Distribution Function in Center of Dark Matter Halo

N-body simulations of dark matter halos show that the density profiles of halos behave as $\rho(r)\propto r^{-\alpha(r)}$, where the density logarithmic slope $\alpha \simeq 1\sim1.5$ in the center and $\alpha \simeq 3\sim 4$ in the outer parts of halos. However, some observations are not in agreement with simulations in the very central region of halos. The simulations also show that velocity dispersion anisotropy parameter $\beta\approx 0$ in the inner part of the halo and the so called "pseudo phase-space density" $\rho/\sigma^3$ behaves as a power-law in radius $r$. With these results in mind, we study the distribution function and the pseudo phase-space density $\rho/\sigma^3$ of the center of dark matter halos and find that they are closely-related.