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We investigate whether the late-time (at $z\leq 100$) velocity dispersionexpected in Warm Dark Matter scenarios could have some effect on the cosmic web(i.e., outside of virialized halos). We consider effective hydrodynamicalequations, with a pressurelike term that agrees at the linear level with theanalysis of the Vlasov equation. Then, using analytical methods, based onperturbative expansions and the spherical dynamics, we investigate the impactof this term for a 1 keV dark matter particle. We find that the late-timevelocity dispersion has a negligible effect on the power spectrum onperturbative scales and on the halo mass function. However, it has asignificant impact on the probability distribution function of the densitycontrast at $z \sim 3$ on scales smaller than $0.1 h^{-1}$Mpc, which correspondto Lyman-$\alpha$ clouds. Finally, we note that numerical simulations shouldstart at $z_i\geq 100$ rather than $z_i \leq 50$ to avoid underestimatinggravitational clustering at low redshifts.

Impact of a Warm Dark Matter late-time velocity dispersion on large-scale structures