THE NONLINEAR CORRELATION-FUNCTION AND DENSITY PROFILES OF VIRIALIZEDHALOES

The correlation function xi(r) of matter in the non-linear regime is a ssumed to be determined by the density profiles rho(r) and the mass di stribution 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 xi(r) on small scales are related to the in itial power spectrum of density fluctuations. The case of clustering f rom scale-free initial conditions is treated in detail. If n is the sl ope of the initial power spectrum of density fluctuations, then stable clustering requires that xi(r) proportional to r(-gamma), where gamma is a known function of n. If halo-halo correlations can be neglected, then rho(r) proportional to r(-epsilon), where epsilon = (gamma + 3)/ 2 = 3(4 + n)/(5 + n). For all values of n of current interest, this sl ope is steeper than the value 3(3 + n)/(4 + n) that was obtained by Ho ffman & Shaham in their treatment of the shapes of the outer regions o f collapsed haloes. Our main result is a prediction for the amplitude of the non-linear correlation function. The predicted amplitude and it s dependence on n are in good quantitative agreement with N-body simul ations of self-similar clustering. If stable clustering is a good appr oximation only inside the half-mass radii of Press-Schechter haloes, t hen the density contrast required for the onset of stable clustering c an be estimated. This density contrast is in the range similar to 300 - 600 and increases with the initial slope n, in agreement with estima tes from N-body simulations.