Is backreaction in cosmology a relativistic effect? On the need for an extension of Newton's theory to non-Euclidean topologies

Cosmological backreaction corresponds to the effect of inhomogeneities of structure on the global expansion of the Universe. The main question surrounding this phenomenon is whether or not it is important enough to lead to measurable effects on the scale factor evolution, eventually explaining its acceleration or the Hubble tension. One of the most important results on this subject is the Buchert-Ehlers theorem [T. Buchert and J. Ehlers, Astron. Astrophys. 320, 1 (1997)] stating that backreaction is exactly zero when calculated using Newton's theory of gravitation, which may not be the case in general relativity. It is generally said that this result implies that backreaction is a purely relativistic effect. We will show that this is not necessarily the case, in the sense that this implication does not apply to a universe which is still well-described by Newton's theory on small scales but has a non-Euclidean topology. The theorem should therefore be generalized to account for such a scenario. In a heuristic calculation where we construct a theory which is locally Newtonian but defined on a non-Euclidean topology, we show that the backreaction is nonzero, meaning that it might be nonrelativistic depending on the topological class of our Universe. However, that construction is not unique and remains to be justified from a nonrelativistic limit of general relativity.