[Abridged] We investigate trends between the recent star formation history and black hole growth in galaxy bulges in the Sloan Digital Sky Survey (SDSS). The galaxies lie at 0.01<z<0.07 where the fibre aperture covers only the central 0.6-4.0kpc diameter of the galaxy. We find strong trends between black hole growth, as measured by dust-attenuation-corrected OIII luminosity, and the recent star formation history of the bulges. We conclude that our results support the popular hypothesis for black hole growth occurring through gas inflow into the central regions of galaxies, followed by a starburst and triggering of the AGN. However, while this is a significant pathway for the growth of black holes, it is not the dominant one in the present-day Universe. More unspectacular processes are apparently responsible for the majority of this growth. In order to arrive at these conclusions we have developed a set of new high signal-to-noise ratio (SNR) optical spectral indicators, designed to allow a detailed study of stellar populations which have undergone recent enhanced star formation. Working in the rest-frame wavelength range 3750-4150AA, ideally suited to many recent and ongoing spectroscopic surveys at low and high redshift, the first two indices are equivalent to the previously well studied 4000AA break strength and Hdelta equivalent width. The primary advantage of this new method is a greatly improved SNR for the latter index, allowing the present study to use spectra with SNR-per-pixel as low as 8.
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