The mass of super massive black holes at the centre of galaxies is tightly correlated with the mass of the galaxy bulges which host them. This observed correlation implies a mechanism of joint growth, but the precise physical processes responsible are a matter of some debate. Here we report on the growth of black holes in 400 local galactic bulges which have experienced a strong burst of star formation in the past 600Myr. The black holes in our sample have typical masses of 10^6.5-10^7.5 solar masses, and the active nuclei have bolometric luminosities of order 10^42-10^44erg/s. We combine stellar continuum indices with H-alpha luminosities to measure a decay timescale of ~300Myr for the decline in star formation after a starburst. During the first 600Myr after a starburst, the black holes in our sample increase their mass by on-average 5% and the total mass of stars formed is about 1000 times the total mass accreted onto the black hole. This ratio is similar to the ratio of stellar to black hole mass observed in present-day bulges. We find that the average rate of accretion of matter onto the black hole rises steeply roughly 250Myr after the onset of the starburst. We show that our results are consistent with a simple model in which 0.5% of the mass lost by intermediate mass stars in the bulge is accreted by the black hole, but with a suppression in the efficiency of black hole growth at early times plausibly caused by supernova feedback, which is stronger at earlier times. We suggest this picture may be more generally applicable to black hole growth, and could help explain the strong correlation between bulge and black hole mass.
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