also that our average relation remains close to a power-law, at variance, e.g., with the Bower et al. (2005) model, which predicts a steepening of the relation for large $\sigma$; in fact, such a steepening is due to the second ”mode” of BH growth implemented in that model, associated with the smooth accretion of hot gas and which ensures that the luminosity of the BH is sufficient to quench cooling in the most massive haloes.

Since the time behavior of the energy injection from AGNs is crucial in determining the effects on the galaxy properties, and in our model such injection is related to the active QSO phase, we test in fig. 2 the evolution of the QSO luminosity distribution by comparing it with the observations from $z=0$ up to $z=4$. Note that our model is able to match the observed decline in the population of bright sources from $z=2$ to the present; this is due to 1) the decline of merging events refueling the cold gas reservoir of the galaxies; 2) the exhaustion of such a reservoir, due to its earlier conversion in stars; 3) the decline in the rate of interactions (triggering the active accretion phase); 4) the 