Impact of a Warm Dark Matter late-time velocity dispersion on
  large-scale structures
  We investigate whether the late-time (at $z\leq 100$) velocity dispersion
expected in Warm Dark Matter scenarios could have some effect on the cosmic web
(i.e., outside of virialized halos). We consider effective hydrodynamical
equations, with a pressurelike term that agrees at the linear level with the
analysis of the Vlasov equation. Then, using analytical methods, based on
perturbative expansions and the spherical dynamics, we investigate the impact
of this term for a 1 keV dark matter particle. We find that the late-time
velocity dispersion has a negligible effect on the power spectrum on
perturbative scales and on the halo mass function. However, it has a
significant impact on the probability distribution function of the density
contrast at $z \sim 3$ on scales smaller than $0.1 h^{-1}$Mpc, which correspond
to Lyman-$\alpha$ clouds. Finally, we note that numerical simulations should
start at $z_i\geq 100$ rather than $z_i \leq 50$ to avoid underestimating
gravitational clustering at low redshifts.
