Cosmological growth history of supermassive black holes and demographics in the high-z universe:: Do Lyman break galaxies have supermassive black holes?

We study the demographics of supermassive black holes (SMBHs) in the local and high-z universe. We use the continuity equation of the population of SMBHs as the leading principle. We consider three physical processes for the growth of SMBHs: mass accretion, mergers, and direct formation of SMBHs. The mass accretion history of SMBHs is estimated by the hard X-ray luminosity functions (HXLFs) of active galactic nuclei (AGNs). First, we compare the mass accretion history at z > 0 with optical luminosity functions (OLFs) of QSOs previously studied and that with HXLFs. We conclude that the constraints on parameters of mass accretion (energy conversion efficiency, epsilon, and Eddington ratio, f(Edd)) based on the continuity equation appear to be adequate using HXLFs rather than OLFs. The sub-Eddington case (f(Edd) < 1) is allowed only when we use HXLFs. Next, we extend the formulation and can obtain the upper limit of the cumulative mass density of SMBHs at any redshift. For an application, we examine if Lyman break galaxies (LBGs) at z similar to 3 already have SMBHs in their centers, which is suggested by recent observations. We tentatively assume the presence of SMBHs in LBGs and that their mass, M-BH, is proportional to the stellar mass of LBGs, M-*, with the mass ratio xi = M-BH/M-*. If most LBGs already have massive SMBHs at z similar to 3, the resultant mass density of SMBHs at z similar to 0 should exceed the observational estimate because such SMBHs should further grow by accretion. Therefore, we can set the upper limit of the probability that one LBG has SMBHs. Since the merger rates and direct formation rates of SMBHs are uncertain, we consider two limiting cases: (1) mergers and/or direct formations are not negligible compared with mass accretion, and (2) mass accretion is the dominant process to grow the SMBHs. The special conditions should be met in order that a large part of LBGs have SMBHs in both cases. In case 1, we may assume the constant parameters of mass accretion of AGNs for simplicity. Then, large energy conversion efficiency and frequent mergers and/or direct formations at z > 3 are needed so that a large part of LBGs have SMBHs with xi = 0. 002-0.005. In case 2, on the other hand, energy conversion efficiency should be mass dependent and the constraint is strict; the fraction of LBGs that have SMBHs must be less than 10%-40%. In both cases, the possibility that nearly all LBGs have SMBHs with large mass ratio, such as xi greater than or equal to 0.005, is reliably ruled out.