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

Abstract

We explore the structure around 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 10th order to 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 locally the phase-space sheet at the quadratic level. To test LPT predictions beyond shell-crossing, we employ a ballistic approximation, which assumes that the velocity field is frozen just after shell-crossing. In the generic case, where the amplitudes of the sine waves are all different, high-order LPT predictions match very well the exact solution, 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. It is also noticed 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, that are still correct from the qualitative point of view, but where multiple foldings of the phase-space sheet produce very high density contrasts, hence a strong backreaction of the gravitational force.

Analytical growth functions for cosmic structures in a $Λ$CDM Universe

“…Buchert 1989;Moutarde et al 1991;Bouchet et al 1992;Buchert 1992;Bouchet et al 1995;Ehlers & Buchert 1997;Rampf & Buchert 2012;Zheligovsky & Frisch 2014), can be straighforwardly applied to generate initial conditions for cosmological 𝑁-body simulations (Klypin & Shandarin 1983;Efstathiou et al 1985;Scoccimarro 1998;Michaux et al 2021;Hahn et al 2021). Secondly, in Lagrangian coordinates it is fairly straightforward to detect the breakdown of the standard perturbative framework, which is linked to the crossing of particle trajectories (Rampf & Frisch 2017;Saga et al 2018;Saga et al 2021). Thirdly, Lagrangian coordinates are optimal to describe advection of perfectly cold fluids and, as a consequence, LPT solutions generically converge much faster than the PT solutions in Eulerian coordinates at fixed truncation order (but see Uhlemann et al 2019 for an exception within a semi-classical context).…”