Exact likelihood evaluations and foreground marginalization in low resolution WMAP data

The large scale anisotropies of Wilkinson Microwave Anisotropy Probe (WMAP) data have attracted a lot of attention and have been a source of controversy, with many favorite cosmological models being apparently disfavored by the power spectrum estimates at low l. All the existing analyses of theoretical models are based on approximations for the likelihood function, which are 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 using the WMAP derived maps of these foregrounds. This improves over the template based foreground subtraction used in the original analysis, which can remove some of the cosmological signal and may lead to a suppression of power. As a result we find an increase in power at low multipoles. For the quadrupole the maximum likelihood values are rather uncertain and vary between 140 and 220 μK². On the other hand, the probability distribution away from the peak is robust and, assuming a uniform prior between 0 and 2000 μK², the probability of having the true value above 1200 μK² (as predicted by the simplest cold dark matter model with a cosmological constant) is 10%, a factor of 2.5 higher than predicted by the WMAP likelihood code. We do not find the correlation function to be unusual beyond the low quadrupole value. We develop a fast likelihood evaluation routine that can be used instead of WMAP routines for low l values. We apply it to the Markov chain Monte Carlo analysis to compare the cosmological parameters between the two cases. The new analysis of WMAP either alone or jointly with the Sloan Digital Sky Survey (SDSS) and the Very Small Array (VSA) data reduces the evidence for running to less than 1σ, giving α_s=-0.022±0.033 for the combined case. The new analysis prefers about a 1σ lower value of Ω_m, a consequence of an increased integrated Sachs-Wolfe (ISW) effect 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 parameter estimation from the WMAP data. They are robust in the sense that they do not change significantly with frequency, mask, or details of foreground template marginalization. The marginalization approach presented here is the most conservative method to remove the foregrounds and should be particularly useful in the analysis of polarization, where foreground contamination may be much more severe.