Skewness and Kurtosis in Large-Scale Cosmic Fields

In this paper, I present the calculation of the third and fourth moments of both the distribution function of the large--scale density and the large--scale divergence of the velocity field, $\theta$. These calculations are made by the mean of perturbative calculations assuming Gaussian initial conditions and are expected to be valid in the linear or quasi linear regime. The moments are derived for a top--hat window function and for any cosmological parameters $\Omega$ and $\Lambda$. It turns out that the dependence with $\Lambda$ is always very weak whereas the moments of the distribution function of the divergence are strongly dependent on $\Omega$. A method to measure $\Omega$ using the skewness of this field has already been presented by Bernardeau et al. (1993). I show here that the simultaneous measurement of the skewness and the kurtosis allows to test the validity of the gravitational instability scenario hypothesis. Indeed there is a combination of the first three moments of $\theta$ that is almost independent of the cosmological parameters $\Omega$ and $\Lambda$, $${( -3 ^2) \over ^2}\approx 1.5,$$ (the value quoted is valid when the index of the power spectrum at the filtering scale is close to -1) so that any cosmic velocity field created by gravitational instabilities should verify such a property.