  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-Lemaitre-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.
