The relation between nuclear ($lesssim$ 50 pc) star formation and nuclear galactic activity is still elusive: theoretical models predict a link between the two, but it is unclear whether active galactic nuclei (AGNs) should appear at the same time, before or after nuclear star formation activity is ongoing. We present a study of this relation in a complete, volume-limited sample of nine of the most luminous ($log L_{rm 14-195 keV} > 10^{42.5}$ erg/s) local AGNs (the LLAMA sample), including a sample of 18 inactive control galaxies (6 star-forming; 12 passive) that are matched by Hubble type, stellar mass (9.5 $lesssim$ log M_star/M_sun $lesssim$ 10.5), inclination and distance. This allows us to calibrate our methods on the control sample and perform a differential analysis between the AGN and control samples. We perform stellar population synthesis on VLT/X-SHOOTER spectra in an aperture corresponding to a physical radius of $approx$ 150 pc. We find young ($lesssim$ 30 Myr) stellar populations in seven out of nine AGNs and in four out of six star-forming control galaxies. In the non-star-forming control population, in contrast, only two out of twelve galaxies show such a population. We further show that these young populations are not indicative of ongoing star-formation, providing evidence for models that see AGN activity as a consequence of nuclear star formation. Based on the similar nuclear star-formation histories of AGNs and star-forming control galaxies, we speculate that the latter may turn into the former for some fraction of their time. Under this assumption, and making use of the volume-completeness of our sample, we infer that the AGN phase lasts for about 5 % of the nuclear starburst phase.
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