Gas dynamics of a luminous $z$ = 6.13 quasar ULAS J1319$+$0950 revealed
  by ALMA high resolution observations
  We present new Atacama Large Millimeter/submillimeter Array (ALMA)
observations of the dust continuum and [C II] 158 $\mu$m fine structure line
emission towards a far-infrared-luminous quasar, ULAS J131911.29$+$095051.4 at
$z=6.13$, and combine the new Cycle 1 data with ALMA Cycle 0 data. The combined
data have an angular resolution $\sim$ $0.3$, and resolve both the dust
continuum and the [C II] line emission on few kpc scales. The [C II] line
emission is more irregular than the dust continuum emission which suggests
different distributions between the dust and [C II]-emitting gas. The combined
data confirm the [C II] velocity gradient that we previously detected in lower
resolution ALMA image from Cycle 0 data alone. We apply a tilted ring model to
the [C II] velocity map to obtain a rotation curve, and constrain the circular
velocity to be 427 $\pm$ 55 km s$^{-1}$ at a radius of 3.2 kpc with an
inclination angle of 34$^\circ$. We measure the dynamical mass within the 3.2
kpc region to be 13.4$_{-5.3}^{+7.8}$ $\times 10^{10}\,M_{\odot}$. This yields
a black hole and host galaxy mass ratio of 0.020$_{-0.007}^{+0.013}$, which is
about 4$_{-2}^{+3}$ times higher than the present-day $M_{\rm BH}$/$M_{\rm
bulge}$ ratio. This suggests that the supermassive black hole grows the bulk of
its mass before the formation of the most of stellar mass in this quasar host
galaxy in the early universe.
