The evolution of the mbh-σ relation inferred from the age distribution of local early-type galaxies and active galactic nuclei 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 lsim 6), under the assumption that (1) most of the stars in each nearby spheroid formed in a single episode and, (2) the velocity dispersion σ remained nearly constant afterward. We assume further that a supermassive 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 vprop σβ with β ≈ 4, but with the normalization allowed to evolve with redshift as vprop(1 + z)α. 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 Sołtan 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 α ~ 0.33, with a positive evolution as strong as α gsim 1.3 excluded at more than 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 λ ~ 0.5-1 that is roughly constant within 0 lsim z lsim 3. A strong positive evolution in the M BH-σ 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 σ(z) = σ(0) × (1 + z)-γ, we find a lower limit of γ gsim 0.23 for α gsim 1.5. The latter estimate represents an interesting constraint for galaxy evolution models and can be tested through hydro simulations. This dissipative model, however, also implies a decreasing λ at higher z, at variance with several independent studies.