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.