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

The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmological 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. We derive the asymptotic expansions of the expected Euler characteristic density incorporating skewness and kurtosis, which is a 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 precisely approximates the true Euler characteristic density.