transformation and Perturbation of Gµν .... 225 C.3.2 Background Equations of Motion.............. 226 C.3.3 δGµν and δTµν Fluctuations................. 227 C.3.4 Field Equations........................ 229 C.3.5 Field Equations (Gauge Invariant Form).......... 229 C.3.6 Conservation......................... 230 C.3.7 Higher Derivative Relations................. 233 viii Chapter 1 Introduction In the theory of cosmological fluctuations, the perturbative equations of motion have historically been acknowledged as forming a quite difficult and complex set of coupled partial differential tensor equations. 1 As a result, extensive
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