Kurtosis and large-scale structure

We discuss the non--linear growth of the excess kurtosis parameter of the smoothed density fluctuation field $delta$, $S_4equiv[landelta^{,4} an-3landelta^{,2} an^2]/ landelta^{,2} an^3$ in an Einstein--de Sitter universe. We assume Gaussian primordial density fluctuations with scale--free power spectrum $P(k)propto k^{,n}$ and analyze the dependence of $S_4$ on primordial spectral index $n$, after smoothing with a Gaussian filter. As already known for the skewness ratio $S_3$, the kurtosis parameter is a {it decreasing function} of $n$, both in exact perturbative theory and in the Zel'dovich approximation. The parameter $S_4$ provides a powerful statistics to test different cosmological scenarios.Comment: 11 pages in Latex (plus 1 figure), SISSA 127/93/