Higher Order Moments of the Matter Distribution in Scale--Free Cosmological Simulations with Large Dynamic Range

We calculate reduced moments $\overline \xi_q$ of the matter density fluctuations, up to order $q=5$, from counts in cells produced by Particle--Mesh numerical simulations with scale--free Gaussian initial conditions. We use power--law spectra $P(k) \propto k^n$ with indices $n=-3,~-2,~-1,~0,~1$. Due to the supposed absence of characteristic times or scales in our models, all quantities are expected to depend on a single scaling variable. For each model, the moments at all times can be expressed in terms of the variance $\overline \xi_2$, alone. We look for agreement with the hierarchical scaling ansatz, according to which $\overline \xi_q \propto \overline \xi_2^{~q-1}$. For $n\leq -2$ models we find strong deviations from the hierarchy, which are mostly due to the presence of boundary problems in the simulations. A small, residual signal of deviation from the hierarchical scaling is however also found in $n \geq -1$ models. For the first time, due to our large dynamic range and careful checks of scaling and shot--noise effects, we are able to detect evolution away from the perturbation theory result.