The Statistics of Density Peaks and the Column Density Distribution of the Lyman-Alpha Forest

We develop a method to calculate the column density distribution of the Lyman-alpha forest for column densities in the range $10^{12.5} - 10^{14.5} cm^{-2}$. The Zel'dovich approximation, with appropriate smoothing, is used to compute the density and peculiar velocity fields. The effect of the later on absorption profiles is discussed and it is shown to have little effect on the column density distribution. We introduce an approximation in which the column density distribution is related to a statistic of density peaks (involving its height and first and second derivatives along the line of sight) in real space. We show that the slope of the column density distribution is determined by the temperature-density relation as well as the amount of power on small scales ($k$ between about $2 h Mpc^{-1}$ and $20 h Mpc^{-1}$). An expression relating the three is given. We find very good agreement between the column density distribution obtained by applying the Voigt-profile-fitting technique to the output of a full hydrodynamic simulation and that obtained using the above approximate method for a test model. It is also found that an alternative approximate method which is based on the lognormal approximation tends to underestimate the number of absorption lines at low column densities. Our formalism is applied to study a group of CDM as well as CHDM models. Comparison with high resolution Keck data is made.