Density Profile Asymptotes at the Centre of Dark Matter Halos

For the spherical symmetric case, all quantities describing the relaxed dark matter halo can be expressed as functions of the gravitational potential $\Phi$. Decomposing the radial velocity dispersion $\sigma_r$ with respect to $\Phi$ at very large and very small radial distances the possible asymptotic behavior for the density and velocity profiles can be obtained. If reasonable boundary conditions are posed such as a finite halo mass and force-free halo centre the asymptotic density profiles at the centre should be much less steep than the profiles obtained within numerical simulations. In particular cases profiles like Plummer's model are obtained. The reasons of that seeming discrepancy with respect to the results of N-body simulations are discussed.