Hierarchical correlations in models of galaxy clustering

We present a comparison of the predictions of perturbation theory for the hierarchical J-order correlation amplitudes $S_J={\text{xibar}}_J\mathrm{/}{\text{xibar}}_2^{J-1}$, $J=3-10$, with the results of large numerical simulations of gravitational clustering. We consider two different initial power spectra of density fluctuations, one being flatter than the other. The clustering amplitudes $S_J$ measured in these models are different at all scales. In each case, the perturbation theory predictions give an excellent agreement with the simulations on scales for which the variance is approximately linear, i.e. for scales on which ${\text{xibar}}_2\text{simlt}1$. For cells of radius $R\text{simgt}30\text{Mpc}$, the sampling variance arising from the finite size of our simulations dominates the results. We also find that $S_J$ are roughly independent of time, $\mathrm{\Omega }$, or $\lambda$, as expected. On comparing with the observations, one can use these results to discriminate between models for structure formation.