Far-infrared Herschel SPIRE spectroscopy of lensed starbursts reveals physical conditions of ionised gas

The most intensively star-forming galaxies are extremely luminous at far-infrared (FIR) wavelengths, highly obscured at optical and ultraviolet wavelengths, and lie at $z\ge 1-3$. We present a programme of ${\it Herschel}$ FIR spectroscopic observations with the SPIRE FTS and photometric observations with PACS, both on board ${\it Herschel}$, towards a sample of 45 gravitationally lensed, dusty starbursts across $z\sim 1-3.6$. In total, we detected 27 individual lines down to 3-$\sigma$, including nine $[\rm C{\small II}]$ 158-$\mu$m lines with confirmed spectroscopic redshifts, five possible $[\rm C{\small II}]$ lines consistent with their far-infrared photometric redshifts, and in some individual sources a few $[\rm O{\small III}]$ 88-$\mu$m, $[\rm O{\small III}]$ 52-$\mu$m, $[\rm O{\small I}]$ 145-$\mu$m, $[\rm O{\small I}]$ 63-$\mu$m, $[\rm N{\small II}]$ 122-$\mu$m, and OH 119-$\mu$m (in absorption) lines. To derive the typical physical properties of the gas in the sample, we stack all spectra weighted by their intrinsic luminosity and by their 500-$\mu$m flux densities, with the spectra scaled to a common redshift. In the stacked spectra, we detect emission lines of $[\rm C{\small II}]$ 158-$\mu$m, $[\rm N{\small II}]$ 122-$\mu$m, $[\rm O{\small III}]$ 88-$\mu$m, $[\rm O{\small III}]$ 52-$\mu$m, $[\rm O{\small I}]$ 63-$\mu$m, and the absorption doublet of OH at 119-$\mu$m, at high fidelity. We find that the average electron densities traced by the $[\rm N{\small II}]$ and $[\rm O{\small III}]$ lines are higher than the average values in local star-forming galaxies and ULIRGs, using the same tracers. From the $[\rm N{\small II}]/[\rm C{\small II}]$ and $[\rm O{\small I}]/[\rm C{\small II}]$ ratios, we find that the $[\rm C{\small II}]$ emission is likely dominated by the photo-dominated regions (PDR), instead of by ionised gas or large-scale shocks.