Lattice Universe: examples and problems

TL;DR

This paper investigates lattice universe models with different topologies, revealing issues with gravitational potential solutions for point masses and emphasizing the physical necessity of mass smearing in certain topologies.

Contribution

It analyzes gravitational potentials in lattice universes with various topologies and demonstrates the need for mass smearing to obtain physically meaningful solutions.

Findings

Point masses cause nonphysical potential singularities in T×T×T topology.

Mass smearing is necessary for regular solutions in lattice models.

Other topologies do not admit nontrivial solutions without reducing to T×T×T.

Abstract

We consider lattice Universes with spatial topologies $T\times T\times T$, $\; T\times T\times R\; $ and $\; T\times R\times R$. 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 $T\times T\times T$ 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 $N$-body simulations is not only a technical method but also a physically substantiated procedure. In the cases of $\; T\times T\times R\; $ and $\; T\times R\times R$ 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 $T\times T\times T$ one.