CMB power spectrum estimation with non-circular beam and incomplete sky coverage

Abstract

Over the last decade, measurements of the CMB anisotropy has spearheaded the remarkable transition of cosmology into a precision science. However, addressing the systematic effects in the increasingly sensitive, high resolution, ‘full’ sky measurements from different CMB experiments pose a stiff challenge. The analysis techniques must not only be computationally fast to contend with the huge size of the data, but, the higher sensitivity also limits the simplifying assumptions which can then be invoked to achieve the desired speed without compromising the final precision goals. While maximum likelihood is desirable, the enormous computational cost makes the suboptimal method of power spectrum estimation using Pseudo-Cl unavoidable for high resolution data. The debiasing of the Pseudo-Cl needs account for non-circular beams, together with non-uniform sky coverage. We provide an analytic framework for correcting the power spectrum for the effect of beam noncircularity and non-uniform sky coverage (including incomplete/masked sky maps). The approach is perturbative in the distortion of the beam from non-circularity allowing for rapid computations when the beam is mildly non-circular. When non-circular beam effect is important, we advocate that it is computationally advantageous to employ ‘soft’ azimuthally apodized masks whose spherical harmonic transform die down fast with m.