Evidence of a Flat Outer Rotation Curve in a Starbursting Disk Galaxy at
  $z=1.6$
  Observations of the baryon to dark matter fraction in galaxies through cosmic
time are a fundamental test for galaxy formation models. Recent observational
studies have suggested that some disk galaxies at $z>1$ host declining rotation
curves, in contrast with observations of low redshift disk galaxies where
stellar or HI rotation curves flatten at large radii. We present an
observational counterexample, a galaxy named DSFG850.95 at $z=1.555$ (4.1 Gyr
after the big bang) that hosts a flat rotation curve between radii of
$\sim$6--14 kpc (1.2--2.8 disk scale lengths) and has a dark matter fraction of
$0.44\pm0.08$ at the H-band half light radius, similar to the Milky Way. We
create position-velocity and position-dispersion diagrams using Keck/MOSFIRE
spectroscopic observations of H$\alpha$ and [NII] emission features, which
reveal a flat rotation velocity of $V_{\rm flat}=285\pm12$ km/s and an ionized
gas velocity dispersion of $\sigma_{0}=48\pm4$ km/s. This galaxy has a
rotation-dominated velocity field with $V_{\rm flat}/\sigma_{0}\sim6$.
Ground-based H-band imaging reveals a disk with S\'ersic index of
$1.29\pm0.03$, an edge-on inclination angle of $87\pm2^{\circ}$, and an H-band
half light radius of $8.4\pm0.1$ kpc. Our results point to DSFG850.95 being a
massive, rotationally-supported disk galaxy with a high dark-matter-to-baryon
fraction in the outer galaxy, similar to disk galaxies at low redshift.
