Multifrequency Wiener filtering of cosmic microwave background data with polarization

One goal of cosmic microwave background (CMB) data analysis is to combine d ata at different frequencies, angular resolutions, and noise levels in orde r best to extract the component with a Planckian spectral behaviour. A mult ifrequency Wiener filtering method has been proposed in this context by Bou chet, Gispert & Puget and in parallel by Tegmark Br Efstathiou. As shown by Bouchet & Gispert, this Linear method is also convenient with which to est imate a priori, given a sky model and an experimental description, the resi dual errors on the CMB power spectrum assuming the foregrounds have been re moved with this method. In this paper, we extend the method to the case in which additional polarization data are available. In particular, we derive the errors on the power spectra involving polarization and show numerical r esults for the specifications of the future CMB space missions MAP and Plan ck(1) when it is assumed that the Galactic synchrotron and dust emission ar e respectively about 40 and 10 per cent polarized. We consider two underlyi ng models for our study: we take a standard CDM model with tau = 0.1 for th e extraction of E-mode polarization and ET cross-correlation; for B-mode po larization we consider a tilted CDM model with n(s) = 0.9, n(t) = -0.1 and T/S = 0.7. We find the following results. (1) The resulting fractional erro rs on E-mode polarization and TE crosscorrelation power spectra are less th an or similar to 10-30 per cent for 50 less than or similar to l less than or similar to 1000 for Planck. The fractional errors are between 50 and 150 per cent for l less than or equal to 50. (2) The corresponding fractional errors for MAP are greater than or equal to 300 per cent for most of the l range. (3) The Wiener filtering gives extraction errors of less than or equ al to twice the expected performance for the combined sensitivity of all th e channels of Planck. For MAP, the corresponding degradation is similar or equal to 4. (4) If, instead of individual modes, one considers band-power e stimates with a logarithmic interval Delta l/l = 0.2 then the fractional er ror for MAP drops to less than or similar to 100 per cent at the Doppler pe ak around l similar or equal to 300 for the ET signal. (5) The fractional e rror for B-mode polarization detection is less than or similar to 100 per c ent with Planck for l less than or equal to 100. A band-power estimate with Delta l/l = 0.2 reduces the fractional errors to less than or similar to 2 5 per cent for 20 less than or equal to l less than or equal to 100.