On the evolution of the ionizing emissivity of galaxies and quasars required by the hydrogen reionization

The average rate of emission of ionizing radiation per unit volume ( or emissivity) in the universe can be calculated as the ratio of the intensity of the ionizing background to the mean free path of ionizing photons. The intensity of the background is measured from the mean transmitted flux of the Lyalpha forest, and the mean free path is measured from the abundance of Lyman limit systems, which has been observed so far up to z = 4. This yields an emissivity that is not larger than seven ionizing photons per Hubble time for each atom in the universe at z = 4, which may reasonably arise from QSOs and star-forming galaxies. In order for the reionization to end by z = 6, and assuming that the clumping factor of ionized gas during the reionization epoch is close to unity, this ionizing comoving emissivity cannot decline from z = 4 up to z similar to 9 by more than a factor 1.5. If the clumping factor were much larger than unity, then the emissivity would need to rapidly increase with redshift. Unless the ionizing emissivity increases substantially from z = 4 to z similar to 10-20, the Thomson optical depth to the cosmic microwave background must be in the range 0.045 < τ < 0.09.