The SCUBA-2 Cosmology Legacy Survey: The EGS deep field - III. The evolution of faint submillimeter galaxies at $z<4$

Our sample of faint SMGs is extracted from the 450 and $850~{}\mu\rm m$ catalogues of the SCUBA-2 Cosmology Legacy Survey (S2CLS; Geach et al., 2013, 2017) in the $\sim 70$ arcmin² deep survey of EGS (Zavala et al., 2017), at $\sigma_{450\mu\rm m}=1.2~{}\rm mJy/beam$ and a deeply confused instrumental $\sigma_{850\mu\rm m}=0.2~{}\rm mJy/beam$. From the initial 92 galaxies detected at $450~{}\mu\rm m$ and 108 galaxies detected at $850~{}\mu\rm m$, Zavala et al. (2018) identified robust optical or NIR counterparts for 71 of them using radio (1.4 GHz), $8~{}\mu\rm m$ and $24~{}\mu\rm m$ observations to improve on the astrometry and link the SMGs to their most probable associations. They estimated that 13 per cent of the faint SMGs could have incorrect counterpart associations and discarded six sources with discrepant photometric redshifts derived from optical-IR and FIR data to minimize the impact of potential incorrect identifications. They estimated the IR luminosities ($L_{\rm IR}$), FIR-based star-formation rates ($\rm{SFR_{FIR}}$) and dust temperatures of the sample using the SCUBA-2 and Herschel photometry at the NIR or radio positions of the counterparts and analyzed the evolution of $\rm{SFR_{FIR}}$, dust temperature and morphology through time.

From these 71 galaxies with optical counterparts, we further restrict the main sample of study to 57 galaxies which fall within the footprints of the deep Hubble Space Telescope (HST) imaging where our comparison sample of optically selected star-forming galaxies is extracted. The dusty galaxies have a median $S_{850\rm\mu m}=2.0\pm 1.2~{}\rm mJy/beam$ and $L_{\rm IR}=10^{12.0\pm 0.5}~{}\rm L_{\odot}$, such that 37 per cent have luminous infrared galaxy (LIRG) luminosities ($10^{11}~{}\rm{L_{\odot}}$ $\leq L_{\rm IR}<10^{12}~{}\rm{L_{\odot}}$), 54 per cent have ultraluminous infrared galaxy (ULIRG) luminosities ($10^{12}~{}\rm{L_{\odot}}$ $\leq L_{\rm IR}<10^{13}~{}\rm{L_{\odot}}$) and 9 per cent have luminosities below the LIRG regime ($L_{\rm IR}<10^{11}~{}\rm{L_{\odot}}$).

We extract a comparison sample of optically selected SFGs from the EGS field mapped by the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS - Grogin et al., 2011; Koekemoer et al., 2011). Stefanon et al. (2017) presented the catalogue of 41,457 sources detected in the H-band (F160W) within the deep $\sim 206~{}\rm{arcmin}^{2}$ footprint of the survey. The 3D-HST catalogue (Skelton et al., 2014; Momcheva et al., 2016) of the field contains 41,200 objects. We cross-matched the two catalogues using a search radius of $r=0.5~{}\rm{arcsec}$.

The SFG sample is defined following the method outlined by Fang et al. (2018). We apply the following selection criteria: (i) magnitude in H-band < 24.5 mag to ensure good GALFIT fittings; (ii) SExtractor parameter CLASS_STAR <0.9 to avoid contamination by stars; (iii) Photometric (PhotFlag=0) and structural parameter (GALFIT flag=0) quality flags to exclude spurious sources and ill-constrained GALFIT estimations. We then use the rest-frame $U-V$ vs. $V-J$ diagram to discriminate between star-forming and quiescent galaxies. Figure 1 shows the color-color diagram for galaxies at $1.0<z<2.0$ and Appendix A contains the diagrams for all redshiff bins. We segregate star-forming and quiescent galaxies using the loci defined by Williams et al. (2009). At $z>2.0~{}$ they found that galaxies showed a relation between the rest-frame $V-J$ color and $24~{}\mu\rm m$ flux densities, which implies that their selection criteria, despite being incomplete and classifying some red star-forming systems as quiescent, is still extendable to higher redshifts. Therefore, for $z>2.5$ we use the division relationships as for $2.0<z<2.5$.

The SFG sample contains all $UVJ$-classified star-forming galaxies that are not counterparts of SMGs: 4878 sources at $0.2<z<4$. This sample will allow us to derive the properties of faint SMG counterparts and coeval optically-selected star forming galaxies in the same way, to assess if faint SMGs are unique in any way among star-forming galaxies.

The CANDELS catalogues¹¹1 https://archive.stsci.edu/missions/hlsp/candels/egs/catalogs/v1/ present stellar masses estimated by 10 independent teams. The methodologies are described by Santini et al. (2015), who found that all methods that used the same stellar population templates are in overall good agreement. Mobasher et al. (2015) also showed a comparison of the stellar masses independently derived by the different teams, finding no significant bias between them and a similar scatter of $\sigma(\log M_{\star}/\rm{M_{\odot}})=0.136~{}\rm dex$.

We adopt the stellar masses of the 2a_tau team, who also presents star formation rates. The stellar masses were estimated fitting spectral energy distributions (SED) with the FAST code (Kriek et al., 2009), a Chabrier (2003) Initial Mass Function (IMF), Bruzual & Charlot (2003) stellar population templates and the Calzetti et al. (2000) extinction law.

We will use 3D-HST redshifts (Momcheva et al., 2016) for the SMG and SFG sample. The mean difference between 3D-HST and CANDELS photometric redshifts is $0.013\pm 0.003$. All SMGs have optical-IR photometric redshifts consistent with FIR photometric redshifts (Zavala et al., 2018), as this was used as a criterium to exclude counterparts that were possible misidentifications (see section 2.1).

The SFRs estimated by the 2a_tau team use a combination of ultraviolet (UV) and IR-based tracers (Barro et al., 2019):

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  $\rm{SFR_{UV}}$, the SFR derived from the 2800 Å rest-frame flux density, not accounting for dust extinction.

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  $\rm{SFR_{UV}^{corr}}$, the SFR derived from the 2800 Å rest-frame flux density, corrected by dust extinction.

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  $\rm{SFR_{UV+IR}^{W11}}$, the SFR extrapolating the Spitzer/MIPS $24~{}\mu\rm m$ emission with the SFR template by Wuyts et al. (2011), and co-adding SFR_UV, $\rm{SFR_{UV+IR}^{W11}}=\rm{SFR_{UV}}+\rm{SFR^{W11}_{IR}}$.

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  $\rm{SFR_{UV+IR}^{Herschel}}$, the SFR derived from fitting dust emission templates to Herschel photometry, and co-adding the SFR_UV, $\rm{SFR_{UV+IR}^{Herschel}={SFR_{UV}}+{SFR}^{Herschel}_{\rm IR}}$.

These estimates are available for both the SFG and SMG sample.

We also use in our analysis the $\rm{SFR}_{FIR}$ estimated by Zavala et al. (2018), where they fitted a modified black body with fixed emissivity index $\beta=1.6$ at the redshift of the 3D-HST catalogues (Skelton et al., 2014; Momcheva et al., 2016) to estimate $L_{\rm IR}$ and then calculated SFR_FIR using the Kennicutt (1998) calibration with a Chabrier (2003) IMF.

Since source 850.26 has no $L_{\rm IR}$ reported in Zavala et al. (2018), but has a robust optical counterpart, in this paper we estimate $L_{\rm IR}$ for this galaxy in a similar manner. We fitted a modified black body with the median dust temperature of the sample $T_{\rm{d}}=45~{}\rm K$ and a fixed emissivity index $\beta=1.6$ at the 3D-HST redshift ($z=2.52\pm 0.01$), obtaining $L_{\rm{IR}}=10^{12.16\pm 0.08}~{}\rm L_{\odot}$ and $\rm{SFR_{FIR}}=150\pm 30~{}\rm M_{\odot}~{}yr^{-1}$, which is within the values found for the rest of the SMG sample.

The dust extinction $A_{V}$ presented by Barro et al. (2019) is estimated from the slope of the UV continuum, $\beta_{\rm UV}$, and then corrected using the IRX-$\beta_{\rm UV}$ relation to account for multiple attenuation laws, where IRX is the infrared excess derived from the ratio between IR-based and UV-based SFRs. They found higher IRX values for galaxies with higher $\rm{SFR_{IR}}$ at the same $\beta_{\rm UV}$ value, where galaxies with $\rm SFR_{IR}>70~{}\rm M_{\odot}yr^{-1}$ lie above the IRX-$\beta_{\rm UV}$ relation.

We adopt the structural parameters derived by Van der Wel et al. (2014) from the HST $H$-band images. Using an automated process Van der Wel et al. (2012) fitted a Sérsic model with GALFIT (Peng et al., 2010) to the galaxies in the field.

For those SMG counterparts with flagged GALFIT fits or outlier structural parameters, we replaced the fits of Van der Wel et al. (2014) by those of García-Rivero (2018), who also performed GALFIT fits, individually, carefully masking out any companion galaxies and nearby stars.

We also use the visual morphological classification presented by Huertas-Company et al. (2015), where they trained a Convolutional Neural Network (CNN) with the visual classification of galaxies in the GOODS-S field by Kartaltepe et al. (2015). Huertas-Company et al. (2015) then classified the galaxies in all the CANDELS fields using the same method.

The mean stellar mass of the faint SMG sample studied in this work is $\log(M_{\star}/\rm{M_{\odot}})=10.75\pm 0.07$. This is slightly lower than the $\log(M_{\star}/\rm{M_{\odot}})=10.95\pm 0.03$ value estimated by Zavala et al. (2018), who adopted the Momcheva et al. (2016) 3D-HST catalogue values. The difference between these estimates is mainly driven by 4 galaxies without estimations of stellar masses in the 3D-HST catalogue, and 3 galaxies with different redshift estimations between the CANDELS and 3D-HST catalogues, which result in mass differences $\Delta(\log(M_{\star}/\rm{M_{\odot}}))=1-2~{}\rm dex$.

The distributions of the stellar masses of SFGs and SMGs in 4 redshift bins are presented in Figure 2, and their medians are listed in Table 1. Errors in the medians are estimated through a bootstrap analysis. The median masses of SMGs are systematically larger than those of SFGs. In order to assess the statistical significance of this claim, we apply a Mann-Whitney test (Mann & Whitney, 1947) to analyse whether the medians of both populations could be compatible with that of a single parent distribution. For all redshift bins the probabilities for the null hypothesis to be true are $p<0.05$ (see Table 6), and hence we reject the null hypothesis of statistical identity of the medians. We use the uncertainty in the redshift estimations in the 3D-HST catalogue to assign to each galaxy a new redshift and recalculate the test 5000 times, finding that the result is robust ($p\sim 5\times 10^{-3}$ – $3\times 10^{-13}$ at different redshift bins). We hence conclude that faint SMGs typically have higher masses than optically selected star-forming galaxies, as has been discussed in the literature for classical SMGs (e.g. Blain et al., 2004; Chapman et al., 2005; Yun et al., 2012).

We note that we do not have sufficient statistical significance to claim an increment in the stellar mass of SMGs from the redshift bin $0.2<z<1$ to $1<z<3$. In contrast, the SFG population shows a consistent increase in stellar mass with redshift, due to Malmquist bias.

In order to check if dust extinction could bias our results, we select a sub-sample of SFGs with similar dust extinction $A_{V}$ and $V-J$ colors as the SMG sample: $A_{V}=1.4\pm 0.09$, $V-J=1.38^{+0.03}_{-0.05}$. We still find the statistical difference between stellar masses of the SMG and SFG samples at $0.2<z<3$. At $3<z<4$, however, the null hypothesis of identity cannot fomally be rejected ($p=0.25$). Hence, color selection and dust extinction, as measured from optical-infrared data, cannot account for the differences in mass found between the SFG and SMG samples.

We explore the morphological differences between faint SMGs and SFGs in H-band using the structural parameters derived by Van der Wel et al. (2014). We selected the SFG sample to have only good GALFIT fits with flag=0. Since Van der Wel et al. (2014) estimated that $\sim 5$ per cent of the sample had catastrophic or bad fits, errors in size, redshift or stellar mass, and our SMG sample is small, we reviewed the H-band postage stamps of all SMG counterparts and individually evaluated whether we could accept the morphological parameters provided in the catalogue. We specially examined the cases with bad GALFIT flags, cases with very small or large sizes ($R_{\rm e}<0.6~{}\rm kpc$ or $R_{\rm e}>10~{}\rm kpc$) or Sérsic indices close to the constraints introduced in the automated process ($n=0.2$ and $n=8$). We replaced the structural parameters of the outlier galaxies with those estimated by García-Rivero (2018), who also fitted them with a single Sérsic profile using GALFIT, after masking out any companion galaxies or nearby stars, and found reduced-$\chi^{2}$ values similar to those in Van der Wel et al. (2014). The following galaxies have revised structural parameters:

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  850.028 at $z=2.5^{+0.4}_{-0.1}$ lies close to the diffraction spike of a field star. The catalogue shows a large radius of $R_{\rm e}=23\pm 15~{}\rm kpc$ and disk-like morphology with index $n=1.06\pm 1.22$, and a good-fit flag=0. We adopt the more moderate radius $R_{\rm e}=4.8~{}\rm kpc$ and $n=1.23$.

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  850.007 lies at $z=3.0\pm 0.1$ and the catalogue includes a very large radius $R_{\rm e}=31\pm 17~{}\rm kpc$ and Sérsic index $n=8\pm 6$ with a bad fitting flag. We adopt $R_{\rm e}=1.97~{}\rm kpc$ and $n=0.68$.

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  850.030 and 850.069 are located at $z=1.52\pm 0.06$ and $z=0.560\pm 0.001$, and have very close companions and perturbed morphologies. Their morphological parameters in the catalogue are $R_{\rm e}=21.5\pm 0.3~{}\rm kpc$, $n=1.10\pm 0.03$ and $R_{\rm e}=11.1\pm 0.3~{}\rm kpc$, $n=1.310\pm 0.005$, respectively. We use instead the more conservative values $R_{\rm e}=14.8~{}\rm kpc$, $n=1.25$ and $R_{\rm e}=9.2~{}\rm kpc$, $n=0.71$, respectively.

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  850.044 ($z=3.17\pm 0.14$), 850.56 ($z=2.48\pm 0.01$) and 850.072 ($z=3.36\pm 0.35$) have bad fitting flags in the catalogue. We adopt $R_{\rm e}=2.48~{}\rm kpc$, $n=0.45$; $R_{\rm e}=8.46~{}\rm kpc$, $n=0.49$; and $R_{\rm e}=1.83~{}\rm kpc$, $n=0.32$, for these cases, respectively.

Table 4 lists the median values of the structural parameters of SFGs and SMGs in four redshift bins.

Our analysis is based on the identifications of optical-infrared counterparts presented in Zavala et al. (2018). There is the potential of misidentification, which was estimated to be of 13 per cent for the full sample. To miminize the impact of misidentifications, six sources with discrepant photometric redshifts from optical-IR and FIR-radio methodologies were discarded.

We nevertheless adopt a 13 per cent contamination into our final sample of 57 SMGs and explore if the conclusions derived in sections 4.1 to 4.3 are robust regardless of the possible contamination of misidentified counterparts. In order to do this, we randomly assign to 13 per cent of the faint SMGs the properties of another SMG in the sample with similar FIR-colours (indicative of redshift), irrespective of their brightness. We recomputed all estimations for median stellar mass, star formation rate, size and morphology per redshift bin. We find that the results do not change and our conclusions for the mean properties of the population are robust.

In section 4.1 we found that the stellar masses of SMGs are consistent with an average value of $\log(M_{\star}/\rm{M_{\odot}})=10.75\pm 0.07$ across all redshifts. We checked that dust obscuration as measured by $A_{V}$ was not biasing this result. We also showed, however, that $A_{V}$ was not able to correct for the total obscuration of SMGs alone (section 4.2.3). Hence, the question still remains whether the stellar mass enshrouded by dust associated to the bulk of $L_{\rm IR}$ emission is significant compared to the mass measured through the most complete optical-MIR photometry available.

Michałowski et al. (2014) simulated a sample of SMGs and their synthetic photometry, finding that a double-peaked burst was the best Star Formation History (SFH) to reproduce the measured colors of SMGs. An exponentially declining SFH returns stellar masses that are lower, but still consistent with their inputs. They also found that the correct age plays a more important role in the estimation of the stellar mass than dust extinction $A_{V}$. Therefore, even though $\rm{SFR_{\rm UV}^{corr}}$ cannot account for the total SFR of SMGs, the masses, which depend on the rest-frame infrared flux densities, are not heavily affected. Based on their results, we estimate that the stellar masses of SMGs could be underestimated by 0.3-0.5 dex, which is a factor of $\sim 2$ times the intrinsic scatter between methods found in the CANDELS catalogs (Mobasher et al., 2015).

We tested the effects of this possible bias, considering a stellar mass 0.3 and 0.5 dex larger than the original stellar mass reported in the CANDELS catalog. The sSFRs are consequentely 0.3 and 0.5 dex smaller. The bias hence would displace SMGs in a diagonal line in Figure 3 towards higher $M_{\star}$ and lower sSFR.

The Mann-Whitney results for the $\Delta\log\rm sSFR$ medians of SFGs and SMGs remain the same for a 0.3 dex stellar mass offset. We would find in this situation 71 per cent of faint SMGs above the main sequence (as compared to the original 82 per cent), and 29 per cent of starbursts (instead of 40 per cent). If we adopt a 0.5 dex stellar mass offset, we would find 61 per cent of faint SMGs above the main sequence and 16 per cent of starbursts. The highest fraction of SMGs above the main sequence still is at $2.5<z<3$.

Hence, considering the possible stellar mass bias, we would find more faint SMGs tracing the main sequence and fewer starbursts, while a large fraction of faint SMGs still populate the upper parts of the scatter of the main sequence.

In this paper we derive that 35–40 per cent of faint SMGs are starbursts, based on the $R_{\rm SB}=\rm{SFR/SFR_{MS}}>3$ criterion and our best estimates of stellar mass and SFR (sections 4.2.2 and  4.2.3).

In previous works the fraction of SMGs classified as starburst varies according to the selection wavelength and depth of the catalog.

Zavala et al. (2018) reported that 85 per cent of their sample of 72 faint SMGs lie within the $3\sigma$ scatter of the main sequence of Speagle et al. (2014). After the exclusion of the SMGs without CANDELS and 3D-HST counterparts and discriminating by redshift bins, we find a 96 per cent fraction of faint SMGs within the $3\sigma$ scatter of the main sequence we derived. We note that the main sequence adopted in Zavala et al. (2018) is shallower than that derived in this paper at $z>2$, and their SFR slightly smaller than those adopted here, and hence the location of SMGs with respect to the main sequence is different. We also find that the residual sSFR to the main sequence for these faint SMGs is significantly different to the distribution of SFGs at $z>1$, even when possible stellar mass biases are considered (section 5.1 ).

Da Cunha et al. (2015) followed-up with ALMA a sample of SMGs with $S_{870\mu\rm m}>4~{}\rm mJy$. They found that half of their SMGs are located above the main sequence with $R_{\rm SB}>3$. We find similar fractions of galaxies above the main sequence in our sample of faint SMGs, when we derive the fractions at similar depths.

In a 1.3 mm ALMA survey of 16 bright galaxies with $S_{1.1\rm mm}>3.3~{}\rm mJy$, Miettinen et al. (2017a) estimated 63 per cent of SMGs are starbursts with $R_{\rm SB}>3$. They hence found a much higher fraction of starburst galaxies, which could be due to the selection wavelength and brightness of the sample. In a subsequent study, Miettinen et al. (2017c) imaged a larger sample of the 129 brightest AzTEC sources in the COSMOS field with ALMA at 1.3 mm, finding 57 per cent of galaxies within the main sequence scatter (and below $R_{\rm SB}\sim 3$). From the starburst SMGs 49 per cent are preferentially located at $z>3$. This differs from our finding of the highest fraction of starbursts being located at $2.5<z<3$, but our statistics at the highest redshift bin are poor.

Franco et al. (2020) studied a sample of 35 ALMA 1.1 mm-detected galaxies in the GOODS-S field, finding that 54 per cent of them have an offset to the main sequence over $R_{\rm SB}>3$. The catalogue is created from a sample of 19 sources detected in a blind survey at $S_{1.1\rm mm}\geq 0.7~{}\mu\rm Jy$ and 16 additional galaxies detected at lower S/N using the Spitzer/IRAC and VLA counterparts. Therefore, they reached a deeper selection limit due to the supplementary catalogue, but found a fraction of starbursts consistent with other samples selected at the same wavelength.

Lim et al. (2020a) studied a sample of $450~{}\mu\rm m$-selected faint SMGs in the STUDIES survey with $L_{\rm IR}\sim 10^{11}~{}{\rm L_{\odot}}$. They found that 35 per cent of the SMGs are starburst.

When the different selection wavelengths, depths, definition of star-formation main sequence and the small sizes of the samples are taken into account, we see an overall agreement in the starburstiness of faint SMGs of around 30-50 per cent, while classical SMGs have reported starbustiness in excess of 50 per cent.

In section 4.2 we found that the absence of mid-to-far IR data heavily impacts the determination of the total SFR of faint SMGs due to an extrapolation from the rest-frame UV to NIR estimation of obscuration. Counter-intuitively, we found that sources with the lowest optically-inferred obscuration ($A_{V}$) were also those that had the strongest underestimations of their total SFR, based on $\rm{SFR_{UV}^{corr}}$. This dust extinction was estimated using the UV continuum’s slope, as well as the IRX-$\beta_{UV}$ relation to correct (see section 3). However, as it was indicated in section 4.2.3, there are discrepancies between the CANDELS and S2CLS teams estimation of $L_{\rm IR}$. Our interpretation is that this is the direct result of patchiness in the dust distribution within these galaxies, such that the estimations from UV-optical data come from different regions than the FIR emitting ones.

A similar conclusion was derived by Wuyts et al. (2011), who found that dust-correction methods to infer the SFR of the highest star-forming galaxies ($\rm{SFR>100~{}M_{\odot}~{}yr^{-1}}$, especially at $z>2.5$) fail to recover the total amount of star formation, when compared to the $\rm{SFR_{FIR}}$ derived from Herschel $70-160~{}\mu\rm m$ data. They estimate that the templates enhanced with PAH emission are not enough to reproduce the FIR contribution to the total SFR from $24~{}\mu\rm m$ fluxes for these high star-forming systems, which are affected by patchy dust obscuration. The uneven dust obscuration causes the saturation of the UV-slope derived extinction, $A_{V}$. This effect has also been seen in local LIRGs and starburst galaxies that are located above the $\rm IRX-\beta$ relation (Meurer et al., 1999; Goldader et al., 2002; Howell et al., 2010). Some of our SMGs are indeed located above the IRX-$\beta$ relation (Zavala et al., 2018). This effect could be due to various factors, including differing star-formation histories (Salmon et al., 2016; Calzetti et al., 2021).

Elbaz et al. (2018) observed a sample of massive starburst galaxies selected in Herschel bands and a complementary sample of 1.3 mm-selected galaxies. They found that the FIR and the UV emissions of starburst galaxies with $R_{\rm SB}=\rm{SFR/SFR_{MS}}>3$ had systematic spatial offsets. Their SFRs estimated by fitting the UV-to-near IR fluxes is consistent with $\rm{SFR_{UV+IR}}$ for $R_{\rm SB}<1$, but increasingly discrepant above the main sequence. In our sample of SMGs 40 per cent have $R_{\rm SB}>3$.

This further supports our claim that the optical dust correction underestimates the total SFR for the population of dusty star-forming galaxies.

In section 4.3.5 we found that most SMGs have a disk-like morphology according to the machine classification, and in section 4.2.2 we found that 82 per cent of faint SMGs are above the main sequence relationship between sSFR and stellar mass. The location of galaxies in the main sequence has been linked to the morphology, where main sequence (U)LIRGs are dominated by non-interacting disks, most galaxies classified as starbursts are either irregular disks or mergers, and spheroids and disk+spheroids are below the main sequence (Kartaltepe et al., 2015; Osborne et al., 2020). Considering that 32 per cent of the SMGs in our sample lie within $1\sigma$ of the main sequence we find consistently that $\sim 76$ per cent are disk-like from visual morphology classifications. We find decreasing fractions of merger SMGs and SFGs towards lower $z$, and lower fractions of merger SFGs than SMGs at all redshifts. There are 35 per cent of mergers in the whole sample of faint SMGs, which is similar to the 40 per cent fraction of starburst found in our sample. Among these starburst SMGs we find indeed 88 per cent of them can be classified as mergers or irregular disks, but we also find these morphologies within the main sequence.

Based on Sérsic fits, however, $\sim 95$ per cent of classical SMGs have been found to be best described as massive $\log(M_{\star}/\rm M_{\odot})\sim 11.3$ disk-like galaxies, while only $\sim 25$ per cent could be classified as mergers based on the presence of multiple clumps in $H$-band images (Targett et al., 2013). Visual classification on classical SMG samples finds also a morphological evolution with redshift, such that merging and irregular morphologies give way to more ordered disk morphologies at lower reddshifts (Ling & Yan, 2022).

We further find in our analysis (sections  4.3.2 and 4.3.4) that both SMGs and SFGs with $\log(M_{\star}/\rm M_{\odot})>10$ increase their concentration towards lower redshifts ($n$ and $C$), but they do so at different rates. SMGs are more concentrated at $z<1$, both considering $n$ and $C$, while at $z>2$ SFGs have larger values of the concentration parameters than SMGs. The cross over of concentration between the populations happens at $1<z<2$. At $0.2<z<1.4$ $n$ is significantly larger in SMGs than in SFGs, while at $2<z<3$ $C$ is significantly smaller in SMGs.

Throughout the evolution of both populations, the bulge component is increasing towards lower redshifts (more disk+sph morphologies at lower redshifts). The statistics for SMGs allows to establish that bulge increase at $z<2$ through the machine-assigned categories, but this classification goes in hand with the increment of $n$ and $C$ in the parametric and non-parametric analysis of the images.

The smooth increase in $C$ and $A$ non-parametric indices at a fixed observed band has been found in other samples (Whitney et al., 2021), and these are in part attributed to rest-frame pass-band changes and reduced resolution. Since our goal is to make a direct comparison to SFGs at the same redshift, a correction for these effects is not necessary.

We find a significant difference in the sizes of SMGs and SFGs at $z<3$ (section 4.3.1), that is mainly driven by the difference in mass selection of the SFG and SMG samples. However, when we study the offset of both populations from the $\log R_{\rm e}-\log M_{\star}$ relationship, we find that SMGs are $\sim 50$ per cent larger than SFGs of the same mass at $2<z<3$, which is the redshift bin with the largest fraction of starbursts and high sSFR galaxies (65 per cent above $1\sigma$ of the Main Sequence). Some of the SMGs in this redshift bin have close companions or disturbed morphology.

When studying massive galaxies with $\log(M_{\star}/\rm M_{\odot})>10$, SFGs have on average monotonically increasing sizes towards lower redshifts. SMGs have larger sizes than the SFGs at $z<3$, but not at $z>3$. This could be due to obscuration effects, that only allow the patchy less obscured areas to be revealed in the highest redshift SMGs $H$-band images. In the $H$-band images the emission traced corresponds to the rest-frame $U$-band ($\sim 0.33~{}\mu\rm m$) at $z\sim 4$ and to the rest-frame NIR ($\sim 1.4~{}\mu\rm m$) at $z\sim 0.2$, less susceptible to dust obscuration. Indeed, Chen et al. (2022a) found progresively smaller sizes at longer observation wavelengths (considering $HST$, $Spitzer$ and $JWST$ data) in the analysis of the size using curve of growth for a small sample of faint SMGs, which includes 6 of the galaxies studied in this work.

The median effective radii of the SMGs in our sample are consistent within the uncertainties with the H-band radii derived in bright SMG samples and other studies of dusty galaxies below the classical SMG regime (Targett et al., 2013; Chen et al., 2015; Chang et al., 2018; Lim et al., 2020b; Franco et al., 2020). Furthermore, the resolved FIR emission of bright SMGs has been found to be more compact than the optical counterparts (Gullberg et al., 2019; Hodge et al., 2019).

Hence, while we find significant morphological similarities between faint SMGs and SFGs, like both populations favoring disk-like galaxies, there are significant differences at $2<z<3$, where SMGs are 50 per cent larger than SFGs of similar mass, starbursts are more prominent and also the morphological classification indicates more disturbed morphologies (81 per cent of irregular disks and mergers).

The newly released data of the JWST allows some clarity on the resolved morphology of these type of sources. A lensed grand-design spiral galaxy discovered by ALMA, located in an overdense region at $z=3$, shows evidence of a minor merger and asymmetry (Wu et al., 2022). However, a larger sample of SMGs is needed to establish the possible multiple evolutionary tracks of the dusty star-forming population, as evidenced by (Cheng et al., 2022). We also find that faint SMGs develop (or reveal) a concentration in their $H$-band images at later times than SFGs of the same stellar mass.