Gravitational potential in spherical topologies

Abstract

We study the properties of the Newtonian gravitational potential in a spherical Universe for different topologies. For this, we use the non-Euclidean Newtonian theory developed in Vigneron [On non-Euclidean Newtonian theories and their cosmological backreaction, Classical Quantum Gravity 39, 155006 (2022)]. describing Newtonian gravitation in a spherical or hyperbolic Universe. The potential is calculated for a point mass in all the globally homogeneous regular spherical topologies, i.e., whose fundamental domain is unique and is a platonic solid. We provide the exact solution and the Taylor expansion series of the potential at a test position near the point mass. We show that the odd terms of the expansion can be interpreted as coming from the presence of a nonzero spatial scalar curvature, while the even terms relate to the closed nature of the topological space. A consequence is that, compared to the point mass solution in a 3-torus, widely used in Newtonian cosmological simulations, the spherical cases all feature an additional attractive first order term dependent solely on the spatial curvature. The choice of topology only affects the potential at second order and higher. For typical estimates of cosmological scales (curvature and topology), the strongest topological effect occurs in the case of the Poincar\'e dodecahedral space, but in general the effect of curvature dominates over topology. We also provide the set of equations that can be used to perform $N$-body simulations of structure formation in spherical topologies.