Correlation between the Mean Matter Density and the Width of the Saturated Lyman Alpha Absorption

We report a scaling of the mean matter density with the width of the saturated Lyman alpha absorptions. This property is established using the ``pseudo-hydro'' technique (Croft et al. 1998). It provides a constraint for the inversion of the Lyman alpha forest, which encounters difficulty in the saturated region. With a Gaussian density profile and the scaling relation, a simple inversion of the simulated Lyman alpha forests shows that the one-dimensional mass power spectrum is well recovered on scales above 2 Mpc/h, or roughly k < 0.03 s/km, at z=3. The recovery underestimates the power on small scales, but improvement is possible with a more sophisticated algorithm.