Scalar and Tensor Inhomogeneities from Dimensional Decoupling

We discuss some perturbative techniques suitable for the gauge-invariant treatment of the scalar and tensor inhomogeneities of an anisotropic and homogeneous background geometry whose spatial section naturally decomposes into the direct product of two maximally symmetric Eucledian manifolds, describing a general situation of dimensional decoupling in which $d$ external dimensions evolve (in conformal time) with scale factor $a(\eta)$ and $n$ internal dimensions evolve with scale factor $b(\eta)$. We analyze the growing mode problem which typically arises in contracting backgrounds and we focus our attention on the situation where the amplitude of the fluctuations not only depends on the external space-time but also on the internal spatial coordinates. In order to illustrate the possible relevance of this analysis we compute the gravity waves spectrum produced in some highly simplified model of cosmological evolution and we find that the spectral amplitude, whose magnitude can be constrained by the usual bounds applied to the stochastic gravity waves backgrounds, depends on the curvature scale at which the compactification occurs and also on the typical frequency of the internal excitations.