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

The correlation function xi(r) of matter in the non-linear regime is assumed to be determined by the density profiles rho(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 xi(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 xi(r)\propto r^{-gamma}, where gamma is a known function of n. If halo--halo correlations can be neglected, then rho(r)\propto r^{-epsilon}, where epsilon = (gamma+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.