Cold dark matter protohalo structure around collapse: Lagrangian cosmological perturbation theory versus Vlasov simulations

Abstract

We explore the structure around the shell-crossing time of cold dark matter protohaloes seeded by two or three crossed sine waves of various relative initial amplitudes, by comparing Lagrangian perturbation theory (LPT) up to the tenth order with high-resolution cosmological simulations performed with the public Vlasov code ColDICE. Accurate analyses of the density, the velocity, and related quantities such as the vorticity are performed by exploiting the fact that ColDICE can follow the phase-space sheet locally at the quadratic level. To test LPT predictions beyond the shell-crossing, we employ a ballistic approximation, which assumes that the velocity field is frozen just after the shell-crossing. In the generic case, where the amplitudes of the sine waves are all different, high-order LPT predictions match the exact solution very well, even beyond collapse. As expected, convergence slows down when going from quasi-1D dynamics, where one wave dominates over the two others, to the axial-symmetric configuration, where all the amplitudes of the waves are equal. We also notice that LPT convergence is slower when considering velocity-related quantities. Additionally, the structure of the system at and beyond collapse given by LPT and the simulations agrees very well with singularity theory predictions, in particular with respect to the caustic and vorticity patterns that develop beyond collapse. Again, this does not apply to axial-symmetric configurations, which are still correct from the qualitative point of view, but rather when multiple foldings of the phase-space sheet produce very high density contrasts and hence a strong back-reaction of the gravitational force.