Velocity dispersion profile in dark matter halos

Numerous numerical studies indicate that dark matter halos show an almost universal radial density profile. The origin of the profile is still under debate. We investigate this topic and pay particular attention to the velocity dispersion profile. To this end we have performed high-resolution simulations with two independent codes, ART and {\sc Gadget}. The radial velocity dispersion can be approximated as function of the potential by $\sigma_r^2 = a (\Phi / \Phi_{\rm{out}})^\kappa (\Phi_{\rm{out}} - \Phi)$, where $\Phi_{\rm{out}}$ is the outer potential of the halo. For the parameters $a$ and $\kappa$ we find $a=0.29\pm0.04$ and $\kappa=0.41\pm0.03$. We find that the power-law asymptote $\sigma^2 \propto \Phi^\kappa$ is valid out to much larger distances from the halo center than any power asymptote for the density profile $\rho \propto r^{-n}$. The asymptotic slope $n(r \to 0)$ of the density profile is related to the exponent $\kappa$ via $n=2\kappa/(1+\kappa)$. Thus the value obtained for $\kappa$ from the available simulation data can be used to obtain an estimate of the density profile below presently resolved scales. We predict a continuously decreasing $n$ towards the halo center with the asymptotic value $n \lesssim 0.58$ at $r=0$.