Hamiltonian formalism for cosmological perturbations: fixing the gauge

TL;DR

This paper develops a systematic method to identify and fix pathological gauges in cosmological perturbation theory, ensuring well-defined gauge conditions and enabling proper quantization of perturbations.

Contribution

It introduces a criterion to detect and complete partially-fixed gauges into healthy gauges, and constructs all gauge-invariant variables within the Hamiltonian formalism.

Findings

Identifies conditions under which gauges are pathological.

Provides a procedure to complete partial gauges into healthy ones.

Ensures proper quantization of cosmological perturbations.

Abstract

Cosmological perturbation theory is an example of a gauge theory, where gauge transformations correspond to changes in the space-time coordinate system. To determine physical quantities, one is free to introduce gauge conditions (\ie to work with specific space-time coordinates), and such conditions are often used to simplify technical aspects of the calculation or to facilitate the interpretation of the physical degrees of freedom. Some of the prescriptions introduced in the literature are known to fix the gauge only partially, but it is commonly assumed that the remaining gauge degrees of freedom can be fixed somehow. In this work, we show that this is not necessarily the case, and that some of these gauges are indeed pathological. We derive a systematic procedure to determine whether a gauge is pathological or not, and to complete partially-fixed gauges into healthy gauges when this is possible. In this approach, the Lagrange multipliers (\ie the perturbed lapse and shift in the ADM formalism) cannot appear in the off-shell definition of the gauges, they necessarily arise as on-shell consequences of the gauge conditions. As illustrative applications, we propose an alternative, non-pathological formulation of the synchronous gauge, and we show that the uniform-expansion gauge (as well as any gauge ensuring vanishing lapse perturbations) can hardly be made healthy. Our methodology also allows us to construct all gauge-invariant variables. We further show that our non-pathological criterion for gauges is also the one that ensures Dirac brackets to be properly defined. This allows cosmological perturbations to be quantised in a gauge-fixed way. We finally discuss possible generalisations of our formalism.