Vector and tensor contributions to the curvature perturbation at second order

TL;DR

This paper derives the second order evolution equations for the curvature perturbation in cosmology, considering scalar, vector, and tensor contributions, and identifies conditions for its conservation at large scales.

Contribution

It provides a comprehensive derivation of second order curvature perturbation evolution equations including all perturbation types and compares different gauge-invariant definitions.

Findings

Conservation of the curvature perturbation requires negligible anisotropic stress.

The gauge invariant curvature perturbation based on the inverse metric determinant is exactly conserved.

Results are valid at all scales and include all perturbation contributions.

Abstract

We derive the evolution equation for the second order curvature perturbation using standard techniques of cosmological perturbation theory. We do this for different definitions of the gauge invariant curvature perturbation, arising from different splits of the spatial metric, and compare the expressions. The results are valid at all scales and include all contributions from scalar, vector and tensor perturbations, as well as anisotropic stress, with all our results written purely in terms of gauge invariant quantities. Taking the large-scale approximation, we find that a conserved quantity exists only if, in addition to the non-adiabatic pressure, the transverse traceless part of the anisotropic stress tensor is also negligible. We also find that the version of the gauge invariant curvature perturbation which is exactly conserved is the one defined with the determinant of the spatial part of the inverse metric.