PRUSSIC

Characterising the relationship between dense gas and star formation is critical for understanding the assembly of galaxies throughout cosmic history. However, due to the faintness of standard dense-gas tracers – HCN, HCO + , and HNC – dense gas in high-redshift galaxies remains largely unexplored. We present ALMA and NOEMA observations targeting HCN/HCO + /HNC (3–2) and (4–3) emission lines in 11 (mostly) gravitationally lensed dusty star-forming galaxies (DSFGs) at redshift z = 1.6 − 3.2. We detect at least one line in 10 out of 11 galaxies. Altogether, we detect 34 dense-gas transitions, more than quadrupling the number of extant high-redshift detections. Additionally, in two targets, we detect lower-abundance CO isotopologues 13 CO and C 18 O, as well as CN emission. We derive excitation coefficients for HCN, HCO + , and HNC in DSFGs, finding them to be systematically higher than those in nearby luminous infrared galaxies. Assuming the canonical dense-mass conversion factor ( α HCN = 10), we find that DSFGs have shorter dense-gas depletion times (median 23 Myr) than nearby galaxies (≈60 Myr), with a star-forming efficiency per free-fall time of 1−2%, a factor of a few higher than in local galaxies. We find a wide range of dense-gas fractions, with HCN/CO ratios ranging between 0.01 and 0.15. Finally, we put the first constraints on the redshift evolution of the cosmic dense-gas density, which increases by a factor of 7 ± 4 between z = 0 and z = 2.5, consistent with the evolution of the cosmic molecular-gas density.