The Mass Function of Supermassive Black Holes in the Direct-collapse Scenario

One of the ideas that explains the existence of supermassive black holes (SMBHs) that are in place by z similar to 7 is that there was an earlier phase of very rapid accretion onto direct-collapse black holes (DCBHs) that started their lives with masses similar to 10(4-5)M(circle dot). Working in this scenario, we show that the mass function of SMBHs after such a limited time period, with growing formation rate paired with super-Eddington accretion, can be described as a broken power law with two characteristic features. There is a power law at intermediate masses whose index is the dimensionless ratio alpha lambda/gamma, where lambda is the growth rate of the number of DCBHs during their formation era, and lambda is the growth rate of DCBH masses by super-Eddington accretion during the DCBH growth era. A second feature is a break in the power-law profile at high masses, above which the mass function declines rapidly. The location of the break is related to the dimensionless number beta = gamma T, where T is the duration of the period of DCBH growth. If the SMBHs continue to grow at later times at an Eddington-limited accretion rate, then the observed quasar luminosity function can be directly related to the tapered power-law function derived in this Letter.