Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Foreground Emission
ABSTRACT We present a new estimate of foreground emission in the WMAP data, using a Markov chain Monte Carlo (MCMC) method. The new technique delivers maps of each foreground component for a variety of foreground models, error estimates of the uncertainty of each foreground component, and provides an overall goodness-of-fit measurement. The resulting foreground maps are in broad agreement with those from previous techniques used both within the collaboration and by other authors. We find that for WMAP data, a simple model with power-law synchrotron, free-free, and thermal dust components fits 90% of the sky with a reduced chi-squared of 1.14. However, the model does not work well inside the Galactic plane. The addition of either synchrotron steepening or a modified spinning dust model improves the fit. This component may account for up to 14% of the total flux at Ka-band (33 GHz). We find no evidence for foreground contamination of the CMB temperature map in the 85% of the sky used for cosmological analysis. Comment: accepted by ApJS, 49 pages, 4 tables, 21 figures. PS and PDF versions with high-resolution figures available at http://lambda.gsfc.nasa.gov/product/map/dr3/map_bibliography.cfm
[show abstract] [hide abstract]
ABSTRACT: The C-Band All-Sky Survey (C-BASS) aims to produce sensitive, all-sky maps of diffuse Galactic emission at 5 GHz in total intensity and linear polarization. These maps will be used (with other surveys) to separate the several astrophysical components contributing to microwave emission, and in particular will allow an accurate map of synchrotron emission to be produced for the subtraction of foregrounds from measurements of the polarized Cosmic Microwave Background. We describe the design of the analog instrument, the optics of our 6.1 m dish at the Owens Valley Radio Observatory, the status of observations, and first-look data. Comment: 10 pages, 11 figures, published in Proceedings of SPIE MIllimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy V (2010), Vol. 7741, 77411I-1 - 77411I-1008/2010;
arXiv:0803.0715v2 [astro-ph] 21 Oct 2008
Revised version, accepted by ApJS
Five-Year Wilkinson Microwave Anisotropy Probe (WMAP1)
Galactic Foreground Emission
B. Gold2, C. L. Bennett2, R. S. Hill3, G. Hinshaw4, N. Odegard3, L. Page5, D. N.
Spergel6,7, J. L. Weiland3, J. Dunkley5,7,8, M. Halpern9, N. Jarosik5, A. Kogut4, E.
Komatsu10, D. Larson2, S. S. Meyer11, M. R. Nolta12, E. Wollack4, and E. L. Wright13
We present a new estimate of foreground emission in the WMAP data, using
a Markov chain Monte Carlo (MCMC) method. The new technique delivers maps
of each foreground component for a variety of foreground models with estimates
1WMAP 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.
2Dept. of Physics & Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD
3Adnet Systems, Inc., 7515 Mission Dr., Suite A1C1 Lanham, Maryland 20706
4Code 665, NASA/Goddard Space Flight Center, Greenbelt, MD 20771
5Dept. of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544-0708
6Dept. of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544-1001
7Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544
8Astrophysics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK
9Dept. of Physics and Astronomy, University of British Columbia, Vancouver, BC Canada V6T 1Z1
10Univ. of Texas, Austin, Dept. of Astronomy, 2511 Speedway, RLM 15.306, Austin, TX 78712
11Depts. of Astrophysics and Physics, KICP and EFI, University of Chicago, Chicago, IL 60637
12Canadian Institute for Theoretical Astrophysics, 60 St. George St, University of Toronto, Toronto, ON
Canada M5S 3H8
13PAB 3-909, UCLA Physics & Astronomy, PO Box 951547, Los Angeles, CA 90095–1547
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of the uncertainty of each foreground component, and it provides an overall
goodness-of-fit estimate. The resulting foreground maps are in broad agreement
with those from previous techniques used both within the collaboration and by
We find that for WMAP data, a simple model with power-law synchrotron,
free-free, and thermal dust components fits 90% of the sky with a reduced χ2
of 1.14. However, the model does not work well inside the Galactic plane. The
addition of either synchrotron steepening or a modified spinning dust model
improves the fit. This component may account for up to 14% of the total flux
at Ka-band (33 GHz). We find no evidence for foreground contamination of the
CMB temperature map in the 85% of the sky used for cosmological analysis.
Subject headings: cosmic microwave background — cosmology: observations —
diffuse radiation — Galaxy: halo — Galaxy: structure — ISM: structure
The Wilkinson Microwave Anisotropy Probe (WMAP) produces temperature and linear
polarization radio maps at five frequencies with 1◦or better resolution and tightly constrained
systematic errors. The frequency bands are centered on 22, 33, 41, 61, and 94 GHz; denoted
K, Ka, Q, V, and W, respectively (see Page et al. 2003 for details). While designed to
measure the cosmic microwave background (CMB) radiation it also observes the large-scale
structure of our Galaxy at angular scales and frequencies that are relatively unexplored.
Study of our own Galaxy has had a significant effect on our understanding of galaxies in
Radio emission from galaxies is generally understood as arising from three effects: “non-
thermal” synchrotron emission from relativistic electrons spiraling in large-scale magnetic
fields, “thermal” free-free emission from non-relativistic electron-ion interactions, and emis-
sion from vibrational modes of thermal dust grains. At lower radio frequencies the syn-
chrotron emission is usually dominant, with flux decreasing at higher frequencies approxi-
mately according to a power law14(β ≈ −3). Free-free emission has a flux that is nearly
constant with frequency (β ≈ −2.1), so free-free emission becomes relatively more important
14In this paper we use the notation that flux density is S ∼ ναand antenna temperature is T ∼ νβ,
with the spectral indices related by β = α − 2. Unless otherwise noted, results will be expressed in antenna
temperature. For the pixel size most commonly used in this work (0.◦92×0.◦92), the conversion from antenna
temperature to flux is approximately 4.0(ν/22.5 GHz)2Jy mK−1(Page et al. 2003).
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than synchrotron at higher frequencies. Typically the crossover frequency is near 60 GHz
at higher latitudes, but can be 20 GHz or lower in specific regions in the Galactic plane.
Frequencies above ∼60 GHz begin to probe the tail (β ≈ 2) of vibrational dust emission,
which is dominant around 90 GHz. In addition to these three foregrounds, much recent
work has focused on the possibility of significant emission from rapidly rotating dust grains;
this emission is thought to peak somewhere in the 10–30 GHz range and fall off roughly
exponentially at higher frequencies.
The spectral behavior for diffuse foregrounds is of great interest. The spectrum for syn-
chrotron radiation follows the energy distribution for high-energy electrons, which is not a
pure power law. The highest energy electrons lose energy more quickly and thus are reduced
in regions where they have not been replenished. Such energy loss shows up as a gradual
steepening (dβ/dν < 0) in the power law index by about 0.5 at frequencies above 10–100
MHz. Further, while the overall index as extrapolated from lower frequencies is β ≈ −2.7
(Reich & Reich 1988; Lawson et al. 1987; Reich et al. 2004), higher frequencies may prefer-
entially sample more energetic electron populations and thus have a flatter index (β ≈ −2.5)
(Bennett et al. 2003). Observations of both discrete sources (Green 1988; Green & Scheuer
1992) and external galaxies (Hummel et al. 1991) show a wide variety of synchrotron be-
havior. Free-free emission also does not follow a strict power law, but the physics is well
understood and the variation of the power-law index over WMAP’s bands is so small that it
can be neglected. Finally, the Rayleigh-Jeans tail for vibrational dust emission (i.e. below
∼100 GHz) has never before been accurately measured and the relevant material proper-
ties of the dust grains themselves are not fully understood (Agladze et al. 1994; Meny et al.
The main focus in this work is on foreground emission. Section 2 describes updates to
masks and foreground-fitting procedures used in previous WMAP analyses (Bennett et al.
2003; Hinshaw et al. 2007). A new method to explicitly marginalize over foregrounds for
the low multipole analysis is described in a companion paper (Dunkley et al. 2008). A new
fitting process is described in Section 3, which has the following features:
• The fitting is entirely in real-space with no spherical harmonic decomposition for any
• The spectral indices of the synchrotron and dust emission are not generally assumed
to be constant and are allowed to vary down to the scale of the fit (approximately one
• The fit includes the CMB and automatically generates the full likelihood (including
covariance) for all foreground parameters.
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• The polarization data are included and fit simultaneously with the total intensity data.
This is similar to the technique of Eriksen et al. (2007), however we fit the CMB in pixel-
space, use less smoothing on the maps, and attempt to obtain more information about
The results of the fit are described in Section 4. While the fitting technique used here
delivers a CMB map with error-bars, the map itself has not proved to be any better for
cosmological analysis and so far has been used only as a check. Implications of the fit are
discussed in Section 5. WMAP’s cosmological results do not depend on the fitting process
2. Five-year Foreground Fits
The diffuse foreground masks are updated for the five-year data analysis. The primary
reason is to mask out free-free emission in the areas of the Gum Nebula and ρ Oph, while
keeping a simple method that applies to the whole sky rather than being ad hoc for these
The new masks are based on three-year public WMAP data products15, specifically the
three-year K and Q band-average maps smoothed to one-degree resolution. These maps are
converted to foreground-only maps by subtracting the three-year Internal Linear Combina-
tion (ILC) map. A cumulative histogram is made of the pixels in each foreground map, which
serves as a lookup table to find a flux level used to define a cut over the desired percentage
of the sky.
Cuts are made at intervals of 5% in the proportion of sky admitted by the resulting
mask. The K and Q band cuts at each percentage level are combined. Resulting masks
are inspected and compared with the masks used in the one and three-year WMAP data
analyses. We replace the old Kp2 mask with the combined K and Q 85% masks. This is the
nominal mask for temperature data analysis and is denoted KQ85. We replace the old Kp0
mask with the combined K and Q 75% masks (KQ75).
15The new masks were based on three-year data because they were needed before the five-year maps could
be finalized. The masks are made from flux cuts at high signal-to-noise on smoothed maps, thus the difference
between basing the masks on three-year versus five-year data is minimal. This was verified explicitly once
the five-year maps were finalized.
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Each of the chosen masks is further processed by omitting any masked “islands” con-
taining fewer than 500 pixels at HEALPix (Gorski et al. 2005) Nsideof 512. Each mask is
then combined with a point source mask, which has been updated from that described in
Bennett et al. (2003) and Hinshaw et al. (2007) to include 32 newly detected sources from
a preliminary version of the WMAP five-year point source catalog. Six sources in the final
five-year catalog are not included; these are relatively weak, with fluxes of 1 Jy or lower in
all WMAP bands. The last step combines each mask with the five-year processing cut used
to omit the Galactic plane from the mapmaking. A comparison of old and new masks is
shown in Figure 1.
The three-year polarization mask was based on a cut in K-band polarized intensity
combined with a model of the dust component (Page et al. 2007). The five-year polarization
analysis mask is the same as the three-year version, with the exception that it is combined
with the five-year processing cut.
The MCMC fit described below uses a version of the combined K and Q 95% mask
(denoted KQ95, and which is similar to the old Kp12 mask) to distinguish “inside” from
“outside” the Galactic plane. The mask was enlarged to account for smoothing, leaving
approximately 91% of the sky.
2.2.Internal Linear Combination Method
The Internal Linear Combination (ILC) method is used to produce a CMB map that
is independent of both external data and assumptions about foreground emission. By con-
struction, it leaves unchanged the component that has the spectrum of the CMB and acts as
a foreground fit by filtering out the combined spectral shape that causes the most variance in
the data. As a minimum variance method the ILC is guaranteed to produce a map with good
statistical properties, but the level of remaining contamination can be difficult to assess.
The algorithm used to compute the WMAP five-year Internal Linear Combination map
is the same as that described in the three-year analysis (Hinshaw et al. 2007). We retain
the same number of regional subdivisions of the sky and their spatial boundaries remain
unchanged from the previous definitions. The frequency weights for each region are somewhat
different, however, reflecting the five-year updates to the calibration and beams. The new
ILC regional coefficients are presented in Table 1, and the map itself is available on the
LAMBDA web site16. The coefficients describe a filter that nulls certain spectral shapes. A
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Fig. 1.— Comparison maps of the five-year masks versus the three-year masks. The new
masks cover slightly more of the Galactic plane and cover more regions with low synchrotron
but high free-free emission. The diamond-shaped features arise because the new processing
mask has been defined to correspond to low-resolution (Nside=16) pixels, so that the same
processing mask can be used at all resolutions. Top: comparison of KQ85 with the three-
year Kp2 mask. Middle: comparison of KQ75 with the three-year Kp0 mask. Bottom:
comparison of KQ95 with the three-year Kp12 mask.
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Table 1. ILC coefficients per regiona
aThe ILC temperature (in thermodynamic units) at pixel
p of region n is Tn(p) =?5
ficients above and the sum is over WMAP’s frequency bands.
i=1ζn,iTi(p), where ζ are the coef-
– 8 –
slice in parameter space of the spectra nulled by the ILC is shown in Figure 2. Differences
between new CMB maps and those from the three-year release are further discussed in
2.3.Maximum Entropy Method
The maximum entropy method (MEM) is a spatial and spectral fit using templates that
are intended to distinguish different low-frequency emission sources. By design, the MEM
reverts to templates made from external data where WMAP’s signal is low. One of the
main goals for the MEM was to use high-signal regions to investigate the spectral properties
of the foregrounds. The error properties for MEM maps are complicated and the model is
essentially under-constrained so there is no meaningful goodness-of-fit statistic. The MEM
maps were not used for analysis of the CMB itself.
The five-year MEM analysis is largely unchanged from the three-year analysis (Hinshaw et al.
2007). As before, the analysis is done on sky maps smoothed to a common resolution of 1◦
full width at half maximum in all bands. To improve the signal-to-noise ratio, we now use
maps degraded to HEALPix Nside= 128 pixelization instead of Nside= 256 (the pixel size
for the former is 0.46◦). In the first year and three-year analyses, the logarithmic term that
forces the solution to converge to the priors for low S/N pixels was missing a factor of e
(Cornwell et al. 1999); this has been fixed. The model is fit for each pixel p by minimizing
the functional H = A + λB (Press et al. 1992), where A is the standard χ2of the model
fit, and we now use B =?
emission component c (synchrotron, free-free, dust) in pixel p, and Pc(p) is the prior esti-
mate of Tc(p). The parameter λ controls the relative weight of A (the data) and B (the
prior information) in the fit. An iterative procedure is followed that uses residuals from the
fit at each iteration to adjust the spectrum of the synchrotron component for each pixel.
The MEM procedure was run for 11 iterations before stopping, the same as in the three-year
cTc(p) ln[Tc(p)/(ePc(p))]. Here Tc(p) is the model brightness of
The dust and free-free spectrum coefficients are required to follow power-laws, with
β = +2 for dust and β = −2.14 for free-free. Hence any “anomalous” component, such as
electric dipole emission from spinning dust, will be included in the synchrotron component.
The priors used are also unchanged, using the Haslam 408 MHz map (Lawson et al. 1987)
for the synchrotron map, extinction-corrected Hα (Finkbeiner 2003) for the free-free map,
and Model 8 of Finkbeiner et al. (1999) for the dust map. The MEM maps are available for
public download on the LAMBDA web site. Figure 3 shows a comparison of the five-year
and three-year MEM foregrounds, and the spectrum of components compared to the total
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Fig. 2.— A slice in parameter space of the surface nulled by the ILC coefficients, assuming
a three-component foreground model with power-law spectral behavior, T(ν) = Tsνβs+
Tfν2.14+ Tdνβd. Each line is for a single ILC region, denoted by number. The parameter
space is Tf/Ts, Td/Ts, βs, βd. For this plot the x-axis is βsand the y-axis is Td/Ts. The
parameters Tf/Ts and βd are fixed at 0.7 and 1.8, respectively. Each color is a different
ILC region. Despite the variety amongst ILC coefficients, they often null similar regions of
– 10 –
observed foreground spectrum for 20◦< |b| < 30◦.
Comparison of MEM results from the five-year and three-year analyses shows an increase
in the model brightness of all foreground components at high Galactic latitudes. The changes
are mostly due to differences in the zero levels of the five-year and three-year maps. The
inclusion of the factor of e in the MEM functional also leads to a small contribution. The
method of setting map zero levels has not changed since the first year analysis. The internal
linear combination CMB map is subtracted from the 1◦smoothed map in each frequency
band, and the zero level is set such that a fit to the residual map of the form T(|b|) =
Tpcsc|b| + c, over the range −90◦< b < −15◦, yields c = 0 (Bennett et al. 2003). The
three-year analysis procedure was done using a preliminary three-year ILC map in which the
monopole was nonzero. Offsets of 21.1, 19.4, 19.3, 19.4, and 19.6 µK should be added to the
three-year K, Ka, Q, V, and W band maps, respectively, to give maps that yield csc|b| fit
intercepts of zero when the final three-year ILC map is subtracted.
Available foreground templates are expected to trace the distribution of foreground
emission more reliably than a csc|b| model, so template fitting has been done to check the
zero levels of the five-year maps. Because the MEM is itself a template fit, this is essentially
equivalent to fitting for the zero levels within the MEM procedure. The five-year ILC map
was subtracted from the five-year 1◦smoothed maps, and the residual map for each band was
fit to a linear combination of synchrotron, free-free, and dust templates plus a constant offset.
Uncertainties in the zero levels of the templates were propagated to obtain an uncertainty in
the derived offset value. For the synchrotron template, the 408 MHz map of Haslam et al.
(1982) was used with an offset of 5.9 K subtracted (Lawson et al. 1987). The quoted zero
level uncertainty of this map is ±3 K (Haslam et al. 1982). For the free-free template, the
composite all-sky Hα map of Finkbeiner (2003) was used, with a correction for extinction
(using the dust extinction map of Schlegel et al. 1998) assuming the dust is coextensive
with the emitting gas along each line of sight (Bennett et al. 2003). The adopted zero level
uncertainty is ±1 Rayleigh, as estimated by Finkbeiner for the southern Hα data. For the
dust template, the 94 GHz emission predicted by model 8 of Finkbeiner et al. (1999) was
used. The adopted zero level uncertainty is ±0.2µK, propagated from a zero level uncertainty
of ±0.044 MJy sr−1for the 100 µm dust map of Schlegel et al. (1998).
Fits were done to Nside= 512 pixels that are outside of the combined KQ85 plus point
source mask and have optical depth at Hα less than 0.5, based on the Schlegel et al. (1998)
extinction map. This pixel selection covers 74% of the sky. The offsets from the fits are
−25 ± 19, −5.4 ± 6.8, −2.2 ± 3.9, −2.2 ± 1.5, and −1.5 ± 0.7µK in K, Ka, Q, V, and W
bands, respectively. Thus there is no evidence for significant error in the five-year map zero
levels as determined from the csc|b| fitting. For comparison, northern hemisphere csc|b| fits
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can be used to estimate uncertainties in the zero levels; the northern hemisphere gives offsets
of −9.2, 3.2, 3.5, −2.5, −5.9 µK for K, Ka, Q, V, and W bands, respectively, relative to the
zero levels calculated from the southern hemisphere.
The foreground template subtraction technique used in the five-year analysis is un-
changed from that used in the three-year release. The method is described in Hinshaw et al.
(2007) for temperature cleaning and Page et al. (2007) for polarization cleaning; details are
not repeated here.
In summary, for temperature cleaning a model of the foreground emission is computed
from a simultaneous fit to the five-year Q, V and W-band maps, and that model is then
used to produce foreground-reduced maps suitable for cosmological studies. WMAP has two
differencing assemblies (DAs) for Q and V-bands (labelled Q1, Q2, V1, and V2) and four
for W-band (labelled W1 through W4), for a total of eight maps with independent noise
The model takes the form
M(ν,p) = b1(ν)(TK(p) − TKa(p)) + b2(ν)IHα(p) + b3(ν)Mdust(p) (1)
where p indicates the pixel, the frequency dependence is entirely contained in the coefficients
bi, and the spatial templates are the WMAP K-Ka temperature difference map (TK−TKa),
the Finkbeiner (2003) composite Hα map with an extinction correction applied (IHα), and
the Finkbeiner et al. (1999) dust model evaluated at 94 GHz (Mdust). All of these spatial
templates are available on LAMBDA.
The Hα map and dust template are based on external data and have not changed since
the three-year analysis. The first template, however, has changed slightly (at the ∼ 10 µK
level) due primarily to changes in the gain calibration since the three-year release, see Figure
5 of Hinshaw et al. (2008) for details. Because this template has contributions from both
synchrotron and free-free emission, foreground parameters are a mixture of b1(ν) and b2(ν).
For free-free emission, the ratio of K-band radio temperature to Hα intensity is
Sff(ν) − 0.552b1(ν)
where Sff(ν) is the free-free emission spectrum converted to thermodynamic temperature
units and is assumed to be a power-law with β = −2.14. The synchrotron spectral index
– 12 –
Fig. 3.— Comparison of MEM foreground modeling results from the WMAP three-year and
five-year analyses. The first three panels show latitude profiles of antenna temperature for the
individual foreground model components. The last panel compares the observed foreground
emission spectrum (diamonds) with spectra of the total MEM model and the individual
model components (line segments between WMAP frequencies), averaged over 20◦< |b| <
30◦. The differences between the three-year and five-year model results are mainly due to
differences in zero levels between the three-year and five-year maps, and are consistent with
the three-year year estimated error of ∼ 4µK . The mean model brightness exceeds the mean
observed brightness at the higher frequencies because the observed brightness is negative for
some pixels and the model is constrained to be positive for each pixel. This is less apparent
in the five-year results because there are fewer negative pixels.
– 13 –
(relative to K-band) is found via
where a(ν) is the conversion factor from antenna temperature to thermodynamic units.
The coefficients of the model fit to the five-year data are presented in Table 2. Small
changes in the five-year coefficients compared to the three-year values (Table 5 of Hinshaw et al.
2007) reflect the five-year updates to absolute calibration and beam profiles. The new tem-
plate maps are shown in Figure 4.
For polarization cleaning the maps are degraded to low resolution (Nside= 16). The
model has the form
[Q(ν,p),U(ν,p)]model= a1(ν)[Q(p),U(p)]K+ a2(ν)[Q(p),U(p)]dust
The templates used are the WMAP K-band polarization for synchrotron ([Q,U]K), and
a low resolution version of the dust template used above with polarization direction de-
rived from starlight measurements ([Q,U]dust). While the dust polarization template maps
are unchanged since the three-year release, further WMAP observations have improved the
signal-to-noise ratio for synchrotron polarization template maps. The coefficients of the
model fit to the five-year data are in Table 3. For polarization, the template maps are as-
sumed to have a one-to-one correspondence to foreground emission, so the spectral indices
for synchrotron and dust are simply the power-law slopes of the coefficients a1(ν) and a2(ν).
As was the case for the three-year data, a fit fixing the synchrotron spectral index was found
to have no influence on cosmological conclusions and was not used for analysis.
3.Markov Chain Monte Carlo Fitting
The analysis is carried out with band-averaged maps at each frequency, which are cal-
ibrated in antenna temperature, smoothed to a one-degree Gaussian beam, and pixelized
using an Nside = 64 HEALPix grid. This makes the fit computationally manageable and
ensures that pixel-pixel correlations are small, simplifying the error description. The maps
use the csc|b| fit process described above to determine the zero-point.
Next we parameterize the emission in each pixel with a physical model. The model
depends on the parameters in a non-linear way and the parameters can be highly correlated.
A Monte Carlo chain is run for each pixel to determine the probability distribution for the
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Fig. 4.— Five-year temperature maps with foregrounds reduced via template cleaning. All
maps have had the five-year ILC estimate for the CMB subtracted, and have been degraded
to Nside= 32. Frequency bands shown are Q, V, and W. Compare to Figure 10 of Hinshaw
et al. (2007). Outside the Galactic mask, the template cleaning reduces foregrounds to
∼ 15µK or less.
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Table 2. Template cleaning temperature coefficients
aWMAP has two differencing assemblies (DAs) for Q and
V-bands and four for W-band; the high signal-to-noise in
total intensity allows each DA to be fitted independently.
bPower law slope relative to K-band, as derived from b1;
W-band values are less than -4.
cFree-free to Hα ratio at K-band, as derived from b1and
b2. The expected value for an electron temperature of 8000
K is 11.4 µK R−1(Bennett et al. 2003).
Table 3. Template cleaning polarization coefficients
aThe aicoefficients are dimensionless and pro-
duce model maps from templates.
bThe spectral indices refer to antenna temper-