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 $\dot{M}\simη\,M_{\rm gas}/t_{\rm dyn}$ in terms of gas mass $M_{\rm gas}$, dynamical time $t_{\rm dyn}$, and some $η$. 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 $(1-f_{\rm wind})/f_{\rm wind}\sim\bar{a}_{\rm grav}/\langle\dot{p}/m_{\ast}\rangle\sim Σ_{\rm eff}/Σ_{\rm crit}$ 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, $η\rightarrow η\,(1-f_{\rm wind})$, 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 $M-σ$ 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 $M_{\rm tot}/π\,r^{2}<Σ_{\rm crit}\sim3\times10^{9}\,M_{\odot}\,{\rm kpc^{-2}}$ (where $Σ_{\rm crit}=\langle\dot{p}/m_{\ast}\rangle/(π\,G)$), BH mass is predicted to specifically trace host galaxy properties above a critical surface brightness $Σ_{\rm crit}$ (B-band $μ_{\rm B}^{\rm crit}\sim 19\,{\rm mag\,arcsec^{-2}}$). This naturally explains why BH masses preferentially reflect bulge properties or central surface-densities ($Σ_{1\,{\rm kpc}}$), not 'total' galaxy properties.

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