Why do Black Holes Trace Bulges (& Central Surface Densities), Instead of Galaxies as a Whole?

Previous studies of fueling black holes (BHs) in galactic nuclei have argued (on scales ~0.01-1000pc) accretion is dynamical with inflow rates $\dot{M}\sim\eta\,M_{\rm gas}/t_{\rm dyn}$ in terms of gas mass $M_{\rm gas}$, dynamical time $t_{\rm dyn}$, and some $\eta$. But these models generally neglected expulsion of gas by stellar feedback, or considered extremely high densities where expulsion is inefficient. Studies of star formation, however, have shown on sub-kpc scales the expulsion efficiency $f_{\rm wind}=M_{\rm ejected}/M_{\rm total}$ scales with the gravitational acceleration as $(1-f_{\rm wind})/f_{\rm wind}\sim\bar{a}_{\rm grav}/\langle\dot{p}/m_{\ast}\rangle\sim \Sigma_{\rm eff}/\Sigma_{\rm crit}$ where $\bar{a}_{\rm grav}\equiv G\,M_{\rm tot}(<r)/r^{2}$ and $\langle\dot{p}/m_{\ast}\rangle$ is the momentum injection rate from young stars. Adopting this as the simplest correction for stellar feedback, $\eta \rightarrow \eta\,(1-f_{\rm wind})$, we show this provides a more accurate description of simulations with stellar feedback at low densities. This has immediate consequences, predicting e.g. the slope and normalization of the $M-\sigma$ and $M-M_{\rm bulge}$ relation, $L_{\rm AGN}-$SFR relations, and explanations for outliers in compact Es. Most strikingly, because star formation simulations show expulsion is efficient ($f_{\rm wind}\sim1$) below total-mass surface density $M_{\rm tot}/\pi\,r^{2}<\Sigma_{\rm crit}\sim3\times10^{9}\,M_{\odot}\,{\rm kpc^{-2}}$ (where $\Sigma_{\rm crit}=\langle\dot{p}/m_{\ast}\rangle/(\pi\,G)$), BH mass is predicted to specifically trace host galaxy properties above a critical surface brightness $\Sigma_{\rm crit}$ (B-band $\mu_{\rm B}^{\rm crit}\sim 19\,{\rm mag\,arcsec^{-2}}$). This naturally explains why BH masses preferentially reflect bulge properties or central surface-densities ($\Sigma_{1\,{\rm kpc}}$), not 'total' galaxy properties.
Comment: 8 pages, 3 figures, updated to match published MNRAS version