  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 anal- ysis. 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.
