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The virial mode $k_v$ approach to Structure Formation with Warm Dark Matter

The small scale structure opens a window to constrain the dynamical properties of Dark Matter. Here we study the clustering of warm dark matter (WDM) in a semi-analytical approach and compared the linear power spectrum of WDM with cold dark matter (CDM) employing a new transfer function $T_v(a,k)$ in terms of the viral wave number $k_v=2π/λ_v$ corresponding to a structure with a viral radius $r_v=λ_v/2$, half the size of the free streaming scale radius $r_v=r_{fs}/2=λ_{fs}/4$. The virial mass $M_v$ contained in this structure corresponds to the lightest structure formed for a WDM particle becoming non-relativistic at the scale factor $a_{nr}$ with the corresponding $λ_{fs}$. The viral transfer function $T_v(a,k)=[1+ \left(k/k_v \right)^{β_v }]^{γ_v}$ is given in terms of the viral mode $k_v$ and two constant parameters $β_v$ and $γ_v$. We compare $T_v(a,k)$ with the Boltzmann code CLASS for WDM in the mass range 1-10 keV and we obtain the constraint $β_vγ_v=-18$ with $ν=1.020 \pm 0.025$. In the standard approach the transfer function is given by $T(a,k)=[1+\left(α\, k \right)^β]^γ$ \cite{Viel:2005qj} where $α$ encodes the dynamical properties of WDM and must be numerically adjusted by means of a Boltzmann code. In contrast, in our viral approach the physical quantity $k_v$ is simply given in terms of the free streaming scale $λ_{fs}$ and can be analytically determined. Our viral proposal has a good agreement with CLASS and improves slightly the results from the standard transfer function. To conclude, we have proposed a new physically motivated transfer function $T_v(a,k)$ where the properties of WDM are encoded in the viral wave number $k_v$, is straightforward to determine and improves the prediction of WDM clustering properties.