  I apply the well controlled Hydro-PM approximation of Gnedin & Hui to model
the column density distribution of the Lyman-alpha forest for 25 different flat
cosmological scenarios, including variants of the standard CDM, tilted CDM, CDM
with a cosmological constant, and CHDM models. I show that within the accuracy
of the HPM approximation the slope of the column density distribution reflects
the degree of nonlinearity of the cosmic gas distribution and is a function of
the rms linear density fluctuation at the characteristic filtering scale only.
The amplitude of the column density distribution, expressed as the value for
the ionizing intensity, is derived as a function of the cosmological parameters
(to about 40% accuracy). The observational data are currently consistent with
the value for the ionizing intensity being constant in the redshift interval
z~2-4.
