Lagrangian Cosmological Perturbation Theory at Shell Crossing

We consider the growth of primordial dark matter halos seeded by three crossed initial sine waves of various amplitudes. Using a Lagrangian treatment of cosmological gravitational dynamics, we examine the convergence properties of a high-order perturbative expansion in the vicinity of shell crossing by comparing the analytical results with state-of-the-art high resolution Vlasov-Poisson simulations. Based on a quantitative exploration of parameter space, we study explicitly for the first time the convergence speed of the perturbative series and find, in agreement with intuition, that it slows down when going from quasi-one-dimensional initial conditions (one sine wave dominating) to quasitriaxial symmetry (three sine waves with same amplitude). In most cases, the system structure at collapse time is, as expected, very similar to what is obtained with simple one-dimensional dynamics, except in the quasitriaxial regime, where the phase-space sheet presents a velocity spike. In all cases, the perturbative series exhibits a generic convergence behavior as fast as an exponential of a power law of the order of the expansion, allowing one to numerically extrapolate it to infinite order. The results of such an extrapolation agree remarkably well with the simulations, even at shell crossing.