Correlations between Maxwell's multipoles for gaussian random functions on the sphere

TL;DR

This paper investigates the correlations between Maxwell's multipoles for Gaussian random functions on the sphere, revealing asymptotic behaviors and potential applications to cosmic microwave background analysis.

Contribution

It introduces a method to compute multipole correlations for Gaussian functions on the sphere and connects these results to known asymptotic forms and CMB analysis.

Findings

2-point correlation function matches Hannay's form at high 

Mapping spherical functions to random polynomials facilitates analysis

Results have implications for cosmic microwave background studies

Abstract

Maxwell's multipoles are a natural geometric characterisation of real functions on the sphere (with fixed $\ell$). The correlations between multipoles for gaussian random functions are calculated, by mapping the spherical functions to random polynomials. In the limit of high $\ell,$ the 2-point function tends to a form previously derived by Hannay in the analogous problem for the Majorana sphere. The application to the cosmic microwave background (CMB) is discussed.