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I apply the well controlled hydro-PM (HPM) approximation of Gnedin & Hui to model the column density distribution of the Ly a forest for 25 different flat cosmological scenarios, including variants of the standard cold dark matter (CDM), tilted CDM, CDM with a cosmological constant, and cold + hot dark matter (CHDM) models. I show that, within the accuracy of the HPM approximation, the slope of the column density distribution reflects the degree of non-linearity 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 per cent accuracy). The observational data are currently consistent with the value for the ionizing intensity being constant in the redshift interval z 2-4.
Nickolay Y. Gnedin (Tue,) studied this question.