The standard model of cosmology currently allows for three types of geometries for our Universe (Euclidean, spherical or hyperbolic), each of these types corresponding to a set of different (multiconnected)-topologies. Among these geometries, Newton’s theory of gravity is only defined on the Euclidean one. Still, extending the validity of this theory to the other two cases, with the aim that this should constitute the limit of a relativistic theory of gravity, could provide a strong theoretical tool to probe non-relativistic effects of topology in our Universe.
The first part of this presentation will aim at constructing such a theory, using the concept of Galilean structures. We will see that it allows to quantify the effects of topology on the gravitational potential, structure formation and on the global backreaction of inhomogeneities.
In the second part, I will show that Einstein’s equation is necessarily incompatible with such an extension of Newton’s theory. However, as a physical hypothesis for our Universe, if we require the non-relativistic regime to exist regardless of the global geometry and topology, then Einstein’s equation should be modified. This can be done by adding a `topological term’ arising from the introduction of a second non-dynamical metric in the theory. The main consequence of this modification is that the expansion law of a homogeneous and isotropic solution no longer features the spatial curvature (i.e. $\Omega = 1, \ \forall \Omega_k$), asking for a reevaluation of that parameter from the current cosmological data.