to grow during galaxy mergers both by merging with other BHs and through gas accretion, the latter being described in terms of tunable scaling laws. The inclusion of AGN feeedback in these models allows to suppress the cooling in massive haloes (a long-standing problem of hierarchical models); however, such a feedback must be still at work at low redshift to continuously suppress star formation in massive haloes at $z\lesssim 1$. Since the QSO activity drops sharply for $z\lesssim 2.5$, these authors assume the energy feedback responsible for the suppression of the cooling to be effective only in halos that undergo quasi-static cooling, and is associated to a continual and quiescent accretion of hot gas onto the SMBHs. In these models the feedback mechanism is thus associated only to a smooth accretion process which is not the main driver of BH growth, so they do not focus on the evolution of the luminous properties of QSOs and of the bright AGN sources.

Here we aim to investigate the effect on galaxy formation of the feedback directly associated with the observable QSOs and luminous AGNs. To this aim we include the AGN feedback model developed by Lapi, Cavaliere, Menci (2005) in our 