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

We calculate reduced moments ${\bar{\xi }}_q$ 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 ${\bar{\xi }}_2$, alone. We look for agreement with the hierarchical scaling ansatz, according to which $\overline \xi_q \propto \overline \xi_2^{~q-1}$\mathrm{.}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 $n\ge -1$ 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.