We investigate the co-evolution of black-hole-accretion-rate (BHAR) and star-formation-rate (SFR) in $1.5<z<2.5$ galaxies displaying a greater diversity of star-forming properties compared to previous studies. We combine X-ray stacking and far-IR photometry of stellar mass-limited samples of normal star-forming, starburst and quiescent/quenched galaxies in the COSMOS field. We corroborate the existence of a strong correlation between BHAR (i.e. the X-ray luminosity, L_X), and stellar mass (M*) for normal star-forming galaxies, although find a steeper relation than previously reported. We find that starbursts show a factor of 3 enhancement in BHAR compared to normal SF galaxies (against a factor of 6 excess in SFR), while quiescents show a deficit of a factor 5.5 at a given mass. One possible interpretation of this is that the starburst phase does not coincide with cosmologically relevant BH growth, or that starburst-inducing mergers are more efficient at boosting SFR than BHAR. Contrary to studies based on smaller samples, we find the BHAR/SFR ratio of main sequence (MS) galaxies is not mass invariant, but scales weakly as M*^(0.43pm0.09}, implying faster BH growth in more massive galaxies at $zsim2$. Furthermore, BHAR/SFR during the starburst is a factor of 2 lower than in MS galaxies, at odds with the predictions of hydrodynamical simulations of merger galaxies that foresee a sudden enhancement of L_X/SFR during the merger. Finally, we estimate that the bulk of the accretion density of the Universe at $zsim2$ is associated with normal star-forming systems, with only 6(+/-1)% and 11(+/-1)% associated with starburst and quiescent galaxies, respectively.
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