# Second Order Gauge Invariant Perturbation Theory and Conserved Charges in Cosmological Einstein's Gravity

## TL;DR

This paper develops a gauge-invariant second order perturbation theory framework in cosmological Einstein's gravity, analyzing conserved charges and the gauge properties of perturbations, with implications for understanding gravitational dynamics in cosmology.

## Contribution

It provides a detailed gauge-invariant formulation of first and second order perturbations and clarifies the gauge behavior of conserved charges in cosmological Einstein's gravity.

## Key findings

- Linearized Einstein tensor is gauge-invariant at first order.
- Second order gauge invariance is more complex due to non-invariance of the Einstein tensor.
- The approach relies on decomposing metric perturbations into gauge-variant and gauge-invariant parts.

## Abstract

Recently a new approach in constructing the conserved charges in cosmological Einstein's gravity was given. In this new formulation, instead of using the explicit form of the field equations a covariantly conserved rank four tensor was used. In the resulting charge expression, instead of the first derivative of the metric perturbation, the linearized Riemann tensor appears along with the derivative of the background Killing vector fields. Here we give a detailed analysis of the first order and the second order perturbation theory in a gauge-invariant form in cosmological Einstein's gravity. The linearized Einstein tensor is gauge-invariant at the first order but it is not so at the second order, which complicates the discussion. This method depends on the assumption that the first order metric perturbation can be decomposed into gauge-variant and gauge-invariant parts and the gauge-variant parts do not contribute to physical quantities.