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We use the largest sample of z~6 galaxies to date from the first four HubbleFrontier Fields clusters to set constraints on the shape of the z~6 luminosityfunctions (LFs) to fainter than Muv=-14 mag. We quantify, for the first time,the impact of magnification uncertainties on LF results and thus provide morerealistic constraints than other recent work. Our simulations reveal that forthe highly-magnified sources the systematic uncertainties can become extremelylarge fainter than -14 mag, reaching several orders of magnitude at 95%confidence at ~-12 mag. Our new forward-modeling formalism incorporates theimpact of magnification uncertainties into the LF results by exploiting theavailability of many independent magnification models for the same cluster. Onepublic magnification model is used to construct a mock high-redshift galaxysample that is then analyzed using the other magnification models to constructa LF. Large systematic errors occur at high magnifications (mu>30) because ofdifferences between the models. The volume densities we derive for faint (>-17mag) sources are ~3-4x lower than one recent report and give a faint-end slopealpha=-1.92+/-0.04, which is 3.0-3.5sigma shallower (including or not includingthe size uncertainties, respectively). We introduce a new curvature parameterdelta to model the faint end of the LF and demonstrate that the observationspermit (at 68% confidence) a turn-over at z~6 in the range -15.3 to -14.2 mag,depending on the assumed lensing model. The present consideration ofmagnification errors and new size determinations raise doubts about previousreports regarding the form of the LF at >-14 mag. We discuss the implicationsof our turn-over constraints in the context of recent theoretical predictions.

The z~6 Luminosity Function Fainter than -15 mag from the Hubble Frontier Fields: The Impact of Magnification Uncertainties