The non - linear correlation function and density pro les ofvirialized

The correlation function (r) of matter in the non-linear regime is assumed to be determined by the density prooles (r) and the mass distribution n(M) of virialized halos. The Press{Schechter approach is used to compute n(M), and the stable clustering hypothesis is used to determine the density prooles of these Press{Schechter halos. Thus, the shape and amplitude of (r) on small scales is related to the initial power spectrum of density uctuations. The case of clustering from scale-free initial conditions is treated in detail. If n is the slope of the initial power spectrum of density uctuations, then stable clustering requires that (r) / r ? , where is a known function of n. If halo{halo correlations can be neglected, then (r) / r ? , where = (+3)=2 = 3(4+n)=(5+n). For all values of n of current interest, this slope is steeper than the value 3(3 + n)=(4 + n) that was obtained by Hooman & Shaham in their treatment of the shapes of the outer regions of collapsed halos. Our main result is a prediction for the amplitude of the non-linear correlation function. The predicted amplitude and its dependence on n are in good quantitative agreement with N-body simulations of self-similar clustering. If stable clustering is a good approximation only inside the half-mass radii of Press{ Schechter halos, then the density contrast required for the onset of stable clustering can be estimated. This density contrast is in the range 300 ? 600 and increases with the initial slope n, in agreement with estimates from N-body simulations.