Dirac procedure and the Hamiltonian formalism for cosmological perturbations in a Bianchi I universe

TL;DR

This paper develops a canonical Hamiltonian formalism for cosmological perturbations in an anisotropic Bianchi I universe, addressing gauge fixing, observables, and gravitational wave definitions, paving the way for quantization.

Contribution

It introduces a fully canonical Hamiltonian approach to cosmological perturbations in Bianchi I universes, including gauge fixing and gravitational wave analysis, extending to multi-field scenarios.

Findings

Revised definition of gravitational waves in anisotropic backgrounds

Formalism applicable to single and multi-field cosmologies

Framework suitable for quantization of perturbations

Abstract

We apply the Dirac procedure for constrained systems to the Arnowitt-Deser-Misner formalism linearized around the Bianchi I universe. We discuss and employ basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian, canonical isomorphism between different gauge-fixing surfaces and spacetime reconstruction. We relate this approach to the gauge-fixing procedure for non-perturbative canonical relativity. We discuss the issue of propagating a basis for the scalar-vector-tensor decomposition as, in an anisotropic universe, the wavefronts of plane waves undergo a nontrivial evolution. We show that the definition of a gravitational wave as a traceless-transverse mode of the metric perturbation needs to be revised. Moreover there exist coordinate systems in which a polarization mode of the gravitational wave is given entirely in terms of a scalar metric perturbation. We first develop the formalism for the universe with a single scalar field and then extend it to the multi-field case. The obtained fully canonical formalism will serve as a starting point for a complete quantization of the cosmological perturbations and the cosmological background.