Model independent foreground power spectrum estimation using WMAP 5-year data

In this paper, we propose and implement on WMAP 5 yr data a model independent approach of foreground power spectrum estimation for multifrequency observations of the CMB experiments. Recently, a model independent approach of CMB power spectrum estimation was proposed by Saha et al. 2006. This methodology demonstrates that the CMB power spectrum can be reliably estimated solely from WMAP data without assuming any template models for the foreground components. In the current paper, we extend this work to estimate the galactic foreground power spectrum using the WMAP 5 yr maps following a self-contained analysis. We apply the model independent method in harmonic basis to estimate the foreground power spectrum and frequency dependence of combined foregrounds. We also study the behavior of synchrotron spectral index variation over different regions of the sky. We use the full sky Haslam map as an external template to increase the degrees of freedom, while computing the synchrotron spectral index over the frequency range from 408 MHz to 94 GHz. We compare our results with those obtained from maximum entropy method foreground maps, which are formed in pixel space. We find that relative to our model independent estimates maximum entropy method maps overestimate the foreground power close to galactic plane and underestimates it at high latitudes.

Figure 1

The top left panel shows the Internal Linear Combination Map obtained by the WMAP Team. The top right panel shows the average of 48 clean maps obtained by the model independent method. The bottom left and right panel show the difference between the ILC and MI clean map over the full sky and outside the KQ85 mask, respectively.

Figure 2

Top left panel: The model independent estimate of angular power spectrum for combined foregrounds at WMAP frequency starting from the K band to the W band. In this case, $C_l^F$ is smoothed by a 1° beam and expressed in terms of antenna temperature. Top right panel: Plot of relative difference in estimated power $(C_l^{\mathrm{MEM}}-C_l^{\mathrm{MI}})/C_l^{\mathrm{MEM}}$ with the multipole $l$, where “MI” stands for the foreground power spectrum obtained from model independent analysis. Foreground power spectrum at the K and Ka bands is consistent with the WMAP team. But for the Q, V, and W bands we obtain slightly less power compared to the MEM method. Bottom left panel: The angular power spectrum of combined foregrounds outside the KQ85 mask at WMAP frequencies starting from the K band to the W band expressed in antenna temperature. Bottom right panel: Plot of relative power difference $(C_l^{\mathrm{MEM}}-C_l^{\mathrm{MI}})/C_l^{\mathrm{MEM}}$ with the multipole $l$ outside KQ85. The plot clearly shows that at low multipoles MEM underestimates the foreground power for a masked sky.

Figure 3

The total foreground emission (black dots) spectra obtained using model independent analysis compared to the sum of foreground components (deep blue line connecting black dots) over the full sky. The average synchrotron spectrum from the K to W bands is $-2.6$ assuming the free-free spectral index ${\beta }_f=-2.14$ and dust spectral index ${\beta }_d=1.8$ are constant parameters. The average synchrotron spectral index from the K-Ka band and the Ka-Q band are$-3.0$ and$-2.91$, respectively.

Figure 4

The first column is the combined foreground maps for the K, Ka, Q, V, and W maps from top to bottom obtained by model independent analysis. The second column represents the combined foregrounds maps for the K, Ka, Q, V, and W bands obtained by the MEM method, and the third column is MEM minus model independent foreground maps. Model independent foreground maps for K and Ka are obtain from 3-channel clean maps, whereas the model independent foreground maps for the Q, V, and W bands are obtained from 4-channel clean maps.

Figure 5

Synchrotron spectral index variation over different positions of the sky obtained by model independent foreground analysis. The spectral index behavior clearly shows that ${\beta }_s$ is $-3.5$ at high latitudes and $-2.5$ close to the galactic plane, which is consistent with the WMAP team. For example, the fit for 3 regions is shown in the plot. The region 126 (latitude range [$10°<\ell <29.5°]$, longitude range $[248°<b<269°]$, synchrotron spectral index, $[{\beta }_s=-2.82]$), region 137 ($-59.1°<\ell <-30.7°$, $23.1°<b<44.5°$, ${\beta }_s=-3.70$), region 165 ($-53.3°<\ell <-30.3°$, $248°<b<267°$, ${\beta }_s=-3.33$).