The Extreme Faint End of the UV Luminosity Function at $z\sim6$ Through
  Gravitational Telescopes: a comprehensive assessment of strong lensing
  uncertainties
  With the Hubble Frontier Fields program, gravitational lensing has provided a
powerful way to extend the study of the ultraviolet luminosity function (LF) of
galaxies at $z \sim 6$ down to unprecedented magnitude limits. At the same
time, significant discrepancies between different studies were found at the
very faint end of the LF. In an attempt to understand such disagreements, we
present a comprehensive assessment of the uncertainties associated with the
lensing models and the size distribution of galaxies. We use end-to-end
simulations from the source plane to the final LF that account for all lensing
effects and systematic uncertainties by comparing several mass models. In
addition to the size distribution, the choice of lens model leads to large
differences at magnitudes fainter than $M_{UV} = -15~$ AB mag, where the
magnification factor becomes highly uncertain. We perform MCMC simulations that
include all these uncertainties at the individual galaxy level to compute the
final LF, allowing, in particular, a crossover between magnitude bins. The best
LF fit, using a modified Schechter function that allows for a turnover at faint
magnitudes, gives a faint-end slope of $\alpha = -2.01_{-0.14}^{+0.12}$, a
curvature parameter of $\beta = 0.48_{-0.25}^{+0.49}$, and a turnover magnitude
of $M_{T} = -14.93_{-0.52}^{+0.61}$. Most importantly our procedure shows that
robust constraints on the LF at magnitudes fainter than $M_{UV} = -15~$ AB
remain unrealistic. More accurate lens modeling and future observations of
lensing clusters with the James Webb Space Telescope can reliably extend the UV
LF to fainter magnitudes.
