Averaging in cosmology

In this paper we discuss the effect of local inhomogeneities on the global expansion of nearly Friedmann-Lemaitre-Robertson-Walker (FLRW) universes, in a perturbative setting. We derive a generic linearized averaging operation for metric perturbations from basic assumptions, and we explicate the issue of gauge invariance. We derive a gauge invariant expression for the back reaction of density inhomogeneities on the global expansion of perturbed FLRW spacetimes, in terms of observable quantities, and we calculate the effect quantitatively. Since we do not adopt a comoving gauge, our result incorporates the back reaction on the metric due to scalar velocity and vorticity perturbations. The results are compared with the results by other authors in this field.