We present a model for the cosmological evolution of quasars ( QSOs ) under the assumption that they are powered by massive accreting black holes .
Accretion flows around massive black holes make a transition from high radiative efficiency ( \sim 10 \% ) to low efficiency , advection-dominated flows when { \dot { M } } / { \dot { M } } _ { Edd } falls below the critical rate \sim 0.3 \alpha ^ { 2 } \sim 10 ^ { -2 } where { \dot { M } } is the mass accretion rate , { \dot { M } } _ { Edd } \propto M is the usual Eddington rate with the nominal 10 \% efficiency , and \alpha ( \leq 1 ) is the dimensionless viscosity parameter .
We identify this transition with the observed break at a redshift \sim 2 in the QSOs ’ X-ray luminosity evolution .
Growth of black holes through accretion could naturally lead to such a transition at a critical redshift z _ { c } \sim 1 - 3 , provided that most of high redshift QSOs appear with near Eddington luminosities at z \sim 3 - 4 and the accretion rates decline over the Hubble time in a roughly synchronous manner .
Before the transition , the QSOs ’ luminosities ( with a high efficiency ) slowly decrease and after the transition at z _ { c } , the QSO luminosities evolve approximately as \propto ( 1 + z ) ^ { K ( z ) } where K ( z ) gradually varies from z = z _ { c } to z \sim 0 around K \sim 3 .
The results depend on the details of the QSO X-ray emission mechanism .
We discuss some further implications .