A Universal Profile of the Dark Matter Halo and the Two‐Point Correlation Function

We have investigated the relation between the two-point spatial correlationfunction and the density profile of the dark matter halo in the stronglynon-linear regime. It is well known that when the density fluctuation growsinto dark matter halo whose density profile is $\rho \proptor^{-\epsilon}$(${3/2}<\epsilon<3$) on almost all mass scales, the two-pointspatial 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 thetwo-point spatial correlation function, which does not obey the power law whenthe 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 byNavarro, Frenk & White (1996,1997). By using the BBGKY equation in the stronglynon-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 $\epsilon > 3/2$ andthen the two-point spatial correlation function can be regarded as the powerlaw. 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$\epsilon < 3/2$.Comment: 15 pages with no figures, submitted to Astrophysical Journa