The three-point correlation function in an ensemble of three-dimensional simulations

We evaluate the three-point function in Fourier space for an ensemble of three-dimensional 128 exp 3 numerical simulations with initial power spectra characterized by spectral index n = +1, 0, -1, -2, -3, with no high-frequency cutoff and with cutoff k(c) = 16 or k(c) = 4. To remove dependences on scale and on time, we present results as the reduced amplitude Q in the hierarchical model as a function of the dimensionless variable kd(rms), where d(rms) is the mean square displacement of a particle from its initial position. For scale-free initial conditions, there is no evolution in Q. For initial conditions with a cutoff, Q evolves until the scale of the cutoff is in the nonlinear regime; the results afterwards are no different from those with no initial cutoff. The transition from quasi-linear to nonlinear regimes is followed. In the quasi-linear regime, our results agree well with gravitational perturbation theory predictions, including a marked dependence on the shape of the configuration. In the nonlinear regime, the value of Q for scale-invariant initial conditions is remarkably independent of evolution epoch, of scale, and of configuration shape, and depends on spectral index roughly as Q = 3/(3 + n).