Phase-space structure of protohalos: Vlasov versus Particle-Mesh
  The phase-space structure of primordial dark matter halos is revisited using
cosmological simulations with three sine waves and Cold Dark Matter (CDM)
initial conditions. The simulations are performed with the tessellation based
Vlasov solver ColDICE and a Particle-Mesh (PM) $N$-body code. The analyses
include projected density, phase-space diagrams, radial density and
pseudo-phase space density. Particular attention is paid to force and mass
resolution. Because the phase-space sheet complexity, estimated in terms of
total volume and simplices count, increases very quickly, ColDICE can follow
only the early violent relaxation phase of halo formation. During the latter,
agreement between ColDICE and PM simulations having one particle per cell or
more is excellent and halos have a power-law density profile, $\rho(r) \propto
r^{-\alpha}$, $\alpha \in [1.5,1.8]$. This slope, measured prior to any merger,
is slightly larger than in the literature. The phase-space diagrams evidence
complex but coherent patterns with clear signatures of self-similarity in the
sine wave simulations, while the CDM halos are somewhat scribbly. After
additional mass resolution tests, the PM simulations are used to follow the
next stages of evolution. The power-law progressively breaks down with a
convergence of the density profile to the well known "NFW"-like universal
attractor, irrespectively of initial conditions, that is even in the three-sine
wave simulations. This demonstrates again that mergers do not represent a
necessary condition for convergence to the dynamical attractor. Not
surprisingly, the measured pseudo phase-space density is a power-law $Q(r)
\propto r^{-\alpha_Q}$, with $\alpha_{\rm Q}$ close to the prediction of
secondary spherical infall model, $\alpha_{\rm Q} \simeq 1.875$. However this
property is also verified during the early relaxation phase, which is non
trivial.
