The non-linear correlation function and the shapes of virialized halos

The correlation function ξ(r) of matter in the non-linear regime is assumed to be determined by the density profiles ρ(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 profiles of these Press–Schechter halos. Thus, the shape and amplitude of ξ(r) on small scales is related to the initial power spectrum of density fluctuations. 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 fluctuations, then stable clustering requires that ξ(r) ∝ r , where γ is a known function of n. If halo–halo correlations can be neglected, then ρ(r) ∝ r, where ǫ = (γ+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 Hoffman & 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.