What Do Statistics Reveal About the MBH-Mbulge Correlation and Co-Evolution?

Observational data show that the correlation between the masses of supermassive black holes M_BH and galaxy bulge masses M_bulge follows a nearly linear trend, and that the correlation is strongest with the bulge rather than the total stellar mass M_gal. With increasing redshift, the ratio Γ=M_BH/M_bulge relative to z = 0 also seems to be larger for M_BH≳10^8.5 M_⊙. This study looks more closely at statistics to see what effect it has on creating, and observing, the M_BH-M_bulge correlation. It is possible to show that if galaxy merging statistics can drive the correlation, minor mergers are responsible for causing a convergence to linearity most evident at high masses, whereas major mergers have a central limit convergence that more strongly reduces the scatter. This statistical reasoning is agnostic about galaxy morphology. Therefore, combining statistical prediction (more major mergers ⟹ tighter correlation) with observations (bulges = tightest correlation), would lead one to conclude that more major mergers (throughout an entire merger tree, not just the primary branch) give rise to more prominent bulges. Lastly, with regard to controversial findings that Γ increases with redshift, this study shows why the luminosity function (LF) bias argument, taken correctly at face value, actually strengthens, rather than weakens, the findings. However, correcting for LF bias is unwarranted because the BH mass scale for quasars is bootstrapped to the M_BH-σ_* correlation in normal galaxies at z = 0, and quasar-quasar comparisons are mostly internally consistent. In Monte-Carlo simulations, high Γ galaxies are indeed present: they are statistical outliers (i.e., “under-merged”) that take longer to converge to linearity via minor mergers. Additional evidence that the galaxies are undermassive at z≳2 for their M_BH is that the quasar hosts are very compact for their expected mass.