On non-Euclidean Newtonian theories and their cosmological backreaction

TL;DR

This paper develops non-Euclidean Newtonian theories on spherical and hyperbolic topologies to explore the impact of global topology on cosmological structure formation, proposing two models with different backreaction properties.

Contribution

It introduces two non-Euclidean Newtonian theories based on Galilean manifolds, extending Newtonian gravity to non-Euclidean topologies for cosmological applications.

Findings

First model exhibits non-zero backreaction but includes gravitomagnetism.

Second model has no backreaction and allows exact N-body calculations.

Both models provide frameworks for studying topology effects in cosmology.

Abstract

Constructing an extension of Newton's theory which is defined on a non-Euclidean topology (in the sense of Thurston's decomposition), called a non-Euclidean Newtonian theory, corresponding to the zeroth order of a non-relativistic limit of general relativity is an important step in the study of the backreaction problem in cosmology and might be a powerful tool to study the influence of global topology on structure formation. After giving a precise mathematical definition of such a theory, based on the concept of Galilean manifolds, we propose two such extensions, for spherical or hyperbolic topologies, using a minimal modification of the Newton-Cartan equations. However as for now we do not seek to justify this modification from general relativity. The first proposition features a non-zero cosmological backreaction, but the presence of gravitomagnetism and the impossibility of performing exact $N$-body calculations make this theory difficult to be interpreted as a Newtonian-like theory. The second proposition features no backreaction, exact $N$-body calculation is possible and no gravitomagnetism appears. In absence of a justification from general relativity, we argue that this non-Euclidean Newtonian theory should be the one to be considered, and could be used to study the influence of topology on structure formation via $N$-body simulations. For this purpose we give the mass point gravitational field in $\mathbb{S}^3$.