Full-Data Results of Hubble Frontier Fields: UV Luminosity Functions at
  $z\sim6-10$ and a Consistent Picture of Cosmic Reionization
  We present UV luminosity functions of dropout galaxies at $z\sim6-10$ with
the complete Hubble Frontier Fields data. We obtain a catalog of $\sim450$
dropout-galaxy candidates (350, 66, and 40 at $z\sim6-7$, 8, and 9,
respectively), whose UV absolute magnitudes reach $\sim-14$ mag, $\sim2$ mag
deeper than the Hubble Ultra Deep Field detection limits. We carefully evaluate
number densities of the dropout galaxies by Monte-Carlo simulations, including
all lensing effects such as magnification, distortion, and multiplication of
images as well as detection completeness and contamination effects in a
self-consistent manner. We find that UV luminosity functions at $z\sim6-8$ have
steep faint-end slopes, $\alpha\sim-2$, and likely steeper slopes,
$\alpha\lesssim-2$ at $z\sim9-10$. We also find that the evolution of UV
luminosity densities shows a non-accelerated decline beyond $z\sim8$ in the
case of $M_\mathrm{trunc}=-15$, while an accelerated in the case of
$M_\mathrm{trunc}=-17$. We examine whether our results are consistent with the
Thomson scattering optical depth from the Planck satellite and the ionized
hydrogen fraction $Q_\mathrm{HII}$ at $z\lesssim7$ based on the standard
analytic reionization model. We find that there exist reionizaiton scenarios
that consistently explain all the observational measurements with the allowed
parameters of $f_{\rm esc}=0.17^{+0.07}_{-0.03}$ and $M_\mathrm{trunc}>-14.0$
for $\log\xi_\mathrm{ion}/[\mathrm{erg}^{-1}\ \mathrm{Hz}]=25.34$, where
$f_{\rm esc}$ is the escape fraction, $M_\mathrm{trunc}$ is the faint limit of
the UV luminosity function, and $\xi_\mathrm{ion}$ is the conversion factor of
the UV luminosity to the ionizing photon emission rate. The length of the
reionization period is estimated to be $\Delta z=3.9^{+2.0}_{-1.6}$ (for
$0.1<Q_\mathrm{HII}<0.99$), consistent with the recent estimate from Planck.
