Gauge-invariant perturbations around symmetry-reduced sectors of general relativity: applications to cosmology

We develop a gauge- invariant canonical perturbation scheme for perturbations around symmetry- reduced sectors in generally covariant theories, such as general relativity. The central objects of investigation are gauge- invariant observables which encode the dynamics of the system. We apply this scheme to perturbations around a homogeneous and isotropic sector ( cosmology) of general relativity. The background variables of this homogeneous and isotropic sector are treated fully dynamically which allows us to approximate the observables to an arbitrary high order in a self- consistent and fully gauge- invariant manner. Methods to compute these observables are given. The question of back- reaction effects of inhomogeneities onto a homogeneous and isotropic background can be addressed in this framework. We illustrate the latter by considering homogeneous but anisotropic Bianchi- I cosmologies as perturbations around a homogeneous and isotropic sector.