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We develop a relativistic framework to investigate the evolution ofcosmological structures from the initial density perturbations to the highlynon-linear regime. Our approach involves proposing a procedure to match'best-fit', exact Bianchi IX (BIX) spacetimes to finite regions within theperturbed Friedmann-Lemaitre-Robertson-Walker universe characterized by apositive averaged spatial curvature. This method enables us to approximatelytrack the non-linear evolution of the initial perturbation using an exactsolution. Unlike standard perturbation theory and exact solutions with a highdegree of symmetry (such as spherical symmetry), our approach is applicable toa generic initial data, with the only requirement being positive spatialcurvature. By employing the BIX symmetries, we can systematically incorporatethe approximate effects of shear and curvature into the process of collapse.Our approach addresses the limitations of both standard perturbation theory andhighly symmetric exact solutions, providing valuable insights into thenon-linear evolution of cosmological structures.

Relativistic modeling of cosmological structures with Bianchi IX spacetimes