The Evolution of the M_BH-sigma relation Inferred from the Age Distribution of Local Early-Type Galaxies and AGN Evolution

We utilize the local velocity dispersion function (VDF) of spheroids, together with their inferred age--distributions, to predict the VDF at higher redshifts (0<z<6), under the assumption that (i) most of the stars in each nearby spheroid formed in a single episode, and (ii) the velocity dispersion sigma remained nearly constant afterward. We assume further that a supermassive black hole (BH) forms concurrently with the stars, and within ~1 Gyr of the formation of the potential well of the spheroid, and that the relation between the mass of the BH and host velocity dispersion maintains the form M_BH ~ sigma^{beta} with beta~4, but with the normalization allowed to evolve with redshift as ~(1+z)^{alpha}. We compute the BH mass function associated with the VDF at each redshift, and compare the accumulated total BH mass density with that inferred from the integrated quasar luminosity function (LF; the so--called Soltan argument). This comparison is insensitive to the assumed duty cycle or Eddington ratio of quasar activity, and we find that the match between the two BH mass densities favors a relatively mild redshift evolution, with alpha ~ 0.26, with a positive evolution as strong as alpha>1.3 excluded at the 99% confidence level. A direct match between the characteristic BH mass in the VDF--based and quasar LF--based BH mass functions also yields a mean Eddington ratio of lambda ~ 0.5-1 that is roughly constant within 0<z<3. A strong positive evolution in the M_BH-sigma relation is still allowed by the data if galaxies increase, on average, their velocity dispersions since the moment of formation, due to dissipative processes. If we assume that the mean velocity dispersion of the host galaxies evolves as sigma(z)=sigma(0)*(1+z)^{-gamma}, we find a lower limit of gamma>0.23 for alpha>1.5. abridged