The Lindquist-Wheeler formulation of lattice universes

Abstract

This paper examines the properties of `lattice universes' wherein point masses are arranged in a regular lattice on space-like hypersurfaces; open, flat, and closed universes are considered. The universes are modelled using the Lindquist-Wheeler (LW) approximation scheme, which approximates the space-time in each lattice cell by Schwarz\-schild geometry. Extending Lindquist and Wheeler's work, we derive cosmological scale factors describing the evolution of all three types of universes, and we use these scale factors to show that the universes' dynamics strongly resemble those of Friedmann-Lema\^itre-Robertson-Walker (FLRW) universes. In particular, we use the scale factors to make more salient the resemblance between Clifton and Ferreira's Friedmann-like equations for the LW models and the actual Friedmann equations of FLRW space-times. Cosmological redshifts for such universes are then determined numerically, using a modification of Clifton and Ferreira's approach; the redshifts are found to closely resemble their FLRW counterparts, though with certain differences attributable to the `lumpiness' in the underlying matter content. Most notably, the LW redshifts can differ from their FLRW counterparts by as much as 30\%, even though they increase linearly with FLRW redshifts, and they exhibit a non-zero integrated Sachs-Wolfe effect, something which would not be possible in matter-dominated FLRW universes without a cosmological constant.