We have developed a consistent analytical model to describe the observed evolution of the quasar luminosity function .
Our model combines black hole mass distributions based on the Press - Schechter theory of the structure formation in the Universe with quasar luminosity functions resulting from a physics-based emission model that takes into account the time-dependent phenomena occurring in the accretion disks .
Quasar evolution and CDM models are mutually constraining , therefore our model gives an estimation of the exponent , n , of the power spectrum , P ( k ) , which is found to be -1.8 \leq n \leq - 1.6 .
We were able to reject a generally assumed hypothesis of a constant ratio between Dark Matter Halo and the Black Hole mass , since the observed data could not be fitted under this assumption .
We found that the relation between the Dark Matter Halos and Black Hole masses is better described by M _ { BH } = M _ { DMH } ^ { 0.668 } .
This model provides a reasonable fit to the observed quasar luminosity function at redshifts higher than \sim 2.0 .
We suggest that the disagreement at lower redshift is due to mergers .
Based on the agreement at high redshift , we estimated the merger rate at lower redshift , and argue that this rate should depend on the redshift , like ( 1 + z ) ^ { 3 } .