Evolution of supermassive black holes in light of PTA measurements: Implications for growth by mergers and accretion

We study the growth of supermassive black holes accounting for both accretion and mergers. The former is informed by observations of the quasar luminosity function (QLF) and the latter by the gravitational wave-background (GWB) recently detected by pulsar timing arrays, while estimates of the present-day black hole mass function provide a boundary condition. The GWB is dominated by the most massive black holes ($\ensuremath{\gtrsim}{10}^{9}{M}_{\ensuremath{\bigodot}}$). We show that their evolution can be simplified into a two-step process: mergers dominate at $z\ensuremath{\le}1$, while accretion peaks at $1.4\ensuremath{\le}z\ensuremath{\le}2$. The large amplitude of the observed GWB suggests a significant number of mergers. We show that this generically implies a higher average Eddington ratio for quasars relative to a scenario in which mergers are negligible. In the absence of mergers, matching local estimates of black hole abundance to the QLF implies a radiative efficiency of ${\ensuremath{\epsilon}}_{r}=0.12$ and an Eddington ratio of $\ensuremath{\lambda}=0.2$. With mergers, a progenitor of mass ${M}_{i}$ is boosted to a final total mass ${M}_{f}$, and there is a direct relation between the mass gained in mergers and the average Eddington ratio of the quasar population, given by ${M}_{f}/{M}_{i}=(1+\overline{q}{)}^{N}\ensuremath{\sim}\ensuremath{\lambda}/0.2$, where $\overline{q}$ is the average mass ratio, and $N$ is the average number of mergers. There is thus a tension between the observed GWB, quasar properties, and the BH mass function: estimates of the mass function consistent with Eddington ratios inferred in quasars and ${\ensuremath{\epsilon}}_{r}\ensuremath{\sim}0.1$ underpredict the GWB; multiple/equal mass mergers can boost the GWB but lead to a high Eddington ratio. If the local mass function is on the high end of current estimates, the GWB is more readily explained but requires low efficiencies ${\ensuremath{\epsilon}}_{r}\ensuremath{\sim}{10}^{\ensuremath{-}2}$ not expected in standard luminous accretion models. The significant merger rate implied by the GWB also strongly suggests that the most massive BHs in the local universe have significant spin due to the orbital angular momentum from mergers, perhaps $a\ensuremath{\sim}0.5$.