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The origin of the high SFR observed in high-z dusty star-forming galaxies isstill unknown. Large fractions of dense molecular gas might provide part of theexplanation, but there are few observational constraints on the amount of densegas in high-z systems dominated by star formation. We present the results ofour ALMA program targeting dense-gas tracers (HCN(5-4), HCO+(5-4), andHNC(5-4)) in 5 strongly lensed galaxies from the SPT SMG sample. We detectedtwo of these lines (SNR>5) in SPT-125-47 at z=2.51 and tentatively detected allthree (SNR~3) in SPT0551-50 at z=3.16. Since a significant fraction of ourtarget lines is not detected, we developed a statistical method to deriveunbiased mean properties taking into account both detections andnon-detections. On average, the HCN(5-4) and HCO+(5-4) luminosities of oursources are a factor of ~1.7 fainter than expected, based on the localL'HCN(5-4)-LIR relation, but this offset corresponds to only ~2 sigma. We findthat both the HCO+/HCN and HNC/HCN flux ratios are compatible with unity. Thefirst ratio is expected for PDRs while the second is consistent with PDRs orXDRs and/or mid-IR pumping of HNC. Our sources are at the high end of the localrelation between the star formation efficiency, determined using the LIR/[CI]and LIR/CO ratios, and the dense gas fraction, estimated using the HCN/[CI] andHCN/CO ratios. In SPT0125-47, we found that the velocity profiles of the linestracing dense (HCN, HCO+) and lower-density (CO, [CI]) gas are similar. Inaddition to these lines, we obtained one robust and one tentative detection of13CO(4-3) and found an average I12CO(4-3)/I13CO(4-3) flux ratio of26.1$_{-3.5}^{+4.5}$, indicating a young but not pristine interstellar medium.We argue that the combination of large and slightly enriched gas reservoirs andhigh dense-gas fractions could explain the prodigious star formation in thesesystems.

Dense-gas tracers and carbon isotopes in five 2.5