The Pseudo-$C_l$ method: Cosmic microwave background anisotropy power spectrum statistics for high precision cosmology
Benjamin D. Wandelt, Eric Hivon, Krzysztof M. Gorski
TL;DR
The paper tackles extracting high-precision $C_l$ statistics from partial-sky CMB data with nonuniform noise, where naive estimators bias cosmological inferences. It presents an exact semi-analytic framework to characterize the sampling distribution of $C_l$ under partial sky and anisotropic noise and to derive the couplings between spherical harmonics on the cut sky. The methodology enables fast, unbiased approximate joint estimation of cosmological parameters and suggests extensions such as non-Gaussianity tests and a practical 'poor man's power spectrum estimator'. The framework applies across space-borne, balloon-borne, and ground-based experiments, enhancing the robustness of precision cosmology and inflationary tests.
Abstract
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.