Lagrange perturbation and averaging of a general relativistic inhomogeneous universe

In the framework of general relativity, we discuss Lagrange perturbation approach to describe locally nonlinear inhomogeneities in a globally FLRW-like universe, in which we assume two cases: only the dust matter and a matter accompanied with pressure. Next, we discuss apparent discrepancies among a lot of works on effects of the nonlinear backreaction. By defining the spatially averaged matter energy density as a conserved quantity in the large comoving volume, it is shown that the nonlinear backreaction neither accelerates nor decelerates the cosmic expansion in a matter-dominated universe.