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

TL;DR

The paper develops a gauge invariant canonical perturbation framework for perturbations around symmetry reduced sectors of general relativity, enabling complete observables to be computed to arbitrary order with full gauge invariance. It builds a perturbation scheme around a dynamically treated symmetry-reduced background (e.g., FLRW cosmology) using clock variables and a projection that separates background from fluctuations, so linear perturbations and backreaction emerge as distinct orders in a controlled expansion. First-order scalar perturbations are derived as gauge invariant complete observables, while second-order terms encode backreaction effects; the authors illustrate the method with a solvable Bianchi I model, showing agreement between perturbative results and exact solutions. The framework provides a robust, background-independent approach to cosmological perturbations and offers potential advantages for quantum gravity, including clearer handling of backreaction and clock-choice dependence.

Abstract

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 arbitrary high order in a self--consistent and fully gauge invariant manner. Methods to compute these observables are given. The question of backreaction 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.

Figures (1)

• Figure 1: Bianchi I as a model for perturbations: Here we compare the exact complete observable evaluated on a phase space point $(A_i,E_i,\Phi,\Pi)$ to the zeroth and second order approximation. The phase space point is given by $A=1,\, E=1m^2,\, a_1=a_2=0.1, \, a_3=-0.2,\, e_1=e_2=0.1 m^2, e_3=-0.2 m^2$ (where we set the speed of light to $c=1$ and the coordinates do not carry units). The momentum $\Pi$ is determined by the constraint (\ref{['bianchicons']}). Choosing a value for $\Phi$ defines the time $\tau$ for which the above values are taken as initial values.