A Universal Profile of the Dark Matter Halo and the Two-point Correlation Function

Abstract

We have investigated the relation between the two-point spatial correlation function and the density profile of the dark matter halo in the strongly non-linear regime. It is well known that when the density fluctuation grows into dark matter halo whose density profile is $\rho \propto r^{-\epsilon}$(${3/2}<\epsilon<3$) on almost all mass scales, the two-point spatial correlation function obeys a power law with the power index $\gamma = 2\epsilon -3$ in the strongly non-linear regime. We find the form of the two-point spatial correlation function, which does not obey the power law when the power index $\epsilon$ is smaller than 3/2, such as the density profile $\rho \propto r^{-1}$ around the center of the halo which is proposed by Navarro, Frenk & White (1996,1997). By using the BBGKY equation in the strongly non-linear regime, it is also found that velocity parameter $h \equiv - < v > / \dot{a}x$ is not a constant even in the strongly non-linear regime ($\tilde{x} \equiv x/x_{nl} \to 0$) although it is a constant when $\epsilon > 3/2$ and then the two-point spatial correlation function can be regarded as the power law. The velocity parameter $h$ becomes 0 at the non-linear limit of $\tilde{x} \to 0$, that is, the stable clustering hypothesis cannot be satisfied when $\epsilon < 3/2$.