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Hollow cores in Warm Dark Matter halos from the Vlasov-Poisson equation

We report the results of extended high--resolution numerical integrations of the Vlasov--Poisson equation for the collapse of spherically symmetric WDM halos. For thermal relics with mass $m=1\,$keV/$c^2$, we find collapsed halos with cores of size from $0.1$ to $0.6\,$kpc. The typical core is hollow, with the mass density decreasing towards the core center by almost three orders of magnitude from its maximum near the core radius $r_{\rm c}$. The core is in equilibrium with the diffused part of the halo but far from virialization. These properties are rooted in the conservation of the squared angular momentum and in the original excess, proper of WDM initial conditions, of kinetic energy in the core region. In a sample of more than one hundred simulated collapses, the values of $r_{\rm c}$ and of the core density $ρ_{\rm c}$ are in the range typical of dwarf spheroids, while the maximal circular velocities $V_{\rm max}$ are proper of small disk galaxies. The product $μ_{\rm c}=ρ_{\rm c}r_{\rm c}$ takes values between $116\,M_\odot/$pc$^2$ and $283\,M_\odot/$pc$^2$, while the surface density $μ_0$, as determined from a Burkert fit, is roughly three times larger. From these data and data obtained at smaller values of $m$, we extrapolate for one particular halo $μ_{\rm c}=263(308)\,M_\odot/$pc$^2$ and $μ_0=754(855)\,M_\odot/$pc$^2$ at $m=2(3.3)\,$keV/$c^2$, to be compared with the observed value $140^{+83}_{-52} M_\odot$/pc$^2$. In view of the many improvements and enhancements available, we conclude that WDM is a viable solution for explaining the presence and the size of cores in low mass galaxies.