Size--luminosity relations and UV luminosity functions at $z=6-9$ simultaneously derived from the complete Hubble Frontier Fields data

We construct $z\sim6-7$, 8, and 9 faint Lyman break galaxy samples (334, 61, and 37 galaxies, respectively) with accurate size measurements with the software $\texttt{glafic}$ from the complete Hubble Frontier Fields (FF) cluster and parallel fields data. These are the largest samples hitherto and reach down to the faint ends of recently obtained deep luminosity functions. At faint magnitudes, however, these samples are highly incomplete for galaxies with large sizes, implying that derivation of the luminosity function (LF) sensitively depends on the intrinsic size--luminosity (RL) relation. We thus conduct simultaneous maximum-likelihood estimation of LF and RL relation parameters from the observed distribution of galaxies on the RL plane with help of a completeness map as a function of size and luminosity. At $z\sim6-7$, we find that the intrinsic RL relation expressed as $r_\textrm{e} \propto L^\beta$ has a notably steeper slope of $\beta=0.46^{+0.08}_{-0.09}$ than those at lower redshifts, which in turn implies that the LF has a relatively shallow faint-end slope of $\alpha=-1.86^{+0.17}_{-0.18}$. This steep $\beta$ can be reproduced by a simple analytical model in which smaller galaxies have lower specific angular momenta. The $\beta$ and $\alpha$ values for the $z\sim8$ and 9 samples are consistent with those for $z\sim6-7$ but with larger errors. For all three samples there is a large, positive covariance between $\beta$ and $\alpha$, implying that the simultaneous determination of these two parameters is important. We also provide new strong lens mass models of Abell S1063 and Abell 370 as well as updated mass models of Abell 2744 and MACS J0416.1$-$2403.