  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 quasi triaxial 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
quasi-triaxial 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.
