Chul-Moon Yoo, Hiroyuki Abe, Ken-ichi Nakao, Yohsuke Takamori
We numerically construct an one-parameter family of initial data sets of an expanding inhomogeneous universe which is composed of regularly aligned black holes with an identical mass. They are initial data of vacuum solutions for the Einstein equations. We call this universe model the "black hole universe" and analyze the structure of these initial data sets by searching for the trapped surfaces. Giving definitions of an effective Hubble parameter and an effective energy density, we show that these quantities asymptotically satisfy the Hubble equation for the Einstein-de Sitter universe in the limit of a large separation between neighboring black holes, although the energy density is always larger than that estimated from the mass of the black hole only. The accuracy of the cosmological Newtonian approximation is also discussed. The deviation of the spatial metric obtained by the cosmological Newtonian approximation from that obtained by the full relativistic calculation is found to be smaller than about 1% if the separation length between neighboring black holes is 10 times larger than the Schwarzschild radius of a black hole, although the deviation of the Hubble parameter defined in the the cosmological Newtonian approximation scheme from that defined in the relativistic scheme is not so small.
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http://arxiv.org/abs/1204.2411
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