Vlasov-Poisson in 1D for initially cold systems: post-collapse
  Lagrangian perturbation theory
  We study analytically the collapse of an initially smooth, cold,
self-gravitating collisionless system in one dimension. The system is described
as a central "S" shape in phase-space surrounded by a nearly stationary halo
acting locally like a harmonic background on the S. To resolve the dynamics of
the S under its self-gravity and under the influence of the halo, we introduce
a novel approach using post-collapse Lagrangian perturbation theory. This
approach allows us to follow the evolution of the system between successive
crossing times and to describe in an iterative way the interplay between the
central S and the halo. Our theoretical predictions are checked against
measurements in entropy conserving numerical simulations based on the waterbag
method. While our post-collapse Lagrangian approach does not allow us to
compute rigorously the long term behavior of the system, i.e. after many
crossing times, it explains the close to power-law behavior of the projected
density observed in numerical simulations. Pushing the model at late time
suggests that the system could build at some point a very small flat core, but
this is very speculative. This analysis shows that understanding the dynamics
of initially cold systems requires a fine grained approach for a correct
description of their very central part. The analyses performed here can
certainly be extended to spherical symmetry.
