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tokens: But note that, as discussed above in Section 4.1 , a direct comparison of our results with the models of Madau Haardt ( 2015 ), Khaire Srianand ( 2015 ), and Puchwein et al. ( 2019 ) is difficult because of the inhomogenous redshift-dependent integration limits adopted by these authors. Before considering the deficit in the photoionization rate obtained from AGN at low redshifts relative to measurements from the Ly α forest as a photon underproduction crisis, it is worthwhile to recall various assumptions entering our derivation. With a spectral index of −1.7, our ionization rate estimate falls short of the Ly α forest measurements by a factor of ∼2 with an integration limit of M 1450 = −18 on the luminosity function. Madau Haardt ( 2015 ) implicitly assume an integration limit of M 1450 = −14 at ∼ 0 (see also Khaire Srianand 2015 ; Puchwein et al. 2019 ) as this limit corresponds to M * + 5 for the Schulze et al. ( 2009 ) luminosity function measurement. It is unclear if the assumption of unit LyC escape fraction is valid for such faint AGN, but we find that even with this extremely faint integration limit, the photoionization rate deficit is reduced by only about 25 per cent for a spectral index of −1.7. While this result does not change significantly if we assume a spectral index of −1.4 as suggested by Stevans et al. ( 2014 ), assuming a much harder spectrum with a spectral index of −0.56 (Scott et al. 2004 ) for faint ( M 1450 > −23) AGN completely alleviates the deficit for an integration limit of M 1450 = −18. One should also note that our ionization rate model ignores the large uncertainties in the H i column density distribution at the Lyman limit (Ribaudo, Lehner Howk 2011 ; Gaikwad et al. 2017a ; Shull et al. 2017 ). Alternatively, it may be possible to balance the ionization rate deficit by the contribution from low-redshift LyC-leaking galaxies (Izotov et al. 2016 , 2018a , b ). 4.5 Helium reionization We now consider the implications of our AGN luminosity function models for He ii reionization. The time evolution of the volume-averaged He iii fraction is given by (e.g. Haardt Madau 2012 ) where is the emission rate of ≥4 Ry photons per unit proper volume, 〈 n He 〉 is the average proper helium number density, and 〈 t rec,He 〉 is the average recombination time-scale for He iii . With our quasar emissivity model from Section 4.1 , the emission rate can be written as with the quasar spectral index α ν = −1.7 and the comoving 912 Å emissivity ε 912 () is obtained from the fits shown in Fig. 8 . The recombination time-scale in equation ( 29 ) is given by where 〈 n H 〉 = 1.881 × 10 −7 (1 + ) 3 cm −3 is the average proper hydrogen number density, α B is the Case B He iii recombination coefficient (Hui Gnedin 1997 ), and χ = 0.079 is the cosmic number fraction of helium for a cosmic helium mass fraction of Y He = 0.24. We assume that the clumping factor C for helium is the same as for hydrogen, for which Shull et al. ( 2012a ) obtained