![Fig. 25.—Constraints on the amplitude of the four-point function. The measured amplitude of the four-point function (expressed in terms of a non-Gaussian amplitude defined in eq. [23]) is compared to the same statistic computed for simulated Gaussian random fields. The yellow line shows the results for Q, V, and W bands, and the red histogram shows the distribution of the results of the Monte Carlo realizations. Note that in both the simulations and the data A is greater than 0 due to the inhomogeneous noise. The excess in Q is may be due to point source contamination.](/figures/fig-25-constraints-on-the-amplitude-of-the-four-point-1t36zw4c.webp)
THREE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS:
IMPLICATIONS FOR COSMOLOGY
D. N. Spergel,
1,2
R. Bean,
1,3
O. Dore
´
,
1,4
M. R. Nolta,
4,5
C. L. Bennett,
6,7
J. Dunkley,
1,5
G. Hinshaw,
6
N. Jarosik,
5
E. Komatsu,
1,8
L. Page,
5
H. V. Peiris,
1,9,10
L. Verde,
1,11
M. Halpern,
12
R. S. Hill,
6,13
A. Kogut,
6
M. Limon,
6
S. S. Meyer,
9
N. Odegard,
6,13
G. S. Tucker,
14
J. L. Weiland,
6,13
E. Wollack,
6
and E. L. Wright
15
Received 2006 March 16; accepted 2007 January 12
ABSTRACT
A simple cosmological model with only six parameters (matter density,
m
h
2
, baryon density,
b
h
2
, Hubble con-
stant, H
0
, amplitude of fluctuations,
8
, optical depth, , and a slope for the scalar perturbation spectrum, n
s
) fits not
only the 3 year WMAP temperature and polarization data, but also small-scale CMB data, light element abundances,
large-scale structure observations, and the supernova luminosity/distance relationship. Using WMAP data only, the best-
fit values for cosmological parameters for the power-law flat cold dark matter (CDM) model are (
m
h
2
;
b
h
2
;
h; n
s
;;
8
) ¼ (0:1277
þ0:0080
0:0079
;0:02229 0:00073;0:732
þ0:031
0:032
;0:958 0:016;0:089 0:030; 0:761
þ0:049
0:048
). The 3 year
data dramatically shrink the allowed volume in this six-dimensional parameter space. Assuming that the primordial
fluctuations are adiabatic with a power-law spectrum, the WMAP data alone require dark matter and favor a spectral
index that is significantly less than the Harrison-Zel’dovich-Peebles scale-invariant spectrum (n
s
¼ 1; r ¼ 0). Adding
additional data sets improves the constraints on these components and the spectral slope. For power-law models, WMAP
data alone puts an improved upper limit on the tensor-to-scalar ratio, r
0:002
< 0:65 (95% CL) and the combination of
WMAP and the lensing-normalized SDSS galaxy survey implies r
0:002
< 0:30 (95% CL). Models that suppress large-
scale power through a running spectral index or a large-scale cutoff in the power spectrum are a better fit to the WMAP
and small-scale CMB data than the power-law CDM model; however, the improvement in the fit to the WMAP data
is only
2
¼ 3 for 1 extra degree of freedom. Models with a running-spectral index are consistent with a higher
amplitude of gravity waves. In a flat universe, the combination of WMAP and the Supernova Legacy Survey (SNLS)
data yields a significant constraint on the equation of state of the dark energy, w ¼0:967
þ0:073
0:072
.Ifweassumew ¼1,
then the deviations from the critical density,
K
, are small: the combination of WMAP and the SNLS data implies
k
¼0:011 0:012. The combination of WMAP 3 year data plus the HST Key Project constraint on H
0
implies
k
¼0:014 0:017 and
¼ 0:716 0:055. Even if we do not include the prior that the universe is flat, by com-
bining WMAP, large-scale structure, and supernova data, we can still put a strong constraint on the dark energy equation
of state, w ¼1:08 0:12. For a flat universe, the combination of WMAP and other astronomical data yield a con-
straint on the sum of the neutrino masses,
P
m
< 0:66 eV (95%CL). Consistent with the predictions of simple infla-
tionary theories, we detect no significant deviations from Gaussianity in the CMB maps using Minkowski functionals,
the bispectrum, trispectrum, and a new statistic designed to detect large-scale anisotropies in the fluctuations.
Subject headinggs: cosmic micr owave background — cosmology: observa tions
1. INTRODUCTION
The power-law CDM model fits not only the Wilkinson
Microwave Anisotropy Probe (WMAP) first-year data, but also a
wide range of astronomical data (Bennett et al. 2003; Spergel et al.
2003). In this model, the universe is spatially flat, homogeneous,
and isotropic on large scales. It is composed of ordinary matter,
radiation, and dark matter and has a cosmological constant. The
primordial fluctuations in this model are adiabatic, nearly scale-
invariant Gaussian random fluctuations ( Komatsu et al. 2003).
Six cosmological parameters (the density of matter, the density
of atoms, the expansion rate of the universe, the amplitude of the
primordial fluctuations, their scale dependence, and the optical
depth of the universe) are enough to predict not only the statis-
tical properties of the microwave sky, measured by WMAP at
several hundred thousand points on the sky, but also the large-
scale distribution of matter and galaxies, mapped by the Sloan
Digital Sky Survey (SDSS) and the 2dF Galaxy Redshift Survey
(2dFGRS).
With 3 years of integration, improved beam models, better un-
derstanding of systematic errors (Jarosik et al. 2007), tempera-
ture data ( Hinshaw et al. 2007), and polarization data (Page et al.
2007), the WMAP data have significantly improved. There have
also been significant improvements in other astronomical data
1
Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544-1001; dns@astro.princeton.edu.
2
Visiting Scientist, Cerro-Tololo Inter-American Observatory.
3
Cornell University, Ithaca, NY 14853.
4
Canadian Institute for Theoretical Astrophysics, University of Toronto, ON
M5S 3H8, Canada.
5
Department of Physics, Jadwin Hall, Princeton University, Princeton, NJ
08544-0708.
6
NASA Goddard Space Flight Center, Greenbelt, MD 20771.
7
Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, MD 21218-2686.
8
Department of Astronomy, University of Texas, Austin, TX.
9
Deptartments of Astrophysics and Physics, KICP and EFI, University of
Chicago, Chicago, IL 60637.
10
Hubble Fellow.
11
Department of Physics, University of Pennsylvania, Philadelphia, PA.
12
Department of Physics and Astronomy, University of British Columbia,
Vancouver, BC V6T 1Z1, Canada.
13
Science Systems and Applications, Inc. (SSAI ), Lanham, MD 20706.
14
Department of Physics, Brown University, Providence, RI 02912-1843.
15
UCLA Astronomy, Los Angeles, CA 90095-1562.
377
The Astrophysical Journal Supplement Series, 170:377 Y 408, 2007 June
# 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.
sets: analysis of galaxy clustering in the SDSS (Tegmark et al.
2004a; Eisenstein et al. 2005) and the completion of the 2dFGRS
(Cole et al. 2005); improvements in small-scale CMB measure-
ments (Kuo et al. 2004; Readhead et al. 2004a, 2004b; Grainge
et al. 2003; Leitch et al. 2005; Piacentini et al. 2006; Montroy
et al. 2006; O’Dwyer et al. 2005); much larger samples of high-
redshift supernova (Riess et al. 2004; Astier et al. 2005; Nobili
et al. 2005; Clocchiatti et al. 2006; Krisciunas et al. 2005); and
significant improvements in the lensing data (Refregier 2003;
Heymans et al. 2005; Semboloni et al. 2006; Hoekstra et al.
2006).
In x 2, we describe the basic analysis methodology used, with
an emphasis on changes since the first year. In x 3, we fit the
CDM model to the WMAP temperature and polarization data.
With its basic parameters fixed at z 1100, this model predicts
the properties of the low-redshift universe: the galaxy power
spectrum, the gravitational lensing power spectrum, the Hubble
constant, and the luminosity-distance relationship. In x 4, we
compare the predictions of this model to a host of astronomical
observations. We then discuss the results of combined analysis of
WMAP data, other astronomical data, and other CMB data sets. In
x 5, we use the WMAP data to constrain the shape of the power
spectrum. In x 6, we consider the implications of the WMAP data
for our understanding of inflation. In x 7, we use these data sets to
constrain the composition of the universe: the equation of state of
the dark energy, the neutrino masses, and the effective number of
neutrino species. In x 8, we search for non-Gaussian features in the
microwave background data. The conclusions of our analysis are
described in x 9.
2. METHODOLOGY
The basic approach of this paper is similar to that of the first-
year WMAP analysis: our goal is to find the simplest model that
fits the CMB and large-scale structure data. Unless explicitly
noted in x 2.1, we use the methodology described in Verde et al.
(2003) an d applied in Spergel et al. (2003). We use Bay esian sta-
tistical techniques to explore the shape of the likelihood func-
tion, we use Monte Carlo Markov chain methods to explore the
likelihood surface, and we quote both our maximum-likeli hood
parameters and the marginalized expectation value for eac h pa-
rameter in a given model:
i
hi¼
Z
d
N
L(dj)p()
i
¼
1
M
X
M
j¼1
j
i
; ð1Þ
where
j
i
is the value of the ith parameter in the chain and j in-
dexes the c hain element. The number of elements (M )inthe
typical merged Markov chain is at least 50,000 and i s always
long enou gh to satisfy the Gelman & Rubin (1992) convergence
test with R < 1:1. In addition, we use the spectral convergence
test describe d in Dunkley et al. (2005) to c onfirm convergence.
Most merged chains have o ver 100,000 elements. We use a uni-
form prior on cosmological p arameters, p(), unless otherwise
specified. We refer to h
i
i as the best-fit value for the para meter
and the peak of the likelihood f unction as the best-fit model.
The Markov chain outputs and the marginalized values of the
cosmological parameters listed in Table 1 are available online
16
for all of the models discussed in the paper.
2.1. Chan
ges in Analysis Techniques
We now use not only the measurements of the temperature
power spectrum (TT) and the temperature polarization power
spectrum (TE), but also measurements of the polarization power
spectra (EE) and (BB).
TABLE 1
Cosmological Parameters Used in the Analysis
Parameter Description Definition
H
0
........................................... Hubble expansion factor H
0
¼ 100h Mpc
1
km s
1
!
b
........................................... Baryon density !
b
¼
b
h
2
¼
b
/1:88 ; 10
26
kg m
3
!
c
............................................ Cold dark matter density !
c
¼
c
h
2
¼
c
/18:8 yoctograms m
3
f
............................................. Massive neutrino fraction f
¼
/
c
P
m
...................................... Total neutrino mass (eV)
P
m
¼ 94
h
2
N
........................................... Effective number of relativistic neutrino species
k
........................................... Spatial curvature
DE
........................................ Dark energy density For w ¼1,
¼
DE
m
.......................................... Matter energy density
m
¼
b
þ
c
þ
w............................................. Dark energy equation of state w ¼ p
DE
/
DE
2
R
.......................................... Amplitude of curvature perturbations R
2
R
(k ¼ 0:002 Mpc
1
) 29:5 ; 10
10
A
A ............................................. Amplitude of density fluctuations (k ¼ 0:002 Mpc
1
) See Spergel et al. (2003)
n
s
............................................ Scalar spectral index at 0.002 Mpc
1
............................................. Running in scalar spectral index ¼ dn
s
/dlnk (assume constant)
r.............................................. Ratio of the amplitude of tensor fluctuations to scalar potential
fluctuations at k ¼ 0:002 Mpc
1
n
t
............................................ Tensor spectral index Assume n
t
¼r/8
............................................. Reionization optical depth
8
............................................ Linear theory amplitude of matter
fluctuations on 8 h
1
Mpc
s
........................................... Acoustic peak scale (deg) See Kosowsky et al. (2002)
A
SZ
.......................................... SZ marginalization factor See Appendix A
b
SDSS
...................................... Galaxy bias factor for SDSS sample b ¼½P
SDSS
(k; z ¼ 0)/P(k)
1/2
(constant)
C
TT
220
........................................ Amplitude of the TT temperature power spectrum at l ¼ 220
z
s
............................................. Weak lensing source redshift
Note.—The Web site http://lambda.gsfc.nasa.gov lists the marginalized values for these parameters for all of the models discussed in this paper.
16
See http://lambda.gsfc.nasa.gov.
SPERGEL ET AL.378 Vol. 170
At the lowest multipoles, a number of the approximations used
in the first-year analysis were suboptimal. Efstathiou (2004) notes
that a maximum-likelihood analysis is significantly better than a
quadratic estimator analysis at l ¼ 2. Slosar et al. (2004) note that
the shape of the likelihood function at l ¼ 2 is not well approxi-
mated by the fitting function used in the first-year analysis (Verde
et al. 2003). More accurate treatments of the low-l likelihoods
decrease the significance of the evidence for a running spectral
index (Efstathiou 2004; Slosar et al. 2004; O’Dwyer et al. 2004 ).
Hinshaw et al. (2007) and Page et al. (2007) describe our approach
to addressing this concern: for low multipoles, we explicitly com-
pute the likelihood function for the WMAP temperature and po-
larization maps. For the analysis of the polarization maps, we use
the resolution N
side
¼ 8N
1
matrices. This pixel-based method is
used for C
TT
l
for 2 l 30 and polarization for 2 l 23. For
most of the analyses in the paper, we use a N
side
¼ 8 version of the
temperature map for the analysis of the low- l likelihood that uses
a pixel-based version for l 12. For the WMAP CDM only case,
we use the more time-consuming N
side
¼ 16 version of the code.
Hinshaw et al. (2007) compares various approaches toward com-
puting the low-l likelihood. In Appendix A, we discuss various
choices made in the maximum-likelihood code. For the CDM
model, we have computed the best-fit parameters using a range
of assumptions for the amplitude of point source contamination
and different treatments of the low-l likelihood.
There are several improvements in our analysis of high-l tem-
perature data (Hinshaw et al. 2007): better beam models, im-
proved foreground models, and the use of maps with smaller
pixels ( N
side
¼ 1024). The improved foreground model is sig-
nificant at l < 200. The N
side
¼ 1024 maps significantly reduce
the effects of subpixel CMB fluctuations and other pixelization ef-
fects. We found that N
side
¼ 512 maps had higher
2
than N
side
¼
1024 maps, particularly for l ¼ 600Y 700, where there is signifi-
cant signal-to-noise and pixelization effects are significant.
We now marginalize over the amplitude of Sunyaev-Zel’dovich
(SZ) fluctuations. The expected level of SZ fluctuations ( Refregier
et al. 2000; Komatsu & Seljak 2001; Bond et al. 2005) is l(l þ
1)C
l
/(2) ¼ 19 3 K
2
at l ¼ 450Y 800 for
m
¼ 0:26,
b
¼
0:044, h ¼ 0:72, n
s
¼ 0:97, and
8
¼ 0:80. The amplitude of
SZ fluctuations is very sensitive to
8
(Komatsu & Kitayama
1999; Komatsu & Seljak 2001). For example at 60 GHz, l(l þ
1)C
l
/(2) ¼ 65 15 K
2
at l ¼ 450Y 800 for
8
¼ 0:91, which
is comparable to the WMAP statistical errors at the same multi-
pole range. Since the WMAP spectral coverage is not sufficient to
be able to distinguish CMB fluctuations from SZ fluctuations
(see discussion in Hinshaw et al. 2007), we marginalize over its
amplitude using the Komatsu & Seljak (2002) analytical model
for the shape of the SZ fluctuations. We impose the prior that the
SZ signal is between 0 and 2 times the Komatsu & Seljak (2002)
value. Consistent with the analysis of Huffenberger et al. (2004)
we find that the SZ contribution is not a significant contaminant
to the CMB signal on the scales probed by the WMAP experiment.
We report the amplitude of the SZ signal normalized to the
Komatsu & Seljak (2002) predictions for the cosmological pa-
rameters listed above with
8
¼ 0:80. A
SZ
¼ 1 implies that the
SZ contribution is 8.4, 18.7, and 25.2 (K)
2
at l ¼ 220, 600, and
1000, respectively. We discuss the effects of this marginalization
in Appendix A. We have checked that gravitational lensing of the
microwave background, the next most significant secondary
effect after the thermal SZ effect ( Lewis & Challinor 2006) does
not have a significant effect on parameters.
We now use the CAMB code (Lewis et al. 2000) for our analysis
of the WMAP power spectrum. The CAMB code is derived from
CMBFAST (Zaldarriaga & Seljak 2000) but has the advantage of
running a factor of 2 faster on the Silicon Graphics, Inc. (SGI), ma-
chines used for the analysis in this paper. For the multipole range
probed by WMAP, the numerical uncertainties and physical uncer-
tainties in theoretical calculations of multipoles are about 1 part in
10
3
(Seljak et al. 2003), significantly smaller than the experimental
uncertainties. When we compare the results to large-scale struc-
ture and lensing calculations, the analytical treatments of the growth
of structure in the nonlinear regime are accurate to better than 10%
on the smallest scales considered in this paper (Smith et al. 2003).
2.2. Parameter Choices
We consider constraints on the hot big bang cosmological sce-
nario with Gaussian, adiabatic primordial fluctuations as would
arise from single field, slow-roll inflation. We do not consider the
influence of isocurvature modes nor the subsequent production
of fluctuations from topological defects or unstable particle decay.
We parameterize our cosmological model in terms of 15
parameters:
p ¼f!
b
;!
c
;;
; w;
k
; f
; N
;
2
R
; n
s
; r; dn
s
=d ln k; A
SZ
; b
SDSS
; z
s
g; ð2Þ
where these parameters are defined in Table 1. For the basic power-
law CDM model, we use !
b
, !
c
,exp(2),
s
, n
s
,andC
TT
l¼220
,
as the cosmological parameters in the chain, A
SZ
as a nuisance
parameter, and unless otherwise noted, we assume a flat prior on
these parameters. Note that is the optical depth since reioniza-
tion. Prior to reionization, x
e
is set to the standard value for the
residual ionization computed in RECFAST (Seager et al. 2000).
For other models, we use these same basic seven parameters plus
the additional parameters noted in the text. Other standard cos-
mological parameters (also defined in Table 1), such as
8
and h,
are functions of these six parameters. Appendix A discusses the
dependence of results on the choice of priors.
With only 1 yr of WMAP data, there were significant degen-
eracies even in the CDM model: there was a long degenerate
valley in n
s
- space, and there was also a significant degeneracy
between n
s
and
b
h
2
(see Fig. 5 in Spergel et al. 2003). With the
measurements of the rise to the third peak (Hinshaw et al. 2007)
and the EE power spectrum (Page et al. 2007), these degeneracies
are now broken (see x 3). However, more general models, most
notably those with nonflat cosmologies and with richer dark ener gy
or matter content, have strong parameter degeneracies. For models
with adiabatic fluctuations, the WMAP data constrain the ratio of
the matter density/radiation density, effectively,
m
h
2
, the baryon
density,
b
h
2
, the slope of the primordial power spectrum and
the distance to the surface of last scatter. In a flat vacuum energy-
dominated universe, this distance is a function only of and h,
so the matter density and Hubble constant are well constrained.
On the other hand, in nonflat models, there is a degeneracy be-
tween
m
, h and the curvature (see x 7.3). Similarly, in models
with more complicated dark energy properties (w 6¼1), there is
a degeneracy between
m
, h, and w. In models where the number
of neutrino species is not fixed, the energy density in radiation is
no longer known so that the WMAP data only constrain a combi-
nation of
m
h
2
and the number of neutrino species. These degen-
eracies slow convergence as the Markov chains need to explore
degenerate valleys in the likelihood surface.
3. CDM MODEL: DOES IT STILL FIT THE DATA?
3.1. WMAP Only
The CDM model is sti ll an excellent fit to the WMAP data.
With longer integration times and smaller pixels, the errors in the
WMAP 3 YEAR IMPLICATIONS FOR COSMOLOGY 379No. 2, 2007
high-l temperature multipoles have shrunk by more than a factor
of 3. As the data have improved, the likelihood function remains
peaked around the maximum-likelihood peak of the first-year
WMAP value. With longer integration, the most discrepant high-l
points from the first-year data are now much closer to the best-
fit model (see Fig. 2). For the first-year WMAP TT and TE data
(Spergel et al. 2003), the reduced
2
eA
was 1.09 for 893 degrees
of freedom (dof ) for the TT data and was 1.066 for the combined
TT and TE data (893 þ 449 ¼ 1342 dof ). For the 3 year data,
which has much smaller errors for l > 350, the reduced
2
eA
for
982 dof (l ¼ 13Y 1000; 7 parameters) is now 1.068 for the TT data
and 1.041 for the combined TT and TE data (1410 dof, includ-
ing TE l ¼ 24Y 450), where the TE data contribution is evaluated
from l ¼ 24Y 500.
For the T, Q, and U maps using the pixel-based likelihood we
obtain a reduced
2
eA
¼ 0:981 for 1838 pixels (corresponding to
C
TT
l
for l ¼ 2Y 12 and C
TE
l
for l ¼ 2Y 23). The combined re-
duced
2
eA
¼ 1:037 for 3162 degrees of freedom for the com-
bined fit to the TT and TE power spectrum at high l and the T, Q,
and U maps at low l.
While many of the maximum-likelihood parameter values
(Table 2, cols. [3] and [7], and Fig. 1) have not changed signifi-
cantly, there has been a noticeable reduction in the marginalized
value for the optical depth, , and a shift in the best-fit value of
m
h
2
. (Each shift is slightly larger than 1 ). The addition of the
EE data now eliminates a large region of parameter space with
large and n
s
that was consistent wi th the first-year data. With
only the first-year data set, the likelihood surface was very flat. It
TABLE 2
Power-Law CDM Model Parameters and 68% Confidence Intervals
Parameter First-Year Mean WMAPext Mean 3 Year Mean ( No SZ ) 3 Year Mean 3 Year + ALL Mean
100
b
h
2
................... 2:38
þ0:13
0:12
2:32
þ0:12
0:11
2.23 0.08 2.229 0.073 2.186 0.068
m
h
2
........................ 0:144
þ0:016
0:016
0:134
þ0:006
0:006
0.126 0.009 0:1277
þ0:0080
0:0079
0:1324
þ0:0042
0:0041
H
0
............................. 72
þ5
5
73
þ3
3
73:5 3:273:2
þ3:1
3:2
70:4
þ1:5
1:6
............................... 0:17
þ0:08
0:07
0:15
þ0:07
0:07
0:088
þ0:029
0:030
0.089 0.030 0:073
þ0:027
0:028
n
s
.............................. 0:99
þ0:04
0:04
0:98
þ0:03
0:03
0.961 0.017 0.958 0.016 0.947 0.015
m
............................ 0:29
þ0:07
0:07
0:25
þ0:03
0:03
0.234 0.035 0.241 0.034 0.268 0.018
8
.............................. 0:92
þ0:1
0:1
0:84
þ0:06
0:06
0:76 0:05 0:761
þ0:049
0:048
0:776
þ0:031
0:032
Parameter First-Year ML WMAPext ML 3 Year ML ( No SZ) 3 Year ML 3 Year + ALL ML
100
b
h
2
................... 2.30 2.21 2.23 2.22 2.19
m
h
2
......................... 0.145 0.138 0.125 0.127 0.131
H
0
............................. 68 71 73.4 73.2 73.2
............................... 0.10 0.10 0.0904 0.091 0.0867
n
s
.............................. 0.97 0.96 0.95 0.954 0.949
m
............................ 0.32 0.27 0.232 0.236 0.259
8
.............................. 0.88 0.82 0.737 0.756 0.783
Notes.—The 3 Year fits in the columns labeled ‘‘No SZ’’ use the likelihood formalism of the first-year paper and assume no SZ contribution,
A
SZ
¼ 0, to allow direct comparison with the first-year results. Fits that include SZ marginalization are given in the last two columns of the upper
and lower parts of the table and represent our best estimate of these parameters. The last column includes all data sets.
Fig. 1.—Improvement in parameter constraints for the power-law CDM model (model M5 in Table 3). The contours show the 68% and 95% joint 2D marginalized
contours for the (
m
h
2
;
8
) plane ( left)andthe(n
s
;)plane(right). The black conto urs repr esent the first-ye ar WMAP data (wit h no prior on ). The red contours show
the first-year WMAP data combined with CBI and ACBAR (WMAPext in Spergel et al. 2003). The bl ue contours represent the three year WMAP data only with the SZ
contribution set to 0 to maintain consistency with the first-year analysis. The WMAP measurements of EE power spec trum provide a strong cons traint on th e value of .
The models with no reionization ( ¼ 0) or a scale-invariant sp ectrum (n
s
¼ 1) are both di sfavored at
2
eA
> 6 for five parameters (see Table 3). Improvements in the
measurement of the amplitude of the third pea k yield better constraints on
m
h
2
.
SPERGEL ET AL.380 Vol. 170
covered a ridge in -n
s
over a region that extended from ’
0:07 to nearly ¼ 0:3. If the optical depth of the universe were
as large as ¼ 0:3 (a value consistent with the first-year data),
then the measured EE signal would have been 10 times larger
than the value reported in Page et al. (2007). On the other hand,
an optical depth of ¼ 0:05 would produce one-quarter of the
detected EE signal. As discussed in Page et al. (2007) the reion-
ization signal is now based primarily on the EE signal rather than
the TE signal. See Figure 26 in Page et al. (2007) for the like-
lihood plot for : note that the form of this likelihood function is
relatively insensitive to the cosmological model (over the range
considered in this paper).
There has also been a significant reduction in the uncertainties
in the matter density,
m
h
2
. With the first year of WMAP data, the
third peak was poorly constrained (see the light gray data points in
Fig. 2). With 3 years of integration, the WMAP data better con-
strain the height of the third peak: WMAP is now cosmic variance
limited up to l ¼ 400, and the signal-to-noise ratio exceeds unity
up to l ¼ 850. The new best-fit WMAP-only model is close to the
WMAP (first-year) + CBI + ACBAR model in the third peak re-
gion. As a result, the preferred value of
m
h
2
now shifts closer to
the ‘‘WMAPext’’ value reported in Spergel et al. (2003). Figure 1
shows the
m
h
2
8
likelihood surfaces for the first-year WMAP
data, the first-year WMAPext data and the 3 year WMAP data. The
accurately determined peak position constrains
0:275
m
h ( Page
et al. 2003a), fixes the cosmological age, and determines the di-
rection of the degeneracy surface. With 1 year data, the best-fit
value is
0:275
m
h ¼ 0:498. With 3 years of data, the best fit shifts to
0:492
þ0:008
0:017
. The lower third peak implies a smaller value of
m
h
2
and because of the peak constraint, a lower value of
m
.
The best-fit value of
8
(marginalized over the other parameters)
is now noticeably smaller for the 3 year data alone, 0:761
þ0:049
0:048
than for the first-year WMAP data alone, 0:92 0:10. This lower
value is due to a smaller third peak height, which leads to a lower
value of
m
, and less structure growth and a lower best-fit value
for . The height of the third peak was very uncertain with the
first-year data alone. In Spergel et al. (2003) we used external
CMB data sets to constrain the third peak and with these data,
the m aximum-likelihood value was 0.84. With 3 years of data,
the third peak is better determined and its height is close to the
value estimated from the ground-based data. With the EE mea-
surements, we have eliminated most of the high- region of pa-
rameter space. Since higher values of imply a higher amplitude
of primordial fluctuations, the best-fit value of is proportional to
exp (). For a model with all other parameters fixed, ¼ 0:10 im-
plies a 7% lower value of
8
. As discussed in x 4.1, this lower
value of
8
is more consistent with the X-ray measurements but
lower than the best-fit value from recent lensing surveys. Lower
8
implies later growth of structure.
In the first-year data, we assumed that the SZ contribution to
the WMAP data was negligible. Appendix A discusses the change
in priors and the change in the SZ treatment and their effects on
parameters: marginalizing over SZ most significantly shifts n
s
and
8
by 1% and 3%, respectively. In Table 2 in the column labeled
No SZ and Figure 1, we assume A
SZ
¼ 0 and use the first-year
likelihood code to make a consistent comparison between the first-
year and 3 year results. As in the first-year analysis, we use a flat
prior on the logarithm of the amplitude and a flat prior on
s
and
a flat prior on . The first column of Table 5 list the parameters
fitted to the WMAP 3 year data with A
SZ
allowed to vary between
0 and 2. In the tables, the ‘‘mean’’ value is calculated according
to equation (1) and the ‘‘maximum-likelihood (ML)’’ value is the
value at the peak of the likelihood function. In the last two col-
umns, we provide our current best estimate of parameters includ-
ing SZ marginalization and using the full N
side
¼ 16 likelihood
code to compute the TT likelihoods. In subsequent tables and fig-
ures, we will allow the SZ contribution to vary and quote the ap-
propriate marginalized values. Allowing for an SZ contribution
lowers the best-fit primordial contribution at high l;thus,thebest-
fit models with an SZ contribution have lower n
s
and
8
values.
However, in other tables, we use the faster N
side
¼ 8 likelihood
code unless specifically noted. In all of the tables, we quote the
68% confidence intervals on parameters and the 95% confidence
limits on bounded parameters.
3.2. Reionization History
Since the Kogut et al. (2003) detection of , the physics of re-
ionization has been a subject of extensive theoretical study (Cen
2003; Ciardi et al. 2003; Haiman & Holder 2003; Madau et al.
2004; Oh & Haiman 2003; Venkatesan et al. 2003; Ricotti &
Ostriker 2004; Sokasian et al. 2004; Somerville & Livio 2003;
Wyithe & Loeb 2003; Iliev et al. 2005). Page et al. (2007) provides
a detailed discussion of the new polarization data: while the best-
fit value for has not changed significantly, the new EE data, com-
bined with an improved treatment of the TE data, imply smaller
marginalized maximum-likelihood value. The 3 year data favor
’ 0:1, consistent with the predictions of a number of simula-
tions of CDM models. For example, Ciardi et al. (2003) CDM
simulations predict ¼ 0:104 for parameters consistent with the
WMAP primordial power spectrum. Tumlinson et al. (2004) use
the nucleosynthetic data to derive and construct an initial mass
function ( IMF) for reionization and find 0:1. Chiu et al.
(2003) found that their joint analysis of the WMAP and SDSS
quasar data favored a model with ¼ 0:11,
8
¼ 0:83, and n ¼
0:96, very close to our new best-fit values. Wyithe & Cen (2006)
predict that if the product of star formation efficiency and escape
fraction for Population III stars is comparable to that for Popula-
tion II stars, ¼ 0:09Y 0:12 with reionization histories charac-
terized by an extended ionization plateau from z ¼ 7Y 12: They
argue that this result holds regardless of the redshift where the
intergalactic medium ( IGM) becomes enriched with metals.
Measurements of the EE and TE power spectrum are a power-
ful probe of early star formation and an important complement
to other astronomical measurements. Observations of galaxies
(Malhotra & Rhoads 2004), quasars (Fan et al. 2006), and gamma-
ray bursts ( Totani et al. 2006) imply that the universe was mostly
Fig. 2.—Comparison of the predictions of the different best-fit models to the
data. The black line is the angular power spectrum predicted for the best-fit
3 year WMAP only CDM model. The red line is the best fit to the 1 year WMAP
data. The orange line is the best fit to the combination of the 1 year WMAP data,
CBI and ACBAR (WMAPext in Spergel et al. 2003). The solid data points
represent the 3 year data and the light gray data points the first-year data.
WMAP 3 YEAR IMPLICATIONS FOR COSMOLOGY 381No. 2, 2007