Light propagation and the average expansion rate in near-FRW universes

Understand

We consider universes that are close to Friedmann-Robertson-Walker in the sense that metric perturbations, their time derivatives and first spatial derivatives are small, but second spatial derivatives are not constrained.

• We show that if we in addition assume that the observer four-velocity is close to its background value and close to the four-velocity which defines the hypersurface of averaging, the redshift and the average expansion rate remain close to the FRW case.
• However, this is not true for the angular diameter distance.
• The four-velocity assumption implies certain conditions on second derivatives of the metric and/or the matter content.