Multi-Black-Hole Configurations as Models for Inhomogeneous Cosmologies

Abstract

On the largest scales, the Universe is homogeneous and isotropic, whereas on smaller scales, various structures immediately begin to emerge. The transition from an inhomogeneous spacetime to the homogeneous and isotropic Friedmann universe is not sufficiently understood yet. Modern cosmology rests on the hypothesis that the LambdaCDM-model applies and, indeed, this model is very successful. On the other hand, as the precision of observations steadily increases, it is more than likely that inhomogeneities will no longer be negligible in the future. For this reason, the study of inhomogeneous cosmological models is reasonable. In this thesis, we consider the question which Friedmann universe is the best fit to a particular given inhomogeneous spacetime, which is known as the fitting problem. We consider models in which matter is replaced by a discrete configuration of black holes, that is, we concentrate on vacuum solutions to Einstein's equations. Since the full system of the field equations is too complicated to find an exact time-dependent solution for the whole spacetime, we restrict ourselves to approximative models as well as solutions to the initial value problem. In the former case, we reconsider Swiss-cheese and Lindquist-Wheeler models. In both models, the spacetime around a mass is described by the Schwarzschild metric. In the latter case, we determine the spatial metric of a space-like hypersurface. We limit our attention to time-symmetric initial data characterised by the vanishing of the extrinsic curvature. In this case, we are able to find a solution for an arbitrary number of black holes using the conformal method. Clearly, it is not reasonable to assume that every configuration of black holes leads to a spacetime which may be approximated well by a Friedmann solution. Such an approximation should be possible if the masses are distributed somehow uniformly. The aim of this thesis is to clarify this statement and to provide criteria which allow quantitative statements about the degree of uniformity. We determine the parameters of the fitted dust universe, in particular the scale factor. Our considerations are supported by several example configurations. In particular, we provide a new method based on Lie sphere geometry to construct various configurations with a high degree of uniformity in a surprisingly simple fashion. Moreover, we provide a generalisation to an approximative inhomogeneous model given by Lindquist and Wheeler. In this case, it is possible to determine the parameters of the fitted Friedmann universe even if we do not know the exact solution. Under certain conditions, this model becomes similar to a Swiss-cheese model, allowing us to formulate first expectations on the time evolution, which is otherwise mostly disregarded within the framework of this thesis.