Properties of observed Ly-$\alpha$ forest
  The main observed properties of Ly-$\alpha$ absorbers are investigated on the basis of theoretical model of formation and evolution of DM structure elements. This model is generally consistent with simulations of absorbers formation and with statistical description of structure evolution based on the Zel'dovich theory. The analysis of redshift variations of comoving linear number density of absorbers was performed in our previous paper.   We show that the observed characteristics of Doppler parameter can be related to the size of DM structure elements what allows us to explain the observed distribution of Doppler parameter. This distribution is found to be consistent with the Gaussian initial perturbations. The observed characteristics of entropy and column density, $N_{HI}$, confirm that merging of pancakes is the main evolutionary process at redshifts $z\geq 2$. The observed sample of absorbers characterizes mainly the matter distribution within large low density regions and therefore it is difficult to reconstruct the density field from the distribution of absorbers. 
M. Demianski, A. Doroshkevich, & V.Turchaninov