The galaxy correlation hierarchy in perturbation theory

The evolution of cosmological density correlation functions is calculated to lowest nonvanishing order in perturbation theory for an initially random Gaussian distribution. The three-point function so obtained scales with size as in the continuous-hierarchy model, but there is a residual nontrivial dependence on shape of the reduced three-point amplitude Q. The average value Q = 34/21 - (1/6) gamma is consistent with observations in the nonperturbative regime. The four-point function is also hierarchical in form, with amplitudes Ra = (34 /21)2 and Rb = 682/189. The perturbation expansion in fact gives kappa(N) proportional to xi to the (N-1) for the reduced correlation function kappa(N) to all orders N. A graphical technique enumerates the terms which appear in kappa(N).