Accepted by the Astrophysical Journal

First Year Wilkinson Microwave Anisotropy Probe (WMAP1 ) Observations:

Preliminary Maps and Basic Results
C. L. Bennett 2 , M. Halpern 3 , G. Hinshaw 2 , N. Jarosik 4 , A. Kogut 2 , M. Limon 2,5 , S. S. Meyer 6 , L.
Page 4 , D. N. Spergel 7 , G. S. Tucker 2,5,8 , E. Wollack 2 , E. L. Wright 9 , C. Barnes 4 , M. R. Greason 10 , R.
S. Hill 10 , E. Komatsu 7 , M. R. Nolta 4 , N. Odegard 10 , H. V. Peiris 7 , L. Verde 7 , J. L. Weiland 10
arXiv:astro-ph/0302207v3 5 Jun 2003

Charles.L.Bennett@NASA.gov

ABSTRACT

We present full sky microwave maps in five frequency bands (23 to 94 GHz) from the WMAP
first year sky survey. Calibration errors are <0.5% and the low systematic error level is well
specified. The cosmic microwave background (CMB) is separated from the foregrounds using
multifrequency data. The sky maps are consistent with the 7◦ full-width at half-maximum
(FWHM) Cosmic Background Explorer (COBE) maps. We report more precise, but consistent,
dipole and quadrupole values.
The CMB anisotropy obeys Gaussian statistics with −58 < fN L < 134 (95% CL). The
2 ≤ l ≤ 900 anisotropy power spectrum is cosmic variance limited for l < 354 with a signal-to-
noise ratio >1 per mode to l = 658. The temperature-polarization cross-power spectrum reveals
both acoustic features and a large angle correlation from reionization. The optical depth of
reionization is τ = 0.17 ± 0.04, which implies a reionization epoch of tr = 180+220
−80 Myr (95% CL)
+10
after the Big Bang at a redshift of zr = 20−9 (95% CL) for a range of ionization scenarios. This
early reionization is incompatible with the presence of a significant warm dark matter density.
A best-fit cosmological model to the CMB and other measures of large scale structure works
remarkably well with only a few parameters. The age of the best-fit universe is t0 = 13.7±0.2 Gyr
old. Decoupling was tdec = 379+8 −7 kyr after the Big Bang at a redshift of zdec = 1089 ± 1. The
thickness of the decoupling surface was ∆zdec = 195 ± 2. The matter density of the universe
is Ωm h2 = 0.135+0.008 2
−0.009 , the baryon density is Ωb h = 0.0224 ± 0.0009, and the total mass-
energy of the universe is Ωtot = 1.02 ± 0.02. It appears that there may be progressively less

1 WMAP is the result of a partnership between Princeton University and NASA’s Goddard Space Flight Center. Scientific
guidance is provided by the WMAP Science Team.
2 Code 685, Goddard Space Flight Center, Greenbelt, MD 20771
3 Dept. of Physics and Astronomy, University of British Columbia, Vancouver, BC Canada V6T 1Z1
4 Dept. of Physics, Jadwin Hall, Princeton, NJ 08544
5 National Research Council (NRC) Fellow
6 Depts. of Astrophysics and Physics, EFI and CfCP, University of Chicago, Chicago, IL 60637
7 Dept of Astrophysical Sciences, Princeton University, Princeton, NJ 08544
8 Dept. of Physics, Brown University, Providence, RI 02912
9 UCLA Astronomy, PO Box 951562, Los Angeles, CA 90095-1562
10 Science Systems and Applications, Inc. (SSAI), 10210 Greenbelt Road, Suite 600 Lanham, Maryland 20706
 –2–

fluctuation power on smaller scales, from WMAP to fine scale CMB measurements to galaxies
and finally to the Ly-α forest. This may be accounted for with a running spectral index of scalar
fluctuations, fit as ns = 0.93 ± 0.03 at wavenumber k0 = 0.05 Mpc−1 (lef f ≈ 700), with a slope
of dns /d ln k = −0.031+0.016
−0.018 in the best-fit model. (For WMAP data alone, ns = 0.99 ± 0.04.)
This flat universe model is composed of 4.4% baryons, 22% dark matter and 73% dark energy.
The dark energy equation of state is limited to w < −0.78(95% CL).
Inflation theory is supported with ns ≈ 1, Ωtot ≈ 1, Gaussian random phases of the CMB
anisotropy, and superhorizon fluctuations implied by the TE anticorrelations at decoupling. An
admixture of isocurvature modes does not improve the fit. The tensor-to-scalar ratio is r(k0 =
0.002 Mpc−1 ) < 0.90 (95% CL). The lack of CMB fluctuation power on the largest angular scales
reported by COBE and confirmed by WMAP is intriguing. WMAP continues to operate, so
results will improve.

Subject headings: cosmic microwave background, cosmology: observations, early universe, dark
matter, space vehicles, space vehicles: instruments, instrumentation: detectors, telescopes

1. INTRODUCTION

The cosmic microwave background (CMB) radiation was first detected by Penzias & Wilson (1965).
After its discovery, a small number of experimentalists worked for years to better characterize the CMB as
they searched for temperature fluctuations. A leader of this effort, and of the WMAP effort, was our recently
deceased colleague, Professor David T. Wilkinson of Princeton University. He was also a leading member
of the Cosmic Background Explorer (COBE) mission team, which accurately characterized the spectrum of
the CMB (Mather et al. 1990, 1999) and first discovered anisotropy (Smoot et al. 1992; Bennett et al. 1992;
Kogut et al. 1992; Wright et al. 1992). The MAP was recently renamed WMAP in his honor.
The general recognition that the CMB is a primary tool for determining the global properties, content,
and history of the universe has led to the tremendous interest and growth of the field. In addition to the
characterization of the large-scale anisotropy results from COBE (Bennett et al. 1996; Hinshaw et al. 1996a,b;
Kogut et al. 1996b,c,a; Górski et al. 1996; Wright et al. 1996a), a host of experiments have measured the
finer scale anisotropy (Benoit et al. 2003; Grainge et al. 2003; Pearson et al. 2002; Ruhl et al. 2003; Kuo et al.
2002; Dawson et al. 2001; Halverson et al. 2002; Hanany et al. 2000; Leitch et al. 2000; Wilson et al. 2000;
Padin et al. 2001; Romeo et al. 2001; Harrison et al. 2000; Peterson et al. 2000; Baker et al. 1999; Coble et al.
1999; Dicker et al. 1999; Miller et al. 1999; de Oliverira-Costa et al. 1998; Cheng et al. 1997; Hancock et al.
1997; Netterfield et al. 1997; Piccirillo et al. 1997; Tucker et al. 1997; Gundersen et al. 1995; de Bernardis
et al. 1994; Ganga et al. 1993; Myers et al. 1993; Tucker et al. 1993). As a result of these tremendous efforts,
the first acoustic peak of the anisotropy power spectrum has been unambiguously detected (Knox & Page
2000; Mauskopf et al. 2000; Miller et al. 1999) and CMB observations have placed important constraints on
cosmological models. Recently, Kovac et al. (2002) reported the first detection of CMB polarization arising
from the anisotropic scattering of CMB photons at decoupling, ushering in a new era of CMB polarization
measurements.
The WMAP mission was designed to advance observational cosmology by making full sky CMB maps
with accuracy, precision, and reliability, as described by Bennett et al. (2003a). The instrument observes the
temperature difference between two directions (as did COBE) using two nearly identical sets of optics (Page
et al. 2003c,a). These optics focus radiation into horns (Barnes et al. 2002) that feed differential microwave
 –3–

radiometers (Jarosik et al. 2003a). We produce full sky maps in five frequency bands from the radiometer
data of temperature differences measured over the full sky. A CMB map is the most compact representation
of CMB anisotropy without loss of information.
In this paper we present the maps, their properties, and a synopsis of the basic results of the first-
year of observations. In §2 we give a brief overview of the WMAP mission. In §3 we summarize the data
analysis, calibration, and systematic errors of the experiment, which are discussed in much greater detail
in the companion papers by Hinshaw et al. (2003a), Page et al. (2003a), Jarosik et al. (2003b), and Barnes
et al. (2003). In §4 we present the maps and their sampling properties, and we compare the WMAP and
COBE maps. In §5 we summarize the foreground analyses of Bennett et al. (2003a). In §6 we establish the
Gaussian nature of the WMAP anisotropy, determined in the companion paper of Komatsu et al. (2003). In
§7 we present the dipole and quadrupole moments and summarize analyses of the angular power spectrum
(Hinshaw et al. 2003b; Verde et al. 2003). In §8 we highlight the WMAP polarization results, including a
detection of the reionization of the universe (Kogut et al. 2003). In §9 we summarize some of the cosmological
implications of the WMAP results (Page et al. 2003c; Spergel et al. 2003; Peiris et al. 2003). Finally, in §10
we discuss the availability of the WMAP data products.

2. OBSERVATIONS

The 840 kg WMAP Observatory was launched aboard a Delta II 7425-10 rocket (Delta launch number
286) on 30 June 2001 at 3:46:46.183 EDT from Cape Canaveral. WMAP executed three phasing loops in
the Earth-Moon system before a lunar-gravity-assist swing-by, a month after launch, catapulted WMAP
to an orbit about the second Lagrange point of the Sun-Earth system, L2 . Station-keeping is performed
approximately four times per year to maintain the Observatory in a Lissajous orbit about the L2 point with
the Earth-WMAP vector within about ∼ 1◦ − 10◦ of the Sun-Earth vector. The phasing loop maneuvers
and station-keeping are executed using the WMAP propulsion system of blow-down hydrazine and eight
thrusters.
The central design philosophy of the WMAP mission was to minimize sources of systematic measure-
ment errors (Bennett et al. 2003a). The COBE mission proved the effectiveness of a differential design in
minimizing systematic errors. Therefore, the WMAP instrument was designed with a back-to-back optical
system with 1.4 m × 1.6 m primary reflectors to provide for differential measurements of the sky. The pri-
mary and secondary reflectors direct radiation into two focal planes, with ten feed horns in each, as described
by Page et al. (2003c).
The beams have a gain pattern, G, which is neither symmetric nor Gaussian. We define the beam solid
R
angle as G(Ω)/Gmax dΩ. The beam size can be expressed as the square root of the beam solid angles, giving
0.22◦ , 0.35◦, 0.51◦ , 0.66◦, and 0.88◦ for W-band though K-band, respectively. Alternately, the beams can
be expressed in terms of a full-width at half-maximum (FWHM) for each band, given in Table 1. Detailed
analyses of the WMAP beams are discussed by Page et al. (2003a,c).
The feed horns are attached to orthomode transducers (OMTs) that split the polarization of the incoming
signal into a differential correlation radiometer system with High Electron Mobility Transistor (HEMT)
amplifiers. There are ten “differencing assemblies” each consisting of two “radiometers” with two “channels”
each (Jarosik et al. 2003a; Bennett et al. 2003a). There are four W-band (∼ 94 GHz), two V-band (∼61 GHz),
two Q-band (∼ 41 GHz), one Ka-band (∼ 33 GHz), and one K-band (∼ 23 GHz) differencing assemblies.
We usually refer to these bands by the generic designations K, Ka, Q, V, and W because there are multiple
 –4–

Table 1. Approximate Observational Properties by Band

Item K-Band Ka-Band Q-Band V-Band W-Band

Wavelength, λ (mm) 13 9.1 7.3 4.9 3.2

Frequency, ν (GHz) 22.8 33.0 40.7 60.8 93.5
Ant./therm. conversion factor, ∆T/∆TA 1.014 1.029 1.044 1.100 1.251
−1/2
Noise, σ0 (mK) σ = σ0 Nobs 1.424 1.449 2.211 3.112 6.498
Beam width θ( FWHM)
◦
0.82 0.62 0.49 0.33 0.21
No. of Differencing Assemblies 1 1 2 2 4
No. of Radiometers 2 2 4 4 8
No. of Channels 4 4 8 8 16

Table 2. Data Flagging Summary

Category K-Band Ka-Band Q-Band V-Band W-Band

Rejected or Lost Data

Lost Data (%) 0.27 0.27 0.27 0.27 0.27
Spacecraft thermal change (%) 0.87 0.87 0.87 0.87 0.87
Gain or baseline step (%) 0.00 0.13 0.12 0.00 0.22
—— —— —— —— ——
Total Lost or Bad Data (%) 1.04 1.27 1.26 1.14 1.36
Data Not Used in Maps
Planet flag (%) 0.11 0.11 0.11 0.11 0.11
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radiometers in each band, whose precise frequencies are not identical. Also, the effective frequency of a
radiometer depends on the spectrum of the emission it detects. Precise frequencies for the radiometers for
a CMB anisotropy spectrum are given by Jarosik et al. (2003a). Polynomials are given to determine the
effective frequency of the radiometers depending on the emission frequency spectrum. See Table 1 for a
summary of radiometer properties.
Undesirable 1/f noise is minimized by the design of the WMAP radiometers (Jarosik et al. 2003a). All
radiometers have 1/f knees below 50 mHz; 18 of 20 are below 10 mHz; and 10 of the 20 are below 1 mHz
Jarosik
√ et al. (2003b). (The 1/f knee is defined as the frequency where the noise power spectral density
is 2 times higher than its high frequency value.) Jarosik et al. (2003a) demonstrate that all radiometer
outputs have Gaussian noise, which “integrates down” with time as expected.
The radiometers are passively cooled to ∼ 90 K with no mechanical refrigerators. In addition, no
actively cycling heaters were permitted anywhere on the WMAP spacecraft. These design features helped
to ensure a mechanically, thermally, and electronically quiet platform that minimizes the driving forces of
systematic measurement errors.
In addition to the differential design, the COBE mission also demonstrated the importance of scanning
large areas of the sky in a short period of time with a complex scan pattern. WMAP follows the COBE
example with a three-axis (three reaction wheel) control system that maintains the Observatory in a nearly
constant survey mode of operations. (The Observatory is in constant survey mode, except for only ∼ 1 hr
for each of ∼ 4 station-keeping maneuvers per year.) In survey mode, the optical boresight sweeps out a
complex pattern on the sky (Bennett et al. 2003a). Approximately 30% of the sky is observed each hour.
The Observatory spins at 0.464 rpm (∼7.57 mHz) and precesses at 1 rev hr−1 (∼ 0.3 mHz).
Six months are required for L2 to orbit half way around the Sun, allowing for full sky coverage. The
observations presented in this and companion papers include a full orbit about the Sun, thus containing two
sets of full sky observations. By 10 August 2001, WMAP was sufficiently stable in its L2 orbit for CMB
data-taking to commence. One year of observations, completed on 9 August 2002, were analyzed. Data
taken beyond this date will be the subject of future analyses.

3. DATA PROCESSING, CALIBRATION, & SYSTEMATIC ERRORS

Time-ordered telemetry data from the Observatory are down-linked via NASA’s Deep Space Network
(DSN) to the WMAP Science and Mission Operations Center (SMOC) at the Goddard Space Flight Center.
The data are then transferred to the WMAP Science Team for analysis. All of the instrument data are down-
linked to the ground without any on-board flight data processing, thus allowing full insight into potential
systematic effects.
Only a fraction of a percent of data was lost in the flow from the Observatory to the SMOC. About 1%
of the received data were not used due to systematic error concerns (e.g., data taken during near station-
keeping maneuvers). Of the ∼ 99% good data, the processing pipeline flagged observations where bright
planets were in the beams so that these data would not used be used in making maps. The statistics on lost,
bad, and flagged data are given in Table 2.
An overview of the data flow is shown in Figure 1. The heart of the data analysis efforts center on studies
of systematic measurement errors (Hinshaw et al. 2003b). Components of spurious signals at the spin period
are the most difficult to distinguish from true sky signals. The Observatory was designed to minimize all
 –6–

Fig. 1.— An overview of the WMAP data flow. The references are: (1) Barnes et al. (2003) (2) Bennett
et al. (2003b) (3) Bennett et al. (2003c) (4) Hinshaw et al. (2003b) (5) Hinshaw et al. (2003a) (6) Jarosik
et al. (2003b) (7) Kogut et al. (2003) (8) Komatsu et al. (2003) (9) Page et al. (2003a) (10) Page et al.
(2003b) (11) Peiris et al. (2003) (12) Verde et al. (2003).
 –7–

thermal and voltage variations and all susceptibilities to these variations, especially at the spin period, as
discussed in §2 and by Bennett et al. (2003a). In addition, high precision temperature monitors on the
Observatory provide the data needed to verify that systematic errors from thermal variations are negligible.
Jarosik et al. (2003b) report that in-flight spin-synchronous effects from the radiometers are < 0.17 µK rms
in the time-ordered-data (TOD), based on flight thermal variations multiplied by upper limits on component
susceptibilities measured in ground testing. Analysis of flight data without use of characterizations derived
from ground-based testing give < 0.14 µK rms from all sources (not just the radiometers). This is a factor
of > 50 times smaller than the requirement that was set in the mission’s systematic error budget. Thus, no
corrections to the first year WMAP data are required for spin-synchronous systematic errors.
The core of the processing pipeline calibrates the data and converts the differential temperatures into
maps. The data are calibrated based on the Earth-velocity modulation of the CMB dipole. A gain model of
the radiometers was derived and fit by Jarosik et al. (2003b). The model is based on the constancy of the
dipole signal on the sky, the measured physical temperature of the front-end radiometer components, and
on the time-averaged RF-bias (total power) of the radiometer outputs. This relatively simple model closely
matches the gains derived from the hourly measurements of the amplitude of the dipole and is used in WMAP
data processing. Calibration is achieved within 0.5% accuracy, dominated by the statistical uncertainty in
the absolute calibration.
Low levels of 1/f noise create stripes in the maps that affect the angular power spectrum and other
statistics derived from the maps. A pre-whitening filter is applied to the TOD to minimize these artifacts.
An estimate of the magnitude of the striping is given in Hinshaw et al. (2003b) for the maps, and by Hinshaw
et al. (2003a) for the power spectrum.
The differential temperature data are formed into maps based on the technique introduced by Wright
et al. (1996b). HEALPix11 is used to define map pixels on the sky in Galactic coordinates. Various levels of
resolution are specified by a “resolution level” with an integer (r = 0, 1, 2, ...). With Nside = 2r , the number
2
of pixels in the map is Npix = 12 Nside . The area per pixel is Ωpix = 4π/Npix and the separation between
1/2
pixel centers is θpix = Ωpix . For example, HEALPix resolution level r = 9 (used in WMAP map-making)
corresponds with Nside = 512, Npix = 3 145 728, Ωpix = 3.99 × 10−6 sr, and θpix = 0.◦ 115 = 6.87 arc-min.
WMAP observes the sky convolved with the beam pattern. This is equivalent to the the spatial transform
of the sky multiplied by the instrument’s “window function.” The beam patterns are measured in-flight from
observations of Jupiter (Page et al. 2003a). Uncertainties in our knowledge of the beam pattern, although
small, are a significant source of uncertainty for WMAP since they imply imperfect knowledge of the window
function. A small difference between the A-side and B-side optical losses was derived based on dipole
observations and corrected in the processing. Far sidelobes of the beam patterns, determined by ground
measurements and in-flight using the Moon, have been carefully examined (Barnes et al. 2003). A small
far-sidelobe correction is applied only to the K-band map. We now describe the maps.

4. THE MAPS

We combine the radiometer results within each band and present the five full sky maps at effective
CMB anisotropy frequencies 23, 33, 41, 61, and 94 GHz in Figures 2a, 2b, 2c, 2d, 2e. The maps are shown

11 http://www.eso.org/science/healpix/
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Fig. 2a.— WMAP K-band sky map in Mollweide projection in Galactic coordinates.
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Fig. 2b.— WMAP Ka-band sky map in Mollweide projection in Galactic coordinates.
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Fig. 2c.— WMAP Q-band sky map in Mollweide projection in Galactic coordinates.
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Fig. 2d.— WMAP V-band sky map in Mollweide projection in Galactic coordinates.
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Fig. 2e.— WMAP W-band sky map in Mollweide projection in Galactic coordinates.
 – 13 –

in the Mollweide projection in units of CMB thermodynamic temperature. The number of independent
observations that contribute to each pixel form the sky pattern in Figure 3. Figure 4 provides an overall
guide to some of the more prominent features of the maps as well as point sources detected by WMAP, as
described by Bennett et al. (2003b).
Figure 5 shows the K-band and Ka-band maps, with the Ka-band map smoothed to K-band resolution.
Note both the significant decrease in Galactic signal from K-band to Ka-band and the high Galactic latitude
similarities of the CMB between the maps. Likewise, Figure 6 shows the Q-band, V-band, and W-band maps
with the latter two smoothed to Q-band resolution. Higher Galactic contamination in Q-band is apparent.
Both Figures 5 and 6 highlight the consistency of the high Galactic latitude CMB anisotropy pattern from
band to band.
Comparisons of data between WMAP radiometers, and between WMAP and COBE, are important
indicators of systematic error levels. Figure 7 illustrates the enormous improvement in angular resolution
from COBE to WMAP. Features in the maps appear to be generally consistent, but the consistency is better
addressed by a more direct comparison. To do this we take a combination of the WMAP Q-band and V-band
maps and smooth it to mimic a COBE-DMR 53 GHz map (see Figure 8). We then examine the difference
between the COBE and WMAP pseudo-map. Figure 9 shows the difference map along with a simulated
map of the noise. With the exception of a feature in the Galactic plane, the agreement is clearly at the noise
level. The Galactic plane feature is likely to be a result of the spectral index uncertainty of combining the
Q-band and V-band maps to make a 53 GHz equivalent map.

5. FOREGROUND ANALYSES

An understanding of diffuse Galactic emission and extragalactic point sources is necessary for CMB
analyses. The WMAP mission carries radiometers at five frequencies for the purpose of separating the
CMB anisotropy from foreground emission based on their different spectra. Figure 10 illustrates the spectral
difference between the CMB and foregrounds. The WMAP bands were selected to be near the frequency
where the ratio of the CMB anisotropy to the contaminating foreground is at a maximum.

5.1. Masks

For CMB analyses it is necessary to mask out regions of bright foreground emission. Bennett et al.
(2003c) present a recipe for foreground masks based on K-band temperature levels. Since foreground con-
tamination is most severe in K-band, it is used as the best tracer of contamination. The contamination
morphology is similar enough across all five WMAP bands that masks based on the temperature levels in
other bands would be redundant and unnecessary. Standard names are given for the mask levels. For ex-
ample, the Kp0 mask cuts 21.4% of sky pixels while the Kp2 mask cuts 13.1%. See Bennett et al. (2003c)
for further detail. An extragalactic point source mask is also constructed based on selections from source
catalogs. An additional 2% of pixels are masked due to these ∼ 700 sources.
 – 14 –

Fig. 3.— The number of independent observations per pixel in Galactic coordinates. The number of obser-
vations is greatest at the ecliptic poles and in rings around the ecliptic poles with diameters corresponding
to the separation angle of the two optical boresight directions (approximately 141◦). The observations are
the most sparse in the ecliptic plane. Small area cuts are apparent where Mars, Saturn, Jupiter, Uranus, and
Neptune data are masked so as not to contaminate CMB analyses. Jupiter data are used for beam mapping.
The histogram of the sky sampling shows the departures from uniform sky coverage.
 – 15 –

Fig. 4.— A guide to the microwave sky for reference. This picture shows the large-scale emission from the
Milky Way galaxy, including some of its notable components such as the Cygnus complex, the North Polar
Spur, the Gum region, etc. The small circles show positions of the microwave point sources detected by
WMAP (Bennett et al. 2003c). The brighter sources are labeled for reference.
 – 16 –

Fig. 5.— A comparison of the K-band map with the Ka-band map smoothed to K-band resolution, both
in thermodynamic temperature, shows the dramatic reduction of Galactic contamination with increased
frequency. The comparison also shows the similarity of the CMB fluctuation features at high Galactic
latitude.
 – 17 –

Fig. 6.— A comparison of the Q-band, V-band, and W-band maps. All three maps are smoothed to Q-band
resolution and are in thermodynamic temperature. The reduction of Galactic contamination relative to K-
band and Ka-band (Figure 5) is apparent. The maps show that the constant features across bands are CMB
anisotropy while the thermodynamic temperature of the foregrounds depends on the band (frequency).
 – 18 –

Fig. 7.— A comparison of the COBE 53 GHz map (Bennett et al. 1996) with the W-band WMAP map.
The WMAP map has 30 times finer resolution than the COBE map.
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Fig. 8.— The COBE-DMR 53 GHz map (Bennett et al. 1996) is shown along with a map made with a linear
combination of the Q-band and V-band WMAP maps to mimic a 53 GHz map. Note the strong similarity
of the maps.
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Fig. 9.— The difference map is shown between the COBE-DMR 53 GHz map and the combination Q-
band/V-band maps from Figure 8. This is compared with a map of the noise level. The maps are consistent
with one another with the exception of a feature in the galatic plane. This discrepancy is likely to be due to
a spectral index that is sufficiently different from the assumed CMB spectrum used to combine the WMAP
Q-band and V-band maps to mimic a 53 GHz map.
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Fig. 10.— False color images represent the spectral information from mutliple WMAP bands. Q-band is
red, V-band is green, and W-band is blue. A CMB thermodynamic spectrum is grey. (top) A three color
combination image from the Q-, V-, and W-band maps. The dipole and high Galactic latitude anisotropy
are seen. (bottom) A similar false color image but with the dipole subtracted.
 – 22 –

5.2. Diffuse Galactic emission

Beyond the use of masks, one technique for reducing the level of foreground contamination is to form
a linear combination of the multifrequency WMAP data that retains unity response for only the emission
component with a CMB spectrum. This technique was introduced for COBE by Bennett et al. (1992). With
five WMAP bands instead of the three on COBE, and with a somewhat more elaborate approach for WMAP,
Bennett et al. (2003c) arrive at the internal (WMAP data only) linear combination map seen in Figure 11
of this paper. The foregrounds are removed to a remarkable degree; however, the statistics of this internal
linear combination map are complex and inappropriate for most CMB analyses.
Below, we use the notation convention that flux density is S ∼ ν α and antenna temperature is TA ∼ ν β ,
where the spectral indices are related by β = α − 2. In general, the CMB is expressed in terms of thermo-
dynamic temperature, while Galactic and extragalactic foregrounds are expressed in antenna temperature.
Thermodynamic temperature differences are given by ∆T = ∆TA [(ex − 1)2 /x2 ex ], where x = hν/kT0 , h is
the Planck constant, ν is frequency, k is the Boltzmann constant, and T0 = 2.725 K is the CMB temperature
(Mather et al. 1999). Values of ∆T /∆TA for the WMAP bands are given by Jarosik et al. (2003b) and can
be found in Table 1.
Bennett et al. (2003c) identify the amplitudes and spectral indices of the individual emission components.
A maximum entropy method (MEM) approach is adopted where priors are used for component amplitudes
and spectral indices, except for free-free emission, which has a fixed spectral index (β = −2.15 in the WMAP
bands). An iterative fit is performed, where the pixel-by-pixel amplitudes are updated in accordance with
the MEM residuals until low (< 1%) residuals are achieved. The process results in a map of each emission
component for each of the five WMAP bands. The derived maps of thermal emission from dust give a
uniform spectral index across the sky of βd ≈ 2.2. The derived map of free-free emission is reasonable given
the amplitude and morphology of Hα measurements. The other radio component fit should include the
combined emission of synchrotron and spinning dust. It shows the synchrotron spectrum steepening with
increasing frequency, as would be expected for a spectral break due to synchrotron losses at ≈ 20 GHz. There
is no indication of the less steep or flattening spectral index that would result from spinning dust emission.
The spinning dust emission is limited to < 5% of the total Ka-band foreground. Reports of dust-correlated
microwave emission from COBE data analyses are understood as an admixture of the fraction of synchrotron
emission (with β ≈ −3) that is traced by a dust template, and thermal dust emission (β ≈ 2.2), giving a
combined spectral index of β ≈ −2.2 between the WMAP Ka-band and V-band, approximating the COBE
31 GHz and 53 GHz bands.
While the MEM method is useful for understanding the nature of the foreground emission components,
these results can not be directly used in CMB analyses due to the complex noise properties that result from
the MEM process and its simultaneous use of multifrequency maps. This is because the multifrequency maps
are smoothed, different weights are used in different regions of the sky and these weights are smoothed, all of
which complicates the noise correlations. For the CMB analyses we use a mask to exclude pixels where the
Galactic emission is strong, combined with template fitting (using external data only) where the foregrounds
can be adequately corrected. This approach does not complicate the noise properties of the maps. The Kp2
cut is used for all analyses except for limits on non-Gaussianity and the temperature-polarization correlation
function, where the more severe Kp0 cut is used.
Bennett et al. (2003c) describe the template fitting in detail. Thermal dust emission has been mapped
over the full sky in several infrared bands, most notably by the COBE and IRAS missions. A full sky
template is provided by Schlegel et al. (1998), and is extrapolated in frequency by Finkbeiner et al. (1999).
 – 23 –

Fig. 11.— This “internal linear combination” map combines the five band maps in such as way as to maintain
unity response to the CMB while minimizing foreground foreground contamination. For a more detailed
description see Bennett et al. (2003c). For the region that covers the full sky outside of the inner Galactic
plane, the weights are 0.109, −0.684, −0.096, 1.921, −0.250 for K, Ka, Q, V, and W bands, respectively. Note
that there is a chance alignment of a particularly warm feature and a cool feature near the Galactic plane. As
discussed in Bennett et al. (2003c), the noise properties of this map are complex, so it should not generally
be used for cosmological analyses.
 – 24 –

The mostly synchrotron emission map of Haslam et al. (1981) at 408 MHz is used as a radio template. The
free-free ionized gas is traced by the Hα map assembled by Finkbeiner (2003) from the Wisconsin H-Alpha
Mapper (WHAM), the Virginia Tech Spectral-Line Survey (VTSS), and the Southern H-Alpha Sky Survey
Atlas (SHASSA) (Dennison et al. 1998; Haffner et al. 2002; Reynolds et al. 2002; Gaustad et al. 2001). The
Haslam map resolution is not as high as that of the WMAP maps, and the Haslam map has artifacts from
experimental effects such as striping from spatial calibration variations. The striping in the Haslam map
is along the survey scan lines and was corrected to first order by the application of a Wiener filter. (The
filtered version of the Haslam map is publicly available on the LAMBDA web site.) The remaining adverse
effects of the Haslam map are mitigated by two effects. First, the template fit calls only for a small Haslam
correlation (see §6 of Bennett et al. (2003c)). Since the correction is small, the error on the correction is
negligible. Second, the foreground contamination is most significant only on the largest angular scales so
the Haslam resolution limit and small-scale map artifacts are not significant sources of error. The MEM
solution only uses the Haslam map as a prior and the spinning dust limit only uses the full sky median of
the Haslam map. Thus the spinning dust limit is insensitive to residual striping in the Haslam map.
The MEM results are used to assess the degree of foreground emission remaining after the template
subtraction. The result is < 7 µK rms at Q-band and < 3 µK rms at both V-band and at W-band for l < 15.
This remaining foreground emission constitutes < 2% of the CMB variance (up to l = 200) in Q-band, and
<
∼ 1% of the CMB variance in V- and W-bands. Figures 3 and 4 of Hinshaw et al. (2003b) demonstrate that
this small residual foreground level has a negligible effect on the cosmological results.

5.3. Point sources

A search was made for point sources in the WMAP data. A catalog of 208 detected sources (with 98%
reliability) is provided by Bennett et al. (2003c). Statistically, five sources are expected to be false detections.
Five of the 208 sources do not have low frequency radio counterparts; these sources are likely to be the false
detections. We include ∼700 sources in our mask, despite having only detected ∼200 sources at the 5σ level,
because sources below this detection level still contribute an undesirable statistical contamination to the
maps. Even beyond masking the 700 sources, we still need to make a statistical correction to the power
spectrum for residual source contamination (Hinshaw et al. 2003b). The derived source counts give a power
spectrum level of C src = (15 ± 3) × 10−3 µK2 sr at Q-band. This is consistent with the level found in the
bispectrum analysis of the maps (Komatsu et al. 2003) and the level found in fits to the map power spectra
(Hinshaw et al. 2003b). We have confidence that the point source level is understood since it is independently
derived using three different methods.

5.4. Sunyaev-Zeldovich (SZ) effect

Hot gas in clusters of galaxies imparts energy to the CMB photons, causing a temperature decrement
in the WMAP bands (the Sunyaev-Zeldovich Effect). The Coma cluster is expected to have the most
pronounced signature. For the highest resolution maps, Bennett et al. (2003c) get −0.34 ± 0.18 mK in W-
band and −0.24 ± 0.18 mK in V-band in the direction of Coma. Use of the XBACS catalog of X-ray clusters
as a template results in −0.36 ± 0.14. This verifies that the Sunyaev-Zeldovich Effect is barely detectable
in even a matched search of the WMAP sky maps and therefore it is not a significant “contaminant” to the
WMAP data.
 – 25 –

6. LIMITS ON NON-GAUSSIANITY

Maps of the sky are the most complete and compact representation of the CMB anisotropy. Cosmological
analyses are based on statistical properties of the maps with the power spectrum as one of the most commonly
derived statistics. The power spectrum is a complete representation of the data only if the CMB anisotropy is
Gaussian. Also, the most common cosmological models predict that the CMB anisotropy should be consistent
with a Gaussian random field (at least at levels that are currently possible to measure). Therefore, we test
the Gaussianity of the anisotropy, both to interpret the power spectrum (and other statistical derivatives of
the maps) and to test cosmological paradigms.
There is no single best test for Gaussianity. Specific tests can be more or less sensitive to different
assumed forms of non-Gaussianity. Komatsu et al. (2003), in a companion paper, search for non-Gaussianity
in the WMAP CMB anisotropy maps using Minkowski functionals and a bispectrum estimator.
Minkowski functionals (Minkowski 1903; Gott et al. 1990) quantify topological aspects of the CMB
maps. Anisotropy is examined via contours at different temperature levels, and the number and areas of
regions enclosed by these contours are computed. Three Minkowski functionals are the area represented by
hot and cold spots, the contour length around these areas, and the difference between the number of these
areas (the “genus”).
It is widely believed that the CMB anisotropy arises from Gaussian linear fluctuations in the gravitational
potential. Komatsu & Spergel (2001) suggest that non-Gaussian anisotropy be considered in terms of the cur-
vature perturbation. The simplest expression for the overall primordial gravitational curvature perturbation,
2
Φ(x), is a sum of a linear ΦL (x) and weak nonlinear components: Φ(x) = ΦL (x) + fN L [Φ2L (x) − hΦL (x)i ]
where ΦL is the linear Gaussian portion of the curvature perturbation and fN L is a nonlinear coupling con-
stant. Then, fN L = 0 corresponds to the purely linear Gaussian case. Since the CMB bispectrum measures
the phase correlations of the anisotropy, it can be used to solve for fN L . The Minkowski functional results
can also be expressed in terms of fN L .
For the Minkowski functionals, Komatsu et al. (2003) find fN L < 139 (95% CL). From the bispectrum,
Komatsu et al. (2003) find −58 < fN L < 134 (95% CL). The two results are consistent. The CMB anisotropy
is thus demonstrated to follow Gaussian statistics. This is a significant result for models of the early universe.
It also means that we can construct and interpret CMB statistics (e.g. the angular power spectrum) from
the maps in a straightforward manner.

7. MULTIPOLES

The temperature anisotropy, T (n), is naturally expanded in a spherical harmonic basis, Ylm , as
X
T (n) = alm Ylm (n). (1)
l,m

The angular power spectrum, Cl , is a cosmological ensemble average given by

Cl = h|alm |2 i (2)

and is observed for our actual sky as

1 X
Clsky = |alm |2 . (3)
2l + 1 m
 – 26 –

Assuming random phases, the temperature anisotropy for each multipole moment, ∆Tl , can be associated
with the angular spectrum, Cl , as q
∆Tl = Clsky l(l + 1)/2π. (4)
The correlation function is
+l
1 X X
C(θ) = Cl Pl (cos θ) (5)
4π
l m=−l
1 X
= (2l + 1)Cl Pl (cos θ) (6)
4π
l

where Pl is the Legendre polynomial or order l.

In practice, an instrument adds noise and spatially filters the sky signal due to the beam pattern and
any other experimental limitations on the sampling of all angular scales. The experimental transfer function,
bl , for each multipole moment depends upon the specific beam pattern of the experiment. WMAP samples
all angular scales limited only by the beam pattern. The window function for the signal power between
channels i and j is
wlij = bil bjl p2l , (7)
where p2l is the pixel window function. The angular power spectrum then becomes

Cl = Clij bil bjl p2l + Nli δij (8)

where Nli is the power spectrum noise that results from the instrument noise, which is assumed to be
uncorrelated between channels i and j.
Note that an auto-power spectrum has twice the noise variance as a cross-power spectrum
D withEthe
corresponding noise in each map. This follows from the property of Gaussian noise that Nli 2 Nlj 2 =
2 D E D E2
2 2
2 Nli 2 δij + Nli 2 Nlj 2 , with Nli Nlj = Nli 2 δij , and Nli 4 = 3 Nli 2 .





The use of a sky mask for foreground suppression breaks the orthogonality of the spherical harmonics
on the sky and leads to mode coupling. Hinshaw et al. (2003b) discuss how WMAP handles this, and other
complexities.

7.1. l = 1 dipole

COBE determined the dipole amplitude is 3.353 ± 0.024 mK in the direction (l, b) = (264.26◦ ±
0.33 , 48.22◦ ± 0.13◦ ), where l is Galactic longitude and b is Galactic latitude (Bennett et al. 1996). This
◦

dipole was subtracted from the WMAP data during processing. Examination of the WMAP maps allow
for the determination of a residual dipole, and thus an improvement over the COBE value. Note that this
does not have any effect on WMAP calibration, which is based on the Earth’s velocity modulation of the
dipole, and not on the dipole itself. The WMAP -determined dipole is 3.346 ± 0.017 mK in the direction (l,
b)= (263.◦ 85 ± 0.◦ 1, 48.◦ 25 ± 0.◦ 04). The uncertainty of the dipole amplitude is limited by the WMAP 0.5%
calibration uncertainty, which will improve with time.
 – 27 –

7.2. l = 2 quadrupole

The quadrupole is the l = 2 term of the spectrum ∆Tl2 = l(l + p 1)Cl /2π, i.e. ∆T
2
p l=2 = (3/π)Cl=2 .
Alternately, the quadrupole amplitude can be expressed as Qrms = (5/4π) Cl=2 = 5/12 ∆Tl=2 . The
4-year COBE quadrupole is Qrms = 10+7−4 µK with the peak of the likelihood in the range 6.9 µK < Qrms <
10 µK, as shown in Figure 1 of Hinshaw et al. (1996a).
The WMAP quadrupole, Qrms = 8 ± 2 µK or ∆T22 = 154 ± 70 µK2 , is consistent with COBE but
with tighter limits because of better measurements and understanding of foregrounds. We determine the
quadrupole value by computing the power spectrum of the internal linear combination map and individual
channel maps, with and without foreground corrections, for a range of Galactic cuts. The final l = 2 value
corresponds to a full sky estimate with an uncertainty that encompasses a range of foreground-masked or
foreground-corrected solutions.” The foreground level is still the leading uncertainty. (The small kinematic
quadrupole is not removed from the maps nor accounted for in this analysis.) The quadrupole value is low
compared with values predicted by ΛCDM models that fit the rest of the power spectrum. ΛCDM models,
in particular, tend to predict relatively high quadrupole values due to the enhanced, Λ-driven, integrated
Sachs-Wolfe effect.

7.3. n-poles

A central part of the task of computing multipole information from the maps is the evaluation and
propagation of errors and uncertainties. This largely involves arriving at an adequate representation of the
Fisher matrix, which is the inverse covariance matrix of the data. The Fisher matrix must take into account
mode-coupling from the sky cut, beam (window function) uncertainties, and noise properties. The fact that
the WMAP data use a nearly azimuthally symmetric cut, and have a nearly diagonal pixel-pixel covariance,
greatly simplifies the evaluation of the Fisher matrix.
Two approaches to computing the angular power spectrum have been used by Hinshaw et al. (2003b):
a quadratic estimation based on Hivon et al. (2002); and a maximum likelihood estimate based on Oh
et al. (1999). The quadratic estimator is used in the final WMAP spectrum analyses, while the maximum
likelihood technique is used as a cross-check.
The K-band and Ka-band beam sizes are large enough that these bands are not used for CMB analysis
since they have the most foreground contamination and probe the region in l-space that is cosmic variance
limited by the measurements at the other bands. These bands are invaluable, however, as monitors of Galactic
emission. The two Q-band, two V-band, and four W-band differencing assemblies are the source of the prime
CMB data. The matrix of auto- and cross-correlations between the eight Q-, V-, and W-band differencing
assemblies has eight diagonal (auto-correlations) and 28 unique off-diagonal elements (cross-correlations).
Since auto-correlations are difficult to assess due to the noise bias (see equation 8), we included only the
28 unique off-diagonal (cross-correlations) in the WMAP power spectrum analysis. In dropping p the auto-
correlations, each of which has twice the noise variance of a cross-correlation, we lose only 1− 56/(56 + 8) =
6% of the ideally achievable signal-to-noise ratio. In Hinshaw et al. (2003b), we show that the power
spectrum computed from the auto-correlation data is consistent with the angular power spectrum from the
cross-correlation data. We anticipate using the auto-correlation data in future analyses.
The cross-power spectrum from the 28 pairs is considered in four l ranges. For l ≤ 100 we use uniform
pixel weighting of only V- and W-band data. This reduces the Galactic contamination where measurement
 – 28 –

errors are well below the cosmic variance. For 100 < l ≤ 200 we use uniform pixel weighting of the
combined 28 pairs. For 200 < l ≤ 450 all 28 cross-power pairs are used with a transitional pixel weighting.
The transitional pixel weighting, defined and discussed in detail in Appendix A of Hinshaw et al. (2003b),
smoothly transitions the weighting from the uniform pixel weights in the signal-dominated l < 200 multipole
regime to inverse-noise-variance weighting in the noise-dominated l > 450 multipole regime. For l > 450 all
28 pairs are used with inverse noise weighting. Our Monte Carlo simulations show that this approach is a
nearly optimal scheme.
The angular power spectrum is shown for the WMAP data in Figure 12. The WMAP power spectrum
agrees closely with COBE at the largest angular scales, and with CBI and ACBAR at the finer angular
scales. We highlight the CBI and ACBAR results because they are a useful complement to WMAP at the
smaller angular scales. The acoustic pattern is obvious. Page et al. (2003b) find that the first acoustic peak
is ∆Tl = 74.7 ± 0.5 µK at l = 220.1 ± 0.8. The trough following this peak is 41.0 ± 0.5 µK at l = 411.7 ± 3.5
and the second peak is 48.8 ± 0.9 µK at l = 546 ± 10.
ΛCDM models predict enhanced large angle power due to the integrated Sachs-Wolfe effect. The WMAP
and COBE data, on the other hand, have the opposite trend. The conflict is also seen clearly in the correlation
function, C(θ), shown in Figure 13. The WMAP correlation function is computed using the Kp0 cut on a
combination of the Q-band, V-band, and W-band maps with the MEM Galactic model removed. The COBE
correlation function is computed on the “custom cut” sky (Bennett et al. 1996). The best-fit ΛCDM model
is shown with a grey band indicating one standard deviation as determined by Monte Carlo simulations.
The lower two plots in Figure 13 display the correlation function of the difference between the COBE-
DMR and WMAP maps with a |b| = 10◦ Galactic plane cut. A synthesis of the WMAP Q- and V-band maps
was made to approximate a 53 GHz map to compare with the COBE-DMR 53 GHz map. The COBE-DMR
90 GHz map is compared directly, without corrections, to the WMAP W-band map. These plots emphasize
the consistency of the WMAP and COBE measurements. The slightly higher than expected deviations at
53 GHz are likely to be due to Galactic contamination, arising from outside the cut regions and from the
construction of the synthesized WMAP 53 GHz map.
The model is an excellent fit to the WMAP full power spectrum except, perhaps, at l ∼ <
6. Since only
a small fraction of the total number of measured multipoles are involved, the statistical contribution of the
l∼
<
6 points to the overall power spectrum fit is small and does not greatly drive the overall best-fit model.
The correlation function emphasizes the low l signal because these modes contribute to C(θ) at all angular
separations. The discrepancy at θ < ∼ 30
◦
reflects the average lack of power in the data relative to the model
at l < 6. More significantly, the lack of power at θ > ∼ 60
◦
relative to the model reflects the special shape of
the power spectrum from 2 < l < 5 seen in both the WMAP (and COBE) maps. This result is generically
true for ΛCDM models, independent of the exact parameters of the model. There is very little large scale
CMB anisotropy power in our sky. This fact, first seen by COBE is confirmed by WMAP. The probability
of so little C(θ > 60◦ ) anisotropy power is ∼ 2 × 10−3 , given the best-fit ΛCDM model. The lack of power
is seen both in C(θ) and the behavior of the low order (l = 2, 3, 4, and 5) multipoles.

8. CMB POLARIZATION & THE DETECTION OF REIONIZATION

Each differencing assembly measures the sky in two orthogonal linear polarizations. As the Observatory
spins, precesses, and orbits the Sun, the instrument observes the sky over a range of polarization angles.
The range of angles observed is neither complete nor uniform, but it is sufficient to provide valuable new
 – 29 –

Fig. 12.— The WMAP angular power spectrum. (top:) The WMAP temperature (TT) results are consistent
with the ACBAR and CBI measurements, as shown. The TT angular power spectrum is now highly con-
strained. Our best fit running index ΛCDM model is shown. The grey band represents the cosmic variance
expected for that model. The quadrupole has a surprisingly low amplitude. Also, there are excursions from a
smooth spectrum (e.g., at ℓ ≈ 40 and ℓ ≈ 210) that are only slightly larger than expected statistically. While
intriguing, they may result from a combination of cosmic variance, subdominant astrophysical processes, and
small effects from approximations made for this first year data analysis (Hinshaw et al. 2003b). We do not
attach cosmological significance to them at present. More integration time and more detailed analyses are
needed. (bottom:) The temperature-polarization (TE) cross-power spectrum, (l + 1)Cl /2π. (Note that this
is not multiplied by the additional factor of l.) The peak in the TE spectrum near l ∼ 300 is out of phase
with the TT power spectrum, as predicted for adiabatic initial conditions. The antipeak in the TE spectrum
near l ∼ 150 is evidence for superhorizon modes at decoupling, as predicted by inflationary models.
 – 30 –

Fig. 13.— (top): CMB temperature correlation function of the WMAP and COBE data. The WMAP
correlation function is computed using a combination of the Q-band, V-band, and W-band maps with the
Kp0 cut sky and the MEM Galactic model subtracted. The COBE correlation function is computed using
the “custom cut” sky. The running index ΛCDM model that is fit to the power spectrum is shown with a
Monte Carlo determined grey band indicating one standard deviation. The model is, overall, an excellent fit
to the WMAP power spectrum. However, a correlation plot emphasizes the low l power. The discrepancy
between the model and data illustrates that there is surprisingly little anisotropy power in the WMAP and
COBE maps at large angles. (bottom): The lower two plots display the correlation function of the difference
between the COBE-DMR and WMAP maps with a |b| = 10◦ Galactic plane cut. A synthesis of the WMAP
Q and V band maps was made to approximate a 53 GHz-like map to compare with the COBE-DMR 53
GHz map. The COBE-DMR 90 GHz map is compared directly, without corrections, to the WMAP W-band
map. These plot emphasize the consistency of the WMAP and COBE measurements. The slightly higher
than expected deviations at 53 GHz are likely to be due to Galactic contamination, arising from outside the
cut regions and from the construction of the synthesized WMAP 53 GHz map.
 – 31 –

CMB polarization results.

We express polarization in term of the standard I, Q, and U Stokes parameters, in units of thermody-
namic temperature. Polarization coordinates are not rotationally invariant so a coordinate system is defined
by Kogut et al. (2003) for expressing information. By appropriate summing and differencing of the time-
ordered-data from the pairs of corresponding radiometers, we iteratively make maps in unpolarized intensity,
I, and in each polarized component, Q and U (Hinshaw et al. 2003b). The Q and U maps not only have
much lower signal-to-noise than the I maps, but they are more susceptible to systematic errors since the
differencing between radiometer outputs occurs on the ground, not in the front-end of the radiometers.
The U polarization maps are expected to be most affected by systematic errors. This is due to the
orientation of the observed polarization angles relative to the spin axis of the Observatory, causing systematics
not fixed on the sky to preferentially go into the U map.
Taurus A, an extremely strong source in the WMAP bands, is polarized and its polarization amplitude
and direction are known (Flett & Henderson 1979). Polarization measurements from WMAP are consistent
with previous observations of Taurus A, providing a useful check on the operation of the WMAP hardware
and software. However, the polarization systematic errors have not yet been fully quantified. We chose to
release the I data and results as soon as possible, rather than postpone their publication and release until
the systematic measurement errors of the Q and U data are fully assessed. All data, including the Q and U
maps will be released when the characterization of their instrumental signatures are complete. The dominant
systematic effect in the low signal-to-noise polarization data arises from correlated noise in the radiometers.
By using only cross correlations of temperature maps with polarization maps, generated from independent
radiometer combinations that have uncorrelated noise, we mitigate the leading systematic error. For this
reason, results of temperature-polarization correlations between the I and Q maps, i.e. TE correlations,
are much less sensitive to systematic effects than polarization signals alone, so we are able to report those
results.
Figure 12 shows the temperature-polarization (TE) cross power spectrum. Kogut et al. (2003) report a
detection of TE correlations on both large angular scales (from reionization) and on small scales (from the
adiabatic fluctuations). The TE power spectrum, shown in Figure 12, is discussed in detail by Kogut et al.
(2003). In the TE angular power spectrum the antipeak is −35 ± 9 µK2 at l = 137 ± 9 and the peak is
105 ± 18 µK2 at l = 329 ± 19 (Page et al. 2003b).
The detection of the reionization of the universe corresponds to an optical depth τ = 0.17 ± 0.04
(0.09 ≤ τ ≤ 0.28 at 95% confidence). Although WMAP measures the integrated optical depth, the epoch
and redshift of reionization can be derived from the integral optical depth within the context of a model of
the reionization process. A single instantaneous step function in the ionization fraction from zero to a steady
fixed value is physically unlikely. For more likely models, Kogut et al. (2003) conclude that the redshift of
reionization is zr = 20+10 +220
−9 (95% CL), corresponding to an epoch of reionization of tr = 180−80 Myr (95%
CL) after the Big Bang. Cen (2002) presents a detailed model of a reionization process and predicts an
integral value of the optical depth, τ = 0.10 ± 0.03. This is low compared with the newly measured value,
but would be higher and consistent if the assumed cosmological parameters in the model were adjusted to
the values of the new WMAP best-fit parameters (see §9.1).
The measured optical depth means that reionization suppressed the acoustic peak amplitudes by 1 −
e−2τ ≈ 30%. While accounted for in our model fits (§9, below), this suppression was not accounted for in
previous CMB parameter determinations. The anticorrelations observed in the TE power spectrum directly
imply superhorizon fluctuations, a new result in support of inflation-like theories, as discussed in §9.3.
 – 32 –

9. COSMOLOGICAL INTERPRETATION

In this section we summarize the cosmological intepretation of WMAP first year results, which are
discussed in more detail by Spergel et al. (2003), Peiris et al. (2003), Page et al. (2003b), and Kogut et al.
(2003). The methodology used in the model fits is described by Verde et al. (2003).
Spergel et al. (2003) show that a cosmological model with a flat universe, seeded with a scale-invariant
spectrum of adiabatic Gaussian fluctuations, with reionization, is an acceptable fit not only to the WMAP
data but also to a host of astronomical data. These data are: smaller angular scale CMB anisotropy data
from ACBAR (Kuo et al. 2002) and CBI (Pearson et al. 2002); the HST key project value of H0 (Freedman
et al. 2001); the accelerating Universe seen in Type Ia SNe (Riess et al. 2001); the shape and amplitude
of the large scale structure seen in clusters and superclusters of galaxies (Percival et al. 2001; Verde et al.
2003); and the linear matter power spectrum seen in the Lyman α forest (Croft et al. 2002). There has been
mounting evidence in the direction of this model for years (Peebles 1984; Bahcall et al. 1999). The optical
depth since reionization is a new, but not surprising component of the model. The WMAP data establish
this model as the standard model of cosmology by testing the key assumptions of the model and by enabling
a precise determination of its parameters.
The WMAP data test several of the key tenets of the standard model. The WMAP detection of
temperature-polarization correlations (Kogut et al. 2003) and the clear detection of acoustic peaks (Page
et al. 2003b) implies that the primordial fluctuations were primarily adiabatic: the primordial ratio of
dark matter/photons and the primordial ratio of baryons/photons do not vary spatially. The analysis of
the WMAP temperature data demonstrates Gaussianity (Komatsu et al. 2003). The WMAP data, when
combined with any one of the following three external data sets: HST Key Project measurement of H0
(Freedman et al. 2001), the 2dFGRS measurement of the matter density (Percival et al. 2001; Verde et al.
2003) or the Type Ia supernova measurements (Riess et al. 2001) imply that the radius of curvature of the
universe, R = cH0−1 |1 − Ωtot |−1/2 , must be very large, Ωtot = 1.02 ± 0.02 and 0.99 < Ωtot < 1.05 (95% CL).
These measurements also require that the dark energy be the dominant constituent of Ωtot . The WMAP
data alone rule out the standard Ωm = 1 CDM model by > 7σ.

9.1. Best Fit Cosmological Model

While an acceptable fit, the model described above is not our best fit model. In the discussion below we
concentrate on our best fit model, which adds a scale-dependent primordial spectral index. This cosmological
model is a flat universe with a baryon fraction of Ωb = 0.044 ± 0.004, a matter fraction of Ωm = 0.27 ± 0.04,
and a dark energy fraction of ΩΛ = 0.73±0.04, seeded with a scale-dependent spectrum of adiabatic Gaussian
fluctuations. This model has Cl=10 = 46.0 µK2 , consistent with the COBE measurement of Cl=10 = 44.4
µK2 .
The WMAP data alone enable accurate determinations of many of the key cosmological parameters
(Spergel et al. 2003). But a combination of the WMAP data with the COBE determination of the CMB
temperature (Mather et al. 1999), the CBI (Pearson et al. 2002) and the ACBAR (Kuo et al. 2002) CMB
measurements, and the 2dFGRS survey determination of the power spectrum of the local galaxy fluctuations
(Percival et al. 2001), yields the best fit cosmological parameters listed in Table 3. Verde et al. (2003)
describes our methodology for determining these parameters and Spergel et al. (2003) describes the best fit
models for different combinations of data sets.
 – 33 –

Table 3. “Best” Cosmological Parameters

Description Symbol Value + uncertainty − uncertainty

Total density Ωtot 1.02 0.02 0.02

Equation of state of quintessence w < −0.78 95% CL —
Dark energy density ΩΛ 0.73 0.04 0.04
Baryon density Ωb h 2 0.0224 0.0009 0.0009
Baryon density Ωb 0.044 0.004 0.004
Baryon density (cm−3 ) nb 2.5 × 10−7 0.1 × 10−7 0.1 × 10−7
Matter density Ωm h 2 0.135 0.008 0.009
Matter density Ωm 0.27 0.04 0.04
Light neutrino density Ων h 2 < 0.0076 95% CL —
CMB temperature (K)a Tcmb 2.725 0.002 0.002
CMB photon density (cm−3 )b nγ 410.4 0.9 0.9
Baryon-to-photon ratio η 6.1 × 10−10 0.3 × 10−10 0.2 × 10−10
Baryon-to-matter ratio Ωb Ω−1m 0.17 0.01 0.01
Fluctuation amplitude in 8h−1 Mpc spheres σ8 0.84 0.04 0.04
Low-z cluster abundance scaling σ8 Ω0.5
m 0.44 0.04 0.05
Power spectrum normalization (at k0 = 0.05 Mpc−1 )c A 0.833 0.086 0.083
Scalar spectral index (at k0 = 0.05 Mpc−1 )c ns 0.93 0.03 0.03
Running index slope (at k0 = 0.05 Mpc−1 )c dns /d ln k −0.031 0.016 0.018
Tensor-to-scalar ratio (at k0 = 0.002 Mpc−1 ) r < 0.90 95% CL —
Redshift of decoupling zdec 1089 1 1
Thickness of decoupling (FWHM) ∆zdec 195 2 2
Hubble constant h 0.71 0.04 0.03
Age of universe (Gyr) t0 13.7 0.2 0.2
Age at decoupling (kyr) tdec 379 8 7
Age at reionization (Myr, 95% CL)) tr 180 220 80
Decoupling time interval (kyr) ∆tdec 118 3 2
Redshift of matter-energy equality zeq 3233 194 210
Reionization optical depth τ 0.17 0.04 0.04
Redshift of reionization (95% CL) zr 20 10 9
Sound horizon at decoupling (◦ ) θA 0.598 0.002 0.002
Angular size distance (Gpc) dA 14.0 0.2 0.3
Acoustic scaled ℓA 301 1 1
Sound horizon at decoupling (Mpc)d rs 147 2 2

a from COBE (Mather et al. 1999)

b derived from COBE (Mather et al. 1999)
cl ≈ 700
ef f

dℓ −1
≡ πθA θA ≡ rs d−1
A a
 – 34 –

A power spectrum of primordial mass fluctuations with a scale invariant spectral index is given by
P (k) = Ak ns with ns = d ln P/d ln k. Inflationary models predict a running spectral index (Kosowsky &
Turner 1995), and our best fit model uses a power spectrum of primordial mass fluctuations with a scale-
dependent spectral index:
 ns (k0 )+(1/2)(dns /d ln k) ln(k/k0 )
k
P (k) = P (k0 ) . (9)
k0
As in the scale-independent case, we define
d ln P
ns (k) ≡ , (10)
d ln k
so  
dns k
ns (k) = ns (k0 ) + ln (11)
d ln k k0
(with d2 ns /d ln k 2 = 0). The definition for ns used here includes a factor of (1/2) difference from the
Kosowsky & Turner (1995) definition. Peiris et al. (2003) explore the implications of this running spectral
index for inflation. The best fit values of A, ns and dns /d ln k are in Table 3 for k0 = 0.05 Mpc−1 . A is
the normalization parameter in CMBFAST version 4.1 with option UNNORM. The amplitude of curvature
2
fluctuations at the horizon crossing is |∆R (k0 )| = 2.95×10−9A. We discuss the implications of the measured
values of these parameters in §9.3 and in Peiris et al. (2003).
The WMAP data constrains the properties of both the dark matter and the dark energy in the following
ways:

(a) The WMAP detection of reionization at z ∼ 20 is incompatible with the presence of significant warm
dark matter density. Since the warm dark matter moves too fast to cluster in small objects, the first
objects do not form in this scenario until z ∼ 8 (Barkana et al. 2001).

(b) The running spectral index implies a lower amplitude for mass fluctuations on the dwarf galaxy scale.
Dark matter simulations of models (Ricotti 2002) find that the dark matter mass profiles depend upon
the spectral index on the relevant mass scale. Thus, the shallower spectral index implied by our best fit
model may solve the CDM dark matter halo profile problem (Moore et al. 1998; Spergel & Steinhardt
2000).

(c) While the WMAP data alone are compatible with a wide range of possible properties for the dark
energy, the combination of the WMAP data with either the HST key project measurement of H0 , the
2dFGRS measurements of the galaxy power spectrum or the Type Ia supernova measurements requires
that the dark energy be 73% of the total density of the Universe, and that the equation of state of the
dark energy satisfy w < −0.78(95% CL).

9.2. Best Fit Parameters

WMAP’s measurements of the baryon density, Hubble constant, and age of the universe strengthen the
cosmic consistency that underlies the Big Bang model.
ATOMIC DENSITY (Ωb h2 ): WMAP measures the atomic density at recombination to an accuracy of
4% through the shape of the angular power spectrum, and particularly through the ratio of the heights of
 – 35 –

the first to second peak (Page et al. 2003b; Spergel et al. 2003). Our best fit value is Ωb h2 = 0.0224 ± 0.0009.
The baryon density is also probed via abundance measurements of [D]/[H] (O’Meara et al. 2001; Pettini &
Bowen 2001; D’Odorico et al. 2001). It is impressive that Ωb h2 is the same at z = 1089 as measured via
the CMB as it is at z = 109 from Big Bang nucleosynthesis. Thus we find cosmic consistency of the baryon
density throughout cosmic time and measurement technique.
HUBBLE CONSTANT (H0 ): The WMAP measurements of the age and Ωm h2 yield a measurement of
H0 = 71+4
−3 km s
−1
Mpc−1 that is remarkably consistent with the HST Key Project value of H0 = 72±3±7 km
s−1 Mpc−1 (Freedman et al. 2001), but with smaller uncertainty. Recent measurements of the Hubble
constant from gravitational lens timing and the Sunyaev-Zeldovich effect yield independent estimates that are
generally consistent, but with larger uncertainties at present. Through a variety of measurement techniques
that sample different cosmic times and distances we find cosmic consistency on H0 .
AGE OF THE UNIVERSE (t0 ): The first acoustic peak in the CMB power spectrum represents a
known acoustic size (rs = 147 ± 2 Mpc) at a known redshift (zdec = 1089 ± 1). From these, WMAP measures
the age of the universe (t0 = 13.7 ± 0.2 Gyr) to an accuracy of ∼ 1% by determining the CMB light travel
time over the distance determined by the decoupling surface (dA = 14.0+0.2 −0.3 Gpc) and the geometry of the
universe (i.e., flat). The age of the universe is also estimated via stars in three ways:

(i) the main sequence turn-off in globular clusters yielding a cluster age of 12 ± 1 Gyr (Reid 1997);

(ii) the temperature of the coldest white dwarfs in globular clusters yielding a cluster age of 12.7 ± 0.7 Gyr
(Hansen et al. 2002), and

(iii) nucleosynthesis age dating yielding an age of 15.6 ± 4.6 Gyr (Cowan et al. 1999).

These stellar ages are all consistent with age of the universe found by WMAP.
MATTER DENSITY (Ωm h2 ): The matter density affects the height and shape of the acoustic peaks.
The baryon-to-matter ratio determines the amplitude of the acoustic wave signal and the matter-to-radiation
ratio determines the epoch, zeq , when the energy density of matter equals the energy density of radiation. The
amplitude of the early integrated Sachs-Wolfe effect signal is sensitive to the matter-radiation equality epoch.
From these effects WMAP measures the matter density, Ωm h2 ,to an accuracy of ∼ 5%. Large scale structure
observations measure Ωm h through the shape of the power spectrum. When combined with estimates of h,
this yields Ωm h2 . Large scale velocity field measurements yield Ω0.6
m b
−1
, where b is the bias in how the galaxy
2
power spectrum traces the matter power spectrum (Pgal = b P (k)). Galaxy bispectrum measurements yield
b, allowing for estimates of Ωm . From the galaxy data, Verde et al. (2002) find Ωm = 0.27 ± 0.06, which is
consistent with the WMAP result of Ωm = 0.27 ± 0.04.
Cluster lensing observations yield measurements of the total mass in the cluster. X-ray measurements
give both the baryonic mass and the total mass through the gravitational potential. Sunyaev-Zeldovich effect
observations give a different determination of the baryonic mass in clusters. The combined X-ray and SZ
+0.009 −1
measurements give a value of Ωb Ω−1m = 0.081−0.011 h (Grego et al. 2001), which is reasonably consistent
with 0.17 ± 0.01 from WMAP.

9.3. Implications for Inflation

WMAP data tests several of the key predictions of the inflationary scenario (see Peiris et al. (2003) for
 – 36 –

further discussion):

(a) Inflation predicts that the universe is flat. As noted in §9.1 and discussed in detail in Spergel et al.
(2003), the combination of WMAP data with either H0 , Type Ia SNe, or large scale structure data
constrains |1 − Ωtot | < 0.03.

(b) Inflation predicts Gaussian random phase fluctuations. Komatsu et al. (2003) shows that the CMB
fluctuations have no detectable skewness and place strong constraints on primordial non-Gaussianity.
Komatsu et al. (2003) also shows that the Minkowski functionals of the WMAP data are consistent
with the predictions of a model with Gaussian random phase fluctuations.

(c) Inflation predicts fluctuations on scales that appear to be superhorizon scales in a Friedman-Robertson-
Walker (FRW) cosmology. The WMAP detection of an anti-correlation between polarization and
temperature fluctuations on scales of ∼ 1◦ − 2◦ (Kogut et al. 2003) confirms this prediction and rules
out subhorizon causal mechanisms for generating CMB fluctuations (Peiris et al. 2003).

(d) Inflation predicts a nearly scale invariant spectrum of fluctuations, as seen by WMAP.

The WMAP data, in combination with complementary cosmological data, not only test the basic ideas
of the inflationary scenario but also rule out broad classes of inflationary models, and therefore the data
guide us towards a specific workable inflationary scenario. The WMAP data place significant constraints on
r, the tensor-to-scalar ratio, ns , the slope of the scalar fluctuations and dns /d ln k, the scale dependence of
these fluctuations. The addition of an admixture of isocurvature modes does not improve the WMAP model
fits.
The best fit model to the combination of the WMAP, ACBAR, CBI, 2dFGRS and the Lyman-α forest
data has a spectral index that runs from n > 1 on the large scales probed by WMAP to n < 1 on the small
scales probed by the 2dFGRS and the Lyman-α forest data. Only a handful of inflationary models predict
this behavior. The Linde & Riotto (1997) hybrid inflationary model is one example. The data, however, do
not yet require n > 1 on large scales: our best fit model has ns = 1.03 ± 0.04 at k = 0.002 Mpc−1 .
Our analysis of inflationary models (Peiris et al. 2003) marks the beginning of precision experimental
tests of specific inflationary models. With the addition of on-going WMAP observations and future improved
analyses, WMAP will be able to more accurately constrain τ and hence ns on large scales. When other CMB
experiments are calibrated directly to the WMAP sky maps, they will provide improved measurements of
the temperature angular power spectrum for l > 700. The upcoming release of the Sloan Digital Sky Survey
(SDSS) power spectrum will provide an improved measurement of the galaxy power spectrum. The SDSS
Lyman-α forest data are expected to be a significant improvement over the data used in our current analysis.
Looking further towards the future, ESA’s Planck mission will provide improved measurements on the CMB
angular power spectrum on smaller angular scales and should be able to improve constraints on r.

10. DATA PRODUCTS

All of the WMAP data will be released. In addition, several ancillary and analyzed data sets are released.
These include beam patterns, angular spectra, etc. Some software tools are also provided. An Explanatory
Supplement provides detailed information about the WMAP in-flight operations and data products (Limon
et al. 2003). All WMAP data products are distributed through the Legacy Archive for Microwave Background
 – 37 –

Data Analysis (LAMBDA) at http://lambda.gsfc.nasa.gov. This is a new NASA data center dedicated to
the rapidly growing field of microwave background data archiving and analysis.

11. SUMMARY AND CONCLUSIONS

(1) WMAP has produced high quality full sky maps in five widely separated frequency bands. These
maps can be used to test cosmological models and serve as the primary legacy of the mission.
(2) We have characterized and placed stringent limits on systematic measurement errors. The calibration
is based on the modulation of the CMB dipole, and is accurate to better than 0.5%.
(3) We have demonstrated the ability to separate the CMB anisotropy from Galactic and extragalactic
foregrounds. We provide masks for this purpose. In addition, we have produced CMB maps where the
Galactic signal is minimized.
(4) We have a new determination of the dipole. It is 3.346 ± 0.017 mK in the direction (l, b)= (263.◦ 85 ±
0.◦ 1, 48.◦ 25 ± 0.◦ 04).
(5) We have a new determination of the quadrupole amplitude. It is Qrms = 8±2 µK or ∆T22 = 154±70
2
µK .
(6) We have placed tight new limits on non-Gaussianity of the CMB anisotropy. The coupling coefficient
of a quadratic non-Gaussian term is limited to −58 < fN L < 134 (95% CL) (Komatsu et al. 2003).
(7) We have produced an angular power spectrum of the anisotropy with unprecedented accuracy and
precision. The power spectrum is cosmic variance limited for l < 354 with a signal-to-noise ratio >1 per
mode to l = 658.
(8) We have, for the first time, observed the angular power spectrum of TE temperature-polarization
correlations with sufficient accuracy and precision to place meaningful limits on cosmology.
(9) We have detected the epoch of reionization with an optical depth of τ = 0.17 ± 0.04. This implies a
reionization epoch of tr = 180+220 +10
−80 Myr (95% CL) after the Big Bang at a redshift of zr = 20−9 (95% CL)
for a range of ionization scenarios. This early reionization is incompatible with the presence of a significant
warm dark matter density.
(10) We have fit cosmological parameters to the data. We find results that are consistent with the
Big Bang theory and inflation. We find that the addition of a running spectral index, while not required,
improves the fit at the ∼ 2σ level. We provide values and uncertainties for a host of parameters based on
this non-power-law inflationary model. Our “best” values for cosmic parameters are given in Table 3.
(11) WMAP continues to collect data and is currently approved for 4 years of operations at L2 . The
additional data, and more elaborate analyses, will help to further constrain models. The addition of other
continuously improving CMB and large scale structure observations is essential for progress towards the
ultimate goal of a complete understanding of the global properties of the universe.

The WMAP mission is made possible by the support of the Office of Space Sciences at NASA Head-
quarters and by the hard and capable work of scores of scientists, engineers, technicians, machinists, data
analysts, budget analysts, managers, administrative staff, and reviewers. We are grateful to the National
Radio Astronomy Observatory, which designed and produced the HEMT amplifiers that made WMAP pos-
 – 38 –

sible. We are grateful to A. Riess for providing the likelihood surfaces for the supernova data. D. Finkbeiner
supplied us with his full sky composite map of Hα emission in advance of publication. LV is supported by
NASA through a Chandra Fellowship issued by the Chandra X-ray Observatory Center, operated by the
Smithsonian Astrophysical Observatory. ML and GT are supported by the National Research Council.

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