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

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.

arXiv commentReferences added and some changes in the introduction. This new version appears in Classical and Quantum Gravity

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• The Chaplygin Gas Equation of State for the Quantized Free Scalar Field on Cosmological Spacetimes
• Applications of the quaternionic Jordan form to hypercomplex geometry
• Almost Abelian Lie groups, subgroups and quotients
• Global and Concrete Quantizations on General Type I Groups
• Explicit construction of Hadamard states for Quantum Field Theory in curved spacetimes
• States of Low Energy on Bianchi I spacetimes