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Open Elliotnelson-Dissertation.Pdf The Pennsylvania State University The Graduate School The Eberly College of Science NON-GAUSSIAN STATISTICS AS A PROBE OF THE EARLY UNIVERSE A Dissertation in Physics by Elliot Luke Nelson © 2015 Elliot Luke Nelson Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2015 The dissertation of Elliot Luke Nelson was reviewed and approved∗ by the following: Sarah Shandera Assistant Professor of Physics Dissertation Advisor, Chair of Committee Martin Bojowald Professor of Physics Donghui Jeong Assistant Professor of Astronomy & Astrophysics Eugenio Bianchi Assistant Professor of Physics Nitin Samarth Professor of Physics Head of the Department of Physics ∗Signatures are on file in the Graduate School. ii Abstract We study effects from mode coupling or non-Gaussianity in the primordial curvature perturbations. For local-type non-Gaussianity, which couples fluctuations on very different scales, we study the influence of long-wavelength background modes on the statistical properties of short-scale modes in a finite volume such as the observable universe. We show that background modes can introduce a shift or bias to observed parameters, 2 and quantify this influence for the primordial power spectrum ∆ζ , non-Gaussianity parameter fNL, spectral index ns, and dark matter halo power spectrum. We show that a non-Gaussian field with sufficiently strong coupling of modes on long and short scales will appear nearly Gaussian in typical subvolumes that are sufficiently small, indicating the naturalness of the weakly non-Gaussian local ansatz. In light of these results, we discuss the implications of observing the universe in a finite volume for our understanding of inflation. In addition to studying the statistical properties of curvature perturbations on a fixed spatial slice at the end of inflation, we also consider an example of mode coupling arising from nonlinear inflationary dynamics. A non-vacuum initial state at the onset of inflation can allow scalar curvature perturbations to couple strongly to long-wavelength tensor perturbations, resulting in an anisotropic contribution to the power spectrum of density perturbations. iii Table of Contents List of Figures vii List of Symbols ix Acknowledgments xii Chapter 1 Introduction1 1.1 Overview of Cosmic History...................... 1 1.2 Primordial Perturbations........................ 3 1.3 Cosmological Collider Physics..................... 8 1.4 Cosmic Variance ............................ 8 1.5 Observing a Finite Volume....................... 9 1.6 Super Cosmic Variance......................... 10 1.7 Non-Gaussianity ............................ 12 1.8 Statistical Formalism for Primordial Perturbations.......... 13 1.8.1 Gaussian Statistics....................... 14 1.8.2 Non-Gaussian Statistics .................... 16 1.8.3 Local-type non-Gaussianity .................. 17 1.9 Long Wavelength Modes in Cosmology................ 19 1.9.1 When Are Infrared Modes Relevant? ............. 19 1.9.2 Long Wavelength Backgrounds in Inflationary Models . 20 1.9.3 Long-Wavelength Modes in Large Scale Structure . 22 1.9.4 Lyth’s Study of ζ(x) in a Subvolume............. 23 1.9.5 Recent Work on Super Cosmic Variance ........... 24 1.9.6 Primordial Gravitational Waves and Non-Vacuum Initial States 25 1.10 Outline of Chapters2-4......................... 25 Chapter 2 Super Cosmic Variance from Local Non-Gaussianity 29 iv 2.1 Introduction............................... 29 2.2 Statistics of ζ in a Subvolume with Local Non-Gaussianity . 32 2.2.1 Long and Short Wavelength Split............... 33 2.2.2 The Long Wavelength Background .............. 36 2.2.3 Spatial Curvature from Long-Wavelength Modes . 38 2.3 Example I: Weakly Non-Gaussian Initial Conditions......... 39 2.3.1 Biasing of Parameters with Local Non-Gaussianity . 39 2.3.2 Implications for Model Builder................. 43 2.4 Example II: Strongly Non-Gaussian Initial Conditions . 44 2.5 Example III: Two-field Initial Conditions............... 49 2.5.1 Case 1: σ Is Weakly Non-Gaussian .............. 50 2.5.2 Case 2: σ Is Strongly Non-Gaussian.............. 52 2.6 General Local-type Non-Gaussianity ................. 55 2.6.1 Running of Non-Gaussian Parameters............. 56 2.7 Behavior of N-Point Function Shapes. ................ 57 2.8 Conclusions............................... 60 Chapter 3 Super Cosmic Variance with Scale-Dependence 64 3.1 Introduction............................... 64 3.2 Subsampling the local ansatz with scale-dependence in single- and multi-source scenarios.......................... 66 3.2.1 The power spectrum ...................... 66 3.2.2 The bispectrum and the level of non-Gaussianity . 69 3.3 Observational consequences ...................... 72 3.3.1 The shift to the power spectrum................ 73 3.3.2 The shift to the spectral index, ∆ns . 74 3.3.3 The shift to the scale dependence of the bispectrum . 88 3.3.4 Generalized local ansatz and single source vs. multi source effects .............................. 89 3.4 Mode coupling effects from a non-local factorizable bispectrum . 90 3.5 Tensor mode running as a test of inflation?.............. 93 3.6 Discussion and conclusions....................... 95 Chapter 4 Fossilized Gravitational Wave Relic from Squeezed Limit Coupling 98 4.1 Introduction............................... 98 4.2 A generalized initial state and the gravitational fossil . 100 4.2.1 Scalar-scalar-tensor bispectrum . 101 v 4.2.2 Enhanced fossil signature in locally anisotropic scalar fluctu- ations ..............................105 4.3 General condition for scalar-scalar-fossil correlation . 112 4.4 Quadrupolar anisotropy and superhorizon modes . 113 4.5 Conclusion................................115 Chapter 5 Discussion of Results 118 5.1 Summary of Thesis Work........................118 5.2 Super Cosmic Variance and Inflation . 119 5.2.1 Inflationary Trajectories for Finite Volumes . 120 5.2.2 Implications for Inflation....................121 5.2.3 Statistical Naturalness for Inflationary Trajectories . 123 5.2.4 Statistical and Inflationary Landscapes . 124 5.3 Conclusion and Outlook ........................125 Appendix A Diagrammatic Representations of N-point Functions 127 A.1 Feynman-like Rules for Momentum Space Diagrams . 131 Appendix B Behavior of Local-type N-point Functions 134 Appendix C Mapping Between Statistics in Vl and Statistics in Vs 141 Bibliography 144 vi List of Figures 1.1 A gaussian random field......................... 6 1.2 The causal past of the observable universe............... 10 1.3 Correlation of long and short scales for a non-Gaussian random field. 11 2 2.1 The variance hζGli of the long-wavelength background. 37 2.2 Probability distribution for the global power spectrum, conditioned on an observed power spectrum..................... 41 2.3 Probability distribution for the global fNL, conditioned on an ob- obs served fNL . ............................... 42 2.4 Variation between the global and observed levels of non-Gaussianity, fNL∆ζ ................................... 45 obs 2.5 Dependence of observed parameter fNL on superhorizon scales and global spectral index........................... 46 obs 2.6 Probability distribution for the global power spectrum and fNL , conditioned on observed parameters, with two sources for curvature perturbations and weak non-Gaussianity................ 52 obs 2.7 Probability distribution for the global power spectrum and fNL , conditioned on observed parameters, with two sources for curvature perturbations and strong non-Gaussianity .............. 54 2 3.1 Parameter space (fNL, hζGli) of scale-invariant local non-Gaussianity with linear and quadratic terms..................... 78 2 3.2 Parameter space (fNL(kpiv), hζGli) for local non-Gaussianity with scale-dependent power and fNL(k). .................. 81 3.3 Probability distribution for the shift or bias to the locally observed spectral index, in terms of the amplitude of the long-wavelength background, global fNL(kpiv) on observable scales, and running of fNL(k)................................... 84 2 2 obs 3.4 Global power spectrum ∆ζ (k) and locally observed spectrum (∆ζ (k)) , with a shifted spectral index on observable scales........... 85 vii 2 3.5 Parameter space (fNL(kpiv), hσGli) for scale-dependent local non- Gaussianity with two sources φG and σNG............... 87 4.1 Contour plot for survey size needed to detect primordial gravitational waves, in terms of tensor power spectrum and parameter β for non- vacuum initial state. ..........................109 4.2 Contour plot for survey size needed to detect primordial gravitational waves from scalar-scalar-tensor bispectrum with stronger-than-local squeezed limit, in terms of tensor power spectrum and bispectrum amplitude.................................114 B.1 Correction factor for the power spectrum Pζ (k) and other polyspec- tra in local non-Gaussianity, from a nonzero fNL. 139 viii List of Symbols Used primarily in Chapters1–3: ζ(x) The (non-Gaussian) curvature perturbation. ζ(k), ζk A Fourier mode of the curvature perturbation. Pζ (k) The power spectrum of ζ. 2 ∆ζ (k) The dimensionless power spectrum or variance of fluctu- 2 3 2 ations for ζ, ∆ζ (k) ≡ k Pζ (k)/(2π ). ζG A Gaussian random field used to generate ζ. PG(k) The power spectrum of ζG. 2 ∆G(k) The dimensionless power spectrum for ζG. 2 ns The scalar spectral index ns − 1 = d ln ∆ /d ln k. Bζ (k1, k2, k3) Bispectrum for curvature perturbations. Ωk, H0 The spatial curvature and Hubble scale today. Vl The volume over which ζNG is defined, e.g. the
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