Microwave background temperature and polarization observations are a powerful way to constrain cosmological parameters if the likelihood function can be calculated accurately. The temperature and polarization fields are correlated, partial-sky coverage correlates power spectrum estimators at different $l$, and the likelihood function for a theory spectrum given a set of observed estimators is non-Gaussian. An accurate analysis must model all these properties. Most existing likelihood approximations are good enough for a temperature-only analysis, however they cannot reliably handle temperature-polarization correlations. We give a new general approximation applicable for correlated Gaussian fields observed on part of the sky. The approximation models the non-Gaussian form exactly in the ideal full-sky limit and is fast to evaluate using a precomputed covariance matrix and set of power spectrum estimators. We show with simulations that it is good enough to obtain correct results at $l\gtrsim 30$ where an exact calculation becomes impossible. We also show that some Gaussian approximations give reliable parameter constraints even though they do not capture the shape of the likelihood function at each $l$ accurately. Finally we test the approximations on simulations with realistically anisotropic noise and asymmetric foreground mask.

• Received 8 January 2008

DOI:https://doi.org/10.1103/PhysRevD.77.103013