Kurtosis in large-scale structure as a constraint on non-Gaussian initial conditions

We calculate the kurtosis of a large-scale density field which has undergone weakly non-linear gravitational evolution from arbitrary non-Gaussian initial conditions. It is well known that the weakly evolved skewness is equal to its initial value plus the term induced by gravity, which scales with the rms density fluctuation in precisely the same way as for Gaussian initial conditions. As in the case of skewness, the evolved kurtosis is equal to its initial value plus the contribution induced by gravity. The scaling of this induced contribution, however, turns out to be qualitatively different for Gaussian versus non-Gaussian initial conditions. Therefore, measurements of the kurtosis can serve as a powerful discriminating test between the hypotheses of Gaussian and non-Gaussian nature of primordial density fluctuations.