We present UV luminosity functions of dropout galaxies at z \sim 6 - 10 with the complete Hubble Frontier Fields data .
We obtain a catalog of \sim 450 dropout-galaxy candidates ( 350 , 66 , and 40 at z \sim 6 - 7 , 8 , and 9 , respectively ) , whose UV absolute magnitudes reach \sim - 14 mag , \sim 2 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 \sim 6 - 8 have steep faint-end slopes , \alpha \sim - 2 , and likely steeper slopes , \alpha \lesssim - 2 at z \sim 9 - 10 .
We also find that the evolution of UV luminosity densities shows a non-accelerated decline beyond z \sim 8 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 \lesssim 7 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 _ { 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 _ { 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 .