On the Radiative Efficiencies, Eddington Ratios, and Duty Cycles of Luminous High-Redshift Quasars

We investigate the characteristic radiative efficiency ε, Eddington ratio λ, and duty cycle P0 of high-redshift active nuclei (AGNs), drawing on measurements of the AGN luminosity function at z = 3–6 and, especially, on measurements of quasar clustering at z = 3–4. 5 from the Sloan Digital Sky Survey. The free parameters of models are ε, λ, and the normalization, scatter, and redshift evolution of the relation between black hole (BH) MBH and halo virial velocity Vᵥir. We compute the luminosity function from the implied growth of the BH function and the quasar correlation length from the bias of the host halos. We test our adopted formulae for the mass function and halo bias against measurements from the large N-body simulation developed by the MICE. The strong clustering of AGNs observed at z = 3 and, especially, at z = 4 implies that massive BHs in rare, massive dark matter halos. Reproducing the observed luminosity function then requires high efficiencyε and/or low Eddington ratio λ, with a lower limit (based on 2σ agreement with the measured z = 4 correlation) ε ~> 0. 7λ/ (1 + 0. 7λ), implying ε ~> 0. 17 for λ > 0. 25. Successful models predict high duty cycles, ₀ ~ 0. 2, 0. 5, and 0. 9 at z = 3. 1, 4. 5, and 6, respectively, and they require that the fraction of halo baryons locked in central BH is much larger than the locally observed value. The rapid drop in the abundance of themassive and rare halos at z > 7 implies a proportionally rapid decline in the number density of luminous quasars, much stronger simple extrapolations of the z = 3–6 luminosity function would predict. For example, our most successful predicts that the highest redshift quasar in the sky with true bolometric luminosity L > 10⁴7. 5 erg s^−1 should at z ~ 7. 5, and that all quasars with higher apparent luminosities would have to be magnified by lensing.