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Building and Testing Models of Cosmic Inflation with Modified Gravity

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Building and Testing Models of Cosmic Inflation with Modified Gravity Building and testing models of cosmic inflation with modified gravity Chris Longden Submitted for the degree of Doctor of Philosophy School of Mathematics and Statistics August 11, 2018 supervised by Prof. Carsten van de Bruck Abstract Cosmic inflation, the idea that the very early universe underwent a dramatic accelerating expan- sion of space, has found great success in explaining aspects of the universe that were previously poorly understood. As a result, it has gained popularity and traction in the scientific main- stream in recent decades. However, it is still unclear exactly how inflation could have occurred; nothing in the established laws of physics can explain it. Now, in the modern era of precision cosmology, experimental data capable of probing and testing the details of this epoch has be- come available. With this, a deeper understanding of the physics of inflation may be possible, and it may prove to be the key to unlocking some of the greatest unsolved mysteries in theo- retical physics. In this thesis, models of beyond-standard-model physics, with a particular focus on those in- spired by modified theories of gravity (those that extend Einstein's theory of General Rela- tivity), are studied with the goal of understanding their inflationary consequences and hence establish how feasible these exotic theories are as descriptions of the early universe from this perspective. Additionally, some thought is given to present and future tests of inflation as well as how new data, or improvements in the presently available data, will increase cosmologists' ability to discriminate between different theories of inflation and hence move closer to answering the question of what caused it once and for all. Preface to the thesis This is a thesis consisting of research on the general theme of models of cosmic inflation and modifications of Einstein's theory of gravitation, general relativity. In an effort to be largely self-contained, this thesis is divided into two major parts. There will be a series of introductory chapters reviewing the background and key equations/results that will be used and referred to later in the subsequent series of chapters presenting my original research. To give an overview of the chapters comprising the thesis, we have: Background Review • 1. Introduction 2. Gravity 3. Cosmology Original Research • 4. Disformally Coupled Inflation 5. Inflation and the Gauss-Bonnet term 6. Testing Inflation with the Running of the Running of the Spectral Index This will be followed with some brief concluding remarks. 7. Conclusions The content presented in this thesis is based on the original work of the author except in a few specific cases where results produced by collaborators are used. In particular, it is noted that the derivation of equation (5.3.17) was done by Charlotte Owen of Lancaster University, as was the application of our results described in Chapter5 to obtaining the data in Table 5.1. Acknowledgements I first give thanks to the scientists and fellow students I have collaborated with on research projects, as well as those with whom I have enjoyably discussed my work throughout my postgraduate studies. My interactions with Konstantinos Dimopoulos, Tomi Koivisto, Jurgen Mifsud, Charlotte Owen, Laura Paduraru and Mat Robinson have helped shape my growth as a researcher, as a physicist, and as a mathematician over the past four years. On a similar note, I thank all the lecturers and teachers who provided my formal education leading up to this, as well as my parents for providing the environment which allowed me the freedom to pursue and dedicate myself to my educational and intellectual goals, from childhood to adulthood. To Marysia, I give my thanks for helping me find the mental strength needed to come this far, and for endlessly and unconditionally1 supporting me with sound advice. Finally, it is an understatement to say that these acknowledgements would not be complete without a mention of Carsten who I have been privileged to have as a supervisor and friend. On both a personal and professional level, I don't think I could have asked for anything better. Whether discussing the intricacies of cosmological perturbation theory, or simply talking about life in general, I have enjoyed and benefited from all our meetings and conversations, and as a result I have emerged from my postgraduate studies as a more capable, accomplished and confident person. Truly, thank you. 1At worst, subject to the occasional bribe with chocolate. \Much human ingenuity has gone into finding the ultimate Before. The current state of knowl- edge can be summarized thus: In the beginning, there was nothing, which exploded. " - Sir Terry Pratchett, Lords and Ladies, 1992 CONTENTS 1. Introduction1 1.1. Conventions and Notation........................7 2. Gravitation8 2.1. General Relativity............................9 2.2. Modified Gravity............................. 16 2.2.1. f(R) theories........................... 17 2.2.2. Scalar-tensor theories....................... 18 2.2.3. Conformal and disformal transformations............ 19 2.2.4. Further Modifications of GR................... 22 2.3. String Theory............................... 24 2.3.1. Supersymmetry and Superstrings................ 27 2.3.2. Open strings, and D-branes................... 29 3. Cosmology 31 3.1. The Expanding Universe......................... 31 3.1.1. The Cosmological Principle................... 31 3.1.2. Cosmology, Relativity and Gravitation............. 32 3.1.3. Curvature and Density of the Universe............. 36 3.1.4. Realistic, Multi-Fluid Models.................. 38 3.1.5. Cosmological Constant and Dark Energy............ 40 Contents 3.1.6. History of the Universe and Thermodynamics......... 45 3.2. Inflation.................................. 48 3.2.1. Problems with the Hot Big Bang................ 48 3.2.2. Inflationary Cosmology...................... 51 3.2.3. Resolving the HBB Problems: Accelerating Expansion.... 51 3.2.4. How to Inflate a Universe.................... 54 3.2.5. How to Stop a Universe Inflating................ 62 3.3. Cosmological Perturbations and Inflation................ 76 3.3.1. Perturbation Theory....................... 77 3.3.2. Inflationary Perturbations and the Primordial Power Spectrum 80 3.3.3. Quantisation........................... 83 3.3.4. Power Spectrum of Curvature Perturbations.......... 84 3.3.5. Testing Inflation with the Primordial Power Spectrum.... 86 3.3.6. Primordial Spectra in Slow-Roll Inflation............ 87 3.3.7. Observable modes......................... 89 3.3.8. Extensions............................. 91 4. Disformally Coupled Inflation 98 4.1. The model................................. 98 4.2. Inflationary Dynamics.......................... 100 4.2.1. Background cosmology...................... 101 4.2.2. A short detour: the kinetic structure of multi-field theories.. 105 4.2.3. Kinetic structure of disformally coupled inflation: sound speeds108 4.2.4. Results............................... 110 4.3. Perturbations............................... 117 4.3.1. Results............................... 119 4.4. Non-Gaussianity............................. 129 4.4.1. Relating non-Gaussianity of fields to non-Gaussianity of cur- vature............................... 130 4.4.2. Three point functions of the fields................ 132 4.4.3. Total Non-Gaussianity...................... 137 Contents 4.4.4. Results and Discussion...................... 138 5. Inflation and the Gauss-Bonnet term 142 5.1. Dynamics of the Gauss-Bonnet-coupled inflaton............ 142 5.1.1. Gauss-Bonnet and Conformal Transformations......... 144 5.1.2. Slow-roll power spectra...................... 147 5.2. Effects on the end of inflation and reheating.............. 152 5.2.1. Late time behaviour of the Gauss-Bonnet coupled inflaton.. 153 5.2.2. Implications for reheating.................... 158 5.2.3. Minimal extension of model................... 159 5.2.4. Effects on spectra from non-standard reheating........ 165 5.3. Towards Quintessential Inflation with the Gauss-Bonnet term.... 168 5.3.1. Inflation.............................. 170 5.3.2. Late time behaviour and reheating............... 172 5.3.3. Post-reheating evolution of field................. 175 5.3.4. Producing dark energy...................... 179 5.3.5. Constraints from cosmology................... 181 5.3.6. Tests of modified gravity as further constraints......... 184 6. Testing inflation with the running of the running of the spectral index 186 6.1. Motivations to study the Running of the Running........... 187 6.1.1. Forecasts for the Runnings.................... 189 6.2. Spectral runnings in minimal single-field inflation........... 190 6.2.1. Slow-roll inflation with a sound speed.............. 191 6.3. Multi-field scenarios and Runnings................... 192 6.3.1. Analysis and Limiting Cases................... 195 6.3.2. Particular Models......................... 197 6.4. Alternatives to Inflationary Cosmology................. 202 6.4.1. VSL and Runnings........................ 202 6.4.2. Quantum Gravity and Runnings................. 203 6.4.3. Alternative Parametrisation of the Power Spectrum...... 205 Contents 7. Conclusions 208 A. Appendices 211 A.1. Useful derivatives of the P function of Disformally Coupled Inflation. 211 A.1.1. Second order symmetrised derivatives.............. 211 A.1.2. Third order symmetrised derivatives.............. 213 A.2. Perturbation Coefficients in Disformally Coupled Inflation...... 215 A.2.1. Coefficients in the Perturbed Einstein Equations........ 215 A.2.2. Coefficients in
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