Linear relation between HI circular velocity and stellar velocity
  dispersion in early-type galaxies, and slope of the density profiles
  We report a tight linear relation between the HI circular velocity measured
at 6 $R_{\rm e}$ and the stellar velocity dispersion measured within 1 $R_{\rm
e}$ for a sample of 16 early-type galaxies with stellar mass between $10^{10}$
and $10^{11}$ $\mathrm{M}_\odot$. The key difference from previous studies is
that we only use spatially resolved $v_\mathrm{circ}$(HI) measurements obtained
at large radius for a sizeable sample of objects. We can therefore link a
kinematical tracer of the gravitational potential in the dark-matter dominated
outer regions of galaxies with one in the inner regions, where baryons control
the distribution of mass. We find that $v_\mathrm{circ}$(HI) = 1.33
$\sigma_\mathrm{e}$ with an observed scatter of just 12 percent. This indicates
a strong coupling between luminous and dark matter from the inner- to the outer
regions of early-type galaxies, analogous to the situation in spirals and dwarf
irregulars. The $v_\mathrm{circ}$(HI)-$\sigma_\mathrm{e}$ relation is shallower
than those based on $v_\mathrm{circ}$ measurements obtained from stellar
kinematics and modelling at smaller radius, implying that \vcirc\ declines with
radius -- as in bulge-dominated spirals. Indeed, the value of
$v_\mathrm{circ}$(HI) is typically 25 percent lower than the maximum
$v_\mathrm{circ}$ derived at $\sim0.2\ R_\mathrm{e}$ from dynamical models.
Under the assumption of power-law total density profiles $\rho \propto
r^{-\gamma}$, our data imply an average logarithmic slope
$\langle\gamma\rangle=2.18\pm0.03$ across the sample, with a scatter of 0.11
around this value. The average slope and scatter agree with recent results
obtained from stellar kinematics alone for a different sample of early-type
galaxies.
