Imperial College London Department of Theoretical Physics Numerical Calculation of Inflationary Non-Gaussianities Jonathan Horner September 14, 2015 Supervised by Prof. Carlo Contaldi Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Theoretical Physics of Imperial College London and the Diploma of Imperial College London 1 Declaration All the work presented in this thesis is my own original work unless refer- enced otherwise. Specifically chapters 5, 6 and 7 are based on papers [1{3] written entirely by me with the collaboration of Carlo Contaldi. Chapter 8 is based on a paper mostly written by Carlo Contaldi in collaboration with myself [4]. The central focus of that paper is a program written completely by myself for the work described in chapters 5, 6 and 7. The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 2 Acknowledgements Throughout my PhD I have received help and advice from countless people. Firstly, and most importantly, I would like to thank my supervisor Carlo Contaldi for taking me on as a student, initiating the project in the first place and his unlimited patience and support over the last few years. I would like to thank all my fellow students for making my PhD so en- joyable and providing countless hours of physics debates. In particular Ali Mozaffari and Dan Thomas for all the advice they gave me over the last few years. I would like to thank my family for all their support and without them I would never have gotten this far. And last but not least I would like to thank my wife, Suna, for all her moral support and being there for me during stressful times. 3 Abstract In this thesis the numerical calculation of non-Gaussianity from inflation is discussed. Despite a strong interest in non-Gaussianity from inflation models in recent years, not much attention has been devoted to its numerical computation. Calculating the inflationary bispectrum in an efficient and accurate manner will become more important as observational constraints on primordial non-Gaussianity continue to increase. Despite this, attention given to numerically calculating the primordial bispectrum has been relatively low. The approach presented here differs from previous approaches in that the Hubble Slow-Roll (HSR) parameters are treated as the fundamental parameters. This allows one to calculate the bispectra for a variety of scales and shapes in the out-of-slow-roll regime and makes the calculation ideally suited for Monte-Carlo sampling of the bispectrum. The work is further extended to include potentials with features and non- canonical kinetic terms, where the standard squeezed limit consistency re- lation is demonstrated even for models which produce large fNL in the equilateral limit. The method presented here is also independent of the standard field redefinition used in analytic calculations, removing the need for delicate cancellations in the super-horizon limit used in other numerical methods. 4 Contents 1. A Brief History of Nearly Everything 11 2. Review of basic cosmology ideas 16 2.1. FRW metric . 16 2.2. The Stress-Energy Tensor . 18 2.3. The Einstein Equation . 20 2.4. Problems with the cosmological model . 23 2.4.1. The Flatness Problem . 24 2.4.2. The Horizon Problem . 25 2.4.3. The Monopole Problem . 27 3. Inflation as the solution of the cosmological problems 29 3.1. Accelerated Expansion . 29 3.2. A scalar field . 31 3.3. The Power Spectrum . 35 3.3.1. ADM formalism and the Scalar Power Spectrum . 35 3.3.2. Tensor Power Spectrum . 43 3.4. Formalising slow-roll . 45 3.4.1. Correspondence with a potential and some analytical solutions . 47 3.4.2. Anisotropies from the Primordial Power Spectrum . 52 4. What is non-Gaussianity? 54 4.1. A probe of inflationary models . 55 4.1.1. Defining fNL ....................... 57 4.2. The Bispectrum from Inflation . 61 4.2.1. The In-In Formalism . 62 4.2.2. The Third Order Action . 64 4.2.3. The tree-level calculation . 70 5 5. Non-Gaussian signatures of general inflationary trajectories 74 5.1. Introduction . 74 5.2. Hamilton Jacobi approach to inflationary trajectories . 78 5.2.1. Monte Carlo generation of HJ trajectories . 81 5.3. Computational method . 82 5.3.1. Computation of the power spectrum . 82 5.3.2. Computation of the bispectrum . 86 5.4. Results . 93 5.5. Discussion . 102 6. BICEP's Bispectrum 103 6.1. Introduction . 103 6.2. Computation of the scalar power spectrum . 104 6.3. Computation of the bispectrum . 107 6.4. Results . 109 6.5. Discussion . 110 7. Sound-Speed Non-Gaussianity 114 7.1. Introduction . 114 7.2. Monte-Carlo approach to sampling trajectories . 116 7.3. Computational method . 118 7.3.1. Computation of the power spectrum . 118 7.3.2. Computation of the bispectrum . 125 7.4. Results . 133 7.4.1. Shape dependence . 133 7.4.2. cs dependence . 134 7.4.3. Monte Carlo Plots . 134 7.5. Discussion . 137 8. Planck and WMAP constraints on generalised Hubble flow inflationary trajectories 138 8.1. Introduction . 138 8.2. Hubble flow equations . 140 8.2.1. Hubble flow measure . 143 8.2.2. Potential reconstruction . 144 8.3. Calculation of observables . 144 8.3.1. Power spectrum . 144 6 8.3.2. Non-Gaussianity . 148 8.4. Constraints on Hubble Flow trajectories . 150 8.4.1. Base parameters . 150 8.4.2. Derived parameters . 156 8.4.3. Inflaton potential . 158 8.5. Discussion . 159 9. Conclusion 162 A. Appendix 164 A.1. More details for perturbing the ADM formalism . 164 7 List of Figures 3.1. Typical time dependence of ζ................... 43 5.1. Typical time dependence of fNL. 75 5.2. Convergence of fNL with respect to sub-horizon start times. 76 5.3. Dependence of fNL with respect to integration split point X. 95 5.4. Shape and scale dependence of fNL. 97 5.5. n2 vs r scatter plot. 98 5.6. ns vs fNL scatter plot. 100 5.7. fNL histogram for different ensembles. 101 6.1. and η for potentials with a feature. 107 6.2. P (k); ns(k); r(k) for potentials with a feature. 108 6.3. fNL(k) for a potential with a feature. 112 6.4. Shape and scale dependence of fNL(k) for a potential with a feature. 113 7.1. Dependence of fNL on the damping factor δ when n = 1. 120 7.2. Dependence of fNL on the damping factor δ when n = 3. 121 7.3. Dependence of fNL on δ for different shapes and sound speed. 124 7.4. Shape and scale dependence of fNL for cs 6= 1. 126 7.5. cs dependence of fNL. 129 7.6. fNL Monte-Carlo plots comparing small sound speeds with cs = 1 for different shapes. 132 7.7. fNL Monte-Carlo plots for very small sound speed cs 1. 136 8.1. Evolution of typical trajectories used in Monte-Carlo sampling.142 8.2. Hubble flow proposal densities projected into the space of derived parameters ns, r, and fNL. 145 8.3. Comparison of 1d marginalised posteriors in the overlapping parameters between the reference PLANCKr run and the Hubble flow case. 151 8 8.4. 1d marginalised posteriors for the Hubble flow parameters. 152 8.5. The 2d marginalised posterior for ξ and η. 153 8.6. The 2d marginalised posterior for ξ and η at reference scale k?.154 8.7. The 2d marginalised posterior for ns and r. 156 8.8. The 2d marginalised posterior for ns and fNLat the picot scale k?...............................157 8.9. Sample of the best-fitting primordial curvature power spectra. 158 8.10. Sample of the best fitting potentials V ('). 161 9 List of Tables 8.1. Uniform MCMC priors for cosmological parameters and a short description of each parameter. 146 8.2. Parameter constraints from the marginalised posteriors for both Hubble flow `max = 2 and PLANCKr runs. 155 10 1. A Brief History of Nearly Everything Compound interest is the most powerful force in the universe. { Albert Einstein Cosmology is the study of the history and structure of the universe. From the first moments of the Big Bang, throughout its 13.6 billion year history most of it can be explained by well understood and tested physics. Most of it, except for the first fractions of a second and it is these earliest moments that will be the subject of this thesis. Looking at the visible universe one of it's most striking features is it's large scale isotropy, that the universe looks roughly the same in all direc- tions. As we have no reason to believe we're in a special place either, we naturally conclude that the universe appears isotropic to observers in other galaxies too. This isotropy and homogeneity only holds for scales larger than about 100 Mpc [5, 6]. On scales smaller than this, such as the size of individual galaxies and planets, the universe is obviously not homogeneous and isotropic. Clearly, attempting to explain all the objects in the universe on the small- est scales is outside the bounds of reality. Therefore Cosmology tends to focus on only the largest scales in the universe (those greater 100 Mpc) and many of the non-linearities can be neglected.