CMB multipole measurements in the presence of foregrounds

Most analysis of cosmic microwave background spherical harmonic coefficients a_ has focused on estimating the power spectrum C_=⟨|a_|^2⟩ rather than the coefficients themselves. We present a minimum-variance method for measuring a_ given anisotropic noise, incomplete sky coverage and foreground contamination, and apply it to the Wilkinson Microwave Anisotropy Probe (WMAP) data. Our method is shown to constitute lossless data compression in the sense that the widely used quadratic estimators of the power spectrum C_ can be computed directly from our a_-estimators. As the Galactic cut is increased, the error bars a_ on low multipoles go from being dominated by foregrounds to being dominated by leakage from other multipoles, with the intervening minimum defining the optimal cut. Applying our method to the WMAP quadrupole and octopole as an illustration, we investigate the robustness of the previously reported ``axis of evil'' alignment to Galactic cut and foreground contamination.