We examine possible phenomenological constraints for the joint evolution of supermassive black holes ( SMBH ) and their host spheroids .
We compare all the available observational data on the redshift evolution of the total stellar mass and star formation rate density in the Universe with the mass and accretion rate density evolution of supermassive black holes , estimated from the hard X-ray selected luminosity function of quasars and active galactic nuclei ( AGN ) for a given radiative efficiency , \epsilon .
We assume that the ratio of the stellar mass in spheroids to the black hole mass density evolves as ( 1 + z ) ^ { - \alpha } , while the ratio of the stellar mass in disks + irregulars to that in spheroids evolves as ( 1 + z ) ^ { - \beta } , and we derive constraints on \alpha , \beta and \epsilon .
We find that \alpha > 0 at the more than 4-sigma level , implying a larger black hole mass at higher redshift for a given spheroid stellar mass .
The favored values for \beta are typically negative , suggesting that the fraction of stellar mass in spheroids decreases with increasing redshift .
This is consistent with recent determinations that show that the mass density at high redshift is dominated by galaxies with irregular morphology .
In agreement with earlier work , we constrain \epsilon to be between 0.04 and 0.11 , depending on the exact value of the local SMBH mass density , but almost independently of \alpha and \beta .