A Census of Star-formation and Gas Mass Tracers in Two Lensed z ∼ 4 Dusty Star-forming Galaxies

We present new and archival Atacama Large Millimeter/submillimeter Array observations of two strongly lensed dusty star-forming galaxies (DSFGs) selected from the South Pole Telescope survey, SPT0418-47 (z = 4.225) and SPT2147-50 (z = 3.760). We study the [C ii], CO(7–6), [C i](2–1), and, in SPT0418-47, p–H2O emission, which, along with the underlying continuum (rest-frame 160 and 380 μm), are routinely used as tracers of gas mass and/or star-formation rate (SFR). We perform a pixel-by-pixel analysis of both sources in the image plane to study the resolved Kennicutt–Schmidt relation, finding generally good agreement between the slopes of the SFR versus gas mass surface density using the different tracers. Using lens modeling methods, we find that the dust emission is more compact than the line emission in both sources, with CO(7–6) and [C i](2–1) similar in extent and [C ii] the most extended, reminiscent of recent findings of extended [C ii] spatial distributions in galaxies at similar cosmic epochs. We develop the [C i](2–1)/ CO(7–6) flux-density ratio as an observable proxy for gas depletion timescale (τdep), which can be applied to large samples of DSFGs, in lieu of more detailed inferences of this timescale, which require analysis of observations at multiple wavelengths. Furthermore, the extended [C ii] emission in both sources, compared to the total continuum and line emission, suggests that [C ii], used in recent years as a molecular gas mass and SFR tracer in high-z galaxies, may not always be a suitable tracer of these physical quantities.