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

The correlation function xi(r) of matter in the non-linear regime is assumedto 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), andthe stable clustering hypothesis is used to determine the density profiles ofthese Press--Schechter halos. Thus, the shape and amplitude of xi(r) on smallscales is related to the initial power spectrum of density fluctuations. The case of clustering from scale-free initial conditions is treated indetail. If n is the slope of the initial power spectrum of densityfluctuations, then stable clustering requires that xi(r)\propto r^{-gamma},where gamma is a known function of n. If halo--halo correlations can beneglected, 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 steeperthan the value 3(3+n)/(4+n) that was obtained by Hoffman & Shaham in theirtreatment of the shapes of the outer regions of collapsed halos. Our mainresult is a prediction for the amplitude of the non-linear correlationfunction. The predicted amplitude and its dependence on n are in goodquantitative agreement with N-body simulations of self-similar clustering. If stable clustering is a good approximation only inside the half-mass radiiof Press--Schechter halos, then the density contrast required for the onset ofstable clustering can be estimated. This density contrast is in the range~300-600 and increases with the initial slope n, in agreement with estimatesfrom N-body simulations.Comment: 8 pages, uuencoded, gzipped, postscript, submitted to M