Stacked Average Far-Infrared Spectrum of Dusty Star-Forming Galaxies from the Herschel/SPIRE Fourier Transform Spectrometer

We present stacked average far-infrared spectra of a sample of 197 dusty, star-forming galaxies (DSFGs) at $0.005 < z < 4$ using close to 90% of the SPIRE Fourier Transform Spectrometer (FTS) extragalactic data archive from the Herschel Space Observatory based on 3.5 years of science operations. These spectra explore an observed-frame $\rm 447\,GHz-1568\,GHz$ ($\rm 191\,\mu m-671\,\mu m$) frequency (wavelength) range allowing us to observe the main atomic and molecular lines emitted by gas in the interstellar medium. The sample is sub-divided into five redshift bins at $0.005 < z < 0.05$, $0.05 < z < 0.2$, $0.2 < z < 0.5$, $0.8 < z <2$, and $2 < z < 4$. To study the dependence of observed spectral lines on total infrared luminosity, the sources in a subset of the redshift bins are stacked in luminosity bins. These stacked spectra are used to determine the average properties of the interstellar medium and dense molecular gas properties of DSFGs, in particular, the fine-structure line ([CII] 158 $\mu$m and [OI] 63 $\mu$m) luminosity ratios, and the line to far-IR luminosity ratios are used to model the gas density and radiation field strength in the photodissociation regions (PDRs). For the low-redshift sample, we additionally present the average spectral line energy distributions (SLED) of CO and $\rm{H_2O}$ rotational transitions and also consider PDR conditions based on a combination of [CI] 370 $\mu$m and 609 $\mu$m and $\rm CO (7-6)$ lines. For the high-z ($0.8 < z < 4$) sample PDR models suggest a molecular gas distribution in the presence of a radiation field that is at least a factor of 10$^3$ larger than the Milky-Way and with a neutral gas density of roughly 10$^3$ to 10$^5$ cm$^{-3}$. The corresponding PDR models for the low-z sample suggest a UV radiation field and gas density comparable to those at high-z.