Astrophysical component separation

CMB and foreground separation

The component-separation papers, Planck-2013-XII and Planck-2015-A09, give details of these processing steps. Four separate component-separation methods are used, which we now describe in turn.

NILC

NILC is a linear method for combining the input frequency channels. It implements an "internal linear combination" method with weighting coefficients varying over the sky and over the multipole range up to ℓ=3200, and it does so using "needlets," which are spherical wavelets. A special procedure is used for processing the coarsest needlet scale, which contains the large-scale multipoles.

In practice, our NILC processing depends on several implementation choices, as follows.

Input channels

In this implementation, the NILC algorithm is applied to all Planck channels from 44 to 857 GHz, omitting only the 30-GHz channel.

Pre-processing of point sources

Identical to the SMICA pre-processing.

Masking and inpainting

The NILC CMB map is actually produced in a three-step process. In a first step, the NILC weights are computed from covariance matrices evaluated using a Galactic mask removing about 2 % of the sky (and apodized at °). In a second step, those NILC weights are applied to needlet coefficients computed over the complete sky (except for point source masking/subtraction), yielding a NILC CMB estimate over the full sky (except for the point source mask). In other words, the weights are computed over a masked sky, but are applied to a full sky (excluding point sources). In a final step, the pixels masked due to point source processing are replaced by the values of a constrained Gaussian realization ("inpainting").

Spatial localization

The boundaries of the zones used for spatial localization are obtained as iso-level curves of a low resolution map of Galactic emission.

Beam control and transfer function

As in the SMICA processing, the input maps are internally re-smoothed to a 5 arcmin resolution, so the resulting CMB map is automatically synthesized with an effective Gaussian beam of 5 arcmin, according to the unbiased nature of the ILC.

Using SMICA recalibration

In our current implementation, the NILC solution uses the values determined by SMICA for the CMB spectrum.

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 polarization 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, for intensity, we have cleaned the 100-, 143-, and 217-GHz maps using four templates. Three or them are constructed as the difference of the following Planck channels (smoothed to a common resolution to remove the CMB contribution): [30GHz – 44GHz]; [44GHz – 70GHz]; and [545GHz – 353GHz]. A fourth template is given by the 857-GHz channel (smoothed at the resolution of the 545-GHz channel). For polarization we clean maps at frequencies of 70-, 100- and 143-GHz using three templates for each channel. In particular, we use [30GHz – 44GHz] smoothed to a common resolution, [353GHz – 217GHz] at 10 arcmin, and [217GHz – 143GHz] at 1° resolution to clean the 70- and 100-GHz maps. To clean the 143-GHz channel, the last template is replaced by [217GHz – 100GHz] at 1° resolution. Before constructing the templates, for both intensity and polarization, we perform inpainting at the positions of detected point sources to reduce contamination in the final map.

A linear combination of the templates is then subtracted from the Planck sky map at the considered frequency, in order to produce the clean CMB map. The coefficients of the linear combination are obtained by minimizing 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). Inpainting of point sources is also carried out in the clean maps.

The final CMB intensity map has then been constructed by combining the 143- and 217-GHz cleaned maps by weighting them 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 arcmin at N_side=2048. The final CMB polarization map has been obtained by combining the 100- and 143-GHz clean maps at N_side=1024 and has a resolution of 10 arcmin.

SMICA

SMICA reconstructs a CMB map as a linear combination in the harmonic domain of N_chan input frequency maps with weights that depend on multipole ℓ. Given the N_chan× 1 vector x_ℓm of spherical harmonic coefficients for the input maps, it computes coefficients s_ℓm for the CMB map as

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

where the N_chan× 1 vector w_ℓ , which contains the multipole-dependent weights, is built to give 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}^\dagger \mathbf{R}_\ell^{-1} \mathbf{a}}, [/math]

where vector a is the emission spectrum of the CMB evaluated at each channel (allowing for possible inter-channel recalibration factors) and R_ℓ is the N_chan × N_chan spectral covariance matrix of x_ℓm. Taking R_ℓ in the second equation to be the sample spectral covariance matrix &#344_ℓ 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}^\dagger[/math]

would implement a simple harmonic-domain ILC. However, this is not what SMICA does. As discussed below, we instead use a model R_ℓ (θ) and determine the covariance matrix to be used in the second equation by fitting R_ℓ(θ) to &#344_ℓ. 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 as a superposition of CMB, noise, and foregrounds. The latter are not parametrically modelled; instead, we represent the total foreground emission by d templates with arbitrary frequency spectra, angular spectra and correlations, i.e.,

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

where C_ℓ is the angular power spectrum of the CMB, A is a N_chan × d matrix, P_ℓ is a positive d × d matrix, and N_ℓ is a diagonal matrix representing the noise power spectrum. The parameter vector θ contains all or part of the quantities in the above equation.

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

The actual implementation of SMICA includes the following steps:

Inputs

All nine Planck frequency channels from 30 to 857 GHz, harmonically transformed up to ℓ = 4000.

Fit

In practice, the SMICA fit (i.e., the minimization of the fourth equation above), is conducted in three successive steps. We first estimate the CMB spectrum by fitting all model parameters over a clean fraction of sky in the range 100 ≤ ℓ ≤ 680 and retaining the best fit value for vector a. In the second step, we estimate the foreground emissivity by fixing a to its value from the previous step and fitting all the other parameters over a large fraction of sky in the range 4 ≤ ℓ ≤ 150 and retaining the best fit values for the matrix A. In the last step, we fit all power spectrum parameters; that is, we fix a and A to their previously found values and fit for C_ℓ and P_ℓ at each ℓ.

Beams

The discussion thus far has assumed that all input maps have the same resolution and effective beam. Since the observed maps actually vary in resolution, we process the input maps in the following way. To the ith input map with effective beam b_i(ℓ) and sampled on a HEALPix grid with Nⁱ_side, the CMB sky multipole s_ℓm actually contributes s_ℓma_ib_i(ℓ) p_i(ℓ), where p_i(ℓ) is the pixel window function for the grid at Nⁱ_side. Seeking a final CMB map at 5-arcmin resolution, the highest resolution of Planck, we work with input spherical harmonics re-smoothed to 5 arcmins, ~x_ℓ; that is, SMICA operates on vectors with entries ~xⁱ_ℓm = xⁱ_ℓmb₅(ℓ) / b_i(ℓ) / p_i(ℓ), where b₅(ℓ) is a 5 arcmin Gaussian beam function. By construction, SMICA then produces a CMB map with an effective Gaussian beam of 5 arcmin (without the pixel window function).

Pre-processing

We start by fitting point sources with S/N > 5 in the PCCS catalogue in each input map. If the fit is successful, the fitted point source is removed from the map; otherwise it is masked and the hole inpainted. This is done at all frequencies except 545 and 857 GHz, where all point sources with S/N > 7.5 are masked and inpainted.

Masking and inpainting

In practice, SMICA uses a small Galactic mask, leaving 97% of the sky. However, we deliver a full-sky CMB map in which the masked pixels (Galactic emission and point sources) are replaced by a constrained Gaussian realization.

Binning

In our implementation, we use binned spectra.

High ℓ

Since there is little point trying to model the spectral covariance at high multipoles (because the sample estimate is sufficient), SMICA implements a simple harmonic ILC at ℓ > 1500; i.e., it applies the filter (second equation above) with R_ℓ = Ř_ℓ.

Viewed as a filter, SMICA can be summarized by the weights w_ℓ applied to each input map as a function of multipole. In this sense, SMICA is strictly equivalent to co-adding the input maps after convolution by specific axisymmetric kernels directly related to the corresponding entry of w_ℓ. The SMICA weights used here are shown below for input maps in units of K[math]_\rm{RJ}[/math]. They show, in particular, the (expected) progressive attenuation of the lowest resolution channels with increasing multipole.

Weights w_ℓ given by SMICA to the input maps, after they are re-smoothed to 5 arcmin and expressed in K_RJ, as a function of multipole.

Commander-Ruler

The Commander-Ruler (C-R) approach implements Bayesian component separation in pixel space, fitting a parametric model to the data by sampling the posterior distribution for the model parameters. For computational reasons, the fit is performed in a two-step procedure: First, both foreground amplitudes and spectral parameters are found at low-resolution using MCMC/Gibbs sampling algorithms (Jewell et al. 2004; Wandelt et al. 2004; Eriksen et al. 2004, 2007, 2008). Second, the amplitudes are recalculated at high resolution by solving the generalized least squares system (GLSS) per pixel with the spectral parameters fixed to their values from the low-resolution run. For the CMB-oriented analysis presented in this paper, we use only the seven lowest Planck frequencies, i.e., from 30 to 353 GHz. We first downgrade each frequency map from its native angular resolution to a common resolution of 40 arcminutes and re-pixelize at HEALPix N[math]_\rm{side}[/math] = 256. Second, we set the monopoles and dipoles for each frequency band using a method that locally conserves spectral indices (Wehus et al. 2013, in preparation). We approximate the effective instrumental noise as white with an RMS per pixel given by the Planck scanning pattern and an amplitude calibrated by smoothing simulations of the instrumental noise including correlations to the same resolution. For the high-resolution analysis, the important pre-processing step is the upgrading of the effective low-resolution mixing matrices to full Planck resolution: this is done by repixelizing from N[math]_\rm{side}[/math] = 256 to 2048 in harmonic space, ensuring that potential pixelization effects from the low-resolution map do not introduce sharp boundaries in the high-resolution map.