  Supermassive black hole (BH) masses (MBH) are strongly correlated with galaxy
stellar bulge masses (Mbulge) 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 (MBH-Mgal) 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 MBH-Mgal 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 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 merging statistics naturally show that MBH would
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 MBH-Mbulge
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 pre-existing, linear, correlation. This idea may be generalized
to other gravitationally bound systems (dark matter halo, compact nuclear
objects) that retain their physical identities after experiencing mergers.
