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 $k_v=2\pi/\lambda_v$ corresponding to a structure with a viral radius $r_v=\lambda_v/2$, half the size of the free streaming scale radius $r_v=r_{fs}/2=\lambda_{fs}/4$. 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 $\lambda_{fs}$. The viral transfer function $T_v(a,k)=[1+ \left(k/k_v \right)^{\beta_v }]^{\gamma_v}$ is given in terms of the viral mode $k_v$ and two constant parameters $\beta_v$ and $\gamma_v$. 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 $\beta_v\gamma_v=-18$ with $\nu=1.020 \pm 0.025$. In the standard approach the transfer function is given by $T(a,k)=[1+\left(\alpha\, k \right)^{\beta}]^{\gamma}$ \cite{Viel:2005qj} where $\alpha$ 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 $\lambda_{fs}$ 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.