A unified mode decomposition method for physical fields in homogeneous cosmology

TL;DR

This paper develops a comprehensive framework for mode decomposition of vector bundle fields in homogeneous cosmological spacetimes, extending previous scalar-focused methods to more general fields and geometries.

Contribution

It introduces a unified approach combining mode decomposition, harmonic analysis, and Fourier analysis for vector bundle fields on general homogeneous spacetimes, including explicit propagator decomposition.

Findings

Mode decomposition limits and uniqueness are characterized.

Weak solutions can be decomposed under analytical conditions.

Invariant bi-distributions are described in Fourier space.

Abstract

The methods of mode decomposition and Fourier analysis of classical and quantum fields on curved spacetimes previously available mainly for the scalar field on Friedman- Robertson-Walker (FRW) spacetimes are extended to arbitrary vector bundle fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier anal- ysis. The limits of applicability and uniqueness of mode decomposition by separation of the time variable in the field equation are found. It is shown how mode decomposition can be naturally extended to weak solutions of the field equation under some analytical assumptions. It is further shown that these assumptions can always be fulfilled if the vector bundle under consideration is analytic. The propagator of the field equation is explicitly mode decomposed. A short survey on the geometry of the models considered in mathematical cosmology is given and it is concluded that practically all of them can be represented by a semidirect homogeneous vector bundle. Abstract harmonic analytical Fourier transform is introduced in semidirect homogeneous spaces and it is explained how it can be related to the spectral Fourier transform. The general form of invariant bi-distributions on semidirect homogeneous spaces is found in the Fourier space which generalizes earlier results for the homogeneous states of the scalar field on FRW spacetimes.