SEVEN-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP^*) OBSERVATIONS: GALACTIC FOREGROUND EMISSION

There are three primary mechanisms for diffuse Galactic radio emission. Relativistic electrons interact with the Galactic magnetic field to produce synchrotron emission, for which the standard template is 408 MHz data compiled by Haslam et al. (1981). Less energetic electrons scatter from each other and ionized nuclei to produce free–free radiation (also known as thermal Bremsstrahlung), which can be traced with Hα line emission (Finkbeiner 2003). Finally, dust grains emit a modified blackbody spectrum through excitation of their vibrational modes, for which the standard template is the fit of Finkbeiner et al. (1999) to data from the Infrared Astronomical Satellite (IRAS) and the Cosmic Background Explorer (COBE). Dust grains may also emit radiation through rotational modes or other excitations (Draine & Lazarian 1998a, 1998b, 1999).

WMAP was designed to measure the frequency range where the ratio of the CMB anisotropy to the rms fluctuations of all three foregrounds is at its maximum, to minimize foreground contamination. This also implies that two or more foreground components will be of comparable amplitude and that they will be relatively weak. Foreground templates, however, are best made by observing a foreground process at a frequency where it dominates the total emission. Hence, there will always be some extrapolation involved when attempting to account for foregrounds on top of CMB observations.

So how well does the extrapolation work? Simple power-law extrapolation of the 408 MHz synchrotron template from Haslam et al. (1981) does not explain very much of the observed emission at 20–40 GHz. Whether this is due to a new low-frequency emission process, errors in the extrapolation due to spatial variation in the spectral index, or both, is difficult to determine. Targeted observations of individual regions (Scaife et al. 2009; Tibbs et al. 2010) suggest a spinning dust-like component, but a model consistent across size scales and data sets remains elusive.

Free–free emission is extrapolated from a map of Hα, corrected for dust extinction using a reddening map based on 100 μm data (Schlegel et al. 1998; Bennett et al. 2003). Variations in electron temperature cause some uncertainty in this extrapolation, but the larger effect is likely uncertainty in the reddening correction. The overall ratio of radio to Hα brightness comes out lower than expected (Bennett et al. 2003); nevertheless, the template otherwise matches quite well with observations at 30–60 GHz, where free–free emission from the Galactic disk is particularly dominant.

The dust extrapolation has so far been tested least precisely by CMB experiments. While the model of Finkbeiner et al. (1999) incorporates COBE FIRAS data all the way down to 60 GHz, the uncertainty at those frequencies is large; most of the dust model comes from information at 100 and 240 μm. While the spectral index of dust at frequencies below 300 GHz has not yet been measured to enough accuracy to challenge the model, the morphology matches observations at lower frequencies, though some experiments suggest overall brightness levels different from the predictions (Veneziani et al. 2010; Culverhouse et al. 2010).

Analysis of data from previous WMAP releases has shown that CMB maps from different foreground removal techniques agree to within 11 μK (Gold et al. 2009) on average in the low Galactic emission regions used for CMB anisotropy measurements, though this does not provide an absolute limit to the amount of contamination. Even when templates are not directly used for foreground removal, they provide an important guide for the construction of masks and other foreground cleaning methods.

Several systems of units are in use throughout this work. Point sources are reported in flux units (Jansky), where the power-law index is denoted by α such that flux follows S ∼ ν^α. Foreground modeling is most easily done in units of antenna temperature, defined by using the Rayleigh–Jeans limit of a blackbody spectrum (for which S ∼ ν²) to convert flux per solid angle to a temperature. In these units the power-law index is denoted as β = α − 2. WMAP's frequency range is not quite in the Rayleigh–Jeans limit for a 2.7 K blackbody, so there is a frequency-dependent conversion factor a(ν) = (e^x − 1)²/x²e^x (where x = hν/k_BT_cmb) to convert antenna temperatures to thermodynamic temperatures convenient for CMB analysis.

Foreground removal always has some uncertainty, so it is useful to mask part of the sky where foregrounds are too bright for CMB analysis. As in the five-year analysis, the starting points for the masks are K- and Q-band-average maps smoothed to 1 deg resolution. The maps are then converted to foreground-only maps by subtracting off an estimate of the CMB using the internal linear combination (ILC) method (see Hinshaw et al. 2007 and Section 2.2). A cumulative histogram in each band is formed to find the flux level above which a given percentage of sky can be cut, and the union of the pixels cut from each band at a given flux level is used to define a mask. We used two masks for most further analysis, based on cuts which leave 75% and 85% of the sky; these are denoted as KQ75 and KQ85, respectively.

For the seven-year analysis, the diffuse foreground masks have been extended based on a χ² analysis of residuals after foreground subtraction. Starting with foreground-reduced maps, differences are taken between bands (Q – V and V – W in thermodynamic units), eliminating any CMB signal. Ideally the only thing left in the resulting maps would be noise; in practice there are visible residuals near the Galactic plane. Given the noise per pixel of the maps, it is possible to compute a map of the χ² for each pixel.

After degradation to HEALPix N_side = 32 (see Gorski et al. 2005 for a description of this pixelization scheme), regions of 4 or more contiguous pixels with χ² higher than four times that of the polar caps are identified and used to define two new masks, one from each difference map. These are then combined with the previous KQ75 and KQ85 masks (Gold et al. 2009) used for the five-year analysis. After promotion back to full resolution, edges of the mask are smoothed with a 3° FWHM Gaussian. The resulting changes to the final mask are primarily around the edge of the Galactic cut, particularly in the Gum and Ophiuchus regions. The additional sky fraction cut from the KQ85 masked sky is 3.4%, and from the KQ75 masked sky is 1.0%.

These expanded masks are then combined with the point source mask as in previous releases, which has been updated with newly detected sources. Also, point sources brighter than 5 Jy have had the radius of their cut extended from 06 to 12, in order to minimize confusion at low frequencies where the instrument beam is large. The new masks, which we denote as KQ75y7 and KQ85y7, are shown in Figure 1 and are available on the LAMBDA Web site.¹⁵ In total 70.6% (KQ75y7) and 78.3% (KQ85y7) of the sky now remains after the masking process.

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The ILC method implemented by WMAP is a technique largely blind to assumptions about the frequency spectrum of foreground emission, which produces CMB maps with little visible foreground contamination. The ILC is a weighted combination formed from all five frequency bands, which are smoothed to a common 1° FWHM Gaussian beam using the symmetrized beam window functions produced by the beam analysis (Jarosik et al. 2011). The coefficients used to weight each individual frequency band are those that minimize the variance of the resulting map under the constraint that the sum of the coefficients is unity, which ensures that the CMB portion of the signal is passed through unaltered.

The details of the algorithm used to compute the WMAP seven-year ILC map are the same as that described in the three-year analysis (Hinshaw et al. 2007). In particular, we perform a bias correction step which uses simulations to estimate and correct for the tendency of the ILC method to produce CMB maps anti-correlated with foreground fluctuations (for an overview of potential ILC pitfalls, see Vio & Andreani 2009). We have found this technique to be robust when applied to WMAP data: the variance between the ILC map and CMB maps made with other techniques is less than 116 μK² (Gold et al. 2009). Similar techniques by other authors have given CMB maps consistent with WMAP's best-fit cosmological results (Kim et al. 2008).

Rather than use a single set of coefficients for the whole sky, to allow for variations in Galactic composition we subdivide the sky into 12 regions and find the ILC coefficients for each, shown in Table 1. All but region 0 lie along the Galactic plane. We retain the same number of regional subdivisions of the sky, and their spatial boundaries remain unchanged from the previous years (for details see Hinshaw et al. 2007). The frequency weights for each region are slightly different, however, reflecting the most recent updates to the calibration and beams. Figure 2 shows the difference between the seven-year and five-year ILC maps, which is dominated by a small change in the dipole. The seven-year ILC map is available on the LAMBDA Web site.

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The maximum entropy method (MEM) is a spatial and spectral fit that uses external templates, intended to distinguish between different emission sources. By design, the MEM output tends to revert to these templates in regions of low signal-to-noise. Thus, the MEM results are most interesting in regions with higher signal.

The seven-year MEM analysis is largely unchanged from previous work (Hinshaw et al. 2007; Gold et al. 2009). As before, the analysis is done in all bands on the 1° smoothed sky maps, with the ILC map subtracted. The zero level of each map is set such that a $\csc |b|$ fit, for HEALPix N_side = 512 pixels at b < −15° and outside of the KQ85y7 mask, yields a value of zero for the intercept. The maps are degraded to HEALPix N_side = 128 pixelization, and a model is fit for each pixel p. Rather than simply minimize χ², the MEM minimizes a function

$$\begin{equation} H = \chi ^2 + \lambda \sum _c T_c(p) \ln \left[\frac{T_c(p)}{eP_c(p)}\right]. \end{equation} \tag{ 1 }$$

Here T_c and P_c are the model brightness and template prior for foreground component c (e is the base of natural logarithms). The second term is what enforces the prior when the signal-to-noise becomes low, and the parameter λ sets the threshold for the transition from signal-dominated to noise-dominated behavior. The spectra of the free–free and dust components are fixed power laws, with β = −2.14 for free–free and β = +2.0 for dust. An iterative procedure uses residuals from the fit at each iteration to adjust the spectrum of the synchrotron component for each pixel. Hence, any anomalous component such as electric dipole emission from spinning dust is included in the synchrotron component. The adopted priors are unchanged from previous analyses and are based on the 408 MHz map of Haslam et al. (1981) with an extragalactic brightness of 5.9 K subtracted (Lawson et al. 1987) for synchrotron, an extinction-corrected Hα map (Finkbeiner 2003; Bennett et al. 2003) for free–free, and model 8 of Finkbeiner et al. (1999) for dust.

The prior map and output map are shown in Figure 3 for each foreground component. The zero level of the output synchrotron map is slightly lower (∼50 μK) than that of the synchrotron prior. This reflects the difference between the zero level of the K-band $\csc |b|$ normalized map and that of the prior. For comparison, the 1σ uncertainty in the prior zero level, based on the quoted uncertainty in the 408 MHz map zero level, is 27 μK. Also, there is a dependence on the adopted extragalactic brightness at 408 MHz. If the ARCADE 2 value (Fixsen et al. 2009) were used, the zero level of the prior would be ∼37 μK below that of the $\csc |b|$ normalized map. Figure 3 can be compared with Figure 5 of Hinshaw et al. (2007) to see the improvement in signal-to-noise ratio (S/N) of the output maps between the three-year and seven-year analyses.

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Differences between seven-year MEM maps and five-year MEM maps are shown in Figure 4. The seven-year MEM foreground component maps tend to be slightly brighter than the five-year versions at mid to high northern Galactic latitudes. This is due to small dipole differences between the seven-year and five-year sky maps, which are caused by a combination of a change in the calibration dipole and small (less than 0.2%) changes in radiometer calibrations between the seven-year and five-year analyses. The seven-year and five-year foreground component maps are in better agreement at southern Galactic latitudes because this is where zero level normalization of the sky maps is determined by $\csc |b|$ fitting. The MEM maps are available on the LAMBDA Web site.

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WMAP continues to use a template cleaning method to produce the foreground-reduced maps used for power spectrum analysis (Hinshaw et al. 2007; Page et al. 2007). For temperature maps, the templates are a K−Ka difference map, an extinction-corrected Hα map, and a dust map (Finkbeiner et al. 1999). For polarization, the templates are the K-band map for synchrotron, and a dust model described in detail below.

The temperature cleaning is applied to seven-year Q-, V-, and W-band maps (K and Ka are used for a template). The model has the form

$$\begin{eqnarray} M(\nu,p) &=& b_1(\nu)[T_{\rm K}(p) - T_{\rm Ka}(p)]\nonumber\\ &&+ b_2(\nu) I_\mathrm{H\alpha }(p) + b_3(\nu) M_\mathrm{dust}(p), \end{eqnarray} \tag{ 2 }$$

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). Because the first 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

$$\begin{equation} h_\mathrm{ff} = \frac{b_2(\nu)}{S_\mathrm{ff}(\nu) - 0.552\, b_1(\nu)}, \end{equation} \tag{ 3 }$$

where S_ff(ν) is the free–free emission spectrum converted to thermodynamic temperature units, normalized to unity at the K band and is assumed to be a power law in antenna temperature with β = −2.14. The synchrotron spectral index (relative to the K band) is found via

$$\begin{equation} \beta _s = \frac{ \log \left[ 0.67\, b_1(\nu) a(\nu) \right] }{ \log (\nu /\nu _K) }, \end{equation} \tag{ 4 }$$

where a(ν) is the conversion factor from antenna temperature to thermodynamic units.

The coefficients of the model fit to the seven-year data are presented in Table 2. Small changes in the seven-year coefficients compared to previous values reflect small changes in the absolute calibration and beam profiles.

For polarization cleaning the maps are degraded to low resolution (N_side = 16). The model has the form

$$\begin{eqnarray} [Q(\nu,p), U(\nu,p)]_\mathrm{model} &=& a_1(\nu) [Q(p), U(p)]_K\nonumber\\ &&+ a_2(\nu) [Q(p), U(p)]_\mathrm{dust}. \end{eqnarray} \tag{ 5 }$$

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 derived from starlight measurements ([Q, U]_dust) and a geometric suppression factor to account for the magnetic field geometry (Page et al. 2007). The coefficients of the model fit to the seven-year data are in Table 3. For polarization, the template maps are assumed to have a one-to-one correspondence with foreground emission, so the spectral indices for synchrotron and dust are simply the power-law slopes of the coefficients a₁(ν) and a₂(ν). If the dust model is correct then the ratio a₂/b₃ gives the polarization fraction; for the W band this is ∼6%.

The full-resolution (N_side = 512) foreground-reduced Stokes Q and U maps were produced using the same cleaning coefficients as derived for the low-resolution maps, but with full-resolution templates. The K band and dust intensity templates can be produced at full resolution from available data, and the starlight polarization map used to determine polarization direction was upgraded to full resolution using nearest-neighbor sampling. The templates subtracted from WMAP data are smoothed to 1° FWHM, potentially leaving artifacts in the foreground-reduced maps due to small-scale power or beam asymmetries. In practice, we find no sign of these effects, as discussed in Section 4.1 and Section 5. All data sets used for templates are available on the LAMBDA Web site.

We again perform a pixel-based Markov chain Monte Carlo (MCMC) fitting technique to the five bands of WMAP data. Our method is similar to that of Eriksen et al. (2007), but we focus more on Galactic foregrounds rather than CMB. The fit results of the five-year release have been reproduced, with the "base" model, which uses three power-law foregrounds, producing virtually the same reduced χ² per pixel. The MCMC fitting has benefited from further understanding of the zero point of the maps. We have used the 408 MHz map of Haslam et al. (1981) with a zero point determined using the same $\csc |b|$ method as for the WMAP data and investigated the effect on the fit of error in this zero point.

The MCMC fit is performed on 1 deg smoothed maps downgraded to HEALPix N_side = 64. An MCMC chain is run for each pixel, where the basic model is

$$\begin{equation} T(\nu) = T_{s} \left(\frac{\nu }{\nu _K} \right)^{\beta _s(\nu)} + T_{f} \left(\frac{\nu }{\nu _K} \right)^{\beta _f} + a(\nu) T_\mathrm{cmb} + T_{d} \left(\frac{\nu }{\nu _W}\right)^{\beta _d} \end{equation} \tag{ 6 }$$

for the antenna temperature. The subscripts s, f, d stand for synchrotron, free–free, and dust emission, ν_K and ν_W are the effective frequencies for the K and W bands (22.5 and 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). The fit always includes polarization data as well, where the model is

$$\begin{equation} Q(\nu) = Q_{s} \left(\frac{\nu }{\nu _{\rm K}} \right)^{\beta _s(\nu)} + Q_{d} \left(\frac{\nu }{\nu _{\rm W}}\right)^{\beta _d} + a(\nu) Q_\mathrm{cmb} \end{equation} \tag{ 7 }$$

$$\begin{equation} U(\nu) = U_{s} \left(\frac{\nu }{\nu_{\rm K}} \right)^{\beta _s(\nu)} + U_{d} \left(\frac{\nu }{\nu _{\rm W}}\right)^{\beta _d} + a(\nu) U_\mathrm{cmb} \end{equation} \tag{ 8 }$$

for Stokes Q and U parameters. Thus, there are a total of 15 pieces of data for each pixel (T, Q, and U for five bands).

As for the five-year release, the noise for each pixel at N_side = 64 is computed from maps of N_obs at N_side = 512. To account for the smoothing process, the noise is then rescaled by a factor calculated from simulated noise maps for each frequency band. The MCMC fit treats pixels as independent and does not use pixel–pixel covariance, which leads to small correlations in χ² between neighboring pixels. This has negligible effect on results as long as goodness of fit is averaged over large enough regions.

We fit three categories of models. All use the K band as a template for the polarization angle of synchrotron and dust emission, so U_s and U_d are not independent parameters, identical to the previous analysis. All models also fix the free–free spectral index to β_f = −2.16, a slight change from β_f = −2.14 used in the previous analysis. This change was motivated as an attempt to better match the effective spectral index at Q and V bands, due to their use in cosmological analysis, but was not found to make a difference to the fits.

The "base" model uses three power-law foregrounds, where the synchrotron spectral index β_s(ν) is taken to be independent of frequency but may vary spatially, and the dust spectral index β_d is allowed to vary spatially. We assume the same spectral indices for polarized synchrotron and dust emission as for total intensity emission. This model has a total of 10 free parameters per pixel: T_s, T_f, T_d, T_cmb, β_s, β_d, Q_s, Q_d, Q_cmb, and U_cmb.

A steepening synchrotron model uses the same three foregrounds but allows for a steepening of the synchrotron spectral index by adding a new parameter c_s, defined by

$$\begin{equation} \beta _s(\nu) = \left\lbrace \begin{array}{@{}ll} \beta _s & \nu < \nu _{\rm K} \\ \ms \beta _s + c_s \ln \left(\dsty\frac{\nu }{\nu _{\rm K}}\right) & \nu > \nu _{\rm K} \end{array}\right.. \end{equation} \tag{ 9 }$$

For the steepening model the dust spectral index is fixed¹⁶ to β_d = +2.0. Therefore, this model also has 10 free parameters per pixel.

For models with a spinning dust component, another term is added to Equation (6):

$$\begin{equation} T_{\rm sd}(\nu) = A_{\rm sd} \frac{(\nu /\nu _{\rm sd})^{\beta _d+1}}{\exp (\nu /\nu _{\rm sd})-1}. \end{equation} \tag{ 10 }$$

The spinning dust component is assumed to have negligible polarization, as theoretical expectations for the polarization fraction are low compared to synchrotron radiation (Lazarian & Draine 2000), and the polarization data thus far show no evidence that such a component is necessary (see Section 4.5). The spinning dust amplitude A_sd was allowed to vary spatially as a new parameter. Both β_s and β_d were fixed to −3.0 and +2.0, respectively, to avoid degeneracies from having too many parameters in the fit. Allowing ν_sd to spatially vary was not found to result in any improvement of the fit, but fits were performed with different global values of ν_sd to find the best overall value. Thus, with fixing of the spectral indices, this model has nine free parameters per pixel.

MCMC fits for the seven-year release were performed with the addition of the 408 MHz data compiled by Haslam et al. (1981). The error on the zero point for this data was estimated in that work to be ±3 K, with an overall calibration error of 10%. Lawson et al. (1987) use a comparison with 404 MHz data to find a uniform (presumably extragalactic) component with a brightness of 5.9 K. As the MCMC method treats all input maps equally, for consistency we estimate and subtract off a nominal zero-point offset of 7.4 K, as determined by the same $\csc |b|$ method we use for the WMAP sky maps. However, the 408 MHz data resemble a $\csc |b|$ behavior much less than the WMAP data, due to the increased relative prominence of large-scale features such as the Northern Polar Spur. Therefore, we attempted the $\csc |b|$ fitting procedure on different hemispheres and with different cuts, and estimate the uncertainty in procedure to be ±4 K. MCMC fits were run for each model with zero points of 3.4 K and 11.4 K in addition to the nominal value, and the effect of these on foregrounds is discussed in Section 4.4. A full set of maps and MCMC variance estimates for the three models is available on the LAMBDA Web site.

As a test of the template-based foreground subtraction process, power spectra of difference maps were made. Figure 6 shows the power spectrum of the difference between the foreground-reduced Q-band and W-band maps, with the point source contribution to the power spectrum subtracted off. Averaging over bins of Δℓ = 50, no bin with more than 120 μK² of power is seen, with an upper limit of ∼220 μK² in power (15 μK in amplitude), and the results are consistent with zero within the expected error. For comparison, CMB power in the range 30 < ℓ < 500 is 1000 μK² or more (Larson et al. 2011). Differences between foreground-reduced V band and W band were also computed, and the power in that case was even smaller.

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While Galactic foregrounds are not fully described by a two-point function (i.e., an angular power spectrum), due to the importance of the CMB it is often useful to examine the power spectrum of foregrounds. Specifically, the relevant quantity to calculate is the contribution of foreground emission to the angular power spectrum in a particular patch of sky of interest for CMB analysis.

A general trend of ℓ(ℓ + 1)C_ℓ ∼ ℓ^−0.6 was found from examination of raw polarization data outside the P06 mask (Page et al. 2007), as the result of a combined fit to WMAP data in both multipole and frequency space. With the MCMC fitting procedure it is possible to separate polarized synchrotron from dust and examine the two components individually. The results, shown in Figure 7, show behavior largely consistent with the previous analysis.

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In detail, MCMC maps from the "base" model including Haslam data were used. Power spectra from the spinning dust model MCMC maps were also inspected and found to be nearly identical at large scales. A union of the polarization analysis mask and the mask of pixels flagged by the MCMC was applied, and the C^EE_ℓ and C^BB_ℓ spectra were computed for both synchrotron and dust. As the MCMC process uses 1 deg smoothed maps, an appropriate correction for the beam window function was applied. Each power spectrum was then fit with a model consisting of a power law plus a pixel noise term

$$\begin{equation} \ell (\ell +1)C_\ell ^{{\rm XX}}/2\pi = {\cal B}_c \ell ^m + \ell (\ell +1)N^2, \end{equation} \tag{ 11 }$$

where ${\cal B}_c$ is the amplitude for foreground component c, m is the power-law index, and N is the noise amplitude.

Values for the fit parameters and an estimate of their errors can be found in Table 4. Because the power spectra are taken from highly processed maps, detailed error propagation is difficult. We used the diagonal portion of the published C_ℓ Fisher errors plus cosmic variance to perform the fit; covariance between multipoles will cause the true errors to be somewhat larger. If all foreground power spectra are assumed to have the same power-law behavior, then the weighted mean m = −0.67 ± 0.24.

Table 4. Foreground Power Spectrum Parameters

Component	${\cal B}_c$ (μK2)a	ma	N (μK)a
Synchrotron EE	271 ± 31	−0.73 ± 0.04	0.109 ± 0.001
Synchrotron BB	130 ± 8.6	−0.61 ± 0.02	0.107 ± 0.001
Dust EE	17.7 ± 2.5	−1.13 ± 0.06	0.065 ± 0.001
Dust BB	6.41 ± 1.1	−0.65 ± 0.06	0.066 ± 0.001

Note. ^aQuoted errors are only statistical uncertainty from the fitting process.

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That the ratio of radio brightness to Hα intensity from the MEM fits is consistently lower than the expected value has long been of concern. The MCMC fits offer some insight, though unfortunately do not resolve the difference. The most important difference between the MEM and the MCMC fits in this case is that the MEM uses the Hα template as a prior in low signal-to-noise regions, while the MCMC fit does not. The result is that in regions of low signal, the degeneracy between synchrotron and free–free causes the MCMC uncertainty in free–free brightness to be large enough to accommodate a large range of possible radio to Hα ratios. Therefore, it becomes necessary to exclude low signal-to-noise regions when calculating the ratio from the MCMC maps.

Due to the uncertainty in the reddening correction to the Hα map itself, it is also customary to exclude regions where the Hα optical depth due to reddening is greater than some value. This unfortunately excludes regions that would otherwise have high signal-to-noise. These two cuts together exclude much of the MCMC maps as unsuitable for analysis. The remaining portion of sky contains bright, mostly discontiguous free–free regions which are also low on dust (and therefore Hα extinction). The largest of these is a region around Gum nebula.

Starting with the free–free maps made from the MCMC process, we define a S/N map as the free–free amplitude divided by the square root of the MCMC variance. We then keep only pixels with S/N>10, τ < 1, and no MCMC error flags. The pixels that remain are largely concentrated in three regions, the Gum nebula, the Ophiuchus complex, and the Orion/Eridanus bubble. The Gum region contains nearly half of the pixels surviving the cut, so for simplicity we restrict our attention to this region, defining it to be any pixel within 30° of (260°, 0°) in Galactic coordinates. Summing all free–free emission in this region and dividing by the total Hα intensity in this region, we estimate that the ratio of radio brightness to Hα intensity h_ff is 9.3 ±  3.2 μKR⁻¹ at the K band for the spinning dust fit, with similar values for the other MCMC models. The uncertainty comes from the variance of the ratio from pixel to pixel; increasing the signal-to-noise threshold decreases the uncertainty somewhat but does not significantly affect the central value. While the central value is consistent with the prediction of 11.4 μKR⁻¹ within this uncertainty, it is also compatible with a reduced electron temperature of 5500 K, an overestimation of the reddening correction by Δτ = 0.3, or some combination of the two.

We find that in order to best fit the 408 MHz data, the spinning dust fit from the five-year MCMC process needs to have its peak frequency adjusted downward by 14% from ν_sd = 4.9 GHz to ν_sd = 4.2 GHz, nearly independent of the offset used for the map. For this value, the frequency at which the flux from the spinning dust component alone peaks is 21 GHz. We have not found any improvement in the fit from including "warm" spinning dust with a peak near 40 GHz, as found by Dobler & Finkbeiner (2008b). The 408 MHz data also introduce some tension, such that the spinning dust model no longer is such a large improvement inside the Galactic plane; in this region χ²_ν = 1.80 for the spinning dust model, compared to χ²_ν = 2.61 for the "base" fit, for 8.7 effective number of degrees of freedom (see Kunz et al. 2006; Gold et al. 2009 for detailed description of effective dof). These values are for the fitted offset of 7.4 K for the 408 MHz map; using a larger offset value of 11.4 K provides slightly better fits (Δχ²_ν = 0.008) for the spinning dust models, while a smaller offset value of 3.4 K provides slightly better fits (Δχ²_ν = 0.074) for models without spinning dust.

The steepening synchrotron model fits the combination of WMAP and 408 MHz data nearly equally as well as the spinning dust model, with χ²_ν = 1.81 in the Galactic plane. The amplitude of c_s is large in the Galactic plane, implying a change of spectral index greater than one per e-fold increase in frequency. This is a sharper change than models of synchrotron steepening predict from aging effects, and so the physical motivation for the model is unclear.

The ARCADE data are not directly comparable to the MCMC fits, due to their greatly different beam and sky coverage. The spinning dust component of the fits for the two regions in the Galactic plane, however, does peak in flux at 22 GHz, consistent with the location of the MCMC peak. The relative amplitude is more difficult to ascertain. For ARCADE, spinning dust is 29% (at l = 33.8) or 43% (at l = 93) of the total flux at 22 GHz, but with the large beam it is impossible to say whether the spinning dust component is relatively diffused or localized on the Galactic plane. For the MCMC fits to WMAP data, the mean spinning dust fraction is considerably lower, at 18% inside the KQ85y7 mask, which suggests that spinning dust may be patchy. Outside of the KQ85y7 mask, the MCMC fits show a mean level of spinning dust consistent with zero within the uncertainty of the fit.

In its low-frequency bands, WMAP observes an excess of emission above what was predicted by scaling the 408 MHz to higher frequencies using the expected spectral index for synchrotron emission. Determining the exact nature of this emission has proven difficult; WMAP has generally treated it as a hard (flatter spectrum) synchrotron component without attempting to explain the origin of such a component. Other suggestions have involved combinations of different types of spinning dust (Finkbeiner 2004; Dobler & Finkbeiner 2008b), though there is typically still a residual "haze" even after those components are fit out (Dobler & Finkbeiner 2008a).

It has been argued that this remainder low-frequency emission has an ellipsoidal shape and is consistent with hard synchrotron emission, possibly from dark matter annihilation in the core of the Galaxy (Hooper et al. 2007). There has been tentative detection of a haze in gamma rays using preliminary data from the Fermi telescope (Dobler et al. 2010).

Interpretation of polarization information toward the center of the Galaxy is difficult, as depolarization through line-of-sight changes in the orientation of the magnetic field can affect the signal significantly. Nonetheless, we search for a hard component in the polarization data using a simplified version of the low-resolution MCMC fit of Dunkley et al. (2009a), shown in Figure 8. We do not detect any significant change of synchrotron spectral index as a function of Galactocentric distance.

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This special fit was done at HEALPix N_side = 16 using only WMAP polarization data, so as to be insensitive to any uncertainties regarding the presence or absence of spinning dust. The fit attempts to model the sky as a sum of three power-law foregrounds: a soft synchrotron component with β = −3.1, a hard synchrotron component with β = −2.39, and a dust component with β = +2.0. These power-law indices were those suggested by the work of Dobler & Finkbeiner (2008a).

The results of the fit are shown in Figure 8. Residuals after the fit are small compared to the noise, and over all bands the mean reduced χ² per pixel is 1.1. For comparison, the synchrotron and dust templates used for polarization cleaning are shown in the right column of the figure. The MCMC result for the soft synchrotron template appears to be essentially a noisy version of the synchrotron template, indicating that the K band is indeed a good proxy for polarized synchrotron emission. For dust, the MCMC and template results differ somewhat. The MCMC hard synchrotron results show no spatial structure beyond WMAP's noise pattern and are consistent with the level of noise bias expected in a map of $P=\sqrt{Q^2+U^2}$.

Figure 9 shows the frequency spectrum of polarized emission for elliptical regions around the Galactic center. In these regions the polarization direction is nearly vertical, and so the Stokes U parameter is negligible for bands K through V and small for W band. The spectra for three different regions are shown, sized 10° × 5°, 20° × 10°, and 30° × 15°. We find no evidence for emission other than soft synchrotron (β = −3.2) and dust (β = +2.0), in particular, no "haze" component appears to be necessary for polarization.

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A full simulation of the WMAP instrument was used to test the MCMC and template cleaning methods, in order to investigate the interaction of systematics in both time-domain data and sky maps. Starting with a set of synthetic sky inputs for the CMB and foregrounds (described below), the scanning of the instrument was applied to the inputs to produce a time stream of data, which was then put through the same entire calibration and map-making pipeline as used for real data.

A random CMB realization was created, starting from the publicly available best-fit cosmological parameters of a ΛCDM model to the combination of five-year WMAP data with supernovae and baryon acoustic oscillations. The CAMB software package (Lewis et al. 2000) was used to generate a model power spectrum and then synfast (Gorski et al. 2005) was used to generate the random sky realization.

Several foregrounds were then added, using high-resolution templates. A synchrotron intensity template was constructed from the 408 MHz data of Haslam et al. (1981), and scaled to higher frequencies with a spectral index with both spatial variations and steepening, in order to test the effects of fitting a simpler model to complicated synchrotron spectral features. A free–free template was made from an extinction-corrected version of the Hα map of Finkbeiner (2003), with a few bright high-latitude sources removed and assuming a spectral index of β = −2.15. The dust template is the 94 GHz prediction of model 8 of Finkbeiner et al. (1999), scaled to other WMAP frequencies with a spectral index of β = 2.0.

Once the simulation inputs were generated, they were passed through a simulation of WMAP's scan strategy, including such effects as thermal gains and baselines in the time-ordered data, loss imbalance and bandpass mismatches, and detector noise with a 1/f component. This simulated time-ordered data was then processed and analyzed in exactly the same way as real observations.

Figure 10 shows a comparison between the "true" simulated input sky maps and the output maps after the map-making process. These are used to test the template cleaning method, as the simulated input foregrounds are generated with structure on scales smaller than the templates used for cleaning. However, no effects due to residual foreground contamination are seen; the cosmological parameters used as input are recovered. Figure 11 then compares the results of the MCMC foreground fit to the input foreground behavior. The largest difference found between the input and output maps from the simulation is in the Galactic plane. This difference is a fraction of a percent of the total intensity, and is entirely consistent with the expected uncertainty in the gain reconstruction.

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The MCMC reconstructs the foregrounds to within the MCMC error, which includes large covariance between synchrotron and free–free brightness. The most important systematic deviation was in the reconstructed synchrotron spectral shape, parameterized with β_s and c_s. This is largely because the simulated model spectrum was more complicated than a power law with constant steepening. This resulted in a bias in the recovered β_s bias of approximately +0.2 in the Galactic plane. This bias was still within the MCMC errors.

Over the course of a full year, the WMAP satellite's scan pattern is such that most points on the sky are observed with a nearly uniform distribution of orientations. The distribution is most symmetric at the ecliptic poles, and least symmetric on the ecliptic plane. Fortuitously, the Galactic center lies near the plane of the ecliptic, with a large angle between the planes of the Galaxy and the ecliptic. The result is that the year can be divided into halves, where WMAP's scanning direction when observing the inner Galactic plane is rotated 180° between the two halves.

This means that maps made from such six-month segments of data are sensitive to beam asymmetries, particularly those where the beam is not equal to itself rotated 180°. This effect is largest in the K band. Figure 12 shows the measured difference of the beam between the six-month intervals, a simple beam model which recreates the effect, and sky maps of the residuals between six-month sky maps and a full year of observation.

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We used this effect to investigate the sensitivity of foreground fitting to beam systematics. For the first five years of data, each year was divided into six months of one scan direction relative to the Galactic center, and six months where the scan direction was reversed. These were then stacked to produce two sets of five-year maps, where the scan directions along the ecliptic have the greatest relative asymmetry. The MCMC foreground fitting was then run for both sets of maps.

The result is shown in Figure 13. Since the largest beam difference is in the K band, low-frequency foregrounds are most strongly affected. The spectral index inferred for synchrotron shows a small gradient across the Galactic plane, with amplitude ∼±0.1 for |b| < 5°. The effect on the CMB is limited; variance outside the KQ85y7 mask is less than 480 μK² (an order of magnitude smaller than intrinsic variance of the CMB), and most of this is from MCMC variations in the dust model. We also emphasize that the six-month intervals were chosen to maximize this asymmetry, which is not seen when full years of data are used to make maps.

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