A unified mode decomposition method for physical fields in homogeneous cosmology

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 analysis. 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.

The Chaplygin Gas Equation of State for the Quantized Free Scalar Field on Cosmological Spacetimes

“…In the second part of the second section, we will specify the introduced notions to the case of a spatially flat FRW background. That is, based on former results by Lüders & Roberts [37] and Schlemmer [46] on the mode decomposition of homogeneous and isotropic states as well as their renormalization, as described by Eltzner & Gottschalk [18], we explicitly calculate the renormalized quantized EMT for homogeneous and isotropic Hadamard (HIH) states in terms of a plane wave mode decomposition − note that a similar mode decomposition for states of spin-0 and spin-1 fields on homogeneous spacetimes was recently achieved by Avetisyan [1] and Avetisyan & Verch [2].…”

Hadamard States from Light-like Hypersurfaces

“…This construction has been generalised to encompass almost equilibrium states in [Ku08] and expanding spacetimes with less symmetry in [TB13]. Noteworthy in this context is the thorough analysis of the mode expansion presented in [Av14].…”

Constructing Hadamard States via an Extended Møller Operator

“…where x = (t, x), y = (t ′ , y) while j = 1, 2 is introduced to distinguish between the values m 1 and m 2 of the mass, which are our starting and arrival point respectively. Here each ψ k is an eigenfunction of the positive elliptic operator K, that is Kψ k = λ 2 k ψ k , λ k ∈ R and dµ(k) is a suitable spectral measure -see [Ave14]. At the same time each T k,j (t) obeys to the ODE:…”