Tensor perturbations in quantum cosmological backgrounds

In the description of the dynamics of tensor perturbations on a homogeneous and isotropic background cosmological model, it is well known that a simple Hamiltonian can be obtained if one assumes that the background metric satisfies Einstein classical field equations. This makes it possible to analyse the quantum evolution of the perturbations since their dynamics depends only on this classical background. In this paper, we show that this simple Hamiltonian can also be obtained from the Einstein - Hilbert Lagrangian without making use of any assumption about the dynamics of the background metric. In particular, it can be used in situations where the background metric is also quantized, hence providing a substantial simplification over the direct approach originally developed by Halliwell and Hawking.