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 \hbox to 0.0 pt { \lower 2.5 pt \hbox { $ \sim$ } } \raise 1.5 pt \hbox { $ < $ } 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 \pm 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 } \propto \sigma ^ { \beta } with \beta \approx 4 , but with the normalization allowed to evolve with redshift as \propto ( 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 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 \alpha \sim 0.26 , with a positive evolution as strong as \alpha \gtrsim 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 \sim 0.5 - 1 that is roughly constant within 0 \lesssim z \lesssim 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 ) \times ( 1 + z ) ^ { - \gamma } , we find a lower limit of \gamma \gtrsim 0.23 for \alpha \gtrsim 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 \lambda at higher z , at variance with several independent studies .