Gravitational potentials and forces in the Lattice Universe: a slab

TL;DR

This paper investigates how a slab topology of the universe influences gravitational potentials and forces, deriving two solution forms and identifying the more computationally efficient Yukawa-type sum for current cosmological conditions.

Contribution

It introduces two alternative solutions for gravitational potentials in a slab universe topology, highlighting the practical advantages of the Yukawa-type sum for numerical calculations.

Findings

Yukawa-type potential sum converges faster for current universe parameters.

Fourier series expansion provides an alternative analytical form.

The study offers practical methods for modeling gravity in slab topologies.

Abstract

We study the effect of the slab topology $T\times R\times R$ of the Universe on the form of gravitational potentials and forces created by point-like masses. We obtain two alternative forms of solutions: one is based on the Fourier series expansion of the delta function using the periodical property along the toroidal dimension, and another one is derived by direct summation of solutions of the Helmholtz equation for the source particle and all its images. The latter one takes the form of the sum of Yukawa-type potentials. We demonstrate that for the present Universe the latter solution is preferable for numerical calculations since it requires less terms of the series to achieve the necessary precision.