We examine the fraction of massive ($M_{*}>10^{10} M_{odot}$), compact star-forming galaxies (cSFGs) that host an active galactic nucleus (AGN) at $zsim2$. These cSFGs are likely the direct progenitors of the compact quiescent galaxies observed at this epoch, which are the first population of passive galaxies to appear in large numbers in the early Universe. We identify cSFGs that host an AGN using a combination of Hubble WFC3 imaging and Chandra X-ray observations in four fields: the Chandra Deep Fields, the Extended Groth Strip, and the UKIDSS Ultra Deep Survey field. We find that $39.2^{+3.9}_{-3.6}$% (65/166) of cSFGs at $1.4<z<3.0$ host an X-ray detected AGN. This fraction is 3.2 times higher than the incidence of AGN in extended star-forming galaxies with similar masses at these redshifts. This difference is significant at the $6.2sigma$ level. Our results are consistent with models in which cSFGs are formed through a dissipative contraction that triggers a compact starburst and concurrent growth of the central black hole. We also discuss our findings in the context of cosmological galaxy evolution simulations that require feedback energy to rapidly quench cSFGs. We show that the AGN fraction peaks precisely where energy injection is needed to reproduce the decline in the number density of cSFGs with redshift. Our results suggest that the first abundant population of massive, quenched galaxies emerged directly following a phase of elevated supermassive black hole growth and further hints at a possible connection between AGN and the rapid quenching of star formation in these galaxies.
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