We investigate the rapid growth phase of supermassive black holes (BHs) within the hydrodynamical cosmological eagle simulation. This non-linear phase of BH growth occurs within $sim$$L_{*}$ galaxies, embedded between two regulatory states of the galaxy host: in sub $L_{*}$ galaxies efficient stellar feedback regulates the gas inflow onto the galaxy and significantly reduces the growth of the central BH, while in galaxies more massive than $L_{*}$ efficient AGN feedback regulates the gas inflow onto the galaxy and curbs further non-linear BH growth. We find evolving critical galaxy and halo mass scales at which rapid BH growth begins. Galaxies in the low-redshift Universe transition into the rapid BH growth phase in haloes that are approximately an order of magnitude more massive than their high-redshift counterparts (M{200} $approx 10^{12.4}$~Msol at $z approx 0$ decreasing to M{200} $approx 10^{11.2}$~Msol at $z approx 6$). Instead, BHs enter the rapid growth phase at a fixed critical halo virial temperature ($T_{mathrm{vir}} approx 10^{5.6}$~K). We additionally show that major galaxy--galaxy interactions ($mu geq frac{1}{4}$, where $mu$ is the stellar mass ratio) play a substantial role in triggering the rapid growth phase of BHs in the low-redshift Universe, whilst potentially having a lower influence at high redshift. Approximately 40% of BHs that initiate the rapid BH growth phase at $z approx 0$ do so within $pm 0.5$ dynamical times of a major galaxy--galaxy merger, a fourfold increase above what is expected from the background merger rate. We find that minor mergers ($frac{1}{10} leq mu < frac{1}{4}$) have a substantially lower influence in triggering the rapid growth phase at all epochs.
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