Precision Predictions of Primordial Power Spectra
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PRECISION PREDICTIONS OF PRIMORDIAL POWER SPECTRA By DANIEL BROOKER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2018 © 2018 Daniel Brooker I dedicate this to my beautiful wife Elizabeth without whom I would be hopelessly lost. ACKNOWLEDGMENTS This work would not have been possible without my wonderful advisor Richard Woodard as well as our incredible collaborators Nick Tsamis and Sergei Odintsov. In the long course of research that went into this thesis and the works which comprise it we profited at various times from conversations with: P. K. S. Dunsby, L. Patino, M. Romania, S. Shandera, M. Sloth, J. Garcia-Bellido, and M. Sasaki. This work was supported by NSF grant PHY1506513, by a travel grant from the UF International Center, by the UF’s McLaughlin fellowship, and by the UF’s Institute for Fundamental Theory. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................... 4 LIST OF TABLES ...................................... 7 LIST OF FIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1 INTRODUCTION, MOTIVATION, AND BACKGROUND .............. 11 1.1 Introduction ................................... 11 1.2 Cosmic Preliminaries ............................... 11 1.2.1 Basic Facts and Definitions ....................... 11 1.2.2 The Need for Inflation .......................... 13 1.2.2.1 The horizon problem ..................... 13 1.2.2.2 The flatness and monopole problems ............. 13 1.3 An Easy Way To Produce Inflation ....................... 14 1.4 Generic Predictions Of Inflation ......................... 17 1.4.1 Spontaneous Particle Production During Inflation ............ 17 1.4.2 Quantum Fields and Mode Equations .................. 18 1.4.3 The Constant Epsilon Solution ..................... 21 1.4.4 How Valid is Constant ϵ(t) as an Approximation? ............ 23 2 MODE FUNCTIONS AND SPECTRA ........................ 27 2.1 Our Evolution Equation ............................. 27 2.1.1 An Evolution Equation for the Norms Squared of u(t; k) and v(t; k) .. 27 2.1.2 Factoring Out the Constant ϵ Part ................... 29 2.1.3 Simplifications .............................. 30 2.2 Simple Analytic Approximations ......................... 36 2.3 Improvements to Our Approximation for the Scalar Power Spectrum ...... 40 2.4 Concluding Remarks on the Power Spectra ................... 42 3 RECONSTRUCTION ................................. 44 3.1 Developing The Reconstruction Proceedure ................... 44 3.2 Using The Reconstruction Procedure on Novel Spectra ............. 51 4 IMPROVING THE SINGLE SCALAR CONSISTENCY RELATION .......... 54 2 2 4.1 Constructing ∆h(k) from ∆R(k) ........................ 55 4.2 Comparison Using Simulated Data ....................... 60 4.3 Discussion .................................... 63 5 5 PRECISION PREDICTIONS FOR PRIMORDIAL POWER SPECTRA FROM F (R) MODELS OF INFLATION ............................... 65 5.1 Introduction ................................... 65 5.2 Numerical Equality but Form Dependence ................... 67 5.2.1 The Model in the Jordan Frame ..................... 67 5.2.2 The Model in the Einstein Frame .................... 70 5.2.3 Relating Backgrounds and Perturbation Fields ............. 72 5.2.4 Starobinsky Inflation ........................... 74 5.3 Constructing Models from Power Spectra .................... 76 5.3.1 Reconstructing a Scalar Potential Model ................ 76 5.3.2 Reconstructing an f(R) Model ..................... 78 5.4 Comparing Analytic and Numerical Results ................... 80 2 2 5.4.1 How We Compute ∆R(k) and ∆h(k) .................. 80 5.4.2 The Two Models ............................. 83 5.4.3 Power Spectra of the Two Models .................... 84 5.4.4 Comparison with Analytic Results .................... 85 5.5 Discussion .................................... 87 6 OUTLOOK AND CONCLUSIONS .......................... 89 REFERENCES ........................................ 95 BIOGRAPHICAL SKETCH ................................. 103 6 LIST OF TABLES Table page 1-1 Simple solutions to the Friedmann equation ..................... 12 4-1 Simulated Data .................................... 61 4-2 Predicted Results ................................... 62 7 LIST OF FIGURES Figure page 2ϵ 1 2 2 1 ϵ − 1−ϵ 1-1 Graph of π (1+ϵ) Γ ( 2 + 1−ϵ )[2(1 ϵ)] as a function of ϵ. ............. 23 1-2 The PLANCK 2015 strength of temperature variations against their angular sizes. 24 2-1 Graphs of the local slow roll correction factor C(ϵ) versus a better approximation. 35 2-2 The ϵ = 0 Green’s function and the coefficient of "00(n) in the small ϵ form for Sh(n; k). ........................................ 37 0 2 0 2-3 The coefficients of [" (n)] and " (n) in the small ϵ form for Sh(n; k). ........ 37 2-4 The Hubble parameter and the first slow roll parameter for a model with features. 39 2-5 The scalar power spectrum compared with the local slow roll approximation and the scalar power spectrum compared with our analytic approximation. ........ 40 2-6 Our approximation for the scalar power spectrum of the step model with the non-linear and extra source terms included. ........................... 42 2-7 The tensor power spectrum for the step model compared with the local slow roll approximation. ..................................... 43 3-1 Numerical values of exponents 1, 2 and 4 for the step model. ............ 45 3-2 Numerical values of exp3(n) and exp5(n) for the step model. ............ 46 3-3 Various choices for the left hand side of the first pass reconstruction equation for the step model. .................................... 46 3-4 Numerical reconstructions of ln[ϵ(n)] for the power spectrum of Figure 2-5. ..... 48 3-5 Numerical evaluation of the integral I(s) and various approximations. ........ 49 2 3-6 The spectrum ∆R whose geometry we will reconstruct. ............... 51 3-7 The values of ln [ϵ(n)] which we have reconstructed. ................. 52 3-8 The spectrum from the reconstructed geometry of Figure 3-7 compared with the mock spectrum we started with. ............................ 52 4-1 The first slow roll parameter for the step model, the resulting scalar power spectrum and the result of our analytic approximation (4–14). ................. 57 5-1 Comparison of first slow roll parameter in the two frames for Starobinsky inflation. 76 5-2 Comparison of the potentials V (') for Starobinsky inflation and the exponential model. ......................................... 82 8 5-3 Comparison of the scalar power spectrum for Starobinsky inflation and the exponential model. ......................................... 84 5-4 Comparison of the tensor power spectrum and the tensor-to-scalar ratio r(k) for Starobinsky inflation and the exponential model. ................... 85 5-5 The various spectra for Starobinsky inflation in the Jordan frame and the Einstein frame. ......................................... 85 5-6 The various spectra for Starobinsky inflation in the Jordan frame and the Einstein frame. ......................................... 85 5-7 The various spectra for the exponential model in the Jordan frame and the Einstein frame. ......................................... 86 5-8 The various spectra for the exponential model in the Jordan frame and the Einstein frame. ......................................... 86 5-9 Comparison of the exact results with the leading slow roll approximation for Starobinsky inflation. ........................................ 86 5-10 Fractional error of our linearized approximation to the scalar and tensor power spectra for Starobinsky inflation. ................................ 87 9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRECISION PREDICTIONS OF PRIMORDIAL POWER SPECTRA By Daniel Brooker August 2018 Chair: Richard Woodard Major: Physics In this work we develop a powerful new formalism for computing the mode functions of scalars and tensors both numerically and analytically in an arbitrary inflating geometry. The formalism is used to derive simple, accurate, and analytic expressions for the primordial power spectra from a given model of inflation as well as for reconstructing an inflationary model from primordial data. Our formalism is used to derive an improved version of the single scalar consistency relation allowing us to test inflationary scenarios with sparse tensor data. Finally, our work is extended to include f(R) modified gravity models of inflation. 10 CHAPTER 1 INTRODUCTION, MOTIVATION, AND BACKGROUND 1.1 Introduction The theory of inflation, simply put, predicts an era of rapid growth in the expansion ofthe early universe. This theory was first proposed as a means of solving three problems plaguing big-bang models of cosmology at the time: the flatness problem, the horizon problem, and the lack of monopoles. Inflation has since been supported experimentally and has become apart of the ‘standard model’ of cosmology [1–15]. In spite of being now over 35 years old however, there remain problems with the lack of more detailed predictions and the vast multiplicity of models which fit the current data for inflationary cosmology. In this work we will attempt to resolve some of these issues by developing new analytic schemes for better understanding the predictions of a