Evolution of skewness and kurtosis of cosmic density fields

We perform numerical simulations of the evolution of the cosmic web for the conventional $\Lambda$CDM model in box sizes $L_0=256,~512,~1024$~\Mpc. We calculate models, corresponding to the present epoch $z=0$, and to redshifts $z=1,~3,~5,~10,~30$. We calculate density fields with various smoothing levels to find the dependence of the density field on smoothing. We calculate PDF and its moments -- variance, skewness and kurtosis. The dimensionless skewness $S$ and the dimensionless kurtosis $K$ characterise symmetry and flatness properties of the 1-point PDF of the cosmic web. Relations $S =S_3 \sigma$, and $K=S_4 \sigma^2$ are now tested in standard deviation $\sigma$ range, $0.015 \le \sigma \le 10$, and in redshift $z$ range $0 \le z \le 30$. Reduced skewness $S_3$ and reduced kurtosis $S_4$ described in log-log format. Data show that these relations can be extrapolated to earlier redshifts $z$, and to smaller $\sigma$, as. well as to smaller and larger smoothing lengths $R$. Reduced parameters depend on basic parameters of models. The reduced skewness: $S_3 = f_3(R) +g_3(z)\,\sigma^2$, and the reduced kurtosis: $S_4 = f_4(R) +g_4(z)\,\sigma^2$, where $f_3(R)$ and $f_4(R)$ are parameters, depending on the smoothing length, $R$, and $g_3(z)$ and $g_4(z)$ are parameters, depending on the evolutionary epoch $z$. The lower bound of the amplitude parameters are, $f_3(R) \approx 3.5$ for reduced skewness, and $f_4(R) \approx 16$ for reduced kurtosis, for large smoothing lengths, $R\approx 32$~\Mpc. With decreasing smoothing length $R$ the skewness and kurtosis values for given redshift $z$ turn upwards.