Precision cosmology with exact inhomogeneous solutions of General Relativity: the Szekeres models

Abstract: The exact Friedman solution to the field equations of General Relativity (GR) describing a homogeneous and isotropic universe together with its linear and higher-order perturbations are the theoretical roots of the current standard model of cosmology. However, despite its global successes, this standard model currently faces a number of tensions and anomalies, which occur mostly from a mismatch between early and late cosmic time region depictions. Actually, in the era of precision cosmology, the late inhomogeneities in the densities of the Universe components can no more be neglected. Moreover, since GR is fundamentally nonlinear, any linear perturbation theory is doomed to fail at reproducing precisely its features, while higher-order perturbed FLRW models cannot either claim the status of exact solution to Einstein's field equations. Fortunately, other GR solutions exist which are best suited for this purpose, e. g., exact inhomogeneous solutions able to represent a matter-cosmological constant dominated universe region. Among these, the Szekeres solution, which is devoid of any symmetry, appears as a proper tool for this purpose. Moreover, since these solutions possess the FLRW model as a homogeneous limit, they can be smoothly matched to the standard representation at the inhomogeneity-homogeneity transition. In this paper, the Szekeres solution and its main interesting properties, as well as the equations needed to use this solution in a cosmological context are presented. Then the prospects for a broader use of its abilities are sketched out. In particular, the use of neural networks is proposed to allow, in the future, the fitting of the huge amount of data becoming available to constrain the arbitrary functions and the constant parameters characterizing the Szekeres model, running for representing our late Universe with an increased precision.