Non Gaussian Minkowski functionals and extrema counts for 2D sky maps

TL;DR

This paper develops a rigorous method to compute non-Gaussian Minkowski functionals and extrema counts for 2D spherical sky maps, such as CMB data, accounting for curvature and finite size effects.

Contribution

It extends the theory of non-Gaussian geometrical measures to spherical sky maps, including curvature effects, providing tools for analyzing weakly non-Gaussian fields like the CMB.

Findings

Derived Minkowski functionals with first non-Gaussian correction for spherical fields.

Presented exact treatment of curvature and finiteness effects on 2D sky maps.

Applicable to weakly non-Gaussian CMB analysis.

Abstract

In the conference presentation we have reviewed the theory of non-Gaussian geometrical measures for the 3D Cosmic Web of the matter distribution in the Universe and 2D sky data, such as Cosmic Microwave Background (CMB) maps that was developed in a series of our papers. The theory leverages symmetry of isotropic statistics such as Minkowski functionals and extrema counts to develop post- Gaussian expansion of the statistics in orthogonal polynomials of invariant descriptors of the field, its first and second derivatives. The application of the approach to 2D fields defined on a spherical sky was suggested, but never rigorously developed. In this paper we present such development treating effects of the curvature and finiteness of the spherical space $S_2$ exactly, without relying on the flat-sky approximation. We present Minkowski functionals, including Euler characteristic and extrema counts to the first non-Gaussian correction, suitable for weakly non-Gaussian fields on a sphere, of which CMB is the prime example.