tokens: redshift z 2 to the present day, the black holes switch to the low-accretion regime again, remaining in a self-regulated state in which both the stellar and AGN feedback balance cooling. The relative importance of AGN over stellar feedback depends on halo mass. While stellar feedback dominates in haloes up to the size of the Milky Way, more massive haloes are mainly regulated through AGN feedback. The stellar mass fraction as a function of halo mass (Fig. 9 ) clearly shows the decrease in star formation efficiency with halo mass at the massive end, in good agreement with observations. We find it useful to compare the stellar mass including the diffuse intracluster light in the high-mass end in observations and theory. In this way, we are less sensitive to the choice of the aperture within which stellar masses are estimated. At the high halo-mass end, the data shown here are in reasonable agreement with Kravtsov, Vikhlinin Meshscheryakov ( 2014 ) who pointed out the importance of outer stellar profiles in high-mass haloes in this type of analysis. Additionally, we plot both the halo mass from the full physics simulation as well as the halo mass of the corresponding halo in the dark matter only run, where the latter, i.e. the dashed line and open circles, should be compared to the results from abundance-matching. Fig. 4 shows the ratio of these two masses as a function of halo mass. In particular at around M 200, c 10 13 M , it turns out to be crucial to take this effect into account. Figure 9. Stellar mass fraction as a function of halo mass for the high-resolution simulation. The stellar mass is calculated as the mass of all star particles within 10 stellar half-mass radii that do not belong to a subhalo. The black line is the corresponding fit to observations from Behroozi etal. ( 2013a , b ) including the intracluster light. We use the simulation values of b and m for both simulation and literature data. The dashed line and open circles correspond to the same simulation data, but the halo mass M 200, c is taken from the corresponding halo in the dark matter only simulation (see Munshi etal. 2013 , for a discussion). One of the conjectured effects of AGN is that they can prevent the most massive galaxies from being blue and star forming, instead making them red and having an old stellar population. We use BV colour and mean stellar age (Fig. 10 ) as a measure for the efficiency of the feedback in the low-accretion state to accomplish this. To probe the relevance of the kinetic feedback mode for this, we compare our fiducial simulation with a simulation without the kinetic mode. The panels of Fig. 10 clearly show the need for this efficient mode to get red and dead galaxies with old stellar populations on the massive end of the galaxy population. Figure 10. Top panel: BV colours as a function of stellar mass within twice the stellar half-mass radius. The red dots are from the fiducial simulations. The blue dots represent the haloes in the run
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