We derive the distribution of neutral and ionized gas in high redshift clouds which are optically thick to hydrogen ionizing radiation , using published data on Lyman-limit and Damped Lyman- \alpha absorption systems in the redshift range 1.75 \leq z < 3.25 .
We assume that the distribution of the hydrogen total ( HI+HII ) column density in the absorbers , N _ { H } , follows a power law KN _ { H } ^ { - \alpha } , whereas the observed HI column density distribution deviates from a pure power law as a result of ionization from a background radiation field .
We use an accurate radiative transfer code for computing the rapidly varying ratio N _ { H } / N _ { HI } as a function of N _ { H } .
Comparison of the models and observations gave excellent fits with Maximum Likelihood solutions for the exponent \alpha and for X , the value of log ( N _ { H } / N _ { HI } ) when the Lyman-limit optical depth along the line of sight is \tau _ { LL } = 1 .
The slope of the total gas column density distribution with its relative 3 \sigma errors is \alpha = 2.7 ^ { +1.0 } _ { -0.7 } and X = 2.75 \pm 0.35 .
This value of X is much lower than what would be obtained for a gaseous distribution in equilibrium under its own gravity .
The ratio \eta _ { 0 } of dark matter to gas density is however not well constrained since log ( \eta _ { 0 } ) = 1.1 \pm 0.8 .
An extrapolation of our derived power law distribution towards systems of lower column density , the Lyman- \alpha forest , tends to favour models with log ( \eta _ { 0 } ) \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } 1.1 and \alpha \sim 2.7 to 3.3 .
With \alpha appreciably larger than 2 , Lyman-limit systems contain more gas than Damped Lyman- \alpha systems and Lyman- \alpha forest clouds even more .
Estimates of the cosmological gas and dark matter density due to absorbers of different column density at z \sim 2.5 are also given .