Perturbation of the expected Minkowski functional for weakly non-Gaussian isotropic fields on a bounded domain
share
The Minkowski functionals (MF) including the Euler characteristic (EC) statistics are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we consider the Minkowski functional of the excursion set for a weakly non-Gaussian isotropic smooth random field on an arbitrary dimensional compact domain. The weak non-Gaussianity is represented by the N-point correlation function of the order O(ν^N-2), where ν≪ 1 is a non-Gaussianity parameter. We obtain the perturbation expansions of the expected Euler characteristic and the Minkowski functional of the excursion set up to O(ν^2) including the skewness and kurtosis. The resulting formula reveals the local power property of the Minkowski functional as a statistic for testing Gaussianity. Moreover, up to an arbitrary order in ν, the perturbation formula for the expected Minkowski functional is shown to be a linear combination of the Euler characteristic density function multiplied by the Lipschitz-Killing curvature of the index set, which has the same form as the Gaussian kinematic formula (GKF). The application of the obtained perturbation formula in cosmic research is discussed.
READ FULL TEXT