How mergers may affect the mass scaling relation between gravitationally bound systems

Supermassive black hole (BH) masses (M-BH) are strongly correlated with galaxy stellar bulge masses (M-bulge), and there are several ideas to explain the origin of this relationship. This study isolates the role of galaxy mergers from considerations of other detailed physics to more clearly show how a linear BH-galaxy mass relation (M-BH-M-gal) can naturally emerge, regardless of how primordial BHs were seeded inside galaxies, if the galaxy mass function declines with increasing mass. Under this circumstance, the M-BH-M-gal relation is a passive attractor that eventually converges to a tight linear relation because of two basic statistical effects: a central-limit-like tendency for galaxy mergers, which is much stronger for major mergers than for minor mergers, and a convergence toward a linear relation that is due mainly to minor mergers. A curious consequence of this thought experiment is that if galaxy bulges are formed by major mergers, then merger statistics naturally show that M-BH will correlate more strongly with bulge-dominated galaxies, because of stronger central-seeking tendencies, than with disk-dominated galaxies. Even if some other physics is ultimately responsible for causing a linear M-BH-M-bulge relationship, this thought experiment shows that, counter to intuition, random merging of galaxies that harbor random BH masses tends to strengthen rather than weaken a preexisting, linear, correlation. This idea may be generalized to other gravitationally bound systems (dark matter halos, compact nuclear objects) that retain their physical identities after experiencing mergers.