The Rest-frame Submillimeter Spectrum of High-redshift, Dusty, Star-forming Galaxies from the SPT-SZ Survey

This work utilizes data from the South Pole Telescope Sunyaev–Zel'dovich (SPT-SZ) DSFG catalog, and the full details of the target selection and ALMA 3 mm observations can be found in Reuter et al. (2020). We will briefly summarize the key points in this section.

The SPT DSFG catalog was selected from the 2500 deg² field of the SPT-SZ survey (Vieira et al. 2010; Everett et al. 2020) with S_{1.4 mm} > 20 mJy. In order to refine their positions, the sources were also required to have been detected with the Large Apex BOlometer CAmera (LABOCA; Siringo et al. 2009) with S_{870 μm} > 25 mJy, bringing the number of sources in the flux-limited selection to 81. Over 70% of the sources in the sample are strongly gravitationally lensed by galaxies (Spilker et al. 2016), and with a median magnification factor of μ_{870 μm} = 5.5. The remainder of the sample are either lensed by galaxy clusters or are unlensed protoclusters (e.g., SPT2349-56; see Miller et al. 2018; Hill et al. 2020; Wang et al. 2021). The sources span redshifts from z = 1.9 to 6.9, apparent IR luminosities from 1.3 to 29 × 10¹³ L_⊙ (integrated from 8 to 1000 μm), and dust temperatures from 21 to 79 K, with medians at 〈z〉 = 3.9 ± 0.2, apparent 〈L_IR〉 = 8.2 × 10¹³ L_⊙, and 〈T_dust〉 = 54 K.

The ALMA 3 mm observations were obtained over several different ALMA cycles, ranging from Cycle 0 to Cycle 7. ¹⁸ The first observations were conducted in ALMA Cycle 0, and spectra were obtained for 26 targets (Weiß et al. 2013). Follow-up surveys conducted in ALMA Cycles 1, 3, and 4 targeted additional sources, giving a total of 78 spectral scans obtained with ALMA (Strandet et al. 2016; Reuter et al. 2020). While there are 81 sources in the complete SPT DSFG sample, spectroscopic redshifts for three sources (SPT0538-50, SPT0551-48, and SPT2332-53; Greve et al. 2012; Strandet et al. 2016) were obtained with Z-Spec (Bradford et al. 2009) on the Caltech Submillimeter Observatory and no additional 3 mm data were obtained with ALMA. These sources are therefore not included in the analysis presented in this work.

Despite the changing capabilities of ALMA during our observations, the observing strategy remained the same for all of the 3 mm observations. The data were obtained by conducting a spectral sweep of ALMA Band 3 in five tunings, scanning the range between 84.2 and 114.9 GHz. The brightest sources were targeted in ALMA Cycle 0 and were scanned for roughly 5 minutes on source, with between 14 and 17 antennas. The remaining fainter sources were scanned with both longer integration times and additional antennas in subsequent ALMA cycles. Sources scanned in Cycles 1, 3, and 4 were scanned for an average of 10, 6, and 13 minutes on source, with between 28–40, 34–41, and 38–46 antennas, respectively. Finally, three sources did not exhibit any emission lines in their initial 3 mm scans, so deeper observations were obtained in ALMA Cycle 7. These scans observed each target for 45–91 minutes on source with 42–49 antennas.

Data were processed using the Common Astronomy Software Application (CASA; McMullin et al. 2007). Because the integration times and number of antennas increased with each ALMA cycle, the typical noise levels vary according to the ALMA cycle and were 1.5 mJy beam⁻¹, 0.91 mJy beam⁻¹, 0.73 mJy beam⁻¹, 0.58 mJy beam⁻¹, and 0.24 mJy beam⁻¹ over a 62.5 MHz bandwidth for Cycles 0, 1, 3, 4, and 7, respectively. The typical rms values for individual SPT-SZ DSFGs can be found in Table E1 of Reuter et al. (2020). The spectral resolution of the data, ∼1.5 km s⁻¹ over a 2 GHz baseband, is much higher than a typical galaxy line width. This allows many channels to be averaged over for each line, increasing significance.

As previously noted, this work utilizes the 3 mm spectra obtained in Weiß et al. (2013), Strandet et al. (2016), and Reuter et al. (2020). These spectra are central to not only this paper but also for papers currently in preparation. The 3 mm line fluxes from mid-J CO lines will be combined with the line fluxes from low-J CO observations with the Australia Telescope Compact Array (ATCA). SLEDs will be fitted for all sources, which will be examined in a following paper that is in preparation.

Because the details of creating a stacked spectrum are similar to those outlined in Spilker et al. (2014), we summarize the key points here and highlight the differences between the two methods.

The stacked spectrum presented in this paper is a composite of 3 mm spectral scans for the 78 sources observed with ALMA. Calibrated data cubes were created with CASA's TClean package and the spectra were extracted. The continuua of the spectra were fitted with a first-order polynomial, excluding channels with signal-to-noise (S/N) > 3σ, and then subtracted from each spectrum. The spectra are then rescaled to a common rest frame (z = 3; as in Equation (1) of Spilker et al. 2014). The choice to scale to z = 3 is intended to represent the typical redshift of DSFGs. Since surveys of unlensed DSFGs are largely selected at shorter wavelengths, they tend to be at lower redshift (z ∼ 2.3–2.9; Koprowski et al. 2014; Simpson et al. 2014; Miettinen et al. 2015; Brisbin et al. 2017; Danielson et al. 2017; Michałowski et al. 2017; Simpson et al. 2017), while the median redshift of the SPT-SZ DSFGs is 3.9 (Reuter et al. 2020).

Because our sample is known to contain a large number (>70%; Spilker et al. 2016) of gravitationally lensed sources, it is necessary to account for the magnification. Ideally, the magnification would be calculated by constructing gravitational lens models of the sources (Spilker et al. 2016). However, in the absence of a complete set of gravitational lens models, the individual spectra are normalized by their respective apparent dust masses as a proxy for size, then multiplied by the average apparent dust mass (〈M_dust〉 = 8.7 × 10⁹ M_⊙). It should be noted that a radius could also be derived from the same spectral energy density fit (e.g., Equations (1) and (2) from Weiß et al. 2007). Radius and dust mass are therefore very strongly correlated, with some intrinsic scatter from the simultaneous fitting of dust temperature. Since normalizing by the apparent values for radius and dust mass give similar results, we choose to normalize by the apparent dust mass throughout this work. The values for apparent dust mass used throughout this work can be found in Appendix A.

A variety of other source properties could also have been used in order to normalize the ALMA spectra, such as apparent L_IR, ¹²CO line luminosity, or scaled 1.4 mm spectral energy distribution (SED) flux density (as in Spilker et al. 2014). While the line ratios derived in Spilker et al. (2014) were robust within ∼15% to the choice of normalization, the ¹²CO line ratios for the complete sample can vary from ∼25% to as much as 65%. The large disparity is in part due to the large intrinsic scatter in population statistics, shown in Figure 1 for a few choices of normalization. It should be noted that normalization by IR luminosity will affect the shape of the ¹²CO SLED, since the excitation is dependent on IR luminosity (e.g., Narayanan et al. 2008). Since the systems at high IR luminosity will be scaled to compare to lower luminosity, this will lead to a lower excitation, as seen in the top panel of Figure 1.

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The 1.4 mm flux density (rest frame 350 μm) is taken from the modified blackbody models in Reuter et al. (2020) and represents a compromise between the apparent L_IR and apparent dust mass normalizations. Because any results derived from a stacked spectrum are dependent on normalization, we acknowledge and emphasize that the stacked spectrum presented in this work is not uniquely defined.

To create the final composite spectrum of the SPT DSFGs, we employ the same methodology as Spilker et al. (2014). That is, the spectrum of each source is interpolated onto a grid spanning the rest frequencies of the ALMA 3 mm spectra, 250–770 GHz. The grid has a uniform spacing of 500 km s⁻¹, which roughly corresponds to the typical FWHM of observed ¹²CO transitions. A weighted average is then performed on each frequency channel, with each contributing source weighted by the inverse standard deviation (Boogaard et al. 2021) rather than the inverse variance used in Spilker et al. (2014). Because the data were collected in different ALMA cycles, the noise of individual spectra over a 62.5 MHz bandwidth can vary from the 1.4 to 2 mJy beam⁻¹ typically observed in Cycle 0 to 0.2–0.3 mJy beam⁻¹ for the three sources reobserved in ALMA Cycle 7 (details in Reuter et al. 2020). The average 3 mm per-channel noise for each source is shown in the top panel of Figure 2 and is also given in Appendix A. Though normalization by dust mass (shown in the bottom panel) does mitigate some of the differences in ALMA cycle, demonstrated by the average weights shown in the middle panel of Figure 2, ¹⁹ the choice to weight by inverse standard deviation is a compromise between optimizing S/N and accounting for the range of noises due to the varying ALMA cycles.

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It should be noted that each frequency channel of the contributing 3 mm spectrum is weighted individually and the average noise across the 3 mm bandwidth is shown in Table A1. Weighting by the inverse variance of noise was used in various composite spectra (Spilker et al. 2014) in order to optimize for S/N. However, as noted in Section 2.2, if the noise is nonhomogeneous (e.g., if there are very different numbers of antennas between observations), weighting by the inverse variance of noise can be sensitive to outliers. In order to mitigate the presence of outliers, one can choose to weight instead by inverse standard deviation (e.g., Boogaard et al. 2021, and this work) or use the median rather than the average (e.g., Bothwell et al. 2013; Birkin et al. 2021). A test of the statistical noise properties is given in Appendix A, and demonstrates the noise is Gaussian across all ALMA cycles.

For ∼32% of the SPT DSFG sample, two or more CO emission lines fall within the 3 mm observation window and are detected in the blind survey (Reuter et al. 2020). Low-J CO lines were obtained from an ATCA CO survey detailed in Aravena et al. (2016). In addition to the published observations (17 sources), data were obtained after publication, giving a total of 33 observations for the J_up = 1−2 lines. Sources that had multiple CO line observations either through ALMA or through the combination of ATCA and ALMA data are so-called "double-lined" sources. These sources enable a comparison of the relative strengths of observed ¹²CO emission lines within individual systems, shown in Figure 3 as histograms, with the optically thick thermalized emission limit represented by the thick black line.

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The ¹²CO emission line ratios from individual SPT systems also provide a direct comparison sample to the composite spectrum obtained from stacking, which is represented by the dashed line in Figure 3. With the exception of CO(5–4)/CO(4–3), the emission line ratios obtained from the composite spectrum are in agreement with the individual ratios and below the optically thick thermalized emission limit.

While the composite spectrum necessarily contains all of the individual line ratios represented in these histograms, it differs in two key respects: each observation is normalized by dust mass and weighted by the inverse of the noise, and also the stack contains many more observations of single emission lines. This is especially apparent in the CO(5–4)/CO(4–3) panel of Figure 3, where the composite line ratio is above both the individual line ratios and the thermalized emission line. One reason for this apparent discrepancy is that because of the fixed observation window, the single-lined sources being compared are necessarily at different redshifts. While Reuter et al. (2020) did not find a statistically significant evolution of dust temperature as a function of redshift, two populations of single-lined sources being compared likely have different effective dust temperatures and luminosities. Additionally, some of the faintest and lowest mass sources observed in Cycle 7 (e.g., SPT0112-55, SPT0457-49, and SPT2340-59) are single-lined sources for this analysis as they contain a single ¹²CO emission line, while a high dust mass outlier (SPT0155-62) is a double-lined source. It is likely that the population of sources is simply not large enough yet to overcome the effects of individual source variations, in addition to the effects of potential redshift evolution of the ¹²CO line properties given in the fixed observed frequency range.

Using the ¹²CO fluxes taken from the composite spectrum, we construct a SLED, shown in Figure 5, in comparison with both well-sampled SLEDs of individual objects as well as other statistical SLEDs. The Milky Way inner disk (Fixsen et al. 1999) and Antennae Galaxies (Zhu et al. 2003) represent objects in our local Universe. However, the high-redshift, gravitationally lensed objects H1413+117 (the "Cloverleaf" quasar at z = 2.56; Bradford et al. 2009), SMM J2135-0102 (the "Cosmic Eyelash" DSFG at z = 2.32; Danielson et al. 2011), and APM 0827+5255 (a quasar at z = 3.91; Weiß et al. 2007) are more analogous to the DSFGs in the SPT sample. While the full SPT sample normalized by L_IR (see bottom panel of Figure 5) shows an apparent flattening around J = 5, analogous to the local starburst galaxies M82 or NGC 253 (Bradford et al. 2004; Panuzzo et al. 2010), normalization by IR luminosity will affect the overall shape of the ¹²CO SLED, since the excitation is dependent on IR luminosity. The intrinsic scatter of the full SPT sample is denoted by the shaded region around the dust mass weighting, and suggests that while there is a large variation in excitation, the excitation is similar to quasi-stellar objects.

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Rather than observe a single object for as many CO transitions as possible, statistical samples can be built through observing a large sample of sources with only a few or even single CO transitions. There are a wide variety of ways to combine these data (see Section 2.2 for a full discussion) in order to create a composite SLED. The first composite SLEDs were compiled in Bothwell et al. (2013) and Spilker et al. (2014), and have since been done for a variety of samples, including Yang et al. (2017), Valentino et al. (2020), Boogaard et al. (2021), and Birkin et al. (2021).

However, because the SLEDs shown are constructed with different normalizations, it is difficult to compare the differences directly. While Bothwell et al. (2013) and Spilker et al. (2014) both utilize populations of DSFGs, they normalize by far-IR luminosity and 1.4 mm flux, respectively. As discussed in Section 2.2, the normalization quantity can make a 25%–65% difference in the line ratio. While Yang et al. (2017), Boogaard et al. (2021), and Birkin et al. (2021) all normalize by L_IR, Yang et al. (2017) is largely composed of lensed DSFGs, while Boogaard et al. (2021) and Birkin et al. (2021) are primarily comprised of unlensed DSFGs. The sample of sources presented in Kirkpatrick et al. (2019) contains between 58% and 75% strongly lensed sources (μ > 5; percentage dependent on J transition), which is similar to the ∼70% of strongly lensed sources (μ > 8; Spilker et al. 2016) found in the SPT sample. In spite of the choice of normalization and the large intrinsic scatter of the SPT sample, the average of the SPT DSFGs does appear more energetic than many other DSFG SLEDs in the literature.

The excitation of the bright ¹²CO emission lines has been shown to correlate well with dust temperature (Rosenberg et al. 2015). In order to test the effects of dust temperature on the shape of the SPT composite SLED, the SPT sample was split into three equal subsamples according to dust temperatures from Reuter et al. (2020). The same stacking procedure was used for each subsample, and the resulting SLEDs are shown in Figure 6 in comparison with the full sample. Because DSFGs are some of the most extreme star-forming galaxies, the hottest sources in the SPT sample exhibit much more excitation in the higher-J transitions than the lower dust temperature sources, which show an apparent flattening around J = 5, like the local starburst galaxies.

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One important caveat is that the lower-J lines tend to be populated by lower redshift sources at lower dust temperature, while the higher-J lines tend to have higher redshift and higher dust temperature sources. While Reuter et al. (2020) found a slight correlation of dust temperature with redshift, the intrinsic scatter was sufficiently large that this effect was not statistically significant. However, the difference between the high and low dust temperature SLEDs can also be observed at intermediate dust temperature (shown in Figure 6 as purple circles) and through J = 3–5, where there is a more equal distribution of sources between the different temperature bins.

Though ¹²CO and its isotopologues, ¹³CO and C¹⁸O, are all thought to originate from the same volume (e.g., Henkel & Downes 2010; Zhang et al. 2018), ¹²CO is optically thick even at moderate densities (${n}_{{{\rm{H}}}_{2}}\lesssim {10}^{3}\,{\mathrm{cm}}^{-3}$), due to its high relative abundance and low dipole moment. However, the rare ¹³CO and C¹⁸O isotopologues are typically optically thin due to their low abundance and are therefore capable of probing the total molecular column density of cold gas regions (Carilli & Walter 2013). While both ¹²C and ¹³C are formed in and ejected from stars, ¹²C is generally produced in high-mass stars at rapid timescales. In contrast, ¹³C is a "secondary" species, since it is produced in longer-lived, low- to intermediate-mass stars and its creation involves enriching ¹²C seed nuclei (Wilson & Rood 1994). Since the creation of ¹³C requires the ISM to be enriched by metals from previous generations of stars, the relative abundances of ¹²C and ¹³C are indications of the star formation history of a system. The emission strengths of the ¹²CO and ¹³CO lines are related to the abundances of their respective carbon isotopes, given a known optical depth. ¹³CO is expected to be optically thin under most conditions, except the most dense, star-forming cores. Less is known about the formation of ¹⁸O, though it is likely formed through the CNO cycle in massive stars and is enhanced in galaxies with recent massive star formation (Henkel & Mauersberger 1993). Due to its lower abundance and optical depth compared to ¹³CO, C¹⁸O can be used to probe different gas column densities, provided the elemental abundances remain constant.

In the SPT DSFG composite spectrum, we detect ¹³CO at >3σ for the J_up = 3−5 transitions and set limits for C¹⁸O. While the exact luminosities and significances can be found in Table B1, we find that the line ratio between the J = 3−2 and J = 4−3 lines increases, with $L{{\prime} }_{{}^{12}\mathrm{CO}}/L{{\prime} }_{{}^{13}\mathrm{CO}}\sim 20$ and $L{{\prime} }_{{}^{12}\mathrm{CO}}/L{{\prime} }_{{}^{13}\mathrm{CO}}\sim 30$ for J = 3−2 and J = 4−3, respectively. This is likely due to a lower excitation in ¹³CO due to less line trapping, which is related to lower optical depth. Past J = 4−3 however, the $L{{\prime} }_{{}^{12}\mathrm{CO}}/L{{\prime} }_{{}^{13}\mathrm{CO}}$ ratio seems to remain constant within errors. However, our measurements are only sensitive enough to constrain until the J = 5−4 transition.

The ¹²C/¹³C abundance ratios have been studied in the Milky Way using a variety of different molecules and have generally been found to steadily increase from ∼20 near the Galactic center to ∼70 in our solar system (e.g., Langer & Penzias 1990; Aalto et al. 1991, 1995; Wilson 1995), likely due to differences in optical depth as one moves from the Galactic center (Milam et al. 2005). In regions of high star formation, such as giant molecular clouds, the ¹²C/¹³C ratio is reported to be between ∼3 and 5, with more ¹³C present in the clouds' central regions (Solomon et al. 1979; Polk et al. 1988). A recent survey of the J_up = 1 line for 147 nearby main-sequence galaxies revealed an integrated intensity ratio of 10.9 with a standard deviation of 7.0 (Morokuma-Matsui et al. 2020), while local star-forming and starbursting galaxies exhibit higher ratios of 16.1 ± 2.5 (Mendez-Hernandez et al. 2020). Taken together, these surveys imply that galaxies with increasing levels of star formation exhibit increasingly suppressed levels of ¹³C, but there is a large intrinsic scatter in the sample. Indeed, some highly star-forming systems have been found to have brightness temperature ¹²CO/¹³CO ratios of ∼40 or even as high as ≥60 (Sliwa et al. 2017; Young et al. 2021). Additional studies have shown that the ¹²C/¹³C ratio is sensitive to environmental conditions such as the density of nearby galaxies (Alatalo et al. 2015) and star formation rate surface density (Davis 2014).

At high redshift (z > 2), samples with ¹³CO and C¹⁸O are limited to observations of the bright CO transitions in gas- and metal-rich galaxies amplified with gravitational lensing (Henkel & Downes 2010; Danielson et al. 2013; Spilker et al. 2014; Béthermin et al. 2018; Zhang et al. 2018). Typical values of the ¹²C/¹³C ratio range from ∼20 to 40 for the most commonly observed J = 4−3 transition, and the J = 4−3 ratio obtained from the composite spectrum, $L{{\prime} }_{{}^{12}\mathrm{CO}}/L{{\prime} }_{{}^{13}\mathrm{CO}}\sim 30$, is consistent with that picture.

Because C¹⁸O is also associated with high-star-formation-rate regions, ¹³CO/C¹⁸O line ratios have been shown to differ in galaxies with differing star formation rates. In the local universe, typical ¹³CO/C¹⁸O ratios range from ∼4 to 8 (e.g., Henkel & Mauersberger 1993). However, in local ultraluminous infrared galaxies (ULIRGs) and starbursts, ratios typically range from ∼1 to 2 (Greve et al. 2009; Sliwa et al. 2017; Mendez-Hernandez et al. 2020). While we do not formally detect C¹⁸O, the limits on the ¹³CO/C¹⁸O ratio range from ∼2 to 3 with an average ratio of 3.2 across all transitions. While these values are smaller than the typical local ¹³CO/C¹⁸O ratios, they are slightly higher than those found in high-redshift ULIRGs and starbursts (e.g., Danielson et al. 2013). However, one cannot directly use the observed abundance ratios alone to draw conclusions about the underlying abundances. Detailed radiative-transfer calculations are needed to further interpret these results.

Though ¹²CO traces the total molecular gas, observations of nearby galaxies have found a close link between the dense gas and star formation activity (e.g., Gao & Solomon 2004a; Jiménez-Donaire et al. 2019). Molecules with higher critical densities and excitation energies such as CN, HCN, HNC, and HCO⁺ have been used as probes of the warm, dense molecular gas where stars form. In the SPT DSFG composite spectrum, we detect the J = 4−3 transitions of HCN, HNC, and HCO⁺, along with the J = 5−4 transition of HNC and the J = 3−2 transition of CN. While the J = 4−3 transition of the CN radical was detected in Spilker et al. (2014), it was only marginally detected with a significance of 2.9σ in this work. Along with the differences in stacking method detailed in Section 2.2, this line was also blended with the J = 5−4 transition of HCN (the deblending process is discussed in Section 3), which also contributed to the decreased significance.

The flux densities of the dense gas tracers HCN, HNC, and HCO⁺ are shown in Figure 7 compared with the ¹²CO flux density as a function of the intrinsic IR luminosity from Reuter et al. (2020). The apparent IR luminosity was determined by fitting the photometry published in Reuter et al. (2020) according to a modified blackbody with an additional power-law component. The intrinsic IR luminosity was then obtained by using either the magnification values from detailed lens models (Spilker et al. 2016) or by using the median magnification factor of μ_{870 μm} = 5.5. Comparison samples include nearby luminous infrared galaxies (LIRGs) and ULIRGs from a variety of surveys (Baan et al. 2008; Zhang et al. 2014) and a few measurements from high-redshift lensed objects (Guélin et al. 2007; Riechers et al. 2007, 2010; Danielson et al. 2013; Béthermin et al. 2018; Cañameras et al. 2021), as well as average values obtained from stacking (Spilker et al. 2014). The rotational ground levels of the dense gas molecules compared to CO are proxies for the fractional value of dense gas to total mass in the system, while the IR luminosity is proportional to the star formation rate (e.g., Murphy et al. 2011). Though some excited rotational ground levels are also shown in Figure 7, it is difficult to compare because of the different excitation requirements for ¹²CO and the dense gas tracers shown. Recent work (Rybak et al. 2022) has found that (U)LIRGs at high redshift have a lower HCN/CO than the local universe for the J_up = 1 transition. Though our results seem to also show lower HCN/CO ratios at high redshift, the SLED of HCN is a subject of ongoing study in the local universe (e.g., Saito et al. 2018) and has not been explored at high redshift.

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In the local universe, it has been shown that the ground-state HCN emission strongly correlates with IR luminosity over many orders of magnitude (Gao & Solomon 2004a; Usero et al. 2015; Bigiel et al. 2016), but it is unknown if this correlation continues at high redshift. At Cosmic Noon (2 < z < 4), the star formation rate was ∼10× higher than its present-day level and is generally not well represented by the modes of star formation seen in nearby galaxies (Genzel et al. 2011; Madau & Dickinson 2014), except perhaps in rare local ULIRGs. It is unclear if the prodigious star formation rates observed in high-redshift galaxies are due to increased gas accretion, or if an increase in the efficiency of the star formation process is required. One possibility is that at high redshift the star formation rate still scales linearly with IR luminosity, but the dense gas fraction is enhanced, driven by some combination of gravitational instabilities from mergers and gas accretion, and extreme stellar feedback from supernovae or radiation pressure (Bournaud et al. 2010). However, recent work (Rybak et al. 2022) indicates that the dense gas fraction is suppressed rather than enhanced. Another possibility is that the efficiency with which dense gas forms stars depends on the local conditions of the galaxy (Usero et al. 2015; Bigiel et al. 2016). More observations, especially of HCN, are necessary to more fully understand the relationship between dense gas and star formation at high redshift.

The ratios between dense gas tracers can also constrain the physical conditions of the gas. For instance, the HNC/HCN and HCO⁺/HCN ratios have been suggested to provide a discriminator between photodissociation regions (PDRs) and X-ray-dominated regions (XDRs) or more complex dynamics (Aalto et al. 2007; Meijerink et al. 2007), though X-ray-driven chemistry can change abundance ratios (e.g., Juneau et al. 2009). For the J = 4−3 transition, we find a HNC/HCN ratio of ∼0.6 and a HCO⁺/HCN ratio of ∼0.9, which are lower than the previously obtained values in Spilker et al. (2014) and for individual sources in Béthermin et al. (2018). However, the obtained ratios are in agreement with values for local ULIRGs (HNC/HCN ∼0.5 and HCO⁺/HCN ∼0.8 for J = 1−0 detections; Jiménez-Donaire et al. 2019). Averaging over all transitions gives ratios of ∼0.9 and ∼1.0 for HNC/HCN and HCO⁺/HCN, respectively, in agreement with both previously obtained SPT values and measurements from other high-redshift lensed galaxies (Cañameras et al. 2021). For the HNC/HCN ratio, measurements are in the typical range of PDRs (HNC/HCN ≃0.1–1.0; Hirota et al. 1998; Hacar et al. 2020), and the differences between the overall HNC/HCN ratio and the individual transitions suggest that the HNC/HCN ratio has some dependence on the rotational quantum number. In contrast, since the J = 4−3 and averaged measurements for the HCO⁺/HCN ratio agree within errors and are consistent with other measurements at high redshift, the HCO⁺/HCN ratio does not appear to exhibit any frequency dependence.

As in Section 4.1, we again order the SPT DSFG sample by the dust temperatures from Reuter et al. (2020) and split the sample into two bins according to temperature, as shown in Figure 8. While the majority of the transitions of the optically thin ¹³CO and C¹⁸O exhibit differences between the so-called "hot" and "cold" DSFG populations, these populations are consistent within errors to the full sample. The dense gas tracers HCN, HNC, and HCO⁺ for hotter DSFGs are potentially more energetic in some transitions than their cooler counterparts. Though more observations are needed to increase the statistical robustness of the sample, this could indicate the presence of more excited dense gas or a more massive dense gas reservoir in hotter DSFGs. Additionally, the "hot" and "cold" DSFG populations show no difference for the cyanide radical, CN, and all three populations show indications of absorption in the J_up = 6 transition. It should be emphasized that these results are speculative since many of these transitions have not been detected in this work and are also dependent on the normalization (see Section 2.2 for details). However, as orders-of-magnitude more sources are expected to be discovered with SPT-3G, these results could highlight interesting future areas of study.

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Like the molecules discussed in Section 4.3, H₂O emission arises from warm gas in dense molecular filaments and structures. In the SPT DSFGs, the H₂O(2_1,1–2_0,2) and H₂O(4_1,4–3_2,1) emission lines (ν_rest = 752 GHz, E_up = 137 K and ν_rest = 380 GHz, E_up = 324 K, respectively) are detected, along with the H₂O(1_1,0–1_0,1) absorption line (ν_rest = 557 GHz, E_up = 61 K; see Table B1 for details). The H₂O(2_1,1–2_0,2) emission line is excited through the absorption and pumping of the 101 μm transition line and traces the IR field. Studies (Omont et al. 2013; Yang et al. 2013; González-Alfonso et al. 2014) show that this line is correlated with IR luminosity over several orders of magnitude in both local and high-z LIRGs and could also be used to trace star formation (e.g., Jarugula et al. 2019). Because the H₂O(4_1,4–3_2,1) emission line is a higher energy transition, it has been detected in very dense regions such as in the circumnuclear disk around a quasar (e.g., Stacey et al. 2020). Since this line was found to be spatially coincident with the CO(11–10) emission line, it is likely that H₂O(4_1,4–3_2,1) is found in highly excited regions, like high-J CO lines. Radiative-transfer models (e.g., Liu et al. 2017) also suggest that the H₂O(4_1,4–3_2,1) requires extreme (T_dust ∼ 100–200) conditions in order to be excited. Given the highly excited CO line fluxes shown in Figure 5, at least a subset of the SPT DSFGs exhibit such conditions.

Though the H₂O(4_2,3-3_3,0) emission line (at 448 GHz) was detected in a strongly lensed DSFG at z = 3.6 (G09v1.97; Yang et al. 2020), it has not been observed in Spilker et al. (2014) or this work. Given the detection of the H₂O(2_1,1–2_0,2) emission line (at 752 GHz) in both Yang et al. (2020) and in the SPT DSFG sample, and the inferred line ratios from Yang et al. (2020), the H₂O(4_2,3–3_3,0) emission line (at 448 GHz) should have been detected in this work if G09v1.97 was representative of the larger population. Additionally, the SPT DSFG sample is on average more intrinsically luminous than G09v1.97 (the SPT median intrinsic luminosity is 〈L_IR〉 = 1.5 × 10¹³ L_⊙ compared to L_IR = 5 × 10¹² L_⊙), which could change the excitation levels in the water transitions.

A deep absorption feature due to the H₂O(1_1,0–1_0,1) ground-state transition was also observed in the SPT DSFG stack. Though we perform a continuum subtraction in order to obtain the stacked spectrum, we measure the continua of the input spectra beforehand and create a stacked continuum in order to determine the equivalent width. We measure an equivalent width of 199 ± 24 km s⁻¹ for the H₂O(1_1,0–1_0,1) transition measured in our stacked spectrum (see Figure 9). This is the only absorption line securely detected in our stack, and indicates the presence of some low-excitation gas backlit against the continuum along the line of sight. This absorption line has also been seen in Riechers et al. (2022) for a DSFG at z = 6.34, where the absorption feature was detected at the ∼2σ level, relative to the CMB, indicating the presence of cold water vapor. However, more study would be needed to determine if this effect is also occurring in the individual DSFGs present in our stacked spectrum.

The carbon hydride CH has been observed locally in both dense photodissociation regions (e.g., Habart et al. 2010) and diffuse molecular clouds (e.g., Chastain et al. 2010). Because CH exhibits constant abundance ratios relative to molecular hydrogen (e.g., Liszt & Lucas 2002), it is considered to be a reliable tracer of molecular hydrogen. Additionally, because neutral hydrides like CH are most abundant in gas with a large molecular fraction (Gerin et al. 2016), measurements of this molecule can be combined with other diagnostic lines to determine the molecular fraction. CH was first detected at high redshift in Spilker et al. (2014) by combining all transitions in the 3 mm window. It has again been detected in the stacked spectrum presented in this work and also at 532 and 536 GHz. It has been suggested that the CH doublet is sensitive to both the fine-structure constant and electron mass ratio (de Nijs et al.2012), and measurements of CH at 532 and 536 GHz could provide a test of these fundamental constants with time. However, precise calibration with another line (e.g., H₂O(1_1,0–1_0,1) at 557 GHz) would be needed to study these effects.