  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*^(0.43\pm0.09}, implying faster BH growth in more massive galaxies at
$z\sim2$. 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 $z\sim2$ is associated with normal star-forming systems, with only
6(+/-1)% and 11(+/-1)% associated with starburst and quiescent galaxies,
respectively.
