High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere

TL;DR

This paper investigates the high-frequency asymptotic behavior of Lipschitz-Killing curvatures of excursion sets on the sphere, providing precise estimates useful for analyzing cosmological data such as the CMB.

Contribution

It introduces a novel asymptotic analysis of geometric functionals for nonlinear transforms of spherical random fields, including spherical harmonics and wavelet coefficients.

Findings

Derived accurate asymptotic formulas for Euler-Poincaré characteristics

Applied results to cosmological data analysis, especially CMB

Enhanced understanding of excursion probabilities on the sphere

Abstract

In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler-Poincar\'{e} characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the investigation of asymmetries and anisotropies in cosmological data. The statistics we focus on are also suitable to deal with spherical random fields which can only be partially observed, the canonical example being provided by the masking effect of the Milky Way on Cosmic Microwave Background (CMB) radiation data.