Explicit harmonic and spectral analysis in Bianchi I-VII type cosmologies

TL;DR

This paper provides a comprehensive spectral analysis of Bianchi I-VII groups, explicitly computing their dual spaces, Plancherel formulas, and Laplace spectra, with applications to mode decomposition in cosmological models.

Contribution

It offers explicit calculations of dual spaces, spectral properties, and eigenfunctions for Bianchi I-VII groups, facilitating mode analysis in cosmology.

Findings

Explicit dual space characterizations for Bianchi I-VII groups

Derivation of explicit Plancherel formulas

Spectral and eigenfunction analysis of the Laplace operator

Abstract

The solvable Bianchi I-VII groups which arise as homogeneity groups in cosmological models are analyzed in a uniform manner. The dual spaces (the equivalence classes of unitary irreducible representations) of these groups are computed explicitly. It is shown how parameterizations of the dual spaces can be chosen to obtain explicit Plancherel formulas. The Laplace operator $\Delta$ arising from an arbitrary left invariant Riemannian metric on the group is considered, and its spectrum and eigenfunctions are given explicitly in terms of that metric. The spectral Fourier transform is given by means of the eigenfunction expansion of $\Delta$. The adjoint action of the group automorphisms on the dual spaces is considered. It is shown that Bianchi I-VII type cosmological spacetimes are well suited for mode decomposition. The example of the mode decomposed Klein-Gordon field on these spacetimes is demonstrated as an application.