Empirical relations defining the growth of supermassive black holes:
  Implications for the origins of black hole seeds
  We compiled available mass and redshift z data for supermassive black holes
(SMBHs) at z >7.5, ranging in mass over 2 orders of magnitude and in age by
nearly 300 million years. The data reveal that a large subset covering the
entire age spectrum has markedly similar masses. The most likely implication is
that SMBHs within the subset had seeds with similar masses and formed
essentially concurrently. Based on this inference, the data of the subset are
used derive quantitative empirical relations that provide insights into the
origins of black hole seeds and constraints for models of seed formation. The
relationships are tested and applied to thousands of SMBHs at nearly all
redshifts. The results show that the masses of SMBHs > a million solar masses
are accounted for with seeds formed at or near z=30 and ranging from Sun's mass
to about 50-thousand solar masses. Apparently, the seeds grew at an
exponentially increasing accretion rate that reached a maximum near z=7 and
decreased thereafter. From z=30 to 15, the average accretion rate either
exceeded the Eddington limit by a factor of 2 or less, or the radiative
efficiency was less than its canonical value of 0.1 and increased thereafter.
About half of the growth apparently occurs from z=30 to 3.5 at an average rate
of about 614 solar masses per million years per unit solar seed mass, and the
rest in the succeeding 12-billion years. The maximum mass that a black hole can
accrete is about 2.35-million times the seed mass, and the largest observable
black hole should not exceed about 100-billion solar masses. The seed of
Sagittarius A* is inferred to have had a mass a few times Sun's mass and may
have stopped accreting recently having achieved its maximum growth potential.
