tokens: late He ii reionization). Therefore, they should not yet be interpreted as evidence in favour of the abundant high- z AGN models considered here. Combined with direct searches to better characterize the high- z faint AGN population, future measurements of the Ly forest opacity and of the IGM temperature will more tightly constrain the contribution of AGNs to the ionizing background, as well as their contribution to reionization. APPENDIX A: MODEL UNCERTAINTIES In this section, we discuss caveats for our models of the Ly forest opacity fluctuations considered in Section 3 . Below we argue that our models do not account for a number of effects that would, in combination, likely lead to a net decrease in the amplitude of eff fluctuations. We do not include the effects of inhomogeneous heating from patchy H i reionization. These temperature inhomogeneities would suppress fluctuations in eff because the overdense regions that were reionized earlier would tend to be cooler than average by z 5.5 (i.e. after reionization), with enhanced equilibrium H i densities from the T 0.7 dependence of the recombination coefficient. However, the ionizing background in those regions would tend to be stronger owing to the bias of the sources. The competition between these two effects would smooth out fluctuations in eff (recall that ). If H i reionization ends at z 6, this competition could have a significant effect on the width of the eff distribution (DAloisio etal. 2015 ). On the other hand, we also do not account for inhomogeneous heating during He ii reionization, which would likely work in the direction of enhancing the eff fluctuations, since the regions of strong ionizing background surrounding AGNs would also be the most recently photoheated. We do not take into account the finite lifetimes of AGNs. With a duty cycle of unity, our models effectively assume an infinite AGN lifetime. We argue that modelling the finite lifetimes could act to suppress eff fluctuations in at least two ways. First, for a fixed AGN abundance (which is set by the assumed luminosity function and magnitude limit), a smaller duty cycle would require that AGNs are hosted in haloes that are less massive and therefore less biased than in our models. This would lead to the AGN being less clustered overall, which would yield somewhat smaller eff fluctuations. Secondly, consider a region with an overdensity of bright AGNs in our fiducial model (see e.g. the lower left-hand corners of the panels in Fig. 3 ). These regions produce large segments of the Ly forest with low optical depth. Suppose now that only a fraction of these AGNs are active at a given redshift. In this case, the Ly opacity in this large-scale region would be less coherent an effect that would make the eff distribution narrower. On the other hand, we note that the proximity zone remnants left by AGNs that recently became dormant might mitigate these effects. We do not account for the anisotropy of emissions from AGNs. Our
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