Hamiltonian approach to 2nd order gauge invariant cosmological perturbations

TL;DR

This paper develops a Hamiltonian formalism to define and analyze second-order gauge-invariant tensor modes in cosmological perturbation theory, enabling a clearer understanding of their properties without gauge fixing.

Contribution

It introduces a Hamiltonian approach using Faddeev-Jackiw reduction to derive gauge-invariant variables up to second order and relates different slicings without additional gauge fixing.

Findings

Derived gauge-invariant variables up to second order

Reduced third order action without gauge fixing

Clarified relation between uniform-φ and Newtonian slicings

Abstract

In view of growing interest in tensor modes and their possible detection, we clarify the definition of tensor modes up to 2nd order in perturbation theory within the Hamiltonian formalism. Like in gauge theory, in cosmology the Hamiltonian is a suitable and consistent approach to reduce the gauge degrees of freedom. In this paper we employ the Faddeev-Jackiw method of Hamiltonian reduction. An appropriate set of gauge invariant variables that describe the dynamical degrees of freedom may be obtained by suitable canonical transformations in the phase space. We derive a set of gauge invariant variables up to 2nd order in perturbation expansion and for the first time we reduce the 3rd order action without adding gauge fixing terms. In particular, we are able to show the relation between the uniform-$\phi$ and Newtonian slicings, and study the difference in the definition of tensor modes in these two slicings.