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 [2022, Class. & Quantum Gravity, 39, 155006] 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 non-zero 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 Poincaré 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.

arXiv commentAccepted in Physical Review D, 12 pages, 4 tables, calculation source code at https://codeberg.org/boud/topoaccel

References in corpus (29)

• Planck evidence for a closed Universe and a possible crisis for cosmology
• Cosmic Topology
• Curvature tension: evidence for a closed universe
• Circles in the Sky: Finding Topology with the Microwave Background Radiation
• The evidence for a spatially flat Universe
• Investigating Cosmic Discordance
• Planck 2013 results. XXVI. Background geometry and topology of the Universe
• Planck 2015 results. XVIII. Background geometry & topology
• Topological Lensing in Spherical Spaces
• An exact quantification of backreaction in relativistic cosmology
• Einstein's Universe: Cosmological structure formation in numerical relativity
• On Determining the Topology of the Observable Universe via 3-D Quasar Positions
• Well-proportioned universes suppress CMB quadrupole
• Exact Evolution of Discrete Relativistic Cosmological Models
• Measuring the scale of cosmic homogeneity with SDSS-IV DR14 quasars
• Constraining spatial curvature with large-scale structure
• An improved cosmic crystallography method to detect holonomies in flat spaces
• A weak acceleration effect due to residual gravity in a multiply connected universe
• COBE and Global Topology: An Example of the Application of the Identified Circles Principle
• The residual gravity acceleration effect in the Poincare dodecahedral space
• Non-Euclidean Newtonian Cosmology
• Gauss-Bonnet-Chern approach to the averaged Universe
• The 1+3-Newton-Cartan system and Newton-Cartan cosmology
• Effect of the spatial curvature of the Universe on the form of the gravitational potential
• Deep redshift topological lensing: strategies for the T^3 candidate
• On the comoving distance as an arc-length in four dimensions
• The perihelion of Mercury advance and the light bending calculated in (enhanced) Newton's theory
• On non-Euclidean Newtonian theories and their cosmological backreaction
• Is backreaction in cosmology a relativistic effect? On the need for an extension of Newton's theory to non-Euclidean topologies

Cited by in corpus (1)

• Promise of Future Searches for Cosmic Topology