WISDOM project -- VI. Exploring the relation between supermassive black
  hole mass and galaxy rotation with molecular gas
  Empirical correlations between the masses of supermassive black holes (SMBHs)
and properties of their host galaxies are well-established. Among these is the
correlation with the flat rotation velocity of each galaxy measured either at a
large radius in its rotation curve or via a spatially-integrated emission line
width. We propose here the use of the de-projected integrated CO emission line
width as an alternative tracer of this rotation velocity, that has already been
shown useful for the Tully-Fisher (luminosity-rotation velocity) relation. We
investigate the correlation between CO line widths and SMBH masses for two
samples of galaxies with dynamical SMBH mass measurements, with respectively
spatially-resolved and unresolved CO observations. The tightest correlation is
found using the resolved sample of 24 galaxies as $\log
(M_\mathrm{BH}/\mathrm{M_\odot})=(7.5\pm0.1)+(8.5\pm0.9)[\log(W_\mathrm{50}/\sin
i \,\mathrm{km\,s}^{-1})-2.7]$, where $M_\mathrm{BH}$ is the central SMBH mass,
$W_{50}$ the full-width at half-maximum of a double-horned emission line
profile, and $i$ the inclination of the CO disc. This relation has a total
scatter of $0.6\,$dex, comparable to those of other SMBH mass correlations, and
dominated by the intrinsic scatter of $0.5\,$dex. A tight correlation is also
found between the de-projected CO line widths and the stellar velocity
dispersions averaged within one effective radius. We apply our correlation to
the COLD GASS sample to estimate the local SMBH mass function.
