The dense molecular gas in the rm zsim6 QSO SDSS J231038.88+185519.7 resolved by ALMA

We present ALMA observations of the CO(6-5) and [CII] emission lines and the sub-millimeter continuum of the $z\sim6$ quasi-stellar object (QSO) SDSS J231038.88+185519.7. Compared to previous studies, we have analyzed a synthetic beam that is ten times smaller in angular size, we have achieved ten times better sensitivity in the CO(6-5) line, and two and half times better sensitivity in the [CII] line, enabling us to resolve the molecular gas emission. We obtain a size of the dense molecular gas of $2.9\pm0.5$ kpc, and of $1.4\pm0.2$ kpc for the 91.5 GHz dust continuum. By assuming that CO(6-5) is thermalized, and by adopting a CO--to--$H_2$ conversion factor $\rm \alpha_{CO} = 0.8~ M_{\odot}~K^{-1}~ (km/s)^{-1} ~pc^{2}$, we infer a molecular gas mass of $\rm M(H_2)=(3.2 \pm0.2) \times 10^{10}\rm M_{\odot}$. Assuming that the observed CO velocity gradient is due to an inclined rotating disk, we derive a dynamical mass of $\rm M_{dyn}~sin^2(i) = (2.4\pm0.5) \times 10^{10}~ M_{\odot}$, which is a factor of approximately two smaller than the previously reported estimate based on [CII]. Regarding the central black hole, we provide a new estimate of the black hole mass based on the C~IV emission line detected in the X-SHOOTER/VLT spectrum: $\rm M_{BH}=(1.8\pm 0.5) \times 10^{9}~ M_{\odot}$. We find a molecular gas fraction of $\rm \mu=M(H_2)/M^*\sim4.4$, where $\rm M^*\approx M_{dyn} - M(H_2)-M(BH)$. We derive a ratio $v_{rot}/\sigma \approx 1-2$ suggesting high gas turbulence, outflows/inflows and/or complex kinematics due to a merger event. We estimate a global Toomre parameter $Q\sim 0.2-0.5$, indicating likely cloud fragmentation. We compare, at the same angular resolution, the CO(6-5) and [CII] distributions, finding that dense molecular gas is more centrally concentrated with respect to [CII]. We find that the current BH growth rate is similar to that of its host galaxy.