Post-collapse Lagrangian perturbation theory in three dimensions

The gravitational collapse of collisionless matter leads to shell-crossing singularities that challenge the applicability of standard perturbation theory. Here, we present the first fully perturbative approach in three dimensions by using Lagrangian coordinates that asymptotically captures the highly nonlinear nature of matter evolution after the first shell-crossing. This is made possible essentially thanks to two basic ingredients: (1) We employ high-order standard Lagrangian perturbation theory to evolve the system until shell-crossing, and (2) we exploit the fact that the density caustic structure near the first shell-crossing begins generically with pancake formation. The latter property allows us to exploit largely known one-dimensional results to determine perturbatively the gravitational backreaction after collapse, yielding accurate solutions within our post-collapse perturbation theory (PCPT) formalism. We validate the PCPT predictions against high-resolution Vlasov-Poisson simulations and demonstrate that PCPT provides a robust framework for describing the early stages of post-collapse dynamics.