Relativistic modeling of cosmological structures with Bianchi IX spacetimes

We develop a relativistic framework to investigate the evolution of cosmological structures from the initial density perturbations to the highly nonlinear regime. Our approach involves proposing a procedure to match ``best-fit, '' exact Bianchi IX (BIX) spacetimes to finite regions within the perturbed Friedmann-Lema\^tre-Robertson-Walker universe, characterized by a positive averaged spatial curvature. This method enables us to approximately track the nonlinear evolution of the initial perturbation using an exact solution. Unlike standard perturbation theory and exact solutions with a high degree of symmetry (such as spherical symmetry), our approach is applicable to generic initial data, with the only requirement being positive spatial curvature. By employing the BIX symmetries, we can systematically incorporate the approximate effects of shear and curvature into the process of collapse. Our approach addresses the limitations of both standard perturbation theory and highly symmetric exact solutions, providing valuable insights into the nonlinear evolution of cosmological structures.