Exact likelihood evaluations and foreground marginalization in low resolution WMAP data

The existing analysis of WMAP data is based on approximations for the likelihood function, which is likely to be inaccurate on large scales. Here we present exact evaluations of the likelihood of the low multipoles by direct inversion of the theoretical covariance matrix for low resolution WMAP maps. We project out the unwanted galactic contaminants rather than subtract them, which can remove some of the cosmological signal and may lead to a suppression of power. We find an increase in power at low multipoles. For the quadrupole the maximum likelihood values are rather uncertain and vary between 140-220\muK^2. On the other hand, the probability distribution away from the peak is robust and, assuming a uniform prior between 0 an 2000\muK^2, the probability of having the true value above 1200\muK^2 (as predicted by the simplest Lambda CDM model) is 10%, a factor of 2.5 higher than predicted by WMAP likelihood code. We develop a fast likelihood evaluation routine that can be used instead of WMAP routines for low l values. We apply it to the MCMC analysis to compare the cosmological parameters between the two cases. The new analysis of WMAP either alone or jointly with SDSS and VSA reduces the reported 2-sigma evidence for running to less than 1-sigma, giving \alpha_s=-0.022\pm 0.033 for the combined case. The new analysis prefers about 1-sigma lower value of Omega_m, a consequence of an increased ISW contribution required by the increase in the spectrum at low l. These results suggest that the details of foreground removal and full likelihood analysis are important for the parameter estimation from WMAP data and these methods should be particularly useful in the analysis of polarisation, where foreground contamination may be much more severe.