Exact Initial Data for Black Hole Universes with a Cosmological Constant
  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.
