ALMA 1.3 mm Survey of Lensed Submillimeter Galaxies Selected by Herschel: Discovery of Spatially Extended SMGs and Implications

To discover and study a significant sample of gravitationally lensed SMGs, we have conducted an extensive imaging survey of massive galaxy clusters in the FIR using the Herschel Space Observatory (Pilbratt et al. 2010), known as the HLS (Egami et al. 2010). The target clusters were selected mainly from the samples produced by the following three surveys: (1) the ROSAT All-Sky Survey (RASS) with the X-ray-luminous cluster sample tabulated by H. Ebeling (2021, private communication), (2) the COnstrain Dark Energy with X-ray (CODEX) survey, which utilizes the combination of the RASS X-ray and Sloan Digital Sky Survey (SDSS) optical data (Finoguenov et al. 2020), and (3) the South Pole Telescope (SPT) survey, which selected clusters via the Sunyaev–Zel'dovich (SZ) effect (the SPT-SZ survey; Bleem et al. 2015). The full HLS cluster sample will be presented and described by the forthcoming survey paper (E. Egami et al. 2021, in preparation).

With a substantial number of SMG detections at z_phot ≳ 1, we specifically selected a subset of exceptionally bright sources and obtained ALMA follow-up observations. The selection criteria used were (1) SPIRE color S₅₀₀/S₂₅₀ > 0.4 to ensure the selection of z ≳ 1 sources, (2) FIR continuum peak (S_peak) brighter than 90 mJy if the source is within 1' from the cluster center, (3) S_peak > 150 mJy or with a spectroscopic redshift if the source is beyond 1' from the cluster center, and (4) observable with ALMA (decl. < +30°). The second criterion is designed to select bright sources at L_IR ≳ 10^13.1 μ⁻¹ L_⊙ at z ∼ 2. The third criterion ensures that the resultant sample includes the brightest sources in the HLS data even if some of them may be boosted by a galaxy component on the line of sight. Due to the coarse resolution of SPIRE, the S_peak quoted here is the sum of all submillimeter sources within a radius of ∼15''. Multiple-source systems will be decomposed later using the ALMA data, and these individual sources will not necessarily satisfy the same Herschel selection criteria.

We eventually constructed a sample of 20 sources based on the criteria listed above.²⁶ There are also several other HLS sources satisfying the same brightness criteria, but we do not incorporate them here because they were not observed by ALMA or lie at different redshift range (z ∼ 5; e.g., HLS0918, Combes et al. 2012; Rawle et al. 2014; HLS0257, F. Sun et al. 2021, in preparation; HLS2043, Zavala et al. 2015; G. W. Walth et al. 2021, in preparation).

The HLS has performed far-infrared imaging observations of 581 massive galaxy clusters with a total observing time of 418.7 hr at two typical depths. The HLS-deep survey imaged 54 clusters deeply with PACS (Poglitsch et al. 2010) and SPIRE (Griffin et al. 2010) through an open-time key program (44 targets; PI: Egami; PID: KPOT_eegami_1) and an open-time Cycle 2 program (10 targets; PI: Egami; PID: OT2_eegami_5). Three of the clusters studied here (MACS J1115.8+0129, A2813, and A3088) were observed by HLS-deep, and therefore have a five-band coverage with both PACS (100/160 μm) and SPIRE (250/350/500 μm). All of the PACS 100 and 160 μm observations consist of two orthogonal scan maps, each comprising 18–22 repetitions of 13 parallel 4' scan legs. The SPIRE observations for the two Abell clusters were performed with 20 repetitions in the large scan map mode, each with two 4' scans and cross-scans (1.6 hr scan for three bands simultaneously). MACS 1115 was observed through 11-repetition small scan maps, and each repetition consisted of one scan and one cross-scan of 4' length (0.4 hr scan).

The HLS-snapshot survey obtained shallower SPIRE-only data for 527 clusters through two open-time programs during Cycles 1 and 2 (PI: Egami; PID: OT1_eegami_4, OT2_eegami_6), providing SPIRE data for the remaining 17 clusters studied here. With shallow observations (3–8-minute scans) in the small scan map mode, HLS-snapshot provides nearly confusion-limited images in all three SPIRE bands (typical rms noise ∼10 mJy beam⁻¹ at 250, 350, and 500 μm, compared with the confusion noise levels of 5.8, 6.3, and 6.8 mJy beam⁻¹ measured by Nguyen et al. 2010).

Our SPIRE images were produced via the standard reduction pipeline in HIPE v12.2 (Ott et al. 2010), and the processing routine was detailed in Rawle et al. (2016) for Herschel coverage of the HST Frontier Fields (HFF; Lotz et al. 2017). The observation ID (OBSID) and total scan time of each obtained Herschel/SPIRE observation are summarized in Table 1. HLS-deep PACS images were generated with UniMap (Piazzo et al. 2015) with a pixel scale of 10 at 100 μm and 20 at 160 μm, also detailed in Rawle et al. (2016).

ALMA Band 6 observations were carried out through projects 2015.1.01548, 2016.1.00372, and 2017.1.01658 (PI: Egami) between 2016 April 30 and 2018 September 30. Because four sources exhibit extended structures in the SPIRE images (FWHM > 20'' at 250 μm), we requested multipointing observations for these special cases. We observed all of our 20 targets in one of two spectral window settings. For 15 sources without a previous spectroscopic redshift determination, we performed continuum-only observations with a central frequency at 233 GHz (corresponding to 1.287 mm). For five sources with prior spectroscopic redshift information, we acquired both dust continuum and at least one CO line spectrum, and thus, the final effective frequencies of these continuum products range from 224 to 238 GHz. The diameter of the ALMA field of view (FoV) at the requested frequencies is 25''. A brief summary of our ALMA observations is also presented in Table 1. The data were taken in various weather conditions with a median precipitable water vapor (PWV) of 0.77 mm, with the 16th to 84th percentiles ranging from 0.56 to 1.84 mm. A median angular resolution of 026 was achieved, and the median maximum recoverable scale was 17.

All of the ALMA data were reduced with CASA (McMullin et al. 2007) with pipelines v4.7.2 and v5.4.1 for observations obtained in different cycles. Before the formal reduction work, we first checked the combined continuum image and spectral cube of each target, delivered by the ALMA archive. If any obvious source was detected above a 4σ significance, we would examine the spectral cubes, searching for possible spectral line features, and both line and continuum would be imaged in the natural/Briggs weighting and uv-tapered modes separately. If undetected, we would only produce the continuum images. We performed continuum imaging at four different levels of synthesized beam size: Briggs weighting (robust = 0.5), natural weighting (robust = 2), 1'' uv tapering, and 2'' uv tapering. These settings were used for visualizing both compact and extended emission structures in the SMGs. Interactive cleaning was performed during each imaging process. The noise level of the final continuum products is ${0.11}_{-0.05}^{+0.04}$ mJy beam⁻¹ with a 1'' tapered beam, which we used to obtain photometry for most of the targets.

We obtained Spitzer/IRAC Channel 1 (3.6 μm) and Channel 2 (4.5 μm) images through various programs. The majority of our data were from Program 12095 and 90218 (both PI: Egami), covering 13 of 20 clusters. We also included other data with public access on the Spitzer Heritage Archive.²⁷ This includes programs 80168, 90213 (PI: Bouwens), 12005, 14281 (PI: Bradac), 60099, 80012 (PI: Brodwin), 60034 (PI: Egami), 30344 (PI: Jarvis), 80162, 90233 (PI: Lawrence), 70149 (PI: Menanteau), 80066 (PI: Rawle), 61061 (PI: Sheth), 12123 (PI: Soifer), 40370, 80096 (PI: Stanford), 14061, 60194 (PI: Vieira). These programs provide 3.6/4.5 μm coverages for all the 20 clusters. All of these fields are observed with cycling subpixel dithering patterns with four or five dithering points at least.

We started our IRAC data processing from the archive-delivered level 1 (BCD) products. A standard and automatic MOPEX reduction routine was applied with an output pixel size of 06 pixel⁻¹. We registered the output frames with the Gaia DR2 (Gaia Collaboration et al. 2018) using SExtractor (for catalog extraction; Bertin & Arnouts 1996) and SCAMP (for astrometric computation; Bertin 2006). This achieved a final astrometric error of ≲01 with the produced IRAC images. The median 5σ point-source depth in each field, estimated from the variance of sky background, is presented in Table 1.

For a number of clusters, we also included other ancillary data to improve the quality of analysis, mainly for the optical SED fitting. This provides more accurate extinction and stellar-mass estimates, compared with IRAC-only analyses:

MACS J0553.4–3342—this cluster was observed with the HST treasury program RELICS (Coe et al. 2019), and therefore, we used its seven-band HST data for photometry and SED fitting (ACS/F435W, F606W, F814W, and WFC3-IR/F110W, F125W, F140W, F160W).

MACS J1115.8 + 0129—this cluster was observed with the HST treasury program CLASH (Postman et al. 2012), and therefore, its nine-band HST data were utilized (ACS/F435W, F606W, F775W, F814W, WFC3-IR/F105W, F110W, F125W, F140W, F160W). We directly used the processed data of MACS J0553 and MACS J1115 available on MAST.²⁸

RXC J2332.4–5358—this cluster was observed through the HST SNAP program 12884 (PI: Ebeling). The WFC3-IR/F110W and F140W images were obtained with an integration time of 706 s per filter. We reduced the data with Drizzlepac v2.1.17 (Gonzaga et al. 2012) under the PyRAF environment with an output pixel size of 006 pixel⁻¹.

Aperture photometry of SMGs in HST images is conducted with SExtractor. We do not obtain flux measurements of HLS0553-C (blended with a star ∼5^m brighter than the SMG in the F814W band) and HLS2332-C (blended with an irregular galaxy).

RXC J1314.3–2515—this cluster was observed with WFCAM on UKIRT in both the J and K bands (PI: Walth), and here we only use the NIR photometry of the lensed SMG in this cluster field.

No additional optical/NIR data were included for the analysis of the remaining sources. The Herschel sources in RXC J1314.3–2515, MACS J0455.2+0657, and MACS J0600.1–2008 were observed with JCMT/SCUBA-2 at 850 μm (Cheale et al. 2019), and here we quote the 850 μm flux densities to improve the quality of far-IR SED modeling.

To obtain reliable and complete (≳1 mJy) ALMA detections of lensed SMGs in all 20 observed cluster fields, we used SExtractor v2.19.5 (Bertin & Arnouts 1996) to derive uniform source extraction in all ALMA continuum image products (without primary beam correction). Based on a quick visual inspection of natural-weighted image products without uv tapering (median beam FWHM = 028), we found that all of the obvious sources detected in our data were spatially resolved. Because uv tapering can increase the detectability of extended structures, though at the expense of resolution, we performed the source extraction with the 1'' tapered image products.

We ran SExtractor for source detection and automatic photometry with Kron-like elliptical apertures (PHOT_AUTOPARAMS values of 1.8 and 2.5) in the primary-beam-uncorrected maps where the primary beam response is greater than 0.2. We estimated the photometric errors (σ_aper) based on aperture-to-beam size ratio (Ω_aper/Ω_beam) and continuum rms (σ_rms) according to the following equation:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{aper}}={\sigma }_{\mathrm{RMS}}\cdot \sqrt{\displaystyle \frac{{({{\rm{\Omega }}}_{\mathrm{aper}}/{{\rm{\Omega }}}_{\mathrm{beam}})}^{2}}{1+({{\rm{\Omega }}}_{\mathrm{aper}}/{{\rm{\Omega }}}_{\mathrm{beam}})/2}},\end{eqnarray} \tag{ 1 }$$

which is derived from a simulation of applying random apertures (enclosed area as Ω_aper) on Gaussian-blurred (kernel area as Ω_beam) Gaussian white-noise maps. We also evaluated the photometric uncertainty by directly applying random apertures on source-free regions in ALMA primary-beam-uncorrected maps. We measured the standard deviation of flux densities enclosed within the apertures of identical size, and the results are consistent with the prediction by Equation (1). We then corrected the flux densities and their errors for the gain of the primary beam.

Based on this method, we detected 77 sources at a signal-to-noise ratio (S/N) > 3.0 in all ALMA images. Here, the S/N is defined as the ratio between the aperture-photometry flux density and its uncertainty (f/σ_aper). To eliminate possible false detections, we studied the false detection rate in Appendix A, and found that an S/N cut at 4.0 would ensure the total false detection number to be ≲1. We therefore detected 28 sources at S/N > 4.0 based on the 1'' tapered images. All of the sources were detected in the area where the primary beam response is greater than 0.5, except for HLS2155-A.

We also included another S/N > 4.0 detection in 2'' tapered images, namely HLS0840. This source is exceptionally extended with an ALMA-measured 1.3 mm effective radius R_e,ALMA of 37 ± 17 with a relatively low surface brightness. It is split into multiple S/N ∼ 3 components in our 1'' tapered image but remains as a single S/N = 4.5 source in the 2'' tapered map. We also detected a point-like 0.5 ± 0.1 mJy source at its center in the 02 resolution image. This might represent the core of the galaxy, although it only contributes 13% ± 4% of the total flux density in the 2'' tapered map. We included this source because of the robustness of the detection in the 2'' tapered images, and no other similar example was found in our data.

We also studied the completeness of our source extraction in Appendix A. We conclude that the completeness of point-like sources at S/N = 5.0 in the 1'' tapered map (0.79 mJy, under the assumption of a median continuum rms and a primary beam correction of $\sqrt{2}$) is 80% ± 4%.

In one cluster field, CODEX 52909, we did not detect any significant source. The SPIRE source in CODEX 52909 is extended, and it can be a composite of several ALMA sources at S/N ≃ 3–3.5 with reddened IRAC counterparts. To avoid any confusion of ∼3σ fake sources in this mosaic ALMA FoV, we did not analyze this field any further.

Because of the noise fluctuation in the ALMA maps, the fluxes of sources at low S/N tend to be overestimated, known as the flux boosting effect (e.g., Geach et al. 2017; Stach et al. 2019). We examined this effect for sources detected at relatively low S/N (HLS0546, HLS0840, HLS0043-B, and HLS2155-A; Table 2). Assuming the surface brightness profiles measured in Section 3.3, we simulated the visibility data for these sources 10–15 times each with CASA, and the rms noise of mock data was controlled to match with that of our observations. We then applied the same imaging and source extraction routine, measured the median output flux densities, and compared them with those of input models. Based on our simulations, no conspicuous flux boosting effect was identified.

3.2.1. Herschel

3.2.2. Spitzer/IRAC

Morphological modeling of our sources in the IRAC images was performed along with photometry using GALFIT, as detailed in Section 3.2. For 24 of our sources with successful IRAC surface brightness profile modeling, we measured a median Sérsic index of 0.9 and 1.0 at 3.6 and 4.5 μm, which is close to n_med = 1.2 ± 0.3 reported by Chen et al. (2015) for z ≃ 1–3 SMGs in the HST/WFC3-IR F160W band. Among the six sources whose postage stamp images we display in Figure 1, the IRAC 3.6/4.5 μm radial surface brightness profiles of three representative sources are shown in Figure 2. The median IRAC effective radius is around 08, corresponding to 6.5 kpc at z = 2 in physical scale before lensing correction. For the five sources with heavy blending and complex morphology (Figure 16), we did not measure their structural profiles in either the IRAC or ALMA images. Here, we concentrate on the modeling of the structural profile of the remaining 24 lensed SMGs at 1.3 mm. As already described, we continue to decompose HLS1314 as three sources, so 26 sources are studied in this subsection.

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We obtained structural parameters of our sources in the ALMA data, modeling their visibility data using uvmodelfit (for single-source case) and uvmultifit (for the multiple-source case; Martí-Vidal et al. 2014), both under the CASA environment. We assumed Gaussian models for the surface brightness profile of all sources, consistent with the modeling procedure applied by Puglisi et al. (2019) and Tadaki et al. (2020). We note that Hodge et al. (2016) measured a median Sérsic index of n = 0.9 ± 0.2 for z ∼ 2.5 SMGs, and Gullberg et al. (2019) measured n = 1.0 ± 0.1 at 870 μm. These Sérsic indices are closer to exponential disk profiles (n = 1) rather than Gaussian ones (n = 0.5). Because the R_e modeled by a Gaussian profile is smaller than an exponential one by only ∼0.045 dex in Hodge et al. (2016), less than the median uncertainty of R_e in this work (∼0.063 dex), we keep this Gaussian profile assumption for SMGs in the ALMA data. We used ALMA source positions, flux densities, and morphological parameters, obtained through SExtractor photometry, as the initial guess of source models in the uv plane. Figure 3 shows three representative visibility profiles of lensed SMGs with different values of R_e (length of the semimajor axis) shown in Figure 2. There is no clear difference between the χ² of the best-fit circular Gaussian and exponential models for HLS0546 and HLS0553-C, and the Gaussian profile fits HLS0612 better.

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Two cases required special attention for multiple-source modeling on the uv-plane. HLS0612 was observed to have two minor nearby components in the ALMA high-resolution image (Figure 1), while uvmultifit favored a single-source model. Therefore, we discarded the three-source fitting results. On the contrary, the goodness of a three-source fit with HLS1314 was better than that of a single-source one, so we continued to decompose it into HLS1314-A, B, and C, consistent with the structural fitting in the IRAC bands.

The effective radii and their uncertainty of all the 26 sources (24 SMGs and HLS1314 with three components) are presented in Table 3. We also compared the IRAC (average of 3.6/4.5 μm) and ALMA source centroids and did not find any detectable offset (Appendix B).

All of our measured effective radii at 1.3 mm are smaller than their corresponding maximum recoverable angular scales for given ALMA antenna configurations, except for HLS0840 (Figure 4). As a result, the R_e measurement of HLS0840 has a large error bar (17). To examine the quality of uv-profile modeling for sources with extended R_e or at relatively low S/N, we also simulated and modeled the visibility data for the same sources as described in Section 3.1 for flux boosting evaluation. The median R_e measured from the 10 to 15 sets of simulated visibility data for each source is consistent with that of the input model at the 1σ confidence level, demonstrating the overall validity of our source-size measurements with ALMA.

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We successfully identified the IRAC counterparts of all 29 sources discovered with ALMA at S/ N > 4.0, indicating that our sample is free from contamination by any false detection. Our Monte Carlo simulation also suggests a low probability (< 2%) of random association between ALMA and IRAC sources. We display all of the ALMA and IRAC images of the sources in our sample in Figures 1, 16, and 17. We notice that HLS0840 and HLS0546 are two spatially extended sources (R_e,ALMA > R_e,IRAC ∼ 1'') at 1.3 mm with exceptionally low surface brightness (≲0.1 mJy arcsec⁻²) and only detectable in uv-tapered ALMA maps instead of those at native resolution (Figure 1). The visibility profiles of these two sources are displayed in Figures 3 and 4.

Although all of the sources were discovered in lensing-cluster fields, we identified five sources that are subject to additional boosting by foreground galaxies, i.e., HLS1124-A/B, HLS0455, HLS0505, and HLS0600 as shown in Figure 16. Among the remaining cluster-lensed cases, HLS0553 and HLS2332 are two spectroscopically confirmed triply imaged systems behind the cluster fields of MACS J0553 and RXC J2332, respectively. HLS0553-A/B/C have been reported at z = 1.14 by Ebeling et al. (2017), and HLS2332-A/B/C at z = 2.73 by Greve et al. (2012). We do not find any other multiply imaged SMGs in our sample.

For the sources with known spectroscopic redshifts (e.g., from a CO search with the IRAM 30 m telescope; M. D. Z. Dessauges-Zavadsky et al. 2021, in preparation), our ALMA Band 6 observations also targeted CO lines when possible. We confirmed the redshifts of HLS0553 and HLS2332, and obtained/confirmed redshifts for additional seven sources (F. Sun et al., in preparation): HLS0455 (z = 2.93; Zavala et al. 2015), HLS1314 (z = 1.45), HLS1124-A/B (z = 1.80), HLS0011-A/B (z = 2.27) and HLS0600 (z = 2.89). We did not detect any spectral line feature in the ALMA data of the remaining 16 sources, and thus their redshifts remain unknown.

Among the full sample of 29 ALMA sources, we found that 16 are potentially isolated sources with no detectable companion brighter than 0.6 mJy at 1.3 mm. We also identified six SMGs that exhibit close companions at similar redshifts, namely HLS1124 (observed as four components and grouped as two in Table 2; hereby noted as 4/2 and same later), HLS0111 (2/2), HLS0600 (2/1), HLS0612 (3/1), HLS1314 (3/1) and HLS2104 (2/2). Such grouping is determined by the angular separation and SExtractor deblending threshold (∼3'') on 1'' tapered maps. Source groups HLS0111, HLS0600, HLS1124, and HLS1314 are spectroscopically confirmed within a maximum velocity separation of 800 km s⁻¹ of each other.

We perform SED modeling with the high-z extension of magphys (da Cunha et al. 2008, 2015). We also use the photo-z extension of magphys (Battisti et al. 2019) to estimate the photometric redshift (z_phot) and physical properties simultaneously when the spectroscopic redshift (z_spec) is unknown. magphys assumes a Chabrier initial mass function (Chabrier 2003), a continuous delayed exponential star formation history (SFH), a two-component dust absorption law (Charlot & Fall 2000) and energy balance between dust absorption in the UV and reemission in the infrared. Other key components of the model assumptions in magphys have been detailed in Martis et al. (2019).

We adopt our observed photometric measurements to feed the SED fitting routine without applying any lensing magnification correction. This is due to a lack of detailed lens models for several of the clusters in this study. Note, therefore, that all the derived physical quantities are those including the effects of lensing. We also assume that there is no differential magnification effect (i.e., effective magnification factors change as a function of the source size and therefore a function of wavelength in general).

For the majority of our sample, their SEDs are modeled using six-band photometry from 3.6 μm to 1.3 mm. We also utilize the ancillary data described in Section 2.5 to improve the fitting, especially to constrain the amount of rest-frame optical dust extinction and therefore stellar mass. All the individual best-fit SEDs are displayed in Figure 18, and a summary of the best-fit galaxy properties is presented in Table 4.

Table 4.  Summary of Source Physical Properties Derived from SED Fitting

ID	Redshifta	log(M*)b	log(SFR)b	log(sSFR)	log(LIR)b,c	log(Mdust)b	Tdustd	AV
		(μ−1M⊙)	(μ−1M⊙ yr−1)	(Gyr−1)	(μ−1L⊙)	(μ−1M⊙)	(K)	(mag)
HLS0043-A	${1.61}_{-0.44}^{+0.79}$	11.77 ± 0.35	2.73 ± 0.48	−0.01 ± 0.58	12.87 ± 0.39	9.30 ± 0.39	29.3 ± 7.0	2.3 ± 1.0
HLS0043-B	${3.23}_{-0.59}^{+0.48}$	11.69 ± 0.29	2.66 ± 0.43	−0.02 ± 0.64	12.78 ± 0.23	9.01 ± 0.23	32.3 ± 5.0	2.2 ± 0.7
HLS0111-A	2.27	11.94 ± 0.21	2.55 ± 0.23	−0.38 ± 0.40	12.78 ± 0.10	8.96 ± 0.10	31.5 ± 1.9	5.5 ± 0.5
HLS0111-B	2.27	11.96 ± 0.26	2.84 ± 0.19	−0.12 ± 0.42	12.98 ± 0.10	9.20 ± 0.10	31.5 ± 1.2	2.9 ± 0.5
HLS0114	${1.49}_{-0.27}^{+0.43}$	11.94 ± 0.32	2.71 ± 0.42	−0.21 ± 0.64	12.88 ± 0.27	8.92 ± 0.27	35.4 ± 5.2	2.0 ± 0.9
HLS0307-28-A	${1.19}_{-0.23}^{+0.32}$	11.61 ± 0.39	2.59 ± 0.38	−0.01 ± 0.61	12.79 ± 0.27	8.97 ± 0.27	33.4 ± 4.2	7.1 ± 1.6
HLS0307-28-B	${2.65}_{-0.30}^{+0.54}$	10.70 ± 0.32	2.44 ± 0.14	0.74 ± 0.29	12.63 ± 0.18	9.15 ± 0.18	29.6 ± 3.7	2.0 ± 0.9
HLS0307-50	${1.93}_{-0.39}^{+0.53}$	12.25 ± 0.33	2.86 ± 0.38	−0.38 ± 0.55	13.13 ± 0.25	9.34 ± 0.25	33.5 ± 5.3	5.0 ± 0.9
HLS0455	2.93	11.82 ± 0.21	3.43 ± 0.06	0.62 ± 0.28	13.57 ± 0.06	9.62 ± 0.06	34.4 ± 0.6	2.2 ± 0.4
HLS0505	${1.27}_{-0.22}^{+1.09}$	11.31 ± 0.44	2.61 ± 0.41	0.44 ± 0.52	12.74 ± 0.40	9.62 ± 0.40	24.6 ± 7.1	5.1 ± 1.6
HLS0546	${1.89}_{-0.42}^{+0.50}$	11.85 ± 0.41	2.74 ± 0.36	−0.09 ± 0.65	12.93 ± 0.26	9.02 ± 0.26	33.9 ± 5.8	3.5 ± 1.1
HLS0553-A	1.14	11.45 ± 0.03	2.68 ± 0.10	0.22 ± 0.10	12.85 ± 0.09	8.99 ± 0.09	31.2 ± 1.5	3.8 ± 0.2
HLS0553-B	1.14	11.54 ± 0.07	2.66 ± 0.15	0.07 ± 0.17	12.82 ± 0.14	9.00 ± 0.14	31.1 ± 1.5	3.7 ± 0.3
HLS0553-C	1.14	11.74 ± 0.32	2.91 ± 0.18	0.18 ± 0.47	13.05 ± 0.10	9.25 ± 0.10	31.4 ± 1.1	4.2 ± 0.9
HLS0600	2.87	12.51 ± 0.27	3.72 ± 0.12	0.22 ± 0.38	13.84 ± 0.07	10.29 ± 0.07	29.8 ± 0.3	2.7 ± 0.5
HLS0612	${2.31}_{-0.55}^{+0.59}$	11.76 ± 0.40	3.11 ± 0.29	0.37 ± 0.48	13.23 ± 0.27	9.53 ± 0.27	31.8 ± 5.5	3.0 ± 0.9
HLS0840	${1.97}_{-0.58}^{+0.52}$	11.84 ± 0.36	2.97 ± 0.36	0.12 ± 0.53	13.09 ± 0.30	9.26 ± 0.30	33.3 ± 6.4	2.2 ± 0.9
HLS1115	${1.59}_{-0.01}^{+0.05}$	12.15 ± 0.01	2.83 ± 0.01	−0.32 ± 0.04	12.92 ± 0.03	9.36 ± 0.03	31.2 ± 0.6	3.1 ± 0.1
HLS1124-A	1.80	10.80 ± 0.30	2.36 ± 0.14	0.57 ± 0.40	12.44 ± 0.13	9.03 ± 0.13	27.0 ± 1.5	1.7 ± 0.6
HLS1124-B	1.80	11.70 ± 0.30	3.27 ± 0.12	0.57 ± 0.40	13.35 ± 0.12	9.98 ± 0.12	27.0 ± 0.4	1.5 ± 0.4
HLS1314e	1.45	12.73 ± 0.06	2.44 ± 0.21	−1.32 ± 0.20	12.94 ± 0.07	9.50 ± 0.07	28.4 ± 0.7	2.3 ± 0.1
HLS1623	${0.96}_{-0.45}^{+0.17}$	11.84 ± 0.48	2.26 ± 0.41	−0.60 ± 0.53	12.68 ± 0.34	8.42 ± 0.34	22.0 ± 3.8	9.2 ± 1.4
HLS2104-A	${2.17}_{-0.54}^{+0.71}$	11.73 ± 0.33	2.60 ± 0.41	−0.08 ± 0.56	12.77 ± 0.31	9.10 ± 0.31	31.0 ± 6.3	2.3 ± 0.9
HLS2104-B	${2.13}_{-0.63}^{+0.62}$	11.87 ± 0.30	2.79 ± 0.44	−0.07 ± 0.56	12.91 ± 0.33	9.28 ± 0.33	30.7 ± 6.2	1.5 ± 0.8
HLS2155-A	${1.72}_{-0.30}^{+0.29}$	11.92 ± 0.35	2.63 ± 0.33	−0.31 ± 0.60	12.87 ± 0.20	8.81 ± 0.20	37.0 ± 4.7	5.2 ± 1.1
HLS2155-B	${2.66}_{-0.73}^{+0.94}$	11.72 ± 0.31	2.92 ± 0.37	0.20 ± 0.45	12.99 ± 0.32	9.25 ± 0.32	31.9 ± 7.4	1.6 ± 0.7
HLS2332-A	2.73	11.91 ± 0.09	3.61 ± 0.12	0.72 ± 0.17	13.64 ± 0.12	9.44 ± 0.12	36.7 ± 0.7	3.0 ± 0.2
HLS2332-B	2.73	11.73 ± 0.12	3.64 ± 0.04	0.93 ± 0.15	13.70 ± 0.07	9.51 ± 0.07	36.7 ± 0.6	2.9 ± 0.1
HLS2332-C	2.73	12.49 ± 0.24	3.51 ± 0.13	0.03 ± 0.35	13.64 ± 0.07	9.53 ± 0.07	36.7 ± 0.6	3.8 ± 0.4

Notes.

^aPhotometric redshifts are presented as the median of their likelihood distributions with a 1σ confidence range. ^bThese quantities are not corrected for lensing magnification. ^cDefined as the total IR luminosity integrated from 8 to 1000 μm. ^dModeled with an MBB spectrum with fixed dust emissivity at β = 1.8. ^eCombination of all three clumps (HLS1314-A/B/C) seen in the ALMA map (see Section 4.1).

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The 16th–50th–84th percentile of the redshift distribution for our sample is 1.23–1.93–2.73, and the highest and lowest photometric redshifts are ${z}_{\mathrm{phot}}={0.96}_{-0.45}^{+0.17}$ (HLS1623) and ${3.23}_{-0.59}^{+0.48}$ (HLS0043-B). Here, the z_phot is the median of the likelihood distribution of redshift for each source. We assess the uncertainty of the photometric redshift estimate based on the likelihood distribution obtained with magphys. The typical uncertainty of derived z_phot is ${\rm{\Delta }}z={0.16}_{-0.04}^{+0.05}\,(1+{z}_{\mathrm{phot}})$. HLS1115 exhibits a small Δz of 0.03 because of the existence of nine-band HST data. We also evaluate the far-IR z_phot by matching with the LIRG templates in Rieke et al. (2009), and the derived redshifts are consistent with those by magphys within ∼a 1σ confidence interval.

We also study the dust temperature with far-IR data over a rest-frame wavelength of 50 μm, following Greve et al. (2012). We fit the dust continua of all SMGs with modified blackbody (MBB). The dust absorption coefficient is assumed to be κ = 0.040 × (ν/250 GHz)^β in units of m² kg⁻¹, where ν is the frequency in gigahertz in the rest frame. We assume a fixed dust emissivity of β = 1.8, which is widely adopted in previous studies (e.g., Díaz-Santos et al. 2017; Dudzevičiūtė et al. 2020). We also incorporate the uncertainty of redshift in that of the dust temperature if the z_spec is unknown. The dust masses derived from this MBB fitting are consistent with those from magphys, listed in Table 4, with a 1σ dispersion of 0.08 dex.

HLS1623 was observed to be much fainter at 1.3 mm compared to the prediction from the SPIRE SED. This may indicate that the SPIRE fluxes are contributed by certain ALMA-undetected components in the same FoV. Here, we use its SPIRE-only SED to assess the dust temperature, but we only use its IRAC and ALMA flux densities for magphys SED fitting.

We first compare the FIR colors, namely f₂₅₀/f₅₀₀ and f₃₅₀/f_{1.3 mm}, of all galaxies detected in our survey (Figure 5, left). Here we do not use the color of f₂₅₀/f₃₅₀ or f₃₅₀/f₅₀₀ due to their narrow ranges of distributions. We find the majority of our galaxy exhibit a positive linear correlation between these two color indices. An MCMC fitting through emcee suggests the power-law relation as $\mathrm{log}({f}_{350}/{f}_{1.3\mathrm{mm}})=(1.507\pm 0.247)\,\times \mathrm{log}({f}_{250}/{f}_{500})+(1.13\pm 0.05)$. This is consistent with the unlensed SMG sample presented in Ikarashi et al. (2015), for which we apply a conversion from 1.1 mm flux densities to 1.3 mm by a factor of 0.53. This indicates red f₂₅₀/f₅₀₀ and red f₃₅₀/f_{1.3 mm} colors will occur simultaneously, basically controlled by source redshift and dust temperature.

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We then compare f₃₅₀/f_{1.3 mm} versus source redshifts, and an exponential relation can be identified against both the spectroscopic and photometric redshift samples (Figure 5; right). A similar MCMC routine implies an underlying relation of $\mathrm{log}({f}_{350}/{f}_{1.3\mathrm{mm}})=-(0.376\pm 0.044)\times z+(2.102\pm 0.095)$, with a standard dispersion of 0.16 dex for the measured f₃₅₀/f_{1.3 mm} ratio. This trend is also consistent with the unlensed sample in Ikarashi et al. (2015) at a similar redshift range. Such a correlation suggests that the f₃₅₀/f_{1.3 mm} color of SMGs, at least in this study, is only weakly affected by dust temperature and mainly reflects the redshift.

On the other hand, the IRAC [3.6 μm]–[4.5 μm] color of our SMG sample seems to show a larger dispersion at any given redshift (see the color-coding in the right panel of Figure 5), and thus, no substantial correlation can be derived between the IRAC color and redshift. This reflects the complexity of stellar age and dust absorption among our SMG sample. However, we find that at a given redshift, a red IRAC color is likely to occur simultaneously with a blue f₃₅₀/f_{1.3 mm} color. The red IRAC color can be a signature of high dust extinction and thus a high dust column density and high IR surface brightness (Σ_IR). Because Σ_IR is observed to be correlated with dust temperature at various redshift ranges (e.g., Spilker et al. 2016; Díaz-Santos et al. 2017), this would result in a bluer f₃₅₀/f_{1.3 mm} color as we have shown.

Several z ∼ 1 galaxies were observed with a red IRAC color (∼0.5), namely HLS0307-28-A and HLS1623, which suggests high dust extinction in these systems (A_V ≳ 7) through magphys SED modeling. Such a high A_V is not seen in AS2UDS sources with secure optical/NIR detection at λ_obs ≲ 2.2 μm (but are found in sources with IRAC-only detections; Dudzevičiūtė et al. 2020). Due to the lack of deep rest-frame UV and optical data, it is not clear whether the determination of such high A_V values is reliable. In the case of HLS0553 and HLS2332, where HST photometry exists for two of the three triplet lensed images, the HST photometry obtained at λ ≤ 1.6 μm slightly reduces the estimated A_V (by ∼0.4) and thus the best-fit stellar mass (by ∼0.2 dex; note that the median uncertainty is 0.32 dex if HST data is excluded). This underscores the difficulty of deriving dust absorption in galaxies with only two-band IRAC observations of their stellar continua.

The dust-obscured fraction of SFR in a galaxy has been claimed to be correlated with the stellar mass with no conspicuous evolution found with this correlation from redshift 2.5 to 0 (Whitaker et al. 2017). For a galaxy with M_* = 10¹⁰ M_⊙, 80% of the total star formation is obscured, and for typical z ∼ 2 SMGs with a stellar mass of ∼10¹¹ M_⊙ (e.g., Hainline et al. 2011; Dudzevičiūtė et al. 2020), this fraction is higher than 95%. Without appropriate lensing correction, we cannot accurately determine the intrinsic stellar mass of our lensed SMG sample. However, the median value is 10^11.8 μ⁻¹ M_⊙, still well above 10¹⁰ M_⊙ if a μ = 5 magnification factor correction is applied. Therefore, it is reasonable to assume that the majority of star formation in our sample is dust-obscured and that the observed IR luminosity (and emitting regions) adequately represents the total SFR (and star-forming regions).

We measure a median total IR luminosity of L_IR = 10^12.92±0.07 μ⁻¹ L_⊙ and thus a median SFR of 552 ± 93 ${\mu }^{-1}{M}_{\odot }{{\rm{yr}}}^{-1}$ without lensing corrections through SED fitting. Figure 6 displays the sSFR versus redshift of all the sources in our sample. The distribution of sSFR in redshift space is consistent with those of galaxies on the so-called star-forming "main sequence" (MS) with a stellar mass of 10¹¹ M_⊙ (Speagle et al. 2014), which have a median sSFR of 1.1 Gyr⁻¹ and a 1σ dispersion of 0.46 dex. If the differential magnification is negligible, sSFR will be conserved by lensing, and hence the distribution of sSFR should be the same among lensed and unlensed SMGs.

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We also compare the stellar-mass doubling timescale (t_* = 1/sSFR) with the Hubble time (t_H(z) = 1/H(z)) at the redshifts of SMGs in this work. The ratio between these two quantities (t_*/t_H) can in principle act as an indicator of whether a galaxy is undergoing a significant star formation event (e.g., Tacchella et al. 2018). We find a median ratio of 0.18 ± 0.02, suggesting that the majority of our sample are vigorously star-forming galaxies. This value is consistent with the median of AS2UDS SMGs at 1 < z < 3 (Dudzevičiūtė et al. 2020), regardless of whether the unlensed SMG is brighter than the single-dish SCUBA-2 detection limit at 850 μm (3.6 mJy) or not. Except for HLS1314 (the square point at the bottom of Figure 6), no other galaxy in this study shows t_*/t_H ≥ 1.

Similar to sSFR, the dust-to-stellar-mass ratio (M_dust/M_*; also referred to as the specific dust mass) and dust temperature are key observables of the properties of star-forming galaxies that are conserved in lensing. Because dust is produced through the process of star formation, the ratio between dust and stellar mass should be closely related to the sSFR. Figure 7 plots the dust-to-stellar-mass ratio versus sSFR in all 29 SMGs, color-coded with dust temperature. A positive linear relation can be found between these two quantities, although the normalization is subject to T_dust.

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Although such a correlation is not a surprise, the wide range of the dust-to-stellar-mass ratio and sSFR of this 29 SMG sample is remarkable. Distributed between −3.4 and −1.5, the $\mathrm{log}({M}_{\mathrm{dust}}/{M}_{* })$ span of this sample is similar to that of low-redshift galaxies, whose SFRs are distributed between 10^−1.5 and 10² M_⊙ yr⁻¹ (da Cunha et al. 2010). The upper end of our M_dust/M_* distribution matches with previous SMG literature (e.g., Santini et al. 2010; Dudzevičiūtė et al. 2020), while our sample does have a significant excess at $\mathrm{log}({M}_{\mathrm{dust}}/{M}_{* })\lesssim -3$. We perform a Kolmogorov–Smirnov (K-S) test of $\mathrm{log}({M}_{\mathrm{dust}}/{M}_{* })$ between our sample and AS2UDS SMGs in Dudzevičiūtė et al. (2020; 454 SMGs at 1 < z < 3), and a null hypothesis that SMGs in these two samples share the same M_dust/M_* distribution can be rejected (p value less than 0.01). The lower end of our M_dust/M_* distribution matches with that of low-redshift dusty early-type galaxies (ETGs; Agius et al. 2013), while still being higher than that of dust-poor early-type galaxies (Rowlands et al. 2012, 2015).

The number excess at the lower end of the M_dust/M_* distribution can actually be a signature of evolved systems with lower gas fraction and dust destruction in the post-starburst stage (e.g., Rowlands et al. 2015; Li et al. 2019). Applying a Scoville et al. (2016) conversion from luminosity at a rest frame of 850 μm (L_ν,850) to the molecular gas mass (M_gas), we find a low gas fraction (f_gas = M_gas/(M_gas + M_star)) of 0.17 ± 0.08 for four sources at M_dust/M_* < 10⁻³. We further discuss this issue of dust-to-mass ratio and its implication in Section 5.4.2.

Assuming a gas-to-dust ratio of 100, we calculate the gas depletion timescale (τ_dep = M_gas/SFR) for our sample, and the median value with 1σ dispersion is ${226}_{-73}^{+196}$ Myr. This is consistent with the median value of the AS2UDS sample (∼320 Myr) recomputed with a similar method (Dudzevičiūtė et al. 2020).

SMGs are generally found to host dust continua that are more compact than the stellar ones (e.g., Hodge et al. 2016, 2019; Gullberg et al. 2019; Lang et al. 2019). This can be interpreted as an evolutionary connection from SMGs to cQs at slightly lower redshift: after star formation ceases in a ∼1 kpc-scale region at the galaxy center, a cusp of the stellar component will remain in this region, which can be observed to be compact and quiescent (e.g., Toft et al. 2014; Simpson et al. 2015; Barro et al. 2016b; Lang et al. 2019).

We adopt the half-light radii measured at 1.3 mm ALMA uv-plane (R_e,ALMA) as the effective radius of the dust continuum (noted as R_e,dust) and thus the star-forming region. We note that this ignores any radial gradients of dust temperature or opacity that could alter the measured size of the dust continuum. Based on the FIRE-2 simulation, Cochrane et al. (2019) predicted that the effective radius of dust emission goes up with the observed wavelength in the submillimeter/millimeter. This is because observations at longer wavelengths are more sensitive to the cooler gas and dust components in the outer region. With an effective rest-frame wavelength of ${440}_{-95}^{+137}\,\mu {\rm{m}}$ for our sample, the measured R_e,dust is not expected to vary by more than 10% due to the difference in the sampled wavelength.

With the IRAC data, we define the effective radius as the geometric mean of the effective radii measured at 3.6/4.5 μm (${R}_{{\rm{e}},\mathrm{IRAC}}={R}_{{\rm{e}},[\mathrm{CH}1]}^{1/2}\cdot {R}_{{\rm{e}},[\mathrm{CH}2]}^{1/2}$). At z_med = 1.9, IRAC Channel 1/2 samples the rest-frame J/H bands with a similar angular resolution. Adopting the Calzetti et al. (2000) extinction law, dust extinction in the rest-frame J/H bands (A_J and A_H) will only be 0.3/0.2 of A_V. For a median A_{V, med} = 2.9 in this study, the dust extinction in the IRAC 3.6/4.5 μm bands is 0.9/0.6, which is only ∼8% of the extinction seen in HST J₁₂₅/H₁₆₀ bands for z ≃ 2 SMGs in Lang et al. (2019). Therefore, the intrinsic stellar distribution should be close to the light profile in the IRAC bands, without a significant overestimate of the effective radius of the stellar-mass distribution due to the concentration of dust extinction at the galaxy center. We measure the effective radius ratio between 3.6 and 4.5 μm as ${0.99}_{-0.19}^{+0.14}$ for our sample, and such a consistency between the radii seen in the two bands also suggests the weak influence of dust extinction on light profiles. Therefore, we directly adopt the R_e,IRAC as a representation of the effective radius of stellar mass (noted as R_e,star).

One caveat is that the measured A_V does not account for the stars that are fully dust-obscured, and thus the A_V derived from an energy-balanced SED fitting code does not necessarily represent the extinction to the full stellar-mass component (e.g., Casey et al. 2014b). As pointed out in Lang et al. (2019), it is possible that a compact and obscured stellar component remains undetected at the ALMA/IRAC continuum centroid (e.g., Simpson et al. 2017), resulting in an even more compact configuration of stellar mass than the IRAC light profile.

The comparison of the effective radii of dust and stellar continua is shown in Figure 8. Except for three sources (HLS1623, HLS0546, and HLS0840), 23 out of the 26 sources have more compact ALMA dust continua relative to their stellar components. The median dust-to-stellar-continuum size ratio (R_e,dust/R_e,star) is 0.38, with a 1σ distribution from 0.28 to 0.73. Lang et al. (2019) presented an R_{e,870 μm}/R_e,star of 0.6 ± 0.2, similar to our measurement. This is much smaller than the dust-to-stellar-size ratio of local spiral galaxies (∼1.0; e.g., Hunt et al. 2015).

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Because all of the 26 sources (except for heavily blended galaxy-lensed cases) are resolved in the ALMA 1.3 mm maps, we are able to obtain the surface luminosity of their dust continua. This should also be a conserved quantity independent of lensing magnification. We hereby define the IR surface luminosity as ${{\rm{\Sigma }}}_{\mathrm{IR}}={L}_{\mathrm{IR}}/(2\pi {R}_{{\rm{e}},\mathrm{ALMA}}^{2})$, and thus, Σ_IR is the average IR surface luminosity within a radius of R_e,ALMA. Note that at a rest-frame wavelength of ∼440 μm, the R_e,ALMA is expected to trace the size of cold dust component and star-forming region better than the far-IR emission (e.g., 70 μm continuum size used in Díaz-Santos et al. 2017). Due to the effect of radial T_dust gradient or dust optical depth, the R_e at the wavelength of FIR SED peak would be smaller than the R_e,ALMA (e.g., Cochrane et al. 2019), and thus, the Σ_IR might be underestimated. The conversion factor between IR and 1.3 mm surface brightness (Σ_IR/Σ_1.3mm) is 10^{10.65 ± 0.20} mJy arcsec⁻² ${L}_{\odot }^{-1}\,{{\rm{kpc}}}^{2}$ for sources in our sample, and the factor between surface SFR density and IR luminosity (Σ_SFR/Σ_IR) is ${10}^{-10.17\pm 0.07}\,{{\rm{M}}}_{\odot }\,{\mathrm{yr}}^{-1}\,{{\rm{L}}}_{\odot }^{-1}$.

We plot the dust temperature versus IR surface luminosity in Figure 9, comparing the distribution with those of the GOALS (local LIRG/ULIRGs; Díaz-Santos et al. 2017), AS2UDS (z_med ≃ 2.7 SMGs; Gullberg et al. 2019; Dudzevičiūtė et al. 2020), SPT (z_med ≃ 4.3 SMGs; Spilker et al. 2016; Strandet et al. 2016), and KINGFISH (nearby galaxies; Skibba et al. 2011; Hunt et al. 2015) galaxies. Galaxies in this work exhibit a wide 3 dex range of Σ_IR from 10^9.4 to 10^12.4 L_⊙ · kpc⁻². Such a range of Σ_IR does resemble that of the local (U)LIRGs in the GOALS sample.

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We find that the dust temperature of SMGs in this work is barely correlated with the IR surface luminosity. At Σ_IR < 10^11.5 L_⊙ · kpc⁻², the lensed SMGs exhibits comparable dust temperature as local LIRGs (Díaz-Santos et al. 2017 also assumed β = 1.8). This indicates that the Σ_IR–T_dust relation for LIRGs may not evolve with redshift up to z ∼ 2. Symeonidis et al. (2013) analyzed Herschel-selected LIRGs at z ≲ 1 and suggested no significant evolution of dust temperature at a constant IR luminosity (L_IR ≲ 10^11.5 L_⊙) across z = 0 ∼ 1. However, ULIRGs (L_IR ≥ 10¹² L_⊙) at z ≳ 1 are much cooler than their local analogs or more optically thick. Assuming that Σ_IR is well correlated with the intrinsic L_IR (reported at various redshifts, e.g., Rujopakarn et al. 2011; Lutz et al. 2016; Fujimoto et al. 2017), our result is consistent with Symeonidis et al. (2013) but extending out to z ∼ 2.

At Σ_IR > 10^11.5 L_⊙ · kpc⁻², the lensed SMGs in our sample seem to show lower dust temperature than both local ULIRGs and z ∼ 4 SPT sources (biased toward galaxy-lensed cases), but consistent with z_med = 2.7 SMGs in AS2UDS sample (Σ_IR based on 870 μm continuum size; Gullberg et al. 2019; Dudzevičiūtė et al. 2020). However, such a comparison is limited to the sample size because only two independent sources in our sample (HLS0553 and HLS2332) are at Σ_IR > 10¹² L_⊙ · kpc⁻².

We then investigate how the dust-to-stellar-continuum size ratio is correlated with other physical quantities. Similar to Lang et al. (2019), we do not find any obvious correlation between the size ratio and sSFR on integrated-galaxy scales. However, we do find that this continuum size ratio is well correlated with FIR surface luminosity, plotted in the left panel of Figure 10. We find that with the increase of Σ_IR, the dust-to-stellar continuum size ratio decreases slowly. We fit a linear relation through emcee, and the best-fit relation for our 26 sources is shown as follows:

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\left(\displaystyle \frac{{R}_{{\rm{e}},\mathrm{dust}}}{{R}_{{\rm{e}},\mathrm{star}}}\right) & = & (-0.23\pm 0.03)\cdot \mathrm{log}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{IR}}}{{10}^{10}}\right)\\ & & -(0.10\pm 0.04),\end{array}\end{eqnarray} \tag{ 2 }$$

where Σ_IR is in units of L_⊙ · kpc⁻².

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We also incorporate six compact star-forming galaxies (cSFGs) presented in Barro et al. (2016b) and 14 SMGs presented in Lang et al. (2019) into a combined data set of 46 sources. All of these sources are around z = 2.2 ± 0.3 with accurate ALMA image-plane morphology modeling at 870 μm. The R_e,star of the six cSFGs in Barro et al. (2016b) was measured using HST/WFC3 F160W images without any correction for dust extinction (A_V,SED is as low as 1.3 ∼ 1.6). R_e,star of 14 SMGs in Lang et al. (2019) was also based on F160W imaging but corrected with a pixel-to-pixel A_V map. A similar MCMC fitting to these 46 sources shows the following R_e,dust/R_e,star−Σ_IR relation:

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\left(\displaystyle \frac{{R}_{{\rm{e}},\mathrm{dust}}}{{R}_{{\rm{e}},\mathrm{star}}}\right) & = & (-0.19\pm 0.04)\cdot \mathrm{log}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{IR}}}{{10}^{10}}\right)\\ & & -(0.09\pm 0.04),\end{array}\end{eqnarray} \tag{ 3 }$$

which is consistent with the fitting of 26 HLS sources within 1σ. Such a linear relation is relatively tight over the 3.1 dex range of Σ_IR distribution. We measure the standard deviation of the 46 sources from the best-fit correlation as 0.21 dex, and the typical uncertainty of effective radius ratio is 0.11 ± 0.06 dex.

We also fit the data with the least-squares method and bootstrapping. The derived relation and its uncertainty are consistent with those by MCMC (slope is −0.27 ± 0.04). If L_IR and R_e,star are (i) random variables with a narrow distribution and (ii) independent of R_e,dust distributed over a wide range, then one should expect to derive a linear relation between log(R_e,dust/R_e,star) and log(Σ_IR) with a slope of −0.5. We also generate mock data set of L_IR, R_e,dust, and R_e,star that satisfies our selection criteria (i.e., above certain thresholds of L_IR and Σ_IR) and matches our measurements with respect to the variance, and the resultant slope is −0.46. However, our fittings suggest that such a slope can be ruled out at a confidence of >5σ, indicating that the observed relation is physical and not a direct consequence of the relatively wider distribution range of R_e,dust.

One of the remarkable properties of z ∼ 2 compact quiescent galaxies (cQs) is their inferred ultra-high stellar surface density (≳10¹⁰ M_⊙ · kpc⁻² within effective radius; e.g., van Dokkum et al. 2008; van der Wel et al. 2014). Such a compact configuration of the stellar component indicates a previous SFH at a similarly compact scale. This could be achieved through a nuclear starburst, because galaxy interaction/merger and disk instability can cause gas inflows and thus trigger the central stellar density enhancement (e.g., Hopkins et al. 2008).

We define Σ_star as the average stellar mass within R_e,star, namely ${{\rm{\Sigma }}}_{\mathrm{star}}={M}_{* }/(2\pi {R}_{{\rm{e}},\mathrm{star}}^{2})$, similar to the definition of Σ_IR. The Σ_star of our SMG sample ranges between 10^8.6 and 10^11.0, with a median value of 10^{9.4 ± 0.1} (units: M_⊙ kpc⁻²), similar to that seen for SMGs by Hodge et al. (2016). Just like Σ_IR, Σ_star is a conserved quantity with respect to gravitational lensing.

We first compare the Σ_star of our sample with the general galaxy population in similar redshift and mass ranges. van der Wel et al. (2014) showed that for early- and late-type galaxies at z ∼ 1.75 with a stellar mass of 10¹¹ M_⊙, which is close to that of our SMG sample, the median Σ_star should be 10^{9.6 ± 0.3} and 10^{8.8 ± 0.4} M_⊙ kpc⁻², respectively. Therefore, the majority of galaxies in our sample do match the stellar surface density of early-type galaxies rather than the late type, although a test on the M_*−R_e plane (e.g., Hodge et al. 2016) cannot be performed due to the incompleteness of lensing magnification.

We then color-code the R_e,dust/R_e,star−Σ_IR plot with Σ_star as shown in the left panel of Figure 10. Although a tight linear relation can be found between the two quantities, we notice that the deviation of the continuum size ratios from the best-fit relation seems to be correlated with Σ_star. We then plot this residual of R_e,dust/R_e,star versus Σ_star in the right panel of Figure 10. An MCMC linear fitting under the assumption of equal weighting of all data points suggests a positive relation between the two quantities as

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\left[\mathrm{log}\left(\displaystyle \frac{{R}_{{\rm{e}},\mathrm{dust}}}{{R}_{{\rm{e}},\mathrm{star}}}\right)\right] & = & (0.24\pm 0.07)\cdot \mathrm{log}\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{star}}}{{10}^{10}}\right)\\ & & +(0.17\pm 0.05),\end{array}\end{eqnarray} \tag{ 4 }$$

where Σ_star is in units of M_⊙ kpc⁻².

This indicates that the relation shown as Equation (3) holds for the majority of z ∼ 2 SMGs with a stellar-mass surface density of roughly 10⁹ ∼ 10¹⁰ M_⊙ kpc⁻². With the enhancement of the central stellar-mass surface density above ∼10¹⁰ M_⊙ kpc⁻², SMGs will show larger R_e,dust/R_e,star ratio than the regular relation. This may reflect not only a newly formed cuspy stellar profile in the galaxy center, but also potentially the quenching of concentrated star formation and dissipation of dust remnant through multiple physical process (e.g., stellar or SMBH feedback), which we discuss further in Section 5.3.2.

We also note that the vertical scatter in the right panel of Figure 10 is related with the sSFR at the galaxy-integrated scale, as color-coded in the diagram. At a given Σ_star, the increase of galaxy-wide sSFR will lead to the growth of R_e,dust/R_e,star, which can be a trivial result of the fact that sSFR is proportional to the FIR luminosity and thus R_e,dust when Σ_IR conserves.

One major discovery of this work is the existence of spatially extended SMGs with a low IR surface luminosity (Σ_IR < 10¹⁰ L_⊙ kpc⁻²), e.g., HLS0546 and HLS0840. These galaxies exhibit extended dust continua compared with the stellar ones, and their FIR surface luminosities are ≳1 dex lower than those of typical SMGs at z = 1 ∼ 3 (Σ_IR ≳ 10¹¹ L_⊙ kpc⁻²; e.g., Hodge et al. 2016; Gullberg et al. 2019 and this work).

A galaxy with a similar Σ_IR has been reported by the ALMA Frontier Fields Survey (e.g., A2744-ID05, Σ_IR = 10^9.9 L_⊙ kpc⁻²; González-López et al. 2017; Laporte et al. 2017). However, the dust continuum of this galaxy is relatively compact (R_e,dust = 1.3 kpc) even compared with the size of stellar continuum measured with HST/WFC3-IR (R_e,star = 4.0 kpc). Such a small dust-to-stellar-size ratio is different from the extended nature of the SMGs reported here. HLS0546 and HLS0840 are also different from the spatially extended SMG at z = 2.8, SMM J02399-0136 (Σ_IR = 10^10.2 L_⊙ kpc⁻²; Genzel et al. 2003; Ivison et al. 2010) because of even lower Σ_IR and a lack of merger feature for our sample. Most recently, Tadaki et al. (2020) reported the discoveries of HST-selected massive (M_star > 10¹¹ M_⊙) star-forming galaxies at z ∼ 2 with R_e,dust ≃ 5 kpc. The dust-to-stellar-size ratio and IR surface luminosity of these sources are generally consistent with those of spatially extended SMGs reported here.

It should be noted that currently spatially extended z ∼ 2 SMGs with a low IR surface luminosity density (Σ_IR ≃ 10^9.5 L_⊙ kpc⁻²) can only be detected and resolved through ALMA observations with deep integration (e.g., pointed observations such as Tadaki et al. 2020; blank-field surveys such as the 1.2 mm ASPECS, Aravena et al. 2020; González-López et al. 2020) or of lensing-cluster fields (this work or ALMA Cycle 6 large program ALCS, K. K. Kohno et al. 2021, in preparation). Previous ALMA Band 6 surveys in cosmological deep fields (e.g., HUDF, Dunlop et al. 2017; GOOD-S, Franco et al. 2018; Hatsukade et al. 2018) can only reach a 4σ depth of ≳0.24 mJy arcsec⁻² at 1.3 mm, which is not sufficient to detect these low-surface-brightness galaxies (≲0.1 mJy arcsec⁻²). Without lensing, these extended SMGs would remain barely resolved with a ∼1'' beam, and therefore, further uv-tapering of the data would hardly improve the detectability of these galaxies. Galaxy lensing usually results in the difficulty of source-plane reconstruction, especially in the rest-frame optical due to the existence of a bright foreground lensing galaxy, introducing large uncertainties into multiwavelength comparison at spatially resolved scales. In contrast, cluster-lensed SMGs, often with a magnification factor of ≳ 5 and reduced contamination from foreground objects, are significantly stretched spatially (R_e ≳ 1''). Therefore, the detectability of cluster-lensed intrinsically extended SMGs can be substantially improved by uv tapering (as shown in Figure 1) or high-sensitivity facilities with a larger beam size and recoverable angular scale.

Qualitatively, one simple way to explain the observed correlation between R_e,dust/R_e,star and Σ_FIR seen in Figure 10 (the left panel) is to assume a model in which a compact dust component (R_e,dust ∼ 1 kpc) with a varying surface luminosity (Σ_IR ≃ 0–10^12.5 L_⊙ kpc⁻²) is superposed on an extended component (R_e,dust ∼ 5 kpc, Σ_IR ∼ 10^9.5 L_⊙ kpc⁻²). In this model, the former corresponds to a central starburst while the latter corresponds to galaxy-wide star formation (e.g., a star-forming disk). Such a two-component model was also suggested by Gullberg et al. (2019) based on the morphological evidence from an SMG stacking analysis. In this model, as the compact component (i.e., the central starburst) increases its brightness, Σ_IR increases while R_e,dust/R_e,star decreases, and the trend will reverse when the compact component fades.

In the left panel of Figure 10, we overplot the behavior of this two-component model (the solid magenta line), using a set of representative values for various parameters. More specifically, the effective radii of the compact and extended dust components were fixed to 1 and 5 kpc. The infrared luminosity of the extended component was fixed to 10^12.1 L_☉ while that of the compact component was varied between 0 and 10^12.9 L_☉ to mimic the rise/decline of the central starburst. R_e,dust was then measured for the combined source. For the stellar component, R_e,star and M_* were assumed to be 3.5 kpc and 10¹¹ M_☉. As Figure 10 shows, this simple two-component model reproduces the observed trend well.

5.3.1. MS Offset

To evaluate the star-forming properties of the observed SMGs further, we calculate the offset between the observed SFR and that of the expected on the star-forming MS, which is a function of redshift and stellar mass (e.g., Speagle et al. 2014; Schreiber et al. 2015). This quantity is commonly referred to as the MS offset (${\rm{\Delta }}\mathrm{MS}=\mathrm{log}[\mathrm{SFR}/{\mathrm{SFR}}_{\mathrm{MS}}({M}_{\mathrm{star}},z)]$), and here, the SFR on the MS (SFR_MS) is calculated using the formula given by Speagle et al. (2014). Following Aravena et al. (2020), we also define the boundaries of the MS as ΔMS = ± 0.4 and classify sources above/below the MS as starburst/passive galaxies.

The accurate derivation of ΔMS requires the knowledge of intrinsic SFR and stellar mass and thus the magnification factor (μ). Based on published cluster-mass models of MACS 1115 and MACS 0553 (Oguri 2010; Zitrin et al. 2015; Ebeling et al. 2017), we derive a median magnification factor of ∼5 for HLS1115 and HLS0553-A/B/C. Therefore, we assume a lensing magnification of μ = 5 for all 26 sources uniformly with morphological measurements. We show that for a z ∼ 2 lensed SMGs with an intrinsic stellar mass of 10¹¹ M_⊙, an uncertainty of 0.5 dex (i.e., a factor of ∼3) with a lensing magnification factor would lead to an error of only 0.12 dex with ΔMS. This is smaller than the uncertainty of observed, lensing-boosted M_star and SFR. Therefore, ΔMS is a relatively robust quantity against the uncertainty of the magnification factor.

Figure 11 displays the distribution of MS offsets and stellar masses for the joint 46 source sample. A general agreement between the lensed and unlensed sample is clear, justifying the use of μ = 5 as a representative magnification factor for the lensed sample. One may notice an anticorrelation between the ΔMS and M_star. However, this could be the consequence of a selection bias against relatively low-mass (M_star ≲ 10^10.8 M_⊙) galaxies without starburst.

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Among the full sample, six (nine) sources can be classified as starburst (passive) galaxies. The remaining 31 sources are therefore galaxies on the MS. K-S tests suggest no significant difference among the redshift distribution of the three subsamples.

Figure 12 (the top panel) shows the relation between the IR surface brightness and MS offset. There is a positive correlation between these two quantities, and an MCMC fitting to the joint 46 source sample suggests $\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{IR}})=(1.10\pm 0.19){\rm{\Delta }}\mathrm{MS}\,+(11.42\pm 0.10)$, although the dispersion is considerable (0.62 dex) for the measured IR surface luminosity. Such a large dispersion was also seen in Elbaz et al. (2018), indicating that central starbursts may or may not be present in galaxies that apparently fall near the MS. This relation demonstrates that sources with larger ΔMS are generally galaxies with more vigorous central star-forming activities and thus higher surface density of IR luminosity. Note, however, that the spatially extended SMGs reported in Section 5.1 are on the star-forming MS, indicating that extended dust continua reflect active star formation over the whole galaxy.

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5.3.2. MS Offset versus Dust-to-stellar-size Ratio

5.3.3. Further Test of the Evolutionary Sequence

5.4.1. Trigger Mechanism of SMGs

5.4.2. The Connection between SMGs and cQs

Previous studies have suggested that SMGs are linked to the cQs seen at slightly lower redshift. The evidence includes the matched number densities (e.g., Simpson et al. 2014; Dudzevičiūtė et al. 2020), clustering properties (e.g., Hickox et al. 2012; An et al. 2019), and size/mass similarities (e.g., Barro et al. 2016b; Lang et al. 2019). In this subsection, we discuss the possible evolutionary connection between SMGs and cQs using those physical quantities that are independent of lensing effects.

We show that a subset of SMGs in our sample exhibits similar dust-to-stellar-mass ratios (four at M_dust/M_star < 10⁻³) to early-type galaxies at low redshift (Figure 7). Such a low dust-to-stellar-mass ratio is consistent with the stacked z ∼ 1.8 quiescent galaxies (Gobat et al. 2018). Assuming a gas-to-dust ratio of 100, these galaxies also match with the z ∼ 0.7 post-starburst (PSB) galaxies reported in Suess et al. (2017). One caveat is that the lack of NIR data may lead to an overestimate of dust extinction and thus stellar mass through an energy-balance approach in the SED fitting (e.g., HLS1623), but we show that for HLS1314 with accurate J/K-band photometry, its A_V is tightly constrained and thus its log(M_dust/M_star) can be determined as −3.25 ± 0.10, similar to the values for PSBs and early-type galaxies quoted above.

We also show that the stellar surface density (Σ_star) of SMGs in this work match better with early-type than late-type galaxies at similar redshift (van der Wel et al. 2014), in the right panel of Figure 10. Because the spatial distribution of star-forming regions in SMGs is typically more compact than that of stars, the ongoing intense star formation will lead to an even higher Σ_star in later phases, increasing the difference from the typical value of late-type galaxies at that cosmic age. Such a comparison has been conducted by Barro et al. (2013), Hodge et al. (2016), and Lang et al. (2019) via HST imaging and interpreted as a structural consistency between the stellar components of SMGs and quiescent galaxies, suggesting a possible evolutionary link after the cessation of star formation in SMGs.

We further show that the IR surface luminosity and spatial extent of extended SMGs in this work may match with SMGs in a transitional phase to quiescent galaxies. Gullberg et al. (2019) showed the existence of an extended dust component (R_e,dust ∼ 4 kpc) of typical z ∼ 3 SMGs by stacking 153 of them in the ALMA Band 7. Such an extended component is reported to contribute to ∼13% of the total emission at 870 μm, and the corresponding surface luminosity is Σ_IR ∼ 10^9.9 L_⊙ kpc⁻², under the assumption of a typical SED of an SMG at z ∼ 3. Furthermore, in the sample of Gullberg et al. (2019), the FIR surface brightness of the compact dust component (R_e,dust ∼ 1 kpc) decreases with the decline of total L_IR while it remains the same for the extended component (Figure 13).

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In this work, we do discover SMGs with an IR surface brightness comparable to the faint and extended component of SMGs reported by Gullberg et al. (2019), and such a comparison is presented in Figure 13. Taking HLS0546 (Σ_IR = 10^{9.8 ± 0.3} L_⊙ kpc⁻²) as an example, assuming a reasonable lensing magnification of μ ≃ 5, the physical effective radius of the dust continuum will be 6.3 ± 1.0 kpc. Consider the radial gradient of dust temperature; this result could be consistent with the size of the extended dust continuum of a typical SMG observed at rest-frame 220 μm (Gullberg et al. 2019). This magnification assumption will also lead to a demagnified stellar mass of M_* = 10^{11.0 ± 0.4} M_⊙ and R_e,star = 3.5 ± 0.6 kpc, within a 1σ distribution of early-type galaxies seen at the given redshift (van der Wel et al. 2014). Therefore, it is possible that after the dissipation of central star formation on a compact physical scale (1–2 kpc), SMGs will maintain a low-level IR surface brightness over a more extended galaxy structure (≳4 kpc), as suggested by the evolutionary tracks in Section 5.3.2. At the same time, the previous concentrated star formation would lead to the formation of compact (∼2 kpc) quiescent and spheroidal galaxies within a timescale of ∼300 Myr. Therefore, spatially extended SMGs with large dust-to-stellar-radii ratio in this study may provide possible evidence of the evolutionary connection between typical SMGs and compact quiescent galaxies at a slightly lower redshift.