Hamiltonian formalism and gauge-fixing conditions for cosmological perturbation theory

TL;DR

This paper develops a Hamiltonian formalism for cosmological perturbations using Dirac's constrained system approach, clarifying gauge choices and setting a foundation for quantization of cosmological models.

Contribution

It introduces a Hamiltonian framework for cosmological perturbations with gauge-fixing, extending to multi-fluid universes, facilitating future quantization and higher-order analyses.

Findings

Formalism applied to various gauges

Extended to multi-fluid cosmologies

Provides basis for quantization of perturbations

Abstract

We apply the Dirac procedure for constrained systems to the Arnowitt-Deser-Misner formalism linearized around the Friedmann-Lemaitre universe. We explain and employ some basic concepts such as Dirac observables, Dirac brackets, gauge-fixing conditions, reduced phase space, physical Hamiltonian and physical dynamics. In particular, we elaborate on the key concept which is the canonical isomorphism between different gauge-fixing surfaces. We apply our formalism to describe the reduced phase space of cosmological perturbations in some popular in the literature gauges. Our formalism is first developed for the universe with a single fluid and then extended to the multi-fluid case. The obtained results are a starting point for complete quantization of the cosmological perturbations and the cosmological background. Our approach may be used in future to derive the reduced phase space of higher order perturbations and in more generic cosmological spacetimes.