Astrophysical component separation

CMB and foreground separation

SEVEM

The aim of Sevem is to produce clean CMB maps at one or several frequencies by using a procedure based on template fitting. The templates are internal, i.e., they are constructed from Planck data, avoiding the need for external data sets, which usually complicates the analyses and may introduce inconsistencies. The method has been successfully applied to Planck simulations Leach et al., 2008 and to WMAP polarisation data Fernandez-Cobos et al., 2012. In the cleaning process, no assumptions about the foregrounds or noise levels are needed, rendering the technique very robust.

The input maps used are all the Planck frequency channels. In particular, we have cleaned the 100, 143 GHz and 217 GHz maps using four templates constructed as the difference of the following Planck channels (smoothed to a common resolution): (30-44) GHz, (44-70) GHz, (545-353) GHz and (857-545)GHz.

The templates are constructed by subtracting two neighbouring Planck frequency channel maps, after first smoothing them to a common resolution to ensure that the CMB signal is properly removed. A linear combination of the templates is then subtracted from the Planck sky map at the frequency to be cleaned, in order to produce the clean CMB. The coefficients of the linear combination are obtained by minimising the variance of the clean map outside a given mask. Although we exclude very contaminated regions during the minimization, the subtraction is performed for all pixels and, therefore, the cleaned maps cover the full-sky (although we expect that foreground residuals are present in the excluded areas).

An additional level of flexibility can also be considered: the linear coefficients can be the same for all the sky, or several regions with different sets of coefficients can be considered. The regions are then combined in a smooth way, by weighting the pixels at the boundaries, to avoid discontinuities in the clean maps. In order to take into account the different spectral behaviour of the foregrounds at low and high galactic latitudes, we have chosen to use two regions: the region with the 3 per cent brightest Galactic emission, and the region with the remaining 97 per cent of the sky.

The final CMB map has then been constructed by combining the 143 and 217 GHz cleaned maps by weighting the maps in harmonic space taking into account the noise level, the resolution and a rough estimation of the foreground residuals of each map (obtained from realistic simulations). This final map has a resolution corresponding to a Gaussian beam of FWHM =5 arcminutes.

SMICA

A linear method, SMICA reconstructs a CMB map as a linear combination in the harmonic domain of [math]N_{chan}[/math] input frequency maps with weights that depend on multipole [math]\ell[/math]. Given the [math]N_{chan} × 1[/math] vector [math]\mathbf{x}_{\ell m}[/math] of spherical harmonic coefficients for the input maps, it computes coefficients [math]s_{\ell m}[/math] for the CMB map as

[math]\label{eq:smica:shat} \hat{s}_{\ell m} = \mathbf{w}^†_\ell \mathbf{x}_{\ell m}[/math]

where the [math]N_{chan} × 1[/math] vector [math]\mathbf{w}_\ell [/math] which contains the multipole-dependent weights is built to offer unit gain to the CMB with minimum variance. This is achieved with

[math]\label{eq:smica:w} \mathbf{w}_\ell = \frac{\mathbf{R}_\ell ^{-1} \mathbf{a}}{\mathbf{a}^† \mathbf{R}_\ell ^{-1} \mathbf{a}} [/math]

where vector [math]\mathbf{a}[/math] is the emission spectrum of the CMB evaluated at each channel (allowing for possible inter-channel recalibration factors) and [math] \mathbf{R}_\ell [/math] is the [math]N_{chan} × N_{chan}[/math] spectral covariance matrix of [math]\mathbf{x}_{\ell m}[/math]. Taking [math]\mathbf{R}_\ell [/math] in Eq. \ref{eq:smica:w} to be the sample spectral covariance matrix [math]\mathbf{\hat{R}}_\ell [/math] of the observations:

[math]\label{eq:smica:Rhat} \mathbf{\hat{R}}_\ell = \frac{1}{2 \ell + 1} \sum_m \mathbf{x}_{ \ell m} \mathbf{x}_{\ell m}^†[/math]

would implement a simple harmonic-domain ILC. This is not what SMICA does. As discussed below, we instead use a model [math]\mathbf{R}_\ell (θ)[/math] and determine the covariance matrix to be used in Eq. \ref{eq:smica:w} by fitting [math]\mathbf{R}_\ell (θ)[/math] to [math]\mathbf{\hat{R}}_\ell [/math]. This is done in the maximum likelihood sense for stationary Gaussian fields, yielding the best fit model parameters θ as

[math]\label{eq:smica:thetahat} \hat{θ} = \rm{arg \, min}_θ \sum_\ell (2\ell + 1) ( \mathbf{\hat{R}}_\ell \mathbf{R}_\ell (θ)^{-1} \, +\, log \, det \, \mathbf{R}_\ell (θ)).[/math]

SMICA models the data is a superposition of CMB, noise and foregrounds. The latter are not parametrically modelled; instead, we represent the total foreground emission by [math]d[/math] templates with arbitrary frequency spectra, angular spectra and correlations:

[math] \label{eq:smica:Rmodel} \mathbf{R}_\ell (θ) = \mathbf{aa}^† \, C_\ell \, + \, \mathbf{A P}_\ell \mathbf{A}^† \, + \, \mathbf{N}_\ell [/math]

where [math]C_\ell [/math] is the angular power spectrum of the CMB, [math]\mathbf{A}[/math] is a [math]N_{chan} ×d[/math] matrix, [math]\mathbf{P}_\ell [/math] is a positive [math]d×d[/math] matrix, and [math]\mathbf{N}_\ell [/math] is a diagonal matrix representing the noise power spectrum. The parameter vector [math]θ[/math] contains all or part of the quantities in Eq. \ref{eq:eq3}.

The above equations summarize the founding principles of SMICA; its actual operation depends on a choice for the spectral model [math]\mathbf{R}_\ell (θ)[/math] and on several implementation-specific details.