The standard model of cosmology rests on a homogeneous-isotropic solution of Einstein's laws of gravitation. The so-called "concordance model" with homogeneous geometry fixes the parameters of this model in conformity with available observational data. Accepting this model leads to a number of unresolved issues related to the conjecture of existence of Dark Matter and Dark Energy. To resolve these, a large community either seeks to generalize the laws of gravitation, or assumes the existence of new fundamental fields providing challenges for particle physics.
In this talk we focus on a third possibility that is conservative by not generalizing the laws of Einstein and by not including any new fundamental field. We present and motivate from first principles a set of effective (i.e. spatially averaged) Einstein equations that govern the regional and global dynamics of inhomogeneous cosmological models. In this framework there are new terms that arise from curvature invariants of the inhomogeneous geometry of spatial hypersurfaces. These terms qualitatively play the role of Dark Matter and Dark Energy.
In this talk, I will first recall basic principles that lead to the standard model of cosmology and discuss its governing cornerstones (without assuming prerequisites in general relativity from the audience). I will also recall how a cosmological model is built from the splitting of space-time into spatial hypersurfaces that evolve in time.
By introducing a spatial averaging operation we will arrive at a set of equations that govern general cosmologies. Due to their generality this set of equations is not closed, and I will outline recent work that investigates a topological approach, based on the Gauss-Bonnet-Chern theorem, to achieve closure.