Gravitational potential in spherical topologies

Abstract

Using the non-Euclidean Newtonian theory developed by Vigneron, we calculate the gravitational potential of a point mass in all the globally homogeneous regular spherical topologies, i.e. whose fundamental domain (FD) shape and size are unique, for which the FD is a platonic solid. We provide the Maclaurin expansion 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 non-zero spatial scalar curvature, while the even terms relate to the closed nature of the topological space. 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. In this sense, close to a mass point, the gravitational field would differ between spherical and Euclidean topologies. The correction terms remain isotropic until an order that depends on the choice of spherical topology, the Poincaré space having the potential that remains isotropic to the highest (fifth) order. We expect these corrections to have a negligible effect on scales small with respect to the size of the topological space. However, at scales where the Maclaurin expansion becomes inaccurate, the significance of the effect remains unknown. This motivates future Nbody simulation work with the non-Euclidean Newtonian theory to see if a topology different to the 3-torus has observable effects on structure formation.

On non-Euclidean Newtonian theories and their cosmological backreaction

“…Because of the many similarities of the second proposed theory in section 5 with the (Euclidean) Newtonian theory, we expect it to be the 'right' one, and its equations (equations ( 60)-( 66)) should be used to perform N-body simulations of structure formation in a universe with a non-Euclidean topology. For this purpose, in [22] we calculate the gravitational potential in different spherical topologies using this NEN theory.…”

“…Ξ c c := 0 and DΞ c ca := 0. Then, from equation (28) we have θ = 3χ + D c v c , and the momentum constraint (22) becomes…”

Hyperbolic space in the Newtonian limit: The cosmological constant

“…One such simplification is provided by the Newtonian limit [2,3,15,17,35,36,37], as opposed to the fully relativisitic regime. Further advantage is provided by the duality between the Newtonian cosmological fluid on the one hand, and the probability fluid of nonrelativistic quantum mechanics, on the other.…”