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

December 26, 2012

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.

Keywords:

QFT on curved spacetimes