Kurtosis of large-scale cosmic fields

An attractive and simple hypothesis for the formation of large-scalestructure is that it developed by gravitational instability from primordialfluctuations with an initially Gaussian probability distribution. Non-lineargravitational evolution drives the distribution away from the Gaussian form,generating measurable skewness and kurtosis even when the variance of thefluctuations is much smaller than unity. We use perturbation theory to computethe kurtosis of the mass density field and the velocity divergence field thatarises during the weakly non-linear evolution of initially Gaussianfluctuations. We adopt an Einstein--de~Sitter universe for the perturbativecalculations, and we discuss the generalization to a universe of arbitrary$\Omega$. We obtain semi-analytic results for the case of scale-free, power-lawspectra of the initial fluctuations and final smoothing of cosmic fields with aGaussian filter. We also give an exact analytical formula for the dependence ofthe skewness of these fields on the power spectrum index. We show that thekurtosis decreases with the power spectrum index, and we compare our moreaccurate results for the kurtosis to previous estimates from Monte Carlointegrations. We also compare our results to values obtained from cosmologicalN-body simulations with power-law initial spectra. Measurements of the skewnessand kurtosis parameters can be used to test the hypothesis that structure inthe universe formed by gravitational instability from Gaussian initialconditions.Comment: 29 pp incl. 8 figs, uuencoded compressed postscript, submitted to MNRAS, preprints CAMK/281, IASSNS-AST 94/3