A Study of High-Order Non-Gaussianity with Applications to Massive Clusters and Large Voids

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The statistical meaning of the local non-Gaussianity parameters fₙL and gₙL is examined in detail. Their relations to the skewness and the kurtosis of the probability distribution of density fluctuations are shown to obey simple fitting formulae, accurate on galaxy-cluster scales. We argue that the knowledge of fₙL and gₙL is insufficient for reconstructing a well-defined distribution of density fluctuations. However, by weakening the statistical significance of fₙL and gₙL, it is possible to reconstruct a well-defined pdf by using a truncated Edgeworth series. We give some general guidelines on the use of such a series, noting in particular that 1) the Edgeworth series cannot represent models with nonzero fₙL, unless gₙL is nonzero also, 2) the series cannot represent models with gₙL<0, unless some higher-order non-Gaussianities are known. Finally, we apply the Edgeworth series to calculate the effects of gₙL on the abundances of massive clusters and large voids. We show that the abundance of voids may generally be more sensitive to high-order non-Gaussianities than the cluster abundance.