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

One of the ideas to explain the existence of supermassive black holes (SMBH) that are in place by z~7 is that there was an earlier phase of very rapid accretion onto direct collapse black holes (DCBH) that started their lives with masses ~ 10^4-10^5 M_solar. Working in this scenario, we show that the mass function of SMBH 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 DCBH during their formation era, and {\gamma} 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 \b{eta} = {\gamma} T, where T is the duration of the period of DCBH growth. If the SMBH 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 paper.