(enriched by the blowout of cold galactic gas due to the AGN feedback) is not enhanced over the baryon fraction in groups and clusters; ii) the temperature of the hot gas is not enhanced by the contribution from AGNs up to values exeeding the observations for a given cluster mass. Indeed, the value of $\beta$ (sect. 2.2) stays close to unity for typical values of the energy $\Delta E$ injected by AGNs.

\[3.2\]

The agreement with observations is not unexpected: indeed, while the AGNs are expected to produce significant effects on the cold gas contained in the shallow potential wells of galaxies (and hence to appreciably affect the star formation history of galaxies which entirely depends on the cold gas phase), the relative variation in the hot gas mass (which is an order of magnitude larger than the $m_c$) remains however small, and has little effect on the baryon fraction in groups and clusters of galaxies. In fact, for such structures the ratio $\Delta E/E$ (see sect. 2.2) is much less than unity due to their deep potential wells (large gas binding energy $E$), at least for our assumed value of the AGN feedback efficiency $f\approx 10^{-2}$, so that relative 