IR luminosities as a function of redshift of our low-L_IR-selected SFGs (squares) compared to our compilation of galaxies with CO measurements from the literature (see Sect. 2.2). Symbols in black refer to L_IR as derived from the Herschel far-IR photometry. Our SFGs populate the regime with the lower L_IR reported at z> 1.5: they reach the L^⋆ and below domain as delimited by the hatched area showing the L^⋆ limit of the Herschel far-IR luminosity function as determined by Magnelli et al. (2013, upper band) and Gruppioni et al. (2013, lower band) and extend the blank field studies to fainter luminosities as is shown with the solid line, which defines the minimal L_IR at each redshift that produces a flux ≳ 2 mJy in the Herschel/PACS 160 μm band (~ 3σ detection limit in GOODS-N, Elbaz et al. 2011).