Lattice Universe: examples and problems.

Area of Science:

• Cosmology
• General Relativity
• Computational Astrophysics

Background:

• Lattice universes are theoretical models used to study the large-scale structure of the cosmos.
• The Newtonian limit of General Relativity provides a simplified framework for understanding gravity on cosmic scales.
• The spatial topology of the universe significantly impacts cosmological models.

Purpose of the Study:

• To investigate the behavior of gravitational potential in lattice universes with different spatial topologies.
• To determine the physical implications of point-like versus smeared mass distributions in these models.
• To assess the validity and necessity of mass-smearing techniques in N-body simulations.

Main Methods:

• Solving the Poisson equation for the gravitational potential within various lattice universe topologies.
• Analyzing the gravitational field behavior around point-like mass distributions.
• Comparing results for different spatial topologies ([Formula: see text], [Formula: see text], [Formula: see text]).

Main Results:

• In the [Formula: see text] topology, point-like masses lead to nonphysical, undefined gravitational potentials along lines connecting identical masses.
• Mass-smearing is demonstrated as a physically necessary procedure to obtain regular and meaningful solutions in the [Formula: see text] model.
• The [Formula: see text] and [Formula: see text] topologies do not yield physically reasonable nontrivial solutions and effectively reduce to the [Formula: see text] topology.

Conclusions:

• Mass-smearing in N-body simulations is not merely a numerical technique but a physically justified requirement for lattice universes.
• The simplest lattice topology ([Formula: see text]) is the only one that allows for physically consistent gravitational dynamics when mass-smearing is applied.
• More complex topologies ([Formula: see text], [Formula: see text]) are either unphysical or degenerate to the simplest case, limiting their independent relevance in cosmological simulations.