The Pseudo-$C_l$ method: Cosmic microwave background anisotropy power spectrum statistics for high precision cosmology

As the era of high precision cosmology approaches, the empirically determined power spectrum of the microwave background anisotropy $C_l$ will provide a crucial test for cosmological theories. We present an exact semi-analytic framework for the study of the ampling statistics of the $C_l$ resulting from observations with partial sky coverage and anisotropic noise distributions. This includes space-borne, air-borne and ground-based experiments. We apply this theory to demonstrate its power for constructing fast but unbiased approximate methods for the joint estimation of cosmological parameters. Further applications, such as a test for possible non-Gaussianity of the underlying theory and a ``poor man's power spectrum estimator'' are suggested. An appendix derives recursion relations for the efficient computation of the couplings between spherical harmonics on the cut sky.