We explore scenarios for reionizing the intergalactic medium with low galaxy ionizing photon escape fractions. We combine simulation-based halo-mass dependent escape fractions with an extrapolation of the observed galaxy rest-ultraviolet luminosity functions to solve for the reionization history from z=20 to z=4. We explore the posterior distributions for key unknown quantities, including the limiting halo mass for star-formation, the ionizing photon production efficiency, and a potential contribution from active galactic nuclei (AGN). We marginalize over the allowable parameter space using a Markov Chain Monte Carlo method, finding a solution which satisfies the most model-independent constraints on reionization. Our fiducial model can match observational constraints with an average escape fraction of <5% throughout the bulk of the epoch of reionization if: i) galaxies form stars down to the atomic cooling limit before reionization and a photosuppression mass of log(M_h/Msol)~9 during/after reionization (-13<M_UV,lim<-11); ii) galaxies become more efficient producers of ionizing photons at higher redshifts and fainter magnitudes, and iii) there is a significant, but sub-dominant, contribution by AGN at z < 7. In this model the faintest galaxies (M_UV>-15) dominate the ionizing emissivity, leading to an earlier start to reionization and a smoother evolution of the ionized volume filling fraction than models which assume a single escape fraction at all redshifts and luminosities. The ionizing emissivity from this model is consistent with observations at z=4-5 (and below, when extrapolated), in contrast to some models which assume a single escape fraction. Our predicted ionized volume filling fraction at z=7 of Q_HII=78% (+- 8%) is in ~1-2 sigma tension with observations of Lya emitters at z~7 and the damping wing analyses of the two known z>7 quasars, which prefer Q_HII,z=7~40-50%.
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