Correlation between the mean matter density and the width of the saturated Lyα absorption

We report a scaling of the mean matter density with the width of the saturated Lyalpha absorptions. This property is established using the `pseudo-hydro' technique (Croft et al.). It provides a constraint for the inversion of the Lyalpha forest, which encounters difficulty in the saturated region. With a Gaussian density profile and the scaling relation, a simple inversion of the simulated Lyalpha forests shows that the one-dimensional mass power spectrum is well recovered on scales above 2 h(-1) Mpc, or roughly k less than or similar to 0.03 km(-1) s, at z = 3. The recovery underestimates the power on small scales, but improvement is possible with a more sophisticated algorithm.