Gauge-invariant Formulation of the Second-order Cosmological Perturbations

TL;DR

This paper develops a gauge-invariant framework for second-order cosmological perturbations in a homogeneous isotropic universe, showing how non-linear effects generate vector and tensor modes from scalar modes.

Contribution

It provides a complete gauge-invariant formulation of second-order perturbations without gauge fixing, including derivation of Einstein equations and mode coupling effects.

Findings

Tensor and vector second-order modes can be generated by scalar mode coupling.

The formulation is applicable when first-order vector and tensor modes are negligible.

Non-linear Einstein effects lead to mode generation in cosmological perturbations.

Abstract

Gauge invariant treatments of the second order cosmological perturbation in a four dimensional homogeneous isotropic universe filled with the perfect fluid are completely formulated without any gauge fixing. We derive all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. These equations imply that the tensor and the vector mode of the second order metric perturbations may be generated by the scalar-scalar mode coupling of the linear order perturbations as the result of the non-linear effects of the Einstein equations.