| Summary: | We present [C ii] observations of 20 strongly lensed dusty star-forming galaxies at 2.1 < z < 5.7 using Atacama Pathfinder EXperiment and Herschel. The sources were selected on their 1.4 mm flux (S1.4 mm > 20 mJy) from the South Pole Telescope (SPT) survey, with far-infrared (FIR) luminosities determined from extensive photometric data. The [C ii] line is robustly detected in 17 sources, all but one being spectrally resolved. 11 out of 20 sources observed in [C ii] also have low-J CO detections from Australia Telescope Compact Array. A comparison with mid- and high-J CO lines from Atacama Large Millimeter/submillimeter Array reveals consistent [C ii] and CO velocity profiles, suggesting that there is little differential lensing between these species. The [C ii], low-J CO and FIR data allow us to constrain the properties of the interstellar medium. We find [C ii] to CO(1–0) luminosity ratios in the SPT sample of 5200 ± 1800, with significantly less scatter than in other samples. This line ratio can be best described by a medium of [C ii] and CO emitting gas with a higher [C ii] than CO excitation temperature, high CO optical depth τ_CO(1–0) ≫ 1, and low to moderate [C ii] optical depth τ_[CII]≲ 1. The geometric structure of photodissociation regions allows for such conditions. © 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society. Accepted 2015 February 19. Received 2015 January 27; in original form 2014 September 11. We thank the referee for a very thorough reading of the paper and constructive comments which have significantly improved the paper. We are grateful to Javier Gracia-Carpio for providing us with a sample of low- and high-z sources with both [C II] and CO detections, ideal for comparison with the SPT sample of high-z sources. We also thank Gordon Stacey and Padelis Papadopoulos for lively, educational, and profitable discussions. We thank Göran Pilbratt, Herschel Project scientist for the allocated Herschel ...
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