Abstract Cosmic hydrogen is reionized and maintained in its highly ionized state by the ultraviolet emission attributed to an early generation of stars and quasars. The Lyα opacity observed in absorption spectra of high-redshift quasars permits more stringent constraints on the ionization state of cosmic hydrogen. Based on density perturbation and structure formation theory, we develop an analytic model to trace the evolution of the ionization state in the post-overlap epoch of reionization, in which the bias factor is taken into account. For quasars, we represent an improved luminosity function by utilizing a hybrid approach for the halo formation rate that is in reasonable agreement with the published measurements at 2≤ z ≤6. Comparison with the classic Press-Schechter mass function of dark matter halos, we demonstrate that the biased mass distribution indeed enhances star formation efficiency in the overdense environment by more than 25 per cent following the overlap of ionized bubbles. In addition, an alternative way is introduced to derive robust estimates of the mean free path for ionizing photons. In our model, starforming galaxies are likely to dominate the ionizing background radiation beyond z = 3, and quasars contribute equally above a redshift of z ∼ 2.5. From 5 ≤z ≤6, the lack of evolution in photoionization rate can thus be explained by the relatively flat evolution in star formation efficiency, although the mean free path of ionizing photons increases rapidly. Moreover, in the redshift interval z ∼ 2−6, the expected mean free path and Gunn-Peterson optical depth obviously evolve by a factor of ∼ 500 and ∼ 50 respectively. We find that the relative values of critical overdensities for hydrogen ionization and collapse could be 430% at z ≈ 2 and 2% at z ≈ 6, suggesting a rapid overlap process in the overdense regions around instant quasars following reionization. We further illustrate that the absolute estimates of the fraction of neutral hydrogen computed from theoretical models may not be important because of comparable uncertainties in the computation.

Keywords cosmology: theory — formation: galaxies — quasars: general — diffuse radiation— intergalactic medium

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