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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 theSPIRE Fourier Transform Spectrometer (FTS) extragalactic data archive from theHerschel Space Observatory based on 3.5 years of science operations. Thesespectra explore an observed-frame $\rm 447\,GHz-1568\,GHz$ ($\rm 191\,\mum-671\,\mu m$) frequency (wavelength) range allowing us to observe the mainatomic and molecular lines emitted by gas in the interstellar medium. Thesample 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 dependenceof observed spectral lines on total infrared luminosity, the sources in asubset of the redshift bins are stacked in luminosity bins. These stackedspectra are used to determine the average properties of the interstellar mediumand dense molecular gas properties of DSFGs, in particular, the fine-structureline ([CII] 158 $\mu$m and [OI] 63 $\mu$m) luminosity ratios, and the line tofar-IR luminosity ratios are used to model the gas density and radiation fieldstrength 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 conditionsbased 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 gasdistribution in the presence of a radiation field that is at least a factor of10$^3$ larger than the Milky-Way and with a neutral gas density of roughly10$^3$ to 10$^5$ cm$^{-3}$. The corresponding PDR models for the low-z samplesuggest a UV radiation field and gas density comparable to those at high-z.

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