Exact Initial Data for Black Hole Universes with a Cosmological Constant

Abstract

We construct exact initial data for closed cosmological models filled with regularly arranged black holes in the presence of $\Lambda$. The intrinsic geometry of the 3-dimensional space described by this data is a sum of simple closed-form expressions, while the extrinsic curvature is just proportional to $\Lambda$. We determine the mass of each of the black holes in this space by performing a limiting procedure around the location of each of the black holes, and then compare the result to an appropriate slice through the Schwarzschild-de Sitter spacetime. The consequences of the inhomogeneity of this model for the large-scale expansion of space are then found by comparing the lengths of curves in the cosmological region to similar curves in a suitably chosen Friedmann-Lemaitre-Robertson-Walker (FLRW) solution. Finally, we locate the positions of the apparent horizons of the black holes, and determine the extremal values of their mass, for every possible regular arrangement of masses. We find that as the number of black holes is increased, the large-scale expansion of space approaches that of an FLRW model filled with dust and $\Lambda$, and that the extremal values of the black hole masses approaches that of the Schwarzschild-de Sitter solution.

arXiv comment19 pages, 5 figures, v2: matches published version

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