The z~6 Luminosity Function Fainter than -15 mag from the Hubble
  Frontier Fields: The Impact of Magnification Uncertainties
  We use the largest sample of z~6 galaxies to date from the first four Hubble
Frontier Fields clusters to set constraints on the shape of the z~6 luminosity
functions (LFs) to fainter than Muv=-14 mag. We quantify, for the first time,
the impact of magnification uncertainties on LF results and thus provide more
realistic constraints than other recent work. Our simulations reveal that for
the highly-magnified sources the systematic uncertainties can become extremely
large fainter than -14 mag, reaching several orders of magnitude at 95%
confidence at ~-12 mag. Our new forward-modeling formalism incorporates the
impact of magnification uncertainties into the LF results by exploiting the
availability of many independent magnification models for the same cluster. One
public magnification model is used to construct a mock high-redshift galaxy
sample that is then analyzed using the other magnification models to construct
a LF. Large systematic errors occur at high magnifications (mu>30) because of
differences between the models. The volume densities we derive for faint (>-17
mag) sources are ~3-4x lower than one recent report and give a faint-end slope
alpha=-1.92+/-0.04, which is 3.0-3.5sigma shallower (including or not including
the size uncertainties, respectively). We introduce a new curvature parameter
delta to model the faint end of the LF and demonstrate that the observations
permit (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 of
magnification errors and new size determinations raise doubts about previous
reports regarding the form of the LF at >-14 mag. We discuss the implications
of our turn-over constraints in the context of recent theoretical predictions.
