What Do Statistics Reveal About the Black Hole versus the Bulge Mass Correlation and Co-evolution?

Abstract

Observational data show that the correlation between supermassive black holes (MBH) and galaxy bulge (Mbulge) masses follows a nearly linear trend, and that the correlation is strongest with the bulge rather than the total stellar mass (Mgal). With increasing redshift, the ratio Gamma=MBH/Mbulge relative to z=0 also seems to be larger for MBH >~ 10^{8.5} Msol. This study looks more closely at statistics to better understand the creation and observations of the MBH-Mbulge 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. With regard to controversial findings that Gamma increases with redshift, this study shows why the luminosity function (LF) bias argument, taken correctly at face value, strengthens rather than weakens the results. However, correcting for LF bias is unwarranted because the BH mass scale for quasars is bootstrapped to the MBH-Sigma* correlation in normal galaxies at z=0, and quasar-quasar comparisons are internally consistent. In Monte-Carlo simulations, high Gamma objects are "under-merged" galaxies that take longer to converge to linearity via minor mergers. Another evidence that the galaxies are undermassive at z >~ 2 for their MBH is that the quasar hosts are very compact for their expected mass.