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 constraints on QSO models and BH growth. 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., four times of the Salpeter timescale 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 (abridged).