Normal modes for metric fluctuations in a class of higher-dimensional backgrounds

We discuss a gauge invariant approach to the theory of cosmological perturbations in a higher-dimensonal background. We find the normal modes which diagonalize the perturbed action, for a scalar field minimally coupled to gravity, in a higher-dimensional manifold M of the Bianchi-type I, under the assumption that the translations along an isotropic spatial subsection of M are isometries of the full, perturbed background. We show that, in the absence of scalar field potential, the canonical variables for scalar and tensor metric perturbations satisfy exactly the same evolution equation, and we discuss the possible dependence of the spectrum on the number of internal dimensions.