A Controlled Study of Cold Dust Content in Galaxies from z = 0–2

The GOALS sample comprises 180 luminous IR galaxies (LIRGs; ${L}_{\mathrm{IR}}={10}^{11}\,{L}_{\odot }$) and 22 ULIRGs (${L}_{\mathrm{IR}}\gt {10}^{12}{L}_{\odot }$). Many of these systems are interacting, resulting in 244 individual galactic nuclei with Spitzer IRS spectroscopy. These galaxies are a representative subset of the IRAS Bright Galaxy Survey (Sanders et al. 2003) and were selected to have ${S}_{60}\gt 5.24\,\mathrm{Jy}$. GOALS sources cover the distance range 15 Mpc < D < 400 Mpc, which corresponds to $z\lt 0.088$.

The IRS spectra have been previously analyzed in depth (Díaz-Santos et al. 2010a, 2010b; U et al. 2012; Stierwalt et al. 2014). Here, we make use of the IRS SL staring observations, $\lambda =5.5\mbox{--}14.5$ μm, to identify and remove AGNs from the sample (Section 3.1). The GOALS galaxies have global flux densities from all three Spitzer MIPS bandpasses (J. Mazzarella et al. 2017, in preparation). ${L}_{\mathrm{IR}}(8\mbox{--}1000$ μm) was calculated following the formula in Sanders & Mirabel (1996) using all four IRAS flux densities (Díaz-Santos et al. 2010a).

The UV-IR SEDs were analyzed in detail for a subset of 64 galaxies (see U et al. 2012 for a discussion of the selection of these objects). These galaxies have stellar mass measurements, so they form our initial dust mass sample as well. From the 64 galaxies, 58 sources have either 850 μm photometry from the James Clerk Maxwell Telescope or 250 μm photometry from Herschel (Chu et al. 2017), which we use to calculate ${M}_{\mathrm{dust}}$. Five of these are double nuclei sources that we remove, and seven are hosting an AGN. This leaves a final sample size of 46 galaxies.

5MUSES is a Spitzer IRS mid-IR spectroscopic survey of 330 galaxies selected from the SWIRE and Spitzer Extragalactic First Look Survey fields (details in Wu et al. 2010). It is a flux-limited sample selected at 24 μm, with ${S}_{24}\gt 5$ mJy. Two hundred and eighty sources have optical spectroscopic redshifts (Wu et al. 2010). 5MUSES is a representative sample of galaxies at intermediate redshift (the median redshift of the sample is 0.14) with ${L}_{\mathrm{IR}}\sim {10}^{10}\mbox{--}{10}^{12}\,{L}_{\odot }$, bridging the gap between local (U)LIRGs and high-redshift observations.

For the present study, we make use of the Spitzer IRS SL spectroscopy to quantify AGN emission. Complete details of the Spitzer data reduction are found in Wu et al. (2010). IRS spectra combined with Spitzer MIPS imaging is used to calculate ${L}_{\mathrm{IR}}(5\mbox{--}1000$ μm) in Wu et al. (2010), with a slightly different cosmology (${{\rm{\Omega }}}_{M}=0.27,{\rm{\Lambda }}=0.73$). We adjust the published L_IRs by multiplying by 0.99 to account for the difference in cosmology and 0.95 to scale to ${L}_{\mathrm{IR}}(8\mbox{--}1000$ μm), which we have determined using a purely star-forming template from Kirkpatrick et al. (2013b). We use Herschel SPIRE observations of a subset of 188 sources in the Herschel Multi-Tiered Extragalactic Survey (HerMES; Oliver et al. 2010, 2012; Magdis et al. 2013) to calculate ${M}_{\mathrm{dust}}$.

Of the 188 Herschel galaxies, 40 have 500 μm detections, necessary to calculate ${M}_{\mathrm{dust}}$. Six of these are removed because they host an AGN. We remove an additional four sources where the 500 μm photometry appears blended with a nearby source. This leaves a final sample of 30 galaxies.

We have assembled a multiwavelength data set for a sample of 343 high-redshift ($z\sim 0.3\mbox{--}4.0$) (U)LIRGs in the Great Observatories Origins Deep Survey North (GOODS-N), Extended Chandra Deep Field Survey (ECDFS), and Spitzer xFLS fields. All sources are selected to have mid-IR spectroscopy from Spitzer IRS, which is used to measure the redshift (Kirkpatrick et al. 2012). Our sample contains a range of sources from individual observing programs, each with differing selection criteria. However, the overarching selection criterion is a 24 μm flux limit of 0.9 mJy for the xFLS galaxies, which are taken from a shallower survey, and 0.1 mJy for the GOODS and ECDFS galaxies, compiled from a deep survey. In addition to IRS spectra, these sources all have Spitzer MIPS, and Herschel PACS and SPIRE imaging (Sajina et al. 2012; Kirkpatrick et al. 2013b). We calculate individual ${L}_{\mathrm{IR}}(8\mbox{--}1000$ μm) by fitting a full suite of Spitzer and Herschel photometry with the library of templates from Kirkpatrick et al. (2013b). Redshifts are determined by fitting the main PAH emission features, or by using optical spectroscopy in the case of a featureless MIR spectrum.

In this work, we also utilize 870 μm photometry of GOODS-S from LABOCA on APEX (Weiß et al. 2009), the combined AzTEC+MAMBO 1.1 mm map of GOODS-N (Penner et al. 2011), and MAMBO 1.2 mm imaging of xFLS (Lutz et al. 2005; Sajina et al. 2008; Martínez-Sansigre et al. 2009), which exists for 169 galaxies. For four of these, the source redshift is high enough that there is no data beyond $\lambda =250$ μm, which is our threshold for calculating dust mass. We also remove any sources from the sample where the submillimeter emission is blended with a nearby source (10 sources). We remove 57 galaxies for hosting an AGN. Finally, we remove 47 galaxies for having submillimeter photometry with S/N < 2, leaving a final sample of 51 galaxies.

We are comparing the far-IR/submillimeter properties of three different samples with different selection criteria, so we must understand whether our results are biased due to the selection criteria. The GOALS sample is selected at 60 μm while the 5MUSES sample and the Supersample are selected at observed frame 24 μm; this selection criteria alone could produce samples of galaxies that are warmer or colder for the same ${L}_{\mathrm{IR}}$ range.

We consider the 5MUSES and GOALS sources together to make up our low-z sample. All of the 5MUSES sources with ${L}_{\mathrm{IR}}\gt {10}^{11}$ (same as GOALS) would be selected at rest frame ${S}_{60}\gt 5.24$ Jy, the GOALS selection criterion. Similarly, all GOALS galaxies meet the 5MUSES selection criterion of ${S}_{24}\gt 5$ mJy.

We use the mid-IR spectrum to determine if a 5MUSES source meets the Supersample selection criterion, since at $z=1\mbox{--}2$, S₂₄ covers the PAH features. Only 28% of the 5MUSES sample would be selected at z = 1, and 11% at z = 2. This is not surprising, since the 5MUSES galaxies are on average less luminous. Only 32% of the 5MUSES galaxies overlap in the ${L}_{\mathrm{IR}}$ range spanned by the Supersample. However, of the 60 5MUSES galaxies with ${L}_{\mathrm{IR}}\gt 2\times {10}^{11}\,{L}_{\odot }$ (the same range as the bulk of the Supersample), 50 would be selected as Supersample galaxies at z = 1 and 30 would be selected at z = 2, due to ${L}_{\mathrm{PAH}}/{L}_{\mathrm{IR}}$ ratios similar to $z\sim 1\mbox{--}2$ (U)LIRGs (Pope et al. 2013; Kirkpatrick et al. 2014b; Battisit et al. 2015). Estimating whether the GOALS sources would be included in our Supersample is more nuanced, since we lack global IRS spectra for the GOALS galaxies. Instead, we use the Chary & Elbaz (2001) library of templates, which were derived from the IRAS BGS. At z = 1, templates with $\mathrm{log}{L}_{\mathrm{IR}}\gt 11.11{L}_{\odot }$ meet the selection threshold, but at z = 2, this increases to $\mathrm{log}{L}_{\mathrm{IR}}\gt 12.21\,{L}_{\odot }$.

Next, we estimate what fraction of the Supersample would be selected as a 5MUSES or GOALS galaxy, if the Supersample existed at a different redshift. This is largely an academic exercise, since the Supersample contains the most luminous galaxies, and Symeonidis et al. (2011) demonstrated that the IRAS selection criteria are sensitive to ULIRGs (${L}_{\mathrm{IR}}\gt {10}^{12}\,{L}_{\odot }$) with $T=17\mbox{--}87$ K. Nevertheless, such a check will ensure that the Supersample contains enough warm dust to be luminous at rest-frame 60 and 24 μm. We calculate synthetic 60 μm photometry following the prescription in Section 3.2, for a redshift of z = 0.015, which is the peak of the GOALS redshift distribution in Figure 1. Ninety-seven percent of the Supersample meets the GOALS selection criterion. The 5MUSES selection criterion is ${S}_{24}\gt 5$ mJy. We shift our synthetic rest-frame photometry to z = 0.15 by scaling down by 1.62, a ratio determined using the Kirkpatrick et al. (2013b) template library. Again, 97% would be selected as a 5MUSES galaxy.

From this, we conclude that if cold, massive galaxies, like most of the Supersample, exist at lower redshift, they are easily detectable in current surveys. Similarly, at $z\sim 1$ at least, we should be able to select a population of galaxies with the range of dust masses and temperatures exhibited by local LIRGs, if they exist. However, at $z\sim 2$, galaxies with spectral shapes similar to local LIRGs are undetectable in current 24 μm surveys.

Throughout the remainder of this paper, we dispense with the (U)LIRG nomenclature for clarity. In the local universe, the term ULIRG evokes a massive merging system, where the merger gives rise to a starburst followed by an AGN (Sanders & Mirabel 1996). At earlier times, the correlation between ULIRGs and mergers still exists, but is more tenuous, and the link between mergers and starbursts is even less clear in these systems (Hopkins et al. 2010; Kartaltepe et al. 2012; Hayward et al. 2013). LIRGs and ULIRGs are simply luminosity cuts, so to avoid any preconceived notions between these luminosity cuts and physical properties, we follow the convention in Casey et al. (2014) and refer to objects in all three samples as DSFGs.

The mid-IR spectrum ($\lambda =3\mbox{--}20$ μm) is rich for identifying star formation features and AGNs. The most prominent dust emission complexes are produced by PAHs ($\lambda =6.2,7.7,11.3,12.7$ μm), which are abundant in galaxies with metallicity close to solar, such as high-redshift DSFGs (Magdis et al. 2012). These molecules are preferentially located in the photodissociation regions surrounding star-forming regions (Helou et al. 2005). As such, PAHs are good tracers of ongoing star formation in a galaxy (Peeters et al. 2004). However, the mid-IR spectrum can also exhibit continuum emission due to the torus enveloping the AGN.

We perform spectral decomposition of the mid-IR spectrum ($\sim 5\mbox{--}15\,\mu {\rm{m}}$ rest frame) for each source in all three samples in order to disentangle the AGN and star-forming components. Pope et al. (2008) explain the technique in detail, and we summarize it here. We fit the individual spectra with a model comprising four components: (1) the star formation component is represented by the mid-IR spectrum of the prototypical starburst M82, (2) the AGN component is determined by fitting a pure power law (${\lambda }^{\alpha }$) with the slope and normalization (N) as free parameters, (3, 4) extinction curves from the Weingartner & Draine (2001) dust models for Milky Way (MW) type dust is applied to the AGN component and star-forming component. The full model is then

$$\begin{eqnarray}&&{S}_{\nu }={N}_{\mathrm{AGN}}{\lambda }^{\alpha }\,{e}^{-{\tau }_{\mathrm{AGN}}}+{N}_{\mathrm{SF}}{S}_{\nu }({\rm{M}}82){e}^{-{\tau }_{\mathrm{SF}}}.\end{eqnarray} \tag{ 1 }$$

We fit for ${N}_{\mathrm{AGN}}$, ${N}_{\mathrm{SF}}$, α, ${\tau }_{\mathrm{AGN}}$, and ${\tau }_{\mathrm{SF}}$ simultaneously.

For each source, we quantify the strength of the AGN, $f{(\mathrm{AGN})}_{\mathrm{MIR}}$, as the fraction of the total mid-IR luminosity ($\lambda =5\mbox{--}15$ μm) coming from the power-law continuum component. For the GOALS sample, our AGN identification technique selects the same sources as the ${S}_{15}/{S}_{5.5}$ versus ${S}_{6.2}/{S}_{5.5}$ diagnostic applied in Petric et al. (2011). For the 5MUSES sources, we also identify the same AGNs as Wu et al. (2010), where the authors classify AGNs on the basis of the 6.2 μm PAH feature equivalent width alone.

In Kirkpatrick et al. (2013b), we found that an AGN had a significant contribution to the far-IR emission, quantified through dust temperatures and luminosities, when $f{(\mathrm{AGN})}_{\mathrm{MIR}}\geqslant 0.5$. Recently, Lee et al. (2016) showed that local dust-obscured galaxies hosting an AGN had lower dust masses than their star-forming counterparts. Consequently, we remove all 5MUSES, GOALS, and Supersample galaxies with $f{(\mathrm{AGN})}_{\mathrm{MIR}}\geqslant 0.5$, so that we are only comparing the far-IR/submillimeter properties of strongly star-forming systems. We note that because the GOALS sample is nearby, in some cases IRS observations only cover the central region of the galaxy, in contrast to the IRS observations of 5MUSES and the high-z Supersample. This could introduce a slight bias as we may remove sources that have a nucleus dominated by AGN emission, whereas the galaxy-integrated emission is dominated by star formation. We only remove 13% of the GOALS sample, so the effects of this potential bias are small.

Our AGN identification technique only relies on detecting hot continuum emission in the mid-IR. While we attribute this to an AGN, in principle, a very compact, dust-enshrouded starburst would display the same mid-IR signature. By removing AGNs, we are also then removing these compact starbursts, which will have different ISM properties than galaxies where the star-forming regions are located throughout the galaxy, due to higher gas densities and a harsher radiation field. However, we note that although such compact starbursts exist locally (e.g., Díaz-Santos et al. 2010b), so far such galaxies are rare at high redshift and still have prominent PAH features (Nelson et al. 2014). From the Supersample, we remove 47% for having $f{(\mathrm{AGN})}_{\mathrm{MIR}}\geqslant 0.5$, and 62% of these removed sources have X-ray detections consistent with being an AGN (Kirkpatrick et al. 2013b), while the remaining sources may be Compton thick based on predicted X-ray luminosities (Kirkpatrick et al. 2012). As such, we do not think that we are missing a large population of compact starbursts by removing sources with $f{(\mathrm{AGN})}_{\mathrm{MIR}}\geqslant 0.5$.

In Section 4, we compare the far-IR SED shape of GOALS, 5MUSES, and the Supersample by utilizing far-IR colors. To make a fair comparison between three samples at different redshift ranges, we require synthetic rest-frame photometry in the MIPS and SPIRE photometric bandpasses for 5MUSES and the Supersample. GOALS is at low enough redshift that the offset in rest-frame wavelengths is negligible. We fit templates to the Supersample and 5MUSES from the MIR-based library in Kirkpatrick et al. (2013b). The MIR-based library is a suite of 11 empirical templates that characterize the full shape of the IR SED based on the relative amounts of PAH and continuum emission in a source's mid-IR spectroscopy. We select the appropriate template for each source from the spectral decomposition described above, fit to the far-IR photometry ($\lambda \gt 20$ μm), and then convolve each template with the MIPS and SPIRE transmission filters to create synthetic rest-frame photometry at 70, 160, and 250 μm. We plot the Supersample intrinsic SEDs and estimated rest-frame photometry in Appendix A. The biggest concern when interpolating between data points is whether we have chosen a template with the correct far-IR shape. However, the majority of galaxies have a sufficiently well-sampled far-IR SED to mitigate this uncertainty, as the estimated photometry lies very close to the observed data points.

It is unlikely that an AGN that is not significantly affecting the mid-IR emission would have a strong effect on the far-IR emission. Even so, we correct the rest-frame broadband colors and L_IRs of all SFGs with $0.0\lt f{(\mathrm{AGN})}_{\mathrm{MIR}}\lt 0.5$ in order to account for any scatter introduced by a buried AGN that may contribute to, but not dominate, the IR luminosity. We have calculated these corrections by decomposing empirical templates into their relative star formation and AGN components as described in detail in Kirkpatrick et al. (2013b). The AGN corrections are listed below, for the rest-frame luminosities. In Kirkpatrick et al. (2013b), we discuss ways to estimate $f{(\mathrm{AGN})}_{\mathrm{MIR}}$ in the absence of mid-IR spectroscopy, so these corrections can be applied to sources with only broadband photometry:

$$\begin{eqnarray}{L}_{\mathrm{IR}}^{\mathrm{SF}} & = & {L}_{\mathrm{IR}}\times (1+0.035f{(\mathrm{AGN})}_{\mathrm{MIR}}-0.66f{(\mathrm{AGN})}_{\mathrm{MIR}}^{2})\\ {L}_{24}^{\mathrm{SF}} & = & {L}_{24}\times (1-1.81f{(\mathrm{AGN})}_{\mathrm{MIR}}+0.94f{(\mathrm{AGN})}_{\mathrm{MIR}}^{2})\\ {L}_{70}^{\mathrm{SF}} & = & {L}_{70}\times (1-0.08f{(\mathrm{AGN})}_{\mathrm{MIR}}-0.16f{(\mathrm{AGN})}_{\mathrm{MIR}}^{2})\\ {L}_{160}^{\mathrm{SF}} & = & {L}_{160}\times (1-0.036f{(\mathrm{AGN})}_{\mathrm{MIR}})\\ {L}_{250}^{\mathrm{SF}} & = & {L}_{250}\times (1-0.018f{(\mathrm{AGN})}_{\mathrm{MIR}}).\end{eqnarray} \tag{ 2 }$$

With our three samples, we can compare how ${M}_{\mathrm{dust}}$ evolves in dusty galaxy populations with redshift. Dust mass is calculated as

$$\begin{eqnarray}&&{M}_{\mathrm{dust}}=\displaystyle \frac{{S}_{\nu }{D}_{L}^{2}}{{\kappa }_{\nu }{B}_{\nu }(T)},\end{eqnarray} \tag{ 3 }$$

where ${B}_{\nu }(T)$ is the Planck equation, ${\kappa }_{\nu }$ is the dust opacity, D_L is the luminosity distance, and S_ν is the flux density. For S_ν, we use the longest far-IR/submillimeter data available, which varies with each sample. We only use sources whose rest-frame measurement is $\lambda \geqslant 250$ μm, otherwise the dust emission may not be tracing the coldest dust component (Scoville et al. 2016). When we combine this criterion with the $f{(\mathrm{AGN})}_{\mathrm{MIR}}\lt 0.5$ criterion, we have the final sample sizes of 46 galaxies from GOALS, 30 galaxies from 5MUSES, and 52 galaxies from the Supersample.

We take ${\kappa }_{\nu }$, the dust opacity, from the Weingartner & Draine (2001)¹⁷ models at the rest wavelength of S_ν for each individual source. We assume MW-like dust and R_V = 3.1, although in these models, the far-IR/submillimeter opacities have negligible changes for different R_V and are similar for the MW-, LMC-, and SMC-type models (at 850 μm, the opacity differs by <10%).

Scoville et al. (2016) argue for using fixed ${T}_{\mathrm{cold}}$, since presumably the temperature in the diffuse ISM should be relatively similar for all massive, dusty galaxies, assuming similar radiation fields and dust grain mixtures. Indeed, in Kirkpatrick et al. (2013b) we found that the average cold dust temperature in SFGs in the Supersample was remarkably constant, $T\sim 26$ K, and did not scale with ${L}_{\mathrm{IR}}$. Accordingly, we set ${T}_{\mathrm{cold}}=25\,{\rm{K}}$ for all three samples and derive ${M}_{\mathrm{dust}}$ using Equation (3). As a test of this assumption, we fit each galaxy with a two-temperature modified blackbody with ${T}_{\mathrm{cold}}=25\,{\rm{K}}$ and achieve excellent fits for all sources. We explore how the dust masses change if ${T}_{\mathrm{dust}}$ is allowed to vary in Appendix B and find that all conclusions in this paper hold regardless of the specific method of measuring dust mass as long as these measurements are consistent across all three samples. In this study, we are primarily concerned with comparing the dust masses at high redshift relative to low redshift. By fixing ${T}_{\mathrm{cold}}$ and using the same κ model for all samples, we are able to probe how dust masses change with redshift if the other dust properties of these galaxies remain the same.

We use the ${L}_{\mathrm{IR}}$ compiled from the literature for each sample (see Section 2). In all cases, the ${L}_{\mathrm{IR}}$ was not determined using the submillimeter data that goes into the ${M}_{\mathrm{dust}}$ calculation, so in this way, ${L}_{\mathrm{IR}}$ and dust mass are independent. ${L}_{\mathrm{IR}}$ for the 5MUSES and Supersample galaxies were measured by fitting template libraries, so the main source of concern is how the choice of β used in constructing the templates might affect ${L}_{\mathrm{IR}}$, as ${M}_{\mathrm{dust}}$ also depends on β. We calculate that varying $\beta \in [1.5,2.0]$ changes ${L}_{\mathrm{IR}}$ by 5%, so using different values of β to calculate ${L}_{\mathrm{IR}}$ and ${M}_{\mathrm{dust}}$ will not be a significant source of scatter in the resulting relationships. In the analysis below, we correct ${L}_{\mathrm{IR}}$ for any contribution from nuclear activity as indicated by the presence of a hot dust continuum in the mid-IR spectrum, so that we are actually comparing ${M}_{\mathrm{dust}}$ with ${L}_{\mathrm{IR}}^{\mathrm{SF}}$. The correction is at most 0.07 dex. As all of the GOALS and 5MUSES photometry and redshifts used in this paper are published elsewhere in the literature (Wu et al. 2010; U et al. 2012; Magdis et al. 2013), we only list the ${M}_{\mathrm{dust}}$ and ${L}_{\mathrm{IR}}$ values that we have calculated for the Supersample in Table 1.

Table 1.  Supersample Properties

Name	R.A.	Decl.	z	$\mathrm{log}{L}_{\mathrm{IR}}({L}_{\odot })$ a	$\mathrm{log}{M}_{\mathrm{dust}}({M}_{\odot })$	$\mathrm{log}{M}_{* }({M}_{\odot })$ b
GN_IRS2	12:37:02.74	+62:14:01.0	1.24	12.13	8.74 ± 0.17	10.87
GN_IRS5	12:36:20.94	+62:07:14.0	1.15	12.23	8.64 ± 0.24	10.79
GN_IRS10	12:37:07.21	+62:14:08.1	2.49	12.72	8.85 ± 0.12	11.07
GN_IRS11	12:36:21.27	+62:17:08.0	2.00	12.56 (12.55)	8.84 ± 0.14	11.20
GN_IRS15	12:37:11.37	+62:13:31.0	2.00	12.86 (12.85)	9.20 ± 0.06	11.28
GN_IRS18	12:37:16.59	+62:16:43.0	1.80	12.46 (12.43)	8.67 ± 0.21	11.52
GN_IRS21	12:36:18.33	+62:15:50.0	2.00	12.77 (12.71)	8.88 ± 0.13	10.75
GN_IRS23	12:36:19.13	+62:10:04.0	2.21	12.30	8.44 ± 0.33	11.02
GN_IRS25	12:37:01.59	+62:11:46.0	1.72	12.62	8.79 ± 0.16	11.54
GN_IRS26	12:36:34.51	+62:12:40.0	1.22	12.62	8.45 ± 0.36	10.94
GN_IRS27	12:36:55.94	+62:08:08.0	0.79	12.06 (12.03)	8.37 ± 0.37	11.10
GN_IRS31	12:36:22.66	+62:16:29.0	1.79	12.44	8.54 ± 0.29	11.15
GN_IRS38	12:36:29.10	+62:10:46.3	1.01	12.00	8.75 ± 0.16	11.14
GN_IRS42	12:36:46.72	+62:08:33.9	0.97	12.13 (12.12)	8.75 ± 0.16	11.05
GN_IRS45	12:38:21.76	+62:17:06.0	1.62	12.45 (12.41)	9.02 ± 0.19	11.09
GN_IRS47	12:35:53.81	+62:13:38.0	0.88	11.87 (12.86)	9.01 ± 0.09	10.88
GN_IRS49	12:37:05.49	+62:21:24.0	0.95	11.85	8.62 ± 0.24	11.11
GN_IRS60	12:36:49.72	+62:13:12.9	0.47	11.24	8.44 ± 0.20	9.99
GN_IRS62	12:36:29.54	+62:06:46.5	0.80	11.58	8.48 ± 0.31	10.56
GN_IRS66	12:37:05.85	+62:21:29.8	0.95	11.82	8.65 ± 0.23	10.47
GN_IRS67	12:36:19.14	+62:13:01.8	1.23	11.76	8.55 ± 0.28	10.47
GN_IRS68	12:36:19.50	+62:12:52.6	0.47	11.47	8.25 ± 0.31	10.56
GS_IRS20	03:32:47.58	−27:44:52.0	1.91	12.60 (12.59)	8.55 ± 0.24	10.77
GS_IRS23	03:32:17.23	−27:50:37.0	1.96	12.35	8.78 ± 0.14	10.99
GS_IRS25	03:32:18.70	−27:49:19.0	1.04	11.83	8.44 ± 0.28	10.60
GS_IRS26	03:32:20.70	−27:44:53.0	0.97	11.52	8.38 ± 0.31	10.69
GS_IRS28	03:31:35.26	−27:49:58.0	0.96	11.61	8.52 ± 0.23	10.62
GS_IRS29	03:32:22.53	−27:45:38.0	2.08	12.12 (12.10)	8.47 ± 0.27	10.62
GS_IRS30	03:32:43.78	−27:52:31.0	1.62	11.85	8.33 ± 0.38	11.07
GS_IRS31	03:32:06.00	−27:45:07.0	1.07	11.78	8.36 ± 0.34	10.57
GS_IRS34	03:32:39.00	−27:44:20.0	1.93	11.95	8.39 ± 0.33	10.11
GS_IRS39	03:31:48.18	−27:45:35.0	1.72	12.34 (12.31)	8.79 ± 0.13	⋯
GS_IRS43	03:31:48.93	−27:39:45.0	0.82	11.62 (11.58)	8.57 ± 0.19	10.34
GS_IRS45	03:32:17.45	−27:50:03.0	1.62	12.48	8.64 ± 0.19	10.39
GS_IRS46	03:32:42.71	−27:39:27.0	1.85	12.32	8.70 ± 0.16	⋯
GS_IRS52	03:32:12.52	−27:43:06.0	1.79	12.11	8.50 ± 0.25	10.43
GS_IRS53	03:32:12.13	−27:42:49.0	2.45	12.37 (12.31)	8.74 ± 0.14	10.90
GS_IRS56	03:32:13.58	−27:47:54.0	1.72	11.91	8.41 ± 0.31	11.22
GS_IRS62	03:32:22.48	−27:49:35.0	0.73	11.76	8.56 ± 0.17	10.76
GS_IRS63	03:32:22.59	−27:44:25.9	0.74	11.64	8.42 ± 0.24	10.63
GS_IRS64	03:32:17.28	−27:49:08.0	1.62	11.88	8.50 ± 0.26	10.45
GS_IRS70	03:32:27.71	−27:50:40.6	1.10	11.99 (11.98)	8.41 ± 0.31	10.78
GS_IRS73	03:32:43.24	−27:47:56.2	0.67	11.26	8.17 ± 0.41	10.47
GS_IRS74	03:32:44.32	−27:49:11.9	2.00	12.12	8.56 ± 0.22	⋯
GS_IRS76	03:32:48.83	−27:42:35.0	1.96	12.58 (12.57)	9.16 ± 0.06	⋯
GS_IRS80	03:32:36.52	−27:46:30.7	1.05	11.91 (11.89)	8.69 ± 0.16	⋯
MIPS289	17:13:50.02	+58:56:57.1	1.89	13.10 (13.07)	9.00 ± 0.07	⋯
MIPS8377	17:17:33.53	+59:46:40.4	0.84	12.05	8.44 ± 0.34	⋯
MIPS8493	17:18:05.06	+60:08:32.6	1.80	12.73 (12.70)	8.62 ± 0.24	⋯
MIPS8543	17:18:12.54	+59:39:23.0	0.65	12.03	9.06 ± 0.14	⋯
MIPS22530	17:23:03.33	+59:16:00.1	1.96	13.05 (13.04)	8.97 ± 0.12	⋯
19456000	17:14:29.66	+59:32:33.7	1.97	13.27 (13.26)	8.93 ± 0.07	⋯

Notes.

^a ${L}_{\mathrm{IR}}^{\mathrm{SF}}$ is listed in parentheses for those sources requiring a correction to account for nuclear activity as indicated by the presence of a hot dust continuum in the mid-IR spectrum. ^bM_* was initially derived using a Salpeter IMF (see Kirkpatrick et al. 2012 for details), and we have corrected it to a Chabrier IMF, which we list here.

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Stellar masses are compiled from the literature. For GOALS, U et al. (2012) calculates M_* by fitting Bruzual & Charlot (2003) stellar population models to 10 different broadband UV-NIR photometry points (from GALEX, ground-based observatories, and Spitzer). The star formation history is parameterized as $\mathrm{SFH}={e}^{-t/\tau }$ with $\tau \in [1,30]$ Gyr. Metallicity is a free parameter. The authors choose a Chabrier initial mass function (IMF; Chabrier 2003) with a mass range of $0.1\,{M}_{\odot }$–$100\,{M}_{\odot }$.

Shi et al. (2011) calculate stellar masses for the 5MUSES sample by fitting Bruzual & Charlot (2003) stellar population models to UV-IR data assuming a Chabrier IMF. The SFH has $\tau \in [0.03,22.4]$ Gyr, and metallicity is a free parameter.

For the GOODS and ECDFS sources (we do not have M_* for the six xFLS sources included in the final high-z sample), Kirkpatrick et al. (2012) describes the procedure for calculating M_* by fitting Bruzual & Charlot (2003) models to a suite of photometry from the U band to 4.5 μm (see also Pannella et al. 2015). For the SFH, $\tau \in [0.1,20]$ Gyr and metallicity is fixed to solar. For the Supersample, a Salpeter IMF is used (Salpeter 1955), which results in higher stellar masses than a Chabrier IMF. We convert the Supersample masses to a Chabrier framework using ${M}_{* }^{\mathrm{Cha}}=0.62\,{M}_{* }^{\mathrm{Sal}}$ (Zahid et al. 2012; Speagle et al. 2014).

We are drawing from three different samples fit with three different methods, so we verify the stellar masses by comparing with the rest-frame K-band estimated M_* (Howell et al. 2010). In all cases, the stellar masses are consistent to within 0.6 dex, which is the largest deviation. We further check the consistency of the Supersample masses, since these were initially calculated with a Salpeter IMF and require a conversion. We fit the UV-submillimeter SEDs of 10 randomly selected galaxies with the magphys code, which employs an energy-balance technique to simultaneously fit the UV/optical/IR emission (da Cunha et al. 2008). The magphys-derived stellar masses are on average 0.2 dex larger than the masses from Kirkpatrick et al. (2012), converted to a Chabrier IMF.

All three samples cover a similar range of ${M}_{* }\,=1\times {10}^{10}\mbox{--}3\times {10}^{11}\,{M}_{\odot }$, as can be seen in Figure 8.

Our higher-z sample is also at a higher ${L}_{\mathrm{IR}}$, which can confuse any redshift evolution. We compare ${L}_{160}^{\mathrm{SF}}/{L}_{70}^{\mathrm{SF}}$ with ${L}_{\mathrm{IR}}^{\mathrm{SF}}$ in Figure 4. Figure 4 shows a clear trend between ${L}_{160}^{\mathrm{SF}}/{L}_{70}^{\mathrm{SF}}$ and ${L}_{\mathrm{IR}}^{\mathrm{SF}}$, otherwise known as the ${L}_{\mathrm{IR}}\mbox{--}{T}_{\mathrm{dust}}$ relation (Chapman et al. 2003; Casey et al. 2012; Magnelli et al. 2014; Lee et al. 2016). We overplot the mean of each sample in $\mathrm{log}({L}_{\mathrm{IR}}^{\mathrm{SF}})$ bins of 0.25 as the larger symbols.

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The 5MUSES and GOALS galaxies follow the same general trend, but there is a clear offset between the low-z samples and Supersample, due to evolution of the ${L}_{\mathrm{IR}}\mbox{--}{T}_{\mathrm{dust}}$ relationship with redshift (Chapin et al. 2009; Casey et al. 2012; Magnelli et al. 2014). At a given ${L}_{\mathrm{IR}}^{\mathrm{SF}}$, the difference between low z and high z is ∼0.2 dex, which corresponds to approximately a 5 K temperature difference. The Supersample galaxies have a much larger scatter than the GOALS sample, illustrating the increasing diversity of DSFGs with redshift (Symeonidis et al. 2011). Interestingly, the 5MUSES galaxies with $\mathrm{log}{L}_{\mathrm{IR}}^{\mathrm{SF}}\gt 11.4$ overlap with the Supersample rather than the GOALS sample. These are also the highest redshift 5MUSES galaxies, with $z\gt 0.15$, and they also overlap with the Supersample below in Figure 5. Overall, we see a similar slope in the mean ${L}_{160}^{\mathrm{SF}}/{L}_{70}^{\mathrm{SF}}$ versus ${L}_{\mathrm{IR}}^{\mathrm{SF}}$ points, with a normalization that depends on redshift, in agreement with the findings in Magnelli et al. (2014) and Casey et al. (2012).

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The DSFGs at $z\sim 1\mbox{--}2$ have higher dust masses for a given ${L}_{\mathrm{IR}}^{\mathrm{SF}}$ than local DSFGs. Here, we want to make an important distinction. The observed quantity in our sample is the luminosity density in the Rayleigh–Jeans tail, and it is the luminosity density that is actually higher in high-redshift galaxies (as we have used the same assumptions for all other parameters when calculating ${M}_{\mathrm{dust}}$). The luminosity density in the Rayleigh–Jeans tail is directly proportional to ${M}_{\mathrm{dust}}\times {\kappa }_{\nu }$, meaning that in principle, ${M}_{\mathrm{dust}}$ and ${\kappa }_{\nu }$ are degenerate. Thus far, we have been assuming the same opacity relationship for all galaxies, but in reality, the opacity might be evolving with redshift rather than the ${M}_{\mathrm{dust}}$. Dust opacity has a wavelength dependence

$$\begin{eqnarray}&&{\kappa }_{\nu }={\kappa }_{0}{\left(\displaystyle \frac{\lambda }{{\lambda }_{0}}\right)}^{-\beta },\end{eqnarray} \tag{ 8 }$$

so that either the normalization, ${\kappa }_{0}$, or the slope, β, could be evolving.

Although the Weingartner & Draine (2001) models all have similar ${\kappa }_{\nu }$ values in the submillimeter, grain compositions and distributions that differ considerably from these models can give different submillimeter opacities. The apparent increase in dust mass could possibly be due to a change in the dust grain properties with redshift (Dwek et al. 2014). For example, ice mantles can affect β, causing opacity to increase, and ice mantles should be prevalent in cold star-forming regions (Tielens et al. 1984; Bergin et al. 2000; Spoon et al. 2002). The presence of ice mantles can increase ${\kappa }_{\nu }$ by a factor of four (Preibisch et al. 1993; Pollack et al. 1994) A factor of four increase in κ for the high-redshift galaxies would reconcile the difference in ${M}_{\mathrm{dust}}$ for GOALS and the Supersample. The strength of water ice, silicate absorption, and 3.4 μm PAH emission features has been measured in a handful of Supersample sources and closely resembles what is measured in local GOALS galaxies with similar obscuration, providing some indication that the amount of ice in dusty galaxies does not strongly evolve with redshift (Sajina et al. 2009).

If β in our galaxies differs significantly from β in the Weingartner & Draine (2001) models, for example, varying with redshift, then our assumed ${\kappa }_{\nu }$ will be incorrect. The emissivity measure from the Rayleigh–Jeans tail is observed to be shallower in lower metallicity galaxies (Galametz et al. 2011; Kirkpatrick et al. 2013a), though there is little evidence that metallicity evolves strongly in massive, dusty galaxies out to $z\sim 2$ with similar M_*, SFR, and ${M}_{\mathrm{mol}}$ (Mannucci et al. 2010; Magdis et al. 2012; Bothwell et al. 2016). In fact, when we measure β directly for individual galaxies by fitting a modified blackbody with $T=25$ K, we find $\beta =1.8$ for the Supersample and $\beta =2.02$ for the GOALS sample. We are not fully sampling the Rayleigh–Jeans tail in the Supersample, and longer wavelength observations are required to more accurately measure the effective β. However, with current observations, we see little evidence supporting a drastic increase in opacity to reconcile the dust mass measurements of our samples. Longer wavelength observations, such as with ALMA or the Large Millimeter Telescope, are required to better measure the effective β at $z=1\mbox{--}2$.

Additional insight into the ISM of galaxies can be gained by examining morphologies and merger signatures, which we have for the GOALS and Supersample sources. Galaxies hosting a compact central starburst will have a higher temperature than galaxies at the same ${L}_{\mathrm{IR}}$, which are disks (Chanial et al. 2007). In local LIRGs and ULIRGs, a compact starburst is triggered by a major merger, as the transfer of angular momentum funnels gas to the central kiloparsec of the merging system.

Stierwalt et al. (2013) classified the GOALS sample into merger classifications using HST $B \mbox{-}$, $I \mbox{-}$, and $H \mbox{-}$ band imaging (see also Haan et al. 2011) and IRAC 3.6 μm imaging. All 46 galaxies in the present work also have a merger classification. The merger classifications we use here are 0: no merger or massive neighbor; 1: galaxy pair; 2: early-stage (disk still intact); 3: mid-stage (amorphous disk, tidal tails); and 4: late-stage (two nuclei in common envelope). Morphology analysis of the Supersample is less straightforward due to our combining different surveys. GOODS-S and GOODS-N galaxies have been visually classified with HST WFC3 images as part of the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS; P.I. S. Faber & H. Ferguson). The CANDELS tool gives classifiers a choice of five different interaction stages, similar to those listed above. The morphology catalogs then contain the fraction of classifiers that selected each interaction stage (Kartaltepe et al. 2015). We use these fractions to determine a weighted average interaction stage, from 0 to 4 (see also Rosario et al. 2015, who create a weighted average on a 0–1 scale). Zamojski et al. (2011) classifies 134 xFLS sources into merger classifications using Hubble NICMOS imaging, from 0 to 5. Here, we combine old mergers (5) with advanced mergers (4), to match the categories described above. In total, 17 Supersample galaxies have dust masses and a merger classification. 5MUSES galaxies lack a morphology classification.

The top panel of Figure 7 shows the interaction stage as a function of ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$, and the points are shaded by ${L}_{160}^{\mathrm{SF}}/{L}_{70}^{\mathrm{SF}}$. There is no correlation between the parameters, for either GOALS or the Supersample. Galaxies at a given ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$ occupy all merger classifications, as do galaxies at a given ${L}_{160}^{\mathrm{SF}}/{L}_{70}^{\mathrm{SF}}$. DSFGs exhibit a variety of ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$ ratios and a variety of major merger classifications, indicating that the major merger scenario cannot directly explain all of the far-IR observed properties of these galaxies. In the same vein, using hydrodynamical simulations post-processed with dust radiative transfer, Lanz et al. (2014) found that the effective dust temperature of mergers does not reflect the merger stage except for a very short period ($\lesssim 100$ Myr) during the coalescence-induced starburst.

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The most direct test of how ISM geometry affects ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$ and ${T}_{\mathrm{dust}}$ is to compare the spatial extent of the dusty ISM. Measuring submillimeter sizes on a resolved scale at low and high redshift requires the sensitivity of ALMA. To date, only a handful of DSFGs have resolved ALMA imaging, but we can compare other tracers of the ISM until we have a larger ALMA sample. Rujopakarn et al. (2011) compiles from the literature ISM diameters for the GOALS sample measured from Paα, 8.4 Ghz, and CO (3-2). The authors also compile CO (3-2) and 1.4 GHz sizes for galaxies in the GOODS-N field. The diversity of measurements underscores the need for a homogeneous ALMA comparison. The bottom panel of Figure 7 compares ISM diameter with ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$. The GOALS DSFGs are all much smaller than the high-z DSFGs. However, this does not translate to different ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$ or ${L}_{160}^{\mathrm{SF}}/{L}_{70}^{\mathrm{SF}}$ ratios. This echoes the result in Lee et al. (2016), where the authors find that local dust-obscured galaxies, likely very compact, with ${L}_{\mathrm{IR}}={10}^{11}\mbox{--}{10}^{12}$ have similar dust masses and temperatures as their less obscured, more extended counterparts. Similarly, using SEDs output from hydrodynamical simulations, Martínez-Galarza et al. (2016) recently found that mergers and compactness alone are not able to reproduce the SEDs of local LIRGs, but that increasing the gas fraction was required.

However, without a large sample of measurements of galaxy size in the dust continuum (i.e., with ALMA), it is very difficult to distinguish between size and ${M}_{\mathrm{dust}}$ as the driver of the colder dust temperatures, since these effects are linked. Figure 7 clearly demonstrates that the high-z galaxies are larger. If a galaxy can be represented as a central heating source surrounded by a disk of dust, then naturally a more extended disk will produce a lower ${T}_{\mathrm{dust}}$, as the outskirts of the disk will see a diminished radiation field (e.g., Misselt et al. 2001). But, this is not a realistic description of most galaxies. Instead, to first order, both the SFR and dust densities are correlated with gas density, and so a mixed geometry of SFR regions within the ISM is more likely. In this geometry, the extent of the system does not affect the global effective dust temperature (Misselt et al. 2001; Safarzadeh et al. 2016). How size and increased dust/gas mass are linked is also difficult to unravel. Locally, compaction can lead to a more efficient transformation of atomic gas into molecular gas (Larson et al. 2016), so a decrease in size could actually be the trigger of increased molecular gas masses in many of the GOALS galaxies (Díaz-Santos et al. 2010a, 2010b). Alternately, a smaller size translates to a higher SFR surface density, which would boost ${L}_{\mathrm{IR}}^{\mathrm{SF}}$ without requiring a boost in ${M}_{\mathrm{gas}}$, leading to higher SFE at smaller size (Hayward et al. 2011, 2012). These effects are observed locally and in simulations, but our small sample of high-z galaxies do not appear to show any of the same trends between small size and higher temperatures or ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$. These galaxies also have optical radii measurements from the CANDELS collaboration, and we find no correlation between the optical radius measured in the observed frame H-band and ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$. On the other hand, Scoville et al. (2016) argue that high-z galaxies have more turbulent ISMs, leading to compression in the ISM and enhancing the SFE per unit mass. This efficient mode of star formation can occur throughout the galaxy, so no obvious correlation between galaxy size and dust temperature may be expected.

Local galaxies can be resolved, allowing their ISM geometry to be measured in more detail than high-redshift galaxies (e.g., Barcos-Muñoz et al. 2015). Given the quality and abundance of observations, it is more straightforward to link mergers with compact starbursts with dust heating in the GOALS samples (Díaz-Santos et al. 2010a). However, without resolved observations of the ISM in high-redshift galaxies, we see no obvious link between mergers, size, ${L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}}$, and ${T}_{\mathrm{dust}}$ in our sample. Therefore, we must conclude that a galaxy's global far-IR/submillimeter emission, parameterized through ${L}_{\mathrm{IR}}$, ${T}_{\mathrm{dust}}$, ${M}_{\mathrm{dust}}$, or ${f}_{\mathrm{gas}}$ may not tell observers anything about the ISM geometry, extent, or merger stage of that galaxy.

There are two likely explanations for the observed increase in ${M}_{\mathrm{dust}}$ with redshift. The first is that DSFGs at $z\sim 1\mbox{--}2$ are simply more massive overall, that is, they have a higher stellar mass. M_* is expected to broadly scale with ${M}_{\mathrm{dust}}$, since a higher M_* in a DSFG implies a higher metallicity, supplying more metals to form dust in the ISM.

In Figure 8, we compare ${M}_{\mathrm{dust}}$ with M_*. The three samples span the same range of $\mathrm{log}{M}_{* }\sim 10\mbox{--}11.5\,{M}_{\odot }$. However, the Supersample galaxies have roughly an order of magnitude higher dust masses. The solid, dotted, and dashed lines indicate how the relationship ${M}_{\mathrm{dust}}$ versus M_* is predicted to evolve with redshift using the semi-analytic models of Popping et al. (2016). We find an increase in the ${M}_{\mathrm{dust}}$ with redshift larger than predicted by those models. The Supersample is offset from the GOALS galaxies by $\sim 0.7$ dex, which is the same amount of offset seen in the bottom panel of Figure 5. The Popping et al. (2016) models shown in our comparison are based on a semi-analytic model of galaxy formation in a cosmological context, which includes a standard suite of physical processes (gas accretion and cooling, star formation, stellar feedback, chemical enrichment, etc.). In addition, the Popping et al. (2016) model includes self-consistent tracking of the main processes thought to produce and destroy dust in galaxies, including dust condensation in stellar ejecta, dust growth through accretion in the ISM, dust destruction by supernovae, and ejection of dust by stellar-driven winds. The much milder evolution of ${M}_{\mathrm{dust}}/{M}_{* }$ with redshift predicted by these models relative to our findings is interesting, as it indicates that one or more of the model ingredients need to be revised.

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The main sequence, which is the relationship between SFR and M_*, evolves with redshift (e.g., Whitaker et al. 2012). That is, for a given M_*, galaxies had a higher SFR at $z\sim 1\mbox{--}2$ than today. A higher SFR can be tied to an increase in dust mass indirectly, as the majority of grain growth is predicted to occur in the ISM, and a more gas-rich ISM will lead to higher dust masses and higher SFRs (e.g., Dwek 1998; Draine 2003; Santini et al. 2014; McKinnon et al. 2016a). The increased dust mass in the Supersample might then be a natural consequence of the increase in gas fractions with lookback time. As mentioned above, submillimeter data can also be converted to ${M}_{{{\rm{H}}}_{2}}$, and we use this parameterization to calculate gas fractions ${f}_{\mathrm{gas}}={M}_{{{\rm{H}}}_{2}}/({M}_{* }+{M}_{{{\rm{H}}}_{2}}$). These gas fractions are consistent with what we derive using CO observations for several 5MUSES and Supersample galaxies (Yan et al. 2010; Kirkpatrick et al. 2014b).

We also calculate the distance of these galaxies from the main sequence. The main sequence evolves with redshift, and it flattens at higher M_*. Therefore, we follow the method in Scoville et al. (2017) to calculate the main sequence. We use the relationship between M_* and SFR parameterized in Lee et al. (2015) at z = 1.2:

$$\begin{eqnarray}&&\mathrm{log}\,{\mathrm{SFR}}_{\mathrm{MS}}=1.72-\mathrm{log}\left[1+{\left(\displaystyle \frac{{M}_{* }}{2\times {10}^{10}{M}_{\odot }}\right)}^{-1.07}\right],\end{eqnarray} \tag{ 9 }$$

and then we renormalize depending on redshift (Speagle et al. 2014):

$$\begin{eqnarray}&&{\mathrm{SFR}}_{\mathrm{MS}}(z)={\left(\displaystyle \frac{1+z}{1+1.2}\right)}^{2.9}\times {\mathrm{SFR}}_{\mathrm{MS}}(z=1.2).\end{eqnarray} \tag{ 10 }$$

Then, we calculate $\mathrm{sSFR}=\mathrm{SFR}/{M}_{* }$ for every source and for the main sequence (MS) at each redshift.

We plot ${f}_{\mathrm{gas}}$ versus ${\rm{\Delta }}\mathrm{sSFR}=\mathrm{sSFR}/{\mathrm{sSFR}}_{\mathrm{MS}}$ in the top panel of Figure 9. The Supersample is offset relative to the low-z samples, so that at a given ΔsSFR, the Supersample sources have higher gas fractions, consistent with previous results in the literature (Magdis et al. 2012; Genzel et al. 2015). This means that massive galaxies at $z\sim 1.2$ have $4\mbox{--}5\times$ higher gas fractions and hence gas masses, since the stellar masses are roughly consistent. Scoville et al. (2017) parameterize the evolution of the molecular gas mass as a function of redshift, M_*, and ΔsSFR. In particular, the authors find that ${M}_{{{\rm{H}}}_{2}}$ evolves as ${(1+z)}^{1.84}$. For $\mathrm{log}{M}_{* }=10.7\,{M}_{\odot }$, and z = 0 to z = 1.2, we calculate the gas masses based on Equation (6) in Scoville et al. (2017) and overplot these predictions as the dashed and dotted–dashed lines. The predicted redshift evolution of ${(1+z)}^{1.84}$ is consistent with our measurements, given the uncertainties in our stellar masses. This redshift evolution is best interpreted by also considering SFE, SFE = SFR/${M}_{{{\rm{H}}}_{2}}$. In the case of a constant dust to gas ratio, $\mathrm{SFE}\,\propto \,{L}_{\mathrm{IR}}^{\mathrm{SF}}/{M}_{\mathrm{dust}};$ massive dusty galaxies show little variation in metallicity, even at $z\sim 1\mbox{--}2$, so a constant dust to gas ratio is a fair assumption (Magdis et al. 2012). We plot SFE versus ΔsSFR in the bottom panel of Figure 9. The obvious correlation (Kendall's $\tau =0.45$) is similar to the well-established dependence of the gas depletion timescale (${t}_{\mathrm{dep}}={\mathrm{SFE}}^{-1}$) on sSFR (Saintonge et al. 2011; Magdis et al. 2012; Tacconi et al. 2013; Huang & Kauffmann 2014; Sargent et al. 2014). We overplot the relation derived in Genzel et al. (2015), where the authors use CO observations to calculate molecular gas mass for 500 SFGs from $z=0\mbox{--}3$. There is no offset in any of our samples, consistent with the very mild redshift evolution between ${t}_{\mathrm{dep}}$ and ΔsSFR found by Genzel et al. (2015). However, this is in contrast to Scoville et al. (2017), who measure an increase in $\mathrm{SFE}\propto {(1+z)}^{1.05}$ for a given stellar mass. We plot what this evolution looks like using the arrow in the bottom-right corner for z = 0 to z = 1.2, and $\mathrm{log}{M}_{* }=10.7\,{M}_{\odot }$.

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The clear increase in gas fractions with redshift, without corresponding increases in SFE, suggests that the higher SFRs (corresponding to higher ${L}_{\mathrm{IR}}$) and higher ${M}_{\mathrm{dust}}$ of the Supersample at $z\sim 1\mbox{--}2$ are driven by an increase in the gas supply rather than an increase in SFE or M_*, or a changing mode of star formation with cosmic time (Bouché et al. 2010; Davé et al. 2012; Tacconi et al. 2013; Genzel et al. 2015; Schinnerer et al. 2016). In fact, the bottom panel of Figure 5 demonstrates that at a given ${L}_{\mathrm{IR}}^{\mathrm{SF}}$ (SFR), SFEs are lower at $z\sim 1\mbox{--}2$. This increase in gas supply is also predicted by the underlying cause of the main-sequence evolution, SFR/M_*, with redshift (Bouché et al. 2010; Davé et al. 2012; Tacconi et al. 2013; Genzel et al. 2015). We can directly test whether an increase in ${f}_{\mathrm{gas}}$ fully explains the increase in ${M}_{\mathrm{dust}}$ using the relationship between ${f}_{\mathrm{gas}}$ and z, parameterized as ${f}_{\mathrm{gas}}=0.1{(1+z)}^{2}$ (Geach et al. 2011). If we assume a dust-to-gas ratio of 100 and $\mathrm{log}{M}_{* }=10.7\,{M}_{\odot }$, then at z = 0, $\mathrm{log}{M}_{\mathrm{dust}}=7.75\,{M}_{\odot }$, and at z = 1.2, $\mathrm{log}{M}_{\mathrm{dust}}=8.67\,{M}_{\odot }$. These predictions are plotted as the dark filled symbols in Figure 8, and they agree remarkably well with GOALS and the Supersample.

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