The IR Compactness of Dusty Galaxies Sets Star Formation and Dust Properties at z ∼ 0–2

Mid-IR spectroscopy of galaxies is key for decomposing the IR spectral energy distribution (SED) into the components powered by AGNs versus star formation. Nuclear toroidal dust heated to high temperatures by buried AGNs emits strongly in the mid-IR (e.g., Laurent et al. 2000; Sturm et al. 2000; Tran et al. 2001), whereas star-forming regions are bright in broad PAH emission features and exhibit relatively shallower mid-IR spectral indices (Allamandola et al. 1989; Sajina et al. 2007; Pope et al. 2008). Kirkpatrick et al. (2012) decomposed the mid-IR spectra for a large sample by fitting power-law and star-forming templates to calculate the λ_rest ∼ 5–12 μm AGN fraction (f_AGN,MIR), defined as the fraction of emission within the Spitzer/IRS bandpass (L_MIR) attributed to an obscured AGN such that f_MIR,AGN = L_MIR,AGN/L_MIR (Pope et al. 2008). Following Kirkpatrick et al. (2015), we distinguish between three general f_AGN,MIR categories: star-formation-dominated galaxies (SFG; f_AGN,MIR < 20%), composite galaxies with an intermediary balance between star formation and AGNs (COM; 20% < f_AGN,MIR < 80%), and AGN-dominated galaxies (f_AGN,MIR > 80%). Mid-IR AGNs with f_AGN,MIR > 80% exhibit a warmer SED, but the average dust temperature (T_d ) of the cold component powered by star formation is remarkably constant at all f_AGN,MIR around T_d ∼ 25 K (Kirkpatrick et al. 2015). To test the dust-obscured star formation and dust mass in galaxies over a range of buried AGN strength, we do not select on any f_AGN,MIR threshold. Rather, we use f_AGN,MIR to correct total IR luminosities for the relative contribution from AGNs and star formation. Using Equation (5) of Kirkpatrick et al. (2015), we first convert the mid-IR AGN fraction to a bolometric IR AGN fraction (f_AGN,IR). Next, we determine the IR luminosity attributed to star formation (L_IR,SF) using L_IR,SF = (1 − f_AGN,IR) × L_IR.

Given the unique constraint on dust-obscured AGNs and star formation provided by mid-IR spectroscopy, we select our initial sample of galaxies on the basis of existing Spitzer/IRS spectra. Specifically, Kirkpatrick et al. (2012, 2015) present a parent sample of 151 (ultra-) LIRGs: $\mathrm{log}{L}_{\mathrm{IR}}/{L}_{\odot }$ > 11 (ULIRGs: $\mathrm{log}{L}_{\mathrm{IR}}/{L}_{\odot }$ > 12) at z ∼ 2 with Spitzer/IRS spectra. The original "supersample" includes galaxies in the Great Observatories Origins Deep Survey North/South (GOODS-N/S) and is representative of Herschel+Spitzer colors of S_24μm > 0.1 mJy galaxies with >3σ 250 μm detections (Kirkpatrick et al. 2012; Sajina et al. 2012). Our next selection criterion is on sources that can be observed by ALMA due to their location in the sky, which narrows the candidates from 151 (U)LIRGs across the GOODS fields to 81 (U)LIRGs in GOODS-S. We search for ALMA detections for these 81 GOODS-S galaxies. Galaxies from each f_AGN,MIR classification (i.e., SFG, COM, AGN) can be found spanning the redshift and L_IR range of the parent sample (Kirkpatrick et al. 2015).

We use the following methods to search for ALMA counterparts to the 81 Spitzer targets in GOODS-S. We search through the ALMA archive, which includes several large surveys, namely ASAGAO (Hatsukade et al. 2018) and GOODS-ALMA 2.0 (Gómez-Guijarro et al. 2022b). ASAGAO contains a smaller subset of the GOODS-S field than GOODS-ALMA 2.0 but has greater sensitivity. We take as many detections out of ASAGAO, then move to GOODS-ALMA 2.0, and then search the archive for sources not detected in either of the large surveys. We homogenize all ALMA images used in this work to the same imaging parameters, namely, we adopt natural weighting and do not uv taper the data.

For a given 20'' × 20'' ALMA cutout taken from the archive or the aforementioned large survey maps and centered on the Spitzer coordinates, we first derive a local rms after masking potential source emission. Next, we find all peaks above 2.5σ, which we use as priors to create a segmentation map using photutils.v1.4 (Bradley et al. 2020) detect_sources with a Gaussian smoothing kernel while enforcing a minimum number of 5 connected pixels, typically less than the number of pixels across the beam FWHM and suitable for flagging spatially unresolved and resolved candidates. We then compare the Spitzer/IRAC Channel 4 (IRAC4) coordinates against each source found in the segmentation map, and we take the closest match within 1'' for further analysis. As a final check, we next overlay the ALMA contours on top of IRAC4 and near-IR imaging from either JWST/NIRCam F_150W (JADES; Eisenstein et al. 2023; JADES Team 2023; Rieke & the JADES Collaboration 2023) or the Hubble Space Telescope (HST)/WFC3 F_160W (3D-HST; Grogin et al. 2011; Koekemoer et al. 2011; Skelton et al. 2014) to visually confirm the association. When comparing between ALMA and HST, we correct for a global astrometry offset in GOODS-S of 009 in R.A. and 026 in decl. (Elbaz et al. 2018; Franco et al. 2018). We consider all targets with ≥3σ contours coincident with the IRAC4 coordinates as candidate ALMA detections for our sample.

Our primary goal is to measure IR sizes from the archival ALMA data to constrain the extent of dust-emitting regions in our sample. As discussed in Gómez-Guijarro et al. (2022b) and Franco et al. (2018, 2020), this requires a continuum peak pixel signal-to-noise ratio S/N ≥ 5. Therefore, of our candidate archival matches to our sample, we only consider detections with S/N_peak > 5 in our analysis.

In summary, of the 81 Spitzer targets in GOODS-S, we find 23 candidate matches detected in both or either of ASAGAO and GOODS-ALMA 2.0. Of these, 10 are of sufficient S/N to measure an IR size. From searching the archive for observations within 1'' of our Spitzer targets, we find 10 more observations with ALMA coverage over our sample, of which seven correspond to targets not already detected in the ASAGAO and/or GOODS-ALMA 2.0 maps. Of these seven new matches, five have S/N > 5, sufficient to measure the IR size, and come from the following ALMA programs: 2017.1.01347.S (PI: A. Pope; see McKinney et al. 2020) and 2018.1.00992.S (PI: C. Harrison; see Lamperti et al. 2021). Our final sample with measurements of the submillimeter/millimeter flux and IR size consists of 15 galaxies (10 from ASAGAO+GOODS-ALMA 2.0, five from targeted programs). We tabulate the general properties of each galaxy and ALMA-derived quantities in Table 1. Near-IR image cutouts with ALMA contour overlays are shown in Figure 10.

Table 1. Source Characteristics for Spitzer/IRS Targets Matched to Archival ALMA Observations

ID	R.A.	Decl.	za	$\mathrm{log}\,{{\rm{L}}}_{\mathrm{IR}}$	$\mathrm{log}\,{{\rm{M}}}_{* }$	fAGN	$\mathrm{log}\,{{\rm{L}}}_{6.2\mu m}$	λobs	Sν,int	Sν,peak	Reff	$\mathrm{log}{M}_{\mathrm{dust}}$	Reference
	(J2000)	(J2000)		(L⊙)	(M⊙)		(L⊙)	(mm)	(mJy)	(mJy beam−1)	(kpc)	(M⊙)
GS IRS1	03:32:44.00	−27:46:35.0	2.69	12.69	10.95	39	9.62 ± 0.18	1.233	1.40 ± 0.15	0.89 ± 0.03	0.62 ± 0.16	8.80 ± 0.03	1
GS IRS15	03:32:40.74	−27:49:26.0	2.11	12.17	10.78	39	⋯	1.131	0.76 ± 0.30	0.52 ± 0.10	1.66 ± 0.40	8.44 ± 0.17	2
GS IRS20	03:32:47.58	−27:44:52.0	1.91	12.60	10.77	25	9.90 ± 0.18	1.233	0.89 ± 0.11	0.53 ± 0.05	0.62 ± 0.19	8.61 ± 0.06	1
GS IRS23	03:32:17.23	−27:50:37.0	1.96	12.35	10.99	0	9.11 ± 0.99	1.131	1.41 ± 0.35	1.01 ± 0.15	1.78 ± 0.25	8.71 ± 0.11	2
GS IRS33	03:32:23.43	−27:42:55.0	2.14	12.30	10.75	95	9.10 ± 0.10	0.872	0.50 ± 0.09	0.18 ± 0.02	1.45 ± 0.34	8.00 ± 0.08	3
GS IRS45	03:32:17.45	−27:50:03.0	1.62	12.48	10.39	6	10.19 ± 0.06	1.131	1.05 ± 0.24	0.81 ± 0.09	<3.10	8.46 ± 0.06	2
GS IRS46	03:32:42.71	−27:39:27.0	1.85	12.32	10.59	0	10.30 ± 0.08	0.456	9.10 ± 1.50	6.64 ± 0.61	1.96 ± 1.02	8.70 ± 0.16	4
GS IRS50	03:32:31.52	−27:48:53.0	1.90	12.01	10.82	28	9.84 ± 0.19	1.233	0.07 ± 0.03	1.00 ± 0.06	<0.64	8.32 ± 0.12	1
GS IRS52	03:32:12.52	−27:43:06.0	1.79	12.11	10.43	15	9.62 ± 0.26	0.444	8.65 ± 0.96	5.21 ± 0.49	1.53 ± 0.85	8.50 ± 0.25	4
GS IRS58	03:32:40.24	−27:49:49.0	1.85	12.06	10.86	7	9.51 ± 0.31	0.456	12.20 ± 3.30	4.86 ± 0.93	3.25 ± 1.53	<8.35	4
GS IRS60	03:32:40.05	−27:47:55.0	2.02	12.46	10.88	23	10.12 ± 0.16	1.233	0.65 ± 0.16	0.14 ± 0.02	3.64 ± 0.28	8.48 ± 0.10	1
GS IRS61	03:32:43.45	−27:49:01.0	1.77	12.13	10.69	15	10.06 ± 0.06	0.441	6.70 ± 1.30	3.87 ± 0.54	1.11 ± 0.85	<7.95	4
GS IRS70	03:32:27.71	−27:50:40.6	1.10	11.99	10.78	23	⋯	1.131	0.37 ± 0.14	0.35 ± 0.06	<2.63	8.05 ± 0.10	2
GS IRS73	03:32:43.24	−27:47:56.2	0.67	11.26	10.47	0	8.66 ± 0.09	1.131	0.68 ± 0.24	0.58 ± 0.10	1.30 ± 0.31	8.20 ± 0.15	2
GS IRS81	03:32:38.49	−27:46:31.9	2.55	12.75	10.34	38	9.90 ± 0.40	1.233	0.60 ± 0.12	0.38 ± 0.02	0.58 ± 0.33	8.43 ± 0.03	1

Notes. Columns: (z) Spectroscopic redshifts derived from fits to the broad PAH features detected in mid-IR Spitzer/IRS spectra following Appendix A of McKinney et al. (2020). The typical uncertainty on IRS-derived redshifts is Δz ∼ 0.02. (L_IR) Total IR luminosities derived by fitting Spitzer and Herschel photometry from Kirkpatrick et al. (2012), with systematic uncertainties of ∼10%. (M_*) Stellar masses originally calculated from optical/near-IR photometry assuming a Salpeter IMF (Kirkpatrick et al. 2012), which we have corrected here to a Chabrier IMF. (f_AGN,MIR) Mid-IR AGN fractions calculated by fitting a star-forming and power-law (AGN) template to the IRS spectra (Pope et al. 2008; Kirkpatrick et al. 2012). (L_6.2μm) The luminosity of the 6.2 μm PAH feature measured from fits to the IRS spectra following Appendix A of McKinney et al. (2020). (λ_obs) Observed continuum effective wavelength. (S_ν,int) Source-integrated ALMA flux. (S_ν,peak) Peak continuum ALMA flux. (R_eff) Effective radius containing half of the total integrated flux. (M_dust) Dust mass derived using Equation (1). (Ref) ALMA program from which properties are derived: 1 = ASAGAO (Ueda et al. 2018), 2 = GOODS-ALMA (Gómez-Guijarro et al. 2022b), 3 = 2018.1.00992.S (Lamperti et al. 2021), 4 = 2017.1.03147.S (McKinney et al. 2020).

^a For these objects only detected in ALMA Band 9 (λ_obs ∼ 450 μm), we use single-dish dust mass estimates from Kirkpatrick et al. (2017) where possible. Otherwise, we place 3σ upper limits using local noise properties derived from the target's position within the ASAGAO map.

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Due to the nature of untargeted archival observations at IR wavelengths, we expect our final ALMA-detected sample to be biased toward higher L_IR. To test for such bias, we compare the subset of galaxies with robust flux and size measurements from the ALMA archive against sources covered by archival observations but with no detectable signal (Figure 1). The mean L_IR of our final ALMA-detected catalog is ∼0.2 dex greater than that of the whole GOODS-S sample, and ∼0.5 dex greater than the ALMA nondetections. We detect 84% of all $\mathrm{log}{L}_{\mathrm{IR}}/{L}_{\odot }$ ≥ 12 candidates with coverage in archival observations. While our final catalog is not, on average, representative of L_IR and z in GOODS-S 24 μm-selected galaxies (Sajina et al. 2012; Kirkpatrick et al. 2015), it does span the range of both quantities. As shown in the bottom panel of Figure 1, we do not preferentially detect any particular mid-IR AGN classification. Most importantly, the ALMA archival detection criterion does not impose a bias on the distribution in specific star formation rates (sSFRs) relative to the main sequence, as shown in Figure 2. Roughly 66% of the galaxies in our final ALMA-detected sample are starbursts (sSFR/sSFR_MS > 3.5; Puglisi et al. 2021), comparable to the starburst fraction among the parent sample and ALMA nondetections.

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2.2.1. Atacama Large Millimetre/Submillimetre Array versus Near-IR Morphology and Offsets

2.3.1. Dust Mass

Eleven of the archival ALMA observations span a range in wavelength between 870 and 1250 μm, which probes the Rayleigh–Jeans (RJ) tail of cold dust emission over the range in redshifts spanned by our sample (∼260–550 μm between z ∼ 1–3). This regime is aptly suited to measuring the total dust mass because the emission is optically thin at submillimeter wavelengths (Scoville et al. 2014) and the temperature dependence is linear, meaning uncertainties on the mass-weighted dust temperature (T_d ) have modest impact on the total dust mass (Scoville et al. 2016, 2017a). Following Kirkpatrick et al. (2017), we use the ALMA flux densities (S_ν ) to measure the dust mass using

$$\begin{eqnarray}&&{M}_{\mathrm{dust}}=\displaystyle \frac{{S}_{\nu }{D}_{L}^{2}}{{\kappa }_{\nu }{B}_{\nu }({T}_{d})},\end{eqnarray} \tag{ 1 }$$

where D_L is the luminosity distance, B_ν is the Planck equation, and κ_ν is the dust opacity from Weingartner & Draine (2001) assuming Milky Way (MW)–like dust and R_V = 3.1. ¹² As noted by Kirkpatrick et al. (2017), the variation in κ_ν at longer wavelengths is negligible across common models (e.g., MW, SMC, LMC) and for different R_V . We choose to fix the cold dust temperature to T_d = 25 K because most of the dust is cold with a temperature remarkably constant over (i) the full range of mid-IR AGN fractions when the SED is decomposed into its AGN- and star-formation-powered components using mid-IR spectroscopy (Kirkpatrick et al. 2015; Scoville et al. 2017a), and (ii) redshift for fixed L_IR (Drew & Casey 2022).

Four of the archival ALMA targets (GS IRS 46, 52, 58, and 61) are detected by ALMA at wavelengths below λ_rest ∼ 250 μm. For two of these targets (GS IRS 46 and 52), we use the dust masses derived using 870 μm APEX/LABOCA photometry from Kirkpatrick et al. (2017) under the same assumptions and formula as used for longer-wavelength detections. Two final sources (GS IRS 58 and 61) do not have submillimeter observations along the RJ tail, in which case we place upper limits on the total dust mass using the 3σ rms derived from their positions within the ASAGAO map where they are not detected. All dust masses are listed in Table 1.

Kirkpatrick et al. (2017) present an analysis of the dust masses of galaxies selected from the Kirkpatrick et al. (2015) supersample on the availability of submillimeter/millimeter single-dish photometry. Seven of the galaxies for which we find ALMA archival matches also have single-dish submillimeter detections in Kirkpatrick et al. (2017), which we use to test for systematic differences in the dust mass measurements from single-dish detections and the ALMA interferometer. Flux densities in confusion-limited submillimeter observations are often boosted by the unresolved background as steeply rising source number counts preferentially scatter flux densities upwards (e.g., Hogg & Turner 1998; Scott et al. 2002; Simpson et al. 2015). Indeed, we find that single-dish-derived dust masses tend to be greater than those derived using the ALMA observations by ∼25%–50%, but both agree within 1σ.

2.3.2. Flux Densities and Dust Sizes

The parent sample from which our targets are selected from have robust multiwavelength photometry and mid-IR spectroscopy from Spitzer/IRS. A full description of the IRS observations can be found in Pope et al. (2008) and Kirkpatrick et al. (2012), and a comprehensive discussion of the ancillary Herschel (PACS and SPIRE) and Spitzer (IRAC and MIPS) photometry is presented in Kirkpatrick et al. (2015). We use the stellar masses derived for our sample in Kirkpatrick et al. (2012), who fit 10 optical/near-IR bands between U − 4.5 μm with a composite stellar population-synthesis code assuming an exponentially declining star formation history (Drory et al. 2004, 2009). Kirkpatrick et al. (2012) assume a Salpeter IMF for their stellar masses, which we convert to a Chabrier framework following Kirkpatrick et al. (2017; ${M}_{* }^{\mathrm{Cha}}=0.62{M}_{* }^{\mathrm{Sal}};$ Speagle et al. 2014). Total IR luminosites are derived from fits to Spitzer/MIPS and Herschel/PACS+SPIRE photometry (Kirkpatrick et al. 2012). The Appendix of McKinney et al. (2020) provides a detailed description of how PAH luminosities and spectroscopic redshifts are derived for our sample using a custom Markov Chain Monte Carlo fitting routine. The GOODS-S targets are within the coverage of 3D-HST, which provides deep WFC3/IR imaging (Brammer et al. 2012; Momcheva et al. 2016). Key derived properties from the ancillary data are listed in Table 1. In summary, galaxies in our sample have L_IR in the range of 10^11.6–10^12.8 L_⊙, stellar masses between ∼10¹⁰–10¹¹ M_⊙, and redshifts from z ∼ 0.7–2.7.

We compare our data against local galaxies in the Great Observatories All Sky LIRG Survey (GOALS; Armus et al. 2009), a 60 μm flux-limited sample of local LIRGs with multiwavelength data comparable to the coverage of our targets including Spitzer/IRS mid-IR measurements of PAH emission (Stierwalt et al. 2013, 2014), intrinsic IR sizes from Herschel/PACS at λ_obs = 160 μm (Lutz et al. 2016), and submillimeter photometry (Chu et al. 2017) from which we derive dust masses. Vivian et al (2012) also present dust masses derived from SED fitting; however, we choose to recalculate the total dust mass under the same assumptions and with the same method as applied to the high-redshift galaxies to avoid introducing systematic offsets (Kirkpatrick et al. 2017). We use 850 μm photometry from the James Clerk Maxwell Telescope where possible to measure the dust mass, and 500 μm Herschel/SPIRE 500 μm photometry otherwise (Chu et al. 2017). Dust masses derived from both agree within ∼20% on average. The Spitzer/IRS SL slit usually traces the nuclear region in GOALS (Stierwalt et al. 2013). To estimate galaxy-integrated PAH luminosities in GOALS, we scale luminosity measurements of the PAHs made through the slit by the total-to-slit flux IRAC4 flux ratio derived in Stierwalt et al. (2014). Aperture corrections do not correlate with distance (Stierwalt et al. 2013) or the total dust mass. We use mid-IR AGN fractions in GOALS derived from the 6.2 μm equivalent width (Díaz-Santos et al. 2017), as these most closely resemble our method for measuring f_AGN,MIR at z ∼ 2, where the 6.2 μm PAH anchors the star-forming template during spectral decomposition (Pope et al. 2008).

To contextualize the measurements of PAHs in the ISM for both GOALS and our high SFR targets with the population of more normal star-forming galaxies, we compare against galaxies from the KINGFISH survey (Kennicutt et al. 2011), a sample of nearby (D < 30 Mpc) galaxies spanning a range in SFR between 0.001–7 M_⊙ yr⁻¹. The sizes of galaxies in KINGFISH have been reported at optical and FUV wavelengths (Dale et al. 2007; Kennicutt et al. 2011), but not in the far-IR. Therefore, we download the Herschel/PACS160 maps of KINGFISH targets (Dale et al. 2012) from the NASA/IPAC Infrared Science Archive (KINGFISH Team 2020) and perform a simple aperture-based measurement to derive the effective radii containing 50% of the 160 μm flux. KINGFISH was designed to overlap with existing Spitzer/IRS spectroscopy from the SINGS program (Kennicutt et al. 2003), which we use to extend our analysis of dust to low Σ_IR using PAH line fluxes presented in Smith et al. (2007). We scale the PAH luminosities measured through the IRS slit by the ratio of total L_IR to L_IR measured through the slit (Smith et al. 2007), an approximate aperture correction assuming the extent of PAHs follows the cold dust continuum (e.g., Bendo et al. 2008; Calapa et al. 2014; Gregg et al. 2022). We note that none of the quantities we derive for KINGFISH correlate with distance or the adopted aperture correction. Finally, we measure dust masses in KINGFISH using Herschel/SPIRE 500 μm photometry (Dale et al. 2012, 2017) under the same assumptions made for the other data sets.

PAH line fluxes for GOALS and KINGFISH are measured using spectral decomposition methods, e.g., PAHFIT (Smith et al. 2007) and CAFE (Marshall et al. 2007). Owing to the lower S/N of the z ∼ 2 IRS spectra, we measure PAH line fluxes using a spline continuum fit (Sajina et al. 2007; Pope et al. 2008; McKinney et al. 2020). Smith et al. (2007) demonstrate that the 6.2 μm PAH line luminosities measured with these two techniques differ by a factor of 1.6, owing to where the continuum is drawn. Therefore, we scale L_PAH,6.2 in GOALS and KINGFISH down by a factor of 1.6 to match our spline-derived PAH luminosities at higher redshift. Omitting this scale factor does not change the results of our analysis, as the empirical scatter in L_PAH,6.2 within GOALS, KINGFISH, and z ∼ 2 (U)LIRGs is greater than 60% of the median.

The dust mass is dominated by large grains, which also dominate the far-IR emission, whereas PAHs populate the smaller end of the grain size distribution and emit strong mid-IR features (Draine & Li 2001). Despite this size difference, mid- and far-IR emission tracing the PAHs and cold dust, respectively, is correlated with the spatial extent of star formation in galaxies (e.g., Kirkpatrick et al. 2014; Gregg et al. 2022), which is, among other reasons, why PAHs have been commonly used to trace dust-obscured SFRs (e.g., Genzel et al. 1998; Peeters et al. 2004; Wu et al. 2005; Lutz et al. 2007; Pope et al. 2008). In Figure 3, we show the ratio of L_PAH to total dust mass to empirically trace the PAH mass fraction in the ISM (q_PAH; Draine & Li 2001), which is otherwise commonly inferred in the literature by fitting dust model grids to SEDs (e.g., Draine & Li 2007; Aniano et al. 2020). We use only the 6.2 μm PAH feature because it is isolated from adjacent lines and distant from the strong silicate absorption, making it the cleanest PAH line to measure in low S/N Spitzer/IRS spectra. We find no correlation between L_PAH/M_dust and Σ_IR,SF for (U)LIRGs (p = 0.2), and that all galaxies included in this analysis scatter around an average L_PAH to total dust mass ratio of $\mathrm{log}\,{{L}}_{\mathrm{PAH}(6.2\,\mu {\rm{m}})}/{{M}}_{\mathrm{dust}}\sim 1.2\pm 0.3\,{{L}}_{\odot }/{{M}}_{\odot }$ (Table 2). This includes galaxies with intermediate to strong dust-obscured AGNs, for which spatially resolved JWST/MIRI observations have shown to host strong PAH emission remarkably close to the AGN (Lai et al. 2022). We do find a positive correlation between L_PAH/M_dust in KINGFISH (r_p , p = 0.63, 4.6 × 10⁻⁵) with Σ_IR,SF, which could be driven by the higher metallicities found for warmer, high Σ_IR,SF KINGFISH galaxies because of the increasing trend between PAH mass fraction and metallicity (Aniano et al. 2020). The intensity of PAH emission is also a function of metallicity at z ∼ 1–2 (Shivaei et al. 2017), and therefore comparable L_PAH/M_dust ratios for GOALS, z ∼ 2 (U)LIRGs, and KINGFISH galaxies with $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ > 8.5 may arise from their similar gas-phase metallicities. Given the similar masses of the (U)LIRGs and the mass–metallicity relation, we do not expect any effects of metallicity on the PAH emission for the purposes of comparing GOALS and our z ∼ 2 sample. In any case, the scatter about the best-fit trend in KINGFISH galaxies is substantially lower than for (U)LIRGs at z ∼ 0−2, which could justify the use of PAH emission to trace the total dust mass at $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ < 9.5 if Σ_IR,SF is known. Further studies of high-redshift, low Σ_IR,SF galaxies are needed to test if this correlation holds at earlier cosmic epochs.

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Table 2. Best-fit Parameters and Their Uncertainties for Linear Fits to Data

x	y	m	b	σ	(rp , p)	Sample
$\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$	$\mathrm{log}\ {L}_{6.2}/{M}_{\mathrm{dust}}\,[{L}_{\odot }/{M}_{\odot }]$	0.39 ± 0.08	−2.2 ± 0.7	0.13	(0.63, 4.6 × 10−5)	1
$\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$	$\mathrm{log}\ {L}_{6.2}/{M}_{\mathrm{dust}}\,[{L}_{\odot }/{M}_{\odot }]$	⋯	1.19	0.33	(0.13, 0.20)	2,3
Reff,FIR [kpc]	$\mathrm{log}\ {L}_{\mathrm{IR},\mathrm{SF}}/{M}_{\mathrm{dust}}\,[{L}_{\odot }/{M}_{\odot }]$	−0.15 ± 0.02	3.87 ± 0.04	0.18	(−0.65, 10−15)	1,2
$\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$	$\mathrm{log}\ {L}_{\mathrm{IR},\mathrm{SF}}/{M}_{\mathrm{dust}}\,[{L}_{\odot }/{M}_{\odot }]$	0.28 ± 0.11	0.64 ± 1.2	0.14	(0.67, 0.03)	3
$\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$	$\mathrm{log}\ {L}_{\mathrm{IR},\mathrm{SF}}/{M}_{\mathrm{dust}}\,[{L}_{\odot }/{M}_{\odot }]$	0.25 ± 0.02	1.15 ± 0.18	0.16	(0.79, 10−26)	1,2,3
$\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$	$\mathrm{log}\ \mathrm{sSFR}/{\mathrm{sSFR}}_{\mathrm{MS}}$	0.33 ± 0.04	−2.2 ± 0.4	0.14	(0.67, 10−14)	2
$\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$	$\mathrm{log}\ \mathrm{sSFR}/{\mathrm{sSFR}}_{\mathrm{MS}}$	0.26 ± 0.08	−2.25 ± 0.9	0.09	(0.71, 9 × 10−3)	3
$\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$	$\mathrm{log}\ \mathrm{sSFR}/{\mathrm{sSFR}}_{\mathrm{MS}}$	⋯	0.64	0.27	⋯	3

Notes. Fits take the functional form y = mx + b. Columns: (x) Domain. (y) Range. (m) Slope. (b) y-intercept. (σ) 1σ dispersion about the best-fit trend. (r_p , p) Perason rank coefficient and corresponding p-value. (sample) Samples included in fit: (1) KINGFISH, (2) GOALS, (3) ALMA-detected (U)LIRGs at z ∼ 2 from this work. For fits showing no slope, the best-fit b and σ correspond to the y-column average and dispersion about the mean.

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Fujimoto et al. (2017) presents a systematic analysis of IR sizes using Cycles 0–3 ALMA programs, made public through the ALMA archive. Their blindly selected sample spans a range in SFRs between ∼10 and 1000 M_⊙ yr⁻¹, stellar masses between $\mathrm{log}\,{M}_{* }/{M}_{\odot }=10\mbox{--}11.5$, and z ∼ 0–6. We compare the sizes measured in our ALMA-detected sample to their 10σ-selected sample in Figure 4, as well as the distribution in sizes found for z ∼ 0 GOALS and KINGFISH galaxies. While we have already subtracted the AGN-powered component from L_IR using f_AGN,MIR in GOALS and z ∼ 2 (U)LIRGs, this has minimal impact on the L_IR,SF–R_eff relation because f_AGN,IR reaches a maximum of ∼0.5 for f_AGN,MIR ∼1 (Kirkpatrick et al. 2015). We have also divided the effective radii reported by Fujimoto et al. (2017), defined as R_eff = 0.826 × FWHM, by a factor of 1.6 for consistency with how we measure R_eff to be 0.5 × FWHM. (U)LIRGs from GOALS and our ALMA-detected sample have, on average, larger R_eff than the blindly selected sample of Fujimoto et al. (2017), and cluster about a SFR surface density about an order of magnitude smaller. Three composite galaxies in our sample are much closer to the locus of Fujimoto et al. (2017) than the rest, and these three also happen to have the highest ΔsSFMS and L_IR. This suggests that the offset in sizes between the two samples could be attributed to, in part, observational bias toward brighter objects that preferentially sit above the main sequence. Indeed, the Fujimoto et al. (2017) sample has a median L_IR ∼2× greater than galaxies in our sample.

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As shown in Figure 4, our far-IR size measurements are broadly consistent with the sizes measured for LIRGs and ULIRGs from z ∼ 0–6 spanning a range of SFR surface densities found at all redshifts between Σ_SFR = 1–1000 M_⊙ yr⁻¹ kpc⁻² assuming SFR_IR/[M_⊙ yr⁻¹] = 1.49 × 10⁻¹⁰ L_IR,SF/L_⊙ (Murphy et al. 2011). (U)LIRGs and ALMA-detected galaxies from this work and Fujimoto et al. (2017) are on average smaller in R_eff than KINGFISH galaxies, which tend not to host strong nuclear star formation as can be found in (U)LIRGs. No galaxy in our z ∼ 2 sample or in GOALS exceeds the theoretical Eddington limit of Σ_SFR ∼ 1000 M_⊙ yr⁻¹ kpc⁻² (Andrews & Thompson 2011) even if we omit the AGN correction to L_IR.

In Figure 5, we show the far-IR extent of the ALMA-detected sample, GOALS, and KINGFISH as a function of their total dust masses. Assuming a dust-to-gas mass ratio of 0.01 as is appropriate for massive z ∼ 0–3 galaxies (e.g., Rémy-Ruyer et al. 2014; Shapley et al. 2020; Popping & Péroux 2022; Shivaei et al. 2022), we find that most (U)LIRGs would fall along an average Σ_gas ∼ 1000 M_⊙ pc⁻² at z ∼ 0 and z ∼ 2, as expected from their large SFR surface densities (Kennicutt & Evans 2012). The fact that ${M}_{\mathrm{dust}}\propto {R}_{\mathrm{eff},\mathrm{IR}}^{2}$ is consistent with optically thin dust (Draine & Li 2007; Scoville et al. 2017a). We note that approximately 25% of GOALS galaxies at $\mathrm{log}\,{{\rm{M}}}_{\mathrm{dust}}/{{\rm{M}}}_{\odot }\lt 8$ have upper limits on their IR size, which could push the trend toward higher Σ_gas at low M_dust among the z ∼ 0 LIRG population.

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Popping et al. (2022) derived a simulated distribution in IR size by performing dust radiative transfer on galaxies from the Illustris TNG50 cosmological simulation (Nelson et al. 2019; Pillepich et al. 2019). We compare against their simulated IR sizes and dust masses in Figure 5, which represents galaxies on and above the star-forming main sequence in TNG50. The GOALS galaxies and most of our z ∼ 2 ALMA-detected sample lie above the star-forming main sequence, and fall within the parameter space in M_dust versus R_eff spanned by simulated galaxies with sizes 2σ below the main-sequence trend. Thus, TNG50 reproduces the size and dust mass parameter space observed in (U)LIRGs, but as an outlier population with smaller R_eff for fixed M_dust relative to main-sequence galaxies. While the size analysis of Popping et al. (2022) does not extend to z ∼ 0, the positive correlation betwall IR surfaceeen R_eff and M_dust we find in KINGFISH is comparable to the higher-redshift trends for main-sequence galaxies in the simulation (shaded regions on Figure 5).

Figure 6 shows the ratio of L_IR,SF to M_dust as a function of the effective IR radius, bringing together the quantities shown independently in Figures 4 and 5. The ratio of L_IR,SF to total dust mass is an empirical tracer of the star formation efficiency, which reflects the amount of star formation sustained by a galaxy given its total gas content. We find that the L_IR,SF/M_dust ratios for z ∼ 0 and z ∼ 2 (U)LIRGs are comparable at fixed R_eff, and anticorrelate with the IR size (r_p , p = −0.65, 10⁻¹⁵). The more spatially extended KINGFISH galaxies have, on average, lower L_IR,SF/M_dust than (U)LIRGs, but reach the lower range of the dustier galaxies below R_eff ∼ 2 kpc.

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3.2.1. Far-IR Size as a Function of Mid-IR Active Galactic Nucleus Fraction

Heavily dust-obscured AGNs can produce high IR surface densities as the nuclear torus is heated to high temperatures, emitting predominantly at mid-IR wavelengths. Whether or not colder dust emission emitting at far-IR wavelengths is also powered by AGNs remains uncertain and debated in the literature (e.g., Symeonidis et al. 2016; Scoville et al. 2017b; Stanley et al. 2017; Symeonidis 2017; Symeonidis & Page 2018; Shangguan et al. 2020; McKinney et al. 2021b). Direct heating by AGN and subsequent absorption/reemission of IR photons could increase the central concentration of IR emission and drive the galaxy-scale effective radii down (e.g., McKinney et al. 2021b; Lamperti et al. 2021). Some star-forming galaxies at high z do not necessarily exhibit systematically different far-IR sizes compared to submillimeter luminous quasars (e.g., Chen et al. 2021; Ansarinejad et al. 2022); however, prior works have not explicitly controlled for the AGN strength using mid-IR spectroscopy.

Using the Spitzer/IRS spectral decomposition between star-forming and AGN for our sample (Pope et al. 2008; Kirkpatrick et al. 2012) and GOALS (Stierwalt et al. 2013, 2014; Díaz-Santos et al. 2017), we show in Figure 7 the relationship between Σ_IR and R_eff with f_AGN,MIR from z ∼ 0–2. GOALS galaxies exhibit no systematic correlation between the far-IR size and f_AGN,MIR, consistent with the comparison against Palomar–Green QSOs in Lutz et al. (2016). The trend between f_AGN,MIR and R_eff could be stronger for the z ∼ 2 ALMA-detected sample in this work. The average R_eff decreases by a factor of ∼2 and the average Σ_IR increases by ∼1 dex between f_AGN,MIR = 0–0.4 for our sample. The highest-IR surface density sources are all moderate to strong AGNs, consistent with Fujimoto et al. (2017) and Franco et al. (2020), who find that X-ray AGNs are preferentially found toward greater Σ_SFR. Further spatially resolved far-IR observations at f_AGN,MIR > 0.4 are required to fully test the incidence of small R_eff at high f_AGN,MIR in high-redshift galaxies. Nevertheless, the most compact and highest-IR surface density sources in our sample all have measurable AGN contribution in the mid-IR.

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For fixed L_IR, galaxies at z ∼ 2 exhibit more PAH emission per unit L_IR than local galaxies (Pope et al. 2013). In a small sample of ALMA-selected Spitzer targets, McKinney et al. (2020) demonstrated that this offset disappears for fixed Σ_IR. We expand upon this prior result using a sample three times larger and sizes predominantly measured along the RJ tail of cold dust emission. Figure 8 shows the L_PAH/L_IR,SF ratio as a function of Σ_IR,SF at z ∼ 2, and for GOALS and KINGFISH. We find the same L_PAH/L_IR,SF ratios at high and low redshift for fixed Σ_IR,SF. Moreover, we recover an anticorrelation between L_PAH/L_IR,SF and Σ_IR,SF reminiscent of photometric measures of PAH emission in dusty galaxies (e.g., IR8; Elbaz et al. 2011), and the far-IR fine-structure line deficit observed in low- and high-z dusty galaxies (Díaz-Santos et al. 2017; Zanella et al. 2018; McKinney et al. 2020). PAHs and far-IR lines predominantly arise from photodissociation regions (PDRs) around sites of recent star formation for actively star-forming galaxies (Tielens & Hollenbach 1985; Malhotra et al. 1997, 2001; Tielens 2008; Beirão et al. 2012; Croxall et al. 2017; Díaz-Santos et al. 2017; Sutter et al. 2019), and thus the coincidence in their trends with respect to Σ_IR,SF favors physical interpretations local to the young, dusty star-forming regions that dominate the IR surface density. Indeed, the mean UV interstellar radiation field intensity impinging upon PDRs, G (measured in G₀ = 1.6 × 10⁻³ erg s⁻¹ cm⁻²; Habing 1968), relative to the neutral PDR density n_H, increases by 1 dex over $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR},\mathrm{SF}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ = 9−11 in GOALS as both [C ii]/L_IR and L_6.2μm/L_IR,SF fall (Díaz-Santos et al. 2017).

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The low L_PAH/L_IR,SF ratios at high Σ_IR,SF might indicate a change in the ISM conditions regulating the excitation of both far-IR lines and PAH emission within dusty and young star-forming regions (e.g., Díaz-Santos et al. 2017). The high G/n_H ratios found among GOALS at $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{IR}}/[{L}_{\odot }\,{\mathrm{kpc}}^{-2}]$ > 10.7 indicate that the average star-forming region sees a stronger radiation field, which can modify the PAH photoelectric heating efficiency (Bakes & Tielens 1994; Galliano et al. 2008; Tielens 2008) and lower the radiative coupling between stars and gas (McKinney et al. 2020, 2021a). Changes in the relative heating and cooling could lead to systematically high star formation efficiency if PAHs photoelectrically convert a lower fraction of energy from the stellar radiation field into gas temperatures (Hollenbach & Tielens 1999; McKinney et al. 2021a). Indeed, simulations that include variable photoheating laws find high photoelectric heating rates suppress star formation due to excess heating (Forbes et al. 2016; Inoguchi et al. 2020; Osman et al. 2020). Systematic changes in the photoelectric heating efficiency might leave imprints on the mid-IR spectra as the ionization and/or grain size distribution of PAHs is modified (Draine & Li 2001; Maragkoudakis et al. 2020). The overlap between GOALS and z ∼ 2 (U)LIRGs along canonical diagnostic plots of PAH grain properties suggests that the physical mechanisms observed at z ∼ 0 are likely in place and playing an important role at higher z (Figure 3; see also McKinney et al. 2020).

In Figure 9 (left) we show the L_IR,SF/M_dust ratio versus Σ_IR,SF. The L_IR,SF/M_dust ratio is an empirical tracer of the global star formation efficiency, which may systematically evolve with redshift (Scoville et al. 2017a). As stated, Kirkpatrick et al. (2017) demonstrated that, for fixed distance above the main sequence, the star formation efficiencies of (U)LIRGs fall along the same relation between z ∼ 0 and 2. We find a similarly redshift-independent result, where high IR luminosity surface density galaxies exhibit more efficient star formation. Combined with KINGFISH, which probes galaxies at lower SFRs and more extended sizes, the correlation persists linearly over ∼five orders of magnitude in Σ_IR,SF, with a comparatively shallow increase in star formation efficiency by one order of magnitude. Kirkpatrick et al. (2017) find no correlation between L_IR,SF/M_dust and the ISM extent measured from CO, radio, or Paα for a handful of high-z galaxies and local LIRGs in GOALS (Rujopakarn et al. 2011). As shown in Figure 6 and Table 2, we find a statistically significant correlation between L_IR,SF/M_dust and the uniformly measured effective radii from far-IR wavelengths in local galaxies, but not at z ∼ 2. A much stronger trend manifests at all redshifts using Σ_IR,SF, consistent with multiple studies in the literature demonstrating that luminosity surface densities more accurately reflect the gas and star formation conditions in galaxies rather than the total luminosity or size alone (e.g., Rujopakarn et al. 2011; Díaz-Santos et al. 2017; Elbaz et al. 2018; McKinney et al. 2020; Díaz-Santos et al. 2021).

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The star formation efficiencies are comparable between GOALS and the z ∼ 2 (U)LIRGs in this study; however, the samples are different in terms of their position relative to main-sequence star formation for their corresponding epochs (Lutz et al. 2016; Kirkpatrick et al. 2017). Using the main-sequence parameterization from Speagle et al. (2014), we show in Figure 9 (right) the ratio of each galaxy's specific star formation rate (sSFR ≡ SFR/M_*) to the main-sequence sSFR for the corresponding stellar mass and redshift against Σ_IR,SF.

GOALS galaxies are typically a factor of ∼10 above the main sequence compared to a factor of ∼4 in ALMA-detected z ∼ 2 (U)LIRGs. (U)LIRGs at z ∼ 2 span a range of nearly two orders of magnitude in Σ_IR,SF with a similar dispersion in distance from the main sequence, ∼0.2 dex, as found locally, albeit shifted down. High-z (U)LIRGs can exhibit high and low dust-obscured SFR surface densities for fixed position along the main sequence, whereas Σ_IR,SF in local (U)LIRGs correlates with distance from the main sequence (Elbaz et al. 2011; Lutz et al. 2016). Indeed, Gómez-Guijarro et al. (2022a) even find high star formation surface densities in main-sequence, millimeter-selected galaxies at z ∼ 1–3 that also exhibit high star formation efficiencies. Such galaxies could remain on the main sequence for 300 Myr–1 Gyr (Ciesla et al. 2023). The comparable star formation efficiencies (Figure 9, left) and dust properties (Figure 8) show that the star-forming cores of GOALS galaxies are good local analogs to high-z (U)LIRGs in terms of their general star formation properties despite both populations occupying a fundamentally different region with respect to the main sequence.

The relation between L_IR,SF/M_dust and Σ_IR,SF is consistent with the Σ_IR,SF dependence of both L_PAH/M_dust (Figure 3) and L_PAH/L_IR,SF (Figure 8). While these scaling relations illustrate some link between the total dust mass, PAH emission, and star formation surface densities, the underlying physical association is not yet clear. This is especially true for the complex role played by dust grains in the ISM. Constant L_PAH/M_dust suggests that, relative to the total dust mass, PAHs are not systematically destroyed in (U)LIRGs at high IR surface densities by strong radiation fields. The low L_PAH/L_IR,SF ratios at high Σ_IR,SF could be causally linked to the high star formation efficiencies if the PAHs consistently trace the total dust at all IR surface densities (i.e., L_PAH ∝ M_dust ⇒ L_IR/M_dust ∼ L_IR/L_PAH ∝ SFE). Indeed, PAH line ratios tracing the ionization state of grains exhibit little scatter among z = 0 (U)LIRGs (e.g., Stierwalt et al. 2014), which could otherwise change the PAH mass-to-light ratio; however, the scatter in the same ratios is higher at z ∼ 2 (McKinney et al. 2020), and some change in the size distribution of grains is apparent when accounting for the 3.3 μm PAH feature at z = 0 (McKinney et al. 2021a). JWST/MIRI medium-resolution spectroscopy could test the PAHs in further detail through high S/N mid-IR spectroscopy. A complementary approach would be to follow-up Spitzer/IRS targets with ALMA to measure far-IR lines like [C ii] and investigate the potential link between low [C ii]/PAH ratios and dust grain properties. This would clarify whether or not changes to the grain properties (i.e., size, charge, PAH mass fraction) are important in regulating the formation of stars within distant and dusty galaxies.