On non-Euclidean Newtonian theories and their cosmological backreaction
  Constructing an extension of Newton's theory which is defined on a
non-Euclidean topology, called a non-Euclidean Newtonian theory, such that it
can be retrieved by 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 classes of 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 N-body calculations makes this theory difficult to
be interpreted as a Newtonian-like theory. The second proposition features no
backreaction, 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
$\mathcal{S}^3$.
