I will also address an inherent ambiguity when it comes to evaluating certain functionals (e.g. density, spatial curvature, expansion) within the Lagrangian framework, coming from the fact that there is more than one definition of these fields in terms of the governing equations. One possible work-around is to compare the perturbative expectations with some exact relativistic solutions. I will put into perspective such comparison for spherically symmetric dust solutions, the Lemaître-Tolman-Bondi metrics, with an emphasis on the evaluation of spatial curvature, extending previous works focusing mostly on the density field.

The goal of this presentation is to introduce the first steps towards relativistic simulations in any topology. After reviewing the current status of relativistic cosmological simulations, I will go into the numerical methods in general relativity and how they could be adapted depending on the topology.

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.

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 spacetime 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.

In the second part, I will present two examples highlighting the application component. The first example concerns theoretical Gaussian random field models with power-law power spectra. We find that a topological characterization through Betti numbers and persistence diagrams provide information that is missed by traditional topological measures like the Euler characteristic and the more familiar geometric Minkowski functionals. The second example concerns the analysis of the topological characteristics of the temperature fluctuations in the Cosmic Microwave Background (CMB) from the temperature anisotropy maps measured by the Planck satellite. We find that the observed maps differ significantly from the simulations modeled as isotropic, homogeneous Gaussian random fields.