Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields

TL;DR

This paper derives asymptotic expansions for the Euler characteristic density of isotropic central limit random fields, revealing non-Gaussian features in Minkowski functionals relevant to cosmology.

Contribution

It provides the first asymptotic expansion of Minkowski functionals for non-Gaussian isotropic fields using 3- and 4-point correlations.

Findings

Asymptotic formulas accurately approximate Euler characteristic density.

Non-Gaussian features are identified that Minkowski functionals cannot capture.

Application to chi-square fields confirms the expansion's effectiveness.

Abstract

The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, the $k$-point correlation functions ($k$th order cumulants) of which have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-square random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.