Backreaction and continuum limit in a closed universe filled with black holes

Abstract

We discuss the continuum limit of the initial data for a vacuum, closed cosmological model with black holes as the only sources of the gravitational field. The model we consider is an exact solution of the constraint equations and represents a vacuum universe with a number of black holes placed on a spatial slice of $S^3$ topology considered at the moment of its largest expansion when the black holes are momentary at rest. We explain how and under what conditions the FLRW metric arises as the continuum limit when the number of black holes contained in the model goes to infinity. We also dicuss the relation between the effective cosmological parameters of the model, inferred from the large scale geometry of the spacetime, and the masses of individual black holes. In particular, we prove an estimate for the difference between the total effective mass of the system and the sum of the masses of all black holes, thus quantifying the effects of the inhomogeneities in the matter distribution or the cosmological backreaction.