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Constraining the epoch of recombination with recent observations of the cosmic microwave background

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Constraining the Epoch of Recombination with Recent
Observations of the Cosmic Microwave Background
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Angela M. Linn, B.S., M.S.
*****
The Ohio State University
2003
Dissertation Committee:
Approved by
Professor Robert J. Scherrer, Adviser
Professor Terrence P. Walker
Professor Gregory W. Kilcup
Professor Charles H. Pennington
Adviser
Department of Physics
UMI Number: 3124361
________________________________________________________
UMI Microform 3124361
Copyright 2004 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
____________________________________________________________
ProQuest Information and Learning Company
300 North Zeeb Road
PO Box 1346
Ann Arbor, MI 48106-1346
ABSTRACT
The radiation in the Cosmic Microwave Background (CMB) last scattered off of
matter during the epoch of recombination, when nuclei captured electrons for the
first time, and its predicted anisotropy power spectra are sensitive to the details
of recombination. In light of recent high quality observations of the CMB, it is
worthwhile to re-examine the constraints that can be placed on recombination. We
use a very general extension to the standard model of recombination characterized
by two parameters, a and b, which multiply the overall rates of recombination and
ionization and the binding energies of hydrogen respectively. We utilize WMAP
temperature and TE cross-correlation data, along with temperature data at smaller
scales from ACBAR and CBI, to place constraints on these parameters. The range
of recombination histories which are the best match to the data fall within the range
−0.4 ≤ log(a) ≤ 0.4 and 0.9 ≤ b ≤ 1.1, forming a diagonal region from low-a/high-b
to high-a/low-b. We find zrec = 1055 ± 25 and ∆z = 84 ± 2 at 68% confidence for the
range of recombination models which are best able to reproduce the observed CMB
power spectra. Standard recombination is an entirely acceptable fit to the data. We
conclude that there is still leeway for non-standard recombination, but the window is
narrowing.
ii
I must down to the seas again, to the lonely sea and the sky,
And all I ask is a tall ship and a star to steer her by....
John Masefield (1878-1967)
To my parents,
who taught me to build tall ships
And to my grandparents,
who showed me the stars
iii
ACKNOWLEDGMENTS
I would first like to thank my adviser, Bob Scherrer, for his guidance and direction
in this project, for always making time to see me, and for many useful discussions.
I would also like to thank the rest of the Astrophysics group at Ohio State. In
particular, I would like to thank David Weinberg for working with me on previous
projects. Also, thanks to the people in the Astrophysics Student Seminar for providing
a useful forum for discussions and practice talks.
I am grateful to a host of people at other institutions for answering my e-mails:
Robert Caldwell, Mike Chu, Jon Goldstein, Manoj Kaplinghat, Jim Kneller, Danny
Marfatia, Tim Pearson, Max Tegmark, and Licia Verde.
Thanks to the Ohio Supercomputer center for providing most of the computing
resources used in this project. I would also like to acknowledge that I was supported
in part by a grant from the Department of Energy.
I would especially like to thank my family for their unwavering faith in me. Their
love and support has made all of this much easier.
I would also especially like to thank Dan Nelson for his example, for his encouragement throughout this process, and most of all for always being there for me.
Finally, I would like to thank God, both for creating such a fascinating universe
and for providing me with the ability and opportunity to study it.
iv
VITA
April 23, 1974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born
Abilene, Kansas, USA
1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Physics, B.S. Astronomy
University of Kansas
Lawrence, Kansas, USA
2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Physics
The Ohio State University
Columbus, Ohio, USA
1997 - 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . William A. Fowler Fellow
Department of Physics
The Ohio State University
1997 - 1999, 2001 - 2002 . . . . . . . . . . . . . . . . . . . . Dean’s Distinguished University Fellow
The Ohio State University
1998 - 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Teaching Associate
Department of Physics
The Ohio State University
2000 - 2001, 2002 - present . . . . . . . . . . . . . . . . . Graduate Research Associate
Department of Physics
The Ohio State University
PUBLICATIONS
Research Publications
J. Kujat, A. M. Linn, R. J. Scherrer and D. H. Weinberg, “Prospects for Determining
the Equation of State of the Dark Energy: What can be Learned from Multiple
Observables?” The Astrophysical Journal, 572, 1 (2002).
v
R. J. Splinter, A. L. Melott, A. M. Linn, C. Buck and J. Tinker, “The Ellipticity
and Orientation of Clusters of Galaxies in N-Body Experiments.” The Astrophysical
Journal, 479, 632 (1997).
FIELDS OF STUDY
Major Field: Physics
Studies in Theoretical Astrophysics and Cosmology: Professor Robert Scherrer
vi
TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Chapters:
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2.
The CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1 Origin of the CMB . . . . . . . . . . . . . . . . . . . . . .
2.2 Temperature Anisotropies in the CMB . . . . . . . . . . .
2.2.1 Sources of Temperature Anisotropies . . . . . . . .
2.2.2 Statistical Description of Temperature Anisotropies
2.3 Polarization Anisotropies in the CMB . . . . . . . . . . .
2.3.1 Sources of Polarization Anisotropies . . . . . . . .
2.3.2 Statistical Description of Polarization Anisotropies
2.4 Recent Experiments . . . . . . . . . . . . . . . . . . . . .
vii
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3
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12
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3.
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5.
6.
The Epoch of Recombination . . . . . . . . . . . . . . . . . . . . . . . .
23
3.1 Recombination . . . . . . . . . . . . . . . .
3.2 Relation to CMB . . . . . . . . . . . . . . .
3.2.1 The Visibility Function . . . . . . .
3.2.2 Effects on Temperature Anisotropies
3.2.3 Effects on Polarization Anisotropies
3.3 Ionization Fraction . . . . . . . . . . . . . .
3.3.1 Equilibrium Calculation . . . . . . .
3.3.2 Non-Equilibrium Calculation . . . .
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23
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28
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30
31
Variations on Standard Recombination . . . . . . . . . . . . . . . . . . .
35
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Our Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
36
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
5.1
5.2
5.3
5.4
5.5
5.6
5.7
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52
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
6.1 Effects of Varying the Recombination History . . .
6.1.1 Ionization Fraction xe . . . . . . . . . . . .
6.1.2 Visibility Function g(z) . . . . . . . . . . .
6.1.3 CMB Power Spectra . . . . . . . . . . . . .
6.2 Allowed Regions of a-b Parameter Space . . . . . .
6.2.1 Analysis of TT Data . . . . . . . . . . . . .
6.2.2 Analysis of TT+TE Data . . . . . . . . . .
6.3 Characteristics of Allowed Regions . . . . . . . . .
6.3.1 Free Parameter Distributions . . . . . . . .
6.3.2 Recombination Histories of Allowed Regions
6.3.3 Power Spectra . . . . . . . . . . . . . . . .
54
54
59
59
68
68
74
74
74
88
93
Overview . . . . . . . . . . .
Data Sets . . . . . . . . . . .
CMBFAST . . . . . . . . . . . .
Calculation of χ2 . . . . . . .
Parameter Choices and Priors
Minimization Routines . . . .
Confidence Regions . . . . . .
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viii
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7.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
Appendices:
A.
Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
ix
LIST OF TABLES
Table
Page
5.1
Priors on the free parameters and their values. . . . . . . . . . . . . . . .
49
5.2
∆χ2 values for a joint estimation of m = 2 parameters. . . . . . . . . . .
52
6.1
For each data set, the number of degrees of freedom DOF , the location of
the point (a, b)min with the lowest χ2 value χ2min , the χ2 value for standard
recombination χ2std rec , and the corresponding minimum confidence region
that would just include standard recombination (1 − A) std rec . Recall that
being included in the 1σ region corresponds to a probability (1−A) ≤ 68.27%. 73
6.2
Free parameter values for standard recombination, the point with lowest χ 2 ,
and the average of the 1σ region for each data set. . . . . . . . . . . . . .
x
79
LIST OF FIGURES
Figure
2.1
2.2
2.3
2.4
2.5
Page
The temperature anisotropy power spectrum C `T T for a ΛCDM model. Data
from WMAP, ACBAR, and CBI are shown. The first peak is a density
maxima, the second a density minima, etc., while the troughs are regions
with maximum velocity. . . . . . . . . . . . . . . . . . . . . . . . . . .
7
The same temperature anisotropy power spectrum C `T T as shown in Figure
2.1, but plotted logarithmically along the `-axis to show the Sachs-Wolfe
plateau at low `. This plateau is only approximate. The Late ISW effect, due
to a cosmological constant, causes it to tilt up at the very smallest `, while
at the high ` end the plateau tilts up due to the Early ISW effect, caused by
the decay of gravitational potentials before matter-radiation equality, and
transitions smoothly into the first acoustic peak. . . . . . . . . . . . . . .
11
Radiation with a polarization Ê incident on an electron will preferentially scatter in a direction perpendicular to the direction of incidence,
but with polarization parallel to the original polarization. . . . . . . .
13
Quadrupolar anisotropies in the temperature field are required to produce polarization anisotropies in the CMB. If the observer is viewing
radiation scattered along the positive z-axis, the incident radiation
preferentially came from the x-y plane. Unpolarized radiation incident from the y direction will yield scattered light polarized in the x
direction, and vice-versa. If the temperature is isotropic, the resulting scattered light will be unpolarized. However, a quadrupole in the
temperature leads to a net polarization in the scattered radiation. . .
14
The E-mode polarization anisotropy power spectrum C `EE for a ΛCDM
model. The first peak is a region with maximum infall velocity, the second with maximum expansion velocity, etc., and the troughs are density
maxima and minima. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
xi
2.6
2.7
2.8
3.1
6.1
6.2
6.3
6.4
6.5
6.6
E-mode (left) and B-mode (right) polarizations for a single plane wave.
The direction of E-mode polarization is parallel or perpendicular to the
direction of greatest change, while the direction of B-mode polarization is
at 45◦ compared to the direction of greatest change. . . . . . . . . . . . .
18
Correlation of polarization with temperature anisotropies. Left: superposition of modes leads to tangential E-mode polarization patterns (solid lines)
around hot spots and radial patterns (dashed lines) around cold spots.
Right: superposition of modes leads to no correlation of B-mode polarization with temperature, due to the parity of the B-modes. . . . . . . . .
18
The TE cross-correlation anisotropy power spectrum C `T E for a ΛCDM
model. Data from WMAP are shown. . . . . . . . . . . . . . . . . . . .
20
The width of the last scattering surface determines the minimum scale for
intrinsic temperature variations because regions smaller than the width of
the LSS are averaged over (bottom line of sight). . . . . . . . . . . . . .
27
xe (z) vs. z for varying a, fixed b. As a increases, recombination has a shorter
duration. The parameters used to create these curves (as well as those in the
following figures) are Ωm = 0.29, ωb = 0.020, h = 0.72, ns = 1.0, τRI = 0.15.
55
xe (z) vs. z for varying b, fixed a. As b increases, recombination starts at
earlier redshifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
g(z) vs. z for varying a, fixed b. As a increases, the width of the visibility
function (and therefore ∆z) decreases and the position of the peak of the
visibility function (and therefore z rec ) shifts to earlier times. . . . . . . .
57
g(z) vs. z for varying b, fixed a. As b increases, the position of the peak
of the visibility function (and therefore z rec ) shifts to earlier times and the
width of the visibility function (and therefore ∆z) increases. . . . . . . .
58
C`T T vs. ` for varying a, fixed b. As a decreases, the power at small scales is
decreased and the acoustic peaks are suppressed. Also, as a decreases, the
acoustic features shift toward slightly larger scales. . . . . . . . . . . . .
61
C`T T vs. ` for varying b, fixed a. As b decreases, the acoustic features shift
to larger scales. Notice also that as b decreases, the relative amplitudes of
the even and odd peaks change. . . . . . . . . . . . . . . . . . . . . . .
62
xii
C`T E vs. ` for varying a, fixed b. As a decreases, the power at small scales is
decreased and the acoustic peaks are suppressed. Also, as a decreases, the
acoustic features shift toward slightly larger scales. . . . . . . . . . . . .
63
C`T E vs. ` for varying b, fixed a. As b decreases, the acoustic features shift
toward larger scales. Notice also that as b decreases, the relative amplitudes
of the even and odd peaks change. . . . . . . . . . . . . . . . . . . . . .
64
C`EE vs. ` for varying a, fixed b. As a decreases, the power at small scales is
decreased and the acoustic peaks are suppressed. Also, as a decreases, the
acoustic features shift toward slightly larger scales. . . . . . . . . . . . .
65
6.10 C`EE vs. ` for varying b, fixed a. As b decreases, the acoustic features move
to larger scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6.7
6.8
6.9
6.11 Allowed region of a-b space from the pre-WMAP TT data set. Standard
recombination lies solidly in the 2σ region. The point with the lowest value
of χ2 is log(a) = −1.1, b = 1.35, with χ2 = 17.50 . . . . . . . . . . . . . .
69
6.12 Allowed region of a-b space from the WMAP TT data set. Standard recombination lies just outside of the 1σ region. The point with the lowest value
of χ2 is log(a) = −0.2, b = 1.0, with χ2 = 975.2 . . . . . . . . . . . . . .
70
6.13 Allowed region of a-b space from the WMAP TT+TE data set. Standard
recombination lies inside the 1σ region. The point with the lowest value of
χ2 is log(a) = −0.3, b = 1.05 with a value of χ 2 = 1430.6. . . . . . . . . .
71
6.14 Allowed region of a-b space from the WMAP+ TT+TE data set. Standard
recombination lies inside the 1σ region. The point with the lowest value of
χ2 is log(a) = −0.2, b = 1.0 with a value of χ 2 = 1435.80. . . . . . . . . .
72
6.15 Distribution of free parameters ϑ for recombination histories falling inside
the pre-WMAP TT 1σ region. The error bars across the top represent the
range for the prior for each parameter. . . . . . . . . . . . . . . . . . . .
75
6.16 Distribution of free parameters ϑ for recombination histories falling inside
the WMAP TT 1σ region. The error bars across the top represent the range
for the prior for each parameter. . . . . . . . . . . . . . . . . . . . . . .
xiii
76
6.17 Distribution of free parameters ϑ for recombination histories falling inside
the WMAP TT+TE 1σ region. The error bars across the top represent the
range for the prior for each parameter. . . . . . . . . . . . . . . . . . . .
77
6.18 Distribution of free parameters ϑ for recombination histories falling inside
the WMAP+ TT+TE 1σ region. The error bars across the top represent
the range for the prior for each parameter. . . . . . . . . . . . . . . . . .
78
6.19 How the parameters ϑ change with a and b for recombination histories falling
inside the pre-WMAP TT 3σ region. The solid squares have a value for the
given parameter that is more than one standard deviation above the mean,
ϑi > ϑ̄i + σi , the open squares with an × have values within one standard
deviation of the mean, ϑ̄i − σi < ϑi < ϑ̄i + σi , and the open squares have
values more than one standard deviation below the mean, ϑ i < ϑ̄i − σi . . .
82
6.20 How the parameters ϑ change with a and b for recombination histories falling
inside the WMAP TT 3σ region. The solid squares have higher values, ϑ i >
ϑ̄i +σi , the open squares with an × have average values, ϑ̄i −σi < ϑi < ϑ̄i +σi ,
and the open squares have lower values, ϑ i < ϑ̄i − σi . . . . . . . . . . . .
83
6.21 How the parameters ϑ change with a and b for recombination histories falling
inside the WMAP TT+TE 3σ region. The solid squares have higher values,
ϑi > ϑ̄i + σi , the open squares with an × have average values, ϑ̄i − σi <
ϑi < ϑ̄i + σi , and the open squares have lower values, ϑ i < ϑ̄i − σi . . . . .
84
6.22 How the parameters ϑ change with a and b for recombination histories falling
inside the WMAP+ TT+TE 3σ region. The solid squares have higher
values, ϑi > ϑ̄i + σi , the open squares with an × have average values,
ϑ̄i − σi < ϑi < ϑ̄i + σi , and the open squares have lower values, ϑ i < ϑ̄i − σi . 85
6.23 How the parameters ϑ change with a and b for recombination histories falling
inside the 3σ region of the WMAP TT+TE τ RI = 0 set. The solid squares
have higher values, ϑi > ϑ̄i + σi , the open squares with an × have average
values, ϑ̄i − σi < ϑi < ϑ̄i + σi , and the open squares have lower values,
ϑi < ϑ̄i − σi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
6.24 xe (z) vs. z for recombination histories falling within the 1σ, 2σ, and 3σ
regions of the WMAP+ TT+TE analysis. The thicker black line is standard
recombination, and in this data set it falls within the 1σ region. It is shown
on the other two graphs for comparison purposes. . . . . . . . . . . . . .
xiv
89
6.25 g(z) vs. z for recombination histories falling within the 1σ, 2σ, and 3σ
regions of the WMAP+ TT+TE analysis. The thicker black line is standard
recombination, and in this data set it falls within the 1σ region. It is shown
on the other two graphs for comparison purposes. . . . . . . . . . . . . .
90
6.26 zrec vs. ∆z for recombination histories falling within the 1σ, 2σ, and 3σ
regions of the WMAP+ TT+TE analysis. . . . . . . . . . . . . . . . . .
92
6.27 C`T T vs. ` for recombination histories falling inside the WMAP+ TT+TE
1σ region. The thicker black line is standard recombination. . . . . . . . .
94
6.28 C`T E vs. ` for recombination histories falling inside the WMAP+ TT+TE
1σ region. The thicker black line is standard recombination. . . . . . . . .
95
6.29 C`EE vs. ` for recombination histories falling inside the WMAP+ TT+TE
1σ region. The thicker black line is standard recombination. . . . . . . . .
96
xv
CHAPTER 1
INTRODUCTION
The detection of anisotropies in the Cosmic Microwave Background (CMB) has
revolutionized the science of cosmology in recent years. Once a field relatively lacking in observational data, cosmology has greatly benefited from the abundance of
increasingly precise measurements of these variations. [1], [2]
Radiation from an era when the universe was in thermal equilibrium, the CMB
provides a snapshot of the early universe. The light of the CMB last interacted
directly (i.e., non-gravitationally) with matter during the epoch of recombination,
the period when the universe became cool enough for atomic nuclei to capture and
hold electrons for the first time. Although the universe was nearly isotropic and
homogeneous at that time, small perturbations in the density of matter – the seeds
of the stars, galaxies, and clusters of galaxies we observe today – were present and
left an imprint in the form of corresponding perturbations, or anisotropies, in the
temperature and polarization of the CMB.
It is possible to predict, based on fundamental physics, the statistical properties
of these anisotropies. This prediction is extremely model dependent, being sensitive
to the value of a number of cosmological parameters. This can be turned to our
1
advantage with good data. Today’s precise measurements of the CMB can be used
to reverse the process and place constraints on the model’s parameters.
In recent years, the CMB has been used, either alone or in combination with
other experiments, to place constraints on various cosmological parameters such as
the density of matter, the curvature of space, the Hubble constant, the number of
neutrino species, etc.
In this work we place constraints on how recombination might have proceeded. A
number of authors have looked at this issue in the past, including [3–6]. However,
we are re-examining this question in the light of recent, high-quality data from the
WMAP satellite [7], and several other recent experiments [8, 9]. Temperature data
from WMAP is substantially better than data used in previous calculations, and
significantly tightens the resulting constraints. In addition, we are including the
analysis of polarization data in the search for constraints on recombination. The only
other analysis to use the recent WMAP data is [6], however they use a different model
to characterize changes in recombination.
The remainder of this paper proceeds as follows. The Cosmic Microwave Background is discussed in greater detail in Chapter 2. In Chapter 3 we examine the
epoch of recombination and in Chapter 4 we explore a way to generalize the standard
equations governing recombination to include a variety of models. The methods used
to constrain these models are explained in Chapter 5, and the results of this analysis
are presented in Chapter 6. Chapter 7 explores the conclusions that can be drawn
from the results.
2
CHAPTER 2
THE CMB
2.1
Origin of the CMB
According to the Big Bang theory, the universe originated approximately fourteen
billion years ago, expanding rapidly from an initial singularity. The early universe
was extremely hot and dense, but cooled as it expanded. As time progressed, the
quark-gluon plasma coalesced into baryons. This occurred about 10−5 seconds after
the Big Bang, when the universe had cooled to a temperature of around 1012 K or
1 GeV. By the time the universe was about one second old, it had cooled to around
1010 K or 1 MeV, allowing the baryons to combine into nuclei. Mostly hydrogen and
helium nuclei were produced in what is known as Big Bang Nucleosynthesis (BBN),
but small amounts of other light elements such as D, 3 He, and 7 Li were produced as
well.
During this period, when the universe was filled with charged particles, radiation
interacted strongly with matter at a rate controlled by the Thomson cross section
σT . The photons and baryons were in thermal equilibrium and behaved as a single
fluid. The large interaction cross section between matter and radiation insured that
the mean free path of the radiation was very short. The universe was opaque.
3
The universe continued to expand and cool, and around 300,000 years after the
Big Bang, when the temperature was about 3000 K or a few eV, the nuclei began
to capture the previously free electrons and form neutral atoms. This process —
recombination — is the origin of the CMB. The rate for neutral atoms to scatter
radiation, a process called Rayleigh scattering, is several orders of magnitude smaller
than the Thomson scattering rate. Therefore, as recombination proceeded, the mean
free path of radiation increased until the universe became transparent and radiation
was able to propagate freely.
This radiation is what we observe today as the CMB. Because it was in thermal
equilibrium with matter and has had little interaction since, the radiation exhibits an
almost perfect thermal, or blackbody, spectrum. The temperature of the radiation
has decreased as the universe expands, but the thermal shape of the spectrum is
unchanged. The best current determination of the CMB’s temperature is TCM B =
2.725 ± 0.002 K [10].
2.2
Temperature Anisotropies in the CMB
The temperature of the CMB is not perfectly isotropic. There are small variations
in the temperature on the order of one part in 105 , or around 10 µK. The average
variation in temperature at different angular scales can be predicted from cosmological
models; however, this prediction depends critically on the values of the parameters
chosen for the model. In this section we will first examine some of the physical
mechanisms that generate changes in the CMB temperature and then present the
standard statistical treatment of the temperature anisotropies. For a description of
4
the basic cosmological model and some of our underlying assumptions, the reader is
referred to Appendix A.
2.2.1
Sources of Temperature Anisotropies
The temperature anisotropies in the CMB can be divided into two categories based
on their source: primary or intrinsic anisotropies are imprinted on the CMB at the
time of recombination, while secondary anisotropies are caused later, as the radiation
travels through space.
Intrinsic temperature anisotropies are primarily caused by fluctuations in the density field ρ(x) of the early universe. Regions which are overdense have a Newtonian
gravitational potential ψ = ψ̄ + δψ (where ψ̄ is the average gravitational potential),
which in general relativity translates into a change in the metric of space-time. Photons originating in an overdense region will be gravitationally redshifted and undergo
time dilation as they climb out of the potential well. For adiabatic perturbations,
this leads to a change in temperature of ∆T /T = −δψ/3 [11]. This is known as the
Sachs-Wolfe effect [12]. The Doppler effect is also a source of intrinsic anisotropies.
Radiation emitted from regions with a bulk velocity will appear either redshifted or
blueshifted, depending on the direction of the flow.
Overdense regions in the tightly coupled photon-baryon fluid undergo acoustic
oscillations (compression and rarefaction) prior to recombination. These acoustic oscillations provide some of the most striking features of the temperature anisotropy
power spectrum. An overdense region will begin to collapse due to gravitation. However, at a certain point, the radiation pressure becomes great enough to counteract
5
the gravitational infall, and the region undergoes a “bounce,” expanding until the
radiation pressure drops enough for it to collapse once again .
The Jeans length LJ determines the minimum size of density perturbations which
are able to collapse. In a region of size L < LJ , the radiation pressure is great
enough to prevent collapse. While matter and radiation are tightly coupled, the
Jeans length is roughly the same scale as the horizon size, both of which increase
with time. Therefore, as time progresses, regions of increasing size will begin to
collapse, and regions of different sizes will be at different phases in their oscillation
at any given time. Recombination occurs fairly quickly compared to the timescale of
the oscillations, so the anisotropies are “frozen” into the CMB at different phases of
oscillation.
Regions which are more or less dense than average are hotter or colder than average, because the particles gain kinetic energy when near the center of a potential well
and lose energy when climbing away from it. Regions which are at maximum compression or rarefaction introduce the largest intrinsic temperature variations into the
CMB. Observationally, we will see a peak in ∆T /T (and in the power spectrum C`T T ,
described in Section 2.2.2) at angular scales corresponding to the size of regions which
were at maximum compression or rarefaction. These are known as acoustic peaks.
At angular scales corresponding to regions halfway between maximum compression
or rarefaction — that is, regions having the greatest velocity — the temperature
anisotropy is due to the Doppler effect, which is 90◦ out of phase with the compression and rarefaction peaks. Because the anisotropies due to the Doppler effect are
weaker than those arising from the regions of maximum compression or rarefaction,
these correspond to minima in the power spectrum (see Figure 2.1).
6
Figure 2.1: The temperature anisotropy power spectrum C `T T for a ΛCDM model. Data
from WMAP, ACBAR, and CBI are shown. The first peak is a density maxima, the second
a density minima, etc., while the troughs are regions with maximum velocity.
7
A full discussion of secondary anisotropies is beyond the scope of this paper; however, a brief description of several sources is relevant. If a photon travels through a
gravitational potential well, the blueshifting and redshifting cancel unless the gravitational potential changes during the time it takes the photon to traverse it. If the
potential ψ does evolve, then radiation from that region will have a net temperature
R
change of ∆T /T = −2 (dψ/dt)dl, where the integral is over the photon path [13].
This is known as the integrated Sachs-Wolfe (ISW) effect. The Early ISW effect occurs from the decay of gravitational potentials shortly after recombination. While the
universe is radiation dominated, potentials at scales smaller than the sound horizon
tend to decay away due to photon pressure, leading to a blueshift of the radiation.
Because this effect ends when the universe becomes matter dominated, an event which
occurs coincidentally close to the time of recombination in standard models, it creates
maximum anisotropy at a slightly larger scale than that of the first acoustic peak.
The Late ISW effect occurs at the end of matter domination, if the universe is open
or has a significant dark energy component, when a similar decay of gravitational
potentials occurs. Finally, the Sunyaev-Zeldovich effect [14] occurs when radiation
passes through a locally ionized region, such as a cluster of galaxies. The radiation
interacts with the ionized gas in these regions through Compton scattering, resulting
in a change in the photon energy.
2.2.2
Statistical Description of Temperature Anisotropies
The effects described above yield differences in temperature that characteristically
cover different sized regions of the sky. In this section we will present the usual
statistical description of these temperature anisotropies as a function of angular size.
8
The difference in temperature between a point on the sky defined by the direction
vector n̂ and the average temperature T , relative to T , is given by ∆T (n̂)/T . Because
we are looking at the temperature projected onto a sphere, it is useful to decompose
this quantity into spherical harmonics:
∞
∆T (n̂) X X̀ T
=
a`m Y`m (n̂),
T
`=2 m=−`
(2.1)
where Y`m (n̂) are the spherical harmonics, and the coefficients aT`m give the relative
power of each spherical harmonic. Each multipole ` corresponds to some angular
scale θ, with larger ` values corresponding to smaller angular scales, and vice versa.
For example, ` = 1 corresponds to the dipole, θ = 180◦ , and scales of θ = 1◦ are
equivalent to ` ≈ 100. The sum starts at ` = 2 because the dipole moment aT`=1
is much larger than the others and has a different source. The dipole temperature
anisotropy comes almost entirely from the Doppler shift of the CMB radiation due
to our motion relative to the frame in which the CMB was emitted, and thus is
subtracted off.
The coefficients are random independent variables having a zero mean, haT`m i = 0,
where the brackets denote an ensemble average over all possible realizations of the
CMB for a given model. Assuming the anisotropies are statistically isotropic, their
variance has the form
C`T T δ``0 δmm0 = ha∗`mT aT`0 m0 i
(2.2)
or
C`T T =
X̀
1
|aT |2 .
2` + 1 m=−` `m
C`T T is the angular power spectrum of the temperature anisotropies.
9
(2.3)
If the underlying density fluctuations are Gaussian in nature, as predicted by
inflation, then the statistical properties of the temperature anisotropies are completely
specified by the angular two-point correlation function C(θ), which is related to the
power spectrum C`T T by
∆T (n̂) ∆T (n̂0 )
C(θ) =
T
T
X 2` + 1
=
C`T T P` (n̂ · n̂0 ).
4π
`
(2.4)
(2.5)
Here n̂ and n̂0 are two direction vectors separated by an angle θ, and P` (n̂ · n̂0 ) =
P` (cos θ) is the Legendre polynomial of order `. The C`T T ’s describe the relative power
of temperature anisotropies on various angular scales.
At small scales, the temperature power spectrum C`T T is dominated by the characteristic peaks and troughs of the acoustic oscillations. For scales larger than that
of the first acoustic peak (` . 40), the temperature power spectrum is dominated by
the Sachs-Wolfe effect. Because in an inflationary cosmology these scales were not in
causal contact before recombination, they are directly related to the power spectrum
of initial density fluctuations P (k) (see Appendix A). If the initial density fluctuations are scale invariant (ns = 1) and the Early ISW effect is small, the power per
logarithmic interval is approximately constant (see Figure 2.2), and can be directly
related to ∆T /T at these scales:
`(` +
1)C`T T
≈
∆T
T
2
(θ) ≈ constant,
(2.6)
where θ ≈ 180◦ /` [15]. For this reason, the power per logarithmic interval
C`T T = `(` + 1)C`T T /(2π)
is frequently plotted versus `.
10
(2.7)
Figure 2.2: The same temperature anisotropy power spectrum C `T T as shown in Figure 2.1,
but plotted logarithmically along the `-axis to show the Sachs-Wolfe plateau at low `. This
plateau is only approximate. The Late ISW effect, due to a cosmological constant, causes
it to tilt up at the very smallest `, while at the high ` end the plateau tilts up due to the
Early ISW effect, caused by the decay of gravitational potentials before matter-radiation
equality, and transitions smoothly into the first acoustic peak.
11
As a final note, there is a fundamental limit to our ability to determine the
anisotropy power spectrum, the problem of “cosmic variance.” The processes which
produce the CMB are stochastic in nature, and our predictions produce an ensemble
average over all possible realizations of the CMB. However, we have only one realization of the CMB to observe, and we should not expect that the observed sky would
exactly follow the predicted average. We can make the assumption that averaging
over space is equivalent to averaging over ensembles, but for each mode `, we are still
only able to obtain 2` + 1 independent values of a`m due to the fact that we can only
observe the CMB from one vantage point in space. Thus, the cosmic variance in C`
is
∆C` /C` =
r
2
.
2` + 1
(2.8)
This equation assumes that we are able to observe the whole sky; in real experiments
the limited sky coverage causes even larger variance [15].
2.3
Polarization Anisotropies in the CMB
2.3.1
Sources of Polarization Anisotropies
Polarization in the CMB is generated through Thomson scattering. The cross
section for Thomson scattering of photons off of free electrons depends on the polarization as
3 0
dσ
=
|Ê · Ê|2 σT ,
dΩ
8π
(2.9)
where dσ/dΩ is the differential cross section with Ω being the solid angle, σT is the
Thomson scattering cross section, and Ê and Ê0 are the directions of the electric field,
and thus the directions of the polarization, of the incident and scattered light. The
incident photon causes the electron to oscillate in the same direction as Ê. The result
12
y
x
z
e-
e-
Figure 2.3: Radiation with a polarization Ê incident on an electron will preferentially
scatter in a direction perpendicular to the direction of incidence, but with polarization
parallel to the original polarization. [16]
(Figure 2.3) is that the intensity of the scattered radiation is greatest in the direction
normal to that of the incident radiation, and the polarization of the scattered light is
parallel to that of the incident light [16].
Incident beams of unpolarized radiation separated by 90◦ yield orthogonal polarizations. If the radiation field is isotropic, then incident light from two right-angle directions will yield polarizations in the scattered light which exactly cancel each other.
However, a quadrupolar anisotropy in the temperature of the incident field will result
in linear polarization of the scattered radiation. In a quadrupolar anisotropy, intensity maxima are separated from intensity minima by 90◦ . Therefore the intensity of
incident light from directions separated by 90◦ would not exactly cancel, leading to a
net polarization in the resulting radiation (Figure 2.4). If the light is re-scattered after this, the coherence in the polarization is lost, so the CMB polarization is due only
13
y
x
z
e-
Figure 2.4: Quadrupolar anisotropies in the temperature field are required to produce polarization anisotropies in the CMB. If the observer is viewing radiation scattered along the
positive z-axis, the incident radiation preferentially came from the x-y plane. Unpolarized
radiation incident from the y direction will yield scattered light polarized in the x direction,
and vice-versa. If the temperature is isotropic, the resulting scattered light will be unpolarized. However, a quadrupole in the temperature leads to a net polarization in the scattered
radiation. [16]
to quadrupole variations in temperature during recombination, when the radiation is
last scattered [16].
The acoustic oscillations of the photon-baryon fluid also show up in the polarization power spectrum. Bulk velocity flows are capable of generating quadrupole
anisotropies. In the case of polarization, then, the intensity peaks are at scales where
the velocity flow is at a maximum, and the intensity troughs correspond to density
maxima and minima. Therefore the polarization power spectrum’s peaks are 90◦ out
of phase with those of the temperature power spectrum (Figure 2.5).
14
Figure 2.5: The E-mode polarization anisotropy power spectrum C `EE for a ΛCDM model.
The first peak is a region with maximum infall velocity, the second with maximum expansion
velocity, etc., and the troughs are density maxima and minima.
15
Because polarization anisotropies are created only by scattering, unlike the temperature anisotropies which can grow due to gravitational effects, there are no “secondary” polarization anisotropies. All of the anisotropy in polarization was established during recombination, making polarization a direct probe of the universe at
the time of recombination.
2.3.2
Statistical Description of Polarization Anisotropies
The statistical description of the polarization of the CMB is very similar to that
of the temperature. Here we primarily follow the derivations of [17] and [18].
If we define the electric field vector E of the CMB radiation, we can then describe
the observations of the CMB in terms of a temperature fluctuation matrix decomposed
into Stokes parameters (Θ, Q, U, V ) [19]:
P = c0 hE(n̂)E† (n̂)i
(2.10)
= Θ(n̂)I + Q(n̂)σ3 + U (n̂)σ1 + V (n̂)σ2 ,
(2.11)
where I is the identity matrix and the Pauli matrices are
σ1 =
0 1
1 0
, σ2 =
0 −i
i 0
,
σ3 =
1 0
0 −1
.
(2.12)
The scalar term, Θ, is the familiar temperature anisotropy, Θ = ∆T /T , and the
constant of proportionality c0 is chosen so that the Stokes parameters have units
of temperature. Circular polarization is described by the parameter V , but it is
assumed to be zero because it cannot be generated through Thomson scattering. The
remaining Stokes parameters Q and U describe the linear polarization. They are not
invariant under a rotation about the direction of observation by some angle φ, but
16
transform into each other as
Q0 =
Q cos(2φ) + U sin(2φ)
U 0 = −Q sin(2φ) + U cos(2φ).
(2.13)
(2.14)
The linear polarization anisotropies can be decomposed into spin-2 spherical harmonics,
±2 Y`m
, much like the temperature anisotropy analysis presented above:
Q ± iU =
Here again the coefficients
±2 a`m
m=`
X
±2 a`m ±2 Y`m (n̂).
(2.15)
m=−`
give the relative power in each mode.
These linear combinations of the Stokes parameters are commonly called E- and
B-mode polarizations. Just as a vector field can be decomposed into a part which is
the gradient of a scalar field and a part that is the curl of a vector field, the E-mode
polarization is the “gradient” portion of the polarization, while the B-mode is the
“curl” contribution [20] (see Figure 2.6):
Q + iU = E
(2.16)
Q − iU = B.
(2.17)
Introducing a rotationally invariant linear combination of the coefficients,
aE
`m = (+2 a`m +−2 a`m )/2
(2.18)
aB
`m = (+2 a`m −−2 a`m )/2i,
(2.19)
we can create E and B anisotropy power spectra as follows:
C`EE δ``0 δmm0 = ha∗`mE aE
`m i
(2.20)
C`BB δ``0 δmm0 = ha∗`mB aB
`m i.
(2.21)
17
Figure 2.6: E-mode (left) and B-mode (right) polarizations for a single plane wave. The
direction of E-mode polarization is parallel or perpendicular to the direction of greatest
change, while the direction of B-mode polarization is at 45 ◦ compared to the direction of
greatest change [16].
Figure 2.7: Correlation of polarization with temperature anisotropies. Left: superposition
of modes leads to tangential E-mode polarization patterns (solid lines) around hot spots and
radial patterns (dashed lines) around cold spots. Right: superposition of modes leads to no
correlation of B-mode polarization with temperature, due to the parity of the B-modes [16].
The reader is referred to [16] for a more extensive explanation of how to gain an intuitive
understanding of polarization.
18
Additionally we can create cross-correlation power spectra,
C`T E δ``0 δmm0 = ha∗`mT aE
`m i
(2.22)
C`T B δ``0 δmm0 = ha∗`mT aB
`m i
(2.23)
C`EB δ``0 δmm0 = ha∗`mE aB
`m i,
(2.24)
though the EB and TB cross-correlations are expected to be zero. Figure 2.7 shows
the correlation between hot or cold spots on the sky and polarization.
Again, we can express the power per logarithmic interval as
0
0
C`XX = `(` + 1)C`XX /2π,
(2.25)
where X and X 0 are one of (T, E, B). Example polarization power spectra are shown
in Figures 2.5 and 2.8.
Polarization is created only by quadrupole anisotropies in the temperature field.
There are three sources, geometrically, of these quadrupole temperature anisotropies:
scalar, vector and tensor perturbations. Scalar perturbations are compressional in
nature. They are caused by perturbations in the density field ρ(x), and grow through
gravitational infall. Scalar perturbations produce only E-mode polarization. Vector
perturbations are caused by vortical flows in the velocity field v of the matter and
radiation, where ∇×v is nonzero, and produce primarily B-mode polarization. There
is no mechanism for generating these types of velocity fields cosmologically, and we
will ignore them. Finally tensor perturbations are due to gravitational waves, which
“stretch” space in a quadrupolar fashion as they pass through and produce roughly
equal amounts of E-mode and B-mode polarization. Tensor perturbations are also
capable of contributing to the temperature anisotropy power spectrum. However,
19
Figure 2.8: The TE cross-correlation anisotropy power spectrum C `T E for a ΛCDM model.
Data from WMAP are shown.
20
in the interests of simplicity, we have neglected tensor contributions throughout this
work.
2.4
Recent Experiments
The CMB was first detected by Penzias and Wilson at Bell Labs in 1965 [21]. In
1992 the first detections of temperature anisotropies in the CMB were made by the
Cosmic Microwave Background Explorer’s (COBE) Differential Microwave Radiometers (DMR) [22]. COBE was a satellite launched by NASA that performed an all-sky
measurement of the CMB with an angular resolution of ∼ 10◦ , or ` . 30. COBE
was the first among many experiments to measure the temperature anisotropy of the
CMB. An explosion of ground-based [8, 9, 23–38] balloon-borne [39–49] and satellite
experiments [50, 51] have followed.
COBE’s direct successor is the Wilkinson Microwave Anisotropy Probe (WMAP),
also a satellite launched by NASA, but capable of measuring the anisotropy on scales
down to ∼ 0.3◦ . The WMAP team released its first-year results to the public in
February 2003. The WMAP results were revolutionary for two main reasons. First,
WMAP measured the temperature anisotropy power spectrum through the first peak
to an unprecedented degree of precision [50]. The measurement uncertainties, at least
for low `, are . 1%. For ` . 350, their measurement errors are smaller than that due
to cosmic variance. For this reason, with the exception of a few recent experiments
[8, 9] that probe smaller scales than WMAP is capable of reaching, it is not worthwhile
to combine WMAP data with previous measurements.
21
Second, WMAP was the first,1 and at present only, experiment to measure the
polarization anisotropy power spectrum [51]. Because the CMB is only polarized
at about the 10% level, the polarization anisotropies are at a level of one part in
106 , making their detection far more challenging. Although E-mode and B-mode
power spectra are not yet available, the TE cross-correlation power spectrum has
been measured up to ` ≈ 500.
We make use of two other experiments in our analysis, both of which provide temperature anisotropy data at smaller scales than WMAP can probe: the Arcminute
Cosmology Bolometer Array Receiver (ACBAR) [9], a ground based experiment located in Antarctica, and the Cosmic Background Imager (CBI) [8, 52], a ground based
experiment located in northern Chile. ACBAR has an angular resolution of ∼ 40 − 50 ,
and is sensitive to anisotropies on scales from 150 < ` < 3000. Uncertainties are on
the order of 30% or better. CBI has an angular resolution of ∼ 60 − 150 . It detected
anisotropies in the range 300 < ` < 3000. For ` . 1700 the signal to noise ratio is
again on the order of ∼ 30% or better, but for ` & 1700 the uncertainty is on the
same order as the signal.
Figures 2.1 and 2.8 display the data from WMAP, ACBAR and CBI that we use
in our analysis.
1
The Degree Angular Scale Interferometer (DASI) [38] was the first experiment to detect CMB
polarization in the fall of 2002, but their detection was not strong enough to produce an estimation
of the power spectrum.
22
CHAPTER 3
THE EPOCH OF RECOMBINATION
3.1
Recombination
Let us now take a closer look at the epoch of recombination. Recombination is
the process of atomic nuclei capturing and keeping electrons for the first time, so that
the universe becomes filled with neutral particles.
We shall be primarily concerned with hydrogen recombination, as the density of
the other elements, with the exception of helium, is low enough that they can be
safely ignored. Even helium, because of its higher ground level binding energy, is
largely finished recombining by the time hydrogen recombination starts and thus can
be dealt with independently.
Recombination begins when the temperature of the photon-baryon fluid is low
enough that when a nucleus captures an electron, there is very little radiation around
energetic enough to reionize it. One might naively expect hydrogen recombination
to begin when the temperature of the radiation falls below kB T = 13.6 eV, the
binding energy of the ground state of hydrogen. However, the temperature of thermal
radiation is defined by the peak of the blackbody curve. Recombination must instead
wait until there are very few ionizing photons left even in the high energy tail of the
23
distribution. In a standard model, the peak temperature of such a distribution is
about kB T ≈ 0.3 eV.
Once recombination begins, the cross section for interaction between matter and
radiation drops and they begin to decouple. The “end” of recombination is generally
considered to be the point at which the fraction of ionized hydrogen falls to less than
1%. The remaining fraction is known as residual ionization, and it continues to fall
until fairly recently, though it does not completely vanish.
Finally, at late times, the universe (or at least intergalactic hydrogen clouds) becomes partially reionized. Observations of quasars at redshifts z ≈ 5 − 6 indicate that
the intergalactic medium has been reionized, probably due to radiation from active
galactic nuclei (AGN’s), starbursting galaxies, or a very early population of stars. In
the standard cosmological picture, the optical depth τRI due to this reionization is
believed to be significantly less than one. The ionized material is capable of scattering the CMB radiation, damping anisotropies by a factor of e−2τRI on small and
intermediate scales (` & 50) [53]. Incidentally, even disregarding reionization, the
optical depth τ due to the residual ionization fraction is a few percent and will also
damp the CMB power spectra by a factor of approximately e−2τ [54].
3.2
3.2.1
Relation to CMB
The Visibility Function
Recombination happens fairly quickly, but not instantaneously, on cosmological
timescales. Because of the travel time of light, events happening at earlier times also
happen at greater distances and therefore at greater redshifts z. It is thus common to
24
measure the time of recombination as occurring at a particular redshift, zrec . Similarly,
the duration of recombinations is measured as a width ∆z.
Because the radiation of the CMB last scattered off of matter during recombination, the electrons and baryons that the CMB last scattered off of create a “last
scattering surface” (LSS). From an observational viewpoint, this surface of last scattering lies in a sphere, where zrec gives the distance to this sphere and ∆z gives its
thickness.
Mathematically, zrec and ∆z are defined through the visibility function, g(z). The
visibility function is
g(z) = e−τ dτ /dz,
(3.1)
where
τ =−
Z
z
cσT ne (z)(dt/dz)dz
(3.2)
0
is the optical depth due to Thomson scattering, and ne (z) is the number density of
free electrons [55]. The quantity g(z)dz is the probability that a given photon was
last scattered within dz of the redshift z. We define the redshift of recombination zrec
as the redshift where the visibility function peaks. The width of the last scattering
surface ∆z is calculated by approximating the visibility function with a Gaussian by
matching it at the peak and at the points where the value is half that of the peak.
∆z is then identified as the width σ of this Gaussian [56].
3.2.2
Effects on Temperature Anisotropies
Both the width ∆z and position zrec of the LSS have a direct effect on the temperature anisotropy power spectrum C`T T .
25
The position zrec influences the temperature anisotropies primary in three ways.
First, delaying recombination, or decreasing zrec , corresponds to moving the surface of
last scattering closer to us. Systems of the same physical size would then correspond
to a larger angular size on the sky, so features of the power spectrum would be
systematically moved to smaller values of `.
Second, the Jeans length LJ , which determines the size of the largest objects that
can start to collapse (Section 2.2.1), increases with time. A delay in recombination
would result in an increased LJ ; thus, larger scales of density perturbations would
have started to collapse and all of the smaller scales would have had more time to
oscillate. This would again have the effect of shifting the acoustic features of the
spectrum to lower values of `.
Finally, delayed recombination would also change the relative amplitudes of the
even and odd peaks [3]. The ratio of the heights of the even and odd peaks depends
on the ratio R = (3ρb /4ργ ) at the time of decoupling, where ρb is the baryon energy
density and ργ is the photon energy density. As R increases, the baryons contribute
more drag to the photon-baryon fluid, inhibiting rarefaction. Since ρb scales as (1+z)3
and ργ scales as (1 + z)4 , R ∝ (1 + z)−1 . Thus, a lower zrec will increase R, thereby
decreasing the height of the even peaks [4].
The width ∆z of the last scattering surface affects the temperature anisotropy
power spectrum through an effect known as diffusion damping or Silk damping [57].
The width ∆z corresponds to a co-moving thickness, and thus to an angular size.
Intrinsic fluctuations that are of a scale smaller than the scale of this width are
erased. Fluctuation at larger scales also suffer damping, with more damping occurring
26
Figure 3.1: The width of the last scattering surface determines the minimum scale for
intrinsic temperature variations because regions smaller than the width of the LSS are
averaged over (bottom line of sight) [58].
at scales closer to the scale of ∆z [11]. Since the width of the last scattering surface
is not large, this primarily affects anisotropies at large `.
The mechanism for this effect is easy to understand qualitatively. The erasure
occurs because the observed temperatures in regions smaller than ∆z will be reduced
by averaging over different regions within the surface of last scattering (see Figure
3.1) [58].
27
3.2.3
Effects on Polarization Anisotropies
The polarization anisotropies are also affected by the details of recombination.
Delaying recombination (decreasing zrec ) shifts the features of the polarization power
spectrum C`EE to smaller `, just as with the temperature power spectrum. This is
again due to objects which are the same physical size having a larger angular size
when the LSS is moved closer to the observer and because of the longer amount of
time collapsing regions have had to undergo acoustic oscillations.
With increasing width ∆z, the polarization anisotropies are suppressed by diffusion damping at small scales. Anisotropies at scales smaller than ∆z are severely
damped, just as with the temperature anisotropies. Also, in order to be polarized,
the radiation has to have last scattered in regions which are optically thin. Thus the
amount of polarized radiation depends on ∆z, and a larger value of ∆z will decrease
the height of the peaks relative to the Sachs-Wolfe part of the spectrum.
3.3
Ionization Fraction
Recombination is described through the evolution of the ionization fraction. We
define the ionization fraction xe as the fraction of ionized hydrogen atoms:
xe =
nH II
ne
=
.
n
n
(3.3)
Here ne is the number density of free electrons, and n is the number density of all
hydrogen atoms, both ionized and neutral. Since the universe is electrically neutral,
the number of electrons and protons (ionized hydrogen nuclei) must be equal, ne =
nH II , if we ignore the other elements such as helium or assume that they have already
undergone recombination and are thus completely neutral.
28
The principle obstacle in the recombination process is that a recombination directly to the ground state emits a photon capable of ionizing hydrogen. To first
order, each photon produced by a direct-to-ground-state recombination will ionize a
neutral hydrogen atom, resulting in no net change in the ionization fraction. Most
(net) recombinations occur by the capture of an electron in an excited state, with
the electron subsequently cascading to the ground state. Therefore it is the population of the excited states, especially the first excited state, that controls the rate of
recombination.
The relevant timescale in determining the rate of recombination is whether an
excited electron will decay via radiative emission to the ground state (controlled by
quantum effects) faster than a photon will hit it and ionize the atom (controlled by
astrophysical conditions). Atoms with electrons in an excited state are easy to ionize,
because while a ground state electron requires a photon of energy B1 = 13.6 eV or
greater to ionize it, the binding energy of an excited electron is a quarter of that
amount or less, and low energy photons are far more abundant than high energy
ones. However, as the universe expands and both the photon density and the energy
of individual photons decreases, it becomes more favorable for the de-excitation rate
to beat the reionization rate, allowing more recombinations.
While a few electrons may transition to the ground state from a very excited
state, the majority of recombinations involve a transition from the first excited state
to the ground state. For this reason, and because the standard calculation of the
ionization fraction utilizes an effective three-level atom with a ground state, a first
excited state, and the continuum (discussed more in Section 3.3.2), it is worthwhile
to discuss de-excitation from the first state here.
29
The first excited state, n = 2, where n is the principle quantum number, is
divided into two states labeled by the angular momentum quantum number l. In
the first excited state, l can have two values, l = 0 (the 2s state) or l = 1 (the 2p
state). Although the 2s and 2p states have the same energy in the hydrogen atom,
transitions to the ground state vary depending on the angular momentum quantum
number. The selection rule for an electric dipole transition requires δl = ±1, which
a 2p − 1s transition satisfies, but a 2s − 1s transition does not. Therefore, to decay
to the ground state, an electron in the 2p state will emit a single photon (a Ly-α
photon) of energy ∼ (3/4)B1 where B1 is the binding energy of the ground state of
hydrogen. An electron in the 2s state may transition via a much suppressed magnetic
dipole transition, or more likely, through two-photon emission, where both photons
emitted are, obviously, less energetic than a Ly-α photon.
The 2p − 1s transition has one obstacle to overcome before a net recombination
occurs: the Ly-α photons produced will excite ground state hydrogen, making it
easier for it to be ionized. The primary way by which the 2p − 1s route succeeds in
a net recombination is for the Ly-α photon to be redshifted out of the range where
it can excite other hydrogen atoms. Therefore the redshifting of Ly-α photons is the
controlling timescale for this route of recombination. The controlling timescale for
the 2s − 1s route, on the other hand, is the competing two-photon excitation rate.
3.3.1
Equilibrium Calculation
If matter and radiation are in equilibrium, then the matter temperature Tm and
the radiation temperature Tr are equal, T ≡ Tr = Tm . Under equilibrium conditions,
the relative abundance of neutral atoms and ions is given by the Saha equation. In
30
the case of hydrogen, the ratio of the number density of ionized hydrogen nH II to
neutral hydrogen nH I can be written
xe
1
nH II
=
=
nH I
1 − xe
ne
me k B T
2π~2
3/2
e−B1 /kB T ,
(3.4)
where me is the electron mass, kB is the Boltzmann constant, ~ is Planck’s constant,
and B1 is the binding energy of ground state hydrogen.
The assumption of equilibrium is reasonable at the beginning of recombination,
but is increasingly less valid as recombination proceeds and matter and radiation
become decoupled. The actual evolution of the ionization fraction with time varies
increasingly from the equilibrium solution as the production of neutral hydrogen
progresses.
In most models helium recombination is complete by the time hydrogen recombination begins. Therefore, the universe is in thermal equilibrium during the whole
period of helium recombination, and the Saha equation applies to it throughout.
3.3.2
Non-Equilibrium Calculation
Non-equilibrium calculations of the ionization fraction are considerably more involved than the equilibrium calculation. A full treatment of the problem involves
calculating ionization and recombination rates to each level of each atomic species
present, as well as accounting for transitions between atomic levels and feedback
effects from resonance line radiation (see e.g. [54]).
The approach we are taking, however, is an approximation which models the
hydrogen atom as an effective three-level atom. The effective three-level atom has a
ground state, a first excited state, and the continuum. The other levels are accounted
for (thus making this an effective three level atom and not just a three-level atom)
31
by changing the recombination and ionization coefficients for the first excited state
into “total” recombination and ionization coefficients. Other elements are ignored.
[59] and [60] independently derived equations for the time evolution of xe in nonequilibrium conditions using the effective three-level atom. Various corrections and
improvements on their equations have been made since then, but the effective threelevel atom is a very good approximation and for the most part these are small effects.
The derivation presented here closely follows that of [59].
Since net recombinations to the ground state are negligible, only recombinations
to the excited states are considered. The net rate of recombination is obtained by
comparing the recombination rate to the ionization rate:
dxe X αnl n2e βnl nnl
.
=
+
−
dt
n
n
n>1
(3.5)
The first term on the right side is the recombination rate, where αnl is the coefficient
for recombination to the (n, l) level of the atom. The second term is the ionization
rate, and βnl is the corresponding ionization coefficient. nnl is the number density of
hydrogen atoms with an electron in the (n, l) level, and ne and n are defined above.
The total recombination coefficient used in the three-level atom approximation is
obtained by summing over all states, and is defined as
α=
X
αnl ,
(3.6)
n>1
and similarly, the total ionization coefficient is
β=
X
(2l + 1)βnl e−(B2 −Bn )/kB T .
(3.7)
n>1
In deriving Equation (3.7), the Boltzmann equation relating the number density of
atoms with an electron in level (n, l) to the number density in state 2s was used:
nnl = n2s (2l + 1)e(B2 −Bn )/kB T .
32
(3.8)
It is usually assumed that the upper levels of the hydrogen atom, n > 1, are in
thermal equilibrium with the radiation background, although the ground state is not.
The basis for this assumption is that there are few higher energy photons capable of
coupling to the ground state, but the excited states are very close to each other and
to the continuum and the more plentiful lower energy photons are sufficient to insure
equilibrium. This assumption of thermal equilibrium allows the ionization rate to be
related to the recombination rate:
β = αe
−B2 /kB T
me k B T
2π~2
3/2
.
(3.9)
Using Equations (3.6) and (3.7), (3.5) can be rewritten
−
αn2e βn2s
dxe
=
−
.
dt
n
n
(3.10)
If the ground state were in thermal equilibrium with the excited states, n2s could be
related to n1s via the Boltzmann equation and the result would be
2
αne βn1s −(B1 −B2 )/kB T
dxe
=
−
e
.
−
dt
n
n
(3.11)
However, the ground state is not in equilibrium with the excited states. In order
to calculate the correction for this, the redshifting of Ly-α photons and the rate of
two-photon decay must be taken into account. When this is done, the equation for
the ionization fraction becomes
2
dxe
αne βn1s −(B1 −B2 )/kB T
−
= CP
−
e
.
dt
n
n
(3.12)
CP is a correction factor introduced by Peebles. It is the fraction by which recombination is self-inhibited due to the resonance-line radiation produced in the recombination
process:
CP =
1 + KΛ2s1s n1s
.
1 + K(Λ2s1s + β)n1s
33
(3.13)
Here, Λ2s1s = 8.22 s−1 is the two-photon decay rate from 2s to 1s, K = λ3α /8πH(z)
describes the redshifting of Ly-α photons, where λα = 121 nm is the wavelength of
a Ly-α photon and H(z) is the Hubble constant as function of redshift. The time
dependence of n, T , and H can be calculated from the Friedmann equation (Appendix
A).
Making the approximation that the number of hydrogen atoms with electrons in
the ground state is approximately equal to the total number of neutral hydrogen
atoms, n1s ≈ nH I = (n − ne ), and making a change of variables to xe = ne /n, we can
rewrite Equation (3.12) as
−
dxe
= CP αnx2e − β(1 − xe )e−(B1 −B2 )/kB T .
dt
(3.14)
This is the equation governing the change in the ionization fraction with time for
“standard recombination.”
34
CHAPTER 4
VARIATIONS ON STANDARD RECOMBINATION
4.1
Motivation
Recombination clearly has a fundamental effect on the shape of the CMB power
spectra. Changes to the position and width of the surface of last scattering induce
corresponding changes to both the temperature and polarization power spectra. Given
the current precision of the data available, it is worthwhile to re-examine this epoch
and place stronger constraints on it.
This issue has been investigated in the past and previous authors have suggested
a variety of mechanisms which could cause deviations from standard recombination.
Delaying recombination through the introduction of a source of ionizing and/or
Ly-α photons into the universe was suggested by [3]. Distortions to the recombination
history can be parameterized through efficiency functions i and α , which modify the
rate of production of ionizing and Ly-α photons, respectively:
dni,α /dt = i,α nH(z),
(4.1)
where ni,α are the number density of ionizing or Ly-α photons, n is the number density
of neutral plus ionized hydrogen, and H(z) is the Hubble constant. The efficiency
35
functions i,α can be constants or functions of redshift, depending on the model. These
modifications influence the evolution of the ionization fraction as [6]
dxe
dxe − i CP H(z) − α (1 − CP )H(z).
−
=−
dt
dt std rec
(4.2)
where dxe /dt|std rec is given by Equation (3.14).
Although [3] did not make suggestions as to the source of these extra photons,
several other authors have. Decaying dark matter [5, 61] or decaying neutrinos [62]
have been examined as sources of these photon excesses, although these models tend
to affect reionization rather than recombination per se. The possibility of decaying
primordial black holes was investigated in [63, 64]. As an alternative to decaying particles, [65] investigated a means of accelerating recombination through the clumpiness
of baryonic clouds in the early universe. The various decaying particle and clumpy
baryon models can all be described through the formulation above, with appropriate
definitions of i,α [6, 66].
The possibility of a varying fine structure constant and the effect it would have
on the CMB through the corresponding change in recombination was examined by
[4, 67, 68], and is not described by this formulation. Instead, the fine structure
constant has an effect on recombination via the differential Thomson scattering optical
depth τ̇ (see Section 5.3), primarily through the dependence of the Thomson cross
section σT on the square of the fine structure constant [67].
4.2
Our Model
It is our goal to constrain recombination in a general way, rather than investigating
specific mechanisms. To that end, we follow the model presented in [69], which is a
general extension of recombination.
36
Our approach is to change the equation governing the evolution of the ionization
fraction as a function of time. We modify the ionization history by inserting two
parameters, a and b, into Equation (3.14):
−
dxe
= aCP αnx2e − β(1 − xe )e−b(B1 −B2 )/kB T .
dt
(4.3)
The constant a multiplies both the overall rates of recombination and ionization
while preserving the ratio α/β from Equation (3.9). The constant b multiplies the
binding energies of hydrogen. The values a = 1 (or log(a) = 0) and b = 1 are
the case of standard recombination. We neglect any corresponding change in helium
recombination, as that effect is small and for most models helium recombination is
finished by the time hydrogen recombination begins. Very roughly, a change in a alone
will change the duration of recombination, while keeping the onset of recombination
fixed, while altering b shifts the onset of recombination to earlier or later redshifts.
The changes to the recombination history as a result of altering a and b will be
discussed in detail in Section 6.1.1.
Our work differs from previous efforts in several ways. Unlike most previous
efforts, the model we use is a very general model, not based on a specific mechanism
for altering recombination. In this way the constraints that we obtain are potentially
more general and applicable to a wider variety of mechanisms which might alter
recombination. Also, with the exception of [6], previous work was accomplished prior
to the release of the WMAP data. Our results benefit greatly from the increased
precision of the temperature data. Furthermore, since polarization probes the epoch
of recombination directly, the inclusion of the newly available TE cross-correlation
data greatly increases our ability to constrain recombination. Finally, we present an
37
analysis of the ability of the E-mode polarization power spectrum to further constrain
recombination.
38
CHAPTER 5
METHODS
5.1
Overview
Our goal is to find a range of recombination histories — that is, regions in a-b
parameter space — which are capable of predicting CMB power spectra compatible
with observational evidence.
In order to find this range of recombination histories, we take the following approach. We first create a grid in a-b space. For each point in this space, we allow a
set of five cosmological parameters to vary, as well as the overall normalization of the
power spectrum. The vector of free parameters that we allow to vary is ϑ={Ω m , ωb ,
h, ns , τRI , Q}: the Hubble parameter h, the matter density relative to critical Ωm ,
the baryon matter density ωb ≡ Ωb h2 , the spectral tilt ns , the optical depth to the
surface of last scattering τRI , and the overall normalization of the power spectrum
Q. (For the descriptions of h, Ωm , ωb , and ns see Appendix A. τRI is introduced in
Section 3.1.)
We then use CMBFAST [70] to produce a CMB anisotropy spectrum from our model,
as a function of a, b, and ϑ. Comparing this predicted spectrum to the observed
spectrum, we obtain a value for the goodness of fit parameter χ2 . For each point in
39
a-b space this is done for various combinations of the free parameters. By minimizing
χ2 , we find the combination of free parameters which is best able to reproduce the
observed CMB power spectra, given the fixed values of a and b.
5.2
Data Sets
In this analysis, three different sets of data were used.
A data set consisting of a compilation of all experiments [8, 23–49] with successful
detections of the CMB as of July 2002 was created by [71]. This data set was used
for our early reseach, prior to the release of the WMAP data, and will be referred to
as the pre-WMAP data set. It has the benefit of being easy to use, as the relative
calibrations and beam widths of the different data sets are already taken into account.
The compiled data consists of 25 binned data points out to ` ≈ 1700. We emphasize
that this data set is presented primarily to show the improvements in tightening the
constraints on recombination that have become possible using WMAP data, and we
do not base any of our conclusions on the analysis of this set.
The second data set we use is the WMAP data alone, including both the temperature [50] and TE cross-correlation data [51]. The temperature power spectrum
consists of 899 data points out to ` ≈ 1000, and the TE cross-correlation power
spectrum has 449 data points out to ` ≈ 500.
The data from WMAP is of high enough quality to make including other data
sets at large scales (small `) unneccesary. However, the WMAP temperature data
only goes out to ` ≈ 1000. Thus, for our third data set, we use the WMAP data plus
data from two other CMB experiments that measure the anisotropy at smaller scales:
ACBAR [9, 72] and CBI [8, 52, 73]. This data set will be referred to as WMAP+.
40
Combining the WMAP, ACBAR, and CBI data sets is fairly straightforward, and
we follow the lead of [74] in our procedure for doing so. From CBI, we use the evenbinned version of the mosaic data set. Only three bandpowers are included, those
at `eff = 896, 1100, and 1300, as these are at high enough ` to be independent from
the WMAP data, simplifing the analysis. Data at higher ` was excluded to avoid
complications arising from contamination by the Sunyaev-Zeldovich effect.
From ACBAR, we used the bandpowers at `eff = 842, 986, 1128, 1297, 1426, 1580,
and 1716. Data is included to higher ` from ACBAR because the ACBAR experiment
is less sensitive to the Sunyaev-Zeldovich effect.
Although the data from CBI and ACBAR covers similar ranges in `, the data
sets are independant because they observed different parts of the sky, making it unnecessary to worry about correlations between the two. For ACBAR we assume a
calibration uncertainty of 20%, and for CBI a calibration uncertainty of 10%. Although ACBAR has a 3% beam width uncertainty, this effect is extremely small
compared to the calibration uncertainty and we neglect it. CBI does not have a
beam uncertainty.
The WMAP mission was able to make measurements of both the temperature and
TE cross-correlation power spectra. We do two different analyses: we minimize χ2
using the temperature power spectrum alone, and using the combination of temperature and TE power spectra. The temperature plus TE analysis is repeated for the
WMAP+ data set. The TE power spectrum alone does not contain enough information to constrain all of the parameters and break the resulting degeneracies [75], thus
we do not use it by itself.
41
5.3
CMBFAST
Predicting the CMB power spectrum involves solving coupled differential equations governing the behavior and evolution of the particles present in the universe:
baryons, cold dark matter (CDM), neutrinos, and photons. These equations are the
Boltzmann, Einstein, and fluid continuity equations.
In this analysis, we used the publicly available Fortran code CMBFAST [70] to calculate the C` power spectra for a given model cosmology. CMBFAST takes a variety of
cosmological parameters as input, and computes the TT, EE, and TE cross-correlation
power spectra. Here we present the basic equations that govern the evolution of the
CMB anisotropies, following closely the approach used in [70]. We restrict our analysis
to scalar perturbations.
The temperature anisotropies are ∆T (x, n̂) ≡ ∆T (x, n̂)/T , where we have added
the position vector x as an index, and can be transformed into Fourier modes with
wavevector k as [76]
T
∆ (k, n̂) =
Z
d3 x T
∆ (x, n̂)e−ik·x ,
V
(5.1)
where the integral is normalized to the volume V being integrated over. In linear
perturbation theory, all modes k evolve independently of each other, and the density
perturbations at early times are assumed to be small enough to fall well within the
linear regime.
We expand ∆T (k, n̂) in terms of Legendre polynomials P` (µ), where µ ≡ k̂ · n̂,
similar to the expansion in Equation (2.4):
∆T (k, n̂) =
X
(2` + 1)(−i)` ∆T` P` (µ).
`
42
(5.2)
One can define a corresponding polarization anisotropy of the net linear polarization
∆P (k, n̂) = Q, (U = 0 in this particular frame) which is Legendre expanded as
∆P (k, n̂) =
X
(2` + 1)(−i)` ∆P` P` (µ).
(5.3)
`
The temperature and polarization anisotropies evolve according to the Boltzmann
evolution equations, which for scalar perturbations become
˙ T ) + ikµ∆T = ϕ̇ − ikµψ + τ̇ {−∆T + ∆T0 + iµvb + P2 (µ)Π/2}
(∆
(5.4)
˙ P ) + ikµ∆P = τ̇ {−∆P + [1 − P2 (µ)]Π/2}
(∆
(5.5)
Π = ∆T2 + ∆P2 + ∆P0 .
Overdots denote derivatives with respect to the conformal time η =
(5.6)
R
dt/a, where a
is the scale factor, vb is the baryon velocity, and τ is the optical depth from Thomson
scattering. Here ϕ and ψ are the two scalar potentials that specify the perturbations
to the metric in the longitudinal, or conformal Newtonian, gauge:1 ψ is the Newtonian
potential while (−ϕ) is the perturbation to the spatial curvature [78]. Neutrino
evolution is likewise governed by the Boltzmann equation.
The evolution of cold dark matter and baryons is governed by the continuity and
Euler equations:
δ̇cdm = −kvcdm + 3ϕ̇
(5.7)
v̇cdm = −(ȧ/a)vcdm + kψ
(5.8)
δ̇b = −kvb + 3ϕ̇
(5.9)
v̇b = −(ȧ/a)vb + kψ + c2s kδb + τ̇ (3∆T1 − vb )/R.
1
(5.10)
The longitudinal gauge is defined through the line element ds2 = a2 (η)[−(1 + 2ψ)dη 2 + (1 −
2ϕ)dxi dxi ] [77]
43
Here δcdm , δb , vcdm and vb are the densities and velocities for cold dark matter and
baryons respectively. The baryon sound speed is cs , and R = (3ρ̄b /4ρ̄γ ) where the
average photon and baryon densities are ρ̄γ and ρ̄b .
The evolution of the scalar perturbations ϕ and ψ are governed by the Einstein
equations. These can be solved in the longitudinal gauge to yield equations constraining the energy and momentum:
k 2 ϕ + 3(ȧ/a)[ϕ̇ + (ȧ/a)ψ] = −4πGa2 δρ
k 2 [ϕ̇ + (ȧ/a)ψ] = 4πGa2 δf.
(5.11)
(5.12)
Here G is the gravitational constant, δρ is the total density perturbation and δf is
the momentum density perturbation.
These coupled equations must be evolved to the present time in order to obtain
the anisotropy for a given mode k. All modes and the initial power spectrum of the
metric perturbation Pψ (k) must be integrated over to obtain the CMB power spectra,
C`T
C`P
= (4π)
2
= (4π)
2
Z
Z
k 2 dkPψ (k)|∆T` (k, η0 )|2
(5.13)
k 2 dkPψ (k)|∆P` (k, η0 )|2 .
(5.14)
Note that the effect of recombination comes in principally through the differential
optical depth for Thomson scattering τ , described in Section 3.2.1:
τ̇ ≡
cσT ne
dτ
=
,
dη
H(z)(1 + z)2
(5.15)
where c is the speed of light, H(z) is the Hubble parameter at redshift z, and in our
analysis the evolution of ne is controlled by Equation (4.3).
44
For completeness, I present here all the options we use in CMBFAST. We take scalar
perturbations only. For the pre-WMAP data set, we did not include reionization.2
For the WMAP and WMAP+ data sets, we include reionization, choosing to specify
the optical depth to the surface of last scattering, τRI . This is left as one of the free
parameters in the minimization. Recombination is handled by the Peebles method,
though we have altered the specific subroutine that handles recombination to reflect
our modified recombination equation (4.3). A non-running spectral index was used,
and the dark energy was assumed to be a cosmological constant (w = −1).
5.4
Calculation of χ2
The output of CMBFAST is compared to the data to determine how well the theory
matches the data. The statistic used for this is the likelihood L or the goodness-of-fit
statistic χ2 , which are approximately related by
−2lnL = χ2 .
(5.16)
For uncorrelated data points compared to a theory prediction, a naive χ2 can be
calculated simply as
χ2 =
X (CˆB − CB )2
,
2
σ
B
B
(5.17)
where CB is the theory value, CˆB is observed data, σB is the uncertainty of the
data, and the sum is over all the bins B. CB is a binned version of C` , obtained by
2
The WMAP team found that their best fit models favored a rather higher value for τ RI than
was previously expected. We performed the pre-WMAP analysis prior to this finding, when τ RI was
not perhaps as interesting a parameter as it is now, and therefore we unfortunately neglected it in
that analysis, choosing to set it to zero. Since the pre-WMAP set does not figure strongly in our
conclusions this does not have much of an effect on our analysis.
45
multiplying the continuous power spectrum by the window function WB of bin B:
P
C` WB` /`
.
(5.18)
CB = P`
` WB` /`
P
The window function is normalized such that for each bin B, ` WB` /` = 1.
In practice, a somewhat more sophisticated calculation is performed. First, the
possibility of correlations between neighboring points must be taken into account.
These are defined through the correlation matrix R. For uncorrelated data, R is the
identity matrix. For correlated data, the off-diagonal terms quantify the correlation
between data points.
The correlation matrix can be multiplied by the uncertainties σB of the data points
to obtain the covariance matrix C:
CBB 0 = σB RBB 0 σB 0 .
(5.19)
The inverse of the covariance matrix is the weight matrix or curvature matrix,
M(C) = C−1 ,
(5.20)
which is then used to calculate χ2 :
χ2 =
X
BB 0
(C)
(CˆB − CB )MBB 0 (CˆB 0 − CB 0 ).
(5.21)
We refer to Equation (5.21) as the Gaussian likelihood approximation, as it is
derived under the assumption that the likelihood function has a Gaussian distribution.
This assumption results in a poor approximation to the full likelihood function in some
cases. As suggested in [79] a somewhat better approximation is an offset lognormal
distribution. A change of variables to
ZB = ln(CB + xB )
(5.22)
ZˆB = ln(CˆB + xB ),
(5.23)
46
where the lognormal offset xB is the effective noise in the experiment, results in a
much better estimate of the true likelihood in most cases, because the variable ZB ,
unlike CB , is normally distributed. Defining the local transformation of the weight
matrix into the lognormal variables ZB as
(Z)
(C)
MBB 0 = (CˆB + xB )MBB 0 (CˆB 0 + xB 0 ),
(5.24)
the lognormal offset approximation of χ2 becomes
χ2 =
X
BB 0
(Z)
(ZˆB − ZB )MBB 0 (ZˆB 0 − ZB 0 ).
(5.25)
For the pre-WMAP data set, Equation (5.21) was used to calculate χ2 , as offsets
were not available for all the experiments included in the compilation [80].
Following [74], the χ2 for the WMAP+ data set was calculated by computing the
χ2 for each experiment individually and simply adding: χ2tot = χ2WMAP +χ2CBI +χ2ACBAR .
This is possible because the data used from these three experiments is uncorrelated
(see Section 5.2).
The χ2 for the CBI data was calculated using Equation (5.25).
χ2ACBAR was also calculated using a form of Equation (5.25). However, because
the ACBAR data is decorrelated, the correlation matrix R and hence the covariance
matrix C and its transformation M(Z) are all diagonal. Equation (5.25) can be
simplified in this case to
χ2 =
X (ZˆB − ZB )2 ˆ
e 2 ZB .
2
σ
B
B
(5.26)
For the WMAP data set, a Fortran routine compute likelihood was made available to the public for the comparison of the WMAP data to theory [74, 81]. In this
routine, χ2WMAP is found by using a combination of the Gaussian likelihood approximation and the offset lognormal likelihood approximation, because for the WMAP
47
data the errors of these two approaches tend to occur in opposite directions and
partially cancel:
1
2
χ2 = χ2Gauss + χ2LN ,
3
3
(5.27)
where χ2Gauss is given by Equation (5.21) and χ2LN by Equation (5.25), except that the
local transformation of the weight matrix M(Z) is applied using the theoretical power
spectrum CB rather than the data power spectrum CˆB :
(Z)
(C)
MBB 0 = (CB + xB )MBB 0 (CB 0 + xB 0 ).
5.5
(5.28)
Parameter Choices and Priors
To determine whether a particular recombination history (a, b) is capable of reproducing the observed CMB power spectrum, we minimize χ2 over the vector of
free parameters ϑ. However, the values of several of these parameters is fairly well
known from sources other than the CMB. In our analysis, we use these independent
observations to impose priors on the values of these parameters.
For the Hubble parameter, we impose the prior h = 0.72 ± 0.08, consistent with
estimates obtained from the Hubble Space Telescope (HST) Key Project and observations of Type Ia supernovae (SNe Ia) [82, 83]. The baryon density is well known
from BBN constraints [84]: ωb ≡ Ωb h2 = 0.020 ± 0.002. Our estimate for the total
matter density comes from supernovae studies [85, 86] and x-ray observations of the
gas mass fraction of relatively relaxed galaxies [87]. We combine these, using generous
error bars, to get Ωm = 0.29 ± 0.10. All uncertainties are at the 68% confidence level.
For h, ωb , and Ωm we use Gaussian priors. For the spectral tilt ns , however, we
impose a uniform, or tophat, prior of 0.7 < ns < 1.3. Values of ns beyond this range
are disallowed. The reason for this is that ns is generally not well known, and the
48
Parameter
Ωm
Range
0.29 ± 0.10
Type of Prior
Gaussian
ωb
h
ns
τRI
Q
0.020 ± 0.002
0.72 ± 0.08
0.7 – 1.3
–
–
Gaussian
Gaussian
tophat
none
none
Source
X-ray gas mass fraction of galaxies,
SNe Ia [85–87]
BBN [84]
HST Key project, SNe Ia [82, 83]
–
–
–
Table 5.1: Priors on the free parameters and their values.
best estimates for the value of ns come from the CMB [88–90]. Thus we use generous
bounds which are theoretically motivated, rather than observational evidence that
could prejudice our results. We do not include priors for the optical depth τRI or the
overall normalization of the power spectra Q. We additionally constrain ourselves to a
flat ΛCDM universe (Ωtot = 1) and adiabatic initial density perturbations (Appendix
A).
Including these priors, our determination of χ2 becomes
χ2 = χ2prev +
X (ϑ̂i − ϑi )2
i
σi2
.
(5.29)
Here i denotes h, ωb , or Ωm . ϑi is the value for the given parameter used in the model,
ϑ̂i is the value from observations, and σi is the corresponding 1σ error bar on that
observational value. χ2prev is the χ2 value obtained as detailed in Section 5.4.
5.6
Minimization Routines
In order to find the combination of cosmological parameters that allows a given
recombination history (a, b) to most closely reproduce the observed CMB power
49
spectra, we minimize χ2 over the free parameters ϑ. In this work, we have relied on
two methods: simulated annealing and Powell’s direction set [91].
Simulated annealing takes its name from the process in solid state physics where
a material is slowly cooled, or annealed, into a crystalline state. In this process, the
individual atoms all line up, finding the state with minimal energy. If the material is
instead cooled rapidly, the atoms become frozen in place without a chance to align,
resulting a much higher energy state.
In this minimization routine, an artificial “temperature” is introduced. With N
free parameters, the function is evaluated at a simplex of N + 1 points in the space
spanned by ϑ. The points in the simplex are gradually replaced by new points, which
are chosen for possible inclusion by contractions or reflections of the simplex. If the
new point has a lower χ2 than the point it is replacing, it is accepted into the simplex.
However, a point with higher χ2 can still be accepted with a probability that depends
on the annealing temperature. In this way, the simplex can climb out of local minima
and have a greater chance of finding the global minimum.
Powell’s direction set works by repeatedly doing one-dimensional minimizations
along orthogonal vectors in the N -dimensional parameter space that is being minimized over. At least initially, it is easiest to allow these orthogonal vectors to be
wholly along the coordinate axes. Thus, a routine might find an optimal value for ωb
keeping the other free parameters fixed, then using that value, find a better value for
Ωm , then h, etc.
Powell’s direction set method is both simpler and less robust than simulated annealing. Its main disadvantage is that it is more likely to become trapped in local
50
minima. However, if one already has knowledge of the general location of a global
minimum, it is considerably faster to use Powell’s direction set.
One complication in our situation is that the cosmological parameters are not
allowed to be negative: such values are unphysical. The minimization routines from
[91] however, do not have any method of accounting for this. It was necessary to
alter the off-the-shelf code to reject negative values for the parameters. This step was
taken at the point where the routine selected new values. If the values were negative,
a value between the old value and the rejected negative value was chosen instead.
We used the simulated annealing method on the pre-WMAP data set. The WMAP
data is entirely consistent with earlier experiments, but with much smaller error bars.
Therefore, it was reasonable to assume that the minima found in the pre-WMAP data
sets would not shift significantly when switching to the WMAP data set. With the
WMAP analysis, we used Powell’s direction set, starting with the values of the free
parameters found for the same point in a-b space using the pre-WMAP set. Similarly,
the WMAP+ analysis used the values found in the WMAP analysis as starting points.
Unfortunately, it was not feasible to implement the simulated annealing routine
with the WMAP and WMAP+ data sets; it proved to be overwhelmingly too timeintensive, taking in excess of 60 hours of computer time to find the minimum χ2 for
one point in a-b space, whereas the Powell’s direction set routine generally took only
from 2 to 20 hours. We were thus unable to make a direct comparison to check that
we had not become trapped in a local minimum. The effect of this is that there
are possibly some points in a-b space which have a lower minimum than we found.
However, we believe this is unlikely to significantly affect our results.
51
Confidence Region
1σ
2σ
3σ
(1 − A)%
68.27
95.45
99.73
∆χ2
2.30
6.18
11.83
Table 5.2: ∆χ2 values for a joint estimation of m = 2 parameters [92].
5.7
Confidence Regions
In order to determine confidence regions in a-b parameter space, we use the ∆χ2
method described in Section 31.4.2.3 of [92]. A point is determined to lie within a
given confidence region if it satisfies the following condition:
χ2 (a, b) ≤ χ2min + ∆χ2
(5.30)
Here, χ2min is the lowest value of χ2 found for any point in a-b space. The value of
∆χ2 varies depending on the confidence region desired.
In general, if a confidence level of (1 − A)% is desired for a joint estimation of m
parameters, the following equation can be solved for ∆χ2 :
R ∞ (m/2−1) −t/2
t
e
dt
∆χ2
A=
,
m/2
2 Γ(m/2)
(5.31)
where Γ(x) is defined as
Γ(x) =
Z
∞
tx−1 e−t dt.
(5.32)
0
A closed form for the solution to the integral does not exist, so this equation
must be solved numerically. In our case, we are attempting a joint estimation of two
parameters, a and b. We would furthermore like to look at 1σ, 2σ, and 3σ confidence
levels. The appropriate values for ∆χ2 are displayed in Table 5.2.
52
Note that the value of ∆χ2 is independent of the number of data points or the
degrees of freedom in our minimization procedure. It depends only on the dimensionality of the a-b parameter space, that is, m = 2.
53
CHAPTER 6
RESULTS
6.1
Effects of Varying the Recombination History
As described in Section 4.2, altering a or b from their standard values has a direct
effect on how recombination proceeds. The most direct way to see this is by looking
at the ionization fraction xe as a function of time. This then has an effect on the
visibility function g(z) and the CMB power spectrum C` . Recall that a modifies the
overall rates of recombination and ionization, while b multiplies the binding energies
of hydrogen.
6.1.1
Ionization Fraction xe
Figures 6.1 and 6.2 show the ionization fraction versus redshift for a variety of
recombination histories (a, b). In each case, the free parameters ϑ are {Ωm = 0.29,
ωb = 0.020, h = 0.72, n = 1.0, τ = 0.15}.
In Figure 6.1, the ionization fraction as a function of redshift is examined as a
is changed and b is held fixed. As a increases, note that recombination has a longer
duration, though it starts at about the same time. Beyond a certain limit, roughly
log(a) = 1.5, recombination proceeds as rapidly as possible, and increasing a has no
further effect because in the limit of high a, recombination proceeds in equilibrium.
54
Figure 6.1: xe (z) vs. z for varying a, fixed b. As a increases, recombination has a shorter
duration. The parameters used to create these curves (as well as those in Figures 6.2-6.10)
are Ωm = 0.29, ωb = 0.020, h = 0.72, ns = 1.0, τRI = 0.15.
55
Figure 6.2: xe (z) vs. z for varying b, fixed a. As b increases, recombination starts at earlier
redshifts.
56
Figure 6.3: g(z) vs. z for varying a, fixed b. As a increases, the width of the visibility
function (and therefore ∆z) decreases and the position of the peak of the visibility function
(and therefore zrec ) shifts to earlier times.
57
Figure 6.4: g(z) vs. z for varying b, fixed a. As b increases, the position of the peak of the
visibility function (and therefore z rec ) shifts to earlier times and the width of the visibility
function (and therefore ∆z) increases.
58
Figure 6.2 shows xe versus z for fixed a and varying b. As b increases, the primary
effect is that the onset of recombination shifts to earlier epochs. A secondary effect
is that the duration of recombination changes as well, with longer durations as b
increases.
6.1.2
Visibility Function g(z)
As discussed in Section 3.2, recombination is sometimes described through the
visibility function, g(z). Figures 6.3 and 6.4 show the resulting visibility functions of
the models used in Figures 6.1 and 6.2. Note that since g(z)dz is the probability that
a given photon was last scattered at z ± dz, the area under each curve is normalized
to 1.0.
As a decreases in Figure 6.3, the peak of the visibilty function, which determines
the redshift of recombination zrec , moves to more recent times, and the width of
recombination ∆z increases. The degeneracy of models with sufficiently high values
of a is clearly seen.
In Figure 6.4, as b decreases, the peak of the visibility function and zrec move
toward lower redshifts, just as with decreasing a. However, with decreasing b, the
width ∆z decreases, unlike the previous case. Thus, at the expense of allowing ∆z
to change, changing a and b in opposite directions allows zrec to remain relatively
unchanged.
6.1.3
CMB Power Spectra
Changes to the recombination history have a direct effect on the temperature and
polarization anisotropy spectra of the CMB, as discussed in Section 3.2. Here we
examine what effects changing a and b has on the CMB power spectra.
59
TT
Figures 6.5 and 6.6 show the effects on the temperature power spectrum of varying
a and b, respectively. In Figure 6.5 we see that as a is decreased, the acoustic peaks
at small scales are suppressed through diffusion damping. This effect starts at larger
scales as a is decreased. Also, there is a slight shifting of the spectrum down and
to the left. That is, the features of the spectrum, such as the acoustic peaks and
troughs, generally move towards lower ` and smaller values of C`T T .
As we saw in the previous section, decreasing a has the effect of increasing ∆z,
and to a much lesser extent decreaseing zrec . The increase in the width of the last
scattering surface ∆z explains the damping at large scales. As a decreases, the width
∆z increases, and larger and larger scales of anisotropies are erased through diffusion
damping. Also, for lower values of a, the features of the power spectrum move to
slightly larger scales because recombination occurs closer to us and because larger
sized regions have had time to enter the horizon and begin acoustically osciallating. Note that the relative heights of the even and odd peaks is largely preserved.
This is because zrec is changing only slightly, and the larger increase in ∆z, with its
corresponding damping of higher ` features, dominates.
Figure 6.6 shows C`T T as b varies. From Section 6.1.2 we see that decreasing b
decreases zrec and secondarily decreases ∆z. The power spectrum shifts strongly to
smaller ` as b is decreased both because the surface of last scattering is closer and
because larger regions have had time to begin to oscillate. In this case the relative
amplitudes of the peaks vary considerably, whereas when a is varied, the relative
amplitudes remain fairly constant: as b is increased, the height of the even peaks
relative to the odd peaks is also increased. This is due to the change in the ratio
60
Figure 6.5: C`T T vs. ` for varying a, fixed b. As a decreases, the power at small scales is
decreased and the acoustic peaks are suppressed. Also, as a decreases, the acoustic features
shift toward slightly larger scales.
61
Figure 6.6: C`T T vs. ` for varying b, fixed a. As b decreases, the acoustic features shift to
larger scales. Notice also that as b decreases, the relative amplitudes of the even and odd
peaks change.
62
Figure 6.7: C`T E vs. ` for varying a, fixed b. As a decreases, the power at small scales is
decreased and the acoustic peaks are suppressed. Also, as a decreases, the acoustic features
shift toward slightly larger scales.
63
Figure 6.8: C`T E vs. ` for varying b, fixed a. As b decreases, the acoustic features shift
toward larger scales. Notice also that as b decreases, the relative amplitudes of the even
and odd peaks change.
64
Figure 6.9: C`EE vs. ` for varying a, fixed b. As a decreases, the power at small scales is
decreased and the acoustic peaks are suppressed. Also, as a decreases, the acoustic features
shift toward slightly larger scales.
65
Figure 6.10: C`EE vs. ` for varying b, fixed a. As b decreases, the acoustic features move
to larger scales.
66
of ρb to ργ caused by later recombination. Because ∆z increases only slightly as b
is decreased, we do not see significant diffusion damping at small scales for higher
values of b.
TE
The effect of varying a on the TE cross-correlation power spectrum is shown in
Figure 6.7. A decrease in a primarily increases ∆z, and we see the effects of this in
the diffusion damping of features at small scales. The peaks of the power spectrum
are damped more strongly and at larger scales for smaller values of a, just as with the
temperature power spectrum. Decreasing a also decreases zrec which both moves the
LSS closer to us and allows time for larger regions to begin oscillating. Thus we see
that the features of the power spectrum have been systematically shifted to smaller
values of `. Additionally, the amplitudes of the peaks vary non-monotonically with
increasing a, seen most clearly in the first peak and obscured in the higher peaks by
diffusion damping. There does not appear to be a simple analytic explanation for
this behavior.
In Figure 6.8, we see how the C`T E power spectrum changes with varying b. The
primary effect of decreasing b is to decrease zrec and we see that the features of the
power spectrum in this case are moved to smaller scales both because the LSS is
closer to us and because larger regions have had time to begin to collapse. Also, we
see that the height of the even peaks is suppressed with decreasing zrec due to the
increased ratio of baryon to photon density allowed by a later recombination.
67
EE
Although there are currently no released measurements of the EE power spectrum,
it is possible to predict how variations to standard recombination will affect it. Figure
6.9 shows the effect of changing a and fixed b. The decrease in a and correponding
increase in the width ∆z of the last scattering surface again show up in the diffusion
damping of anisotropies at small scales. As a secondary effect, decreasing a decreasing
zrec , and thus all the features are shifted to larger scales due to the decreased distance
to the LSS and regions of larger sizes having time to begin oscillating. Again there is
a non-monotonic change in the heights of the peaks.
Figure 6.10 shows the C`EE power spectrum with varying b and fixed a. The peaks
and troughs moves to higher ` as b is increased, because of the accompanying decrease
in zrec . Decreasing b also decreases the width ∆z only slightly, and we do not see
significant changes in diffusion damping as b changes.
6.2
Allowed Regions of a-b Parameter Space
By applying the methods of Chapter 5, we are able to obtain a region in a-b
parameter space that is consistant with observational data. Recall that standard
recombination corresponds to log(a) = 0, b = 1.
6.2.1
Analysis of TT Data
Pre-WMAP
Figure 6.11 displays the range of recombination histories that provide an acceptable fit to the pre-WMAP data set. The 3σ region is quite large, and spans a bananashaped region from −0.5 ≤ log(a) ≤ ∞ and 0.7 ≤ b ≤ 2.0. The tail of the 3σ region
68
Figure 6.11: Allowed region of a-b space from the pre-WMAP TT data set. Standard
recombination lies solidly in the 2σ region. The point with the lowest value of χ 2 is log(a) =
−1.1, b = 1.35, with χ2 = 17.50
69
Figure 6.12: Allowed region of a-b space from the WMAP TT data set. Standard recombination lies just outside of the 1σ region. The point with the lowest value of χ 2 is
log(a) = −0.2, b = 1.0, with χ2 = 975.2
70
Figure 6.13: Allowed region of a-b space from the WMAP TT+TE data set. Standard
recombination lies inside the 1σ region. The point with the lowest value of χ 2 is log(a) =
−0.3, b = 1.05 with a value of χ2 = 1430.6.
71
Figure 6.14: Allowed region of a-b space from the WMAP+ TT+TE data set. Standard
recombination lies inside the 1σ region. The point with the lowest value of χ 2 is log(a) =
−0.2, b = 1.0 with a value of χ2 = 1435.80.
72
Data Set
pre-WMAP TT
WMAP TT
WMAP TT+TE
WMAP+ TT+TE
DOF
23
896
1345
1353
(log(a), b)min
(-1.1, 1.35)
(-0.2, 1.00)
(-0.3, 1.05)
(-0.1, 1.00)
χ2min
17.50
975.22
1430.64
1435.17
χ2std rec
20.54
977.59
1431.84
1436.09
(1 − A)std rec
78.13%
69.38%
45.12%
36.88%
Table 6.1: For each data set, the number of degrees of freedom DOF , the location of
the point (a, b)min with the lowest χ2 value χ2min , the χ2 value for standard recombination
χ2std rec , and the corresponding minimum confidence region that would just include standard
recombination (1 − A)std rec . Recall that being included in the 1σ region corresponds to a
probability (1 − A) ≤ 68.27%.
continues to indefinitely high a, as the evolution of xe over time becomes degenerate
for values of log(a) & 1.5 (see Figure 6.1). The 1σ region does not include standard
recombination. It covers a diagonal region from −1.5 ≤ log(a) ≤ −0.4 and from
1.1 ≤ b ≤ 1.6.
Table 6.1 shows, for various data sets, the location of the point with lowest χ2 , its
χ2 value, the χ2 value obtained for standard recombination, and the corresponding
probability of standard recombination.
WMAP
For the WMAP TT data set, the allowed recombination histories are shown in
Figure 6.12. Although this set includes data only up to ` ≈ 1000, the range of allowed
histories has been tightened considerably. The 3σ allowed region is again a crecent
shape spanning −1.0 ≤ log(a) ≤ ∞ and 0.6 ≤ b ≤ 1.3. The 1σ region again excludes
standard recombination, but this time it is very close. The cutoff probability for
the 1σ region is 68.27%, while standard recombination falls just outside of that with
(1 − A) = 69.38%.
73
6.2.2
Analysis of TT+TE Data
WMAP
Including the TE power spectrum in our determination of the allowed parameter
space further contracts the allowed region. Figure 6.13 shows the result of using
both the TT and TE data from the WMAP set. The high-b, low-a tip has been
reduced, and most of the 1σ region from the pre-WMAP set is now excluded at 3σ.
The region is now more self-contained, and the outlying points with high probability
have disappeared. The 3σ region is a slightly curved oval from −0.8 ≤ log(a) ≤ ∞
and 0.75 ≤ b ≤ 1.2. The 1σ region is a diagonal from −0.5 ≤ log(a) ≤ 0.6 and
0.9 ≤ b ≤ 1.1. Standard recombination is solidly within the 1σ region.
WMAP+
In Figure 6.14 we show the results of including higher ` experiments in our data
set. The allowed region of the WMAP+ TT+TE analysis is very small. The 3σ
region extends only from −0.6 ≤ log(a) ≤ 1.0, completely eliminating the indefinitly
extending high a tail. The 3σ range in b is relatively unchanged by the addition of
the high ` data, and runs from 0.8 ≤ b ≤ 1.2. The 1σ regions are −0.4 ≤ log(a) ≤ 0.4
and 0.9 ≤ b ≤ 1.1; a slightly smaller a range, but no significant change in b.
6.3
6.3.1
Characteristics of Allowed Regions
Free Parameter Distributions
The distributions of the free parameters are shown for the different data sets in
Figures 6.15 through 6.18. Only data from the recombination histories in the 1σ
regions are included.
74
Figure 6.15: Distribution of free parameters ϑ for recombination histories falling inside the
pre-WMAP TT 1σ region. The error bars across the top represent the range for the prior
for each parameter.
75
Figure 6.16: Distribution of free parameters ϑ for recombination histories falling inside the
WMAP TT 1σ region. The error bars across the top represent the range for the prior for
each parameter.
76
Figure 6.17: Distribution of free parameters ϑ for recombination histories falling inside the
WMAP TT+TE 1σ region. The error bars across the top represent the range for the prior
for each parameter.
77
Figure 6.18: Distribution of free parameters ϑ for recombination histories falling inside
the WMAP+ TT+TE 1σ region. The error bars across the top represent the range for the
prior for each parameter.
78
Data Set
pre-WMAP
TT
WMAP
TT
WMAP
TT+TE
WMAP+
TT+TE
History
(log(a), b)
lowest χ2
(-1.1, 1.35)
std rec
(0.0, 1.0)
average for
1σ region
lowest χ2
(-0.2, 1.0)
std rec
(0.0, 1.0)
average for
1σ region
lowest χ2
(-0.3, 1.05)
std rec
(0.0, 1.0)
average for
1σ region
lowest χ2
(-0.1, 1.0)
std rec
(0.0, 1.0)
average for
1σ region
Ωm
Free Cosmological Parameters
ωb
h
ns
τRI
0.333
0.0216
0.703
1.151
0a
0.317
0.328
± 0.015
0.0223
0.0216
± 0.0003
0.668
0.71
± 0.03
0.979
1.11
± 0.08
0a
0.319
0.0223
0.648
1.008
0.155
0.320
0.312
± 0.013
0.0236
0.0221
± 0.0008
0.689
0.65
± 0.02
1.025
0.99
± 0.04
0.165
0.153
± 0.007
0.283
0.0219
0.688
0.984
0.102
0.282
0.273
± 0.018
0.0223
0.0216
± 0.0009
0.696
0.68
± 0.03
0.965
0.96
± 0.03
0.149
0.119
± 0.02
0.248
0.0216
0.703
0.965
0.121
0.233
0.252
± 0.019
0.0220
0.0216
± 0.0006
0.735
0.69
± 0.02
0.959
0.97
± 0.02
0.124
0.133
± 0.011
0a
Table 6.2: Free parameter values for standard recombination, the point with lowest χ 2 ,
and the average of the 1σ region for each data set.
a The
pre-WMAP data set was analyzed using only four free cosmological parameters — the optical depth τRI was set to 0 throughout.
79
For the pre-WMAP TT set, shown in Figure 6.15, the parameter distributions
tend to stay within one standard deviation of the value we picked as a prior (their
“canonical” value). Since χ2 increases as these parameters move away from their
canonical values due to the gaussian priors imposed, this is perhaps not surprising.
The values of ns tend to run up against the upper boundary of ns ≤ 1.3, however, and
might have been higher if we had not constriained them to be within 0.7 < ns < 1.3.
The distributions from the WMAP TT data set are shown in Figure 6.16. This
data set prefers values of h which are somewhat below one standard deviation from
the canonical value. Also, values of ωb tend to be a bit higher than the canonical
value. Both these findings are consistant with those found by the WMAP team [89].
The value for ns is now comfortably centered on 1. The values for Ωm are close to the
canonical value and τRI tends to hover around 0.15, again in good agreement with
the WMAP findings.
For the WMAP TT+TE analysis, Ωm and ns both lie within the middle of their
prior range. The values for ωb tend to be about one standard deviation greater than
the canonical value, and although most values of h lie within one standard deviation
of the canonical value, the values tend to be lower than the canonical value. The
values for τRI found within the 1σ region of the WMAP TT+TE range appear to
be bimodal, with several points having values around τRI ≈ 0.10, but several others,
at high-b low-a, favoring τRI ≈ 0.14. We do not attach any significance to this
distribution at present.
The parameter distributions of Ωm , h, and ns for points in the 1σ region of the
WMAP+ TT+TE analysis are all very close to the canonical values. Only ωb is a
80
little higher than the canonical value, but not as much higher as with the previous
data sets. The distribution for τRI is centered around 0.14.
For each data set, Table 6.2 shows the values of the free parameters for the point
with lowest χ2 , the values for standard recombination, and the average values for
recombination histories falling within the 1σ region of that analysis.
Perhaps more interesting is the manner in which the various parameters change
in order to compensate for changes in a and b.
The shape of the CMB power spectrum is sensitive to changes in each of the
cosmological parameters [13]. Increasing the baryon density suppresses the even peaks
with respect to the odd peaks because the increased baryon mass contributes more
gravity to the potential wells, lessening the effect of the rarefaction phase of the
acoustic oscillations. Increasing the Hubble constant will lead to matter-radiation
equality happening earlier and an increased expansion rate, thus the power spectrum
will tend to have suppressed even peaks and move to slightly lower values of `. An
increase in the total matter density keeping the total density fixed at Ωtot = 1, or
equivalently a decrease in the dark energy density, again moves matter-radiation
equality to earlier times, which tends to move the spectra to lower ` and increase
the Late ISW effect. An increase in the scalar spectral index gives more power to
smaller scales, and will tilt the CMB power spectrum, increasing the amplitude at
large ` compared to small `. Finally increasing the optical depth due to reionization
will decrease the power at small scales by a factor e−2τRI .
Figures 6.19 through 6.22 show how the parameters change for points falling within
the 3σ region of each analysis. The solid squares show points whose best-fit parameter
ϑi is higher than average, with values falling above ϑ̄i + σi , the open squares with ×’s
81
Figure 6.19: How the parameters ϑ change with a and b for recombination histories falling
inside the pre-WMAP TT 3σ region. The solid squares have a value for the given parameter
that is more than one standard deviation above the mean, ϑ i > ϑ̄i + σi , the open squares
with an × have values within one standard deviation of the mean, ϑ̄i −σi < ϑi < ϑ̄i +σi , and
the open squares have values more than one standard deviation below the mean, ϑ i < ϑ̄i −σi .
82
Figure 6.20: How the parameters ϑ change with a and b for recombination histories falling
inside the WMAP TT 3σ region. The solid squares have higher values, ϑ i > ϑ̄i + σi , the
open squares with an × have average values, ϑ̄i − σi < ϑi < ϑ̄i + σi , and the open squares
have lower values, ϑi < ϑ̄i − σi .
83
Figure 6.21: How the parameters ϑ change with a and b for recombination histories falling
inside the WMAP TT+TE 3σ region. The solid squares have higher values, ϑ i > ϑ̄i + σi ,
the open squares with an × have average values, ϑ̄i − σi < ϑi < ϑ̄i + σi , and the open
squares have lower values, ϑi < ϑ̄i − σi .
84
Figure 6.22: How the parameters ϑ change with a and b for recombination histories falling
inside the WMAP+ TT+TE 3σ region. The solid squares have higher values, ϑ i > ϑ̄i + σi ,
the open squares with an × have average values, ϑ̄i − σi < ϑi < ϑ̄i + σi , and the open
squares have lower values, ϑi < ϑ̄i − σi .
85
Figure 6.23: How the parameters ϑ change with a and b for recombination histories falling
inside the 3σ region of the WMAP TT+TE τ RI = 0 set. The solid squares have higher
values, ϑi > ϑ̄i + σi , the open squares with an × have average values, ϑ̄i − σi < ϑi < ϑ̄i + σi ,
and the open squares have lower values, ϑ i < ϑ̄i − σi .
86
show points where ϑi is about average, with values falling between ϑ̄i − σi < ϑi <
ϑ̄i + σi , and the open squares show points where ϑi is lower than average, with values
falling below ϑ̄i − σi . Here ϑ̄i is the average value and σi the standard deviation
obtained from the 1σ region of the corresponding data set, shown in Table 6.2.
The baryon density ωb shows a strong tendency to increase both as a is increased
and as b is increased. Similarly, the Hubble constant h increases both with increasing a
and increasing b. The scalar spectral index ns increases with decreasing a. Although
in the pre-WMAP TT set, there is some slight increase in ns with decreasing b,
the opposite effect occurs in the WMAP TT+TE set, while in the WMAP TT and
WMAP+ TT+TE sets there does not appear to be any correlation of ns with changing
b. The optical depth τRI appears to increase with decreasing b, but this effect is not
nearly so monotonic as the correlations in the previous three parameters. There does
not appear to be any significant correlation with a change in a.
For Ωm , only the pre-WMAP TT analysis shows a strong correlation with changes
in a or b. For the pre-WMAP TT case, Ωm decreases with increasing a, but shows
little correlation with b. There is a hint of decreasing Ωm with increasing a in the
WMAP TT case, but the correlation essentially vanishes for the WMAP TT+TE
and WMAP+ TT+TE analyses. We ran a second analysis of the WMAP TT+TE
set with τRI set to 0, as in the pre-WMAP TT case. Interestingly, for the WMAP
TT+TE τRI = 0 case, shown in Figure 6.23, the correlation between increasing a and
decreasing Ωm resurfaced, along with a tendency for Ωm to decrease with increasing
b. Apparently when both τRI and Ωm are allowed to vary, the manner in which they
change to compensate for different recombination histories shows no clear pattern.
87
To restate these correlations in terms of a and b, an increase in a is compensated
for by corresponding increases in ωb and h and corresponding decrease in ns , and
a decrease in Ωm when τRI is held fixed. An increase in b is compensated for by
increases in ωb and h and possibly a decrease in τRI .
The corresponding changes in the cosmological parameters as a and b are changed
are due to competing effects. As a is increased, the peaks at high ` are less damped,
because the width of recombination is decreased, and the peaks are shifted to higher
` because recombination is moved to higher redshift. The decrease in ns lowers the
amplitude of the spectrum at small scales, counteracting the increased power at small
scales due to a narrower LSS. An increase in h tends to move the peaks to larger
scales, counteracting the effect of a higher zrec . Similarly, increasing b will enhance
the height of the even peaks by increasing zrec , thus lowering the ratio of the baryon
to photon density. This is compensated for by a higher values of h and ωb , which
tend to suppress the height of the even peaks. The increased h will again move the
peaks to slightly larger scales, even as an increase in b also increases zrec and shifts
the spectrum to smaller scales.
6.3.2
Recombination Histories of Allowed Regions
It is instructive to look at the evolution of xe (z) for those recombination histories
which provide a good fit to the data. Figure 6.24 shows xe versus redshift for a
selection of recombination histories in the 1σ, 2σ, and 3σ regions of the WMAP+
TT+TE analysis. In each case, the free parameters ϑ used in calculating xe (z) were
those which produced the best fit to the CMB data.
88
Figure 6.24: xe (z) vs. z for recombination histories falling within the 1σ, 2σ, and 3σ regions
of the WMAP+ TT+TE analysis. The thicker black line is standard recombination, and in
this data set it falls within the 1σ region. It is shown on the other two graphs for comparison
purposes.
89
Figure 6.25: g(z) vs. z for recombination histories falling within the 1σ, 2σ, and 3σ regions
of the WMAP+ TT+TE analysis. The thicker black line is standard recombination, and in
this data set it falls within the 1σ region. It is shown on the other two graphs for comparison
purposes.
90
From Figure 6.24, it can be seen that the commonality between recombination
histories capable of producing the observed CBM is their ending point. The recombination histories start at different times and have different durations, but all end
at approximately the same era. For the 1σ region, the average redshift at which
the ionization fraction falls to 1% is z(xe = 0.01) = 856 ± 20. By contrast, the
point where the ionization fraction falls to 50% has a much larger standard deviation:
z(xe = 0.50) = 1237 ± 44. All error estimates presented here and througout this
paper are at the 1σ (68%) confidence level.
We find a similar but smaller effect looking at the visibility function. Figure 6.25
shows the corresponding visibility functions g(z) for the 1σ, 2σ and 3σ regions of
the WMAP+ TT+TE analysis. For the 1σ region, we find that the average redshift
of the peak of the visibility function is zrec = 1055 ± 25, while the average width
is ∆z = 84 ± 2. If we look at the point where the visibility function falls to 10%
of its peak value, we again find a somewhat tighter constraint than at the peak,
z(g/gmax = 0.10) = 811 ± 19, though the difference in the standard deviations is not
quite as great as that found using xe .
The shapes of the visibility functions stray further and further from that of standard recombination in the 2σ and 3σ regions. Although some curves in the 3σ region
have a similar zrec to standard recombination, these curves tend to have a significantly different ∆z. This suggests that we are not placing a constraint on zrec or ∆z
individually, but jointly.
In Figure 6.26, we have taken points from the 1σ, 2σ, and 3σ regions of the allowed
a-b region for the WMAP TT+TE analysis, calculated their visibility functions, and
plotted the resulting values of zrec versus ∆z. Although we are not drawing confidence
91
Figure 6.26: zrec vs. ∆z for recombination histories falling within the 1σ, 2σ, and 3σ
regions of the WMAP+ TT+TE analysis.
92
regions in this zrec − ∆z space, it can be clearly seen that the recombination histories
which are best able to reproduce observations lie in a cluster, with successivly poorer
histories lying further out.
We find that the mean value of zrec for the 1σ region of the WMAP+ TT+TE
analysis is zrec = 1055 ± 25, while the average width is ∆z = 84 ± 2, where the error
bars represent one standard deviation. However, this does not strictly correspond
to a confidence region. Not all of the recombination histories from the 1σ region
have values of zrec and ∆z that fall within one standard deviation of the mean.
Furthermore, there are a few points within the 2σ region that have values of zrec and
∆z which both fall within one standard deviation of their respective means (though
far less than the number in the 1σ region), and several more points in both the 2σ
and 3σ regions where one of the two values is within a standard deviation of its mean.
6.3.3
Power Spectra
Figures 6.27 through 6.29 show the temperature, TE cross-correlation, and Emode polarization power spectra for all of the recombination histories falling within
the 1σ region of the WMAP+ TT+TE analysis. Figures 6.27 and 6.28 show that all
of the power spectra match the data very well, which is unsurprising since that was
our criteria.
The C`T T power spectra are shown in Figure 6.27. The recombination histories all
provide temperature power spectra that agree on the position of the peaks, though
their predictions for the amplitude have a little more leeway. The same holds true
for the C`T E power spectra in Figure 6.28: the positions of the TE cross-correlation
peaks are remarkably consistant, but the amplitudes vary somewhat.
93
Figure 6.27: C`T T vs. ` for recombination histories falling inside the WMAP+ TT+TE 1σ
region. The thicker black line is standard recombination.
94
Figure 6.28: C`T E vs. ` for recombination histories falling inside the WMAP+ TT+TE 1σ
region. The thicker black line is standard recombination.
95
Figure 6.29: C`EE vs. ` for recombination histories falling inside the WMAP+ TT+TE 1σ
region. The thicker black line is standard recombination.
96
Figure 6.29 shows the E-mode power spectra for the same models. Again the
position of the peaks in the C`EE spectrum seem very well constrained, but the height
of the peaks differs somewhat. Given the likely precision of data on the C`EE power
spectrum in the near future, it is unlikely that measurements of the E-mode polarization would be able to break any degeneracies in the allowed a-b parameter space
and significantly improve the constraints on recombination.
97
CHAPTER 7
CONCLUSIONS
We have examined the new data from the WMAP mission, along with data from
higher ` experiments, ACBAR and CBI, in an attempt to constrain recombination.
We have included analyses of both the temperature power spectrum C`T T and the TE
cross-correlation power spectrum C`T E . In the general parameterization that we used,
the 1σ allowed region in a-b space was significantly contracted compared to earlier
attempts with less precise data [69].
The results of the analyses using WMAP data (WMAP TT, WMAP TT+TE, and
WMAP+ TT+TE) are consistent with one another. We use the WMAP+ TT+TE
set in our final analysis because it contains the most observational data and correspondingly produces the tightest constraints. The addition of the TE cross-correlation
data shifts the most favorable region of a-b space slightly up in both variables. It
eliminates points at the high-b, low-a end of the distribution as well as some outlying
points at low-b, high-a. With the addition of the TE cross-correlation data, standard
recombination becomes more favorable and moves closer to the center of the distribution. The addition of higher ` data from ACBAR and CBI shrinks the region a bit
more and cuts off the high-a tail of the 3σ region.
98
We find that the allowed region falls within −0.5 ≤ log(a) ≤ 0.4 and 0.9 ≤ b ≤ 1.1
for the WMAP+ TT+TE analysis. We find that the average width and position of
the surface of last scattering for recombination histories that are in the 1σ region
of the WMAP+ TT+TE analysis are zrec = 1055 ± 25 and ∆z = 84 ± 2. Strictly
speaking, this is not a constraint, as there were other recombination histories whose
position and width of the LSS fell within this range, but were less able to reproduce
the observed CMB power spectra.
Our values for zrec and ∆z are consistent with the WMAP team’s estimations of
WMAP
these quantities. Their value for the position of the LSS is zrec
= 1088+1
−2 . The two
numbers do not mean precisely the same thing. Their number is a measurement of
when recombination occurred, assuming it was standard recombination. Our number
is the average value of zrec for a variety of nonstandard recombination histories which
are able to reproduce the observed data. The fact that the two numbers are very
similar is a reflection of the fact that standard recombination is an excellent fit to
the data and is approximately in the center of the distribution of the allowed variant
recombination histories. Our value of ∆z, is not directly comparable to the WMAP
WMAP
= 194 ± 2, as their value is defined as the full width at
team’s value of ∆zFWHM
half max (FWHM) of the visibility function. However, converting our number to a
FWHM, we obtain ∆zFWHM ≈ 198 ± 5, which is very consistent with their number.
Again, the two numbers are measuring different things, and their similarity is a result
of standard recombination being a good fit to the data.
We find that the distribution of free cosmological parameters in our fits tends
to fall close to the values as constrained by other observations. Thus, all of the
recombination histories that can match CMB observations are also consistent with
99
measurements of the other cosmological parameters. The changes in a and b are
largely compensated for by changes in other parameters. The largest effects are that
an increase in a is matched by corresponding increases in ωb and h and a corresponding
decrease in ns , while an increase in b is compensated for by increases in ωb and h.
Tighter constraints on these parameters, especially h, would allow us to narrow the
range of allowed recombination histories.
The result of these competing effects — changes in a and b being partially compensated for by changes in the free cosmological parameters — is that the anisotropy
power spectra from the 1σ region of the WMAP+ TT+TE analysis all share very
similar features. In general, the position of the peaks is strongly constrained, while
the amplitude of the peaks is much less so. The range in E-mode polarization power
spectra corresponding to recombination histories from this region is very similar to
the range in temperature and TE cross-correlation power spectra, and is unlikely to
be useful in breaking degeneracies in a-b parameter space.
We find that the range of alternate recombination histories allowed by the WMAP,
ACBAR, and CBI temperature and WMAP TE cross-correlation data is very small
compared to that allowed by earlier data, and is approximately centered on standard recombination. While there is still some room for non-standard recombination
histories, the constraints have narrowed considerably with this new data. Standard
recombination fits the data extremely well, and there is no compelling evidence from
this study to suggest that a non-standard recombination is preferable.
100
APPENDIX A
COSMOLOGY
We adopt a standard cosmological framework where the expansion of the universe
is governed by the Friedmann equation:
1 da
= H0
H(z) ≡
a dt
s
X
Ωi,0
i
ρi (z)
.
ρi,0
(A.1)
H0 is the present-day value of the Hubble parameter H(z). The Hubble constant H0 is often expressed in terms of the dimensionless parameter h as H0 = 100h
km/s/Mpc. The scale factor a(t) describes the expansion of the universe, and it is
normalized to a0 = a(t0 ) = 1. It is related to the cosmological redshift by
a=
1
1+z
(A.2)
The density in the ith component of matter is ρi (z). The density of that component relative to the critical density is Ωi,0 ≡ ρi,0 /ρc,0 , where the critical density is the
density required to insure that space is flat, and its value today is ρc,0 = 3H0 /8πG.
In general, for a component with a constant equation of state pi = wi ρi , the energy
density at redshift z is related to the present energy density by
ρi (z)
= (1 + z)3(1+wi ) .
ρi,0
101
(A.3)
For non-relativistic particles, wm = 0, while for relativistic ones wr = 1/3. For a
cosmological constant Λ, wΛ = −1, though other forms of dark energy with different
equations of state are possible.
In this paper we restrict our analysis to a flat universe with a cosmological constant. The geometry of the universe is determined by the total matter density,
P
Ωtot = i Ωi . If Ωtot < 1 the universe is negatively curved like a saddle or a hy-
perbola, if Ωtot > 1 the universe is positively curved like a sphere, and if Ωtot = 1
the universe is flat. The geometry affects the relation between the angular size and
the actual size of observed objects at large distances: objects which have the same
physical size will have a larger angular size in a spherically curved universe and a
smaller angular size in a hyperbolically curved universe.
The most direct evidence for a flat cosmology potentially comes from the CMB
itself, through the position of the acoustic peaks. Since the scale of the first peak
of the acoustic spectrum is directly related to the horizon size (Section 2.2.1), the
position of the first peak is an extremely robust measure of the curvature of space in
√
the absence of nonstandard recombination. Roughly speaking `peak ≈ 200/ Ωtot [93].
However, nonstandard recombination has the capability of altering this relation,
p
giving `peak ∝ (zrec /Ωtot ) [3], therefore we do not use it as a basis for our assumption
of flatness. We confine our attention to flat ΛCDM cosmologies because of theoretical
considerations, consistency with other observations, and simplicity of calculations in
a flat cosmology.
A universe which is flat on large scales is a prediction of most inflationary models
and thus is theoretically favored. Inflation is a period of superluminal expansion of
space very early in the history of the universe, stretching out previously small patches
102
of space to distances greater than the current horizon size, thus insuring that they are
flat. (The horizon size is the distance light could have traveled since the beginning of
the universe, ignoring inflation, and thus defines the maximum size of a region which
is in causal contact with itself since inflation).
We do not have direct evidence that Ωtot = 1, but it is consistent with a variety of observations. Observations of Type Ia supernovae (SNe Ia) indicate that the
expansion of the universe is accelerating, suggesting the presence of a cosmological
constant. With other observations confining the matter density to be low, Ωm ≈ 0.3
(e.g. [87]), the SNe Ia results indicate that Ωm ≈ 0.7 ± 0.2 [85, 86], a result consistent
with a flat universe.
In the case of a flat universe with a cosmological constant, the Friedmann equation
becomes
H(z) = H0
p
Ωm (1 + z)3 + Ωr (1 + z)4 + ΩΛ
(A.4)
where we have dropped the 0 subscripts on the Ωi ’s for brevity. The matter term
includes contributions from baryons and from cold dark matter (CDM):
Ωm = Ωb + Ωcdm .
(A.5)
Because Big Bang Nucleosynthesis places constraints on a combination of the baryon
density and the Hubble parameter, it is common to see the baryon density relative
to critical written as ωb ≡ Ωb h2 . The relativistic term includes contributions from
radiation and neutrinos:
Ωr = Ω γ + Ω ν .
(A.6)
This term is extremely small at the present time compared to the matter and cosmological constant terms, though it dominated prior to recombination.
103
The value for the cosmological constant component is set through the condition
for flatness:
Ωm + Ωr + ΩΛ = Ωtot = 1.
(A.7)
The perturbations in the CMB arise from density fluctuations in the early universe.
Defining the density at a point x at time t as ρ(x, t), the density fluctuations can be
described in terms of the spatial average of the density ρ̄(t) as
δ(x, t) =
ρ(x, t) − ρ̄(t)
.
ρ̄(t)
(A.8)
We can Fourier expand these fluctuations in terms of wave vectors k:
δ(x, t) ∝
X
δk (t)eik·x
(A.9)
k
The power spectrum of these fluctuations, describing the relative power at each scale
k, is then defined as
P (k) ∝ h|δk (t)|2 i
(A.10)
The two generic models for the origin of density perturbations are topological
defect (isocurvature) and inflation (adiabatic) models. They lead to very different
predictions for the anisotropy power spectra of CMB. At present the CMB appears
to favor adiabatic initial conditions, so we confine our interest to inflationary models.
Inflation generally predicts that density perturbations arise from Gaussian initial
fluctuations, and thus the power spectrum is given approximately by a power law:
P (k) ∝ k ns −1 .
(A.11)
Here ns is the scalar spectral index. There is a similar power law for tensors, however
we restrict our attention to scalar fluctuations throughout this paper. For ns = 1, the
104
amplitude in each mode as it crosses the Hubble radius is the same. This is known
as a scale-invariant or Harrison-Zeldovich spectrum.
If the spectral index ns is slowly varying, rather than constant, we can Taylor
expand the power spectrum in terms of some reference wavenumber k0 [94, 95]:
k
ln P (k) = ln P (k0 ) + (ns − 1) ln
k0
1 dns k
2
+
+··· .
ln
2 d ln k k0
k0
(A.12)
We can rewrite this in terms of a power law again as
P (k) = P (k0 )
k
k0
(ns −1)+ 12 ddnlnsk |
k0
ln
k
k0
(A.13)
where the term dns /d ln k is called the “running” of the scalar spectral index.
Although the WMAP team has found some indication that a running spectral
may be mildly favored by the data [89], in this work we only consider constant ns .
105
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