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

The statistical meaning of the local non-Gaussianity parameters f NL and g NL is examined in detail. Their relations to the skewness and the kurtosis of the initial distribution are shown to obey simple fitting formulae, accurate on galaxy–cluster scales. We argue that the knowledge of f NL and g NL is insufficient for reconstructing a well-defined distribution of primordial fluctuations. Requiring the reconstructed probability density function (pdf) to be positive enforces a theoretical lower bound g NL ≳ −1.2 × 10 5 , competitive with the observational bounds in the current literature. By weakening the statistical significance of f NL and g NL , 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 f NL , unless g NL is nonzero, and also (2) the series cannot represent models with g NL < 0, unless some higher-order non-Gaussianities are known. Finally, we apply the Edgeworth series to calculate the effects of g NL 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.