Statistics of statistical anisotropy measures

Abstract

Cosmic Microwave Background (CMB) is a Gaussian random field to a sufficient approximation, and its statistics is completely specified by the 2-point correlation function, which, most generally, can be expanded in Bipolar Spherical Harmonic (BipoSH) basis. Statistical Isotropy (SI) of 2-point correlation function is a common assumption in cosmology, which needs to be tested. Any SI violating signal can be searched in the expansion BipoSH coefficients. We have analytically evaluated the moments and the distribution of these coefficients using characteristic function approach. We have found that coefficients with M = 0 have an exact analytical form for any order moment. For the remaining BipoSH coefficients with M ≠ 0, the moments have to be supplemented with a correction term. We have verified our results with measurements of BipoSH coefficients on numerically simulated statistically isotropic CMB maps.