We investigate a sample of 152 dusty sources at 1:5 < z < 2:5 to understand the connection of enhanced Star-Formation-Rate (SFR) and Black-Hole-Accretion-Rate (BHAR). The sources are Herschel-selected, having stellar masses M*>10^10Msun and SFR (100-1000Msun/yr) elevated(>4?) above the star-forming main sequence, classifying them as Starbursts (SB). Through a multiwavelength fitting approach (including a dusty torus component), we divided the sample into active SBs (dominated by an AGN emission, SBs-AGN, ? 23% of the sample) and purely star-forming SBs (SBs-SFR). We visually inspected their HST/UV-restframe maps: SBs-SFR are generally irregular and composite systems; ? 50% of SBs-AGN are instead dominated by regular compact morphologies. We then found archival ALMA continuum counterparts for 33 galaxies (12 SBs-AGN and 21 SBs-SFR). For these sources we computed dust masses, and, with standard assumptions, we also guessed total molecular gas-masses. SBs turn to be gas rich systems (fgas = Mgas=Mgas/(Mgas+M*) sim 20%-70%), and the gas fractions of the two SB classes are very similar (fgas = 43 +/-4% and fgas = 42+/- 2%). Our results show that SBs are consistent with a mixture of: 1) highly star-forming merging systems (dominating the SBs-SFR), and 2) primordial galaxies, rapidly growing their M* together with their Black Hole (mainly the more compact SBs-AGN). Anyway, feedback effects have not reduced their fgas yet. Indeed, SBs at z = 2, with relatively low bolometric AGN luminosities in the range 10^44 < Lbol(AGN) < 10^46 erg/s (compared to bright optical and X-ray quasars), are still relatively far from the epoch when the AGN feedback will quench the SFR in the host and will substantially depress the gas fractions.
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