Level area of spin random fields: a chaos decomposition

TL;DR

This paper derives an explicit chaos decomposition formula for the level set areas of spin Gaussian fields on SO(3), advancing understanding of their geometric properties and differences from zero spin cases.

Contribution

It provides a novel explicit chaos decomposition for level set areas of spin Gaussian fields, connecting recent chaos analysis to cosmic microwave background modeling.

Findings

Explicit chaos decomposition formula derived

Reveals differences between high frequency and zero spin regimes

Advances understanding of geometric properties of spin fields

Abstract

We study real left-invariant spin Gaussian fields on $SO(3)$, a special class of non-isotropic random fields used to model the polarization of the Cosmic Microwave Background. Leveraging recent results from "New chaos decomposition of Gaussian nodal volumes" (arXiv:2505.22350), we provide an explicit formula for the Wiener-It\^o chaos decomposition of the area measure of level sets of such random fields. Our analysis represents a step forward in the study of second-order asymptotic properties of the Lipschitz-Killing curvatures of excursion sets of spin random fields. Remarkably, our formulas reveal a clear difference between the high frequency regime and the zero spin case.