Flat Rotation Curves found in Merging Dusty Starbursts at $z=2.3$
  through Tilted-Ring Modeling
  The brightest 500$\,\mu$m source in the XMM field, HXMM01, is a rare merger
of luminous starburst galaxies at $z=2.3$ with a dust-obscured star-formation
rate of 2,000$\,M_{\odot}\,{\rm yr}^{-1}$. Here we present high-resolution
spectroscopic observations of HXMM01 with the Atacama Large
Millimeter/submillimeter Array (ALMA). We detect line emission from ${\rm
CO\,{\it J}=7\to6}$, [C I]$\,{{^3P_2}\to{^3P_1}}$, and p-${\rm
H_{2}O}\,{2_{11}}\to{2_{02}}$ and continuum emission at $230\,$GHz. At a
spatial resolution of 0.2" and a spectral resolution of 40$\,\rm km\,s^{-1}$,
the source is resolved into three distinct components, which are spatially and
dynamically associated within a projected radius of 20$\,$kpc and a radial
velocity range of 2,000$\,\rm km\,s^{-1}$. For two major components, our
Bayesian-based tilted-ring modeling of the ALMA spectral cubes shows almost
flat rotation curves peaking at $\sim500\,\rm km\,s^{-1}$ at galactocentric
distances between 2 and 5$\,$kpc. Each of them has a dynamical mass of
$\sim10^{11}\,M_\odot$. The combination of the dynamical masses and the
archival ${\rm CO\,{\it J}=1\to0}$ data places strong upper limits on the
CO$\to$H$_2$ conversion factor of $\alpha_{\rm
CO}\lesssim1.4-2.0\,{M_{\odot}}\,\rm (K\,km\,s^{-1}\,pc^{2})^{-1}$. These
limits are significantly below the Galactic inner disk $\alpha_{\rm CO}$ value
of $4.3\,{M_{\odot}}\,\rm (K\,km\,s^{-1}\,pc^{2})^{-1}$ but are consistent with
those of local starbursts. Therefore, the previously estimated short gas
depletion timescale of $\sim200\,$Myr remains unchanged.
