  An attractive and simple hypothesis for the formation of large-scale
structure is that it developed by gravitational instability from primordial
fluctuations with an initially Gaussian probability distribution. Non-linear
gravitational evolution drives the distribution away from the Gaussian form,
generating measurable skewness and kurtosis even when the variance of the
fluctuations is much smaller than unity. We use perturbation theory to compute
the kurtosis of the mass density field and the velocity divergence field that
arises during the weakly non-linear evolution of initially Gaussian
fluctuations. We adopt an Einstein--de~Sitter universe for the perturbative
calculations, and we discuss the generalization to a universe of arbitrary
$\Omega$. We obtain semi-analytic results for the case of scale-free, power-law
spectra of the initial fluctuations and final smoothing of cosmic fields with a
Gaussian filter. We also give an exact analytical formula for the dependence of
the skewness of these fields on the power spectrum index. We show that the
kurtosis decreases with the power spectrum index, and we compare our more
accurate results for the kurtosis to previous estimates from Monte Carlo
integrations. We also compare our results to values obtained from cosmological
N-body simulations with power-law initial spectra. Measurements of the skewness
and kurtosis parameters can be used to test the hypothesis that structure in
the universe formed by gravitational instability from Gaussian initial
conditions.
