Statistics of Bipolar Representation of CMB maps

TL;DR

This paper analytically derives the statistical distributions of Bipolar Spherical Harmonic coefficients of CMB maps to aid in detecting deviations from isotropy, providing formulas for moments and corrections validated by simulations.

Contribution

It introduces an analytical method to evaluate the moments and distributions of BipoSH coefficients, including correction terms for M≠0, enhancing isotropy violation detection.

Findings

Analytical formulas for moments of BipoSH coefficients with M=0.

Correction terms are necessary for accurate moments at low multipoles.

Validated results with simulations of isotropic CMB maps.

Abstract

Gaussianity of temperature fluctuations in the Cosmic Microwave Background(CMB) implies that the statistical properties of the temperature field can be completely characterized by its two point correlation function. The two point correlation function can be expanded in full generality in the bipolar spherical harmonic(BipoSH) basis. Looking for significant deviations from zero for Bipolar Spherical Harmonic(BipoSH) Coefficients derived from observed CMB maps forms the basis of the strategy used to detect isotropy violation. In order to quantify "significant deviation" we need to understand the distributions of these coefficients. We analytically evaluate the moments and the distribution of the coefficients of expansion($A^{LM}_{l_1 l_2}$), using characteristic function approach. We show that for BipoSH coefficients with M=0 an analytical form for the moments up to any arbitrary order can be derived. For the remaining BipoSH coefficients with $M\neq0$, the moments derived using the characteristic function approach need to be supplemented with a correction term. The correction term is found to be important particularly at low multipoles. We provide a general prescription for calculating these corrections, however we restrict the explicit calculations only up to kurtosis. We confirm our results with measurements of BipoSH coefficients on numerically simulated statistically isotropic CMB maps.