FIRST-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP)
1
OBSERVATIONS:
DETERMINATION OF COSMOLOGICAL PARAMETERS
D. N. Spergel,
2
L. Verde,
2,3
H. V. Peiris,
2
E. Komatsu,
2
M. R. Nolta,
4
C. L. Bennett,
5
M. Halpern,
6
G. Hinshaw,
5
N. Jarosik,
4
A. Kogut,
5
M. Limon,
5,7
S. S. Meyer,
8
L. Page,
4
G. S. Tucker,
5,7,9
J. L. Weiland,
10
E. Wollack,
5
and E. L. Wright
11
Received 2003 February 11; accepted 2003 May 20
ABSTRACT
WMAP precision data enable accurate testing of cosmol ogical models. We find that the emerging standard
model of cosmology, a flat -dominated universe seeded by a nearly scale-invariant adiabatic Gaussian
fluctuations, fits the WMAP data. For the WMAP data only, the best-fit parameters are h ¼ 0:72 0:05,
b
h
2
¼ 0:024 0:001,
m
h
2
¼ 0:14 0:02, ¼ 0:166
þ0:076
0:071
, n
s
¼ 0:99 0:04, and
8
¼ 0:9 0:1. With
parameters fixed only by WMAP data, we can fit finer scale cosmic microwave background (CMB) measure-
ments and measurements of large-scale structure (galaxy surveys and the Ly forest). This simple model is
also co nsistent with a host of other astro nomical measurements: its inferred age of the universe is consistent
with stellar ages, the baryon/photon ratio is consistent with measuremen ts of the [D/H] ratio, and the
inferred Hubble constant is consistent with local observations of the expansion rate. We then fit the model
parameters to a combination of WMAP data with other finer scale CMB experiments (ACB AR and CBI),
2dFGRS measurements, and Ly forest data to find the model’s best-fit cosmological parameters:
h ¼ 0:71
þ0:04
0:03
,
b
h
2
¼ 0:0224 0:0009,
m
h
2
¼ 0:135
þ0:008
0:009
, ¼ 0:17 0:06, n
s
(0.05 Mpc
1
)=0:93 0:03,
and
8
¼ 0:84 0:04. WMAP’s best determination of ¼ 0:17 0:04 arises directly from the temperature-
polarization (TE) data and not from this model fit, but they are consistent. These parameters imply that the
age of the universe is 13:7 0:2 Gyr. With the Ly forest data, the model favors but does not require a slowly
varying spectral index. The significance of this running index is sensitive to the uncertainties in the Ly
forest.
By combining WMAP data with other astronomical data, we constrain the geometry of the universe,
tot
¼ 1:02 0:02, and the equation of state of the dark energy, w < 0:78 (95% confidence limit assuming
w 1). The combination of WMAP and 2dFGRS data constrains the energy density in stable neutrinos:
h
2
< 0:0072 (95% confidence limit). For three degenerate neutrino species, this limit implies that their mass
is less than 0.23 eV (95% confidence limit). The WMAP detection of early reionization rules out warm dark
matter.
Subject headings: cosmic microwave background — cosmological parameters —
cosmology: observations — early uni verse
On-line material: color figure
1. INTRODUCTION
Over the past century, a standard cosmological model has
emerged: with relatively few parameters, the model
describes the evolution of the universe and astronomical
observations on scales ranging from a few to thousands of
megaparsecs. In this model the universe is spatially flat,
homogeneous, and isotropic on large scales, composed of
radiation, ordinary matter (electrons, protons, neutrons ,
and neutrinos), nonbaryonic cold dark matter, and dark
energy. Galax ies and large-scale structure grew gravitation-
ally from tiny, nearly scale-invariant adiabatic Gaussian
fluctuations. The Wilkinson Microwave Anisotropy Probe
(WMAP) data offer a demanding quantitative test of this
model.
The WMAP data are powerful because they result from a
mission that was carefully designed to limit systematic mea-
surement errors (Bennett et al. 2003a, 2003b; Hinshaw et al.
2003b). A critical element of this design includes differential
measurements of the full sky with a complex sky scan pat-
tern. The nearly uncorrelated noise between pairs of pixels,
the accurate in-flight determ ination of the beam patterns
(Page et al. 2003a, 2003c; Barnes et al. 2003), and the well-
understood properties of the radiometers (Jarosik et al.
2003a, 2003b) are invaluable for this analysis.
Our basic approach in this analysis is to begin by
identifying the simplest model that fits the WMAP data and
determining the best-fit parameters for this model using
WMAP data only without the use of any significant priors
on parameter values. We then compare the predictions of
this model to other data sets and find that the model is
1
WMAP is the result of a partnership between Princeton University and
the NASA Goddard Space Flight Center. Scientific guidance is provided by
the WMAP Science Team.
2
Department of Astrophysical Sciences, Princeton University,
Princeton, NJ 08544.
3
Chandra Postdoctral Fellow.
4
Department of Physics, Jadwin Hall, Princeton, NJ 08544.
5
NASA Goddard Space Flight Center, Code 685, Greenbelt,
MD 20771.
6
Department of Physics and Astronomy, University of British
Columbia, Vancouver, BC V6T 1Z1, Canada.
7
National Research Council (NRC) Fellow.
8
Departments of Astrophysics and Physics, EFI, and CfCP, University
of Chicago, Chicago, IL 60637.
9
Department of Physics, Brown University, Providence, RI 02912.
10
Science Systems and Applications, Inc. (SSAI), 10210 Greenbelt
Road, Suite 600, Lanham, MD 20706.
11
Department of Astronomy, UCLA, P.O. Box 951562, Los Angeles,
CA 90095-1562.
The Astrophysical Journal Supplement Series, 148:175–194, 2003 September
# 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
E
175
basically consistent with these data sets. We then fit to
combinations of the WMAP data and other astronomical
data sets and find the best-fit global model. Finally, we place
constraints on alternatives to this model.
We begin by outlining our methodology (x 2). Verde et
al. (2003) describe the details of the approach used here
to compare theoretical predictions of cosmological
models to data. In x 3, we fit a simple, six-parameter
CDM model to the WMAP data set (temperature-
temperature and temperature-polarization angular power
spectra). In x 4 we show that this simple model provides
an acceptable fit not only to the WMAP data, but also
to a host of astronomical data. We use the comparison
with these other data sets to test the validity of the model
rather than further constrain the model parameters. In
x 5, we include large-scale structure data from the Anglo-
Australian Telescope Two-D egree Field Galaxy Redshift
Survey (2dFGRS; Colless et al. 2001) and Ly fores t
data to perform a joint likelihood analysis for the cosmo-
logical parameters. We find that the data favor a slowly
varying spectral index. This seven-parameter model is our
best fit to the full data set. In x 6, we relax some of the
minimal assumptions of the model by adding extra
parameters to the model. We examine nonflat models,
dark energy models in which the properties of the dark
energy are parameterized by an effective equation of
state, and models with gravity waves. By adding extra
parameters we introduce degenerate sets of models con-
sistent wi th the WMAP data alone. We lift these degener-
acies by including additional microwave background data
sets (Cosmic Background Imager [CBI], Arcminute Cos-
mology Bolometer Array Receiver [ACBAR]) and obser-
vations of large-scale structure. We use these combined
data sets to place strong limits on the geometry of the
universe, the neutrino mass, the energy density in gravity
waves, and the properties of the dark energy. In x 7, we
note an intriguing discrepancy between the standard
model and the WMAP data on the largest angular scales
and speculate on its origin. In x 8, we conclude and
present parameters for our best-fit model.
2. BAYESIAN ANALYSIS OF COSMOLOGICAL DATA
The basic approach of this paper is to find the simplest
model consistent with cosmological data. We begin by fit-
ting a simple six-parameter model first to the WMAP
data and then to other cosmological data sets. We then
consider more complex cosmological models and evaluate
whether they are a better description of the cosmological
data. Since Komatsu et al. (2003) found no evidence for
non-Gaussianity in the WMAP data, we assume the pri-
mordial fluctuations are Gaussian random phase
throughout this paper. For each model studied in the
paper, we use a Monte Carlo Markov chain to explore
the likelihood surface. We assume flat priors in our basic
parameters and impose positivity constraints on the
matter and baryon density (these limits lie at such low
likelihood that they are unimportant for the models). We
assume a flat prior in , the optical depth, but bound
<0:3. This prior has little effect on the fits but keeps
the Markov chain out of unphysical regions of parameter
space. For each model, we determine the best-fit
parameters from the peak of the N-dimensional likeli-
hood surface. For each parameter in the model we also
compute its one dimensional likelihood function by
marginalizing over all other parameters; we then quote
the (one-dimensional) expectation value
12
as our best
estimate for the pa rameter:
h
i
i¼
Z
d
N
LðaÞ
i
; ð1Þ
where a denotes a point in the N-dimensional parameter
space (in our application these are points—sets of cosmo-
logical parameters—in the output of the Markov chain),
L denotes the likelihood (in our application the
‘‘ weight ’’ given by the chain to each point). The WMAP
temperature (TT) angular power spectrum and the
WMAP temperature-polarization (TE) angular power
spectrum are our core data sets for the likelihood analy-
sis. Hinshaw et al. (2003b) and Kogut et al. (2003)
describe how to obtain the temperature and temperature-
polarization angular power spectra respectively from the
maps. Verde et al. (2003) describe our basic methodology
for evaluating the likelihood functions using a Monte
Carlo Markov chain algorithm and for including data
sets other than WMAP in our analysis. In addition to
WMAP data we use recent results from the CBI (Pearson
et al. 2002) and ACBAR (Kuo et al. 2002) experiments.
We also use the 2dFGRS measurements of the power
spectrum (Percival et al. 2001) and the bias parameter
(Verde et al. 2002), measurements of the Ly power spec-
trum (Croft et al. 2002; Gnedin & Hamilton 2002), Type
Ia supernova measurements of the angular diameter dis-
tance relation (Garnavich et al. 1998; Riess et al. 2001),
and the Hubble Space Telescope (HST ) Key
Project measurements of the local expansion rate of the
universe (Freedman et al. 2001).
3. POWER-LAW CDM MODEL AND THE
WMAP DATA
We begin by considering a basic cosmological model: a
flat universe with radiation, baryons, cold dark matter and
cosmological constant, and a power-law power spectrum of
adiabatic primordial fluctuations. As we will see, this model
does a remarkably good job of describing WMAP TT and
TE power spectra with only six parameters: the Hubble con-
stant h (in units of 100 km s
1
Mpc
1
), the physical matter
and baryon densities w
m
m
h
2
and w
b
b
h
2
, the optical
depth to the decoupling surface, , the scalar spectral index
n
s
, and A, the normalizat ion parameter in the CMBFAST
code version 4.1 with option UNNORM. Verde et al. (2003)
discuss the relationship between A and the amplitude of
curvature fluctuations at horizon crossing, jDRj
2
¼
2:95 10
9
A.Inx 4, we show that this model is also in
acceptable agreement with a wide range of astronomical
data.
This simple model provides an acceptable fit to both the
WMAP TT and TE data (see Figs. 1 and 2). The reduced
13
2
eff
for the full fit is 1.066 for 1342 degrees of freedom,
which has a probability of 5%. For the TT data alone,
12
In a Monte Carlo Markov chain, it is a more robust quantity than the
mode of the a posteriori marginalized distribution.
13
Here,
2
eff
2lnL and is number of data minus the number of
parameters. We have used 100,000 Monte Carlo realizations of the WMAP
data with our mask-, noise-, and angle-averaged beams and found that the
2lnL=
hi
¼ 1 for the simulated temperature data.
176 SPERGEL ET AL. Vol. 148
2
eff
= ¼ 1:09, which for 893 degrees of freedom has a prob-
ability of 3%. Most of the excess
2
eff
is due to the inability of
the model to fit sharp features in the power spectrum near
‘ 120, the first TT peak and at ‘ 350. In Figure 3 we
show the contribution to
2
eff
per multipole. The overall
excess variance is likely due to our not including several
effects, each contributing roughly 0.5%–1% to our power
spectrum covariance near the first peak and trough: gravita-
tional lensing of the CMB (Hu 2001), the spatial variations
in the effective beam of the WMAP experiment due to varia-
tions in our scan orientation between the ecliptic pole and
plane regions (Page et al. 2003a; Hinshaw et al. 2003a), and
non-Gaussianity in the noise maps due to the 1=f striping.
Including these effects would increase our estimate of the
power spectrum uncertainties and improve our estimate of
2
eff
. Our next data release will include the corrections and
errors associated with the beam asymmetries. The features
in the measured power spectrum could be due to underlying
features in the primordial power spectrum (see x 5 of Peiris
et al. 2003), but we do not yet attach cosmological
significance to them.
Table 1 lists the best-fit parameters using the WMAP data
alone for this mod el and Figure 4 shows the marginalized
probabilities for each of the basic parameters in the model.
The values in the second column of Table 1 (and the subse-
quent parameter tables) are expectation values for the
marginalized distribution of each parameter, and the errors
are the 68% confidence interval. The values in the third
column are the values at the peak of the likelihood function.
Since we are projecting a high dimensional likelihood func-
tion, the peak of the likelihood is not the same as the expect-
ation value of a parameter. Most of the basic parameters
are remarkably well determined within the context of this
model. Our most significant parameter degeneracy (see
Fig. 5) is a degeneracy between n
s
and . The TE data favors
0:17 (Kogut et al. 2003); on the other hand, the low
value of the quadrupole (see Fig. 1 and x 7) and the rela-
tively low amplitude of fluctuations for ‘<10 disfavor high
as reionization produces additional large-scale anisotro-
pies. Because of the combination of these two effects, the
likelihood surface is quite flat at its peak: the likelihood
changes by only 0.05 as changes from 0.11 to 0.19. This
particular shape depends upon the assumed form of the
power spectrum: in x 5.2, we show that models with a scale-
dependent spectral index have a narrower likelihood
function that is more centered around ¼ 0:17.
Fig. 1.—Comparison of the best-fit power-law CDM model to the
WMAP temperature angular power spectrum. The gray dots are the
unbinned data. [See the electronic edition of the Journal for a color version of
this figure.]
Fig. 2.—Comparison of the best-fit power-law CDM model to the
WMAP temperature angular power spectrum.
Fig. 3.—Contribution to 2 ln L per multipole binned at D‘ ¼ 15. The
excess
2
comes primarily from three regions, one around ‘ 120, one
around ‘ 200, and the other around ‘ 340.
TABLE 1
Power-Law CDM Model Parameters: WMAP Da ta Only
Parameter
Mean
(68% Confidence Range)
Maximum
Likelihood
Baryon density,
b
h
2
....... 0:024 0:001 0.023
Matter density,
m
h
2
....... 0:14 0:02 0.13
Hubble constant, h .......... 0:72 0:05 0.68
Amplitude, A .................. 0:9 0:1 0.78
Optical depth, ............... 0:166
þ0:076
0:071
0.10
Spectral index, n
s
............. 0:99 0:04 0.97
2
eff
= .............................. 1431/1342
Note.—Fit to WMAP data only.
No. 1, 2003 WMAP FIRST-YEAR RESULTS: PARAMETERS 177
Since the WMAP data allows us to accurately deter-
mine many of the basic cosmological parameters, we can
now infer a number of important derived quantities to
very high accuracy; we do this by computing these quan-
tities for each model in the Monte Carlo Markov chain
and use the chain to determ ine their expectation values
and uncertainties.
Table 2 lists cosmological parameters based on fitting a
power-law (PL) CDM model to the WMAP data only. The
parameters t
dec
and z
dec
are determined by using the
CMBFAST code (Seljak & Zaldarriaga 1996) to compute
the redshift of the CMB ‘‘ photosphere ’’ (the peak in the
photon visibility function). We determine the thickness of
the decoupling surface by measuring Dz
dec
and Dt
dec
, the full
width at half-maximum of the visibility function. The age of
the uni verse is derived by integrating the Friedm ann
equation, and
8
(the linear theory predictions for the
amplitude of fluc tuations within 8 Mpc h
1
spheres) from
the linear matter power spectrum at z ¼ 0 is computed by
CMBFAST.
Fig. 4.—Likelihood function of the WMAP TT+TE data as a function of the basic parameters in the power-law CDM WMAP model (
b
h
2
,
m
h
2
, h, A,
n
s
and .) The points are the binned marginalized likelihood from the Markov chain, and the solid curve is an Edgeworth expansion of the Markov chain’s
points. The marginalized likelihood function is nearly Gaussian for all of the parameters except for . The dashed lines show the maximum-likelihood values
of the global six-dimensional fit. Since the peak in the likelihood, x
ML
, is not the same as the expectation value of the likelihood function, hxi, the dashed line
does not lie at the center of the projected likelihood.
Fig. 5.—Spectral index constraints. Left: n
s
- degeneracy in the WMAP data for a power-law CDM model. The TE observations constrain the value of
and the shape of the C
TT
l
spectrum constrain a combination of n
s
and . Right: n
s
-
b
h
2
degeneracy. The shaded regions show the joint 1 and 2 confidence
regions.
178 SPERGEL ET AL. Vol. 148
4. COMPARSION WITH ASTRONOMICAL
PREDICTIONS
In this section, we compare the predictions of the best-fit
power-law CDM model to other cosmological observa-
tions. We also list in Table 10 the best-fit model to the full
data set: a CDM model with a running spectral index (see
x 5.2). In particular, we consider determinations of the local
expansion rate (i.e., the Hubble constant), the amplitude of
fluctuations on galaxy scales, the baryon abundance, ages
of the oldest stars, large-scale structure data, and Type Ia
supernova data. We also consider whether our determina-
tion of the reionization redshift is consistent with the predic-
tion for structure formation in our best-fit universe and with
recent models of reionization. In xx 5 and 6, we add some of
these data sets to the WMAP data to better constrain
parameters and cosmological models.
4.1. Hubble Constant
CMB observations do not directly measure the local
expansion rate of the universe rather they measure the con-
formal distance to the decoupling surface and the matter-
radiation ratio through the amplitude of the early integ rated
Sachs-Wolfe (ISW) contribution relative to the height of the
first peak. For our power-law CD M model, this is enough
information to ‘‘ predict ’’ the local expansion rate. Thus,
local Hubb le constant measurements are an important test
of our basic model.
The HST Key Project (Freedman et al. 2001) has carried
out an extensive program of using Cepheids to calibrate
several different secondary distance indicators (Type Ia
supernovae, Tully-Fisher, Type II supernovae, and surface
brightness fluctuations). With a distance modulus of 18.5
for the LMC, their combined estimate for the Hubble con-
stant is H
0
¼ 72 3ðstat:Þ7ðsyst:Þ km s
1
Mpc
1
. The
agreement between the HST Key Project value and our
value, h ¼ 0:72 0:05, is striking, given that the two
methods rely on differen t observables, different underlying
physics, and different model assumptions.
As we will show in x 6, models with equation of state for
the dark energy very different from a cosmological constant
(i.e., w ¼1) fit the WMAP data only if the Hubble con-
stant is much smaller than the HST Key Project value. An
independent determination of the Hubble constant that
makes different assumptions than the traditional distance
ladder can be obtained by combining Sunyaev-Zel’dovich
and X-ray flux measurements of clusters of galaxies, under
the assumption of sphericity for the density and tempera-
ture profile of clusters. This method is sensitive to the Hub-
ble constant at intermediate redshifts (z 0:5), rather than
in the nearby universe. Reese et al. (2002), Jones et al.
(2001), and Mason et al. (2001) have obtained values for the
Hubble constant systematically smaller than the HST Key
Project and WMAP CDM model determinations, but all
consistent at the 1 level. Table 3 summarizes recent
Hubble constant determinations and compares them with
the WMAP CDM model value.
4.2. Amplitude of Fluctuations
The overall amplitude of fluctuations on large-scale struc-
ture scales has been recently determined from weak lensing
surveys, clusters number counts, and peculiar velocities
from galaxy surveys. Weak-l ensing surveys and peculiar
velocity measurements are most sensitive to the combina-
tion
8
0:6
m
, cluster abundance at low redshift is sensitive to
TABLE 2
Derived Cosmological Parameters
Parameter Mean (68% Confidence Range)
Amplitude of galaxy fluctuations,
8
............................................ 0:9 0:1
Characteristic amplitude of velocity fluctuations,
8
0:6
m
.............. 0:44 0:10
Baryon density/critical density,
b
.............................................. 0:047 0:006
Matter density/critical density,
m
.............................................. 0:29 0 :07
Age of the universe, t
0
.................................................................. 13:4 0:3 Gyr
Redshift of reionization,
a
z
r
.......................................................... 17 5
Redshift at decoupling, z
dec
.......................................................... 1088
þ1
2
Age of the universe at decoupling, t
dec
.......................................... 372 14 kyr
Thickness of surface of last scatter, Dz
dec
...................................... 194 2
Thickness of surface of last scatter, Dt
dec
...................................... 115 5 kyr
Redshift at matter/radiation equality, z
eq
.................................... 3454
þ385
392
Sound horizon at decoupling, r
s
................................................... 144 4Mpc
Angular diameter distance to the decoupling surface, d
A
.............. 13:7 0:5 Gpc
Acoustic angular scale,
b
‘
A
........................................................... 299 2
Current density of baryons, n
b
...................................................... ð2:7 0:1Þ10
7
cm
3
Baryon/photon ratio, ................................................................ ð6:5
þ0:4
0:3
Þ10
10
Note.—Fit to the WMAP data only.
a
Assumes ionization fraction, x
e
¼ 1.
b
l
A
¼ d
C
=r
s
.
TABLE 3
Recent Hubble Constant Determinations
Method
Mean
(68% Confidence Range) Reference
HST Key Project....................... 72 3 71
Sunyaev-Zel’dovich+X-ray ...... 60 4
þ13
18
2
66
þ14
11
15 3
WMAP PL CDM model......... 72 5 x 3
References.—(1) Freedman et al. (2001); (2 ) Reese et al. (2002);
(3) Reese et al. (2002).
No. 1, 2003 WMAP FIRST-YEAR RESULTS: PARAMETERS 179
![Fig. 13.—Constraints on the geometry of the universe: m- plane. Two-dimensional likelihood surface for various combinations of data:WMAP (upper left); WMAPext (upper right);WMAPext+HSTKey Project (supernova data [Riess et al. 1998, 2001] are shown but not used in the likelihood in this part of the panel) (lower left);WMAPext+HSTKey Project+supernova (lower right).](/figures/fig-13-constraints-on-the-geometry-of-the-universe-m-plane-3i6yqkz7.webp)