On the trispectrum as a Gaussian test for cosmology

In the standard model for structure formation, bound objects originate from the gravitational collapse of small perturbations arising from quantum flu ctuations with random phases. In other scenarios, based on defects, structu res are seeded by localized energy density. In principle, it is possible to differentiate between these models on the basis of their statistical prope rties; only in the former case is the initial density field an almost-perfe ct random Gaussian held. In this paper, we investigate the use of the tri-s pectrum of the galaxy density field, which is the connected four-point func tion in Fourier space, as a discriminant between Gaussian and non-Gaussian models. It has the advantage of having only weak nonlinear growth. We defin e a related statistic tau which, as a test of the Gaussian hypothesis, is i ndependent of cosmology, the power spectrum, and biasing, in real space, an d which is, in principle, a measure of the departure from Gaussian statisti cs. For galaxy redshift surveys, the statistic depends on cosmology and bia s only through the potentially observable parameter beta. We compute the ex pected errors on the estimate of tau, and demonstrate with numerical simula tions that it can be a useful discriminant of models, with the important pr oviso that any bias is linear on large scales. Whether it is the most effec tive method is uncertain and depends on the nature of the departure from Ga ussianity.