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 ∆ 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 ∆. 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.

Global and concrete quantizations on general type I groups

“…Although the results of the previous section are explicit (if one follows closely the details in [7,8]), they are quite complicated. In certain particular cases it might be easier to rely on some direct treatment, as [2].…”

“…Then the Lie bracket is [(α, β; γ), (α ′ , β ′ ; γ ′ )] = γM (α ′ , β ′ ) − γ ′ M (α, β); 0 . From now on we shall rely on results from [2], in which the authors found easier to apply directly Mackey's induced representation theory for semidirect product than the parametrization methods of Currey. The emphasis is on the orbit structure of the contragredient action M ⊥ of R on the dual R 2 ≡ R 2 .…”

“…The (1-dimensional) algebraic submanifolds Σ , composed of one or several connected components, and its relevant measures dρ(σ) = ρ(σ)dσ, will be indicated below, via a parametrization, case by case. The generic classes of irreducible representations can all be realized on the Hilbert space H σ = L 2 (R) as To be more specific, one has to invoke the Bianchi classification [15,25] and results describing the parametrization spaces from [2]. It can be shown that two such semi-direct product Lie algebras given respectively by M where q = 0, −1 and p ≥ 0 , with corresponding group actions e tM IV = e t 0 te t e t , e tM V = e t 0 0 e t , e tM (q) VI = e t 0 0 e −qt , e tM (p) VII = e pt cos t −e pt sin t e pt sin t e pt cos t .…”

“…The densities γ are indicated up to a strictly positive constant. Details, including more formulae and a picture of the orbits and of the cross-sections Σ , can be found in [2]; they help to understand the splittings. The results are roughly compatible with Currey's theory.…”

States of low energy in homogeneous and inhomogeneous expanding spacetimes

“…The set of spacetimes with homogeneous Cauchy hypersurfaces encompasses the well known Bianchi spacetimes [19]. Hadamard states were recently constructed on these spacetimes by the authors of [20]. We will briefly comment on examples of spacetimes whose Cauchy hypersurfaces are compact and homogeneous.…”