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We build an evolution model of the central black hole that depends on theprocesses of gas accretion, the capture of stars, mergers as well aselectromagnetic torque. In case of gas accretion in the presence of coolingsources, the flow is momentum-driven, after which the black hole reaches asaturated mass; subsequently, it grows only by stellar capture and mergers. Wemodel the evolution of the mass and spin with the initial seed mass and spin in$\Lambda$CDM cosmology. For stellar capture, we have assumed a power-lawdensity profile for the stellar cusp in a framework of relativistic loss conetheory that include the effects of black hole spin, Carter's constant, losscone angular momentum, and capture radius. Based on this, the predicted capturerates of $10^{-5}$--$10^{-6}$ yr$^{-1}$ are closer to the observed range. Wehave considered the merger activity to be effective for $z \lesssim 4$, and weself-consistently include the Blandford-Znajek torque. We calculate theseeffects on the black hole growth individually and in combination, for derivingthe evolution. Before saturation, accretion dominates the black hole growth($\sim 95\%$ of the final mass), and subsequently, stellar capture and mergerstake over with roughly equal contribution. The simulations of the evolution ofthe $M_{\bullet} - \sigma$ relation using these effects are consistent withavailable observations. We run our model backward in time and retrodict theparameters at formation. Our model will provide useful inputs for buildingdemographics of the black holes and in formation scenarios involving stellarcapture.

Cosmic spin and mass evolution of black holes and its impact