Supermassive Black Holes: Connecting the Growth to the Cosmic Star
  Formation Rate
  In this Letter, we present a model connecting the cosmic star formation rate
(CSFR) to the growth of supermassive black holes. Considering that the
evolution of the massive black hole is dominated by accretion (Soltan's
argument) and that the accretion process can be described by a probabilistic
function directly regulated by the CSFR, we obtain the evolution of the black
hole mass density. Then using the quasar luminosity function, we determine both
the functional form of the radiative efficiency and the evolution of the quasar
duty-cycle as functions of the redshift. We analyze four different CSFRs
showing that the quasar duty-cycle, $\delta(z)$, peaks at $z\sim 8.5-11$ and so
within the window associated with the reionization of the Universe. In
particular, $\delta_{\rm max}\sim 0.09-0.22$ depending on the CSFR. The mean
radiative efficiency, $\bar\eta(z)$, peaks at $z\sim 0.1-1.3$ with
$\bar\eta_{\rm max}\sim 0.10-0.46$ depending on the specific CSFR used. Our
results also show that is not necessary a supercritical Eddington accret
