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On the imprints of inflation in the Cosmic Microwave Background

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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 . . . . . . . . .
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VERSUS REIONIZA. . . . . . . . . . . .
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SPECTRUM FEATURES
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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 . . . . . . . . . . . . .
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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
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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. . . . . . . . . . . . .
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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. .
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44
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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
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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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