We present measurements of the specific ultraviolet luminosity density from a sample of 483 galaxies at 6<z<8. These galaxies were selected from new deep near-infrared HST imaging from the CANDELS, HUDF09 and ERS programs. In contrast to the majority of previous analyses, which assume that the distribution of galaxy ultraviolet (UV) luminosities follows a Schechter distribution, and that the distribution continues to luminosities far below our observable limit, we investigate the contribution to reionization from galaxies which we can observe, free from these assumptions. We find that the observable population of galaxies can sustain a fully reionized IGM at z=6, if the average ionizing photon escape fraction (f_esc) is ~30%. A number of previous studies have measured UV luminosity densities at these redshifts that vary by 5X, with many concluding that galaxies could not complete reionization by z=6 unless a large population of galaxies fainter than the detection limit were invoked, or extremely high values of f_esc were present. The observed UV luminosity density from our observed galaxy samples at z=7-8 is not sufficient to maintain a fully reionized IGM unless f_esc>50%. Combining our observations with constraints on the emission rate of ionizing photons from Ly-alpha forest observations at z=6, we can constrain f_esc<34% (2-sigma) if the observed galaxies are the only contributors to reionization, or <13% (2-sigma) if the luminosity function extends to M_UV = -13. These escape fractions are sufficient to complete reionization by z=6. These constraints imply that the volume ionized fraction of the IGM becomes less than unity at z>7, consistent with a number of complementary reionization probes. If faint galaxies dominate reionization, future JWST observations will probe deep enough to see them, providing an indirect constraint on the ionizing photon escape fraction [abridged].
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