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.