The Innermost Mass Distribution of the Gravitational Lens SDP.81 from
  ALMA Observations
  The central image of a strongly lensed background source places constraints
on the foreground lens galaxy's inner mass profile slope, core radius and mass
of its nuclear supermassive black hole. Using high-resolution long-baseline
Atacama Large Millimeter/submillimeter Array (ALMA) observations and archival
$Hubble~Space~Telescope$ ($HST$) imaging, we model the gravitational lens
H-ATLAS J090311.6+003906 (also known as SDP.81) and search for the demagnified
central image. There is central continuum emission from the lens galaxy's
active galactic nucleus (AGN) but no evidence of the central lensed image in
any molecular line. We use the CO maps to determine the flux limit of the
central image excluding the AGN continuum. We predict the flux density of the
central image and use the limits from the ALMA data to constrain the innermost
mass distribution of the lens. For a power-law profile with a core radius of
$0.15^{\prime\prime}$ measured from $HST$ photometry of the lens galaxy
assuming that the central flux is attributed to the AGN, we find that a black
hole mass of $\mathrm{\log(M_{BH}/M_{\odot})} \gtrsim 8.5$ is preferred. Deeper
observations with a detection of the central image will significantly improve
the constraints of the innermost mass distribution of the lens galaxy.
