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

Abstract

We evaluate the three-point function in Fourier space for an ensemble of three-dimensional 128^3^ 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. We are able to follow, for the first time, the transition from quasi-linear to nonlinear regimes. 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).