Kurtosis and Large--Scale Structure

We discuss the non–linear growth of the excess kurtosis parameter of the smoothed density fluctuation field $\delta$, $S_4\equiv[\lan\delta^{,4}\ran-3\lan\delta^{,2}\ran^2]/ \lan\delta^{,2}\ran^3$ in an Einstein–de Sitter universe. We assume Gaussian primordial density fluctuations with scale–free power spectrum $P(k)\propto k^{,n}$andanalyzethedependenceof$S_4$onprimordialspectralindex$n$, 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.