tokens: in mass by up to one or two order of magnitudes from z 8.5 to z 0. Its final mass ranges between 10 6 and 10 7 M . On the other hand, when the AGN feedback energy is entirely supplied to the cold phase, the multiphase ISM is not promoted to outflow and remains close to the BH: as a consequence, the BH experiences a rapid phase of mass growth (in the redshift range 3 z 5) during which it deprives of gas the centre of the host galaxy, and then its growth is suppressed. The BH mass at z 0 can be as high as 10 9 M in this case. Therefore, a prediction of our model is that at least a share of the AGN feedback energy has to couple with the diffuse hot gas. We examined the effect of coupling the AGN feedback energy injected in a multiphase ISM to its phases according to their physical properties. We considered a model where feedback energy coupling is driven by the covering factors of the hot and cold phases, assuming that the larger is the volume occupied by the cold gas clumps, the larger is the amount of energy that the cold gas absorbs. Gas accretion is the process that contributes the most to the BH growth, rather than mergers with other BHs (Section 5.2 ). Throughout BH evolution, the total accretion rate is dominated by cold gas accretion, with respect to hot. Remarkably, this is in line with Gaspari etal. ( 2019 ), even if they derived such a conclusion by means of a phenomenological analysis focused on more massive systems. Our conclusion that gas accretion is a more viable channel for BH growth than BH mergers is true unless cold gas which is supported by rotational velocity is prevented from accreting (see below). The quiet merging history that characterizes our simulated disc galaxies also contributes to the minor role of BH mergers to the BH mass growth. The contribution to the BH growth by BHBH mergers is expected to become more significant for increasing BH mass (Fanidakis etal. 2011 ; Dubois, Volonteri Silk 2014a ; Weinberger etal. 2018 ). We find that when the BH only accretes cold gas, it experiences a growth by gas accretion that is faster than (or at most comparable to) the case in which both cold and hot gas are accreted. As for the galaxy M bulge M BH scaling relation, predictions from simulations are in keeping with observations, considering a BH seed mass as large as 10 5 M . When the accretion of cold gas that is supported by rotational velocity is reduced, the BH mass growth is delayed and the BH mass at z 0 is reduced by up to an order of magnitude with respect to the case in which both hot and cold gas accretion proceed unimpeded. Within the scenario emerging from this model, the SMBH in MW-sized galaxies is prevented from growing by gas accretion below z 2, aside from possible BH mergers. Also, the lower is the viscosity assumed for the cold gas which is accreting (i.e. the larger C visc ), the more the BH mass growth is reduced. Simulations that include AGN feedback produce spiral galaxies with a more extended stellar
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