On the Trispectrum as a Gaussian Test for Cosmology

In the standard model for structure formation, bound objects originate fromthe gravitational collapse of small perturbations arising from quantumfluctuations with random phases. In other scenarios, based on defects,structures are seeded by localized energy density. In principle, it is possibleto differentiate between these models on the basis of their statisticalproperties; only in the former case is the initial density field analmost-perfect random gaussian field. In this paper, we investigate the use ofthe trispectrum of the galaxy density field, which is the connected four-pointfunction in Fourier space, as a discriminant between gaussian and non-gaussianmodels. It has the advantage of having only weak non-linear growth. We define arelated statistic $\tau$ which, as a test of the gaussian hypothesis, isindependent of cosmology, the power spectrum and biasing, in real space, andwhich is, in principle, a measure of the departure from gaussian statistics.For galaxy redshift surveys, the statistic depends on cosmology and bias onlythrough the potentially observable parameter $\beta$. We compute the expectederrors on the estimate of $\tau$, and demonstrate with numerical simulationsthat it can be a useful discriminant of models, with the important proviso thatany bias is linear on large scales. Whether it is the most effective method isuncertain and depends on the nature of the departure from gaussianity.Comment: to appear in ApJ, 28 pages, 5 figure