Spatial conformal flatness in homogeneous and inhomogeneous cosmologies

Abstract

In the past few years there has been a growing interest in cosmological models which are not spatially homogeneous. The assumption of spatial homogeneity simplifies the Einstein equations to ordinary differential equations. If the assumption of spatial homogeneity is relaxed, some other symmetries are needed to make the Einstein equations mathematically tractable. The recently discovered solutions of Szekeres have been found to possess an interesting type of symmetry: The three spaces orthogonal to the fluid flow are conformally flat. Herein, we prove a theorem restricting the possible inhomogeneous cosmologies with conformally flat 3-surfaces. We determine which spatially homogeneous models admit conformally flat 3-surfaces. This information, although interesting in its own right, will serve as a guide in determining those spatially homogeneous models that may be generalized by retaining spatial conformal flatness but relaxing the condition of spatial homogeneity.