Proving almost-homogeneity of the universe: an almost Ehlers-Geren-Sachs theorem

We show that if all fundamental observers measure the cosmic microwave background radiation to be almost isotropic in an expanding universe region, then that universe is locally almost spatially homogeneous and isotropic in that region. This shows the stability of the important exact result of this type of previously proved by Ehlers, Geren, & Sachs (EGS) and formalizes the way in which almost-isentropy of the background radiation gives evidence that the universe is almost a Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. This is the needed foundation for all recent analyses, based on the Sachs-Wolfe effect, of how background radiation anisotropies relate to growing density inhomogeneities in an almost-FLRW expanding universe.