Invariant random fields in vector bundles and application to cosmology

TL;DR

This paper develops a spectral theory for invariant random fields in vector bundles, with applications to cosmology, particularly in analyzing the cosmic microwave background using group representations.

Contribution

It introduces a spectral decomposition framework for invariant random fields in vector bundles induced by Lie group representations, applied to cosmological models.

Findings

Spectral decomposition of invariant random fields in homogeneous vector bundles.

Application of the theory to cosmic microwave background analysis.

A theorem establishing equivalence of different cosmological assumptions.

Abstract

We develop the theory of invariant random fields in vector bundles. The spectral decomposition of an invariant random field in a homogeneous vector bundle generated by an induced representation of a compact connected Lie group $G$ is obtained. We discuss an application to the theory of cosmic microwave background, where $G=SO(3)$. A theorem about equivalence of two different groups of assumptions in cosmological theories is proved.