We report a new analysis of the Hubble Frontier Fields clusters Abell 2744 and MACS 0416 using wavelet decomposition to remove the cluster light, enabling the detection of highly magnified (>50x) galaxies a factor of 10x fainter in luminosity than previous studies. We find 167 galaxies at z > 6, and with this sample we are able to characterize the UV luminosity function to M_UV = -12.5 at z ~ 6, -14 at z ~ 7 and -15 at z ~ 8. We find a steep faint-end slope (alpha <-2), and with our improved statistics at the faint end we reduce the fractional uncertainty on alpha to <2% at z ~ 6 - 7 and 4% at z ~ 8. We also investigate the systematic uncertainty due to the lens modelling by using every available lens model individually and comparing the results; this systematic fractional uncertainty on alpha is <4% at all redshifts. We now directly observe galaxies in the luminosity regime where simulations predict a change in the faint-end slope of the luminosity function, yet our results provide statistically very strong evidence against any turnover in the luminosity range probed, more consistent with simulations in which stars form in lower-mass halos. Thus we find strong support for the extension of the steep luminosity function to M_UV = -13 at z > 6, consistent with the number of faint galaxies needed to reionize the Universe under standard assumptions.
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