The Statistics of Density Peaks and the Column Density Distribution of the Lyα Forest

We develop a method to calculate the column density distribution of theLyman-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 tocompute the density and peculiar velocity fields. The effect of the latter onabsorption profiles is discussed and it is shown to have little effect on thecolumn density distribution. An approximation is introduced in which the columndensity distribution is related to a statistic of density peaks (involving itsheight 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 thetemperature-density relation as well as the power spectrum on scales $2 hMpc^{-1} < k < 20 h Mpc^{-1}$. An expression relating the three is given. Wefind very good agreement between the column density distribution obtained byapplying the Voigt-profile-fitting technique to the output of a fullhydrodynamic simulation and that obtained using our approximate method for atest model. This formalism then is applied to study a group of CDM as well asCHDM models. We show that the amplitude of the column density distributiondepends 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. Theslope of the distribution, on the other hand, can be used to distinguishbetween different models: those with a smaller amplitude and a steeper slope ofthe power spectrum on small scales give rise to steeper distributions, for therange of column densities we study. Comparison with high resolution Keck datais made.Comment: match accepted version; discussion added: the effect of the shape of the power spectrum on the slope of the column density distributio