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

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 ρ ∝ r−ǫ( 2 < ǫ < 3) on almost all mass scales, the two-point spatial correlation function obeys a power law with the power index γ = 2ǫ− 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 ǫ is smaller than 3 2 , such as the density profile ρ ∝ r 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 ≡ −〈v〉/ȧx is not a constant even in the strongly non-linear regime (x̃ ≡ x/xnl → 0) although it is a constant when ǫ > 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 x̃ → 0, that is, the stable clustering hypothesis cannot be satisfied when ǫ < 3/2. Subject headings: cosmology:theory-large scale structures-dark matter halo