tokens: (Barnes et al. 2018 ). 5 THE LINK TO AGN EFFICIENCY The inclusion of ATC leads to an increased CC fraction, increased central cumulative gas fraction, and flattens the CC fraction redshift evolution relative to simulations that neglect it. The temperature profiles and the gas fraction profiles both suggest that the central black hole injects less energy into the ICM when thermal conduction is included. Therefore, we now examine the energy injected by the central black hole and how this energy couples to the ICM. We define the central black hole of each cluster as the most massive black hole within of the potential minimum at z = 0. Its properties are then traced as a function of redshift. During the formation of the ICM ( z ≤ 2) the low accretion rate, kinetic mode feedback dominates (Weinberger et al. 2018 ). In Fig. 9 , we plot the median cumulative energy injected by the central black hole in the kinetic mode feedback for simulations with and without ATC. At z = 0, the inclusion of ATC results in a 24 per cent reduction in the energy injected by the low accretion rate kinetic feedback mode, which agrees with the results from the temperature and gas fraction plots. Figure 9. Median cumulative energy injected in the kinetic mode (solid lines) as a function of redshift for the central black hole compared to the energy lost via radiative cooling within r 500 (dash–dot-dotted lines) for simulations with (red) and without (blue). For simulations with ATC, we plot the conductive heating (short dashed line) within as a function of redshift. The right scale shows the median stellar mass (long-dashed) and black hole mass (dash-dotted) with (dark red) and without (dark blue) ATC. When ATC is included the central black hole injects less energy, but the stellar and black hole masses are unchanged. Conductive heating only contributes a small fraction of the energy difference. The right axis of Fig. 9 shows the median stellar mass of the brightest cluster galaxy, defined as the mass within twice the stellar half-mass radius for the most massive subhalo, and the median central black hole mass as a function of redshift. Despite the reduction in energy produced by the inclusion of ATC, the masses of the galaxy and black hole have a negligible difference for simulations with and without ATC, especially for z ≤ 1. Therefore, the central black hole is still able to self-regulate and control its growth and the growth of the galaxy. One explanation of the difference in injected energy with and without ATC is that there is significant heat transfer from outer radii into the core of the cluster, providing the additional energy required to offset radiative losses. To investigate this we define the conductive heating with a sphere by its conductive luminosity. This is defined as the heat flux integrated over the surface of the sphere where we chose a sphere of radius to account for the collapse and growth of the cluster with cosmic time. Only gas that is
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