Evolution of skewness and kurtosis of cosmic density fields
  Methods. We perform numerical simulations of the evolution of the cosmic web
for the conventional LCDM model. The simulations cover a wide range of box
sizes L = 256 - 4000 Mpc/h, mass and force resolutions and epochs from very
early moments z = 30 to the present moment z = 0. We calculate density fields
with various smoothing lengths to find the dependence of the density field on
smoothing scale. We calculate PDF and its moments - variance, skewness and
kurtosis. Results. We focus on the third (skewness S) and fourth (kurtosis K)
moments of the distribution functions: their dependence on the smoothing scale,
the amplitude of fluctuations and the redshift. During the evolution the
reduced skewness $S_3= S/\sigma$ and reduced kurtosis $S_4=K/\sigma^2$ present
a complex behaviour: at a fixed redshift curves of $S_3(\sigma)$ and
$S_4(\sigma)$ steeply increase with $\sigma$ at $\sigma\le 1$ and then flatten
out and become constant at $\sigma\ge2$. If we fix the smoothing scale $R_t$,
then after reaching the maximum at $\sigma\approx 2$, the curves at large
$\sigma$ start to gradually decline. We provide accurate fits for the evolution
of $S_{3,4}(\sigma,z)$. Skewness and kurtosis approach at early epochs constant
levels, depending on smoothing length: $S_3(\sigma) \approx 3$ and $S_4(\sigma)
\approx 15$. Conclusions. Most of statistics of dark matter clustering (e.g.,
halo mass function or concentration-mass relation) are nearly universal: they
mostly depend on the $\sigma$ with the relatively modest correction to explicit
dependence on the redshift. We find just the opposite for skewness and
kurtosis: the dependence of moments on evolutionary epoch $z$ and smoothing
length $R_t$ is very different, together they determine the evolution of
$S_{3,4}(\sigma)$ uniquely. The evolution of $S_3$ and $S_4$ cannot be
described by current theoretical approximations.
