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Constraining the Lifetime of Quasars with the Present-Day Mass Function of Supermassive Black Holes

Using the theoretical models of quasar (QSO) formation, we can reproduce optical QSO luminosity functions (LFs) at high redshifts (z ≥ 2.2). Two different models can reproduce LFs successfully, though the lifetime of the QSOs, tQ, and the relation between the black hole mass and the host halo mass are different for each model; tQ ~ 106 yr in one model and tQ ≥ 107 yr in the other model. Here we propose a method to break this degeneracy. We calculate the mass function of supermassive black holes (SMBHs) at z = 2.5 and compare the results with the current mass function obtained by Salucci et al. (1999). In the shorter lifetime model, the mass function at z = 2.5 exceeds that of z = 0.0 by 1 order of magnitude, in which case it should be ruled out. We conclude that the lifetime is at least tQ ≥ 107 yr. Next, we examine the difference of the formation epoch of SMBHs existing at z = 3.0 for each model, under the model assumptions. We simply discuss the difference of the formation epoch as another possible model discriminator.