We develop an analytical method based on the lognormal approximation to compute the column density distribution of the Lyman-alpha forest in the low column density limit .
We compute the column density distributions for six different cosmological models and found that the standard , COBE -normalized CDM model can not fit the observations of the Lyman-alpha forest at z = 3 .
The amplitude of the fluctuations in that model has to be lowered by a factor of almost 3 to match observations .
However , the currently viable cosmological models like the slightly tilted COBE -normalized CDM+ \Lambda model , the CHDM model with \Omega _ { \nu } = 0.2 , and the low-amplitude Standard CDM model are all in agreement with observations , to within the accuracy of our approximation , for the value of the cosmological baryon density at or higher than the old Standard Big Bang nucleosynthesis value of \Omega _ { b } h ^ { 2 } = 0.0125 for the currently favored value of the ionizing radiation intensity .
With the low value for the baryon density inferred by Hogan & Rugers \markcite HR96 ( 1996 ) , the models can only marginally match observations .