The non-linear correlation function and density profiles of virialized haloes

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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 haloes. 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 haloes. Thus, the shape and amplitude of ξ(r) on small scales are 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−γ, where γ is a known function of n. If halo-halo correlations can be neglected, then ρ(r) ∝ r−ϵ, where ϵ = (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 haloes. 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 haloes, 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.