Lattice Universe: examples and problems

Abstract

We consider lattice Universes with spatial topologies $$T\times T\times T$$T×T×T, $$ T\times T\times R $$T×T×R, and $$ T\times R\times R$$T×R×R. In the Newtonian limit of General Relativity, we solve the Poisson equation for the gravitational potential in the enumerated models. In the case of point-like massive sources in the $$T\times T\times T$$T×T×T model, we demonstrate that the gravitational potential has no definite values on the straight lines joining identical masses in neighboring cells, i.e. at points where masses are absent. Clearly, this is a nonphysical result, since the dynamics of cosmic bodies is not determined in such a case. The only way to avoid this problem and get a regular solution at any point of the cell is the smearing of these masses over some region. Therefore, the smearing of gravitating bodies in $$N$$N-body simulations is not only a technical method but also a physically substantiated procedure. In the cases of $$ T\times T\times R $$T×T×R and $$ T\times R\times R$$T×R×R topologies, there is no way to get any physically reasonable and nontrivial solution. The only solutions we can get here are the ones which reduce these topologies to the $$T\times T\times T$$T×T×T one.