Evolution of the $M_{\bullet} - \sigma$ relation
  Black holes at the centers of the galaxies grow mainly by the processes of
accretion, mergers, and consumption of stars. In the case of gas accretion with
cooling sources, the flow is momentum driven, after which the black hole
reaches a saturated mass, and subsequently, it grows only by consumption of
stars. In addition, we include the effect of mergers on the growth of black
hole spin and mass and study its evolution as a function of redshift in a
$\Lambda$CDM cosmology using an initial seed mass and spin distribution
functions that we have derived. For the stellar ingestion, we have assumed a
power-law density profile for the galaxy in our framework of a new relativistic
loss cone theory that includes the effect of the black hole spin. We predict
the impact of the evolution on the $M_{\bullet} - \sigma$ relation and compare
it with available observations.
