Locally extracting scalar, vector and tensor modes in cosmological perturbation theory

TL;DR

This paper introduces local differential operators to extract scalar, vector, and tensor modes in cosmological perturbation theory, simplifying the analysis of gauge-invariant quantities at second order.

Contribution

It provides a novel local method for extracting tensor modes, complementing existing local procedures for scalar and vector modes, enhancing second-order perturbation analysis.

Findings

Local scalar mode extraction via divergences

Local vector mode extraction using differential operations

New operators for gauge-invariant second-order quantities

Abstract

Cosmological perturbation theory relies on the decomposition of perturbations into so-called scalar, vector and tensor modes. This decomposition is non-local and depends on unknowable boundary conditions. The non-locality is particularly important at second- and higher-order because perturbative modes are sourced by products of lower-oder modes, which must be integrated over all space in order to isolate each mode. However, given a trace-free rank-2 tensor, a locally defined scalar mode may be trivially derived by taking two divergences, which knocks out the vector and tensor degrees of freedom. A similar local differential operation will return a pure vector mode. This means that scalar and vector degrees of freedom have local descriptions. The corresponding local extraction of the tensor mode is unknown however. We give it here. The operators we define are useful for defining gauge-invariant quantities at second-order. We perform much of our analysis using an index-free `vector-calculus' approach which makes manipulating tensor equations considerably simpler.