Constraints on QSO models from a relation between the QSO luminosity function and the local black hole mass function

QSOs are believed to be powered by accretion onto massive black holes (BHs). In this paper, assuming that each central BH in nearby galaxies has experienced the QSO phase and ignoring BH mergers, we establish a relation between the QSO luminosity function (LF) and the local BH mass function (MF). The QSOLF is jointly controlled by the luminosity evolution of individual QSOs and the triggering history of the accretion onto seed BHs. By comparing the time integral of the QSOLF with that inferred from local BHs, we separate the effect of the luminosity evolution of individual QSOs from the effect of the triggering history. Assuming that the nuclear luminosity evolution includes two phases (first increasing at the Eddington luminosity with growth of BHs and then declining), we find that observations are generally consistent with the expected relation between the QSOLF and the local BHMF and obtain the following constraints on QSO models and BH growth: (i) The QSO mass-to-energy efficiency epsilon should be greater than or similar to0.1. (ii) The lifetime (defined directly through the luminosity evolution of individual QSOs here) should be greater than or similar to4 x 10(7) yr. The characteristic declining timescale in the second phase should be significantly shorter than the Salpeter timescale tau(Sp), and BH growth should not be dominated by the second phase. (iii) The ratio of obscured QSOs/AGNs to optically bright QSOs should be not larger than 7 at M-B similar to 23 and 3 at M-B similar to -26 if epsilon = 0.31, and not larger than 1 at M-B similar to -23 and negligible at M-B similar to -26 if epsilon = 0.1. (iv) It is unlikely that most QSOs are accreting at super-Eddington luminosities. We point out that it is hard to accurately estimate the value of the QSO lifetime from the QSOLF and/or the local BHMF, if it is longer than a certain value (e. g., similar to4tau(Sp) in this study). We discuss the importance of accurate measurements of the intrinsic scatter in the BH mass and velocity dispersion relation of local galaxies and the scatter in the bolometric correction of QSOs. We also discuss some possible applications of the work in this paper, such as to the study of the demography of QSOs and the demography of normal galaxies at intermediate redshift.