We map the co-eval growth of galaxies and their central supermassive black holes in detail by measuring the incidence of Active Galactic Nuclei (AGN) in galaxies as a function of star formation rate (SFR) and redshift (to z~4). We combine large galaxy samples with deep Chandra X-ray imaging to measure the probability distribution of specific black hole accretion rates (LX relative to stellar mass) and derive robust AGN fractions and average specific accretion rates. First, we consider galaxies along the main sequence of star formation. We find a linear correlation between the average SFR and both the AGN fraction and average specific accretion rate across a wide range in stellar mass ($M_* sim 10^{8.5-11.5}M_odot$) and to at least z~2.5, indicating that AGN in main-sequence galaxies are driven by the stochastic accretion of cold gas. We also consider quiescent galaxies and find significantly higher AGN fractions than predicted, given their low SFRs, indicating that AGN in quiescent galaxies are fuelled by additional mechanisms (e.g. stellar winds). Next, we bin galaxies according to their SFRs relative to the main sequence. We find that the AGN fraction is significantly elevated for galaxies that are still star-forming but with SFRs below the main sequence, indicating further triggering mechanisms enhance AGN activity within these sub-main-sequence galaxies. We also find that the incidence of high-accretion-rate AGN is enhanced in starburst galaxies and evolves more mildly with redshift than within the rest of the galaxy population, suggesting mergers play a role in driving AGN activity in such high-SFR galaxies.
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