Five-Year Wilkinson Microwave Anisotropy Probe Observations: Galactic Foreground Emission

The Astrophysical Journal Supplement Series (Impact Factor: 11.22). 02/2009; 180(2):265. DOI: 10.1088/0067-0049/180/2/265
Source: arXiv
ABSTRACT
We present a new estimate of foreground emission in the Wilkinson Microwave Anisotropy Probe (WMAP) data, using a Markov chain Monte Carlo method. The new technique delivers maps of each foreground component for a variety of foreground models with estimates 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 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 χ2 ν 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 the Ka band (33 GHz). We find no evidence for foreground contamination of the cosmic microwave background temperature map in the 85% of the sky used for cosmological analysis.

Full-text

Available from: Edward J. Wollack
arXiv:0803.0715v2 [astro-ph] 21 Oct 2008
Revised version, accepted by ApJS
Five-Year Wilkinson Microwave Anisotropy Probe (WMAP
1
)
Observations:
Galactic Foreground Emission
B. Gold
2
, C. L. Bennett
2
, R. S. Hill
3
, G. Hinshaw
4
, N. Odegard
3
, L. Page
5
, D. N.
Spergel
6,7
, J. L. Weiland
3
, J. D unkley
5,7,8
, M. Halpern
9
, N. Jarosik
5
, A. Kogut
4
, E.
Komatsu
10
, D. Larson
2
, S. S. Meyer
11
, M. R. Nolta
12
, E. Wollack
4
, and E. L. Wright
13
bgold@pha.jhu.edu
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 with estimates
1
WMAP is the result of a par tnership between Princeton University and NASA’s Goddard Space Flight
Center. Scientific guidance is provided by the WMAP Science Team.
2
Dept. of Physics & Astronomy, The Johns Ho pkins University, 3400 N. Charles St., Baltimore, MD
21218-2686
3
Adnet Systems, Inc., 7 515 Mission Dr., Suite A1C1 Lanham, Maryland 20706
4
Code 665, NASA/Godda rd Space Flight Center, Greenbelt, MD 20771
5
Dept. of Physics, Jadwin Hall, Princeton Univers ity, Princeton, NJ 08544-0708
6
Dept. of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544-1001
7
Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544
8
Astrophysics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK
9
Dept. of Physics and Astronomy, University of British Columbia, Vancouver, BC Canada V6T 1Z1
10
Univ. of Texas, Austin, Dept. of Astronomy, 2511 Speedway, RLM 15.3 06, Austin, TX 78712
11
Depts. of Astrophysics and Physics, KICP and EFI, University of Chicago, Chicago, IL 60637
12
Canadian Institute for Theoretical Astr ophysics, 60 St. George St, University of Toronto, Toronto, ON
Canada M5S 3H8
13
PAB 3-909, UCLA P hysics & Astronomy, PO Box 951547, Los Angeles, CA 90095–1547
Page 1
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of the uncertainty of each foreground component, and it provides an overall
goodness-of-fit estimate. The resulting foregro und maps are in broa d agreement
with t hose 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, a nd thermal dust components fits 90% of t he sky with a reduced χ
2
ν
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 f or 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
1. Introduction
The Wilkinson Microwave Anisotropy Probe (WMAP) produces temperature a nd 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
general.
Radio emission from galaxies is generally understood a s 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 g r ains. At lower radio frequencies t he syn-
chrotron emission is usually dominant, with flux decreasing at higher frequencies approxi-
mately according to a power law
14
(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
14
In this paper we use the notation that flux density is S ν
α
and antenna temperature is T ν
β
,
with the spectral indices related by β = α 2. Unless otherwise noted, results will be expressed in antenna
temper ature. Fo r the pixel size most commonly used in this work (0.
92 × 0.
92), the conversion from antenna
temper ature to flux is approximately 4.0(ν/22.5 GHz)
2
Jy mK
1
(Page et al. 2003).
Page 2
3
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 fo llows 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 (/dν < 0) in the power law index by about 0.5 at frequencies above 1 0–100
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. 200 3). 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.
2007).
The main focus in this work is on foreground emission. Section 2 describes updat es 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 fo r
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
component.
The spectral indices of the synchrotron and dust emission a re not generally assumed
to be constant and are allowed to vary down to the scale of the fit (approximately one
square degree).
The fit includes the CMB a nd automatically generates the full likelihood (including
cova r ia nce) for all foreground parameters.
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4
The polarization data are included and fit simultaneously with the total intensity data.
This is similar to the technique of Eriksen et a l. (2007), however we fit the CMB in pixel-
space, use less smoothing on the maps, and attempt to obtain more information a bout
individual foregrounds.
The results of the fit are describ ed 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
used here.
2. Five-year Foreground Fits
2.1. Masks
The diffuse foreground masks are updated for the five-year data a nalysis. 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 ra ther than being ad hoc for these
regions.
The new masks are ba sed on three-year public WMAP data products
15
, specifically the
three-year K and Q band-average maps smoothed to one-degree resolution. These maps are
converted to f oreground-only maps by subtracting the three-year Internal Linear Combina-
tion (ILC) map. A cumulative histogram is made of the pixels in each for egro und 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 interva ls 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 o ne 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 a nd is denoted KQ85. We replace the old K p0
mask with the combined K and Q 75% masks (KQ75).
15
The new masks were based on three-year data bec ause they were needed before the five-year maps could
be finalized. T he masks are made from flux cuts at high signal-to-noise on smoothed maps, thus the difference
between bas ing the masks on three-year versus five-year data is minimal. This was verified explicitly once
the five-year maps were finalized.
Page 4
5
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) N
side
of 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 cata lo g. 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 o mit the Galactic plane f r om the mapmaking. A comparison of old and new masks is
shown in Figure 1.
The three-year polar ization 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) t o distinguish “ inside” from
“outside” the Galactic plane. The mask wa s 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 t he 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 site
16
. The coefficients describe a filter that nulls certain spectral shapes. A
16
http://lambda.gsfc.nasa.gov/
Page 5
6
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 r egions 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 (N
side
=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.
Page 6
7
Table 1. ILC coefficients per region
a
Region K- band Ka-band Q-band V-band W-band
0 0.1336 -0.6457 -0.3768 2.2940 -0.4051
1 -0.0610 -0.1327 -0.1873 1.7691 -0.3880
2 0.0037 -0.2432 -0.3792 1.7956 -0.1768
3 -0.1104 0.2395 -0.6424 1.5032 0.0101
4 -0.0843 0.1271 -0.4584 0.9739 0.4417
5 0.1918 -0.7238 -0.4902 2.4844 -0.4622
6 -0.1052 0.2614 -0.6223 1.0253 0.4407
7 0.0913 -0.3849 -0.6033 2.3288 -0.4319
8 0.2208 -0.5436 -1.0938 3.2084 -0.7918
9 -0.0922 -0.0695 -0.1810 1.2619 0.0808
10 0.1724 -0.9608 0.0350 2.6456 -0.8923
11 0.2374 -0.8975 -0.4897 2.7246 -0.5747
a
The ILC temperature (in thermodynamic units) at pixel
p of region n is T
n
(p) =
P
5
i=1
ζ
n,i
T
i
(p), where ζ are the coef-
ficients above and the sum is over WMAP’s frequency bands.
Page 7
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
Section 5.
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 t o 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 a na lysis 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 N
side
= 128 pixelization instead of N
side
= 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. 19 99); this has been fixed. The model is fit for each pixel p by minimizing
the functional H = A + λB (Press et a l. 1992), where A is the standard χ
2
of the model
fit, and we now use B =
P
c
T
c
(p) ln[T
c
(p)/(eP
c
(p))]. Here T
c
(p) is the model brightness of
emission component c (synchrotron, free-free, dust) in pixel p, and P
c
(p) is the prior esti-
mate of T
c
(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
analysis.
The dust and free-free spectrum coefficients are required to follow p ower-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) f or 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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9
Fig. 2.— A slice in para meter space of the surface nulled by the ILC coefficients, assuming
a three-component foreground model with power-law spectral behavior, T (ν) = T
s
ν
β
s
+
T
f
ν
2.14
+ T
d
ν
β
d
. Each line is for a single ILC region, denoted by number. The parameter
space is T
f
/T
s
, T
d
/T
s
, β
s
, β
d
. For this plot the x-axis is β
s
and the y-axis is T
d
/T
s
. The
parameters T
f
/T
s
and β
d
are fixed at 0.7 and 1.8, respectively. Each color is a different
ILC region. Despite the variety amongst ILC coefficients, they of t en null similar regions of
parameter space.
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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 t o 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|) =
T
p
csc |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, 1 9.3, 19.4, and 19.6 µK should be a dded to t he
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, t his 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
1
for the 100 µm dust map of Schlegel et al. (1998).
Fits were done to N
side
= 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. (1 998)
extinction map. This pixel selection covers 74% of the sky. The offsets f rom t he 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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11
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.
2.4. Template Cleaning
The fo reground 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
properties.
The model takes t he form
M(ν, p) = b
1
(ν)(T
K
(p) T
Ka
(p)) + b
2
(ν)I
Hα
(p) + b
3
(ν)M
dust
(p) (1)
where p indicates the pixel, the frequency dependence is entirely contained in the coefficients
b
i
, and the spatial templates are the WMAP K-Ka temperature difference map (T
K
T
Ka
),
the Finkbeiner (2003) composite Hα map with an extinction correction applied (I
Hα
), and
the Finkbeiner et al. (1999) dust model evaluated at 94 GHz (M
dust
). 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 1 0 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 b
1
(ν) and b
2
(ν).
For free-free emission, the ratio of K-band radio temperature to Hα intensity is
h
=
b
2
(ν)
S
(ν) 0.552 b
1
(ν)
(2)
where S
(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
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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 t emperature 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 f or
some pixels and the model is constrained to be positive for each pixel. This is less apparent
in the five-year results because there a r e fewer negative pixels.
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13
(relative to K-band) is found via
β
s
=
log [0.67 b
1
(ν)a(ν)]
log(ν/ν
K
)
(3)
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 ar e degraded to low resolution (N
side
= 16). The
model has the form
[Q(ν, p), U(ν, p)]
model
= a
1
(ν)[Q(p), U(p)]
K
+ a
2
(ν)[Q(p), U(p)]
dust
(4)
The templates used are the WMAP K-ba nd polarization for synchrot r on ([Q, U]
K
), and
a low r esolution version of the dust template used above with polar ization 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 a
1
(ν) and a
2
(ν).
As was the case fo r the three-year data, a fit fixing the synchrotron spectral index was found
to have no influence on cosmological conclusions and was not used f or analysis.
3. Markov Chain Monte Carlo Fitting
3.1. Description
The analysis is carried out with band-averaged maps at each frequency, which are cal-
ibrated in antenna temperature, smoothed t o a one-degree Gaussian beam, and pixelized
using an N
side
= 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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14
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 N
side
= 32. Frequency bands shown are Q, V, and W. Compare to Figure 10 of Hinshaw
et al. (20 07). Outside the Galactic mask, the template cleaning reduces foregrounds to
15µK or less.
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15
Table 2. Template cleaning temperature coefficients
DA
a
b
1
b
2
(µK R
1
) b
3
β
s
b
h
c
(µK R
1
)
Q1 0.245 0.981 0.201 -3.18 5.99
Q2 0.243 1.009 0.199 -3.22 6.01
V1 0.058 0.666 0.461 -3.44 6.38
V2 0.056 0.647 0.477 -3.43 6.38
W1 0.000 0.398 1.262 · · · 6.62
W2 0.000 0.393 1.277 · · · 6.62
W3 0.001 0.398 1.242 · · · 6.61
W4 0.000 0.395 1.271 · · · 6.62
a
WMAP 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.
b
Power law slope relative to K-band, as derived from b
1
;
W-band values are less than -4.
c
Free-free to Hα ratio at K-band, as derived from b
1
and
b
2
. 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
Band a
1
a
β
s
(ν
K
, ν)
b
a
2
a
β
d
(ν, ν
W
)
b
Ka 0 .3 161 -3.17 0.0165 1.35
Q 0.1765 -3.04 0.0147 1.85
V 0.0 595 -2.96 0.0366 1.58
W 0.0450 -2.35 0.0822 · · ·
a
The a
i
coefficients are dimensionless and pro-
duce model maps from templates.
b
The spectral indices refer to antenna temper-
ature.
Page 15
16
parameters of the model using the Markov chain technique (Gilks et al. 19 96). Because of
parameter correlations, the matrix describing the optimal step size is not diagonal. The
starting points and initial step proposal matrices are generated using a “best guess” from
the data. In cases where the initial guess turns out to be poor, the fitting process is retried
using the existing chain to improve the guess. Any retries or poor ly conditioned proposal
matrices are flag ged. Each chain is checked for convergence using the criteria described in
Dunkley et al. (2005), and any lack of convergence is also flagged.
The basic form of the model for each pixel is
T (ν) = T
s
ν
ν
K
β
s
(ν)
+ T
f
ν
ν
K
β
f
+ a(ν)T
cmb
+ T
d
ν
ν
W
β
d
(5)
for the antenna temperature and
Q(ν) = Q
s
ν
ν
K
β
s
(ν)
+ Q
d
ν
ν
W
β
d
+ a(ν)Q
cmb
(6)
U(ν) = U
s
ν
ν
K
β
s
(ν)
+ U
d
ν
ν
W
β
d
+ a(ν)U
cmb
(7)
for Stokes Q and U parameters. The subscripts s, f, d stand for synchrotron, free-free, a nd
dust emission, ν
K
and ν
W
are the effective frequencies for K and W bands (22.5 a nd 93.5
GHz), and a(ν) accounts for the slight frequency dependence of a 2.725 K blackbody using the
thermodynamic to antenna temperature conversion factors found in Bennett et al. (2003).
For each pixel, the χ
2
of the fit is then calculated in the standard way
χ
2
=
X
ν
D
T
ν
N
1
ν
D
ν
(8)
where D
ν
is the difference between the data vector (T , Q, U) and the model vector at each
frequency. The matrix N
ν
is the no ise covariance matrix, and is derived directly from the
N
obs
maps (rebinned from an N
side
of 512 to 64), with minor modifications discussed below.
Not all parameters in the model are free to vary; a(ν) and β
f
are fixed by known
physics, the Q a nd U parameters for f oregrounds are related using K band as a template
for the polarization angle, and for most of the following β
s
(ν) is assumed to be constant
with f r equency ( tho ugh allowed to vary spatially). The free-free index was fixed at β
f
=
2.14; typical variation in this value at WMAP frequencies is ±0.015 (Oster 1961, see also
Quireza et al. 2006; Itoh et al. 2000 for recent refinements) which is too small for WMAP
to detect. Similarly, results of the fit were not found to depend strongly on whether U
s
and
U
d
were treated as freely independent parameters.
Page 16
17
The “base” fit which allows for spatially varying synchrotron and dust spectral indices
has 10 independent parameters per pixel: T
s
, T
f
, T
d
, T
cmb
, β
s
, β
d
, Q
s
, Q
d
, Q
cmb
, and U
cmb
.
More restricted fits fixed β
s
, β
d
, or both. Other fits allowed for a frequency-dependent β
s
by
defining
β
s
(ν) = β
s
+ c ln(ν
K
) (9)
where the new parameter c can introduce a gradual steepening (or shallowing).
Note that the models used here assume that polarized and unpolarized synchrotron
emission have the same spectral behavior. While this assumption appears to be safe at high
latitudes, it may not be accurate for lines of sight that pass through the Galactic plane. This
is further explored in Section 5.
Finally, some fits allowed for an additional independent component, either using the
exact “cold neutral medium” (CNM) spectrum for spinning dust (Dr aine & Lazarian 19 98),
or using a generalized analytic form
T
sd
(ν) = A
sd
(ν
sd
)
β
d
+1
exp(ν
sd
) 1
(10)
The analytic form is a modified blackbody with the amplitude, low-frequency spectral index,
and turnover frequency explicitly decoupled from one another. Plots of both the exact
and “ shifted” spectra used in the fitting process are shown in Figure 5, as well as a curve
showing that the analytic f orm is indeed a good approximation to the numerically calculated
spectrum. In practice the low-frequency spectral index is irrelevant because the desired shape
for foreground fitting has ν
sd
well below 22 GHz and is thus dominated by the exponential
cutoff. While this form was motiva t ed as an analytic approximation to spinning dust spectra
it could also represent a variety of other physical sources of microwave emission. This
component was assumed to have no significant polarization.
Each chain itself is a multi-step process. The code makes an initial guess for the best-fit
parameters and runs for a burn-in period to find the region of parameter space near minimum
χ
2
. There is then a “pre-chain” to find the approximate moments of the likelihood; these
moments are used t o optimize the proposal distribution for the final chain. Problems at any
stage due to lack of convergence or poorly characterized parameter distributions are flagged
and recorded; only rarely are more than 0.5% of pixels so affected, and most problems are
due to random fluctuations and disappear with longer chains.
Page 17
18
Fig. 5.— The exact cold neutral medium ( CNM) spinning dust spectrum as calculated by
Draine & Lazarian (1998), an analytic fit to the model, and the ad hoc “shifted” model
which better fits radio observations in the Galactic plane. The shifted model may represent
a mix of warm neutral medium and warm ionized medium models, or another emission
process entirely. The vertical axis is in units of antenna temperature, but the overall scale
is arbitrary. Agreement between the exact model and the analytic approximation is better
than 5% over the frequency range from 15 to 35 GHz. This is smaller than the fractional
error from fits that include a spinning dust component.
Page 18
19
3.2. Tests and Sources of Error
The Monte Carlo process (with Metropolis steps) has the advantage that it can sample
the full parameter space and will converge o n the likelihood even if the likelihood is non-
Gaussian or unknown a priori. The disadvantage is that degeneracies in parameter space will
slow the convergence, and cutting off regions of parameter space to improve convergence can
bias the results. The prime example o f this is degeneracy between synchrotron and free-free
emission amplitude. If the synchrotron spectral index is allowed to flatten to the free-free
value then the amplitudes of the two components become degenerate parameters, which can
distort the fit.
We test this with simulated maps where the input foreground is known. To ensure that
the noise properties of the maps are well understood we used extensive simulations. We
combined the high resolution noise information with the low resolution pixel-pixel cova ri-
ance in order to generate noise realizations that are as realistic as possible, which are then
smoothed using the same process used for the real sky maps. We t hen produce mock sky
maps with CMB realizations synthesized from the WMAP “best-fit” ΛCDM model, noise
realizations from the five year noise covariance matrix, and foregrounds generated from a
variety of models.
When degeneracies exist the random fitting process tends to share the amplitude evenly
between degenerate parameters. This can lead to biasing if the true sky does not a lso have
equal contributions from such parameters. This effect can be seen clearly by comparing the
reconstruction of the synchrotron and free-free components (Figure 6). Figure 7 shows his-
tograms of a single pixel chain from the mock sky fit, with the input values of the parameters
marked with a bold cross.
The fitting process uses the error information contained in the N
obs
maps which in-
cludes covariance between the Q a nd U Stokes parameters within a pixel, but covariance
between pixels (due to low-frequency noise or the smoothing process) is not included. The
correlation coefficient between adjacent pixels ranges from less than 0.20 up to 0.45 (for
K-band and W-band, respectively) due to the smoothing process. The fit treats pixels as
independent, essentially marginalizing over all other pixels when fitting, so the main effect
of the correlations is to introduce similarly small pixel-pixel correlations in the χ
2
values.
This has a negligible effect on the results as long as goodness-of-fit is averaged over large
enough regions.
Smoothing a lso reduces the overall noise level, and this has been modeled through direct
noise simulations and accounted for in the fit process. The method used was to generate
many realizations of simulated noise maps based on the original N
obs
information, smo oth
Page 19
20
Fig. 6.— Difference maps between input and output foregrounds for the mock sky recon-
struction. For comparison, the peak temperature for synchrotron plus free-free inputs was
17 mK, and for the input dust the peak was 1.9 mK. The degeneracy between synchrotron
and f ree-fr ee emission means their sum is much better constrained than either component
individually. The scatter is larger than the bias, which is small compared t o the input signal
and within the error estimate given by the fit. These degeneracy issues are also illustrated
in Figure 7. Synchrotron and free-free antenna temperatures are defined as measured at
K-band, and dust as measured at W-band.
Page 20
21
Fig. 7.— Histograms for foreground parameters for a single pixel of mock sky reconstruction.
The mock foregro unds were designed t o have the same statistical behavior as the true sky but
do not match in detail. The same pixel is examined for the true sky in Figure 8. Dashed lines
indicate the mean of the chain, crosses indicate the best-fit point, a nd stars indicate the input
values for the parameters. The strong T
s
T
f
degeneracy and the curved T
s
β
s
degeneracy
are typical for high signal-to-noise pixels. Other parameter correlations are minor. Both
the mean and best-fit parameters of the chain are within the expected error range from the
input values.
Page 21
22
them using the same window functions used to smoo th the real data, and fit a Gaussian
shape to a histogram of the result. From this process an overall multiplicative rescaling
factor was determined for each frequency band and applied to the N
obs
files used for the
final fit. With this correction, the χ
2
per degree of freedom should be close to unity for an
ideal fit; f or the mock fit described above the mean (per pixel) reduced χ
2
was 1.11 with 7.2
degrees of freedom.
Small differences in the beam solid angle from one frequency band to another can distort
the inferred spectral index, especially near bright sources. We used the Jupiter-based beam
maps from Hill et al. (2008) and smoothed them to a common one-degree beam, similar to
how the sky maps are smoothed, and f ound that beam systematics at the known level can
cause deviations of up to ±0.1 in the spectral index. Errors of this type (that are multiplied
by the sky signal) are included in the fit by adding 0.3% of t he antenna temp erature to
the error budget for each pixel. This number was derived from the observed scatter one
beam-width away from bright point sources. Systematics of this type do not average down,
and so can quickly become dominant at low r esolutions.
The smoothing kernel used to match the bands to one-degree resolution uses the sym-
metrized beam profile, and hence does not take into account beam asymmetries. WMAP’s
observational strategy, however, symmetrizes the beam to a large extent. Page et al. (2003)
investigated the extent to which remaining beam asymmetry could affect the beam window
functions and found it to be < 1%. Any effect on the maps due to beam asymmetries should
be weak near the ecliptic poles and for extended emission not aligned along the plane of the
ecliptic.
There remain small uncertainties o f a few µK both in the true zero-level of the maps
and in the dipole subtraction process. The o ff sets primarily affect foreground estimation
by changing the apparent spectral index when averaging over large, very low signal-to -noise
regions. We avoid this by explicitly de-weighting or masking pixels with weak spectral index
constraints when reporting results. Second, as a purely pixel-based method, the MCMC
foreground fitting process we use is free to produce for egro und (or even CMB) maps with
non-zero monopole and dipole contributions. The sensitivity of the fit to offsets was checked
by adding offsets of 100 µK (several times larger than the error as estimated f r om the csc |b|
fits) to the sky maps and repeating the analysis. No foreground component was found to
change by more than 10% for pixels where the signal-to-noise was significant.
Page 22
23
4. Fit Results and Comparisons
4.1. MCMC Fit
Each pixel fit consists of 15 data points (Stokes I, Q, and U for each of the five frequency
bands) and a foreground fitting model can use from 8 to 12 parameters per pixel. The fitting
process produces a χ
2
value for each pixel. Normally the reduced χ
2
is found by dividing by
the number of degrees of freedom. However, the true number of degrees o f freedom in this
case is difficult to determine because neither t he data points nor the fitting parameters are
statistically independent of one another. Using the MCMC chain fo r each pixel, though, it is
possible to use the “Bayesian complexity” (described in a cosmological context in Kunz et al.
2006), defined as the difference between the average χ
2
over the chain a nd the χ
2
of the best
fit. This serves as a measure of the effective number of degrees of freedom, and can then
be used to determine the reduced χ
2
per pixel. Using the simulated skymaps described
above, we have found that the statistical behavior of the reduced χ
2
defined this way is
consistent with that of a χ
2
distribution. This “effective” reduced χ
2
is how we quantify the
goodness-of-fit in the tables and figures.
Pixels with high reduced χ
2
are not being well-fit by the model. Since such pixels
are largely confined t o the plane, the sky was divided into regions outside” and “inside”
the Galactic plane by using progressively smaller masks until t he average “outside” χ
2
was
no longer independent of the mask. Regions near known point sources from Wright et al.
(2008) are excluded from a ll a nalysis, both inside and outside of the plane, leaving 92% of
the full sky. Flagged pixels a r e also not included; for most fits such pixels arise from poorly
conditioned covar ia nce matrices, are uniformly distributed, and make up less than 0.5% of
the sky.
The χ
2
ν
results of several fits are shown in Table 4. Histograms for a single pixel chain
are shown in Figure 8. The mock sky simulations appear to capture the basic behavior of
the parameter correlations. Detailed comparisons of specific results are in the subsections
below. Figure 9 shows the basic results and Figure 10 shows the t emperature residuals of the
best-fit base model subtracted from the data, in units of one-sigma of noise. Figure 11 shows
the difference between the MCMC fit and the five-year MEM fit. The overall fit outside the
complex and troublesome Galactic plane region approaches a χ
2
ν
of 1.14 and the residuals
are mostly randomly distributed, which suggests that the overall fit works reasonably well
and t hat the noise properties have been described properly.
The “base” model uses the 10 parameters described above. Another fit is done including
data from 408 MHz (Haslam et al. 1981), assuming 10% calibration errors. As a check, a
“loose priors” fit is done which allows foreground temperatures to become negative. For t his
Page 23
24
Table 4. Model fits to WMAP t emperature and polarization dat a
# o f Best-fit χ
2
ν
a
model params outside plane
b
inside plane
b
full sky
base 10 1.14 2.23 1.24
base + Haslam 10 1.14 2.36 1.26
loose priors 8 1.09 3.26 1.29
steep 10 1.14 0.97 1.13
exact sd 9 1.21 1.63 1.25
shifted sd 9 1.24 1.00 1.22
β
s
= 3.2, β
d
= 1.7 8 1.16 4.33 1.45
β
s
= 2.6, β
d
= 1.7 8 1.30 3.42 1.50
β
s
variable, β
d
= 1.7 9 1.16 2.92 1.32
β
s
variable, β
d
variable 10 1.14 2.23 1.24
a
Reduced χ
2
averaged over pixels in the region, with effective degrees of
freedom determined by the chain. The statistical errors ar e less than 0.01.
b
The mask used to define these regions is a smoothed version of the 95%
mask, t he 5-year release analogue of the Kp12 mask.
Page 24
25
Fig. 8 .— Histograms for foreground parameters f or a single pixel of the observed sky from
actual WMAP data, not simulation. Dashed lines indicate the mean of the chain, and crosses
indicate the best-fit point. The observed sky shows the same basic behavior as the simulated
sky used for testing.
Page 25
26
Fig. 9.— Temperature maps f or foreground components as determined by the MCMC fitting
process for the “base” model. Maps from other models ar e qualitatively similar. Synchrotron
and free-free temperatures are as measured at K-band, dust is measured at W-band. Gray
pixels are those masked due to point sources or flagged as problematic. Top: synchrotron;
middle: free-free; bottom: dust
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27
Fig. 10.— Residuals of the “base” model foreground fit subtracted from the data. The left
column shows the residuals in units of noise sigma, from -6 to 6. The right column has the
85% mask applied and a scale o f -3 to 3. Frequency bands are (from top to bottom) K, Ka ,
Q, V, W. The main feature is that the model underestimates Galactic flux in Ka band and
overestimates it in Q band by a factor of 3 to 5 times the pixel noise. Outside the analysis
mask the residuals to t he model are consistent with noise at the expected level.
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28
Fig. 11.— Difference between the MCMC fit a nd MEM maps. The top panel is the difference
(at K band) in synchrotron plus free-free emission. Differences between these two components
separately are larger due to the degeneracy direction. The bottom panel is the difference (at
W band) in the dust maps. The differences are roughly one percent of total emission at K
band and a few percent at W band.
Page 28
29
fit to converge the spectral indices must be fixed, however, so it only uses 8 parameters. The
“steep” model fixes the dust spectral index but allows for a synchrotron steepening parameter
c as described above. The “exact sd” model uses the CNM spinning dust spectrum fo r an
additional f oreground, whereas the “shifted sd” model uses the generalized spectrum with
ν
sd
= 4.9 GHz. In both cases, the synchrotron and dust spectral indices are fixed to be
the same at each pixel (again for convergence reasons). Finally, several fits were done with
fixed spectral indices to examine the effect of using different values, shown as the last part
of Table 4. The last entry of the ta ble is the “base” model, repeated for ease of comparison.
4.2. Overall Foreground Features
Figures 9–14 show maps of the results from the “base” fit. Figures 9 and 12 show the
three foregrounds themselves and their errorbars as determined from the parameter variance
in the Markov chains. The maps are in units of antenna temperature as measured at K-band
for synchrotron and free-free emission, and at W-band for dust emission. Figure 13 shows
spectral index maps binned to lower resolution, where color indicates the sp ectral index and
the size o f the circle indicates the significance of the fit result at that location. Figure 14
shows the best χ
2
ν
value achieved at each pixel on the sky.
Almost regar dless of foreground model the fit works quite well outside the Galactic plane,
giving low χ
2
ν
and foreground maps that a re in good agreement with the MEM templates
(Hinshaw et a l. 2007) and other works (Eriksen et al. 2007). Error maps for synchrotron and
free-free emission have similar morphology due to the degeneracy between their a mplitudes.
The overall dust brightness seems to be largely consistent with the template prediction
(Finkbeiner et al. 1999), though the fit appears to prefer a spatial distribution somewhat
less sharply peaked toward the Galactic center (Figure 15). The excess of o bserved emission
compared to that predicted at 9 0 GHz, seen in the original model comparison with COBE
DMR data, is still present. Since the fit in the plane has high χ
2
and is untrustworthy,
the overall preferred spectral index for dust may be < 2.0, but the significance of this is
not high. Weighting with the covariance information f rom the fit and masking low-signal
regions, the average value for β
d
in the “base” fit is 1.8 with ±0.3 from statistical errors and
±0.2 from systematic error depending on how the cuts a r e defined. Fo r comparison, model
8 of Finkbeiner et al. (1999) predicts β
d
= 1.55 ± 0.01 for a comparable sky cut.
The free-free component is consistent with expectations from previous fits and with Hα
observations when dust obscuration is taken into account. Free-free emission is quite high in
the Gala ctic plane and in several regions (including Gum and Orion) appears to be dominant
Page 29
30
Fig. 12.— Error maps for foreground components as determined fr om the marginalized
variance given by the MCMC fitting process. Error maps for synchrotron and f r ee-free
emission are similar due to the parameter degeneracy between them. Synchrotron and free-
free temperatures are as measured at K-band, dust is measured at W-ba nd. Gray pixels
are those masked due to point sources or flagged as problematic. Top: synchrotron; middle:
free-free; bottom: dust
Page 30
31
over synchrotron, even in K-band. The ratio of the “base” fit free-free map to extinction-
corrected Hα map of Finkbeiner (2 003) was used to make a map of h
ff
(the temperature–Hα
intensity conversion factor). A histogram was then made of all pixels with intensities larger
than 5 Rayleighs (t o mask out low-signal regions) and less than one magnitude of extinction
(using the reddening map of Schlegel et al. 1998). A gaussian fit to the peak of the histogram
gives h
ff
= 11.8 ± 8.8 µK R
1
at K-band, comparable to t he value of 11.4 µK R
1
expected
from an electron temperature of 8000 K (Bennett et al. 2003), but also consistent with the
lower values from template cleaning.
While subject to degeneracy with the free-free emission, synchrotron radiation is a
stronger signal in WMAP data than dust emission. Pixel-by-pixel constraints become poor
far from the plane, however there are still constraints on the best-fit spectral index. For
example, by comparing fits with constant spectral index, the Northern Polar Spur and the
Fan region prefer an index of 3.0 or steeper. All fits including total intensity data show
the same preference for shallower spectral index in the plane a s concluded by Bennett et al.
(2003); Figure 16 shows β
s
as a function of latitude for a number of different fits.
From the polarized data, the synchrotron polarization fraction indicates strong depo-
larization toward the Galactic plane consistent with Kogut et al. (2007). Since Faraday
rotation should not be large at these frequencies, this effect is due to multiple magnetic field
orientations along the line of sight. Dust polarization fraction appears to follow a pattern
similar to the synchrotron polarization fraction, though the signal-to-noise rat io is low. This
is physically reasonable, as the polarization fraction is largely affected by the coherence of the
magnetic field along t he line o f sight. This implies that the dust intensity times a constant
fraction may not be the best template to use for dust polarization in the Galactic plane.
4.3. The Galactic Plane
Regions at very low latitudes are not as well fit by the “base” model, and there is
dependence both on foreground model and fit parameters. A map of poorly fit regions
reveals that they are in the brightest parts of the Galaxy, where at t hese frequencies the
free-free emission dominates.
Pixels poorly fit by t he “base” model have some common characteristics. Most are
bright, but this is probably because similar less bright pixels have lower signal-to -noise and
thus lower χ
2
. Many have a K-Ka temperature spectral index similar to what one would
expect from fr ee-free emission, but a considerably steeper Ka-Q sp ectral index. Such pixels
tend not to be highly polar ized, and what polarized emission exists appears to be consistent
Page 31
32
with synchrotron emission with a typical spectral index of β 3. Data for several such
individual pixels are shown in Figure 17.
For the published cold neutral medium spinning dust model (the “exact” model of Table
4), the maximum fraction of Ka-band flux attributable to spinning dust is 17% outside o f
the Galactic plane (using the 85% mask). The maximum full-sky fraction of Ka-band flux
attributable to spinning dust is 20% for this model. However, this model still does not
provide a good fit within the Galactic plane (χ
2
ν
in this region is 1.63).
Allowing the spinning dust spectrum to shift in frequency to obtain a better fit results
in a Ka-band flux fraction of 14% for spinning dust, roughly independent of sky cut. A
map of the spinning dust component from this fit and its error is shown in Figure 20. The
morphology lies somewhat between that of dust and free-free emission, though the details
depend on the specifics of the model. The Galactic plane is equally well-fit by adding a
synchrotron steepening para meter c into the fit. The actual value of c is generally not well-
constrained, but the average value in the plane is 1.8. This very rapid steepening does
not appear to be consistent with cosmic ray models (Strong et al. 2007), but may have some
other physical origin.
Figure 18 shows the low-frequency foregrounds given by the MCMC fit using the steep”
model. Thermal dust emission is indistinguishable from the the “base” fit. Residual maps
from this fit are featureless, as hinted at from the χ
2
information in Table 4. Figure 19 shows
low-frequency foregrounds and residuals in K, Ka, and Q-bands for the “shifted sd” fit which
includes a spinning dust-like component. This model produces a good fit in the plane, but
seems to have some problems with the Ophiuchus and Gum regions.
Since the goodness-of-fit outside the plane is not improved by the addition of a spinning
dust component, and low signal-to-noise regions bias the spinning dust fraction upwards, we
regard the spinning dust fraction of the fits above as an upper bound to the overall amount
of diffuse spinning dust emission present. As with previous WMAP fits, this new fitting
technique continues to find that spinning dust is a subdominant emission process.
5. Discussion
5.1. Effect on CMB and Cosmology
The uncertainties of the fit in the Galactic plane preclude CMB analysis for those re-
gions. Fortunately, such regions appear to be tightly confined to the plane inside a very
narrow sky cut (9% of the total sky) and thus can be excluded without losing much infor-
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33
mation for cosmological analysis. The foreground maps fro m the MCMC fit are similar t o
those f rom the MEM fit and other foreground templates, which means CMB polarization
maps cleaned using such templates will also be similar.
Outside of a narrow band on the Galactic plane the CMB map produced by the fit is
visually identical t o the ILC map. The difference between the “base” fit CMB map and the
ILC map is shown in Figure 21. The t otal variance of this difference map outside of the
KQ85 mask used for power spectrum analysis is 116 µK
2
, much lower than the CMB power.
The variance between the “base” fit CMB map and the “shifted sd” CMB map is 44.1 µ K
2
outside of the KQ85 mask; va r ia nce from one fit to another is generally even smaller for
other combinations. Spherical harmonic decomposition did not show these total variances
to be strongly focused at any particular multipole, and the numbers a re small enough t hat
differences between maps fall within cosmic variance.
The CMB polarization maps produced by the MCMC fit presented in this work are
noisy and show some evidence of synchrotron contamination. Nevertheless, the covariance
maps from the fits can be used to bound t he amount of contamination present, and are
available on t he LAMBDA website. These are produced from the (marginalized) variance of
each parameter over the Markov chain for each pixel. For cosmological analysis a different
method is used to marginalize over polarization foregrounds. For a full description see
Dunkley et al. (2008).
5.2. More Complicated Models
All of the models so far fit assume that the spectral shape of foreground emission in
a 1deg
2
pixel can be described as a sum of power- laws o r other simple shapes. This is
justified if the observed emission is dominated by a few emission mechanisms which simply
combine additively along the line of sight and have minor spatial varia t io n within the beam.
In more complex regions of the Galaxy, however, things may not be so simple.
If two synchrotron regions along the line of sight have their polarization angles oriented
nearly orthogonally, then the total polarized emission will be sharply reduced. If the two
regions have different spectral indices then the cancellation in polarization will be maximized
at the f r equency where the individual polarization amplitudes match, causing a dip in the
polarization spectrum. Thus even with pure synchrotron emission the polarization spectrum
can look quite different from the temperature spectrum.
To assess this effect 100,000 Monte Carlo realizations were made of a superposition of
two independent randomly oriented synchrotron emitting regions. Parameters for the distri-
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34
bution of intensity and spectral index were chosen to roughly correspond t o observatio ns, but
the simulation was meant only to provide a generous estimate of how different temperature
and polarization behavior could be simply due to multiple synchrotron regions along the
line of sight. The mean spectral index difference was small (0.051) but the standard devi-
ation was not insignificant (0.12) and the distribution was non-gaussian with high kurtosis
(4.4). Over 20% of the simulations had an absolute slope difference larger than 0.1. Thus
we cautiously conclude that while differences between temperature and polarization spectral
indices at t he 0.1 level could be quite mundane in origin, consistent differences of 0.25 or
larger a re probably not due to chance alignments in polarization angle and may be caused
by an unpolarized non-thermal temperature component.
For the fit, free-free emission was modeled as a pure power law based on the assumption
that the plasma is optically thin. In reality, H II regions can become dense enough to become
optically thick at frequencies as high as 20 GHz, although such regions are spatially small
and do not contribute significantly to the observed emission for a beam as large as WMAP’s.
Further, to obtain rising flux at K- band requires very high emission measure (10
9
cm
6
pc).
Even if they were somehow significant, such regions can no t explain a steep ening spectrum
past Ka-band.
Synchrotron self-absorption can a lso cause a low-frequency turnover, but the physical
parameters necessary for the turnover frequency to lie in or near the WMAP range imply
conditions typically only f ound in active galactic nuclei or other extreme regions. It may be
physically possible for synchrotron radiation from stellar-mass black hole jets or accretion
disks to become optically thick at WMAP f r equencies, but this phenomenon has yet to
be clearly observed and it is unlikely that such emission would contribute significantly at
WMAP’s resolution.
Synchrotron radiation is suppressed when emitted from a region with a refra ctive index
less than unity, such as a plasma. This is known as the Tsytovitch-Razin effect, and causes
strong suppression of synchrotron emission below 20(n
e
/B) Hz, where n
e
is the electron
density (in cm
3
) and B the perpendicular component of the magnetic field (in G). For
typical Galactic electron densities a nd magnetic fields, this cutoff is in the 3–300 MHz range,
at most. Unless energetic electrons play an unexpectedly large role in diffuse emission,
WMAP should not see significant Tsytovitch-Razin suppression.
Diffusive synchrotron radiation (DSR) differs from an ideal synchrotron spectrum be-
cause of the presence of significant random fluctuations in the magnetic field (Fleishman
2005). In this model lower-energy electrons experience small-scale turbulence in the magnetic
field structure and follow non-circular paths due to the random deflections. In such models
the emission spectrum can turn over from a power law with β 2.1 in the turbulence-
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dominated diffusive regime to a normal synchrotron spectrum at higher frequencies. While
in most models this occurs at low frequencies, there is some indication that for pulsar wind
nebula the turbulence is relevant up to the GHz range and above (Fleishman & Bietenholz
2007). Whether D SR can occur for less compact objects is not understood at this time.
5.3. Other Components
Much has been written on the possible presence of anomalous emission in the lower
frequency band