THE UNIVERSITY OF CHICAGO ON THE IMPRINTS OF INFLATION IN THE COSMIC MICROWAVE BACKGROUND A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS BY CORA DVORKIN CHICAGO, ILLINOIS AUGUST 2011 UMI Number: 3472839 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3472839 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 c 2011 by Cora Dvorkin Copyright All Rights Reserved To my parents, Elena and Eduardo, and my sister Julia. TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Homogeneous Universe . . . . . . . . . . . . . . . . . . . . . 1.3 The Cosmic Microwave Background . . . . . . . . . . . . . . . . . 1.3.1 Cosmic Microwave Background Temperature Anisotropies 1.3.2 Cosmic Microwave Background Polarization Anisotropies . 1.4 The Inflationary Era . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Big Bang theory is incomplete . . . . . . . . . . . . . 1.4.2 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Cosmological Perturbations . . . . . . . . . . . . . . . . . 1.4.4 Slow-Roll Inflation . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Connecting Theory with Observations . . . . . . . . . . . 2 CMB POLARIZATION FEATURES FROM INFLATION TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Inflationary Features . . . . . . . . . . . . . . . . . . . 2.2.1 Inflationary Model . . . . . . . . . . . . . . . . 2.2.2 CMB Power Spectra . . . . . . . . . . . . . . . 2.3 Confirming Features with Polarization . . . . . . . . . 2.3.1 Fiducial Polarization Significance . . . . . . . . 2.3.2 Potential Parameters . . . . . . . . . . . . . . . 2.3.3 Tensors . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Temperature Conditioning . . . . . . . . . . . . 2.4 Reionization Features . . . . . . . . . . . . . . . . . . . 2.4.1 Reionization Principal Components . . . . . . . 2.4.2 Data and Model Optimization . . . . . . . . . . 2.4.3 Reionization Confusion . . . . . . . . . . . . . . 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 3 GENERALIZED SLOW ROLL FOR LARGE POWER 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.2 Generalized Slow Roll . . . . . . . . . . . . . . . 3.2.1 Exact Relations . . . . . . . . . . . . . . . 3.2.2 GSR for Small Deviations . . . . . . . . . iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VERSUS REIONIZA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SPECTRUM FEATURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 6 7 9 14 14 15 17 18 20 21 22 24 24 28 33 33 35 39 40 44 45 46 49 52 55 55 56 58 62 3.3 3.4 3.2.3 GSR for Large Deviations 3.2.4 Power Spectrum Features 3.2.5 Iterative GSR Correction . Applications . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 69 71 76 78 4 CMB CONSTRAINTS ON PRINCIPAL COMPONENTS OF THE INFLATON POTENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Generalized Slow Roll variant used . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Principal components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.1 Basis Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.2 Principal Component Basis . . . . . . . . . . . . . . . . . . . . . . . 86 4.3.3 MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.4 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.1 WMAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4.2 Joint Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 COMPLETE WMAP CONSTRAINTS ON BANDLIMITED INFLATIONARY FEATURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3 MCMC Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.3.1 All Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.3.2 Robustness Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.4.1 Testing Slow Roll and Single Field Inflation . . . . . . . . . . . . . . 120 5.4.2 Constraining Inflationary Models . . . . . . . . . . . . . . . . . . . . 122 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A RELATION TO PRIOR WORK ON STEP POTENTIALS . . . . . . . . . . . . 143 B OTHER GENERALIZED SLOW ROLL VARIANTS . . . . . . . . . . . . . . . . 145 C FAST WMAP LIKELIHOOD EVALUATION . . . . . . . . . . . . . . . . . . . . 152 D MCMC OPTIMIZATION FOR MANY ADDITIONAL PARAMETERS . . . . . 156 D.1 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 D.2 Likelihood corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 v E GENERALIZED SLOW ROLL ACCURACY . . . . . . . . . . . . . . . . . . . . 162 vi LIST OF FIGURES 1.1 Temperature fluctuations in the Cosmic Microwave Background. . . . . . . . . . 7 1.2 Temperature power spectrum measured by WMAP 7-year. . . . . . . . . . . . . 9 1.3 Linear polarization generated by Thomson scattering of radiation with a quadrupole anisotropy off a free electron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Examples of E- and B-mode patterns of polarization. . . . . . . . . . . . . . . . 11 1.5 Predicted E- and B-mode polarization power spectra. . . . . . . . . . . . . . . . 13 2.1 Inflationary potential with a step (upper panel) and slow-roll parameters (middle and lower panels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Primordial curvature power spectra of a model with a step in the inflationary potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Temperature and E-mode polarization transfer functions. . . . . . . . . . . . . . 2.4 Temperature and polarization power spectra of a model with a step in the inflationary potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Temperature and polarization transfer functions for multipoles near the temperature dip (ℓ = 20) and bump (ℓ = 40). . . . . . . . . . . . . . . . . . . . . . . . 2.6 Temperature likelihood contour plot for height vs. width of a step in the inflationary potential with WMAP 5-year data. . . . . . . . . . . . . . . . . . . . . . 2.7 E-mode polarization likelihood contour plot for height vs. width of a step in the inflationary potential for Planck (lower panel) and a cosmic variance limited experiment (upper panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Primordial curvature power spectra for models illustrating projection degeneracies in the temperature for a step in the inflationary potential. . . . . . . . . . . 2.9 Temperature power spectrum relative differences of two step-models with equal temperature amplitude at ℓ ∼ 20 relative to a smooth model (upper panel). Polarization power spectrum relative differences of the same models (lower panel). 2.10 Effect of tensor fluctuations on polarization power spectra for a model with a feature in the curvature power spectrum. . . . . . . . . . . . . . . . . . . . . . . 2.11 E-mode power spectrum constrained to the temperature data for a smooth ∆2R (k) model along with the band representing sample variance per ℓ for the ideal experiment (upper panel). Fractional difference between the average of the constrained realizations and the full ensemble average for both models (lower panel). . . . . 2.12 False positive example of the two-step process to account for reionization uncertainty in polarization significance for an ideal experiment. . . . . . . . . . . . . 2.13 Test of false positives due to reionization for Planck and for an idealized experiment limited by sample variance. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Test of false negatives due to reionization for Planck and for an idealized experiment limited by sample variance. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Upper panel: inflationary potential with a step. Lower panel: conformal time to the end of inflation as a function of the value of the field. . . . . . . . . . . . . . 3.2 Slow-roll parameters ǫH , ηH and δ2 for a step-potential. . . . . . . . . . . . . . 3.3 Source functions for the deviations from slow roll used in the different generalized slow roll (GSR) approximations considered. . . . . . . . . . . . . . . . . . . . . vii 26 28 29 30 32 35 36 37 38 40 43 48 50 51 57 58 60 3.4 Ratio of field solution y of the Mukhanov equation to the scale invariant approximation y0 . Real part in upper panel and imaginary part in lower panel. . . . . 3.5 GSRS approximation to the curvature power spectrum compared to the exact solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Curvature evolution after horizon crossing in the GSRS (upper panel) and GSRL (lower panel) approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 GSRL approximation to the curvature power spectrum. . . . . . . . . . . . . . . 3.8 Fractional difference between |y0 | and |y| for the maximum likelihood model at k values at the dip, node and bump of the feature in the power spectrum. . . . . . 3.9 Contribution of the real part of the y field to the curvature power spectrum. . . 3.10 Fractional difference between the exact (y) and nth order iterative solutions (yn ) for the maximum likelihood step potential model. . . . . . . . . . . . . . . . . . 3.11 Curvature power spectrum in the GSRS approximation when y → yn in the GSRS source compared to the exact solution. . . . . . . . . . . . . . . . . . . . . . . . 3.12 Second order GSRL2 power spectrum correction functions I12 and I2 . . . . . . . 3.13 GSRL2 approximation to the curvature power spectrum. . . . . . . . . . . . . . 3.14 GSRL2 approximation to the CMB temperature power spectrum. . . . . . . . . 3.15 GSRL2 approximation to the CMB E-mode polarization power spectrum. . . . 3.16 Alternate inflationary model with a perturbation in the mass. . . . . . . . . . . 3.17 GSRL2 approximation to the alternate model of Fig. 3.16. . . . . . . . . . . . . 63 65 66 70 71 72 73 74 75 77 78 79 79 80 4.1 The first 5 principal components (PC) of the source function G′ as a function of conformal time based on the WMAP7 specifications. . . . . . . . . . . . . . . . 87 4.2 Predicted RMS error on the PC amplitudes as a function of mode number for WMAP7 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.3 Sensitivity of the curvature power spectrum to the first 5 PC parameters. . . . . 88 4.4 Sensitivity of the temperature power spectrum to the first 5 PC parameters. . . 88 4.5 Predicted RMS errors on running of tilt as a function of the maximum number of PC components included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 Posterior probability distributions of the cosmological and 5 PC parameters using WMAP7 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.7 The 5 PC filtered source function G′5 posterior using WMAP7 data. . . . . . . . 94 4.8 Joint probability distributions of the principal component amplitudes and the cosmological parameters from an MCMC analysis of WMAP7 data. . . . . . . . 95 4.9 Power spectra of the 5 PCs maximum likelihood model compared to power law maximum likelihood model. Top panel: temperature power spectrum. Bottom panel: polarization power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.10 Decomposition of the fractional difference between the PC and power law maximum likelihood models into contributions from specific parameters. . . . . . . . 98 4.11 The 5 PC filtered G′5 posterior using WMAP7 data and additional SN, H0 and BBN constraints in a flat universe. . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.12 Principal component amplitudes for the step function potential model that best fits the glitches in the temperature spectrum at ℓ ∼ 20 − 40 (upper panel), and projected cumulative signal-to-noise (lower panel). . . . . . . . . . . . . . . . . . 104 viii 5.1 The first 20 principal components of the GSR source as a function of conformal time to the end of inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Sensitivity of the nonlinearity parameter I1,max to the amplitude of the first 20 principal components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Constraints on the 20 principal √ component amplitudes from the all-data analysis with a prior of I1,max < 1/ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 Parameter probability distributions from the all-data analysis in a flat universe √ with I1,max = 1/ 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.5 The temperature and E-mode polarization power spectra posterior using the alldata PC constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.6 Parameter decomposition of the temperature power spectrum difference between the power law and PC maximum likelihood models. . . . . . . . . . . . . . . . . 117 5.7 Comparison of the maximum likelihood √ models of the three MCMCs analyzed: the all-data analysis with I1,max = 1/ 2, all-data with I1,max = 1/2, and CMB √ data with I1,max = 1/ 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.8 The temperature quadrupole power C2T T distribution for the all-data analysis √ with I1,max = 1/ 2, all-data with I1,max = 1/2, and CMB data with I1,max = √ 1/ 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.9 The 20 PC filtered source function G′20 posterior from the fiducial all-data analysis.122 5.10 A model with a linear deviation in the source function compared to G′20 . . . . . 124 5.11 Posterior probability distribution of α from a direct MCMC analysis constructed from 5, 20 and 50 PCs compared to the distribution using the χ2 approximation. 125 5.12 Initial curvature power spectrum of a model with running of the tilt compared to a model with a linear deviation in the source function. . . . . . . . . . . . . . . 126 5.13 Upper panel: Constraints on the step potential model parameters height and width using the χ2 approximation compared to the 20 PC posterior. Bottom panel: constraints from the 20 PCs posterior compared to a direct GSR calculation of the step model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.14 The maximum likelihood model of the step potential from the χ2 approximation compared to the maximum likelihood model from the projection onto 20 PCs. . 128 5.15 Top panel: step potential model with a small width represented by the full source function compared to its 20 PC description. Bottom panel: fractional difference between the full GSR description and its 20 PC decomposition. . . . . . . . . . 129 A.1 Observed T T spectrum and best-fit feature models for WMAP3 and WMAP5 data.143 B.1 Curvature power spectrum for the ML and 3ML models. . . . . . . . . . . . . . 146 B.2 Fractional error in the curvature power spectrum for first order GSR variants for the ML model (lower panel) and the 3ML model (upper panel). . . . . . . . . . 147 B.3 Temperature power spectrum for the extreme case of a step with height c = 8cML .148 B.4 Fractional error in the curvature power spectrum for second order GSR variants. 149 C.1 Comparison of the low-ℓ polarization pixel likelihood and the approximate fit as a function of E-mode polarization amplitude in two multipole bands ℓ=(4-6) and ℓ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 ix C.2 Posterior probability distribution of the optical depth τ and the fourth PC amplitude using the exact likelihood and the approximation with WMAP data only. 153 D.1 Power law parameter posteriors from the approximations used to run the MCMC chain, from an independent MCMC with no approximation, from the approximate chain with importance sampling correction, and from the approximate chain without lensing correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 D.2 The m18 probability distributions √ from the approximations used to run the MCMC with all data and I1,max = 1/ 2, from the approximate chain with importance sampling correction, and from the approximate chain without lensing correction (m18 has the largest correction of the PC amplitudes). . . . . . . . . . . . . . . 161 E.1 Fractional difference in temperature power spectra between GSR and the exact inflationary solution for the maximum likelihood model from the all-data analysis, √ and a model that saturates the prior I1,max = 1/ 2. . . . . . . . . . . . . . . . 163 E.2 Likelihood difference between the GSR solution and the full inflationary calculation of a series of step potential models as a function of I1,max . . . . . . . . . . 163 x LIST OF TABLES 2.1 Fiducial parameters of a model with a step in the inflationary potential chosen to maximize WMAP 5-year likelihood. . . . . . . . . . . . . . . . . . . . . . . . 2.2 Parameters used when making forecasts for idealized and Planck-like experiments. 2.3 Likelihood difference in E-mode polarization for false positive and false negative tests comparing models with a smooth power spectrum and a power spectrum with a feature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 E-mode polarization likelihood difference for false positive and false negative tests comparing models with smooth ∆2R (k) and a feature in ∆2R (k), with polarization either unconstrained or constrained to the observed temperature data. . . . . . 2.5 E-mode polarization likelihood difference for tests of false positives and false negatives with ionization histories of the data and model tuned at 6 < z < 50. . 27 34 34 44 45 4.1 Means, standard deviations and maximum likelihood values for ΛCDM and the 5 PCs model with WMAP7 data. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Means and standard deviations of the posterior probabilities of the PC amplitudes with different data sets added to the WMAP7 data. . . . . . . . . . . . . . . . . 100 5.1 Power law parameter results: means, standard deviations and maximum likelihood values with CMB data (WMAP7 + BICEP + QUAD) and all data (+UNION2 +H0 + BBN) in a flat universe. . . . . . . . . . . . . . . . . . . . . 111 5.2 20 principal component parameter results: means, standard deviations and √ maximum likelihood values for the all data analysis with priors I1,max = 1/ 2 and I1,max = 1/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3 20 principal component parameter results: means, standard √ deviations and maximum likelihood values for CMB data set and I1,max = 1/ 2. . . . . . . . . . . 113 xi ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor Wayne Hu for his permanent advice and encouragement over the past years. Wayne’s imagination, enthusiasm for science and deep insight are a source of inspiration for me. He has always pushed me further and urged me to aim high, and I am greatly indebted to him. I would also like to thank Bruce Winstein, who passed away before I finished this thesis. Bruce’s enthusiasm for physics was contagious. From physics discussions to Antonioni movies, Bruce became an academic mentor and a friend, and I will always conserve the fondest memories of him. I wish to give a special thank to Hiranya Peiris, who guided me in my first project in Chicago, and devoted much time to share her vast knowledge of cosmology with me. We collaborated over the years, and I have valued her friendship ever since. I have benefited immensely from talking and working with Kendrick Smith. I am thankful for many outstanding inputs from him during my time at KICP and for his tireless enthusiasm. I would like to thank the members of my thesis committee, Wayne Hu, John Carlstrom, Steve Meyer, and Jeff Harvey for their suggestions, and insightful questions on this manuscript, in particular I thank Wayne again, for going carefully through the thesis and helping me to improve it. I am most grateful with Sasha Belikov, who has been my companion throughout all these years. He has supported me through good and bad times. His integrity and curiosity for science and life in general continue to inspire me. I am very lucky to have been able to share my time in Chicago with him. Thanks to Colin Bischoff, my former officemate, for a daily dose of entertaining conversations, and for always having an answer to each of my infinite IDL questions. Thanks to my current officemates Immanuel Buder, Sam Leitner, and Denis Erkal, for hundreds of conversations about physics and life. xii Along the way, I have met and had insightful conversations with many people. In particular, I would like to thank Dragan Huterer, Chris Gordon, Amol Upadhye, Mark Wyman, Peter Adshead, and Akito Kusaka. A special thank you goes to Michael Mortonson for many interesting insights. Thanks to Nobuko McNeill, who helped me out on all the administrative aspects of my life at the University of Chicago and beyond. I also thank the KICP system administrator, Valeri Galtsev, who has always helped me with computing problems, at any time, any day of the week. I wish to express special thanks to my friends who have enriched my personal life in Chicago, and without whom these years would have not been the same, in particular: Sophia Domokos, Carmen Varela, Ali Brizius, Nico Busca, Mario Camuyrano, Agustin Casas, and Maria Beltran. Thanks to my friends in Argentina, los domingueros, who have accompanied me for the last 17 years, even when we were thousands of kilometers away. I am grateful to the system of public education in Argentina, in particular to Universidad de Buenos Aires, on which my background stands. In addition, I thank my family in Argentina: my grandmothers Raquel and Aida, and my aunt Patricia. I thank my sister Julia for being always there for me, and for being such a good friend. Finally, I cannot thank my parents, Elena and Eduardo, enough for having been such an invaluable source of support, encouragement and advice. From my early stages in life, they have taught me the value of a good education. Certainly, nothing would have been possible without them. I owe them everything. xiii ABSTRACT A major question in cosmology is what sourced the curvature perturbations that grew into the large-scale structure of the universe that we observe today. Under the assumption that cosmological perturbations were generated from quantum fluctuations during inflation, features in the Cosmic Microwave Background (CMB) temperature and polarization power spectra constrain features in the inflationary potential. Currently our best constraints on the shape of the primordial power spectrum at large scales come from observations of the CMB anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. Oscillatory features in the CMB temperature power spectrum have been interpreted as possible evidence for new physics during inflation. It has been shown that a model with a sharp step in the inflationary potential can give rise to these oscillations. In the first part of the thesis, we show that upcoming polarization measurements provide fertile ground for consistency checks on inflationary models proposed to explain these features. As predictions of specific models of inflation, polarization statistics move beyond a posteriori inferences. In the second part of the thesis, we propose an accurate prescription to map constraints from the CMB onto constraints on the shape of the inflationary potential in a model independent manner, allowing for order unity deviations in the slow-roll parameters. In this formalism, there is a single source function that is responsible for the observable features and it is simply related to the local slope and curvature of the inflaton potential. In the final part, we use this formalism to test the hypotheses of single-field and slow-roll inflation. This analysis greatly simplifies the testing of inflationary models in that it can be used to constrain parameters of specific models of inflation without requiring a separate likelihood analysis for each choice. Our results show that there is no significant evidence for deviations from slow roll across the entire range of scales observable to WMAP. As a test of single-field inflation, we present predictions for the polarization power spectrum. Single field inflation makes falsifiable predictions for the acoustic peaks in the polarization, whose xiv violation would require extra degrees of freedom. xv CHAPTER 1 INTRODUCTION 1.1 Thesis Overview The focus of this thesis is to shed light on the physics of inflation using observations of the Cosmic Microwave Background. This first chapter sets the stage for the discussion in the following chapters. We begin discussing the homogeneous background, and we then review aspects of the Cosmic Microwave Background (CMB), in particular the temperature and polarization fluctuations. Later we present puzzles that remain unexplained in the standard Big Bang context, and we show a mechanism proposed to solve these problems: Inflation. We review key aspects of the inflationary solution, and analyze its observational consequences. In Chapter 2, we discuss how Cosmic Microwave Background polarization measurements can be used to constrain specific models of inflation. In particular, we analyze a model with a step in the inflationary potential proposed to explain a dip at ℓ ∼ 20 and a bump at ℓ ∼ 40 observed in the CMB temperature power spectrum. It is debatable whether this feature is a signature of primordial physics or merely a statistical anomaly. The detection of this feature is only marginally significant from temperature data alone. We show that the inflationary feature hypothesis predicts glitches in the E-mode polarization power spectrum with a structure similar to that in the temperature power spectrum. Therefore, measurement of the CMB polarization can be used as a consistency check of this hypothesis. We show that the Planck satellite has the statistical sensitivity to confirm or rule out the model that best fits the temperature features at ℓ ∼ 20 − 40 at 3σ significance, and a cosmic variance limited experiment [7] can improve this significance to 8σ. We also quantify possible sources of degradation of this significance and we find that the main source of confusion with inflationary features at these scales comes from polarization features created by a complex reionization history. 1 In Chapter 3 we introduce a variant of the Generalized Slow Roll (GSR) approximation for calculating the curvature power spectrum. Our approach allows for order unity deviations in power caused by sharp features in the inflaton potential. As an example, we show that predictions for the step potential considered in Chapter 2 are accurate at the percent level. Our analysis shows that to good approximation there is a single source function in the initial curvature power spectrum that is responsible for observable features, and that this function is simply related to the local slope and curvature of the inflaton potential. We use these properties in Chapters 4 and 5 to construct a general method that relates the CMB observables to the shape of the inflationary potential and we apply it to the data. In Chapter 4, we apply the GSR formalism to the best constrained region of WMAP 7-year data (i.e., the first acoustic peak) by means of a principal component (PC) decomposition of the source function, and use it to impose functional constraints on the shape of the inflaton potential. We do a Markov Chain Monte Carlo likelihood analysis keeping only those modes measured to better than 10%. The analysis results in 5 nearly independent Gaussian constraints. Detection of any non-zero component would represent a violation of ordinary slow roll and indicate a feature in the inflaton potential or sound speed. One component (that resembles a local running of the tilt) shows a 95% CL preference for deviations around the 300 Mpc scale at the ∼ 10% level, but the global significance is reduced considering the 5 components examined. This deviation also requires a change in the cold dark matter density which in a flat ΛCDM model is disfavored by current supernova and Hubble constant data. We show that the inflaton potential can be even better constrained with current and upcoming high sensitivity experiments that will measure small-scale temperature and polarization power spectra of the CMB. For this analysis, we have implemented a ∼40× faster WMAP7 likelihood code which we have made publicly available1 . In Chapter 5 we extend this analysis to constrain the inflationary potential across the entire range of angular scales observable to WMAP. We use a complete basis of 20 principal 1. http://background.uchicago.edu/wmap fast 2 components that accounts for order unity features in the slow roll parameters as fine as 1/10 of a decade. Although one component shows a deviation at the 98% CL, it cannot be considered statistically significant given the 20 components tested. The maximum likelihood PC parameters only improves 2∆ ln L by 17 for the 20 parameters associated with known glitches in the WMAP power spectrum at large scales. We make model-independent predictions for the matching glitches in the polarization power spectrum that could soon test their inflationary origin with high resolution ground based experiments and the Planck satellite. Even allowing for the presence of features in the temperature spectrum, single field inflation makes sharp falsifiable predictions for the acoustic peaks in the polarization, whose violation would require extra degrees of freedom. This complete analysis for bandlimited features in the source function of generalized slow roll can be used to constrain parameters of specific models of the inflaton potential without requiring a separate likelihood analysis for each choice. We illustrate its use by placing bounds on the height and width of the step potential introduced in Chapter 2. In Chapter 6 we discuss our results. 1.2 The Homogeneous Universe Under the assumption that the universe is spatially homogeneous and isotropic on large scales, the metric that describes the spacetime of the universe is the Friedmann-RobertsonWalker (FRW) metric: dS 2 = −dt2 + a2 (t) dr 2 2 2 2 2 + r (dθ + sin θdφ ) , 1 − kr 2 (1.1) where the coordinates (r,θ,φ) are comoving, and the parameter k describes the spatial curvature: k is 0 for flat universes (Euclidean universe, where free particles remain parallel), negative for open universes (in which initially parallel particles diverge in their trajectories), and positive for closed universes (in which initially parallel particles converge in their 3 trajectory). In 1929 Hubble found that galaxies are receding from us with a velocity proportional to their distance. This was very strong evidence to believe that the universe is expanding. In other words, the physical distance between two fixed points on a grid is increasing with time. It is convenient to describe this effect by introducing the scale factor a(t), which is defined to be equal to 1 at present times, and whose value is smaller at earlier times. Since the universe is expanding, the wavelength of light emitted by a receding object is stretched out so that the observed wavelength is larger than the emitted one. We will call this factor redshift z and define it in the following way [19]: 1+z ≡ 1 λobs = λemit a (1.2) The Hubble rate (H = ∂t ln a), that measures how rapidly the scale factor changes with time, is determined by the Friedmann equation: H 2 (t) = 8πG X k ρi (t) − 2 , 3 a (t) (1.3) i where the sum runs over all the components of the universe. This equation arises from solving the time-time part of the Einstein’s equations (Gµν = 8πGTµν ) for a FRW universe filled with a perfect fluid, Tµν = diag(−ρ, p, p, p). The evolution of the scale factor depends on the energy density in the universe, and different types of species have different evolutions of their energy density. Evidence from distant supernovae suggests that there is another kind of energy besides ordinary matter and radiation, and it is known as dark energy. Riess et al. (1998) [96] and Perlmutter et al. (1999) [94] measured the apparent magnitudes of Type Ia supernovae, and they inferred their luminosity distances. They saw that the luminosity distance was larger than expected, especially for objects at large redshift, and they used this fact to impose constraints on dark energy. In the simplest models, dark energy remains constant with time, 4 acting as a cosmological constant. According to current measurement, there is approximately 74% of dark energy in the universe. There is also strong evidence for nonbaryonic matter in the universe. This kind of matter is inferred to exist based on gravitational effects on visible matter and gravitational lensing of the CMB, but it does not emit or scatter electromagnetic radiation. We can derive the evolution of the energy density of each species by combining Eq. (1.3) with the space-space component of the Einstein equations in a flat universe: Ḣ + H 2 = ä 4πG =− (ρ + 3p) , a 3 (1.4) We can then write the continuity equation: ρ̇ = −3H (ρ + p) , (1.5) where overdots denote derivative with respect to time, and integrate it to get the following equation that defines the energy density evolution for each component of the universe: ρi ∝ a−3(1+wi ) (1.6) Here wi defines the equation of state of each component, wi = pi /ρi . Radiation satisfies w = 13 , non-relativistic matter has w = 0, and cosmological constant, w = −1. For each species we can define the ratio of its energy density to the critical energy density 3H 2 today in a flat universe ρcrit = 8πG0 : Ωi ≡ ρi ρcrit (1.7) We can then re-express the Friedmann equation in the following way: Ω H 2 X = Ωi a−3(1+wi ) + K H0 a2 i 5 (1.8) where ΩK = −k/(H0 )2 and P i Ωi + ΩK = 1. In the “concordance” model of cosmology, the main components are dark energy (Λ), which has been constrained to ΩΛ ≈ 0.74, “cold” (or non-relativistic) dark mater (CDM) with ΩCDM ≈ 0.22, and baryonic matter with Ωb ≈ 0.04 [63]. 1.3 The Cosmic Microwave Background Penzias and Wilson first detected the Cosmic Microwave Background in 1965 using a horn antenna at the Crawford Hill Laboratory in New Jersey. This discovery gave strong evidence that the universe comes from an early hot and dense state. However, they only took measurements at a single frequency (in the microwave region of the electromagnetic spectrum, at 4.08 GHz). Even stronger evidence for the cosmological origin of the microwave background radiation can be found from its spectrum. If the signal originated from a plasma of protons and electrons in equilibrium with photons, then we should expect to see a black body spectrum. The Far Infrared Absolute Spectrometer (FIRAS) instrument on board of the Cosmic Microwave Background Explorer (COBE) satellite measured the CMB frequency spectrum [75], with a spectrum being an excellent match to a blackbody at T = (2.725 ± 0.002)K. Data from the Cosmic Microwave Background (CMB) [53] (see Fig. 1.1) have further enabled us to understand the physics of the earliest times of the universe and of its subsequent evolution. Today we know that the universe begun as a hot and dense plasma of photons, electrons, protons and dark matter particles in thermal equilibrium. This plasma expands and cools, and when the temperature is sufficiently low (T. 3000K), protons and electrons recombine into hydrogen in a period known as “recombination”. The small density inhomogeneities grow by gravitational interactions to form the large scale structures of the universe that we observe today. The clustering becomes stronger during the period of matter domination. The smallest scales become non linear first and they form gravitationally bound objects. Later, the small scale structures (stars and galaxies) 6 Figure 1.1: Full sky temperature map from the WMAP 7-year data release [53]. The statistical properties of these fluctuations contain information about the background evolution and the initial conditions of the universe. merge into larger structures (cluster of galaxies). Around redshift of z ∼ 10 − 20 [78], the first stars and quasars reionize the universe in a period known as “reionization” and about 10% of the photons re-scatter. 13.7 billion years after the time of “last scattering” we observe these photons and we use these observations to infer the physics of the early universe. 1.3.1 Cosmic Microwave Background Temperature Anisotropies The CMB spectrum is a nearly uniform blackbody with inhomogeneities of order 10−5 which correspond to hot and cold spots in temperature. These variations in temperature were first detected by COBE (Smoot et al., 1992 [105]) and since then, many ground and balloon based experiments and the the Wilkinson Microwave Anisotropy Probe (WMAP) satellite have refined the measurements of the CMB. At the time of writing, the WMAP satellite has finished observations, and the Planck satellite has been taking data for over one year. The spectrum of the anisotropies is measured as a function of angular scale in the sky. Let us denote ∆T (n̂) the CMB temperature relative to the mean temperature T = 2.725 K, at a direction n̂ in the sky. We can decompose the temperature field into spherical harmonics on the sky in the 7 following way: ∆T (n̂) = X aTℓm Yℓm (n̂), (1.9) ℓm where each value of ℓ corresponds to a particular angular scale. Under the assumption of statistical isotropy, for a given ℓ, each aTℓm has the same variance given by: TT haTℓm a∗T ℓ′ m′ i = δℓℓ′ δmm′ Cℓ (1.10) and CℓT T defines the temperature power spectrum. If the temperature field has a Gaussian distribution, then all the information about its statistical properties is contained in the power spectrum (the N-point correlation functions can be determined from the 2-point correlation function using Wick’s theorem). Note, however, that because there are only 2ℓ + 1 modes for each value of ℓ, there is a fundamental uncertainty in the knowledge that we can get about the Cℓ ’s. This uncertainty is called “cosmic variance” and is given by: ∆Cℓ Cℓ = r 2 2ℓ + 1 (1.11) Fig. 1.2 shows the theoretical prediction for the CMB power spectrum (red curve) and the temperature power spectrum measured by the WMAP satellite in its 7th year (black dots) [66]. As we will see later in this thesis, the theoretical curve depends both on the cosmological parameters and on the spectrum of initial fluctuations. The structure of the CMB angular power spectrum can be understood from the physics of the plasma in the early universe. Gravity pulls in matter and increases overdense regions. This process is opposed by photon pressure. These two forces create conditions for oscillations in the initial density perturbations. At the time of recombination, these oscillations cease. At this point, each mode had a particular length of time to evolve, based on the time when it entered the causal horizon. The acoustic peaks in the power spectrum correspond to wavelengths that undergo an integer number of oscillations before recombination. For 8 Figure 1.2: Temperature power spectrum reported in the WMAP 7-year data release (binned in ℓ) [66]. The red curve is the ΛCDM best fit to the data. The error bars include contributions from cosmic variance and instrumental noise. example, the first acoustic peak (at ℓ ≈ 200) corresponds to modes that had just enough time to collapse to their maximum point of compression, resulting in larger fluctuations in power. The second peak (at ℓ ≈ 500) corresponds to modes that collapsed, expanded, and reached a point of maximum rarefaction at recombination, and so on. 1.3.2 Cosmic Microwave Background Polarization Anisotropies In addition to the temperature anisotropies, the Cosmic Microwave Background radiation is also polarized. The power of the polarization spectrum is about two orders of magnitude smaller than the temperature spectrum. The polarization spectrum was first detected by the Degree Angular Scale Interferometer (DASI) [65] in 2002. Polarization is generated by Thomson scattering of photons off free electrons during recombination and reionization. Suppose we have a photon travelling with wavevector parallel to the x̂ axis. Its transverse electric field makes an electron oscillate in the ŷ and ẑ axis. There will be radiation scattered with wavevector along these directions, generating a linear polarization. However, photons are incident from all directions, so the polarization averages to zero. If instead the 9 temperature has a quadrupolar pattern, a net linear polarization can occur. In this case, the two perpendicular components of the scattered light have different temperature (hot and cold, see Fig. 1.3). The polarization spectrum is smaller than the temperature spectrum by two orders of magnitude. This is due to the fact that polarization is generated by quadrupole moments, and the quadrupole is suppressed due to Compton scattering. Figure 1.3: Thomson scattering of radiation with a quadrupole anisotropy off a free electron generates linear polarization. Higher temperature photons are shown in red and lower temperature photons are shown in blue. Figure adapted from Hu&White (1997) [49]. Linear polarization can be described by the Stokes parameters Q and U, and by the analogue of the temperature multipole moments Tℓm : the Eℓm and Bℓm multipole moments, (Q ± iU)(n̂) = − X ℓm (Eℓm ± iBℓm ) ±2 Yℓm (n̂), 10 (1.12) Figure 1.4: Examples of E-mode and B-mode patterns of polarization. The quantities Q and U transform under rotation by an angle ψ as a spin-2 field: (Q ± iU) (n̂) → e∓2iψ (Q ± iU) (n̂) (1.13) Therefore, the harmonic analysis of Q ± iU requires an expansion on the sphere in terms of spin-2 spherical harmonics ±2 Yℓm [49, 115, 58]. E and B modes are the tensor analogues of curl-free and divergence free components of a vector. An E-mode has a polarization direction that is aligned with or orthogonal to the direction that the mode amplitude changes (it is radial around cold spots and tangential around hot spots). A B-mode has this direction rotated by ±45◦ (see Fig. 1.4 for examples of E- and B-mode patterns). Analogously to Eq. (1.10), the E and B-mode power spectra are defined as: EE ∗E haE ℓm aℓ′ m′ i = δℓℓ′ δmm′ Cℓ (1.14) BB ∗B haB ℓm aℓ′ m′ i = δℓℓ′ δmm′ Cℓ (1.15) 11 Note that E- and B-modes behave differently under parity transformations. When reflected about a line going through the center, the E-mode pattern remains unchanged, while the B-mode pattern changes sign. More specifically, under a parity reversing operation n̂ → −n̂, the E- and B-mode components transform as ℓ E aE ℓm → (−1) aℓm ℓ+1 aB aB ℓm → (−1) ℓm (1.16) Therefore, for symmetry reasons: ∗B haE ℓm aℓ′ m′ i = 0 (1.17) When gravity waves source the anisotropy, both E- and B-modes are generated. On the other hand, when density perturbations source the anisotropy, only E-mode polarization is generated. Beyond linear theory, second order effects such as gravitational lensing [87, 44] and inhomogeneous reionization can produce B-modes [29, 27] from density fluctuations. The amplitude of the B-mode signal coming from gravity waves is a direct measure of the energy scale of inflation. Hence, a detection of B-modes would probe the physics of the earliest observable period in our universe. Tensor fluctuations are usually parametrized by r, the ratio of the tensor to the scalar mode power generated by inflation: r≡ ∆2t (k) , ∆2s (k) (1.18) where ∆2s is the amplitude of the scalar fluctuations, and it is measured to be ∆2s ∼ 10−9 . Since ∆2t ∝ H 2 ∝ V , the tensor-to-scalar ratio r is a direct measure of the energy scale of inflation [6]: 12 Figure 1.5: Predicted polarization power spectra. The B-mode power spectrum corresponds to a tensor-to-scalar ratio of r = 0.01. V 1/4 r 1/4 1016 GeV ∼ 0.01 (1.19) Values of the tensor-to-scalar ration of r ≥ 0.01 correspond to inflation occurring at the GUT scale. At the time of writing, there are only upper limits on the B-mode power spectrum, but no detection yet. The current best upper limit on r is r < 0.24 with 95% confidence, coming from WMAP measurement of the temperature power spectrum in combination with measurements of the Baryon Acoustic Oscillations (BAO) and a prior on the value of the Hubble constant [63]. The best constrain coming only from B-modes currently corresponds to the BICEP experiment with r ≥ 0.72 at 95% confidence [14]. On the other hand, E-modes have been already measured, and there are a number of experiments measuring its spectrum with even better accuracy. Fig. 1.5 shows theoretical curves for the predicted E- and B-mode power spectra (a value of r = 0.01 is shown here for illustrative purposes). 13 1.4 The Inflationary Era Inflation has become the main paradigm that explains why the universe is so homogeneous and flat. In addition of explaining the high level of homogeneity of the universe that we observe today, inflation also provides a mechanism for explaining the existence of the perturbations present in the Cosmic Microwave Background. 1.4.1 The Big Bang theory is incomplete The Big Bang model is incomplete in that there are puzzles that it cannot explain. We summarize its problems as follows: • Flatness Problem: According to observations, the universe is nearly spatially flat. Explaining the flatness of space today requires a high level of fine-tuning in a Big Bang cosmology. We can rewrite the Friedmann equation (1.3) as 1 − Ω(a) = −k (aH)2 (1.20) where note that Ω(a) is now defined to be time-dependent. In the standard Big Bang theory, the comoving Hubble radius (aH)−1 grows with time. This implies that |Ω − 1| must diverge with time, and therefore the near flatness observed today requires a high level of fine tuning of Ω being close to 1 in earlier times. • The Horizon Problem: The comoving horizon is the maximum distance a light ray can travel between time 0 and time t, which in a flat FRW spacetime is given by: Z a Z t dt′ d ln a′ (a′ H)−1 η= ′) = a(t 0 0 14 (1.21) In the standard Big Bang theory, the comoving Hubble radius (aH)−1 increases with time, therefore the comoving horizon increases with time as well. Regions that, according to the Big Bang theory, would be causally connected at the time of recombination correspond to an angle of 1◦ in the sky. However, the CMB has been measured to have nearly the same temperature in all directions on the sky. This implies that the universe was extremely homogeneous at the time of last scattering on scales encompassing many regions that should be causally independent. • Relic Problem: Magnetic monopoles are expected to be produced in Grand Unified Theories, and should have persisted to the present day. The absence of monopoles cannot be explained in the context of the Standard Big Bang theory. 1.4.2 Inflation These problems are solved by the assumption that the universe underwent a brief period of accelerated expansion. This period is known as inflation, and it was first proposed by Alan Guth in 1981 [37]. During inflation, the comoving Hubble radius decreases, which implies that the universe is expanding in an accelerated fashion, d(aH)−1 <0 dt d2 a >0 dt2 (1.22) (1.23) With this hypothesis, a flat universe is an attractor solution and the CMB sky was in causal contact in the past. Looking at the second Einstein’s equation (1.4), we see that the condition of being a period of accelerated expansion implies that inflation is sourced by a component with negative 15 pressure. In the simplest models of inflation this component involves a single scalar field (known as the inflaton field) with canonical kinetic term, whose Lagrangian is given by: 1 L = − g µν ∂µ φ∂ν φ − V (φ) 2 (1.24) Assuming that the scalar field acts like a perfect fluid, we can write its density and pressure as: 1 ρ = φ̇2 + V (φ) 2 1 2 p = φ̇ − V (φ) 2 (1.25) (1.26) The equation of motion of the inflaton field (the Klein-Gordon equation) and the dynamics of the Hubble parameter are determined by: dV φ̈ + 3H φ̇ + =0 dφ 1 1 2 2 H = φ̇ + V (φ) , 3 2 (1.27) where the expansion rate of the universe appears as a source of friction in the motion of the field. Here and throughout the thesis we choose units where the reduced Planck mass MPl = (8πG)−1/2 = 1. Accelerated expansion occurs when ǫH < 1, with: d ln H Ḣ ǫH ≡ − 2 = − , dN H (1.28) where dN = d ln a measures the number of e-folds of inflationary expansion. Therefore, the end of inflation (ä = 0) happens when the slow-roll parameter ǫH is equal to unity. 16 1.4.3 Cosmological Perturbations In the previous section, we have learned about the classical dynamics of the scalar field; however, the inflaton field has quantum fluctuations δφ(t, x) around the classical background evolution φ̄(t): φ(t, x) = φ̄(t) + δφ(t, x) (1.29) The quantum fluctuations of the inflaton field imply that different regions of space end inflation at different times. This local delay of the end of inflation caused by quantum fluctuations induces relative density fluctuations, which grow by gravitational instability to form the large-scale structure of the universe that we observe today. The most general first-order perturbation to a spatially flat FRW metric is: dS 2 = −(1 + 2Φ)dt2 + 2a(t)Bi dxi dt + a2 (t)[(1 − 2ψ)δij + Eij ]dxi dxj , (1.30) where Φ is a scalar called the lapse, Bi is a vector called the shift, ψ is a scalar called the spatial curvature perturbation, and Eij is a tensor which is symmetric and traceless. Geometrically, ψ measures the spatial curvature of constant time hypersurfaces, R(3) = 4∇2 ψ/a2 . At this point, it is useful to introduce a gauge-invariant quantity called the comoving curvature perturbation which is conserved outside the horizon: R = ψ + δφ H , φ̇ (1.31) The effective action during inflation for a scalar field with a canonical kinetic term is given by Z √ R 1 2 S= − + (∂φ) − V (φ) −gd4 x, 2 2 where g = det(gµν ) and R is the Ricci scalar. 17 (1.32) Let us define a gauge with spatially flat comoving hypersurfaces (ψ = 0), the spatiallyflat gauge. In this gauge, the comoving curvature perturbation is related to the inflaton fluctuations in the following way: R = δφ H φ̇ (1.33) Under linear perturbation theory, we can rewrite the action in this gauge as 1 S= 2 Z z ′′ 2 ′ 2 2 (u ) − (∂i u) + u dηd3 x, z (1.34) where primes denote derivatives with respect to conformal time η, z = aφ̇/H, u = zR, and u is known as the “Mukhanov potential” [80, 98]. Varying the action, we get the equation of motion of the k-modes of the inflaton field: 1 d2 z d2 u k 2 uk = 0, + k − z dη 2 dη 2 (1.35) and the curvature power spectrum can be written as ∆2R (k) = u 2 k3 k lim 2π 2 kη→0 z (1.36) 1.4.4 Slow-Roll Inflation The de Sitter limit p → −ρ corresponds to ǫH → 0, in which case the potential energy dominates over the kinetic energy. Accelerated expansion will be only sustained for a sufficiently long period of time if |φ̈| << |3H φ̇| |φ̈| << | dV | dφ 18 (1.37) (1.38) This requires that the second slow-roll parameter is small: ηH = − φ̈ 1 dǫH = ǫH − , 2ǫH dN H φ̇ (1.39) and, therefore, ǫH is approximately a constant. The slow-roll parameters are related to the shape of the potential in the following way: V,φ 2 V 2 V,φ,φ 1 Vφ ηH ≈ − , V 2 V 1 ǫH ≈ 2 (1.40) where ,φ denotes derivative with respect to φ. This implies that the slow-roll conditions ǫH , ηH << 1 are only valid for potentials that are sufficiently flat and slowly varying. However, as we will see in this thesis, the slow-roll parameters are not necessarily small or constant during the entire period of inflation. In fact, there are models of inflation in which the slow-roll condition is broken for an e-fold or less, leading to oscillatory features in the CMB temperature power spectrum. Under the approximation that ǫH and ηH are constant, one can analytically compute the curvature power spectrum [110]: ∆2R (k) = " 3 2ν− 2 2 Γ(ν) ν− 12 H (1 − ǫ ) H Γ( 32 ) 2π φ̇ #2 |k=aH , (1.41) where ν= 1 1 − ηH + ǫH + 1 − ǫH 2 (1.42) As we will see in Chapter 3, Eq. (1.41) is not always a valid approximation to the curvature power spectrum. In this thesis we will study a more general solution that allows for order unity deviations from the slow roll condition. 19 In the next section we will review the effect of the initial curvature fluctuations on the Cosmic Microwave Background observables, in particular on the CMB temperature and polarization power spectra. 1.4.5 Connecting Theory with Observations The CMB fluctuations depend on the initial curvature fluctuations R in the following way: aX ℓm = 4π(−i)ℓ Z d3 k X T (k)Rk Yℓm (k̂) (2π)3 ℓ (1.43) The CMB angular power spectra are therefore related to the curvature power spectrum as: ′ ℓ(ℓ + 1)CℓXX = 2π Z ′ d ln k TℓX (k)TℓX (k) ∆2R (k) (1.44) The transfer function TℓX (k) depends on the parameters of the background cosmology, while the curvature perturbation contains information about the inflationary epoch. Therefore, CMB measurements open the possibility of learning about the inflationary initial conditions. In the next chapter, we will constrain a particular model of inflation using CMB temperature and polarization power spectra. We choose to study a model with a sharp step in the inflationary potential proposed to explain features observed in the temperature spectrum and we will make predictions for the polarization field. 20 CHAPTER 2 CMB POLARIZATION FEATURES FROM INFLATION VERSUS REIONIZATION A model with a sharp step in the inflationary potential has been proposed to explain observed glitches in the CMB temperature power spectrum, in particular a dip (at ℓ ∼ 20) and bump (at ℓ ∼ 40). The detection of these features is only marginally significant from temperature data alone. In this chapter we show that the inflationary feature hypothesis predicts a specific shape for the E-mode polarization power spectrum with a structure similar to that observed in temperature at ℓ ∼ 20−40. Measurement of the CMB polarization on few-degree scales can therefore be used as a consistency check of this hypothesis. We show that the Planck satellite has the statistical sensitivity to confirm or rule out the model that best fits the temperature features with 3 σ significance, assuming all other parameters are known. With a cosmic variance limited experiment, this significance improves to 8 σ. For tests of inflationary models that can explain both the dip and bump in temperature, we show that the primary source of uncertainty is confusion with polarization features created by a complex reionization history, which at most reduces the significance to 2.5 σ for Planck and 5 − 6 σ for an ideal experiment. Smoothing of the polarization spectrum by a large tensor component only slightly reduces the ability of polarization to test for inflationary features, as does requiring that polarization is consistent with the observed temperature spectrum given the expected low level of T E correlation on few-degree scales. If polarized foregrounds can be adequately subtracted, Planck will supply valuable evidence for or against features in the primordial power spectrum. A future high-sensitivity polarization satellite would enable a decisive test of the feature hypothesis and provide complementary information about the shape of a possible step in the inflationary potential. 21 2.1 Introduction Our best constraints on the shape of the primordial power spectrum at large scales come from observations of the cosmic microwave background (CMB) anisotropy by the Wilkinson Microwave Anisotropy Probe (WMAP) [8, 43]. The WMAP 5-year data [86, 23] is well described by the simplest inflationary scenario of a single, slowly rolling, minimally coupled scalar field with a canonical kinetic term [62, 92, 60, 4, 70]. Since the 3-year release [107], the WMAP data have indicated a deviation from scale invariance — a red tilt of the scalar spectral index — the significance of which has been debated in the literature from a Bayesian model selection point of view (e.g. [89, 36]). Recent minimally-parametric reconstructions of the primordial power spectrum incorporating some form of penalty for “unnecessary” complexity [114, 9] show some evidence for a red tilt, but no evidence for scale dependence of the spectral index. These methods, as currently implemented, are not very sensitive to sharp, localized features in the primordial power spectrum. However, it has been pointed out ever since the original data release [106, 91] that there are several sharp glitches in the WMAP temperature (T T ) power spectrum. In particular, several model-independent reconstruction techniques that are sensitive to features localized in a narrow wavenumber range have consistently picked out a feature at ℓ ∼ 20−40 that leads to an improvement of ∆χ2 ∼ O(10) over a smooth power-law spectrum [41, 100, 81, 102, 84]. Power spectrum features could arise, in principle, in more general classes of inflationary models where slow roll is momentarily violated. Such an effect can be phenomenologically modeled as a discontinuity or singularity in the inflaton potential [108, 1, 54]. A “step-like” feature [1], in particular, would be a good effective field theory description of a symmetry breaking phase transition in a field coupled to the inflaton in multi-field models [103, 45, 95, 2, 50], which can arise in supergravity [69] or M-theory-inspired [13, 5] contexts. Several analyses have confronted such phenomenological descriptions of features in the inflationary potential with current data [91, 18, 40, 51, 55, 74, 59, 52]. It is debatable whether the large scale feature seen in the WMAP T T spectrum is a signal 22 of exotic primordial physics or merely a statistical anomaly. Currently, our information about the smoothness of the primordial power spectrum is dominated by the temperature data. However, future high fidelity CMB polarization measurements at large scales have the potential to shed light on this question. The importance of polarization data for constraining oscillatory features has been previously discussed in the literature (e.g. [48, 61, 88, 83, 84]) and exploited in particular as a cross-check of the observed low CMB temperature quadrupole [21, 104, 35, 32]. In this chapter, we propose to use the large-scale polarization of the CMB to test the hypothesis that the ℓ ∼ 20−40 glitch is due to a step in the inflaton potential. We exploit the fact that, in the relevant multipole range, the sharpness of the polarization transfer function and lack of contamination by secondary effects (assuming instantaneous reionization) makes polarization a cleaner probe of such features than temperature [48]. We also investigate how our conclusions are affected by relaxing the assumption of instantaneous reionization [47, 78, 77], changing the parameters of the feature, and including large-amplitude tensor fluctuations. Data coming from the Planck satellite [111] promises to greatly increase our knowledge of the large-scale polarization signal. It is also relevant for future dedicated CMB polarization missions [7]. As in Ref. [28], it is our objective to make a prediction for the polarization statistics that will be observed by future CMB experiments, given current temperature data. We present the inflationary model and the numerical procedure used to compute the primordial curvature power spectrum in § 2.2. The polarization consistency tests of the features, both for instantaneous and general reionization histories, are presented in § 2.3 and § 2.4, and we conclude in § 2.5. We discuss in Appendix A the relation of our work to previous analyses of features in the WMAP temperature data. 23 2.2 Inflationary Features We review the inflationary generation of features in the curvature power spectrum from steplike features in the inflaton potential in §2.2.1 and their transfer to the CMB temperature and polarization power spectra in §2.2.2. 2.2.1 Inflationary Model To model a feature in the primordial power spectrum that matches the glitches in the WMAP temperature data at ℓ ∼ 20 − 40, we adopt a phenomenological inflationary potential of the form V (φ) = m2eff (φ)φ2 /2 where the effective mass of the inflaton φ has a step at φ = b corresponding to the sudden change in mass during a phase transition [1]: m2eff (φ) = m2 φ−b 1 + c tanh d , (2.1) with the amplitude and width of the step determined by c and d respectively, assuming that both are positive numbers. We express the potential parameters m, b, and d in units of the reduced Planck mass, MPl = (8πG)−1/2 = 2.435 × 1018 GeV; the step amplitude c is dimensionless. In physically realistic models with a sufficiently small step in the potential, the interruption of slow roll as the field encounters the step does not end inflation but affects density perturbations through the generation of scale-dependent oscillations that eventually die away. The phenomenology of these oscillations is described in Ref. [1]: the sharper the step, the larger the amplitude and width of the “ringing” superimposed upon the underlying smooth power spectrum. Hence we shall see in § 2.3.2 that lowering d increases the width of the feature in ℓ in the CMB power spectra. Standard slow-roll based approaches are insufficient for computing the power spectrum for this potential, and instead the equation of motion must be integrated numerically modeby-mode [68]. 24 The dynamics of the Hubble parameter, described by the Friedmann equation, and the background dynamics of the unperturbed inflaton field, described by the Klein-Gordon equation, can be written respectively as 1 H ′ = − H(φ′ )2 , 2′ H 1 dV ′′ φ + + 3 φ′ + 2 = 0, H H dφ (2.2) (2.3) where ′ = d/d ln a. The solution of the mode equation depends on the background dynamics. With the help of these background equations, the mode equation (1.35) introduced in the previous chapter can be written as u′′k " ( H ′ φ′′ H′ k2 ′ − 2 − 4 + + 1 uk + H H φ′ a2 H 2 #) ′ 2 H′ 1 d2 V H uk = 0 , −5 − 2 2 −2 H H H dφ (2.4) where the term in square brackets is z̈/(za2 H 2 ). We set that the initial conditions for the two orthogonal solutions that contribute to uk well within the horizon and we check that these are free of contamination due to any transient contribution to the background dynamics. The power spectrum is then obtained by continuing the integration until the mode freezes out far outside the horizon, yielding the asymptotic value of |uk /z|. Further details regarding the numerical solution of the coupled system of differential equations can be found in Ref. [1]. To match a given mode to a physical wavenumber k, one must make an assumption about the reheating temperature, but this choice is degenerate with b, corresponding to a translation of the step in φ. To compare our results with those of Refs. [18, 40], we adopt the following prescription for the matching: k⋆ ≡ a⋆ H⋆ = aend e−N⋆ H⋆ , 25 (2.5) Figure 2.1: Upper panel, solid black : Inflationary potential with a step (2.1). The parameters for the potential are chosen to maximize the WMAP5 likelihood and are listed in Table 2.1. The dashed red line shows a smooth m2 φ2 potential (c = 0) with m = 7.120 × 10−6 so that the two models have equal power on small scales (φ ≪ b). Middle and lower panels: slow-roll parameters ǫV and ηV for the two inflationary potentials. where H⋆ is the Hubble scale corresponding to the physical wavenumber k⋆ , which left the horizon N⋆ e-folds before the end of inflation, defined by d2 a/dt2 (aend ) = 0. Following the above authors, we set the pivot scale k⋆ = 0.05 Mpc−1 to correspond to N⋆ = 50 (although there are differences in the implementation of the k-mode matching that we discuss in Appendix A). Figure 2.1 shows our fiducial inflationary potential, with parameters given in Table 2.1 that are chosen to fit the WMAP5 temperature glitches at ℓ ∼ 20 − 40 as we will show in the next section. The number of e-folds of inflation after the step in this potential is Nstep ≈ 54. 26 Parameter m b c d N⋆ Ωb h2 Ωc h2 h τ Value 7.126 × 10−6 14.668 1.505 × 10−3 0.02705 50 0.02238 0.1081 0.724 0.089 Table 2.1: Fiducial feature model parameters chosen to best fit WMAP5 under a flat ΛCDM cosmology, compared in the text with a smooth model with c = 0, m = 7.120 × 10−6 , and the same cosmological parameters, which matches the small scale normalization As (k⋆ ) = 2.137 × 10−9 and tilt ns ≈ 0.96 at the pivot k⋆ = 0.05 Mpc−1 . The slow-roll parameters M2 ǫV = Pl 2 dV /dφ 2 , V 2 ηV = MPl d2 V /dφ2 V (2.6) are plotted in the lower panels of Fig. 2.1. Note that near the step at φ = b, |ηV | & 1 confirming that the slow-roll approximation is not valid. Figure 2.2 shows the inflationary curvature power spectrum ∆2R (k) for this potential, computed by integrating Eqs. (2.2)−(2.4). For comparison, in Figs. 2.1 and 2.2 we also show a smooth, c = 0 potential with the same small-scale amplitude and tilt as the fiducial potential, and its slow-roll parameters and inflationary power spectrum. The smooth spectrum is nearly indistinguishable from a pure power law of ns ∼ 0.96 with amplitude As (k⋆ ) = 2.137 × 10−9. Note that the spectral index is determined by the choices of N⋆ and k⋆ in the matching condition of Eq. (2.5), while the amplitude comes from the inflaton mass m. 27 Figure 2.2: Primordial curvature power spectra for the potentials in Fig. 2.1. 2.2.2 CMB Power Spectra As we saw in § 1.4.5, the mapping between the inflationary curvature power spectrum and the observable CMB angular power spectra ′ XX , ∗ X′ hXℓm ℓ′ m′ i = δℓℓ′ δmm′ Cℓ (2.7) where X, X ′ ∈ T, E, is given by the scalar radiation transfer functions ′ ℓ(ℓ + 1)CℓXX = 2π Z ′ d ln k TℓX (k)TℓX (k) ∆2R (k) . (2.8) In Fig. 2.3, we show the T and E transfer functions for the fiducial cosmological parameters of Table 2.1. For a more extended discussion of the transfer functions and their relationship to features in the inflationary power spectrum, see [48]. The resultant tempera28 Acoustic Feature range ISW SW Acoustic Feature range Reionization (b) Figure 2.3: Transfer function TℓX (k) for the fiducial model with instantaneous reionization. Upper panel: temperature X = T ; lower panel: polarization X = E. Contours are spaced by factors of 2. Dashed lines represent the range of k-modes where features appear in Fig. 2.2. Polarization is a cleaner probe of features in this range and, for instantaneous reionization, is nearly uncontaminated by secondary effects. The temperature and polarization are also only weakly correlated here due to the transition between the Sachs-Wolfe (SW) and acoustic regimes in temperature. 29 ture and polarization angular power spectra from the inflationary power spectra of Fig. 2.2 are plotted in Fig. 2.4. Figure 2.4: Temperature and polarization power spectra for the inflationary power spectra in Fig. 2.2, with solid black curves for the model with a feature and dashed red curves for smooth ∆2R (k). Dotted curves indicate where CℓT E is negative. Blue points with error bars show the 5-year WMAP measurements of CℓT T including sample variance. For both models, the reionization history is assumed to be instantaneous and the cosmological parameters not determined by the inflationary potential are given in Table 2.1. For the wavenumbers of interest, 1 . k/10−3Mpc−1 . 5, the transfer of power to temperature fluctuations transitions between the Sachs-Wolfe and acoustic regimes at high ℓ and carries substantial contributions from the integrated Sachs-Wolfe (ISW) effect at low 30 ℓ. These effects and geometric projection lead to a very broad mapping of power in k to power in ℓ. In particular, the oscillations at the upper range in k are largely washed out, leaving only a single broad dip at ℓ ∼ 20 and bump at ℓ ∼ 40 in the temperature spectrum. Likewise, the power at these multipoles correspond to a wide range in k as shown in Fig. 2.5. Polarization spectra differ notably from the temperature spectra due to the differences in the transfer function shown in Fig. 2.3. For the standard instantaneous reionization history and the upper portion of the range of k affected by the feature, the polarization is dominated by the onset of acoustic effects only. We shall see that this makes the bump in ℓ ∼ 40 a particularly clean test of inflationary features (see Fig. 2.5). Furthermore oscillations from high k at higher ℓ are retained at a significant level in the polarization. On the other hand at k ∼ 10−3 Mpc−1 , the polarization transfer from recombination becomes very inefficient and reionization effects come into play. This leads to a very low level of polarization around ℓ ∼ 20 with features even for a smooth inflationary power spectrum. These properties leave the ℓ ∼ 20 dip vulnerable to external contamination such as tensor contributions (see § 2.3.3) or foregrounds as well as uncertainties in the ionization history (see § 2.4). Finally, the cross correlation between the temperature and polarization fields for the entire range of 20 ≤ ℓ ≤ 40 is very low due to the transition between the Sachs-Wolfe and acoustic-dominated regimes in the temperature field. We shall see in § 2.3.4 that this prevents statistical fluctuations in the observed temperature power spectrum from being repeated in the polarization. These differences in the transfer functions also play a role in defining the region in the potential parameter space that best fits the WMAP T T data versus the region that is best tested by polarization. For the former, we conduct a grid based search over the potential parameters. The mass parameter m determines the amplitude of the spectrum away from the feature and so is mainly fixed by the acoustic peaks at high ℓ. The location of the feature b is also well determined independently of the other parameters [18, 40]. We therefore fix 31 m and b at their best-fit values and search for the best fit in the step amplitude and width parameters c and d. The values of m, b, c, and d given in Table 2.1 specify the maximum likelihood model. This model improves the fit to the 5-year WMAP data by −2∆ ln LT T ≈ −8. We will explore variations in the parameters about the maximum and their relationship to the temperature and polarization power spectra through the transfer functions in § 2.3.2. The improvement is only marginally significant given the 3 extra parameters of the step and the choice of one out of many possible forms. Matching polarization features can therefore provide a critical confirmation or refutation of the inflationary nature of the temperature features. Figure 2.5: Transfer function TℓX (k) for the fiducial model with instantaneous reionization for multipoles near the temperature dip (ℓ = 20) and bump (ℓ = 40) for temperature and polarization. For temperature, the dip multipoles receive a broad range of contributions from k & 10−3 and the bump multipoles from k & 3 × 10−3. The localization of the transfer function is sharper for polarization, especially for ℓ = 40 which is immune to reionization effects. The polarization transfer functions have been scaled by 104 and 105 for convenience. 32 2.3 Confirming Features with Polarization In this section, we discuss the significance with which polarization measurements can confirm or rule out the inflationary features discussed in the previous section under the instantaneous reionization model. We begin in § 2.3.1 with the significance of the best-fit feature model under the simplest set of assumptions. We assess changes in the significance due to variation in the potential parameters in § 2.3.2, and due to the inclusion of tensor E-modes in § 2.3.3. In § 2.3.4, we describe the impact of conditioning polarization predictions on the alreadymeasured temperature spectrum. 2.3.1 Fiducial Polarization Significance To evaluate the significance of discriminating between models, we assume a Gaussian likelihood for the polarization angular power spectrum. In the absence of detector noise, the likelihood LEE of data ĈℓEE given a model power spectrum CℓEE is −2 ln LEE ≈ fsky X (2ℓ + 1) ℓ ĈℓEE CℓEE + ln CℓEE ĈℓEE −1 ! , (2.9) where fsky is the fraction of sky with usable E measurements. If the data ĈℓEE have no inflationary feature and the model CℓEE has the inflationary feature, we call this the significance at which false positives can be rejected. Conversely, if the data have an inflationary feature and the model spectrum has no feature, we call this the significance at which false negatives can be rejected. For forecasts throughout this chapter, we assume that the data are equal to the ensemble average of realizations for a particular model. Therefore, the minimum −2 ln L is zero and −2∆ ln L = −2 ln L. The exception to this is that when we discuss the WMAP T T likelihood, the relevant quantity is −2∆ ln L = −2 ln(L/LML ) where the likelihood of the best fit model is −2 ln LML 6= 0. We make forecasts for an ideal, sample variance limited experiment and for 33 Experiment Ideal Planck ν — 70 GHz 100 GHz 143 GHz (ν) (ν) θFWHM 0 14.0′ 10.0′ 7.1′ ∆P fsky 0 0.8 255.6 0.8 109.0 0.8 81.3 0.8 Table 2.2: Parameters used when making forecasts for idealized and Planck-like experiments. (ν) Here ∆P is in units of µK-arcmin. Experiment Ideal Ideal Planck Planck Test False False False False positive negative positive negative −2∆ ln LEE 64 60 8 9 Table 2.3: −2∆ ln LEE for false positive and false negative tests comparing models with smooth ∆2R (k) and a feature in ∆2R (k). Planck using the experimental specifications in Table 2.2. For the Planck case with a finite noise power Nℓ , Cℓ → Cℓ + Nℓ in Eq. (2.9), where Nℓ is the minimum variance combination of the noise powers of the individual frequency channels (ν) Nℓ (ν) = (ν) ∆P µK-rad 2 (ν) ℓ(ℓ + 1)(θFWHM /rad)2 exp 8 ln 2 (2.10) (ν) using ∆P and θFWHM from Table 2.2 converted to the appropriate units. Table 2.3 lists −2∆ ln LEE for rejecting false positives and false negatives. The signifip cance of false positive or negative rejection in this most optimistic case is −2∆ ln LEE ∼ 8 for the ideal experiment and ∼ 3 for Planck. In the following subsections, we will discuss various effects that can degrade this significance. 34 2.3.2 Potential Parameters Variation in the parameters of the inflationary potential from the best fit model can affect the significance of polarization tests of features. As noted in § 2.2.2, m and b are strongly constrained by the observed CMB temperature spectrum, but the parameters c and d that control the amplitude and width of an inflationary step are less well determined by temperature alone. In terms of the curvature power spectrum, increasing c increases the amplitude of the features. However, decreasing the width of the potential step by lowering d enhances the deviations from slow roll, thereby also amplifying the feature in the power spectrum. -2 -4 0.06 0.05 -6 0.03 -2 -8.3 -8 0.02 -4 0.01 0 -6 -2 10 50 0.002 c 0.001 -6 d 0.04 -2 -4 -4 -2 10 0.003 Figure 2.6: Contour plot of −2∆ ln LT T for parameters c and d using 5-year WMAP data. Other potential parameters are fixed at their fiducial values. The minimum, with −2∆ ln LT T = −8.3 relative to the smooth c = 0 model, is shown with a cross. Figure 2.6 shows a contour plot of the WMAP temperature likelihood −2∆ ln LT T for the parameters c and d (relative to c = 0) and Fig. 2.7 shows −2∆ ln LEE for false positives using simulated polarization data. The similarities and differences between these two plots reflect properties of the temperature and polarization transfer functions. For the temperature case near the minimum, the degeneracy between the two parameters is approximately c ∝ d2 . This line roughly corresponds to keeping the amplitude of the 35 0.06 Ideal 10 4 81 49 64 16 25 36 9 d 4 0.04 0 0.05 14 6 19 0.03 0.02 256 324 400 0.01 900 4 Planck 0.05 9 0.04 d 16 25 0.03 0.01 0 4 1 0.02 9 16 5 6 2 3 49 64 1 100 8 0.001 0.002 c 141496 256 0.003 Figure 2.7: Contour plot of −2∆ ln LEE for the parameters c and d, for tests of false positives with a cosmic variance limited experiment (upper panel ) and Planck (lower panel ). The best fit model to WMAP T T is shown with a cross. enhanced power in ∆2R (k) at k ∼ 3×10−3 Mpc−1 fixed. The preferred value of c corresponds to the best amplitude of the negative dip at k ∼ 2×10−3 Mpc−1 . For the best fit parameters, including the feature in ∆2R (k) improves the fit to 5-year WMAP data by −2∆ ln LT T ≈ −8. Due to the weak significance of the feature detection, the contours become substantially distorted away from the maximum likelihood. In particular, the contours in Fig. 2.6 show a triangular region extending to high d ∼ 0.04. This region corresponds to a lower amplitude in both the first dip and bump in k as shown in Fig. 2.8. Due to projection effects in temperature, the ℓ = 20 dip gets contributions from both the dip and the bump in k (see 36 Fig. 2.5). Consequently, a model with smaller features in k in both the dip and bump can lead to the same amplitude of the dip at ℓ = 20 if the amplitude of the bump is reduced more. 4 109∆2R(k) 3 2 1 0.0001 0.0010 0.0100 k [Mpc-1] 0.1000 Figure 2.8: Primordial curvature power spectra for models illustrating projection degeneracies in the temperature. The parameters of the two models are chosen to have similar temperature dips at ℓ ∼ 20 and equal WMAP likelihoods: (c, d) = (0.00128, 0.043) (solid black ) and (c, d) = (0.0023, 0.028) (dashed red ); other parameters are fixed to the values in Table 2.1 for both models. We illustrate these projection effects in Figs. 2.8 and 2.9 with two models chosen to have the same likelihood improvement of −2∆ ln LT T ≈ −6. For the model with smaller features in k, the temperature enhancement at ℓ ∼ 40 is substantially reduced compared with the best-fit model, while the model with larger features in k overshoots the bump at ℓ = 40 in temperature. Despite these differences, both models have about the same T T amplitude in the ℓ = 20 dip as the best-fit model but a slightly worse overall fit. In particular, for the model with smaller features in k, inflationary features can only explain the observed ℓ = 20 dip in temperature and not the ℓ = 40 bump. The degeneracies in c and d for polarization significance share similarities with, yet have important differences from, those for temperature. The polarization significance remains largely unchanged for small variations in c and d along the constant c/d2 line favored by 37 ∆ClEE/ClEE (smooth) ∆ClTT/ClTT (smooth) 0.4 0.2 0.0 -0.2 -0.4 0.4 0.2 0.0 -0.2 -0.4 20 40 60 80 100 l Figure 2.9: Upper panel: Relative difference in CℓT T , with respect to a smooth power spectrum, of two models with equal T T amplitude at ℓ ∼ 20 and curvature power spectra shown in Fig. 2.8. Lower panel: Due to differences in the polarization transfer function, the models do not have degenerate ℓ ∼ 20 dips and show significant differences at ℓ & 60 as well, leading to a much higher polarization significance for the model plotted with dashed lines. the temperature spectra. Near the maximum, variations along this direction preserve the amplitude of intrinsic features in k (see Fig. 2.6). However within the −2∆ ln LT T = −4 region the significance for a ideal experiment can either drop or rise significantly. The reason is that due to projection effects in temperature, the polarization better separates changes in the overall and relative amplitude of the features in k. In the triangular high d region, where the amplitude of the T T dip remains unchanged but the intrinsic features in k are all reduced, the significance of the polarization difference decreases markedly (see Fig. 2.9). Because of the sharper projection, even the ℓ ∼ 20 dip in polarization is reduced. The net result is that the polarization significance is a stronger function of c, which controls the overall amplitude, than the temperature significance. Note that while the significance can be substantially degraded from our best fit assumptions, this is mainly because of the weak detection of a feature in the temperature spectrum itself. In cases where the polarization significance is greatly reduced, the temperature bump at ℓ ∼ 40 cannot be explained by the inflationary features. In other words, polarization remains a robust probe of the inflationary 38 nature of the ℓ = 40 bump across variations of the potential parameters. 2.3.3 Tensors The m2 φ2 potential with the parameters in Table 2.1 predicts substantial gravitational wave contributions with tensor-to-scalar ratio r ≈ 0.16. Relative to a smooth-∆2R (k) model without tensors, the m2 φ2 model with a feature has extra distinguishing power due to the presence of B-mode polarization. Because other forms for the potential can also be used as the smooth base on which to place the feature [40] we choose not to include tensors for most of our calculations. Moreover, a B-mode detection would not be useful for discriminating features. The B-mode amplitude is insensitive to features since the potential amplitude is left nearly unchanged by the step. Additionally, a small step in the potential is not expected to generate features in the CMB power spectra of tensor modes. Unlike the scalar spectrum whose shape is sensitive to the second slow-roll parameter ηV , the shape of the tensor spectrum depends primarily on ǫV , which remains small at the step (see Fig. 2.1, [40]). In Fig. 2.10 we show the B-mode prediction for r = 0.16 and a pure power law tensor spectrum with tilt nt = −r/8. On the other hand, it is important to assess the possibility of degradation of the E-mode feature from the curvature spectrum due to the nearly smooth tensor E-mode contributions. Due to the shape of the tensor E-mode spectrum, which mimics the B-mode spectrum, the main impact of tensors is to fill in the dip in the polarization spectrum around ℓ ∼ 20 (see Fig. 2.10). Correspondingly, the decrease in significance for the ideal experiment is 13 − 15% p in −2∆ ln LEE , and for Planck, 4%. Planck is less affected since its lower sensitivity limits the accuracy of measurements in the ℓ ∼ 20 dip. Since these degradations are relatively small, we ignore tensors when considering the impact of the reionization history below. Moreover, we shall see that reionization uncertainties are very similar to tensors in that they make the ℓ ∼ 20 dip less useful for distinguishing features through E-mode polarization. 39 Figure 2.10: Effect of tensor fluctuations on polarization power spectra for the model with a feature in ∆2R (k). Solid black: no tensor component. Dashed red: including tensors with r = 0.16. Tensors smooth theEE spectrum near the ℓ ∼ 20 dip. 2.3.4 Temperature Conditioning The usefulness of polarization for providing an independent test of features observed in temperature may also be reduced by the correlation of temperature and polarization: a positive correlation would make observation of polarization features more likely given the WMAP T T data regardless of whether the features have an inflationary or chance statistical origin. We expect the reduction in significance to be small given that CℓT E is small on the relevant angular scales (see Fig. 2.4), and in this subsection we quantify this statement. To assess the impact of conditioning polarization predictions on the WMAP temperature data, it is convenient to replace the likelihood statistic of Eq. (2.9) with a χ2 statistic. This allows us to phrase the impact in terms of the bias and change in variance predicted for the EE power spectrum from the T T measurements. Note that in the absence of the 40 temperature constraint and in the limit of small differences between the model and the data, ! X 2ℓ + 1 Ĉ EE − C EE 2 ℓ ℓ −2∆ ln LEE ≈ fsky EE 2 Cℓ ℓ 2 X ĈℓEE − CℓEE , ≈ EE ) Var( Ĉ ℓ ℓ (2.11) which is equal to a simple χ2 statistic. Now let us include the temperature constraint. First take the idealization that the temperature multipole moments Tℓm have been measured on the full sky with negligible noise. Given a model that correlates the polarization field through the cross correlation coefficient Rℓ = q CℓT E CℓT T CℓEE , (2.12) a constrained realization of the polarization field that is consistent with the temperature field can be constructed as E T q ℓm = Rℓ q ℓm + CℓEE CℓT T q 1 − Rℓ2 gℓm , (2.13) ∗ i = 1, and a where gℓm is a complex Gaussian field with zero mean, unit variance hgℓm gℓm ∗ = (−1)m g real transform gℓm ℓ,−m . The estimate of the power spectrum is then ĈℓEE = 1 X ∗ E E , 2ℓ + 1 m ℓm ℓm (2.14) and its mean over the constrained realizations is biased from the true CℓEE hĈℓEE i − CℓEE CℓEE Ĉ T T − C T T = Rℓ2 ℓ T T ℓ , Cℓ by the fixed observed temperature power spectrum ĈℓT T = 41 (2.15) ∗ m Tℓm Tℓm /(2ℓ P + 1). With a high correlation coefficient, chance features in the temperature spectrum induce similar features in the observed polarization spectrum. For example, if T T fluctuates high, EE will also fluctuate high (on average). The temperature constraint also removes some of the freedom in the variance of the polarization power spectrum: Var(ĈℓEE ) (CℓEE )2 = + 2 2 1 − Rℓ2 2ℓ + 1 Ĉ T T 4 Rℓ2 (1 − Rℓ2 ) ℓT T . 2ℓ + 1 C (2.16) ℓ In the limit that the correlation Rℓ → 0, the variance takes on its usual form for a Gaussian random field. In the limit that Rℓ → 1, there is no uncorrelated piece and the observed temperature spectrum determines the observed polarization spectrum with no variance. Now let us add in detector noise and finite sky coverage. Given a noise power spectrum NℓEE and a fraction of the sky fsky , Var(ĈℓEE ) 2 ≈ fsky EE 2ℓ + 1 (C )2 ℓ + 1 − Rℓ2 + NℓEE !2 CℓEE ! NℓEE ĈℓT T 4 R2 1 − Rℓ2 + EE 2ℓ + 1 ℓ Cℓ CℓT T . (2.17) Figure 2.11 shows, in the upper panel, the E-mode polarization power spectrum for the smooth inflationary spectrum constrained to WMAP5 temperature data for the ideal experiment. For comparison, CℓEE for the best fit feature model is also plotted. Note that even the second dip in the spectrum at ℓ ∼ 60 remains significantly distinct in polarization. In the lower panel, the impact of the temperature power spectrum constraint is plotted as the fractional difference between ĈℓEE and CℓEE for each model. Due to the lack of temperature-polarization correlation in the 10 . ℓ . 60 regime, the impact of the constraint on the polarization features is negligible. 42 EE 2 (ClEE-ClEE)/ClEE l(l+1) Cl /2π [µK ] 1.00 0.10 Smooth Feature 0.01 0.2 0.1 0.0 -0.1 -0.2 20 40 60 80 100 l Figure 2.11: Upper panel: Solid lines show the E-mode power spectrum constrained to the temperature data for the smooth-∆2R (k) model along with the band representing sample variance per ℓ for the ideal experiment. The model with a feature (dashed ) lies significantly outside of the band in the 10 . ℓ . 60 range, making false negatives unlikely. Lower panel: Fractional difference between the average of the constrained realizations hĈℓEE i and the full ensemble average CℓEE for both models. The impact of the constraint is minimal due to the lack of correlation between the temperature and polarization fields in the region of interest. We can quantify these conclusions by generalizing the χ2 statistic in Eq. (2.11) to include the temperature constraint: 2 X ĈℓEE − hĈℓEE i ∆χ2EE ≡ . EE ) Var( Ĉ ℓ ℓ (2.18) As in the likelihood analysis, we assume that the data are a typical draw of the true model (“1”) and that we are testing the significance at which the second model (“2”) can be rejected. EE(1) Then we set ĈℓEE = hĈℓ EE(2) i, hĈℓEE i = hĈℓ EE(2) i, and Var(ĈℓEE )= Var(Ĉℓ ). Note that the bias induced by the temperature constraint enters into both models whereas the change in the variance enters only from model 2. Table 2.4 assesses the χ2 significance of the rejection of false positives and false negatives 43 Experiment Ideal Ideal Planck Planck Test False False False False positive negative positive negative ∆χ2EE w/o T with T 70 63 59 56 8 8 9 8 Table 2.4: ∆χ2EE for false positive and false negative tests comparing models with smooth ∆2R (k) and a feature in ∆2R (k), with polarization either unconstrained or constrained to observed temperature data. for the fiducial feature model. In the last column we have applied the constraint from the WMAP5 temperature data and in the penultimate column we artificially drop the constraint by setting Rℓ = 0 in the evaluation of ∆χ2EE . Even with the constraint, the significance q with which false positives can be rejected is ∆χ2EE = 7.9 for the ideal experiment and q ∆χ2EE = 2.8 for Planck. For the case of false negatives, these numbers become 7.5 for the ideal experiment and 2.8 for Planck. The difference between false positive and false negative significances comes from the dependence of sample variance on the model tested. In all cases, the significance in terms of ∆χ2EE is comparable to −2∆ ln LEE in Table 2.3. The impact of the temperature constraint is to lower the significance of both cases but q only by . 5% in ∆χ2EE . This small difference in significance justifies our choice to omit the constraint to temperature in our exploration of other effects that can degrade the significance. 2.4 Reionization Features A more complicated ionization history can in principle produce features in the polarization spectrum that might mimic or obscure features from the inflationary power spectrum. This is especially true for the dip at ℓ ∼ 20. In this section we search for ionization histories that lead to a higher incidence of false positives and false negatives. Reduced significance of false positives or negatives due to confusion between features from inflation and from reionization can arise in two ways. First, the true reionization history 44 Experiment Ideal Ideal Planck Planck Test False False False False positive negative positive negative −2∆ ln LEE 36 25 6 6 Table 2.5: −2∆ ln LEE for tests of false positives and false negatives with ionization histories of the data and model tuned at 6 < z < 50 to minimize the significance of rejection using the methods described in § 2.4.2. can introduce features in the data that either falsely mimic inflationary features or hide true features. Second, additional reionization freedom in the (false) model we wish to test can allow a better match to data generated from the alternate (true) model assumption. To account for both effects, we use a two-step method in which we first optimize the ionization history of the true model to produce a false positive or negative result, and then vary the ionization history of the false model. We will describe this procedure in § 2.4.2. Table 2.5 summarizes the results of this section. Relative to the significance of rejecting false positives or negatives for instantaneous reionization models, ionization freedom lowers p the significance for an ideal experiment by a factor of 0.64 − 0.75 to −2∆ ln LEE ≈ 5 − 6, p and for Planck by a substantially smaller factor of 0.83 − 0.87 to −2∆ ln LEE ≈ 2.5. Given the amount of freedom we allow in the ionization history these should be viewed as the maximal degradation possible due to reionization. We describe the details of this calculation in the following subsections. 2.4.1 Reionization Principal Components The form of the ionization history, and therefore the shape of the large-scale reionization peak in the polarization spectrum, are only weakly constrained by current observations and theoretical modeling, especially on the scales relevant for inflationary features [79]. We treat the evolution of the mean ionized fraction of hydrogen with redshift, xe (z), as an unconstrained function between z = 6 and some high redshift z = zmax . At lower redshifts, 45 we assume xe ≈ 1 as required by the observed Lyα transmission in quasar spectra at z . 6 (see e.g. [31]). The highest redshift of reionization is less certain, so we take zmax = 50 which is quite conservative for conventional sources of ionizing radiation. We parametrize general reionization histories with a basis of principal components (PCs) Si (z) of the large-scale E-mode polarization [47]. We use the 7 lowest-variance PCs only since the higher-variance PCs have a negligible impact on the polarization power spectrum. Thus the ionization history at 6 < z < 50 is xe (z) = xfid e + 7 X mi Si (z), (2.19) i=1 where we take a constant fiducial ionized fraction of xfid e = 0.07 so that the fiducial model with {mi } = 0 has a total reionization optical depth of τ ≈ 0.09. We vary the PC amplitudes {mi } and compare the resulting CMB power spectra with data simulated for ideal and Planck experiments using Markov Chain Monte Carlo (MCMC) likelihood analysis as we describe in the next section. 2.4.2 Data and Model Optimization We use a two-step optimization process to determine the maximum reduction in significance of false positive or negative rejection that can be caused by reionization features. We categorize models in this section by whether ∆2R (k) is smooth (S) or has a feature (F), and by whether the reionization history is instantaneous (I) or more complex (C) and parametrized by principal components as in Eq. (2.19). For comparisons of models we introduce the notation false:true; for example, FI:SI represents the false positive test for instantaneous reionization models from § 2.3. In the case of false positives, the goal of the optimization is to go from the FI:SI comparison of the previous section to FC:SC, in which both false and true models have complex ionization histories. In particular, we want to find the FC:SC pair that minimizes the dif46 ference between the two models, thus minimizing the significance of false positive rejection. To find this optimal pair of models, we use the following procedure: 1. Optimize true (S) model: FI:(SI→SC) Vary the ionization history of the SI model to find the SC model that best matches FI. 2. Optimize false (F) model: (FI→FC):SC Taking the optimal SC model from step 1 as the true model used to generate simulated data, vary the ionization history of the FI model to find the FC model that matches the best-fit SC. Then the significance of rejecting false positives including reionization freedom is −2∆ ln LEE computed for the optimal FC:SC pair, i.e. the maximum likelihood from step 2. These steps for the false positive tests are illustrated in Fig. 2.12 for the ideal experiment. The process for false negatives can be described by simply swapping which models have features and which are smooth (F ↔ S). Note that even with our conservative choice of zmax = 50, Fig. 2.12 shows that the main impact of reionization on the polarization spectra is limited to ℓ . 30. We will explore the consequences of this restriction to large scales in the following section. To implement the steps described above, we vary ionization histories to minimize −2∆ ln LEE following the methods of Refs. [78, 77, 79], using CosmoMC1 [71] for MCMC likelihood analysis with a version of CAMB [72] modified to include reionization histories parametrized with principal components as in Eq. (2.19). For example, in the FI:(SI→SC) step above, we take the FI model as the simulated polarization data and search over ionization histories of the SC model class. For each optimization step, we run 4 MCMC chains long enough to be well past any initial burn-in phase and stop when the region of parameter space near the best fit is sufficiently well sampled that all 4 chains agree on the maximum likelihood to within ∼ 1% in −2∆ ln L. 1. http://cosmologist.info/cosmomc/ 47 Figure 2.12: False positive example of the two-step process to account for reionization uncertainty in polarization significance for the ideal experiment. Upper panel : FI:(SI→SC) — varying the ionization history from SI to SC (red curves) to match FI (solid black ). Lower panel : (FI→FC):SC — varying the ionization history from FI to FC (black curves) to match SC (dashed red ). All polarization spectra are plotted relative to FI. This joint optimization minimizes the FI:SI difference (shading in upper panel, with fiducial significance given in Table 2.3) at ℓ . 30 using the optimal FC:SC models (shading in lower panel ; Table 2.5). See § 2.4.2 for an explanation of the notation used here. Typically this requires computing an initial chain to estimate the covariance matrix of the reionization PC amplitudes, followed by generating chains with ∼ 104 samples each. The optimal true or false model is taken to be the overall maximum likelihood model from the final 4 chains. All cosmological parameters besides the 7 reionization PC amplitudes are assumed to be fixed by measurements of the temperature spectrum, except for the amplitude of scalar fluctuations As which is varied to keep As exp(−2τ ) fixed, preserving the temperature and polarization power at small scales. Fixed parameters are set to the values in Table 2.1. We use top-hat priors on the PC amplitudes corresponding to 0 ≤ xe ≤ 1 as described 48 in Ref. [78]. Note that the number of PCs used here is larger than the 3 to 5 needed for completeness in Ref. [78] due to our choice of a larger maximum redshift. Although we are interested in the ability of polarization data to test features appearing in the observed temperature spectrum, we include the contributions from the model T T and T E spectra as well as EE in the likelihood for MCMC. Keeping the temperature data in the likelihood ensures that we do not obtain models that fit the polarization spectrum well at the expense of changing the shape of CℓT T . For example, ionization histories with sharp transitions in xe at high redshift can generate polarization power at ℓ ∼ 40 to match inflationary features, but these models also add power to the temperature on similar scales through an enhanced Doppler effect [78, 79]. For the best-fit models, the contribution of temperature data to the likelihood is approximately constant: −2∆ ln LT T ≈ 7. 2.4.3 Reionization Confusion The optimal true and false model spectra obtained from the steps described in the previous section (SC and FC) are plotted in Figs. 2.13 and 2.14 for each of our 4 scenarios (ideal/Planck tests of false positives/negatives). We also plot the corresponding models with instantaneous reionization histories (SI and FI) to show where ionization freedom has the largest effect on the spectra. Table 2.5 lists −2∆ ln LEE for each FC:SC or SC:FC comparison. The ability of reionization to either mimic or obscure the signature of inflationary features is greatest in the low-power 10 . ℓ . 30 regime of CℓEE . For tests of false positives with p an ideal experiment, nearly all of the 25% reduction of −2∆ ln LEE due to reionization comes from ℓ < 30. Planck, on the other hand, has relatively greater sensitivity to small changes in the polarization bump at ℓ ∼ 40 since observations at such scales suffer less from p instrumental noise than at ℓ ∼ 20. The 17% reduction of −2∆ ln LEE for false positive rejection for Planck due to reionization is split equally between ℓ < 30 and ℓ ≥ 30. This fact combined with the weakness of reionization effects at ℓ & 30 makes Planck somewhat less 49 Figure 2.13: Test of false positives due to reionization for an idealized experiment limited by sample variance and for Planck. Thick curves show the true smooth model (long dashed red ) and best-fit false feature model (short dashed black ) for the false positive scenario that would be the most difficult to reject due to freedom in the reionization history. For comparison, the instantaneous reionization polarization spectra from Fig. 2.4 are plotted as thin solid curves. Reionization histories are parametrized by 7 principal components that cover redshifts 6 < z < 50. sensitive to reionization uncertainties than an idealized noise-free experiment. For false negative tests, changes to polarization spectra at ℓ > 30 are generally more important than they are for false positives. In the case of the ideal experiment, if we ignored p multipoles above ℓ = 30 reionization would only reduce −2∆ ln LEE by 20% relative to the instantaneous reionization significance instead of the 36% reduction that we find when including all scales. The false model being tested in this case (smooth ∆2R (k)) has less power and therefore lower sample variance at 30 . ℓ . 50 than the spectrum with a feature, and therefore changes in the polarization spectra on these scales have a greater effect on the significance than they do for false positive tests. Likewise, Planck’s significance is more 50 Figure 2.14: Same as Fig. 2.13, but for tests of false negatives. Here the true model has a feature in ∆2R (k) and the false model is assumed to have smooth ∆2R (k) (c = 0). dependent on the ℓ ∼ 40 bump for testing false negatives than for false positives. In fact, p nearly all of the 13% degradation in −2∆ ln LEE for false negative rejection comes from ℓ > 30 for Planck. Changes in the reionization history at 6 < z < 50 are unable to significantly affect the polarization power spectrum at ℓ & 50. A detection of polarization features on these scales would therefore be robust to reionization uncertainty. Likewise, measurement of a smooth spectrum on these scales would strengthen bounds on the height and width of a step in the inflaton potential. By considering variations in xe (z) up to z = 50, we include a wide variety of ionization histories, many of which may not be physically plausible. In practice, however, the ionization histories of the spectra in Figs. 2.13 and 2.14 have xe . 0.2 at z > 20. Nevertheless, had we chosen to limit ionization variation to lower redshifts the possibility of confusing reionization 51 with inflationary features would be lessened, particularly for tests of false negatives and for Planck, due to the greater reliance on small-scale features in those cases. Note that the effects of optimizing the ionization history and smoothing the polarization spectra with the addition of a large tensor component (§ 2.3.3) are similar: both are able to make a spectrum with inflationary features and a smooth spectrum appear more alike at 10 . ℓ . 30. Due to this similarity, we expect that considering tensors and reionization simultaneously would not further degrade the significance of false positive or negative tests. For variations in the potential parameters discussed in § 2.3.2 that retain only the dip at ℓ ∼ 20 and not the bump at ℓ ∼ 40, the impact of reionization will be greater. In these cases one cannot expect polarization to provide unambiguous confirmation of features without external input on the ionization history. 2.5 Discussion Models with a step in the inflationary potential produce oscillations in the angular power spectra of the CMB that can improve the fit to WMAP temperature data at multipoles ℓ ∼ 20 − 40 at the expense of 3 additional phenomenological parameters controlling the step height, width, and location on the inflaton potential. Such models predict that these oscillations should appear in the E-mode polarization spectrum on similar, few-degree scales. The first precise measurements of the polarization on these scales are expected from the Planck satellite, enabling tests of the inflationary-step hypothesis. Moreover, inflationary features at the upper range of ℓ & 30 (k & 2 × 10−3 Mpc−1 ) that are smoothed out due to projection effects in temperature should be more visible in polarization. For the lower range of ℓ . 30 (k . 2 × 10−3 Mpc−1 ), it becomes important to assess the impact of reionization and tensor mode uncertainties. We have explored in detail the prospects for polarization tests of features, focusing in particular on the risk of errors that can be classified as false positives (falsely confirming an inflationary feature) and false negatives (falsely rejecting an inflationary feature). Under the 52 simplest set of assumptions for large-scale polarization in which we take the best-fit model for the temperature features, neglect tensor fluctuations, and take the reionization history to be instantaneous, polarization measurements from Planck should be able to confirm or exclude the inflationary features that best match current temperature data with a significance p of −2∆ ln LEE ∼ 3. All-sky experiments beyond Planck could potentially increase this p significance to −2∆ ln LEE ∼ 8, providing a definitive test for features from inflation. The estimated significance degrades slightly with the addition of a large-amplitude, smooth tensor component to the E-mode spectrum, which tends to hide the effect of an inflationary step at the largest scales. Assuming that the step modifies an m2 φ2 potential, p −2∆ ln LEE is reduced by 4% for Planck and ∼ 14% for a cosmic variance limited exper- iment. Allowing non-standard reionization histories with arbitrary changes to the ionized p fraction at 6 < z < 50 can lower −2∆ ln LEE by as much as ∼ 15% for Planck and ∼ 30% for cosmic variance limited data. Since tensor fluctuations and reionization have the greatest impact on detectability of inflationary features at similar scales (ℓ ∼ 20), their effects on the significance should not be cumulative. The possible contamination due to tensors or reionization could eventually be mitigated with constraints from other types of observations, e.g. stronger limits on the tensor-to-scalar ratio from the B-mode polarization power spectrum. The B-mode contribution from an m2 φ2 potential is potentially within the reach of Planck [30]. Note, however, that a failure to reject false positives or negatives for inflationary features in E-mode polarization would generally bias the inferred ionization history and reionization parameters such as the optical depth. Such biases would in turn lead to biased constraints on inflationary parameters from tensor B-mode measurements [77]. These estimated significances assume that the parameters of the step in the inflaton potential are those that best fit the WMAP temperature spectrum. Away from this best fit, the polarization significance can either increase or decrease. Cases where the significance substantially decreases correspond to parameter combinations where at most one of the dip 53 (ℓ ∼ 20) and bump (ℓ ∼ 40) temperature features can be explained by the step in the potential. In the case that only the dip is inflationary, Planck will be unable to confirm the feature. We have not computed the impact of foreground removal uncertainties on our results; in general one might expect our forecasts to degrade somewhat upon including them. However, recent studies for Planck [30] and a future dedicated polarization satellite mission [113, 22] indicate that foregrounds will not be a substantial problem in the relevant multipole range. Finally, we do not address the possibility that the features in the WMAP data arise from a systematic effect (cf. Appendix A). Nonetheless, if all of the ℓ = 20 − 40 features in the temperature power spectrum are inflationary, polarization should ultimately provide a statistically significant confirmation. In the following chapters, we will take a phenomenological approach. We will introduce a new formalism that enables us to map constraints from observations of the Cosmic Microwave Background onto constraints on the inflationary potential, ultimately allowing us to constrain a more generic class of models: slow-roll and single-field inflation. 54 CHAPTER 3 GENERALIZED SLOW ROLL FOR LARGE POWER SPECTRUM FEATURES In this chapter we introduce an approximation for calculating the curvature power spectrum that allows for order unity deviations in power caused by sharp features in the inflaton potential. As an example, we show that predictions for the step potential introduced in Chapter 2 are accurate at the percent level. Our analysis shows that to good approximation there is a single source function that is responsible for observable features, and that this function is simply related to the local slope and curvature of the inflaton potential. We will use these properties in Chapter 4 and 5 to impose model-independent constraints on the shape of the inflationary potential using CMB observations. 3.1 Introduction The ordinary slow roll approximation provides a model-independent technique for computing the initial curvature power spectrum for inflationary models where the scalar field potential is sufficiently flat and slowly varying (see Eq. (1.41)). Such models lead to curvature power spectra that are featureless and nearly scale invariant (e.g. [73]). On the other hand, features in the inflaton potential produce features in the power spectrum. As we saw in the previous chapter, glitches in the observed temperature power spectrum of the cosmic microwave background (CMB) [8] have led to recent interest in exploring such models (e.g. [91, 18, 40, 76, 88, 55]). To explain the glitches as other than statistical flukes, these models require order unity variations in the curvature power spectrum across about an e-fold in wavenumber. Such cases are typically handled by numerically solving the field equation on a caseby-case basis (e.g. [1]). For model-independent constraints and model building purposes it is desirable to have a simple but accurate prescription that relates features in the inflaton 55 potential to features in the power spectrum (cf. [50, 38, 56, 57]). The generalized slow roll (GSR) approximation was introduced by Stewart [109] to overcome some of the problems of the ordinary slow roll approximation for potentials with small but sharp features. In this approximation, the ordinary slow roll parameters are taken to be small but not necessarily constant. In this chapter we examine and extend the GSR approach for the case of large features where the slow-roll parameters are also not necessarily small. In §3.2, we review the GSR approximation and develop the variant for large power spectrum features. In Appendix B, we compare this variant to other GSR approximations in the literature [109, 15, 20, 57, 34]. We show that our variant provides both the most accurate results and is the most simply related to the inflaton potential. In §3.3, we show how this technique can be used to develop alternate inflationary models to explain a given observed feature. We discuss these results in §3.4. 3.2 Generalized Slow Roll The GSR formalism was developed to calculate the curvature power spectrum for inflation models in which the usual slow roll parameters ǫH and ηH (see Eqs. (1.28) and (1.39)) are small but ηH (= −δ1 ) is not necessarily constant. In these models, the third slow-roll parameter ... φ , δ2 = H 2 φ̇ (3.1) can be large for a small number of e-folds [109, 15, 20]. We study here the more extreme case where ηH is also allowed to become large for a fraction of an e-fold. These models lead to order unity deviations in the curvature power spectrum. As we shall see, different implementations of the GSR approximation perform very differently for such models. An example of such a case is a step in the inflaton potential of the form V (φ) = 56 Figure 3.1: Upper panel: inflationary potential with a step from Eq. (2.1) with parameters that maximize the WMAP5 likelihood (ML, black/solid) and an m2 φ2 potential that matches the W MAP5 normalization (smooth, red/dashed). Lower panel: conformal time to the end of inflation as a function of the value of the field. m2eff (φ)φ2 /2, where the effective mass of the inflaton potential is given by Eq. (2.1). The potential for the maximum likelihood (ML) parameter values for WMAP5 (in Table 2.1) is shown in Fig. 3.1 (upper panel). For comparison we also show the best fit smooth model (c = 0) with m = 7.12 × 10−6. Since it will be convenient to express results in terms of physical scale instead of field value, we also show in the lower panel the relationship to the Rt conformal time to the end of inflation η = t end dt′ /a. Note that η is defined to be positive during inflation. The two models have comparable power at wavenumbers k ∼ η −1 ∼ 0.02 Mpc−1 . The slow-roll parameters for these models as a function of η are shown in Fig. 3.2. Notice that ǫH remains small in the ML model though its value changes fractionally by order unity. On the other hand, ηH is of order unity and δ2 is greater than unity in amplitude in this model around η ∼ 1 Gpc when the inflaton rolls across the feature. 57 Figure 3.2: Slow-roll parameters ǫH , ηH and δ2 for the two models of Fig. 3.1: ML step model (black/solid) and smooth model (red/dash ed). 3.2.1 Exact Relations It is useful to begin by examining the exact equations and solutions. The exact equation of motion of each k-mode of the inflaton field is given by Eq. (1.35), where z= f , 2πη f = 2π φ̇aη . H (3.2) Following [109], we begin by transforming the field equation into dimensionless variables √ y = 2kuk , x = kη d2 y g(ln x) 2 + 1− 2 y = y, (3.3) 2 dx x x2 where g= f ′′ − 3f ′ . f (3.4) Primes here and throughout are derivatives with respect to ln η. The functions f and g carry information about deviations from the exact de Sitter space ǫH = 0, ηH = 0 and δ2 = 0. Specifically, without assuming that these three parameters are 58 small or slowly varying ǫ f 2 = 8π 2 H2 (aHη)2 , H f′ = −aHη(ǫH − ηH ) + (1 − aHη) , f f ′′ f′ = 3 + 2[(aHη)2 − 1] f f (3.5) +(aHη)2 [2ǫH − 3ηH + 2ǫ2H − 4ηH ǫH + δ2 ] , and the dynamics of the slow-roll parameters themselves are given by dǫH = 2ǫH (ǫH − ηH ) , d ln a dηH 2 −δ . = ǫH ηH + ηH 2 d ln a (3.6) (3.7) Moreover, these quantities are related to the inflaton potential by ( V,φ 2 (1 − ηH /3)2 , ) = 2ǫH V (1 − ǫH /3)2 V,φφ ǫ + ηH − δ2 /3 = H , V 1 − ǫH /3 (3.8) which in the limit of small and nearly constant ηH , ǫH return the ordinary slow roll relations (see Eq. (1.40)). In general, there is no way to directly express the source function g in terms of the potential without approximation. Here we want to consider a situation where the feature in the potential is not large enough to interrupt inflation and hence ǫH ≪ 1, but is sufficiently large to make ηH of order unity for less than an e-fold. By virtue of Eq. (3.7), |δ2 | ≫ 1 during this time. This differs from other treatments which assume |ηH | ≪ 1 and by virtue of Eq. (3.6) a nearly constant ǫH [109]. Even under these generalized assumptions there are some terms in ηH and δ2 that can 59 be neglected. For example, even if ηH is not small, it suffices to take aHη − 1 = ǫH + ǫH O(ηH ) . (3.9) This expression preserves the ordinary slow roll relations when |ηH | ≪ 1. When ηH is not small, this quantity remains of order ǫH and so is negligible compared with bare ηH and δ2 terms. Hence this approximation suffices everywhere. Figure 3.3: Source functions for the deviations from slow roll used in the GSR approximations: 2g/3, 2gV /3 and G′ (see §3.2.3) for the maximum likelihood model. To good approximation g = gV which directly relates the source function to features in the inflaton potential. Likewise G′ ≈ 2gV /3 and is most simply related to the curvature power spectrum for large deviations. Following this logic, we obtain g = gV + ǫH O(ǫH , ηH , δ2 ) , (3.10) where gV is directly related to the potential gV ≡ V,φφ 9 V,φ 2 ( ) −3 2 V V = 6ǫH − 3ηH + δ2 + ǫH O(ǫH , ηH , δ2 ) . 60 (3.11) As shown in Fig. 3.3, this relationship between the source function g and features in the potential V holds even for the ML step potential. Thus, if we can express the functional relationship between g and the curvature power spectrum that is valid for large g we can use features in the power spectrum to directly constrain features in the inflaton potential. To determine this relation note that in the x and y variables the curvature is R = xy/f , and its power spectrum is ∆2R (k) = limx≪1 |R|2 . The LHS of Eq. (3.3) is simply the equation for scale invariant in exact de Sitter space and is solved by i y0 (x) = 1 + eix , x (3.12) and its complex conjugate y0∗ (x). An exact, albeit formal solution to the field equation can be constructed with the Green function technique [109] Z ∞ du g(ln u)y(u)Im[y0∗ (u)y0(x)] . y(x) = y0 (x) − 2 u x (3.13) The solution is only formal since y appears on both the left and right hand side of the equation. The corresponding formal solution for the curvature power spectrum can be made more explicit by parameterizing the source y(u) as y(u) = FR (u)Re[y0 (u)] + iFI (u)Im[y0 (u)] (3.14) so that Z i ∞ du x3 lim (xy) = i − F (u)g(ln u) 3 x u u3 I x≪1 Z i ∞ du W (u)FI (u)g(ln u) + 3 x u Z ∞ du 1 X(u)FR(u)g(ln u) − 3 x u Z x3 ∞ du + W (u)FR (u)g(ln u) + O(x2 ) , 9 x u 61 (3.15) where 3 W (u) ≡ − Im[y0 (u)]Re[y0 (u)] u 3 sin(2u) 3 cos 2u 3 sin(2u) − − , = 2u 2u3 u2 3 X(u) ≡ Re[y0 (u)]Re[y0 (u)] u 3 cos(2u) 3 sin(2u) − = − 2u3 u2 3 cos(2u) 3 + + 3 (1 + u2 ) . 2u 2u (3.16) Note that limu→0 W (u) = 1 and limu→0 X(u) = u3 /3 and we have utilized the fact that 1 u Im[y0 (u)]Im[y0 (u)] = 1 + 2 − X(u) 3 u (3.17) goes to 1/u2 in the limit u → 0. Finally, the curvature power spectrum becomes [Im(y)]2 + [Re(y)]2 ∆2R (k) = lim x2 , x≪1 f2 (3.18) with y given by Eq. (3.15). 3.2.2 GSR for Small Deviations The fundamental assumption in GSR is that one recovers a good solution by setting FI (u) = FR (u) = 1 in the formal solution for the field fluctuations in Eq. (3.15). Equivalently, y(u) → y0 (u) in the source term on the RHS of Eq. (3.3). Note that this does not necessarily require that g itself is everywhere much less than unity. For example, modes that encounter a strong variation in g while deep inside the horizon do not retain any imprint of the variation and hence the GSR approximation correctly describes the curvature they induce. In Fig. 3.4, we show an example of FI and FR for a mode with k = 10−4 Mpc−1 for 62 Figure 3.4: Ratio of field solution y to the scale invariant approximation y0 . Upper panel: real part FR for a smooth case (red/dashed line), and for the maximum likelihood model (black/solid line), both at k = 10−4 Mpc−1 . Lower panel: imaginary part FI for the same models. both the ML and smooth models. For the ML model, this mode is larger than the horizon when the inflaton crosses the feature. Note that even in the smooth model, the two functions deviate substantially from unity at x ≪ 1. In fact, they continue to increase indefinitely after horizon crossing and FR ∝ x−3 diverges to compensate for |Re(y0 )| ∝ x2 . For the ML model, even FI deviates strongly from unity during the crossing of the feature at x ∼ 0.1. The impact that these deviations have on the curvature spectrum can be better understood by reexpressing the various contributions in a more compact form. First note that Z 1 f′ x3 ∞ du −3 , u g(ln u) = − lim u 3 f x≪1 3 x (3.19) and so Eq. (3.15) becomes Z 1 f ′ 1 ∞ du 1 1+ + W (u)g(ln u) , lim |RGSRS | = f 3f 3 x u x≪1 (3.20) where note that we have dropped the Re(xy) contribution since it adds in quadrature to 63 the power spectrum and hence is second order in g. We call this the “GSRS” approximation for the curvature power spectrum ∆2R = limx≪1 |RGSRS |2 given its validity for small fluctuations in the field solution from y → y0 . The choice of x is somewhat problematic [109]. From Fig. 3.4, we see that taking x too small will cause spurious effects since FI increases as x decreases. On the other hand, x cannot be chosen to be too large for the ML model since it will cause some k modes to have their curvature calculated when the inflaton is crossing the feature. Moreover, if x is set to be some fixed conformal time during inflation ηmin , then it will vary with k. We illustrate these problems in Fig. 3.5. For ηmin = 10−1 Mpc (upper panel), GSRS underpredicts power at low k for the smooth model and overpredicts it for the ML model. Agreement for the smooth model is improved by choosing x = 10−2 , i.e. nearer to horizon crossing (cf. Appendix B for variants that take x ≈ 1). On the other hand, the agreement for the ML model becomes worse and has a spurious feature at k ∼ 10−5 Mpc−1 where the inflaton is crossing the feature at x = 10−2 . In the next section, we shall examine the origin of the deviations from the exact solution and how a variant of the GSR approximation can fix most of them. 3.2.3 GSR for Large Deviations When considering large deviations from scale invariance, either due to sharp features in the potential or due to extending the calculation for many e-folds after horizon crossing, the first qualitative problem with the GSRS approximation of Eq. (3.20) is that it represents a linearized expansion for a correction that is not necessarily small. When the correction becomes large, RGSRS can pass through zero leaving nodes in the spectrum. While this is not strictly a problem for the ML model, it is better to have a more robust implementation of GSR for likelihood searches over the parameter space. We can finesse this problem by replacing the linearized expansion 1 + x by ex and write 64 Figure 3.5: GSRS approximation to the curvature power spectrum (dashed lines) compared to the exact solution (solid lines) for a choice of ηmin = 10−1 Mpc (upper panel) and ηmin = 10−2 /k Mpc (lower panel). The ML model is shown in blue and the smooth model in red for GSRS. the power spectrum in the form ln ∆2R (k) Z 2 ∞ dη = G(ln ηmin ) + W (kη)g(ln η) , 3 ηmin η (3.21) where G(ln η) = ln 1 f2 + 2 f′ . 3f (3.22) This procedure returns the correct result at first order since g and f ′ /f are both first order 65 in the slow-roll parameters (see Eq. (3.5)). We shall see below that it can be further modified to match the fully non-linear result for superhorizon modes. Figure 3.6: Curvature evolution after horizon crossing in the GSRS (upper panel) and GSRL (lower panel) approximations, both normalized to the exact solution. The ML model (black/solid line) and smooth model (red/dashed line) are both shown at k = 10−4 Mpc−1 . The more fundamental problem with GSRS is the deviation of the true solution y from the scale invariant solution y0 when the mode is outside the horizon (see Fig. 3.4). The origin of this problem is that the exact solution requires the curvature R = xy/f to be constant outside the horizon, independently of how strongly f evolves. Thus, if f is allowed to vary significantly, either due to the large number of e-folds that have intervened since horizon crossing or due to a feature in the potential, then y must follow suit and deviate from y0 breaking the GSRS approximation. Fig. 3.6 (upper panel) illustrates this problem. Even for the smooth model, the curvature is increasingly underestimated as x → 0 . With the ML model, the crossing of the feature induces an error of the opposite sign. For x ∼ 10−5 these problems fortuitously cancel but not for any fundamental or model independent reason. Given this problem, GSRS actually works better than one might naively expect. For example at k = 10−4 Mpc−1 , even though FI ∼ 1.28 at x = 10−5 , the GSRS approximation 66 gives a ∼ 2.5% difference in the curvature and a ∼ 5% difference in the power spectrum with the exact solution for the smooth model instead of the 28% and the 64% differences one might guess. The main contribution to the GSRS correction from scale invariance is given by the integral term in Eq. (3.20), which is ∼ 0.25 for the smooth case. Given that FI is a linear function in ln η and g is slowly varying, we can approximate enhancement due to FI of the integral term by its average interval (∼ 1.14). With this rough estimate we obtain an approximately (1 + 0.25)2 /(1 + 0.25 × 1.14)2 ∼ 5 − 6% error in power in agreement with the power spectrum result in Fig. 3.5. Furthermore although FR diverges as x−3 in Fig. 3.4, the contribution to the power spectrum of the real part of y remains small. Its absence in the GSRS approximation produces a negligible effect for modes that are larger than the horizon when the inflaton crosses the feature. The integrands for the real contribution contain either the function X, which peaks at horizon crossing x ∼ 1, or x3 W (u) which is likewise suppressed at x ≪ 1. The correction adds in quadrature to the imaginary part and so it is intrinsically a second order correction (see §3.2.5). For k = 10−4 Mpc−1 its contribution to the power spectrum is 0.08% of the power spectrum in the ML model. The fact that integrals over the deviation of y from y0 can remain small even when neither g nor the maximum of y −y0 is small is crucial to explaining why the GSR approximation works so well and why we can extend GSRS with small, controlled corrections. Nonetheless these problems with GSRS are significant and exacerbated by the presence of sharp features in the potential. The fundamental problem with GSRS is that its results depend on an arbitrarily chosen value of x ≪ 1, i.e. R is not strictly constant in this regime. Phrased in terms of Eq. (3.22) the problem is that g is not directly related to G but rather 2 2 g = G′ + 3 3 67 ′ 2 f , f (3.23) where G′ = dG 2 f ′′ f ′ f ′2 = ( − 3 − 2 ). d ln η 3 f f f (3.24) In GSRS, replacing g with 3G′ /2 amounts to a second order change in the source function. In fact even for the ML step function this change is a small fractional change of the source everywhere in ln η: it is small as the inflaton rolls past the feature since |f ′′ /f | ≫ (f ′ /f )2 and it is small before and after this time since |f ′ /f | ≪ 1. In terms of the slow-roll parameters, 2 ∼ O(1) only where |δ | ≫ 1 and g ≈ δ (see this replacement is a good approximation if ηH 2 2 Eqs. (3.10) and (3.11)). 2 2 2 + ǫH O(ǫH , ηH , δ2 ) G′ = g + ηH 3 3 (3.25) Moreover G′ ≈ 2gV /3 and remains directly relatable to the inflaton potential through Eq. (3.11). For comparison we show all three versions of the GSR source function in Fig. 3.3. Nonetheless, the replacement can have a substantial effect on the curvature once the source is integrated over ln η because the difference is a positive definite term in the integral. Moreover, this cumulative effect is exactly what is needed to recover the required superhorizon behavior. Replacing 2g/3 → G′ in the power spectrum expression, we obtain [57] ln ∆2R (k) Z ∞ dη W (kη)G′ (ln η) , = G(ln ηmin ) + ηmin η (3.26) which we call the GSRL approximation. The field solution corresponding to this approximation, valid for x ≪ 1, is given by Z 1 f ′ 1 ∞ du ′ lim |xy| = exp + W (u)G (ln u) . 3f 2 x u x≪1 (3.27) Now any variation in f while the mode is outside the horizon and W (kη) ≈ 1 integrates away and gives the same result as if ln ηmin were set to be right after horizon crossing for 68 the mode in question. This can be seen more clearly by integrating Eq. (3.26) by parts [57] ln ∆2R (k) Z ∞ dη ′ =− W (kη)G(ln η) . ηmin η (3.28) R Since − 0∞ d ln xW ′ (x) = 1 and limx→0 W ′ (x) → 0, the curvature spectrum does not depend on the evolution of f outside the horizon. Moreover, the integral gets its contribution near x ∼ 1 so for smooth functions G(ln η) we recover the slow roll expectation that ln ∆2R (k) ≈ G(ln η) kη≈1 . (3.29) If the slow-roll parameters are all small then the leading order term in Eq. (3.29) returns the familiar expression for the curvature spectrum ∆2R ≈ f −2 ≈ H 2 /8π 2 ǫH at kη ≈ 1. Choe et al. [15] showed that Eq. (3.28) is correct up to second order in g for kη ≪ 1. Here we show that it is correct for arbitrary variations in f and g outside the horizon. The superhorizon curvature evolution for k = 10−4 Mpc−1 corresponding to the GSRL approximation is shown in Fig. 3.6 (lower panel). In the x ≪ 1 domain of applicability of Eq. (3.27), the curvature is now appropriately constant for both the ML and smooth models. The net result is that the curvature power spectrum shown in Fig. 3.7 is now a good match to the exact solution for low k. 3.2.4 Power Spectrum Features We now turn to issues related to the response of the field and curvature for k modes that are inside the horizon when the inflaton rolls across the feature. Fig. 3.7 shows that the GSRL approximation works remarkably well for the ML model despite the fact that the power spectrum changes by order unity there. The main problem is a ∼ 10 − 20% deficit of power for a small range in k near the sharp rise between the trough and the peak. In Fig. 3.8, we show the deviation of the exact solution y from the scale invariant y0 that is at the heart of the GSR approximation. The three modes shown, kdip = 1.8 × 10−3 69 Figure 3.7: GSRL approximation to the curvature power spectrum. Upper panel: approximation compared with the exact solution (solid lines) for the maximum likelihood model. Lower panel: fractional error between the approximation and the exact solution. Mpc−1 , knode = 2.5 × 10−3 Mpc−1 , kbump = 3.2 × 10−3 Mpc−1 , correspond to the first dip, node and bump in the power spectrum of the ML model. The first thing to note is that for higher k, the inflaton crosses the feature at increasing x where the deviations of y from y0 actually decrease. Hence the fundamental validity of the GSR approximation actually improves for subhorizon modes. Combined with the GSRL approximation that enforces the correct result at x ≪ 1, this makes the approximation well behaved nearly everywhere. The small deviations from GSRL appear for modes that cross the horizon right around the time that the inflaton crosses the feature. It is important to note that the step potential actually provides two temporal features in g or G′ displayed in Fig. 3.3. Each mode first crosses a positive feature at high η and x and then goes through a nearly equal and opposite negative feature. The end result for the field amplitude or curvature is an interference pattern of contributions from both temporal features. For example, the peak in power is 70 Figure 3.8: Fractional difference between |y0 | and |y| for the ML model at k values at the dip, node and bump of the feature in the power spectrum (see text). due to the constructive interference between a positive response to the positive feature and a negative response to the negative feature. This suggests that one problem with the GSRL approach is that it does not account for the deviation of the field y from y0 that accumulates through passing the positive temporal feature when considering how the field goes through the negative feature. This is intrinsically a non-linear effect. The final thing to note is that, since g and G′ are of order unity as these modes exit the horizon, the real part of the field solution is not negligible. Moreover, it contributes a positive definite piece to the power spectrum. In Fig. 3.9, we show the result of dropping the real part from the exact solution. Note that the fractional error induced by dropping the real part looks much like the GSRL error in Fig. 3.7 but with ∼ 1/2 the amplitude. 3.2.5 Iterative GSR Correction The good agreement between GSRL and the exact solution even in the presence of large deviations in the curvature spectrum suggests that a small higher order correction may further improve the accuracy. Moreover, the analysis in the previous section implies that 71 Figure 3.9: Contribution of the real part of the y field to the curvature power spectrum. Upper panel: spectrum with and without the real part. Lower panel: fractional error between the two solutions. there are two sources of error: the omission of the field response from inside the horizon x > 1 when computing the response of the field to features at horizon crossing x ∼ 1 and the dropping of the real part of the field solution. Both of these contributions come in at second order in the GSR approximation. All first order GSR variants involve the replacement of the true field solution y with the scale invariant solution y0 in Eq. (3.3). This replacement can be iterated with successively better approximations to y. We begin with the GSRS approximation of replacing y → y0 to obtain the first order solution y1 . We then replace y → y1 in the source to obtain a second order solution y2 , etc. We show the fractional error between the iterative solutions and the exact solution for k = knode in Fig. 3.10, where the error in GSRL is roughly maximized. As in the first order GSRS approach, the accuracy depends on the arbitrary choice of x = kηmin when the curvature is computed. The number of iterations required for a given accuracy increases 72 with decreasing x. We show the curvature spectrum in Fig. 3.11 for the same two choice of ηmin = 10−1 Mpc (upper panel) and ηmin = 10−2 /k Mpc (lower panel) as in Fig. 3.5. Note that in both cases, the result has converged at the 0.5% percent level or better to the exact solution within three iterations. Figure 3.10: Fractional difference between the exact (y) and nth order iterative solutions (yn ) for the ML model at k = knode where the errors in the GSRL approximation are maximized. Unfortunately the iterative GSRS approach is not of practical use in that each iteration requires essentially the same effort as a single solution of the exact approach. On the other hand, rapid convergence in the iterative GSRS approach suggests that a nonlinear correction to GSRL based on a second order expansion might suffice. A second order GSRL approach differs conceptually from the iterative GSRS approach in that it is formally an expansion in g where in our case |g| ≪ 1 is not satisfied. The iterative GSRS approach is exact in g but expands in y − y0 . What makes a second order GSRL approach feasible is that the critical elements involve time integrals over g which can be small even if g is not everywhere small. Our strategy for devising a non-linear correction to GSRL is to choose a form that reproduces GSRL at first order in g, is exact at second order in g, is simple to relate to the inflaton potential, and finally is well controlled at large values of g. The second order in g expressions for the curvature are explicitly given in [15] and come about by both iterating 73 Figure 3.11: Curvature power spectrum in the GSRS approximation for ηmin = 10−1 Mpc (upper panel) and ηmin = 10−2 /k Mpc (lower panel) when y → yn in the GSRS source compared to the exact solution. the integral solution in Eq. (3.15) and dropping higher order terms. Our criteria are satisfied by ∆2R = ∆2R |GSRL 1 1 2 1 2 2 [1 + I1 (k) + I2 (k)] + I1 (k) 4 2 2 (3.30) where Z ∞ dη ′ 1 I1 (k) = √ G (ln η)X(kη) , 2 0 η Z ∞ 1 f′ du [X + X ′ ] F (u) , I2 (k) = −4 u 3 f 0 74 (3.31) with Z ∞ dv f ′ F (u) = . 2 u v f (3.32) We call this the GSRL2 approximation. In Appendix B we discuss alternate forms [15]. In the GSRL2 approach, I1 corrections come half from the first order calculation of the real part of the field and half from iterating the imaginary part to second order. In Fig. 3.12 we show I12 and I2 for the ML model. Note that I12 dominates the correction to the net power as it always enhances power, while I2 is both smaller and oscillates in its correction. Furthermore, both |I12 | ≪ 1 and |I2 | ≪ 1 for the ML model which justifies a second order approach to these corrections. The GSRL2 correction can be taken to be {1 + I12 + I2 } in this limit. Figure 3.12: Second order GSRL2 power spectrum correction functions I12 and I2 for the ML model. We show in Fig. 3.13 how the GSRL2 corrections reduce the power spectrum errors of GSRL in Fig. 3.7 for the ML model. For the full GSRL2 expression the power spectrum errors are reduced from the 10 − 20% level to the . 4% level. We show that the GSRL2 approximation remains remarkably accurate for substantially larger features in Appendix B. Moreover, the errors are oscillatory and their observable consequence in the CMB is 75 further reduced by projection. The temperature and polarization power spectra are shown in Fig. 3.14 and 3.15 and the errors are . 0.5% and . 2% for the respective spectra. Given the intrinsic smallness of I2 and its oscillatory nature, the most important correction comes from the positive definite I1 piece. Note that it is a single integral over the same G′ function as in the linear case. Thus, I1 corrections simply generalize the GSRL mapping between G′ and curvature in a manner that is equally simple to calculate. I2 on the other hand is more complicated and involves a non-trivial double integral with a different dependence on the inflaton potential. We also show in Figs. 3.13-3.15 the results for the GSRL2 expression with I2 omitted. While the curvature power spectrum errors increase slightly to ∼ 5%, the temperature power spectrum errors at . 2% are still well below the ∼ 20% cosmic variance errors per ℓ at ℓ ∼ 30. They are even sufficient for the cosmic variance limit of coherent deviations across the full √ range of the feature (20 . ℓ . 40) 20%/ 20 ∼ 4 − 5% in the ML case. The polarization spectrum has slightly larger errors due to the reduction of projection effects but still satisfies these cosmic variance based criteria. 3.3 Applications In the previous section, we have shown that a particular variant of the GSR approximation which we call GSRL2 provides a non-linear mapping between G′ and the curvature power spectrum. G′ quantifies the deviations from slow roll in the background and moreover is to good approximation directly related to the inflaton potential. These relations remain true even when the slow-roll parameter ηH is not small compared to unity for a fraction of an e-fold. This relationship is useful for considering inflation-model independent constraints on the inflaton potential, as we will see in Chapters 4 and 5. It is likewise useful for inverse or model building approaches of finding inflaton potential classes that might fit some observed feature in the data. 76 Figure 3.13: GSRL2 approximation to the curvature power spectrum. Upper panel: approximation of Eq. (3.30) (red/dashed line) compared to the exact solution (black/solid line). We also show the GSRL2 approximation omitting the I2 term (blue/dashed-dotted line). Lower panel: fractional error between these GSRL2 approximations and the exact solution. Here as a simple example let us consider a potential that differs qualitatively from the step potential but shares similar observable properties through G′ : V (φ) = m2eff φ2 /2 where the effective mass of the inflaton now has a transient perturbation instead of a step h i 2 2 m2eff = m2 1 + Ae−(φ−b) /(2σ ) (φ − b) (3.33) In Fig. 3.16 we show the potential for the choice of parameters b = 14.655, A = 0.0285, σ = 0.025, and m = 7.126 × 10−6 (upper panel) and we also show G′ in the lower panel. For comparison we show the smooth case A = 0. Comparison with Figs. 3.1 and 3.2 shows that this potential, which has a bump and a dip instead of a step, produces a similar main feature in G′ but has additional lower amplitude secondary features. In Fig. 3.17 we compare the GSRL2 approximation with and without the double integral I2 term compared to the exact solution. Notice that GSRL2 performs equally well for this 77 Figure 3.14: GSRL2 approximation to the CMB temperature power spectrum. Upper panel: approximation (red/dashed line) compared to the exact solution (black/solid line). We also show the GSRL2 approximation omitting the I2 term (blue/dashed-dotted line). Lower panel: fractional error between the GSRL2 approximations and the exact solution. very different sharp potential feature. Furthermore, similarity in G′ with the step potential carries over to similarity in the curvature power spectrum. 3.4 Discussion We have shown that a variant of the generalized slow roll (GSR) approach remains percent level accurate at predicting order unity deviations in the observable CMB temperature and polarization power spectra from sharp potential features. Unlike other variants which explicitly require |ηH | ≪ 1, and hence nearly constant ǫH , our approach allows ηH to be order unity, as long as it remains so for less than an e-fold, and hence ǫH to vary significantly. We have tested our GSR variant against the step potential introduced in Chapter 2. Our analysis also shows that to good approximation a single function, G′ (ln η), controls the observable features in the curvature power spectrum even in the presence of large fea78 Figure 3.15: GSRL2 approximation to the CMB E-mode polarization power spectrum. The same as in Fig. 3.14 Figure 3.16: Alternate inflationary model with a perturbation in the mass. Upper panel: comparison of potential in Eq. (3.33) (black/solid line) and the smooth potential (red/dashed line). Lower panel: source function of the deviation from slow roll G′ for the same models. tures. We have explicitly checked this relationship and the robustness of our approximation by constructing two different inflationary models with similar G′ . Therefore observational 79 Figure 3.17: GSRL2 approximation to the alternate model of Fig. 3.16. Upper panel: approximation (red/dashed line) compared to the exact solution (black/solid line) for an effective mass given by Eq. (3.33). We also show the GSRL2 approximation with I2 omitted (blue/dashed-dotted line). Lower panel: fractional error between GSRL2 approximations and the exact solution. constraints from the CMB can be mapped directly to constraints on this function independently of the model for inflation. Moreover, this function is also simply related to the slope and curvature of the inflaton potential in the same way that scalar tilt is related to the potential in ordinary slow roll G′ ≈ 3(V,φ /V )2 − 2(V,φφ /V ). These model independent constraints can then be simply interpreted in terms of the inflation potential. We will explore these applications in Chapters 4 and 5. 80 CHAPTER 4 CMB CONSTRAINTS ON PRINCIPAL COMPONENTS OF THE INFLATON POTENTIAL In this chapter, we apply the Generalized Sow Roll formalism described in Chapter 3 to the best constrained region of WMAP 7-year data by means of a principal component decomposition of the source function, and use it to impose functional constraints on the shape of the inflaton potential. We do a Markov Chain Monte Carlo (MCMC) likelihood analysis keeping only those modes measured to better than 10%. The analysis results in 5 nearly independent Gaussian constraints that may be used to test any single-field inflationary model where such deviations are expected. Detection of any non-zero component would represent a violation of ordinary slow roll and indicate a feature in the inflaton potential or sound speed. One component (that resembles a local running of the tilt) shows a 95% CL preference for deviations around the 300 Mpc scale at the ∼ 10% level, but the global significance is reduced considering the 5 components examined. This deviation also requires a change in the cold dark matter density which in a flat ΛCDM model is disfavored by current supernova and Hubble constant data. We show that the inflaton potential can be even better constrained with current and upcoming high sensitivity experiments that will measure small-scale temperature and polarization power spectra of the CMB. For this analysis, we have implemented a ∼40× faster WMAP7 likelihood code which we have made publicly available. 4.1 Introduction Under the assumption that cosmological perturbations were generated during an inflationary period from quantum fluctuations in a single scalar field, features in the cosmic microwave background (CMB) temperature and polarization power spectra constrain features in the inflaton potential V (φ). In this chapter we take a model independent approach to constraining the shape of the 81 inflaton potential. We have shown in Chapter 3 that even in the presence of large local changes in the curvature of the inflaton potential that can explain the glitches observed in the temperature spectrum, there is to excellent approximation only a single function of the inflaton potential that the observations constrain [25]. Moreover, this function is approximately the same combination of slope and curvature that enters into the calculation of the scalar tilt in the ordinary slow-roll approximation. With it, we can bypass a parameterization of the initial curvature power spectrum (e.g. [48, 11, 67, 99, 90, 85, 93]) and the problem that not all spectra correspond to possible inflationary models. In this chapter, we take a principal components approach [57] to functional constraints on the inflaton potential under the GSR formalism. Principal components constructed a priori from a noise model of the WMAP CMB measurements determine the theoretically best constrained deviations from a featureless potential before examining the actual data. Constraints from the low order principal components thus efficiently encapsulate the expected information content of the data and may be used to test a variety of inflationary models without a reanalysis of the data. We begin in §4.2 with a brief review of the baseline GSR variant for our analysis. In §4.3.2 we develop the principal component analysis of the GSR source function and apply it to the WMAP 7 year (WMAP7) data in §4.4. In §4.5 we consider applications of these derived constraints on the inflaton potential and discuss these results in §4.6. In Appendix C, we present the fast likelihood approach to the WMAP7 data employed in these analyses. 4.2 Generalized Slow Roll variant used In Chapter 3 we have shown that a variant of GSR works well for cases where ηH becomes of order unity for a fraction of an e-fold [25]. In this variant of the GSR approximation, the curvature power spectrum is a non-linear 82 functional ln ∆2R (k) Z ηmax dη ≈ G(ln ηmin ) + W (kη)G′ (ln η) η ηmin " Z ηmax 2 # dη 1 X(kη)G′ (ln η) , + ln 1 + 2 η ηmin (4.1) of the function f ′ f ′2 2 f ′′ −3 − 2) G′ (ln η) = ( 3 f f f (4.2) which is related to the inflaton potential through the background solution f = 2π φ̇aη/H. The window functions 3 sin(2u) 3 cos 2u 3 sin(2u) − − , 2u 2u3 u2 3 cos(2u) 3 sin(2u) 3 cos(2u) X(u) = − − + 2u 2u3 u2 3 + 3 (1 + u2 ) , 2u W (u) = (4.3) define the linear and nonlinear response of the curvature spectrum to G′ respectively. For the models we consider, the nonlinear response is small compared with the linear one. The key property of Eq. (4.1) is that deviations from scale invariance in the curvature spectrum depend only on a single function of time G′ . Moreover, to good approximation, this function is related to the inflaton potential as [25] G′ ≈ 3( V,φφ V,φ 2 ) −2 , V V (4.4) ′ | ≫ |η | when η is large, i.e. that η remains large only for a fraction so long as |ηH H H H of an e-fold. This function is also closely related to the source of corresponding bispectrum features [3]. Finally, if the ordinary slow roll approximation where ǫH and ηH are both small 83 and nearly constant holds, then G′ may be evaluated at horizon crossing η ≈ k −1 and taken out of the integrals in Eq. (4.1). As we shall see below, under this approximation G′ = 1−ns . By allowing G′ to be both time varying and potentially large, we recover ordinary slow roll results where they apply but allow the data themselves to test their validity. 4.3 Principal components The GSR approximation allows us to go beyond specific models of inflation in examining how the data constrain the inflaton potential. The data directly constrain the function G′ and hence the derivatives of the inflaton potential through Eq. (4.4). Given that G′ is related to the curvature spectrum ∆2R by an integral relation and the curvature spectrum itself is related to the observable CMB power spectra by a line-of-sight integration, not all aspects of the function G′ are observable even with perfect data. We therefore seek a basis for an efficient representation of observable properties of G′ . We begin in §4.3.1 with a general description of a basis expansion for G′ and its relationship to the usual normalization and tilt parameters. We then turn in §4.3.2 to principal components (PCs) as the basis which best encapsulates expected deviations from scale-free conditions [57]. 4.3.1 Basis Expansion In general, we seek to represent the function G′ as G′ (ln η) = N X ma Sa (ln η) , (4.5) a=0 where the basis functions Sa for a > 0 are assumed to have compact support in some region between ln ηmin and ln ηmax corresponding to the range probed by the data. We assume S0 = 1 so that within this range m0 represents a constant tilt in the spectrum. 84 We seek functions that obey the orthogonality and completeness relations 1 X Sa (ln η)Sa (ln η ′ ) = δ(ln η − ln η ′ ) , ∆ ln η a Z 1 d ln η Sa (ln η)Sb (ln η) = δab , ∆ ln η (4.6) where ∆ ln η = ln ηmax − ln ηmin . From these relations, the ma amplitudes are related to G′ as 1 ma = ∆ ln η Z d ln η Sa (ln η)G′ (ln η) . (4.7) Note that our normalization differs from that of Ref. [57] in that unit amplitude ma corresponds to unit variance in G′ averaged over the whole range in ln η. Substituting this model into the power spectrum expression (4.1) yields ln ∆2R (k) ≈ G(ln ηmin ) + m0 C −m0 ln(kηmin ) + N X ma Wa (k) (4.8) a=1 2 N 1 X ma Xa (k) , + ln 1 + 2 a=0 where C = 37 − γE − ln 2 ≈ 1.06297 with γE as the Euler-Mascheroni constant, X0 = π/2. The k-space responses to the modes are characterized by Wa (k) = Xa (k) = Z ηmax η Z min ηmax ηmin d ln η W (kη)Sa (ln η) , d ln η X(kη)Sa (ln η) , where kηmin ≪ 1 and kηmax ≫ 1. Note that if ma = 0 for a > 0 ∆2R (k) = eG(ln ηmin )+m0 C (1 + 85 π2 2 m )(kηmin )−m0 8 0 (4.9) from which we infer that the model is a pure power law spectrum. We can therefore choose instead to represent G(ln ηmin ) and m0 by ns and As bringing our parameterization of the power spectrum to ns −1 # X N k 2 + ma Wa (k) ln ∆R = ln As kp a=1 2 N X 1 π + ln 1 + (1 − ns ) + ma Xa (k) . 2 2 " (4.10) a=1 Note that this replacement ensures that the normalization and tilt parameters are defined at a scale kp that is well-constrained by the data. Hence any small and unobservable running of G and G′ from ηmin to η ∼ 1/kp is absorbed into As and ns respectively. Non zero values of ma>0 required by the data thus represent a deviation from purely scale-free initial conditions. Hereafter in this chapter, we consider Eq. (4.10) as the definition of the parameterized curvature power spectrum. In practice we choose kp = 0.05 Mpc−1 , and for reference note that k ≈ 0.02 Mpc−1 for modes contributing to the well-measured first acoustic peak. 4.3.2 Principal Component Basis We choose here to construct the basis functions Sa from the principal components (PCs) of the projected WMAP7 covariance matrix for perturbations in G′ . To define the PCs, we begin with a fiducial flat ΛCDM model with a scale-free initial spectrum. We take the baryon density to be Ωb h2 = 0.02268, cold dark matter density Ωc h2 = 0.1080, cosmological constant ΩΛ = 0.7507, optical depth τ = 0.089, As = 2.41 × 10−9, ns = 0.96 and ma>0 = 0. This model corresponds to a constant G′ = 0.04. We then construct the PCs as the theoretically best constrained non-constant deviations in G′ around this fiducial model. We start by adding a set of perturbations in G′ (ln ηi ) = 0.04 + pi at 50 equally spaced intervals in ln η between 1 < η/Mpc < 105 . From this set, we 86 Figure 4.1: The first 5 principal components (PCs) of G′ as a function of conformal time based on the WMAP7 specifications. The power law model with zero amplitude PCs is shown in red dashed lines. The first 5 PCs represent a local expansion of G′ around η ∼ 102 Mpc. Figure 4.2: Predicted RMS error on the PC amplitudes as a function of mode number for WMAP7 data. PCs are rank ordered from lowest to highest error with the first 5 describing modes with better than ∼ 10% constraints on G′ . define the function G′ (ln η) by a cubic spline. This sampling of 10 per decade or δ ln η = 0.23 across ∆ ln η = 11.5 is sufficient to capture the observable properties of G′ barring unphysical models with both high frequency and high amplitude perturbations. The spline ensures a 87 smooth interpolation between the samples. Figure 4.3: Sensitivity of the curvature power spectrum to the first 5 PC parameters. Low order PCs mainly change the power spectrum at wavenumber in the decade around k ∼ 10−2 Mpc−1 . Red dashed line represents the zero PC amplitude fiducial power law model. Figure 4.4: Sensitivity of the temperature power spectrum to the first 5 PC parameters. Low order PCs represent slowly varying features around the first peak ℓ ∼ 200. Red dashed line represents the zero PC amplitude fiducial power law model. 88 To these parameters pi we add the cosmological parameters pµ = {pi , As , ns , τ, Ωb h2 , Ωc h2 , θ} , (4.11) where the angular extent of the sound horizon θ takes the place of ΩΛ , and construct the Fisher matrix Fµν = ℓX max X ℓ=2 XY,X̃ Ỹ ∂CℓX̃ Ỹ ∂CℓXY −1 C , XY X̃ Ỹ ∂pν ∂pµ (4.12) where the XY pairs run over the observable power spectra T T, EE, T E. For the data covariance matrix, we take CXY X̃ Ỹ = h 1 (CℓX X̃ + NℓX X̃ )(CℓY Ỹ + NℓY Ỹ ) (2ℓ + 1)fsky i +(CℓX Ỹ + NℓX Ỹ )(CℓY X̃ + NℓY X̃ ) , (4.13) where the WMAP7 noise power NℓX X̃ = δX X̃ NℓXX is inferred from the temperature power spectrum errors from the LAMBDA site1 and the assumption that NℓEE = 2NℓT T . We set ℓmax = 1200 and fsky = 1. We then invert the Fisher matrix to form the covariance matrix C = F−1 . Next we take the sub-block Cij and decompose it with the orthonormal matrix Sja , Cij = ∆ ln η X S σ2S δ ln η a ia a ja (4.14) rank ordered from lowest to highest σa . Each eigenvector defines a discrete sampling of the basis function Sa via Sa (ln ηi ) = s ∆ ln η S , δ ln η ia (4.15) with the normalization of Eq. (4.6). The full functions Sa (ln η) are defined by taking a cubic spline through the samples. The first 5 PC components are shown in Fig. 4.1. 1. http : //lambda.gsf c.nasa.gov 89 In the Fisher approximation σa represents the WMAP7 expected errors for ma hma mb i = δab σa2 , (4.16) for a zero mean fiducial model hma i = 0. Figure 4.5: Predicted RMS errors on running of tilt α as a function of the maximum number of PC components included. Note that the errors cease to improve after the first 4 components and most of the improvement comes from the 4th component. These errors are shown in Fig. 4.2. Note that the noise rises by a factor of 4 across the first 5 components and then increases to order unity by the 20th component. Given that the peak amplitudes of Sa lie in the ∼ 4 range, a ∼ 2.5% error corresponds to a ∼ 10% peak variation in G′ or the effective tilt. By keeping the first 5 PCs, we retain all of the constraints on deviations in G′ in the 10% range. In this sense, these 5 modes represent the best set for examining deviations from a scale-free power law model. Note that no actual WMAP7 data goes into the construction and so that the modes are chosen a priori rather than a posteriori. For the first 5 PC components, the Sa basis functions are centered near η ≈ 102 Mpc and reflect the strong WMAP sensitivity to the first peak at ℓ ≈ 200 or k ∼ 0.02 Mpc−1 in the 90 fiducial model. The first 5 components resemble a local decomposition of G′ in the decade surrounding this scale. This fact can be seen more directly by examining the sensitivity of the curvature and temperature power spectra to the 5 PC amplitudes (see Figs. 4.3 and 4.4, respectively). As an example of the utility of retaining only the first 5 PCs, consider a linear model for G′ G′ (ln η) = 1 − ns + α ln(η/η0 ). (4.17) The local slope in the power spectrum in the GSR approximation is d ln ∆2R kη απ I1 , = ns − 1 + α ln( 0 ) − √ d ln k C 2 1 + I12 (4.18) where C = e7/3−γE /2 ≈ 2.895 and i 1 hπ (1 − ns − α ln kη0 ) + 1.67α . I1 = √ 2 2 (4.19) This linear model can be projected onto the first 5 PCs. The variance of α is then given by aX max 1 1 ∂ma 2 = , ∂α σ 2 (α) σa2 (4.20) a=1 where ∂ma 1 = ∂α ∆ ln η Z d ln η Sa (ln η) ln(η/η0 ) . (4.21) By virtue of the marginalization of ns in the construction of the PCs, the Sa functions have nearly zero mean and ∂ma /∂α does not depend on the scale η0 where the effective tilt is defined as G′ (ln η0 ) = 1 − ns . Fig. 4.5 shows the predicted RMS error on α as a function of the maximum PC mode retained. With 5 PCs, σ(α) = 0.0355, while the fully-saturated value of the error with 50 PCs is σ(α) = 0.0327. This should be compared with the projected error using α itself as 91 a Fisher matrix parameter, σ(α) = 0.0328. These results verify the completeness of the 50 PC basis as well as show that most of the information for |α| ≪ 1 is expected to come from the first 5 PCs. In fact most of the information comes from a single mode, the 4th. This mode corresponds to a local variation in the effective tilt G′ with a null near 300Mpc (see Fig. 4.1), or in the power spectrum near ∼ 0.003 Mpc−1 (see Fig. 4.3), and an extent spanning 1-2 decades. Even though the 1st mode has smaller overall errors and better constrains peak variations in G′ , it is not the most effective mode for constraining running of the tilt given its extremely local form. This caveat applies more generally. A given model may have large deviations in G′ in a region where the data does not best constrain G′ . In this case the first 5 PCs no longer represent a complete or efficient representation. We return to this point in §4.5. Figure 4.6: Posterior probability distributions of the cosmological and 5 PC parameters using WMAP7 data. The red dashed line represents the power law results where the first 5 PC parameters are held fixed to ma = 0. 92 4.3.3 MCMC We use a Markov Chain Monte Carlo (MCMC) likelihood analysis to determine joint constraints on the first 5 PC amplitudes and cosmological parameters pµ = {m1 , . . . , m5 , As , ns , τ, Ωb h2 , Ωc h2 , θ} . (4.22) On top of this basic set we also examine the impact of spatial curvature ΩK and tensor-scalar ratio r and the amplitude of a Sunyaev-Zel’dovich contaminant ASZ on a case-by-case basis. The MCMC algorithm samples the parameter space evaluating the likelihood L(x|p) of the data x given each proposed parameter set p (e.g. see [16, 64]). The posterior distribution is obtained using Bayes’ Theorem, P(p|x) = R L(x|p)P(p) , dθ L(x|p)P(p) (4.23) where P(p) is the prior probability density. We place non-informative tophat priors on all parameters in Eq. (5.2). For example, for the PC amplitudes we take P(ma>0 ) = 1 for −1 ≤ ma>0 ≤ 1 and 0 otherwise. The MCMC algorithm generates random draws from the posterior distribution that are fair samples of the likelihood surface. We test convergence of the samples to a stationary distribution that approximates the joint posterior density P(p|x) by applying a conservative Gelman-Rubin criterion [33] of R − 1 < 0.01 across four chains. We use the code CosmoMC [71] for the MCMC analysis2 . For the WMAP7 data [66], we optimize the likelihood code available at the LAMBDA web site as detailed in Appendix C. The net improvement in speed on an 8-core desktop processor is a factor of ∼ 40, which will enable our 20 PC analysis in Chapter 5. 2. http : //cosmologist.inf o/cosmomc 93 Figure 4.7: The 5 PC filtered G′5 posterior using WMAP7 data. The shaded region encloses the 68% CL region and the upper and lower curves show the upper and lower 95% CL limits. The maximum likelihood (ML) G′5 is shown as the thick central curve, and th e power law ML model is shown in red dashed lines. Structure in this representation mainly reflects the form of the PC modes and is dominated by the modes with the largest uncertainties. 4.4 Constraints In this section we present the constraints on the 5 best measured principal components of the GSR source function G′ and implicitly the inflaton potential V (φ). We first examine constraints using the WMAP7 data only and then joint with a variety of cosmological constraints to remove residual parameter degeneracies. 4.4.1 WMAP We begin by considering the WMAP7 data in a flat ΛCDM cosmological context. We show the probability distributions of the first 5 PCs and cosmological parameters in Fig. 4.6. Table 4.1 shows the mean, standard deviation of the posterior probabilities as well as the maximum likelihood parameter values for the power law models vs. the 5PC models. 94 Figure 4.8: Joint probability distributions of the principal component amplitudes and the cosmological parameters from MCMC analysis of WMAP7 data (68% and 95% CL contours). For visualization purposes we show in Fig. 4.7 the functional posterior probability of G′5 (η) ≡ 1 − ns + 5 X ma Sa (η) , (4.24) a=1 which should be interpreted as G′ filtered through the first 5 PCs and not a reconstruction of G′ itself (cf. Fig. 4.11 below). All 5 PCs are tightly constrained, with errors that are comparable to the Fisher matrix projection, nearly Gaussian posteriors and little covariance with each other. The correlation coefficients between two different ma ’s, |Cma mb /σ(ma )σ(mb )| < 0.2. The first component m1 in particular is consistent with zero and places the tightest constraints of . 3% local variations in G′ around η ≈ 102 Mpc. Interestingly, the power law prediction of m4 = 0 lies in the tails of the posterior with as extreme or more values disfavored at 94.8%CL. With the 95 Gaussian approximation m4 = 0 is 1.9σ from the mean and disfavored at 94.5%CL. In Fig. 4.8, we show joint posteriors of the PCs with other parameters. Notably, for the anomalous m4 component there is a degeneracy with Ωc h2 , Ωb h2 and ns which is also reflected in the broadening of the cosmological parameter posteriors in Fig. 4.6 and the shift in means and maximum likelihood values in Table 4.1. The maximum likelihood (ML) model found by the chain is an improvement over the power law case of 2∆ ln L ≈ 5 which is marginal considering the addition of 5 parameters. The intriguing aspect of the ML model, like the ma posteriors, is that the improvement is concentrated in the m4 component. Note that the finite m4 component allows ns = 1 to be a good fit to the data implying that the data can be marginally better fit by a local deviation from scale invariance rather than tilt. An examination of the ML model helps illuminate the degeneracies with cosmological parameters. Fig. 4.9 (top) shows the temperature power spectra of the 5 PC ML model compared to the power law ML model (upper panel), and the fractional difference between the two (lower panel). Note that in the well constrained ℓ ∼ 30 − 800 regime the two spectra agree to ∼ 1% or better. This accidental degeneracy is not preserved beyond ℓ = 1000. Furthermore the E-mode polarization power spectra shown in Fig. 4.9 (bottom) reveal substantially larger fractional deviations of up to ∼ 10% that break the degeneracy in the temperature spectra. Indeed the main improvement of the PC model relative to the PL model actually comes from the ℓ ≥ 24 polarization cross correlation part of the likelihood (MASTER T ET E), where 2∆ ln L = 3.5, with half of this contribution coming from ℓ < 200. The low-ℓ temperature part of the likelihood has an improvement of 2∆ ln L = 1.8 relative to the PL model, and there is a smaller improvement coming from the high-ℓ (MASTER T T ) part with 2∆ ln L = 1.1. Finally, the PC model is worse than the PL model by 2∆ ln L = −1.3 for the low ℓ < 24 polarization. The intermediate ℓ temperature degeneracy exhibited by the two models is further ex96 Figure 4.9: Power spectra of the 5 PCs maximum likelihood model (black curve) compared to power law (PL) maximum likelihood model (re d dashed curve) in the upper panel, and the difference (PC-PL)/PL in the lower panel. Top: temperature power spectrum. Bottom: polarization power spectrum. plored in Fig. 4.10. We show the impact of the cumulative parameter variations between the PL and PC ML models. The addition of m4 carries almost all of the impact of the PCs and 97 Figure 4.10: Decomposition of the fractional difference between the PC and power law (PL) maximum likelihood models (ML) shown in Fig. 4.9 into contributions from specific parameters. Curves show the cumulative effect of adjusting the PL ML parameters to the PC ML values. The main effects come from the change in m4 , Ωc h2 , and ns . is mainly compensated by variations in Ωc h2 to adjust the relative heights of the peaks and ns to tilt the spectrum to small scales. The change in Ωc h2 also changes the physical size of the sound horizon which must be compensated by a change in the distance to recombination reflected in a lower value for H0 and ΩΛ to leave the angular scale θ compatible with the data. These results are robust to marginalizing ΩK given any reasonable prior on H0 , an SZ component through ASZ , or tensors within the B-mode measured limits of the BICEP +0.31 experiment r = 0.03−0.26 [14]. Through an explicit MCMC analysis of these separate cases, we have verified that the shift in the means and change in the errors for all 5 PCs are much smaller than 1σ. The largest effect is from marginalizing tensors where for example m4 = 0.042±0.20. A small improvement in the B-mode limits would eliminate this ambiguity entirely. 98 Parameters 100Ωb h2 Ωc h2 θ τ ns ln[1010 As ] m1 m2 m3 m4 m5 H0 ΩΛ −2 ln L Power Law (PL) 2.220 ± 0.055 2.217 0.1116 ± 0.0053 0.1130 1.0386 ± 0.0026 1.0387 0.088 ± 0.014 0.088 0.9650 ± 0.0136 0.9622 3.083 ± 0.034 3.088 0 0 0 0 0 0 0 0 0 0 70.13 ± 2.38 69.50 0.726 ± 0.028 0.720 7474.97 Principal Components (PC) 2.040 ± 0.196 2.067 0.1308 ± 0.0127 0.1284 1.0361 ± 0.0049 1.0365 0.089 ± 0.017 0.088 0.9916 ± 0.0233 0.9877 3.119 ± 0.041 3.105 0.0014 ± 0.0077 0.0021 0.0015 ± 0.0132 0.0068 −0.0253 ± 0.0197 −0.0264 0.0339 ± 0.0175 0.0337 −0.0033 ± 0.0315 0.0023 61.41 ± 6.08 62.18 0.581 ± 0.116 0.614 7469.82 Table 4.1: Means, standard deviations (left subdivision of columns) and maximum likelihood values (right subdivision of columns) with likelihood values for ΛCDM and the 5 PCs model with WMAP7 data. 4.4.2 Joint Constraints The results of the previous section suggest that other data which measure the high-ℓ temperature spectrum, polarization spectrum, or pin down the cosmological parameters that control the distance to recombination and baryon density can eliminate the remaining degeneracies and enable WMAP7 to better constrain the inflaton potential. We start with adding more CMB information from the QUAD experiment. QUAD helps mainly by reducing the m5 − Ωb h2 degeneracy. Interestingly, most of the impact comes from the polarization measurements rather than the extended range of the temperature constraints. Adding in non-CMB cosmological information helps even more, especially with m4 . We take the UNION2 supernovae data set3 , the SHOES H0 = (74.2 ± 3.6) km/s/Mpc measurement [97] and a big bang nucleosynthesis constraint of Ωb h2 = 0.022 ± 0.002. In a flat ΛCDM cosmology, the degeneracy between m4 and Ωc h2 is nearly eliminated yielding m4 = 0.0191 ± 0.0163, i.e. consistent with power law initial conditions. Of the additional 3. http : //www.supernova.lbl.gov/U nion 99 PC +QUAD +BBN+SN+H0 , flat m1 0.0000 ± 0.0072 0.0045 ± 0.0071 m2 0.0033 ± 0.0123 0.0091 ± 0.0121 m3 −0.0261 ± 0.0184 −0.0120 ± 0.0166 m4 0.0296 ± 0.0168 0.0191 ± 0.0163 m5 0.0149 ± 0.0250 0.0187 ± 0.0249 +BBN+SN+H0 , w/ΩK 0.0027 ± 0.0073 0.0045 ± 0.0125 −0.0208 ± 0.0178 0.0384 ± 0.0197 0.0091 ± 0.0256 Table 4.2: Means and standard deviations of the posterior probabilities of the PC amplitudes with different data sets added to the WMAP7 data. For supernovae (SN) we used the UNION2 dataset, and for H0 the SHOES measurement. data it is the supernovae that drive this improvement by disfavoring the low ΩΛ values required by the increase in Ωc h2 . On the other hand, these improvements require an assumption that the dark energy is a cosmological constant and the Universe is spatially flat. For example if ΩK is marginalized, m4 = 0.0384 ± 0.0197. The addition of spatial curvature allows the freedom to adjust the relative distance to the high-z supernova and recombination. A better measurement of H0 could resolve this degeneracy since the constraints on ΩK are already limited by the SHOES data. Table 4.2 summarizes these results for the constraints on the PC amplitudes. 4.5 Applications The model independent results of the previous section can be used to test a wide variety of inflationary deviations from scale-free conditions. Moreover given that the constraints on the PC amplitudes are uncorrelated and approximately Gaussian, these tests are straightforward to apply. As the simplest example, take the model with a linear deviation in G′ given by Eq. (4.17). 100 Using Eq. (4.7), we obtain m1 = 0.048α , m4 = −0.576α , m2 = −0.079α , m3 = 0.054α , m5 = −0.034α . (4.25) We can then construct the effective χ2 statistic 2 χ (α) = 5 X ma (α) − m̄a 2 σa a=1 . (4.26) With the means and variances taken from Table 4.1 for WMAP7 we obtain α = −0.057 ± 0.029. When we take into account the covariance between the PC amplitudes, we obtain a 3% shift in the mean with the same error: α = −0.059±0.029. Likewise we have verified that using Eq. (4.25) in a separate MCMC analysis gives consistent results α = −0.058 ± 0.030. This result should be compared with the direct analysis of the running of the tilt which gives α = −0.034 ± 0.027 consistent with the analysis from [66]. The mean is shifted from the PC derived mean by ∼ 0.8σ while the errors are 7% higher. As we will see in the next chapter, for large negative values of α, the linear G′ model no longer matches the running of the tilt due to the I1 term in Eq. (4.18). On the other hand, the Fisher expectation in §4.3.2 shows that for an infinitesimal α the first 5 PC components contain nearly all the information. A model with α = −0.057, which fits the intermediate ℓ ∼ 30 − 800 range well, implies a large change across the extended observable range from η ∼ 20 − 5000 Mpc of |δG′ | ∼ |δns | ∼ 0.3. In particular, it overpredicts the suppression of the ℓ < 30 temperature multipoles. For the same amplitude and tilt at the first peak, 2 the amplitude at the horizon is suppressed by ∼ e(α/2) ln (100) = 0.55. This suppression can only be partially compensated by red tilting the spectrum without over suppressing the high ℓ > 800 multipoles. Note that cosmic variance completely dominates the uncertainties in the ℓ < 30 region and decreases with the predicted signal, an effect that is not represented in the Fisher matrix. 101 In other words, the 2σ preference for finite m4 is not completely consistent with a constant running of the tilt but rather points to a more local deviation from scale-free conditions. When we eliminate this preference by adding in the additional SN, H0 and BBN constraints the inferred limits on α from the first 5 PC amplitudes becomes α = −0.033 ± 0.027. Note however that the direct constraints on α also improve with the addition of these data sets to α = −0.013 ± 0.021. In Fig. 4.11 we plot an α = −0.033 model for the 5 PCs filtered G′5 against the posterior constraints from the WMAP7 data and additional SN, H0 and BBN constraints in a flat universe. We overplot the original unfiltered G′ for this α and note that even with the reduced value the deviations become large outside of the region probed by the first 5 PCs. Figure 4.11: The 5 PC filtered G′5 posterior using WMAP7 data and additional SN, H0 and BBN constraints in a flat universe. The shaded region encloses the 68% CL region and the upper and lower curves show the upper and lower 95% CL limits. A model with running of the tilt α = −0.033, the mean value given these constraints, is shown as the thick solid blue curve and the ML PL model as the dashed red curve. The unfiltered G′ of the same α is shown in blue dashed lines for comparison (arbitrary offset). Note that outside the range probed by the first 5 PCs the model deviations continue to grow linearly and oscillating features in G′5 do not necessarily imply features in the underlying G′ . This example points out a caveat to the use of the first 5 PCs as general constraints on models. For a model with features that are substantially larger in a regime away from 102 the well-constrained first acoustic peak, the first 5 PCs may not be the best constraints in terms of signal-to-noise. One can check whether this is the case by examining the predicted G′ or by projecting the model onto the full 50 PC space and checking for large amplitude components. Indeed if the higher components are extremely large compared with the low components, non-linear effects can break the orthogonality of PCs and lead to larger allowed variations in the low components when compensated by the high components. As an example, the full 50 PC decomposition of the the step function potential model with an effective mass given by Eq. (2.1) with maximum likelihood parameters taken from Table 2.1 is shown in Fig. 4.12. Interestingly, m4 = 0.0266 in the step model and has the highest amplitude of the first 5 components. On the other hand, a complete analysis based on signal-to-noise would require ∼ 20 PC components. By keeping only 5 components, the improvement compared with the ML PL model is only ∆χ2 = −1.7. In other words, the step function model is certainly allowed by our 5 PC constraint and even marginally favored but the majority of the improvement is not captured by the truncated analysis. Nevertheless, when interpreted as an upper limit on deviations from scale-free conditions, the 5 PC approach works as a general, albeit typically conservative, method to constrain a wide variety of possible deviations from a single analysis. As the running of the tilt example shows, the results are remarkably close to a direct analysis and differences can be used to expose the self-consistency of the model inferences with independent parts of the data. 4.6 Discussion We have employed a variant of the generalized slow roll approximation (GSR) introduced in Eq. (4.1) to place functional constraints on the GSR source function and implicitly on the inflaton potential. By employing a principal component (PC) decomposition, we isolated the 5 best functional constraints imposed by the WMAP7 data. The analysis is greatly facilitated by our optimization of the WMAP7 likelihood code which we have made publicly 103 Figure 4.12: Principal component amplitudes for the step function potential model [76] that best fits the glitches in P the temperature spectrum at ℓ ∼ 20−40 (upper panel), and projected cumulative (S/N)2 = (ma /σa )2 (lower panel). Given the large values of ma in the high order PC components, ∼ 20 PCs are required to completely characterize this model. available4 . These 5 PCs provide incisive constraints on the inflaton potential around the e-folds of inflation when the scales associated with the first acoustic peak were crossing the horizon. Non-zero values for their amplitudes represent deviations from slow roll and power law initial spectra. The first component implies that deviations are less than ∼ 3% near η ∼ 102 Mpc and the first 5 represent constraints around that scale at better than the 10% level. The result is 5 nearly independent Gaussian constraints that can be applied to any inflationary model where this level of deviation is expected. We have also made the eigenfunctions, which are required to project a given model onto the PC amplitudes, publicly available. These limits are robust to the inclusion of tensor contributions allowed by current B-mode limits, spatial curvature and Sunyaev-Zel’dovich contamination from unresolved clusters. 4. http://background.uchicago.edu/wmap fast 104 Interestingly, for the 4th principal component the null prediction of scale-free initial conditions is disfavored at the 95% CL. However, given the 5 added parameters, this result does not rule out a power law initial spectrum at a significant level. Moreover, the relatively large deviations implied by this anomalous mode are allowed only through correspondingly large variations in the cosmological parameters, mainly the cold dark matter and its effect on the sound horizon and by inference the distance to recombination. Further information from the CMB polarization and high-ℓ temperature power spectrum can break this degeneracy. The QUAD polarization data already have some impact on the constraint and the Planck satellite should definitively resolve this issue. External data also can break this degeneracy. In particular in a flat ΛCDM cosmology, the distance to high redshift supernovae reduce the preference for finite m4 from 1.9σ to 1.2σ. However this improvement disappears if spatial curvature is marginalized. This anomalous 4th PC resembles a local running of the tilt around scales of η ∼ 300Mpc. Direct analysis of a global constant running of the tilt shows that this preference is mainly local, i.e. the low and high multipoles prefer a different and smaller running than the intermediate multipoles that the first 5 PCs probe. The running of the tilt example illustrates the use of the PC constraints both as a technique to constrain inflationary parameters arising from different models with a general analysis and as a method for examining what aspects of the data drive the constraints. The running of the tilt example also illustrates that for models where deviations from scale-free conditions become much larger than ∼ 10% away from the well-constrained region of the acoustic peaks, more principal components are required to ensure a complete and incisive description. We will examine these issues in Chapter 5. 105 CHAPTER 5 COMPLETE WMAP CONSTRAINTS ON BANDLIMITED INFLATIONARY FEATURES In this chapter we constrain the inflationary potential across the entire range of angular scales observable to WMAP. We use a complete basis of 20 principal components that accounts for order unity features in the slow roll parameters as fine as 1/10 of a decade. Although one component shows a deviation at the 98% CL, it cannot be considered statistically significant given the 20 components tested. The maximum likelihood PC parameters only improves 2∆ ln L by 17 for the 20 parameters associated with known glitches in the WMAP power spectrum at large scales. We make model-independent predictions for the matching glitches in the polarization power spectrum that could soon test their inflationary origin with high resolution ground based experiments and the Planck satellite. Even allowing for the presence of features in the temperature spectrum, single field inflation makes sharp falsifiable predictions for the acoustic peaks in the polarization whose violation would require extra degrees of freedom. This complete analysis for bandlimited features in the source function of generalized slow roll can be used to constrain parameters of specific models of the inflaton potential without requiring a separate likelihood analysis for each choice. We illustrate its use by placing bounds on the height and width of the step potential. 5.1 Introduction Most studies of reconstruction of the curvature power spectrum involve parametric, minimally parametric or regularized inverse techniques (e.g. [48, 11, 41, 67, 112, 82, 10, 101, 93, 85, 83]). These studies suffer from two potential issues. Given that fine scale features are observable at high wavenumber, parametric models are not complete unless a very large number of parameters are employed. Secondly, not all curvature power spectra can arise from physical 106 mechanisms in the early universe making parametric models potentially overcomplete and subject to fitting the noise instead of fitting the physics. For example, a delta function in the initial curvature spectrum would be highly observable but not expected to arise in any physical model. In this chapter, we extend our analysis of Chapter 4 to a basis of 20 principal components for the source function of inflationary features. This basis is complete for models where the features vary no more rapidly than 10 per decade of the expansion or about 4 per efold during inflation. In §5.2 we review the principal components technique. In Appendix D, we describe numerical techniques used to reduce the computation time of the analysis. We test the validity of the GSR approximation in Appendix E. We present the results of the WMAP likelihood analysis in §5.3. In §5.4 we develop tests of single field inflation and consider applications to specific classes of potentials. We discuss these results in §5.5. 5.2 Principal Component Analysis The principal components of the WMAP7 Fisher matrix provide an efficient basis with which to decompose the source function ′ G (ln η) = 1 − ns + N X ma Sa (ln η) , (5.1) a=1 where the eigenfunctions Sa sample at a rate of 10 per decade in η or equivalently 4.3 per efold of inflation across η = [1 − 105 ] Mpc. In terms of the width of features in the 1/2 potential, this limit corresponds to ∆φ & ǫH /4.3. This rate is sufficient to capture models that describe the glitches in the WMAP7 power spectrum (see §5.4 for a discussion of the limitations imposed by the sampling). As described in Appendix D we slightly modify the approach in Chapter 4 to improve the convergence properties of the MCMC analysis. Since a constant G′ described by ns is equivalent to tilt in the curvature spectrum and G(ln ηmin ) is equivalent to a normalization 107 parameter we replace them with effective parameters Ḡ′ and Ac . Specifically Ḡ′ is an average of G′ for 30 < η/Mpc < 400 and Ac is the normalization of the temperature power spectrum CℓT T at the first peak ℓ = 220 relative to a fiducial choice that fits the WMAP7 data. From these two phenomenological parameters we can derive constraints for the tilt ns and curvature power spectrum normalization As (see Appendix D). Since the signal-to-noise analysis in Chapter 4 (Fig. 4.12, bottom panel) shows that 20 out of the 50 principal components are required for a complete representation of the WMAP data at our bandlimit [24], we choose N = 20 for our analysis. These first 20 principal components are shown in Fig. 5.1. Note that the first 10 components resemble local Fourier modes around η ≈ 102 Mpc where the well-constrained first acoustic peak gets its power. It is not until components 11-20 that horizon scale features at low multipole or 103 − 104 Mpc are represented. We use the MCMC method to determine joint constraints on the 20 PC amplitudes and cosmological parameters pµ = {m1 , . . . , m20 , Ac , Ḡ, τ, Ωb h2 , Ωc h2 , θ} . (5.2) We place non-informative tophat priors on all parameters in Eq. (5.2). To ensure the validity of the GSR approximation we also place a tophat prior on I1,max = max|I1 (k)|. (5.3) √ As shown in Appendix E, a value of I1,max = 1/ 2 is sufficient to ensure accuracy of the GSR approximation. We call this the GSR condition [26]. Fig. 5.2 shows the maximal contribution to I1 per unit amplitude deviation in each of the first 20 principal components. The higher PCs actually produce a slightly smaller response largely because the frequency of the oscillations in Fig. 5.1 begins to exceed that of the √ nonlinear response function X(kη). Thus a prior of I1,max = 1/ 2 actually allows high PC 108 Figure 5.1: The first 20 principal components of the GSR source G′ as a function of conformal time to the end of inflation, in order of increasing variance from bottom to top. 20 PC components suffice to represent inflationary features observable to WMAP that vary no more rapidly than ∼ 1/4 of an efold. Here and below, dashed red lines represent power law conditions with zero amplitude in the PC components. 109 components to reach order unity and |G′ | to reach ∼ 4 or greater. Figure 5.2: Sensitivity of the nonlinearity parameter I1,max (see Eq. (5.3)) to the amplitude of the first 20 PCs considered individually. This parameter must be less than order unity √ for the GSR approximation to be accurate, and we typically place a prior of I1,max < 1/ 2. For the WMAP7 data [66], we use the optimized approximate likelihood from Appendix C (see Ref. [24]). In addition, we utilize data from BICEP and QUAD which include polarization constraints [14, 12]. We calculate the CMB power spectra with gravitational lensing turned off and the default sparse sampling in ℓ (accuracy boost=1). We correct for these approximations in postprocessing by importance sampling as described in Appendix D before presenting the results in the next section. The main effect is a ∼ 0.5σ upwards shift in the Ωb h2 posterior to compensate the smoothing effect of lensing. In order to ensure that models are compatible with a reasonable cosmology we add nonCMB constraints from the UNION2 supernovae data set1 , the SHOES H0 = (74.2 ± 3.6) km/s/Mpc measurement [97] and a big bang nucleosynthesis constraint of Ωb h2 = 0.022 ± 0.002. These data mainly constrain the energy density components of the universe rather than the inflationary initial conditions. We call the combination of CMB and external data the “all data” analysis. We address the impact of the I1,max prior and the non-CMB data 1. http : //www.supernova.lbl.gov/U nion 110 Parameters 100Ωb h2 Ωc h2 θ τ ns , 1 − Ḡ′ ln[1010 As ] H0 ΩΛ −2 ln L All Data 2.241 ± 0.048 0.1101 ± 0.0040 1.0398 ± 0.0022 0.089 ± 0.014 0.9669 ± 0.9882 3.0808 ± 0.0332 71.23 ± 1.74 0.738 ± 0.0198 8140.06 2.233 0.1098 1.0397 0.086 0.9649 3.0733 71.23 0.739 CMB Only 2.231 ± 0.051 2.229 0.1110 ± 0.0051 0.1116 1.0394 ± 0.0022 1.0394 0.087 ± 0.015 0.085 0.9620 ± 0.0078 0.9622 3.0770 ± 0.0338 3.0746 70.70 ± 2.32 70.40 0.732 ± 0.027 0.730 7608.39 Table 5.1: Power law (PL) parameter results: means, standard deviations (left subdivision of columns) and maximum likelihood values (right subdivision of columns) with CMB data (WMAP7 + BICEP + QUAD) and all data (+UNION2 +H0 + BBN) in a flat universe. H0 and ΩΛ constraints are derived from the other parameters. in §5.3.2 below. Figure 5.3: Constraints on the 20 PC amplitudes from the all-data analysis with an I1,max < √ 1/ 2 prior. The only significant deviation from the ma = 0 PL expectation (red dashed line) is m4 = 0.0427 ± 0.0190. The impact of the prior can be visualized by taking the maximum amplitude of an individual ma that satisfies the prior (solid lines), which implies that only m17 −m20 are significantly prior limited. The maximum likelihood model is shown as starred points. 111 Parameters 100Ωb h2 Ωc h2 θ τ Ḡ′ ln[1010 Ac ] m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 m14 m15 m16 m17 m18 m19 m20 ns ln[1010 As ] H0 ΩΛ 2∆ ln L All Data I1,max 2.279 ± 0.107 0.1127 ± 0.0055 1.0411 ± 0.0030 0.086 ± 0.016 0.0122 ± 0.0268 0.0032 ± 0.0117 0.0048 ± 0.0073 0.0152 ± 0.0122 −0.0120 ± 0.0181 0.0427 ± 0.0190 0.0198 ± 0.0256 −0.0156 ± 0.0325 −0.0061 ± 0.0354 0.0278 ± 0.0486 −0.1239 ± 0.0731 0.0336 ± 0.0609 0.0759 ± 0.0908 −0.0917 ± 0.1027 −0.0947 ± 0.1129 0.1116 ± 0.1616 −0.0199 ± 0.2042 0.1006 ± 0.0975 −0.1253 ± 0.2688 −0.5089 ± 0.2938 0.2239 ± 0.3773 −0.0742 ± 0.4070 1.0299 ± 0.0671 3.0387 ± 0.0582 71.03 ± 2.28 0.730 ± 0.026 16.85 √ = 1/ 2 2.227 0.1101 1.0402 0.096 0.0055 0.0036 0.0060 0.0163 −0.0042 0.0460 0.0050 −0.0089 −0.0015 0.0285 −0.1436 0.0219 0.0225 −0.1604 −0.1895 0.2069 0.0617 0.1318 −0.1953 −0.6131 0.2737 0.0011 1.1296 3.0358 71.22 0.739 All Data I1,max 2.282 ± 0.107 0.1126 ± 0.0056 1.0411 ± 0.0030 0.088 ± 0.016 0.0191 ± 0.0248 0.0032 ± 0.0122 0.0025 ± 0.0071 0.0120 ± 0.0122 −0.0140 ± 0.0179 0.0327 ± 0.0171 0.0168 ± 0.0249 −0.0142 ± 0.0328 −0.0060 ± 0.0333 0.0403 ± 0.0464 −0.0970 ± 0.0670 0.0282 ± 0.0602 0.0599 ± 0.0847 −0.0702 ± 0.0946 −0.0764 ± 0.1036 0.0561 ± 0.1450 0.0191 ± 0.1864 0.0837 ± 0.0964 −0.1094 ± 0.2326 −0.3322 ± 0.2475 0.1524 ± 0.3028 −0.2472 ± 0.3173 1.0075 ± 0.0515 3.0446 ± 0.0573 71.08 ± 2.28 0.731 ± 0.026 14.26 = 1/2 2.410 0.1100 1.0417 0.091 0.0213 0.0098 0.0068 0.0109 −0.0085 0.0481 0.0486 −0.0166 −0.0060 0.0431 −0.1458 0.0462 0.0364 −0.1477 −0.1577 0.2126 −0.0091 0.1102 −0.1302 −0.3798 0.1785 −0.1789 1.0535 3.0654 73.28 0.750 Table 5.2: 20 principal component (PC) parameter results: means, standard deviations (left subdivision of columns) and maximum likelihood (ML) values (right subdivision of columns). √ Fiducial results are for all data and nonlinearity prior I1,max = 1/ 2, left columns, with variations on the nonlinear prior I1,max shown in right columns. Parameters ns − ΩΛ are derived from the chain parameters. The difference in likelihood 2∆ ln L is given for the ML values and taken with respect to the corresponding PL maximum likelihood model in Tab. 5.1. 112 Parameters 100Ωb h2 Ωc h2 θ τ Ḡ′ ln[1010 Ac ] m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12 m13 m14 m15 m16 m17 m18 m19 m20 ns ln[1010 As ] H0 ΩΛ 2∆ ln L CMB Only I1,max 2.160 ± 0.159 0.1297 ± 0.0142 1.0395 ± 0.0032 0.082 ± 0.016 0.0186 ± 0.0283 0.0051 ± 0.0122 0.0021 ± 0.0078 0.0086 ± 0.0137 −0.0151 ± 0.0191 0.0455 ± 0.0195 0.0165 ± 0.0272 0.0062 ± 0.0377 −0.0174 ± 0.0383 0.0174 ± 0.0505 −0.1319 ± 0.0770 0.0150 ± 0.0647 0.0591 ± 0.0966 −0.1100 ± 0.1076 −0.0506 ± 0.1194 0.1507 ± 0.1714 −0.0255 ± 0.2152 0.1481 ± 0.1043 −0.0575 ± 0.2833 −0.4894 ± 0.3083 0.2406 ± 0.3878 −0.1265 ± 0.4065 1.0191 ± 0.0672 3.0684 ± 0.0626 63.86 ± 5.88 0.614 ± 0.105 17.2 √ = 1/ 2 2.110 0.1338 1.0381 0.072 0.0221 0.0056 0.0009 0.0104 −0.0161 0.0583 0.0165 −0.0120 −0.0324 −0.0061 −0.1184 0.0441 0.1339 −0.2137 −0.2300 0.2103 0.0686 0.0772 −0.1376 −0.6610 0.5228 −0.0113 1.0823 3.0726 61.35 0.588 Table 5.3: 20 principal component (PC) parameter results: means, standard deviations (left subdivision of columns) and maximum likelihood (ML) values (right subdivision of columns). √ Results are for CMB data and nonlinearity prior I1,max = 1/ 2. Parameters ns − ΩΛ are derived from the chain parameters. The difference in likelihood 2∆ ln L is given for the ML values and taken with respect to the corresponding PL maximum likelihood model in Tab. 5.1. 113 5.3 MCMC Results In this section, we present the results of the Markov Chain Monte Carlo (MCMC) analysis in the principal component (PC) space of the GSR source function. We discuss the results of our fiducial all data analysis in §5.3.1 and address the impact of priors and non-CMB data in §5.3.2. 5.3.1 All Data For our fiducial results we use the all-data combination of CMB and external data described in the previous section. To establish a baseline for the PC results we start with the ma = 0 power law (PL) case, ∆2R = As (k/kp )ns −1 . Table 5.1 gives the mean, standard deviation of the posterior probabilities, and the maximum likelihood (ML) parameter values for the power law model. For the PC analysis, we take 20 components and a nonlinearity tophat prior of I1,max < √ 1/ 2 (see §5.2). Table 5.2 gives the parameter constraints as well as the maximum likelihood PC model (left columns). The improvement in the ML PC model over the ML power law model is 2∆ ln L = 17 for 20 extra parameters and so is not statistically significant in and of itself. Of course, specific inflationary models may realize this improvement with a smaller set of physical rather than phenomenological parameters (see §5.4.2), and so it is interesting to examine more closely the origin of this improvement. The main improvement comes from ℓ ≤ 60 in the TT part of the WMAP likelihood with a 2∆ ln L = 11.9. We shall see in §5.4.1 that these improvements are largely associated with known features in the WMAP temperature power spectrum. In terms of the principal components, the improvements are localized in only a few of the 20 parameters. Fig. 5.3 plots these ma constraints and ML values. Most of the components are consistent with zero at the ∼ 1σ level. Components m17 − m20 are constrained in part 114 Figure 5.4: Parameter probability distributions from the all-data analysis in a flat universe √ with I1,max = 1/ 2. Dashed lines represent the posteriors with approximations for the low ℓ polarization likelihood and Cℓ accuracy used to run the MCMC (see Appendix); solid lines represent posteriors corrected by importance sampling. Red dashed lines represent corrected posteriors for power law models. by the I1,max prior not just the data. As in the 5 PC analysis of Chapter 4, the single most discrepant parameter between the PL and PC cases is m4 corresponding to a feature centered around η ∼ 300Mpc and resembling a local running of the tilt. Fig. 5.4 shows the posterior probability distributions of the parameters. An m4 value as extreme as the power law value of m4 = 0 is disfavored at 98.2% CL compared with 94.8% for 5 PCs and WMAP7 alone. The increase in significance 115 Figure 5.5: The temperature (left) and E-mode polarization (right) power spectra posterior √ using the all-data PC constraints and a prior of I1,max = 1/ 2. The shaded area encloses the 68% CL region and the upper and lower curves show the upper and lower 95% CL limits. The maximum likelihood (ML) model is shown as the thick black central curve, and the power law ML model is shown in red dashed lines. The blue points with error bars show the 7-year WMAP measurements. by a fraction of a σ arises because of the correlation between m4 and the higher principal components. Perhaps more importantly, freedom in the higher PCs allows large m4 without the need to make large adjustments to the cosmological parameters that would violate nonCMB constraints. On the other hand, one event out of 20 showing a 98% exclusion is not that unlikely. The poorly constrained a > 10 modes allow large amplitude deviations and in fact even marginally prefer them. This explains why including the higher components can change results on the lower components. Large amplitude deviations in the high order components make the modes no longer statistically independent as they would be for infinitesimal deviations. Still the correlation remains relatively small. For example R4a = Cov(m4 , ma )/σm4 σma reaches 0.4 only for one mode, m5 , with more typical correlations in the ±0.1 − 0.2 range. 116 Figure 5.6: Parameter decomposition of the temperature power spectrum difference between the power law (PL) and PC maximum likelihood (ML) models shown in Fig. 5.5 (bottom panel). The curves include cumulative changes in parameters between the models starting with the cosmological parameters, adding the normalization Ac and effective tilt Ḡ′ , m1 . . . m5 , etc. until the full PC ML parameters are utilized. The next most significant deviations are in m9 (with a value of m9 = 0 disfavored at the 89.6% CL) and m18 (with a value of m18 = 0 disfavored at the 91.8% CL). These results are also consistent with the PL null hypothesis of ma = 0, given that there are only 3 events out of 20 where tests of that model exceed the ∼ 90% CL. We can get further insight on the origin of these constraints by examining the maximum likelihood (ML) models. Fig. 5.5 show temperature and polarization power spectra of the ML PL (red dashed lines) and PC (thick solid curve) models respectively. The poorly constrained a > 10 modes create fluctuations in the low order multipoles which marginally fit features in the data better such as the low quadrupole and glitch at ℓ ∼ 20 − 40. These large amplitude modes require small amplitude low order PC variations in order to compensate the broad band residual effects they have. This can be seen by decomposing the difference between the ML PL and PC models into contributions from the various parameters (see Fig. 5.6). Removing the large m10 − m20 components from the model not only removes the low ℓ oscillations but also creates broadband deviations, especially at ℓ . 40, that are 117 compensated by a combination of small amplitude changes in m1 − m5 and effective tilt Ḡ′ . 5.3.2 Robustness Tests In order to test the robustness of the fiducial results of the last section, we run separate MCMC chains with different choices for the nonlinearity prior and data sets. We first examine the impact of our I1,max prior by reanalyzing the all-data case with √ I1,max = 1/2 instead of 1/ 2 (see Tab. 5.2). The main impact of tightening the prior is on m18 − m20 as is expected from Fig. 5.3. These components mainly affect the low ℓ multipoles. In spite of this fact the prior on I1,max has very little impact on the behavior of favored models at low ℓ. In Fig. 5.7, we show the maximum likelihood model with the stronger I1,max prior. Even at low ℓ the differences are much smaller than cosmic variance. In particular the posterior distribution of power in the quadrupole moment for models in the chain shown in Fig. 5.8 differ negligibly. Some of this robustness in the low multipole moments is due to the impact of the nonCMB data. Without the external data, the quadrupole distribution extends to smaller quadrupole moments due to the ability to reduce the integrated Sachs-Wolfe effect by lowering the cosmological constant in the absence of constraints on the acceleration of the expansion (see Fig. 5.8). In this case the data may prefer more extreme inflationary models that further lower the quadrupole that are excluded by our nonlinearity prior on I1,max [17]. The main impact on parameters of removing the non-CMB data is to allow a wider range in Ωc h2 (see Table 5.3). In contrast to the 5PC analysis of Chapter 4, this wider range though has little impact on the PC parameters. In particular the higher order PC components allow compensation of the effects of m4 across the acoustic peaks without the need to vary Ωc h2 substantially. For similar reasons, we expect our flatness prior to have little impact on the PC results aside from weakening the constraints on ΩΛ and Ωc h2 and small shifts of the location of features in G′ with the angular diameter distance degeneracy. 118 Figure 5.7: Comparison of the maximum likelihood √ models of the three MCMCs of Tables 5.2 and 5.3: the all-data analysis with I1,max = 1/ 2 (black curve), all-data with I1,max = 1/2 √ (blue curve), and CMB data with I1,max = 1/ 2 (red curve). The smallness of the differences indicates robustness of our results to the priors and external data sets. Figure 5.8: The temperature quadrupole power C2T T distribution for the all-data analysis √ with I1,max = 1/ 2 (black curve), all-data with I1,max = 1/2 (blue curve), and CMB √ data with I1,max = 1/ 2 (red curve). Without external data to constrain the cosmological constant, the quadrupole can be lowered by reducing the integrated Sachs-Wolfe effect. 119 5.4 Applications Here we discuss applications of the fiducial 20 PC analysis of §5.3.1. In §5.4.1 we place constraints on and devise tests of slow roll and single field inflation in a model independent manner. Alternately, as a complete observational basis for efold bandlimited models, the PC analysis places constraints on any such model that satisfies the GSR condition. We use running of the tilt and a step in the inflaton potential as example test cases in §5.4.2. 5.4.1 Testing Slow Roll and Single Field Inflation Bounds on the PC components can be thought of as functional constraints on G′ itself across the observed range from WMAP. These in turn limit features in the inflaton potential V (φ) through the approximate relation of Eq. (4.4). If the inflaton carries non-canonical kinetic terms then the relationship is modified to include variations in the sound speed [46]. Since the PC decomposition only represents features in G′ across the observable domain, one should consider the constraints on the ma s as defining a PC filtered version of G′ : G′20 (ln η) = 20 X ma Sa (ln η) . (5.4) a=1 Any significant deviation from zero of this function would indicate a violation of ordinary slow roll. We can extract the posterior probability of G′20 by considering its values on a continuous set of samples of η as derived parameters. In Fig. 5.9 we plot both the ML model and the 68% and 95% posterior bands. Note that G′20 = 0 lies within the 95% CL regime for all η. These functional constraints differ from a full reconstruction of G′ in that the PCs filter out deviations at η < 20 Mpc and η > 104 Mpc as well as deviations that are too high frequency to satisfy our bandlimit. In the well-constrained regime of 30 . η/Mpc . 400 constraints are both tight and consistent with G′20 = 0. Only nearly zero mean high frequency deviations are allowed in this regime. Nonetheless, the poorly constrained m10 − m20 components allow, but do not 120 strongly prefer, large oscillatory features between 103 . η/Mpc . 104 . In fact G′20 = 0 lies noticeably outside the 68% CL bands only for the dip and bump between 1000 − 2000 Mpc and a bump at 70 − 100 Mpc. We can associate the most significant features with the corresponding effects on the observable power spectra themselves. Figure 5.5 shows the 68% and 95% range in the power spectra posterior. The 1000 − 2000 Mpc feature in fact corresponds to the ℓ = 20 − 40 dip and bump in the temperature power spectrum. The 70 − 100 Mpc feature corresponds to a glitch at ℓ ∼ 600 − 700 [18, 40]. While the η & 104 Mpc regime is limited by our priors on the amplitude of deviations through I1,max we have shown that the data do not favor a feature corresponding to a low quadrupole ℓ = 2 unless acceleration constraints are omitted (see §5.3.2). Finally, we can examine the posterior distributions of the E-mode polarization. These predictions are not significantly constrained by the polarization data sets employed. Instead these distributions are limited mainly by the common origin of the temperature and polarization spectra from single field inflation. These serve as predictions for future measurements. For example, as we saw in Chapter 2, the low significance features in the temperature power spectrum predict corresponding features in the E-mode polarization which have yet to be measured and can be used to test the hypothesis of their inflationary origin at substantially higher joint significance [76]. In particular, one expects a ∼ 26%+13% enhancement in the −17% EE power spectrum at ℓ = 39 and a ∼ −37%+17% deficit around ℓ = 25. The skew distribu−3% tion in the latter case reflects the difficulty in constructing models with low power out of the principal components rather than the data disfavoring such models. Models that actually explain the low T T power at ℓ = 25 predict low EE power as well. Even in the acoustic regime where the polarization predictions are tight and do not suggest the presence of features, these predictions are of interest. If future observations violate them, then not only will slow-roll inflation be falsified but all single field inflationary models, including those with sound speed variations, as long as they satisfy our weak prior constraint 121 √ on acceptable models: the efold bandlimit and small GSR non-linearity I1,max < 1/ 2. Such a violation might indicate other degrees of freedom breaking the relationship between the temperature and polarization fields, e.g. isocurvature modes in multifield inflation or trace amounts of cosmological defects. For ℓ . 30 violation could alternately indicate a more complicated reionization scenario, as we saw in Chapter 2. Currently these bounds and tests apply to the ℓ < 800 regime measured by WMAP but will soon be extended by high resolution ground based experiments and Planck. Figure 5.9: The 20 PC filtered G′ posterior from the fiducial all-data analysis and I1,max = √ 1/ 2 as a prior. The shaded area encloses the 68% CL region and the upper and lower curves show the upper and lower 95% CL limits. The maximum likelihood is shown as the thick black central curve, and the power law ML model is shown in red dashed lines. 5.4.2 Constraining Inflationary Models We can also apply the model independent principal component analysis to any specific set of √ models that satisfy the GSR condition I1,max < 1/ 2 and bandlimit of features no sharper than about 1/4 efold. To place constraints on the parameters of a model, one projects the 122 source function G′ of the model onto the principal components Z ηmax 1 dη ma = Sa (ln η)G′ (ln η) ln ηmax − ln ηmin ηmin η (5.5) as a function of parameters and compares the result to the joint posterior probability distributions of the components. Likewise one can construct G′20 from the result and compare it with Fig. 5.9. In fact, the means and covariance matrix C of the components ma form a simple but useful representation of the joint PC posteriors. From these, one can construct a χ2 statistic χ2 20 h i X −1 = (ma − m̄a )Cab (mb − m̄b ) , (5.6) a,b=1 or the likelihood L ∝ exp(−χ2 /2) under a multivariate Gaussian approximation to the posteriors. For example the ML PC model gives an improvement of ∆χ2 = −15.36 over PL to be compared with −2∆ ln L = −16.85. As a simple illustration of a concrete model, consider a linear deviation in G′ G′ (ln η) = 1 − n0 + α ln (η/η0 ) . (5.7) The curvature power spectrum for this model has a local tilt of d ln ∆2R = n0 − 1 + α ln d ln k απ I1 , −√ 2 1 + I12 (5.8) i 1 hπ (1 − n0 − α ln kη0 ) + 1.67α . I1 = √ 2 2 (5.9) kη0 C where C = e7/3−γE /2 ≈ 2.895 and For |n0 − 1| ≪ 1 and |α| ≪ 1, the I1 term contributes negligibly and the model gives a linear 123 running of the tilt [24]. Figure 5.10: A model with a linear deviation in G′ with slope α = −0.026 (and arbitrary offset) is shown as the blue curve. The 20 PC filtered source G′20 (in black lines) is compared with the input linear G′ model. 20 PCs captures all of the observable information in α. These models are compatible with the 68% CL region (shaded) for G′20 from the fiducial all data analysis. The 20 PC components are a linear function of α given explicitly by Z ηmax α dη ma (α) = Sa (ln η) ln(η/η0 ) . ln ηmax − ln ηmin ηmin η (5.10) In Fig. 5.10 we show an example with α = −0.026 and compare the original linear G′ to the PC filtered G′20 . The filter introduces features at low and high η that are not present in the actual source. Note that a Fisher analysis of sensitivity to α reveals that most of the signal to noise should lie in the m4 component (see Fig. 4.5) which carries the most significant deviations from zero in the data. The χ2 analysis with all data implies α = −0.039±0.019. We can compare this result to a direct MCMC analysis with α as a parameter constructed from 20 PCs: α = −0.027 ± 0.021. Thus the simple χ2 approximation captures the information on α in the 20 PC posterior to ∼ 0.5σ. 124 We can further test the completeness of the 20 PC decomposition of α by going to 50 PCs. In this case α = −0.026 ± 0.023 showing that 20 PCs completely describe the observable properties of α. In fact, 5 PCs are enough to describe the observable properties of α in this case; a direct MCMC analysis gives α = −0.026 ± 0.020. Fig. 5.11 shows that the full posterior distributions of α for these cases are indistinguishable within the errors. We also show the simple χ2 approximation which is shifted by ∼ 0.5σ as expected. Figure 5.11: Posterior probability distribution of α from a direct MCMC analysis constructed from 50 PCs (black/solid curve), 20 PCs (red curve), and 5 PCs (blue curve). The distribution from the χ2 approximation is shown in black/dashed curve. The posterior distributions are skewed to negative values of α. For example the ML model of the 50 PC chain has α = −0.021 to be compared with a mean of −0.026. For large negative α, the linear G′ model no longer matches a running of the tilt due to the I1 terms in Eq. (5.9). In Fig. 5.12, we show an example with n0 = 0.96 and η0 = 145 Mpc for α = dns /d ln k = −0.09 and −0.02. While the α model closely matches constant dns /d ln k for the smaller value, it produces substantially less deviations at high and low k. This bias explains the difference between constraints on the linear α model and running of the tilt found in Chapter 4. For example, with the same data sets and priors running of the tilt gives dns /d ln k = −0.018 ± 0.019. Note the ML α = −0.021 from the 50PC chain is 125 consistent with this constraint. Figure 5.12: Initial curvature power spectrum of a model with running of the tilt (dns /d ln k = −0.02, −0.09, solid curves) compared to a model with a linear deviation in G′ (α = −0.02, −0.09, dashed curves). For the −0.02 case, the two models are similar whereas for −0.09 the running of the tilt model has larger deviations from scale free conditions at low and high k. Another example is the step potential with an effective mass given by Eq. (2.1). For simplicity, we fix b = 14.668 so that the feature appears at the correct position to explain the glitches. Although we set the smooth part of the potential to correspond to an m2 φ2 model with m = 7.126 × 10−6 for the projection onto PCs, in the analysis we retain the freedom to adjust the amplitude and tilt as usual. This leaves us with 2 additional parameters c and d to control the amplitude and width of the step. The constraints on (c, d) from the χ2 approximation are shown in Fig. 5.13 (top panel). Note that the crude χ2 analysis correctly picks out the favored parameters which can explain the glitches [76]. The minimum χ2 model is c = 0.0015, d = 0.026 and is favored over the PL ma = 0 (or c = 0) model by ∆χ2 = −10.2. Although the χ2 analysis assumes that the joint posterior in ma is a multivariate Gaussian, it does not make that assumption for parameter probabilities. With the distorted shape of the confidence region, the 68% contour corresponds to ∆χ2 = 2.5, 95% contour to 8.6 and 99.7% contour to 13.3, compared with 126 Figure 5.13: Constraints on the step potential model parameters c (height of step) and d width of step. Top panel: the χ2 approximation (black curves) compared to the full 20 PC posterior (blue curves). Bottom panel: constraints from the 20 PCs posterior (blue curves) compared to a direct GSR calculation of the model (black points). the more stringent 2.3, 6.2 and 11.6 obtained for Gaussian distributions in (c, d). Here and below we take a prior of d > 0.005 due to our bandlimit of 1/4 efold (see below). We again compare this with a full analysis of the joint 20 PC posteriors. As in the case of α, the projection onto the two dimensional ma (c, d) space leaves us with too few samples in the original 20 PC chain to reliably extract the posterior via importance sampling. We instead run a direct MCMC analysis on the 20 PC description with ma (c, d). These results are shown in Fig. 5.13 in blue lines. The maximum likelihood model has c = 0.0016, d = 0.025 and is favored over PL ma = 0 (or c = 0) by 2∆ ln L = 9.1. These values are 127 fully consistent with the simple χ2 analysis. This improvement is a substantial fraction of the total of 17 available to the 20 PCs from Tab. 5.2 and is achieved with 3 parameters: c, d and implicitly b, the location of the step. The filtered G′20 source for both the ML and minimum χ2 model are shown in Fig. 5.14 and are consistent with the posteriors of the fiducial all data analysis. Furthermore, the χ2 analysis correctly picks out the best fit region and qualitatively recovers its distorted shape. The main difference is that the confidence region is slightly underestimated. Figure 5.14: The ML model of the step potential from the χ2 approximation is shown in blue dashed lines, and the ML model from the projection onto 20 PCs [ma (c, d)] is shown in black lines. The step potential model captures the main feature seen in the fiducial all-data analysis (shaded 68% CL area). Finally, we test the completeness of the 20 PC description of the step model by conducting a separate MCMC with the full function G′ directly (see Appendix E, Eqs. (E.10)-(E.13) for details). The maximum likelihood model has c = 0.0021, d = 0.029 and is favored over PL by 2∆ ln L = 9.5 As shown in Fig. 5.13 (bottom panel), the main difference is that the models are more tightly constrained at d < 0.01. The features in G′ span less than ∼ 1/4 of an efold for these models and consequently the 20 PC decomposition is not complete. In Fig. 5.15 (top panel) we show a model with d = 9.2 × 10−3 and c = 4.6 × 10−4 represented by the 128 full function G′ (in black lines) compared to its 20 PCs description (in blue/dashed curves). The fractional difference between these two constructions is shown in the bottom panel. In such models, the oscillations in the temperature power spectrum continue to higher ℓ, in this case ℓ ∼ 100, and are not allowed by the data. This example shows that the main limitation of the 20 PC analysis is that it is too conservative for models with high frequency structure in the source: such models tend to be in conflict with the data in ways not represented by the principal components. Figure 5.15: Top panel: step potential model with width d = 9.2 × 10−3 and height c = 4.6 × 10−4 represented by the full source function G′ (in black lines) compared to its 20 PC description (in blue/dashed lines). Bottom panel: fractional difference between the full GSR description and its 20 PC decomposition. The oscillations at ℓ ∼ 100 are not captured by the 20 PCs. 129 5.5 Discussion We have conducted a complete study of constraints from the WMAP7 data on inflationary features beyond the slow roll limit. Using a principal component (PC) basis that accommodates order unity features as fine as 1/10 of a decade across more than 2 decades of the inflationary expansion, we find no significant deviations from slow roll. Although one component shows a deviation at the 98% CL, it cannot be considered statistically significant given the 20 components tested. The maximum likelihood PC parameters only improves 2∆ ln L by 17 for the 20 parameters added. On the other hand, specific inflationary models may access this improvement with fewer physical parameters. Most of the improvement comes from fitting features in the temperature power spectrum at multipoles ℓ ≤ 60 with the known glitch at 20 ≤ ℓ ≤ 40 comprising a large fraction. From our analysis, we also extract predictions for the corresponding features in the polarization power spectrum that can be used to test their inflationary origin independently of a specific choice for the inflaton potential (see Chapter 2). In particular, one expects a ∼ 26% enhancement in the EE power spectrum at ℓ = 39 and a ∼ 37% deficit around ℓ = 25 if the temperature features have an inflationary origin. Outside of the range of these low ℓ features, the predictions are very precise and any violation of them in future observations would falsify single field inflation independently of the potential. Our constraints can also be used to test any single field model that satisfies our conditions. Most of the information from the likelihood analysis is distilled in the means and covariance of the principal components themselves which we make publicly available2 . Two models illustrate this encapsulation: a linear source model that approximates running of the tilt and the step potential model introduced in Chapter 2. A simple χ2 analysis approximates the joint parameter posteriors despite its highly non-Gaussian form for the step parameters. 2. http://background.uchicago.edu/wmap fast 130 This procedure greatly simplifies the testing of inflationary models with features in that parameter constraints on any model that satisfies our conditions can be simply approximated without a case-by-case likelihood analysis. 131 CHAPTER 6 CONCLUSIONS In this thesis we have discussed aspects of inflation that can be revealed by using observations of the Cosmic Microwave Background (CMB). Below we summarize the main points learned. Oscillatory features in the CMB temperature power spectrum have been interpreted as possible evidence for new physics during inflation. Previously [106, 41, 100, 81, 102, 84] it has been shown that a model with a step in the inflationary potential can give rise to these oscillations. In Chapter 2 we show that future CMB polarization measurements have the potential to shed light on this question. The inflationary feature hypothesis predicts features in the E-mode polarization power spectrum with a structure similar to that in the temperature power spectrum. We exploit the fact that, in the relevant multipole range, the sharpness of the polarization transfer function and lack of contamination by secondary effects makes polarization a cleaner probe of such features than temperature. Under the simplest set of assumptions for large-scale polarization in which we take the best-fit model for the temperature features, neglect tensor fluctuations, and take the reionization history to be instantaneous, polarization measurements from Planck should be able to confirm or exclude the inflationary features that best match current temperature data at 3σ significance Allsky experiments beyond Planck could potentially increase this significance to 8σ, providing a definitive test for features from inflation. We also quantify possible sources of degradation of this significance and we find that the main source of confusion with inflationary features comes from polarization features created by a complex reionization history. The estimated significance also degrades slightly with the addition of a large-amplitude, smooth tensor component to the E-mode spectrum, which tends to hide the effect of an inflationary step at the largest scales. On the other hand, there are many possible choices of inflationary potential, and each choice is typically handled by numerically solving the field equation on a case-by-case basis. In Chapter 3, we construct an accurate prescription which generalizes previous studies by 132 directly relating the CMB observables to the shape of the inflaton potential. We develop a variant of the generalized slow roll approach for calculating the curvature power spectrum that is accurate at the percent level for order unity deviations in power caused by sharp features in the inflaton potential. We show that to good approximation there is a single source function that is responsible for observable features and that this function is simply related to the local slope and curvature of the inflaton potential. We first apply this formalism to the best constrained region of WMAP 7-year data by means of a principal component decomposition of the source function (in Chapter 4). Detection of any non-zero component would represent a violation of ordinary slow roll and indicate a feature in the inflaton potential or sound speed. One component (that resembles a local running of the tilt) shows a 95% CL preference for deviations around the 300 Mpc scale at the ∼ 10% level, but the global significance is reduced considering the 5 components examined. This deviation also requires a change in the cold dark matter density which in a flat ΛCDM model is disfavored by current supernova and Hubble constant data. We show that the inflaton potential can be even better constrained with current and upcoming high sensitivity experiments that will measure small-scale temperature and polarization power spectra of the CMB. In Chapter 5, we extend our analysis to the entire range of angular scales observable to the WMAP satellite, and we test for the hypotheses of single-field and slow-roll inflation in a model independent manner. We use a complete basis of 20 principal components that accounts for order unity features in the slow roll parameters as fine as 1/10 of a decade. Although one component shows a deviation at the 98% CL, it cannot be considered statistically significant given the 20 components tested. The maximum likelihood model coming from this analysis only improves 2∆ ln L by 17, and this improvement is associated with known glitches in the WMAP power spectrum at large scales. We provide a model-independent test of the single-field inflationary hypothesis that consists on predictions for the matching features in the CMB polarization power spectrum, that 133 could be soon tested by the Planck satellite and high-resolution ground based experiments. Single field inflation makes falsifiable predictions for the acoustic peaks in the polarization whose violation would require extra degrees of freedom (for example, isocurvature modes in multifield inflation). The constraints coming from this analysis serve as a basis to test any model of single-field inflation with features that are no finer than 1/10 of a decade and that satisfies our “GSR condition” (cf. Chapter 5). This greatly simplifies the testing of models in that constraints of inflationary parameters of models with features can be now approximated without requiring a separate likelihood analysis for each choice. 134 REFERENCES [1] Jennifer A. Adams, Bevan Cresswell, and Richard Easther. Inflationary perturbations from a potential with a step. Phys. Rev., D64:123514, 2001. [2] Jennifer A. Adams, Graham G. Ross, and Subir Sarkar. Multiple inflation. Nucl. Phys., B503:405–425, 1997. [3] Peter Adshead, Wayne Hu, Cora Dvorkin, and Hiranya V. Peiris. Fast Computation of Bispectrum Features with Generalized Slow Roll. 2011. [4] Laila Alabidi and James E. Lidsey. Single-Field Inflation After WMAP5. Phys. Rev., D78:103519, 2008. [5] Amjad Ashoorioon and Axel Krause. Power spectrum and signatures for cascade inflation. 2006. [6] Daniel Baumann. TASI Lectures on Inflation. 2009. [7] Daniel Baumann et al. CMBPol Mission Concept Study: Probing Inflation with CMB Polarization. AIP Conf. Proc., 1141:10–120, 2009. [8] C. L. Bennett et al. First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results. Astrophys. J. Suppl., 148:1, 2003. [9] M. Bridges, F. Feroz, M. P. Hobson, and A. N. Lasenby. Bayesian optimal reconstruction of the primordial power spectrum. 2008. [10] Michael Bridges, A.N. Lasenby, and M.P. Hobson. WMAP 3-year primordial power spectrum. Mon.Not.Roy.Astron.Soc., 2006. [11] S. L. Bridle, A. M. Lewis, J. Weller, and G. Efstathiou. Reconstructing the primordial power spectrum. Mon. Not. Roy. Astron. Soc., 342:L72, 2003. [12] M. L. Brown et al. Improved measurements of the temperature and polarization of the CMB from QUaD. Astrophys. J., 705:978–999, 2009. [13] C. P. Burgess, Richard Easther, Anupam Mazumdar, David F. Mota, and Tuomas Multamaki. Multiple inflation, cosmic string networks and the string landscape. JHEP, 05:067, 2005. [14] H. C. Chiang et al. Measurement of CMB Polarization Power Spectra from Two Years of BICEP Data. Astrophys. J., 711:1123–1140, 2010. [15] Jeongyeol Choe, Jinn-Ouk Gong, and Ewan D. Stewart. Second order general slow-roll power spectrum. JCAP, 0407:012, 2004. [16] Nelson Christensen, Renate Meyer, Lloyd Knox, and Ben Luey. II: Bayesian Methods for Cosmological Parameter Estimation from Cosmic Microwave Background Measurements. Class. Quant. Grav., 18:2677, 2001. 135 [17] Carlo R. Contaldi, Marco Peloso, Lev Kofman, and Andrei D. Linde. Suppressing the lower multipoles in the CMB anisotropies. JCAP, 0307:002, 2003. [18] Laura Covi, Jan Hamann, Alessandro Melchiorri, Anze Slosar, and Irene Sorbera. Inflation and WMAP three year data: Features have a future! Phys. Rev., D74:083509, 2006. [19] S. Dodelson. Modern Cosmology. Academic Press, 2003. [20] Scott Dodelson and Ewan Stewart. Scale dependent spectral index in slow roll inflation. Phys. Rev., D65:101301, 2002. [21] Olivier Dore, Gilbert P. Holder, and Abraham Loeb. The CMB Quadrupole in a Polarized Light. Astrophys. J., 612:81–85, 2004. [22] J. Dunkley et al. CMBPol Mission Concept Study: Prospects for polarized foreground removal. 2008. [23] J. Dunkley et al. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data. Astrophys. J. Suppl., 180:306–329, 2009. [24] Cora Dvorkin and Wayne Hu. CMB Constraints on Principal Components of the Inflaton Potential. Phys. Rev., D82:043513, 2010. [25] Cora Dvorkin and Wayne Hu. Generalized Slow Roll for Large Power Spectrum Features. Phys. Rev., D81:023518, 2010. [26] Cora Dvorkin and Wayne Hu. Complete WMAP Constraints on Bandlimited Inflationary Features. 2011. [27] Cora Dvorkin, Wayne Hu, and Kendrick M. Smith. B-mode CMB Polarization from Patchy Screening during Reionization. Phys. Rev., D79:107302, 2009. [28] Cora Dvorkin, Hiranya V. Peiris, and Wayne Hu. Testable polarization predictions for models of CMB isotropy anomalies. Phys. Rev., D77:063008, 2008. [29] Cora Dvorkin and Kendrick M. Smith. Reconstructing Patchy Reionization from the Cosmic Microwave Background. Phys. Rev., D79:043003, 2009. [30] George Efstathiou, Steven Gratton, and Francesco Paci. Impact of Galactic polarized emission on B-mode detection at low multipoles. 2009. [31] Xiao-Hui Fan, C. L. Carilli, and Brian G. Keating. Observational constraints on Cosmic Reionization. Ann. Rev. Astron. Astrophys., 44:415–462, 2006. [32] Wenjuan Fang et al. Challenges to the DGP Model from Horizon-Scale Growth and Geometry. Phys. Rev., D78:103509, 2008. [33] A. Gelman and D. Rubin. Statistical Science, 7:452, 1992. 136 [34] Jinn-Ouk Gong. Breaking scale invariance from a singular inflaton potential. JCAP, 0507:015, 2005. [35] Christopher Gordon and Wayne Hu. A Low CMB Quadrupole from Dark Energy Isocurvature Perturbations. Phys. Rev., D70:083003, 2004. [36] Christopher Gordon and Roberto Trotta. Bayesian Calibrated Significance Levels Applied to the Spectral Tilt and Hemispherical Asymmetry. Mon. Not. Roy. Astron. Soc., 382:1859–1863, 2007. [37] Alan H. Guth. The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems. Phys. Rev., D23:347–356, 1981. [38] Salman Habib, Andreas Heinen, Katrin Heitmann, Gerard Jungman, and Carmen Molina-Paris. Characterizing inflationary perturbations: The uniform approximation. Phys. Rev., D70:083507, 2004. [39] Jan Hamann. 2009. private communication. [40] Jan Hamann, Laura Covi, Alessandro Melchiorri, and Anze Slosar. New constraints on oscillations in the primordial spectrum of inflationary perturbations. Phys. Rev., D76:023503, 2007. [41] Steen Hannestad. Reconstructing the primordial power spectrum - a new algorithm. JCAP, 0404:002, 2004. [42] G. Hinshaw et al. Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Temperature analysis. Astrophys. J. Suppl., 170:288, 2007. [43] G. Hinshaw et al. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:Data Processing, Sky Maps, & Basic Results. Astrophys. J. Suppl., 180:225– 245, 2009. [44] Christopher M. Hirata and Uros Seljak. Reconstruction of lensing from the cosmic microwave background polarization. Phys. Rev., D68:083002, 2003. [45] Richard Holman, Edward W. Kolb, Sharon L. Vadas, and Yun Wang. Plausible double inflation. Phys. Lett., B269:252–256, 1991. [46] Wayne Hu. Generalized Slow Roll for Non-Canonical Kinetic Terms. 2011. [47] Wayne Hu and Gilbert P. Holder. Model-Independent Reionization Observables in the CMB. Phys. Rev., D68:023001, 2003. [48] Wayne Hu and Takemi Okamoto. D69:043004, 2004. Principal Power of the CMB. Phys. Rev., [49] Wayne Hu and Martin J. White. A CMB Polarization Primer. New Astron., 2:323, 1997. 137 [50] Paul Hunt and Subir Sarkar. Multiple inflation and the WMAP ’glitches’. Phys. Rev., D70:103518, 2004. [51] Paul Hunt and Subir Sarkar. Multiple inflation and the WMAP ’glitches’ II. Data analysis and cosmological parameter extraction. Phys. Rev., D76:123504, 2007. [52] Rajeev Kumar Jain, Pravabati Chingangbam, Jinn-Ouk Gong, L. Sriramkumar, and Tarun Souradeep. Double inflation and the low CMB multipoles. JCAP, 0901:009, 2009. [53] N. Jarosik et al. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Sky Maps, Systematic Errors, and Basic Results. Astrophys. J. Suppl., 192:14, 2011. [54] Minu Joy, Varun Sahni, and Alexei A. Starobinsky. A New Universal Local Feature in the Inflationary Perturbation Spectrum. Phys. Rev., D77:023514, 2008. [55] Minu Joy, Arman Shafieloo, Varun Sahni, and Alexei A. Starobinsky. Is a step in the primordial spectral index favored by CMB data ? JCAP, 0906:028, 2009. [56] Minu Joy, Ewan D. Stewart, Jinn-Ouk Gong, and Hyun-Chul Lee. From the Spectrum to Inflation : An Inverse Formula for the General Slow-Roll Spectrum. JCAP, 0504:012, 2005. [57] Kenji Kadota, Scott Dodelson, Wayne Hu, and Ewan D. Stewart. Precision of inflaton potential reconstruction from CMB using the general slow-roll approximation. Phys. Rev., D72:023510, 2005. [58] Marc Kamionkowski, Arthur Kosowsky, and Albert Stebbins. Statistics of Cosmic Microwave Background Polarization. Phys. Rev., D55:7368–7388, 1997. [59] Masahiro Kawasaki, Fuminobu Takahashi, and Tomo Takahashi. Making waves on CMB power spectrum and inflaton dynamics. Phys. Lett., B605:223–227, 2005. [60] William H. Kinney, Edward W. Kolb, Alessandro Melchiorri, and Antonio Riotto. Latest inflation model constraints from cosmic microwave background measurements. Phys. Rev., D78:087302, 2008. [61] Noriyuki Kogo, Misao Sasaki, and Jun’ichi Yokoyama. Reconstructing the Primordial Spectrum with CMB Temperature and Polarization. Phys. Rev., D70:103001, 2004. [62] E. Komatsu et al. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations:Cosmological Interpretation. Astrophys. J. Suppl., 180:330–376, 2009. [63] E. Komatsu et al. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. Astrophys. J. Suppl., 192:18, 2011. [64] Arthur Kosowsky, Milos Milosavljevic, and Raul Jimenez. Efficient Cosmological Parameter Estimation from Microwave Background Anisotropies. Phys. Rev., D66:063007, 2002. 138 [65] John Kovac et al. Detection of polarization in the cosmic microwave background using DASI. Nature, 420:772–787, 2002. [66] D. Larson et al. Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Power Spectra and WMAP-Derived Parameters. Astrophys. J. Suppl., 192:16, 2011. [67] Samuel M. Leach. Measuring the primordial power spectrum: Principal component analysis of the cosmic microwave background. Mon. Not. Roy. Astron. Soc., 372:646– 654, 2006. [68] Samuel M. Leach and Andrew R. Liddle. Inflationary perturbations near horizon crossing. Phys. Rev., D63:043508, 2001. [69] Julien Lesgourgues. Features in the primordial power spectrum of double D-term inflation. Nucl. Phys., B582:593–626, 2000. [70] Julien Lesgourgues and Wessel Valkenburg. New constraints on the observable inflaton potential from WMAP and SDSS. Phys. Rev., D75:123519, 2007. [71] Antony Lewis and Sarah Bridle. Cosmological parameters from CMB and other data: a Monte- Carlo approach. Phys. Rev., D66:103511, 2002. [72] Antony Lewis, Anthony Challinor, and Anthony Lasenby. Efficient Computation of CMB anisotropies in closed FRW models. Astrophys. J., 538:473–476, 2000. [73] James E. Lidsey et al. Reconstructing the inflaton potential: An overview. Rev. Mod. Phys., 69:373–410, 1997. [74] Jerome Martin and Christophe Ringeval. Superimposed Oscillations in the WMAP Data? Phys. Rev., D69:083515, 2004. [75] John C. Mather, D. J. Fixsen, R. A. Shafer, C. Mosier, and D. T. Wilkinson. Calibrator Design for the COBE Far Infrared Absolute Spectrophotometer (FIRAS). Astrophys. J., 512:511–520, 1999. [76] Michael J. Mortonson, Cora Dvorkin, Hiranya V. Peiris, and Wayne Hu. CMB polarization features from inflation versus reionization. Phys. Rev., D79:103519, 2009. [77] Michael J. Mortonson and Wayne Hu. Impact of reionization on CMB polarization tests of slow- roll inflation. Phys. Rev., D77:043506, 2008. [78] Michael J. Mortonson and Wayne Hu. Model-independent constraints on reionization from large- scale CMB polarization. Astrophys. J., 672:737–751, 2008. [79] Michael J. Mortonson and Wayne Hu. Reionization constraints from five-year WMAP data. Astrophys. J., 686:L53–L56, 2008. [80] V. F. Mukhanov. The quantum theory of gauge-invariant cosmological perturbations. Zhurnal Eksperimental noi i Teoreticheskoi Fiziki, 94:1–11, July 1988. 139 [81] Pia Mukherjee and Yun Wang. Model-Independent Reconstruction of the Primordial Power Spectrum from WMAP Data. Astrophys. J., 599:1–6, 2003. [82] Pia Mukherjee and Yun Wang. Primordial power spectrum reconstruction. JCAP, 0512:007, 2005. [83] Ryo Nagata and Jun’ichi Yokoyama. Band-power reconstruction of the primordial fluctuation spectrum by the maximum likelihood reconstruction method. Phys. Rev., D79:043010, 2009. [84] Gavin Nicholson and Carlo R. Contaldi. Reconstruction of the Primordial Power Spectrum using Temperature and Polarisation Data from Multiple Experiments. JCAP, 0907:011, 2009. [85] Gavin Nicholson, Carlo R. Contaldi, and Paniez Paykari. Reconstruction of the Primordial Power Spectrum by Direct Inversion. JCAP, 1001:016, 2010. [86] M. R. Nolta et al. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectra. Astrophys. J. Suppl., 180:296–305, 2009. [87] Takemi Okamoto and Wayne Hu. CMB Lensing Reconstruction on the Full Sky. Phys. Rev., D67:083002, 2003. [88] Cedric Pahud, Marc Kamionkowski, and Andrew R Liddle. Oscillations in the inflaton potential? Phys. Rev., D79:083503, 2009. [89] David Parkinson, Pia Mukherjee, and Andrew R Liddle. A Bayesian model selection analysis of WMAP3. Phys. Rev., D73:123523, 2006. [90] Paniez Paykari and Andrew H. Jaffe. Optimal Binning of the Primordial Power Spectrum. Astrophys. J., 711:1–12, 2010. [91] H. V. Peiris et al. First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for inflation. Astrophys. J. Suppl., 148:213, 2003. [92] Hiranya V. Peiris and Richard Easther. Primordial Black Holes, Eternal Inflation, and the Inflationary Parameter Space after WMAP5. JCAP, 0807:024, 2008. [93] Hiranya V. Peiris and Licia Verde. The Shape of the Primordial Power Spectrum: A Last Stand Before Planck. Phys. Rev., D81:021302, 2010. [94] S. Perlmutter et al. Measurements of Omega and Lambda from 42 High-Redshift Supernovae. Astrophys. J., 517:565–586, 1999. [95] David Polarski and Alexei A. Starobinsky. Spectra of perturbations produced by double inflation with an intermediate matter dominated stage. Nucl. Phys., B385:623–650, 1992. [96] Adam G. Riess et al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astron. J., 116:1009–1038, 1998. 140 [97] Adam G. Riess et al. A Redetermination of the Hubble Constant with the Hubble Space Telescope from a Differential Distance Ladder. Astrophys. J., 699:539–563, 2009. [98] M. Sasaki. Large Scale Quantum Fluctuations in the Inflationary Universe. Progress of Theoretical Physics, 76:1036–1046, November 1986. [99] Carolyn Sealfon, Licia Verde, and Raul Jimenez. Smoothing spline primordial power spectrum reconstruction. Phys. Rev., D72:103520, 2005. [100] Arman Shafieloo and Tarun Souradeep. Primordial power spectrum from WMAP. Phys. Rev., D70:043523, 2004. [101] Arman Shafieloo and Tarun Souradeep. Estimation of Primordial Spectrum with postWMAP 3 year data. Phys.Rev., D78:023511, 2008. [102] Arman Shafieloo, Tarun Souradeep, P. Manimaran, Prasanta K. Panigrahi, and Raghavan Rangarajan. Features in the Primordial Spectrum from WMAP: A Wavelet Analysis. Phys. Rev., D75:123502, 2007. [103] Joseph Silk and Michael S. Turner. Double Inflation. Phys. Rev., D35:419, 1987. [104] Constantinos Skordis and Joseph Silk. A new method for measuring the CMB temperature quadrupole with an accuracy better than cosmic variance. 2004. [105] George F. Smoot et al. Structure in the COBE differential microwave radiometer first year maps. Astrophys. J., 396:L1–L5, 1992. [106] D. N. Spergel et al. First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. Astrophys. J. Suppl., 148:175– 194, 2003. [107] D. N. Spergel et al. Wilkinson Microwave Anisotropy Probe (WMAP) three year results: Implications for cosmology. Astrophys. J. Suppl., 170:377, 2007. [108] Alexei A. Starobinsky. Spectrum of adiabatic perturbations in the universe when there are singularities in the inflation potential. JETP Lett., 55:489–494, 1992. [109] Ewan D. Stewart. The spectrum of density perturbations produced during inflation to leading order in a general slow-roll approximation. Phys. Rev., D65:103508, 2002. [110] Ewan D. Stewart and David H. Lyth. A more accurate analytic calculation of the spectrum of cosmological perturbations produced during inflation. Phys. Lett., B302:171– 175, 1993. [111] The Planck Collaboration. The Scientific Programme of Planck. April 2006. [112] Domenico Tocchini-Valentini, Marian Douspis, and Joseph Silk. Are there features in the primordial power spectrum? Mon.Not.Roy.Astron.Soc., 359:31–35, 2005. 141 [113] Licia Verde, Hiranya Peiris, and Raul Jimenez. Optimizing CMB polarization experiments to constrain inflationary physics. JCAP, 0601:019, 2006. [114] Licia Verde and Hiranya V. Peiris. On Minimally-Parametric Primordial Power Spectrum Reconstruction and the Evidence for a Red Tilt. JCAP, 0807:009, 2008. [115] Matias Zaldarriaga and Uros Seljak. An All-Sky Analysis of Polarization in the Microwave Background. Phys. Rev., D55:1830–1840, 1997. 142 APPENDIX A RELATION TO PRIOR WORK ON STEP POTENTIALS Our best-fit parameters for the WMAP temperature data (Table 2.1) differ from those found by previous studies of the same inflationary model in Refs. [18, 40]. We explain here the reasons for these discrepancies, which include the addition of data and changes in the likelihood code in going from 3-year to 5-year WMAP data, as well as differences in the computation of the evolution of modes during inflation. Due to small changes in the observed T T spectrum between WMAP 3-year and 5-year data, we find that the best-fit width of the feature increased from d = 0.022 to a value of d = 0.027. The lower value of d agrees with the best-fit value found by Ref. [40], which was based on the 3-year data. Note that a wider feature in k implies a narrower feature in ℓ for the CMB power spectra. Fig. A.1 shows the best fit models for both data sets along with the appropriately binned data. l(l+1) ClTT/2π [µK2] 3000 2500 WMAP5+(c,d)=(0.0015,0.027) WMAP3+(c,d)=(0.0015,0.022) 2000 1500 1000 500 0 20 40 60 80 100 l Figure A.1: Observed T T spectrum (binned in ℓ, with WMAP3 and WMAP5 points offset slightly in ℓ for clarity) and best-fit feature models for WMAP3 (red ) and WMAP5 (black ). Parameters other than c and d are set to the values in Table 2.1 for WMAP5, and to the following values for WMAP3: {m = 6.852 × 10−6 , b = 14.67, Ωb h2 = 0.02222, Ωc h2 = 0.09927, h = 0.753, τ = 0.0817}. 143 Updates to the WMAP likelihood code could also cause small changes in the best-fit potential parameters. In fact, one might be concerned that the feature in the WMAP temperature data is only a systematic effect with some artificial origin in the likelihood calculation. In particular, given the location of a feature, its significance could emerge in some fashion from the transition between the low-ℓ pixel-based T T likelihood code and the high-ℓ harmonic space likelihood code, which happens at ℓ = 32 in the 5-year likelihood code [23]. However, in the original version of the 3-year likelihood code, v2p1, the transition occurred at ℓ = 16, and in the final version, v2p2p2, it was changed to ℓ = 32 [42]. We searched for the best-fit feature model using WMAP3 data with these two versions of the likelihood code and found almost exactly the same values for the potential parameters in both cases, indicating that this particular issue is not the source of a systematic effect. Our best fit value for b is considerably different from Refs. [18, 40] even though we use the same matching condition as they do between e-folds and physical wavenumbers. This is due to a choice of initial conditions for the background evolution of the inflaton by these authors that did not quite satisfy the Friedmann equation, with the result that the subsequent evolution also failed to satisfy it [39]. This essentially translates into a horizontal shift in φ, changing the preferred location of the step b. Ref. [40] discusses relaxing the model dependence of the predicted power spectrum from the chaotic inflation “toy model” adopted here by using a free spectral index that is fit to the data rather than set by the choice of N⋆ . Since the value of ns ≈ 0.96 determined by our matching condition for the chaotic inflation potential as described in § 2.2 is nearly identical to the spectral tilt in the WMAP5 best-fit concordance model (i.e. with smooth ∆2R (k)), we do not carry out this extra step here. However, the form of the underlying potential will be tested by the Planck satellite irrespective of the existence of features; as we note in § 2.3.3, the tensor amplitude predicted by the m2 φ2 potential (which is not affected by the presence of the feature) is within Planck’s reach [30]. 144 APPENDIX B OTHER GENERALIZED SLOW ROLL VARIANTS In this Appendix, we compare various alternate forms discussed in the literature for the curvature power spectrum under the GSR approximation. We test these approximations against the GSRL and GSRL2 approximations introduced in Chapter 3 for the ML model and a more extreme case with c = 3cML = 0.004515 (with other parameters fixed) denoted 3ML (see Fig. B.1). We begin by considering variants that are linear in the GSR approximation and then proceed to second order iterative approaches. The first variant is the original linearized form of GSRS given in [109] (“S02”) ∆2R (k) Z 2 f ′ 2 ∞ du 1 = 2 1+ + W (u)g(ln u) . 3f 3 x u f (B.1) Like GSRS, this approximation depends on an arbitrary choice of x but its impact is exacerbated by the linearization of the correction here. In Fig. B.2 we show the fractional error in this approximation for ηmin = 10−1 Mpc. Note that because of the linearization, the curvature power spectrum can reach the unphysical negative regime (shaded region). A second variant further exploits the relationship between the GSR source functions f , f ′ /f and g and the potential through the slow-roll parameters (see Eq. (3.5)). By further assuming that |ηH | ≪ 1, terms involving V,φ /V can be taken to be constant and evaluated instead at horizon crossing k = aH (see Eq. (3.6)). Finally by rewriting the change in f ′ /f as the integral of (f ′ /f )′ , one obtains [20] (“DS02”) !2 n 1 V,φ 2 V V )( ) |k=aH 1 + (3α − ∆2R (k) = 6 V 12π 2 V,φ Z ∞ V,φφ o du −2 , W (1, u) u θ V 0 145 (B.2) Figure B.1: Curvature power spectrum for the ML and 3ML models. where α ≈ 0.73 and with η ≈ 1/aH, u = k/aH. Here Wθ (u∗ , u) = W (u) − θ(u∗ − u) (B.3) with the step function θ(x) = 0 for x < 0 and θ(x) = 1 for x ≥ 0. Note that limu→0 Wθ (1, u) = 0 and hence the function has weight only near horizon crossing at u ≈ 1. For cases like the ML and 3ML models where ηH is neither small nor smoothly varying, these DS02 assumptions have both positive and negative consequences. They largely solve the problem for superhorizon modes discussed in §3.2.3 by extrapolating the evaluation of the potential terms from kη ≪ 1 to kη ∼ 1. On the other hand, a large ηH means that ǫH evolves significantly. Artifacts of this evolution appear through the prefactor (V /V,φ )2 ∝ 1/ǫH in Eq. (B.2) most notably in the form of a spurious feature at k ∼ 10−3 Mpc−1 in Fig. B.2. Finally, like S02, DS02 does not guarantee a positive definite power spectrum. A third variant is to replace G′ with 2gV /3 in Eq. (3.26) so that the source directly reflects the potential [57] (“KDHS05”) ln ∆2R (k) Z 2 ∞ dη W (kη)gV . = G(ln ηmin ) + 3 ηmin η 146 (B.4) Figure B.2: Fractional error in the curvature power spectrum for first order GSR variants for the ML model (lower) and the 3ML model (upper). As we have seen in §3.2.3, this approximation is actually fairly good locally in ln η and hence 2 terms causes a net error in the locally in k around the feature. However the omission of ηH spectrum for k modes that cross out of the horizon before the inflaton reaches the feature. Hence like the GSRS approximation, KDHS05 overpredicts power at low k for the ML and 3ML models. Fig. B.2 shows a choice with ηmin = 10−1 Mpc. We consider next second order GSR variants. The first variant [15] begins with a second order approach as in GSRL2 but then further assumes that functions such as f ′ /f can be 147 Figure B.3: Temperature power spectrum for c = 8cML (and the other parameters fixed). Note that even in this extreme, observationally unviable, case the temperature power spectrum has < 22% errors everywhere for GSRL2 whereas the linear GSRL and CGS04c approximations substantially under and over predict power respectively. approximated by a Taylor expansion around x∗ to obtain (“CGS04a”) ∆2R ′′ f∗′ π 2 f∗ 1n 2 1 − 2α∗ + −α∗ + = f∗ 12 f∗ f∗2 2 f∗′ 5π 2 + 3α∗2 − 4 + 12 f∗ ′′′ 1 3 π2 4 2 f + − α∗ + α∗ − + ζ(3) ∗ 3 12 3 3 f∗ ′ ′′ f f 7 2 3 + 3α∗ − 8α∗ + π α∗ + 4 − 2ζ(3) ∗ 2∗ 12 f∗ ′ 3 o f +A ∗ , f∗ (B.5) where A = −4α∗3 + 16α∗ − 5/3π 2 α∗ − 8 + 6ζ(3), ζ is the Riemann zeta function, and α∗ = α − ln(x∗ ). This approach is essentially a standard slow roll approximation carried through to third order with the help of an exact solution for power law inflation. For the ML and 3ML models, applying this approximation leads to qualitatively incorrect results as one might expect. We show this variant in Fig. B.4 with x∗ = 1. 148 Figure B.4: Fractional error in the curvature powe r spectrum for second order GSR variants (see text) for the ML model (lower panel) and the 3ML model (upper panel). Note that the error in CGS04a has been divided by a factor of 10 for plotting purposes. A second variant attempts to retain both the generality of GSR and the evaluation of central terms at horizon crossing by implicitly modifying terms of order (f ′ /f )3 and higher 149 when compared with GSRL2 [15] (“CGS04b”) ln ∆2R 2 2 f∗′ 1 1 f∗′ + = ln + 3 f∗ 9 f∗ f∗2 Z 2 ∞ du W (u∗ , u)g(u) + 3 0 u θ Z ∞ 2 du 2 X(u)g(u) + 9 0 u Z Z ∞ 2 ∞ du dv − g(v) X(u)g(u) 3 0 u v2 u Z Z ∞ 2 ∞ du dv Xθ (u∗ , u)g(u) − g(v) , 4 3 0 u u v (B.6) where Wθ was given in Eq. (B.3) and Xθ (u∗ , u) = X(u) − u3 θ(u∗ − u) . 3 (B.7) Here, the subscript ∗ denotes evaluation near horizon crossing. In Fig. B.4 we show the result with u∗ = 1. Notably it performs worse than the first order GSRL approximation for the 3ML model. Finally, the last variant considered takes [15] (“CGS04c”) ln ∆2R Z ∞ 1 du ′ 2 f′ W (u) ln = − + u 3f f2 0 Z ∞ ′ 2 du 1 ′ f +2 X(u) + X (u) u 3 f 0 ′ Z ∞ 1 f du X(u) + X ′ (u) F (u), −4 u 3 f 0 (B.8) where F (u) is given by Eq. (3.32). CGS04c is closely related to GSRL2 as integration by parts shows 2 ∆2R = ∆2R |GSRL eI1 (k)+I2 (k) . (B.9) The main difference is that the second order corrections are exponentiated. This causes a 150 noticeable overcorrection for the 3ML model when compared with GSRL2. In Fig. B.4 we compare the three variants mentioned above. Furthermore, in spite of the 20 − 40% errors in the curvature power spectrum in the 3ML model for GSRL2, the CMB temperature power spectrum has only 1 − 2% errors for ℓ ≥ 20 and a maximum of < 5% errors at ℓ < 20. As discussed in Chapter 3, this level of error is sufficient for even cosmic variance limited measurements at the ℓ . 40 multipoles of the feature. This reduction is due to the oscillatory nature of the curvature errors and projection effects in temperature. In fact for even larger deviations GSRL2 still performs surprisingly well for the temperature power spectrum. In Fig. B.3 we show the temperature power spectra for the GSRL2 approximation, and compare it with GSRL and CGS04c for a very extreme case with c = 8cML (and the other parameters fixed). GSRL2 has a maximum of 22% error in the temperature power spectrum and predicts qualitatively correct features. Finally, the dominant correction is from the term that is quadratic in I1 . The simplified GSRL2 form of ∆2R = ∆2R |GSRL [1 + I12 (k)] , (B.10) works nearly as well. Thus, the curvature power spectrum still depends only on G′ to good approximation even in the most extreme case. 151 APPENDIX C FAST WMAP LIKELIHOOD EVALUATION Figure C.1: Comparison of the low-ℓ polarization pixel likelihood −2 ln Lℓ<24 pol and the approximate fit as a function of E-mode polarization amplitude in two multipole bands pE1 (ℓ = 4 − 6), pE2 (ℓ = 8). Models from a power law chain (red crosses) and from a 5 PCs chain (black crosses) whose likelihood relative to the minimum 1645.84 are (within ±0.1) of 2.29 (68.27% CL) and 6.18 (95.45% CL) are shown with the contours from the fit overplotted (blue curves). In this Appendix, we describe the optimization of the WMAP likelihood code and fast approximate techniques for describing the low-ℓ polarization information. Changes in the initial power spectrum do not require recomputation of the radiation transfer function and are so-called fast parameters for CosmoMC. Hence the WMAP7 likelihood computation is the main bottleneck for the MCMC analysis. We first OpenMP parallelize the likelihood code and remove bottlenecks in the computation of the temperature and high-ℓ polarization likelihood. We obtain a ∼ 2.6Ncore speedup 152 of those parts of the likelihood where Ncore is the number of cores in a shared memory machine. These changes exactly preserve the accuracy of the likelihood evaluation. Figure C.2: Posterior probability distribution of the optical depth τ and the fourth PC amplitude using the exact likelihood (black curves) and the approximation (blue dashed curves) with WMAP data. In place of the computationally expensive low-ℓ polarization pixel likelihood, we seek a fast but accurate approximation. The WMAP team has shown that most of the information in the power law ΛCDM parameter space lies in multipoles ℓ ∼ 2 − 7 as essentially an overall amplitude of power [66]. However, in the broader parameter space allowed by the PCs of G′ , we find that a single amplitude is insufficient to describe the information content of the 153 pixel likelihood. Instead we fit the likelihood function to a two band approximation pEi = 1 ∆ℓi ℓX imax ℓ=ℓimin 1/2 EE ℓ(ℓ + 1)Cℓ 2π , (C.1) where the first band i = 1 has ℓ1min = 4, ℓ1max = 6 and the second band i = 2 has ℓ2min = ℓ2max = 8 and ∆ℓi = ℓimax − ℓimin + 1 normalizes the parameter to reflect the average bandpower. We find that the pixel likelihood is well approximated by a Gaussian in these two bands for the models under consideration T −1 −2 ln Lℓ<24 pol ≈ A + (pE − p̄E ) C (pE − p̄E ) (C.2) with the parameters pE1 = 0.2614 µK, pE2 = 0.01955 µK, A = 1645.84 498.31 −214.23 −2 C−1 = µK . −214.23 190.23 (C.3) In Fig. C.1 we show the accuracy of the fit compared with the pixel likelihood for both power law models and models with additional 5 PCs of G′ . Note that the power law models lie on a 1D curve in this space and can be well parameterized by a single amplitude whereas the 5PC models do not. In fact, in the 2 band space models with low pE1 and high pE2 that populate a direction nearly orthogonal to the power law models are more strongly constrained than the total power at low ℓ would suggest. Since this approximation has trivial computational cost, the net improvement in speed is approximately ∼ 5Ncore. For the cases of interest, the approximation works remarkably well. As an example, we have run an MCMC chain with the exact pixel likelihood for the 5PC chain with WMAP data only. In Fig. C.2, we compare the posterior probability distribution of the optical depth τ and m4 using the full likelihood and the approximation. The difference between the pixel 154 likelihood and the approximation for the 5 PCs maximum likelihood model with WMAP data is likewise negligible: |2∆ ln L| = 0.05. We have also checked that the likelihood approximation remains valid to 10% or better in the (2∆ ln L)1/2 significance of differences between models with varying reionization history as parameterized by ionization principal components [78]. Larger differences can occur for models with sharp, order unity, initial power spectrum features at the horizon scale. These project onto the temperature and polarization spectra differently and leads to qualitatively different results for the temperature-polarization cross spectrum. In this case one can get discrepancies of order unity in 2∆ ln L that err on the side of allowing discrepant models. Even these differences can typically be taken into account via importance sampling at a much smaller computational cost than evaluating the exact pixel likelihood during the MCMC run itself. 155 APPENDIX D MCMC OPTIMIZATION FOR MANY ADDITIONAL PARAMETERS D.1 Parameterization We seek to define amplitude and tilt parameters for the MCMC that are nearly orthogonal to the PC parameters in order to improve the convergence properties of the MCMC chains. A constant G′ is equivalent to tilt ns and hence PC components that have long positive or negative definite stretches become degenerate with tilt and cause problems for MCMC convergence. Instead of a constant tilt, we define a new chain parameter to be the average of G′ across a narrower range that is better associated with the observables Ḡ′ Z η2 1 dη ′ = G , ln η2 − ln η1 η1 η (D.1) where specifically, we choose η1 = 30 Mpc and η2 = 400 Mpc to roughly minimize the variance of Ḡ′ in the chain (see Fig. 5.9). Next, we replace the normalization parameter G(ln ηmin ) with As ≡ ln ∆2R (kp ) , (D.2) where in practice we choose kp = 0.05 Mpc−1 . The effective tilt and normalization parameters bring the model of the power spectrum 156 from Eq. (4.1) to −Ḡ′ # X N k 2 ln ∆R = ln As + ma [W̄a (k) − W̄a (kp )] kp a=1 2 N X 1 π ma X̄a (k) + ln 1 + Ḡ′ + 2 2 " a=1 1 π − ln 1 + Ḡ′ + 2 2 N X a=1 2 ma X̄a (kp ) , (D.3) where Z ηmax dη W̄a (k) = W (kη)(Sa (ln η) − S̄a ) , ηmin η Z ηmax dη X(kη)(Sa (ln η) − S̄a ) , X̄a (k) = ηmin η (D.4) and Z η2 dη 1 Sa . S̄a ≡ ln η2 − ln η1 η1 η (D.5) Note that we can recover the tilt ns , equivalent to the average of Ḡ across the whole range ηmin to ηmax , as ns = (1 − Ḡ′ ) + 20 X ma S̄a , (D.6) a=1 and keep it as a derived parameter in the chain Given the oscillatory nature of the k-space response to the PC eigenfunctions through W̄a and X̄a and the geometric projection from k to angular multipole ℓ, normalization at a given k does not correspond simply to normalization at a given ℓ. Since the observations best constrain the amplitude of the temperature power spectrum near the first acoustic peak at ℓ ∼ 220 it is advantageous to use an ℓ-space normalization in the MCMC and then transform back to As . Let us define a phenomenological parameter Ac which renormalizes the angular power 157 spectra as C T T fid XY C̃ℓ . CℓXY = eln Ac 220 TT C̃220 (D.7) T T fid is the temperature power spectrum at the first peak of a fiducial model that fits Here C220 T T fid = 0.747µK2. Thus if A = 0, C T T = C T T fid regardless the WMAP7 data. We use C220 c 220 220 of the PC parameters. We can recover constraints on the k-space normalization by considering As as a derived parameter. If we compute the original C̃ℓXY with the As = Afid s of the fiducial model, then the true As is given by T T fid/C̃ T T ) + ln Afid . ln As = ln Ac + ln(C220 s 220 (D.8) In summary, we replace the parameters ns and G(ln ηmin ) with Ḡ′ and Ac in order to reduce parameter degeneracies that would otherwise inhibit chain convergence. D.2 Likelihood corrections To speed up the calculation of the WMAP and other CMB likelihoods we employ three approximations when running the chains. Firstly, we use a fitting function for the low ℓ WMAP7 polarization likelihood as described in Appendix C. Secondly, we calculate the CMB power spectra with gravitational lensing artificially turned off. Thirdly, we use the default ℓ-space sampling of CAMB that is designed for smooth underlying power spectra. Each of these approximations produce small errors in the likelihood evaluation that we can correct by importance sampling the chain. The advantage of correcting these approximations in a postprocessing step is twofold. The chains may be thinned due to the high correlation between samples in the chain. Secondly, postprocessing elements of the thinned chains is embarrassingly parallel unlike the running of the original chain. 158 In practice, when we satisfy our convergence criterion described in the main text, we thin the chains by a factor of half of the correlation length. We have tested that with such thinning we reproduce the posteriors of the original chains. Next we compute the CMB power spectra of the thinned chains with lensing turned on and a higher ℓ-space sampling (CAMB “accuracy boost” 2). We use these high accuracy power spectra to correct the chain multiplicity for the change in the likelihood. In Fig. D.1 we show as an example the posteriors coming directly from power law (PL) chains (in blue/solid curves), the chains with all corrections (in blue/dashed lines) and finally all corrections but lensing (in red/dashed lines). These should be compared with results from a separate chain run with all the corrections turned on from the start (in black/solid lines). Importance sampling accurately models the impact of the small corrections for all parameters. The leading correction is on Ωb h2 from lensing. In Fig. D.2, we show the impact of the corrections on the PC chain using m18 as an example with the largest correction. The correction on PC parameters is extremely small and again dominated by lensing. 159 Figure D.1: Power law parameter posteriors from the approximations used to run the MCMC chain (in blue/solid curve), from an independent MCMC with no approximation (in black/solid curve), from the approximate chain with importance sampling correction (in blue/dashed curve), and from the approximate chain without lensing correction (in red/dashed curve). 160 Figure D.2: The m18 probability distributions from the approximations used to run the √ MCMC with all data and I1,max = 1/ 2 (in black lines), from the approximate chain with importance sampling correction (in blue/dashed lines), and from the approximate chain without lensing correction (in red/dashed lines). m18 has the largest correction of the PC amplitudes which is still ≪ 1σ and dominated by the lensing correction. 161 APPENDIX E GENERALIZED SLOW ROLL ACCURACY To test the accuracy of the GSR approximation in the PC space, we need to consider the inverse problem: construct an inflationary model that matches a desired G′ for which we can solve exactly for the curvature power spectrum. In the forward direction, given an inflationary model we can compute the exact curvature spectrum by first evaluating the background behavior of the model through f ′′ f′ 3 g(ln η) = − 3 = G′ + f f 2 ′ 2 f , f (E.1) and then solving the equation g 2 d2 y + 1 − 2 y = 2y , 2 dx x x (E.2) where x = kη, subject to the usual Bunch-Davies initial conditions. The curvature power spectrum is then given by |y|2 ∆2R = lim x2 2 . x→0 f (E.3) Therefore to test the GSR approximation we first need to determine the function g that matches a given G′ (ln η). Transforming variables to r = f ′ /f , we obtain from Eq. (E.1) 3 r ′ − 3r = G′ , 2 (E.4) which has the general solution 3 r = η3 2 Z dη̃ −3 ′ η̃ G + Cη 3 . η̃ 162 (E.5) Figure E.1: Fractional difference in temperature power spectra between GSR and the exact inflationary solution for the maximum likelihood model from the √ all-data analysis (in black lines) as well as a model that saturates the prior I1,max = 1/ 2 from the chain (in blue lines). For reference, the ML model has I1,max = 0.66. Figure E.2: Likelihood difference between the GSR solution and the full inflationary calculation of a series of step potential models as a function of I1,max . Models were chosen from the full GSR chain to be the maximum likelihood in a series of bins in step amplitude c. √ The maximal error is small below I1,max = 1/ 2 (blue dashed line), the prior in the fiducial all-sky analysis. 163 Let us choose the integration constant Z 3 ηmax dη̃ −3 ′ C=− η̃ G , 2 ηmin η̃ (E.6) and assume G′ vanishes outside this range. We then get for η > ηmin and Z 3 3 ηmax dη̃ −3 ′ η̃ G , r=− η 2 η̃ η (E.7) Z 3 3 ηmax dη̃ −3 ′ η̃ G , r=− η 2 ηmin η̃ (E.8) for η < ηmin . With this numerical solution we construct g as 3 g = G′ + r 2 . 2 (E.9) This suffices to specify the source for y in Eq. (E.2). Finally, to get the curvature power −1 . However since this quantity is independent of k, spectrum we need f at some ηlim ≪ kmax it is absorbed into our normalization definition. In Fig. E.1, we take parameters from the all-data chain and use this technique to calculate the temperature power spectra of matching √ inflationary models exactly. Even for the model that saturates the I1,max = 1/ 2 prior, the WMAP likelihood difference between the exact and GSR calculation is |2∆ ln L| = 0.4. Using the step model chain from §5.4.2, we can explore the accuracy of the GSR approximation as a function of I1,max independently of the prior taken in the all data analysis. Specifically, we take a model from the chain that defines G′ and construct the matching full inflationary model as above. Recall that to construct G′ , we solve for the background evolution of φ in the step potential with an effective mass given by Eq. (2.1). This specifies the m2 φ2 model source through Eq. (3.24), which we call G′m . To allow for a retilting of the spectrum, we add an 164 extra constant parameter Ḡ′p to the model source to form the full source G′ (ln η; c, d, Ḡ′p ) = G′m (ln η; c, d) + Ḡ′p . (E.10) The GSR approximation then tells us that the curvature spectrum is given by " ln ∆2R = ln As k kp −Ḡ′p # + Im (k) − Im (kp ) (E.11) with Z ηmax dη Im (k) = W (kη)G′m η ηmin " 2 # Z ηmax dη 1 π ′ Ḡ + . X(kη)G′m + ln 1 + 2 2 p ηmin η (E.12) In Fig. E.2, we compare the impact of taking this power spectrum to a full inflationary calculation with matching source (E.10) on the WMAP likelihood. For the full calculation of Eq. (E.3), we take ∆2R (kp ) = As to define the normalization f . Since I1,max increases monotonically with c, we show models with maximum likelihood parameters in uniform bins of c. Note that the maximal error increases with I1,max but does not exceed order unity at √ I1,max < 1/ 2. For reference, to compute a matching 20 PC representation as in Fig. 5.15 we take the amplitudes of the principal components from Eq. (5.5) and use Eq. (D.1) to define Ḡ′ = Ḡ′p Z η2 dη ′ 1 G (ln η) . + ln η2 − ln η1 η1 η m and keep the normalization As fixed. 165 (E.13)

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