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.