KURTOSIS AND LARGE-SCALE STRUCTURE

We discuss the nonlinear growth of the excess kurtosis parameter of the smoothed density fluctuation field delta, S4 = [[delta4] - 3[delta2]2]/[delta2]3 in an Einstein-de Sitter universe. We assume Gaussian primordial density fluctuations with scale-free power spectrum P(k) is-proportional-to k(n) and analyze the dependence of S4 on primordial spectral index n, after smoothing with a Gaussian filter. As already known for the skewness ratio S3, the kurtosis parameter is a decreasing function of n, both in exact perturbative theory and in the Zel'dovich approximation. The parameter S4 provides a powerful statistics to test different cosmological scenarios.