Tensor Minkowski Functionals for random fields on the sphere

TL;DR

This paper extends tensor Minkowski Functionals to the sphere, providing analytic expressions and numerical methods to analyze the geometry and statistical properties of random fields, including anisotropy and non-Gaussianity, with applications to cosmic microwave background data.

Contribution

It generalizes tensor Minkowski Functionals to the sphere and develops analytic and numerical tools to study their relation to field properties, including anisotropy and non-Gaussianity.

Findings

Analytic expressions for ensemble expectations of tensor Minkowski Functionals for Gaussian and Rayleigh fields.

Demonstration of the method's ability to detect statistical anisotropy in Galactic foreground maps.

Application to Planck data shows the method's effectiveness in real-world cosmological analysis.

Abstract

We generalize the translation invariant tensor-valued Minkowski Functionals which are defined on two-dimensional flat space to the unit sphere. We apply them to level sets of random fields. The contours enclosing boundaries of level sets of random fields give a spatial distribution of random smooth closed curves. We obtain analytic expressions for the ensemble expectation values for the matrix elements of the tensor-valued Minkowski Functionals for isotropic Gaussian and Rayleigh fields. We elucidate the way in which the elements of the tensor Minkowski Functionals encode information about the nature and statistical isotropy (or departure from isotropy) of the field. We then implement our method to compute the tensor-valued Minkowski Functionals numerically and demonstrate how they encode statistical anisotropy and departure from Gaussianity by applying the method to maps of the Galactic foreground emissions from the PLANCK data.