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

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.