Statistics of density peaks and the column density distribution of the Ly alpha forest

We develop a method to calculate the column density distribution of the Ly alpha forest for column densities in the range 10(12.5)-10(14.5) cm(-2). The Zeldovich approximation, with appropriate smoothing, is used to compute the density and peculiar velocity fields. The effect of the latter on absorption profiles is discussed, and it is shown to have little effect on the column density distribution. An approximation is introduced 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 power spectrum, on scales 2 h Mpc(-1) less than or similar to k less than or similar to 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 our approximate method for a test model. This formalism then is applied to study a group of cold dark matter, as well as cold plus hot dark matter, models. We show that the amplitude of the column density distribution depends on the combination of parameters (Omega(b)h(2))(2)T(0)(0.7)J(HI)(-1), which is not well constrained by independent observations. The slope of the distribution, on the other hand, can be used to distinguish between different models: those with a smaller amplitude and a steeper slope of the power spectrum on small scales give rise to steeper distributions, for the range of column densities that we study. Comparison with high-resolution Keck data is made.