A halo model for cosmological Lyman-limit systems

We present an analytical model for cosmological Lyman-limit systems (LLS) that successfully reproduces the observed evolution of the mean free path (lambda(eff)) of ionizing photons. The evolution of the co-moving mean free path is predominantly a consequence of the changing meta galactic photoionization rate and the increase with cosmic time of the minimum mass below which haloes lose their gas due to photoheating. In the model, Lyman-limit absorption is caused by highly ionized gas in the outskirt of dark matter haloes. We exploit the association with haloes to compute statistical properties of lambda(eff) and of the bias, b, of LLS. The latter increases from b similar to 1.5 -> 2.6 from redshifts z = 2 -> 6. Combined with the rapid increase with redshift of the bias of the haloes that host a quasar, the model predicts a rapid drop in the value of lambda(eff) when measured in quasar spectra from z = 5 -> 6, whereas the actual value of lambda(eff) falls more smoothly. We derive an expression for the effective optical depth due to Lyman limit absorption as a function of wavelength and show that it depends sensitively on the poorly constrained number density of LLS as a function of column density. The optical depth drops below unity for all wavelengths below a redshift of similar to 2.5 which is therefore the epoch when the Universe first became transparent to ionizing photons.