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

L. Cardona-Torres,1 I. Aretxaga,1 A. Montaña,1 J. A. Zavala2,3 and S.M. Faber4
1 Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Luis
Enrique Erro 1, Sta. Ma. Tonantzintla, 72840, Puebla, Mexico
2 Department of Astronomy, The University of Texas at Austin, 2515 Speedway
Blvd Stop C1400, Austin, TX 78712, USA
3 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
4 University of California Observatories/Lick Observatory, University of
California, Santa Cruz, CA 95064, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a demographic analysis of the physical and morphological properties
of $450/850~{}\mu\rm m$-selected galaxies from the deep observations of the
SCUBA-2 Cosmology Legacy Survey in the Extended Groth Strip that are detected
below the classical submillimeter-galaxy regime ($S_{850\mu\rm m}\lesssim
6~{}\rm mJy$/beam) and compare them with a sample of optically-selected star-
forming galaxies detected in the Cosmic Assembly Near-infrared Deep
Extragalactic Legacy Survey in the same field. We derive the evolution of the
main sequence of star-forming galaxies, finding a steeper specific star
formation rate versus stellar mass at $z>2.5$ than previous studies. Most
faint submillimeter-galaxies fall within $3\sigma$ of the main sequence, but
40 per cent are classified as starbursts. Faint submillimeter galaxies have 50
per cent larger sizes at $2<z<3$ than optically-selected star-forming galaxies
of the same mass range. This is also the redshift bin where we find the
largest fraction of starbursts, and hence we could be witnessing merging
processes, as confirmed by the preference for visual-morphology
classifications of these systems as irregular disk galaxies and mergers. Both
populations show an increment towards lower redshifts ($z<2$) of their
concentration in $H$-band morphology, but faint submillimeter galaxies on
average show larger concentration values at later times. These findings
support the claim that faint submillimeter galaxies are mostly a population of
massive dust-obscured disk-like galaxies that develop larger bulge components
at later epochs. While the similarities are great, the median sizes, starburst
numbers and $H$-band concentration of faint submillimeter galaxies differ from
those of optically-selected star-forming galaxies of the same stellar mass.

###### keywords:

submillimeter:galaxies – galaxies: high redshift – galaxies: star formation –
galaxies: structure

††pubyear: 2022††pagerange: The SCUBA-2 Cosmology Legacy Survey: The EGS deep
field - III. The evolution of faint submillimeter galaxies at $z<4$–B

##  1 Introduction

The initial studies of the Cosmic Infrared Background with the Cosmic
Background Explorer (COBE, Puget et al., 1996; Dwek et al., 1998) led to the
discovery of a high redshift population of galaxies with strong far infrared
(FIR) emission. These galaxies were first detected at $850~{}\mu\rm m$ with
the James Clerk Maxwell Telescope and were named submillimeter galaxies (SMGs;
Smail et al., 1997; Hughes et al., 1998; Barger et al., 1998). The discovery
indicated that there was a considerable amount of stellar emission obscured by
dust in the high redshift Universe. SMGs are typically located at high
redshifts ($z>1$), have large infrared luminosities ($L_{\rm IR}\gtrsim
10^{12}~{}\rm L_{\odot}$), star formation rates (SFRs$\gtrsim
300~{}\rm{M_{\odot}~{}yr^{-1}}$) and high gas reservoirs
($10^{10-11}~{}\rm{M}_{\odot}$) (e.g. Blain et al., 2004; Chapman et al.,
2005; Aretxaga et al., 2007; Yun et al., 2012; Casey et al., 2014; Da Cunha et
al., 2015; Cowie et al., 2017; Michałowski et al., 2017; Dudzevičiūtė et al.,
2021; Birkin et al., 2021; Chen et al., 2022b). The early submillimeter
surveys traced with single-dish telescopes reached a population of galaxies
with flux densities $S_{850\mu\rm m}\gtrsim 6~{}\rm mJy$, which we will refer
to as classical SMGs (Smail et al., 1997; Hughes et al., 1998; Coppin et al.,
2006; Geach et al., 2017; Simpson et al., 2019).

The counterparts of SMGs were typically found in deep optical, IR and radio
imaging by exploiting the radio-submillimeter correlation (Carilli & Yun,
2000). This allowed the exploration of the multiwavelength properties of SMGs
(i.e Chapman et al., 2005; Targett et al., 2013; Zavala et al., 2018; Lim et
al., 2020a; Dudzevičiūtė et al., 2021), and their role in the cosmic history
of star formation (Madau & Dickinson, 2014). SMGs were initially characterized
as extreme star-forming galaxies. They were associated with major mergers that
enhanced the star-formation activity through their interactions (Tacconi et
al., 2008). On the other hand, there is a population of SMGs that lie within
the scatter of the high-mass end of the star-formation main sequence, but
appear to have enhanced star formation efficiency (Davé et al., 2010). The
exploration of high-resolution FIR surveys have allowed the detection of two
sub-populations of SMGs: starburst and main sequence galaxies with a compact
core component and main sequence galaxies with an extended dimmer component
(i.e. Simpson et al., 2015; Michałowski et al., 2017; Elbaz et al., 2018;
Gullberg et al., 2019; Tadaki et al., 2020; Puglisi et al., 2021). The compact
FIR emission is associated with a post-starburst phase due to the low gas
fraction detected in these galaxies that could be explained with a history of
strong inflow mergers, which enhance the star formation efficiency (French,
2021).

Different studies have used interferometric follow-up of single dish surveys
(e.g. Hodge et al., 2013; Da Cunha et al., 2015; Miettinen et al., 2017b) to
explore SMG propeties at higher resolution. They have shown that there is a
$\sim 26$ per cent chance of finding multiple counterparts to SMGs with fluxes
brighter than $S_{850\mu\rm m}\geq 5~{}\rm mJy$ (Stach et al., 2019). However,
fainter sources suffer to a lesser degree this effect.

New facilities with better resolution and sensitivity allow the exploration of
this fainter population of dusty galaxies. For instance, Ono et al. (2014)
studied a sample of 11 dusty star forming galaxies below the SMG regime
($S_{\rm 1.2mm}\sim 0.1-1.0~{}\rm mJy$) with ALMA, and found that these are
FIR counterparts of UV-selected or K-selected galaxies, like Lyman-break
galaxies (LBGs) or star-forming BzK galaxies. Aravena et al. (2020) analysed a
sample of 32 galaxies detected at 1.2 mm with ALMA as part of the ASPECS Large
Program. They estimated a median redshift of $z=1.85$ (with interquartile
range $1.10-2.57$) and found that 34 per cent of their $S_{1.2\rm{mm}}\sim
0.03-1\ \rm{mJy}$ galaxies are located below the main sequence of star
formation.

Faint SMGs at higher redshifts have also been detected by means of
gravitational lensing magnification effects by either massive galaxies or
galaxy clusters in the foreground (e.g. Chen et al., 2013; Aguirre et al.,
2018). These, however, are generally limited to small samples biased towards
higher redshifts and smaller compact objects (e.g. Bussmann et al., 2013).
Furthermore, accurate modelling of the lensing masses is required to estimate
the intrinsic physical and morphological properties of the lensed sources.

The H-band morphology of SMGs has been previously explored for both bright
SMGs (LESS/ALESS, $S_{870\mu\rm m}>3~{}\rm mJy$; Targett et al., 2013; Chen et
al., 2015) and $450~{}\mu\rm m$-selected faint SMGs (STUDIES, $S_{450\mu\rm
m}=2.8-29.6~{}\rm mJy$; Chang et al., 2018). These studies found that SMGs
have large rest-frame optical sizes, with disk-like and perturbed
morphologies. On the other hand, sub-arcsec angular resolution observations
with ALMA of both bright ($S_{870\mu\rm m}=8-16~{}\rm mJy$; Simpson et al.,
2015) and faint ($S_{1.1\rm mm}>0.5~{}\rm mJy$; Franco et al., 2020) SMGs have
shown that their FIR emission is associated to more compact structures,
suggesting spheroid build-up in these galaxies. The early released data from
the James Webb Space Telescope (JWST) has confirmed the bulge build-up for a
small sample of 7 SMGs in the EGS and UDS fields (Chen et al., 2022a). They
studied the morphology by fitting two component Sérsic models and observed
residual spiral arms structures, as well as tidal remnants and clumps. Some of
these features were also discovered in the JWST data of a lensed grand design
spiral galaxy detected with ALMA (Wu et al., 2022) and 2 SMGs detected in the
SMACS J0723.3–7327 cluster (Cheng et al., 2022).

In paper I of this series Zavala et al. (2017) presented the deepest
submillimeter observations of the SCUBA-2 Cosmology Legacy Survey (S2CLS) in
the Extended Groth Strip (EGS), providing simultaneous maps at both 450 and
$850~{}\mu\rm m$ and the detection of 144 galaxies with flux densities in the
ranges $S_{850\mu\rm m}=0.7-6~{}\rm mJy$ and $S_{450\mu\rm m}=3-17~{}\rm mJy$,
in the flux density regime below classical submillimeter galaxies
($S_{850\mu\rm m}\lesssim 6\rm mJy$). These galaxies are referred to as "faint
SMGs" hereafter. Paper I presented the number counts at flux densities
$S_{850\mu\rm m}>0.9~{}\rm mJy$ and $S_{450\mu\rm m}>4~{}\rm mJy$, and an
estimation of the contribution of the detected galaxies to the Cosmic Infrared
Background at both wavelengths of $\sim 28$ per cent. In paper II, Zavala et
al. (2018) identified robust optical counterparts for 75 per cent of the
galaxies and estimated their infrared luminosities, star formation rates
(SFRs), dust temperatures and the median stellar mass of the population. They
provided an initial comparison to the main sequence of star-forming galaxies
adopting the evolution of the main sequence of a general field, and found that
most faint SMGs lie within the main sequence of star formation and are
predominantly disc-like galaxies, with a transition from irregular discs to
discs+bulges at $z\sim 1.4$, such that the bulge seems to be developing at
later cosmic times.

In this paper we further explore the properties of these faint SMGs and those
of coeval optically-selected massive star-forming galaxies (hereafter referred
to as SFGs) extracted from the same field in order to address some outstanding
questions that were not addressed in paper II: are faint SMGs significantly
different from other massive star forming galaxies in the field? Do faint SMGs
have signs of increased merging or disturbances when compared to other star-
forming galaxies in the field? Does the morphological evolution detected in
faint SMGs happen at the same rate as that of other massive star forming
galaxies? In order to reduce biases in the comparison we consistently estimate
properties for both populations extracted in the same field: stellar mass,
SFRs and morphology are characterized using the same methods for both
populations. The paper is organized as follows: in section 2 we present the
sample selection; in section 3 we describe the ancillary data and catalogs we
will use to explore the comparison between faint SMG and SFG populations. In
section 4, we present the analysis of the star-formation main sequence, the
location of faint SMGs from the main sequence and the evolution of the
morphology of both populations. In section 5 we discuss our results in the
light of other results presented in the literature, and in section 6 we
summarize the conclusions of this work.

We adopt the standard $\Lambda$CDM cosmology with $\Omega_{\Lambda}=0.68$,
$\Omega_{m}=0.32$ and $H_{0}=67~{}\rm{km\,s^{-1}Mpc^{-1}}$ from Planck
Collaboration et al. (2014).

##  2 Sample selection

###  2.1 Faint Submillimeter Galaxies

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 ($L_{\rm IR}<10^{11}~{}\rm{L_{\odot}}$).

###  2.2 Comparison sample of star-forming galaxies

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.

Figure 1: UVJ color-color diagram of $H$-band selected galaxies at $1<z<2$,
where we use rest-frame AB magnitudes, not corrected for dust extinction. The
panels separate the galaxies in narrow bins of stellar mass
($\Delta\log(M_{\star}/\rm{M_{\odot}}))=0.5$). The galaxies are color-coded
according to their sSFR (sSFR=SFR/$M_{\star}$), adopting the SFR that Barro et
al. (2019) estimated with FIR, NIR or UV data and the stellar masses from
Stefanon et al. (2017). The faint SMGs with robust optical counterparts are
represented with empty symbols, indicating whether they comply with the good
quality selection criteria applied to SFGs (triangles), or instead they were
flagged as not complying (squares). The dashed lines divide the quiescent
(upper left) and star-forming regions (Williams et al., 2009). All galaxies
outside of the quiescent region that are not identified as SMG counterparts
are included in our optically-selected star-forming comparison sample.
Eighteen per cent of faint SMGs lie within the quiescent region. These are
hence heavily dust-obscured systems that the UVJ diagram is not able to
properly classify as star-forming galaxies.

##  3 Ancillary data

###  3.1 Stellar Masses

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.

###  3.2 Redshifts

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).

###  3.3 Star Formation Rates

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

  * •

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

  * •

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

  * •

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

  * •

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

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 SFRFIR
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.

###  3.4 Dust extinction

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.

###  3.5 H-band morphologies

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.

##  4 Analysis: comparison between submillimeter and optically selected star
forming galaxies

###  4.1 Stellar masses

Figure 2:  Normalised distributions of stellar mass for optically-selected star-forming galaxies (SFGs, grey) and optical counterparts of submillimeter galaxies (SMGs, orange). The median stellar masses (vertical lines) of faint SMGs are significantly higher than those of SFGs for all redshift bins. Table 1: Median stellar mass of the optically-selected star-forming galaxy (SFG) and faint submillimeter galaxy (SMG) samples. The columns are (1) redshift range; (2) median stellar mass of SFGs; (3) median stellar mass of faint SMGs. The errors of the median values of stellar mass were calculated with a bootstrap. $z$ | $\log M_{\star,\rm{SFGs}}$ | $\log M_{\star,\rm{SMGs}}$
---|---|---
| [$\rm M_{\odot}$] | [$\rm M_{\odot}$]
0.2–1.0 | $9.46\pm 0.01$ | $10.3\pm 0.6$
1.0–2.0 | $9.70\pm 0.01$ | $10.9\pm 0.1$
2.0–3.0 | $9.86\pm 0.02$ | $10.8\pm 0.2$
3.0–4.0 | $10.28\pm 0.05$ | $10.5\pm 0.1$
all $z$ | $9.68\pm 0.01$ | $10.8\pm 0.1$

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.

###  4.2 Star Formation Rates

####  4.2.1 Star-formation main sequence

Figure 3: Specific SFR vs. $M_{\star}$ for the SFG sample (black dots),
adopting the best-estimate of SFR
($\rm{SFR_{UV}^{corr}},\rm{SFR_{UV+IR}^{W11}}$ or
$\rm{SFR_{UV+IR}^{Herschel}}$) by Barro et al. (2019) for SFGs. Red filled
diamonds represent the sSFRs based on $\rm{SFR_{IR}+SFR_{UV}}$ estimations for
the SMGs with optical-NIR counterparts that comply with the good quality
selection criteria in section 2.2. The empty red triangles connected with them
by dashed lines show the corresponding sSFRs based on $\rm{SFR_{FIR+UV}}$.
Similarly, green filled diamonds and empty triangles represent the estimates
for SMGs that were flagged out by the optical-NIR quality selection criteria.
The red solid line is the star-formation main sequence fit for SFGs and the
shaded region the 1$\sigma$ scatter. We also show the main sequences derived
by Fang et al. (2018, blue), Speagle et al. (2014, purple), Whitaker et al.
(2014, green) and Barro et al. (2019, orange). At redshift bins $0.2<z<1.0$
and $1.5<z<2.0$ we represent the main sequences derived by (Fang et al., 2018)
and (Barro et al., 2019) in their own bin definitions: $0.0<z<0.5$,
$0.5<z<1.0$, $1.4<z<1.8$ and $1.8<z<2.2$ for Barro et al. (2019), and
$0.2<z<0.5$ and $0.5<z<1.0$ for (Fang et al., 2018). We find that at $z<2.5$
the main sequences derived in the literature agree well with ours within the
$1\sigma$ scatter, and at $2.5<z<4$ our slope is steeper.

The tight correlation between SFR and $M_{\star}$ for SFGs is referred to as
the main sequence of star-forming galaxies (Noeske et al., 2007; Daddi et al.,
2007; Elbaz et al., 2011; Whitaker et al., 2014; Speagle et al., 2014). This
is often expressed in terms of the specific star formation rate,
sSFR=SFR/$M_{\star}$. Figure 3 shows the main sequence estimated for our
optically-selected SFG sample in 6 redshift bins using the best SFR in the
CANDELS catalog available for each source: $\rm{SFR_{UV+IR}^{Herschel}}$,
$\rm{SFR_{UV+IR}^{W11}}$, or $\rm{SFR_{UV}^{corr}}$, in that order of
preference, which are collectivelly denoted as $\rm{SFR_{IR+UV}}$. We fitted
the $\log{\rm sSFR}-\log M_{\star}$ relation with a linear function and an
iterative 3-step least-squares method, using two $1.5\sigma$ clippings on the
surviving sample. The parameters of the main sequence fits are listed in Table
2.

We note that our main sequence is mostly consistent, within the RMS, with
previously derived main sequences at $z<2.5$ (i.e. Speagle et al., 2014;
Whitaker et al., 2014; Fang et al., 2018; Barro et al., 2019), despite the
fact that these comparison main sequences were derived using samples extracted
at different depths, with different SFR estimations and functional forms for
the fit (e.g. power law, broken power law, mass-time dependant, etc). At
redshifts $z>2.5$, however, our main sequence fit is steeper than the main
sequences derived by Speagle et al. (2014) and Barro et al. (2019).

Table 2: Parameters of the linear fits $\log({\rm sSFR/yr^{-1}})=m~{}\log(M_{\star}/{\rm M_{\sun}})+b$. The columns give: (1) redshift bin, (2) slope, (3) zero point, (4) RMS of the SFR of the galaxies to the best fit sequence at their corresponding stellar mass. $z$ | $m$ | $b$ | $\sigma$
---|---|---|---
0.2–1.0 | $-0.24$ | $-6.99$ | 0.37
1.0–1.5 | $-0.32$ | $-5.80$ | 0.36
1.5–2.0 | $-0.38$ | $-4.99$ | 0.32
2.0–2.5 | $-0.35$ | $-5.22$ | 0.37
2.5–3.0 | $-0.51$ | $-3.47$ | 0.33
3.0–4.0 | $-0.71$ | $-1.14$ | 0.43

####  4.2.2 Location of SMGs with respect to the Main Sequence

Figure 4: Distribution of the sSFR with respect to the star-formation main
sequence for SMGs (orange) and SFGs (grey). We use $\rm{SFR_{IR}+SFR_{UV}}$
for the SMGs and SFGs. The Gaussian distribution marked by the solid line is
centered on the main sequence for each redshift bin and the vertical dashed
lines mark the $3\sigma$ limits. The median
$\Delta\log\rm{sSFR}=\log(\rm{sSFR/sSFR_{MS}})$ of SMGs (orange vertical line)
is significantly larger at $1<z<4$. The redshift bin with the highest fraction
of SMGs above $1\sigma$ is $2.5<z<3$.

Adopting the $\rm{SFR_{IR+UV}}$ estimations for SMGs of the CANDELS catalogs
we find that 82 per cent of SMGs (42 galaxies) are located above the main
sequence of SFGs, 56 per cent of SMGs (32 galaxies) are located above
$1\sigma$, 21 per cent of SMGs (12 galaxies) are above $2\sigma$, and 4 per
cent (2 galaxies) are located above $3\sigma$ (see Figure 3). On the other
hand, 11 per cent of SMGs (6 SMGs galaxies) are below $-1\sigma$ of the main
sequence and one galaxy below $-2\sigma$ of the main sequence.

In Figure 3 we can observe that at all redshifts faint SMGs are located at
higher sSFRs than SFGs for the same stellar mass, indicating more vigorous
star forming activity across the faint SMG population. Figure 4 shows the
normalized distribution of the sSFR differences to the main sequence for SMGs
and SFGs, $\Delta\log\rm{sSFR}=\log(\rm{sSFR/sSFR_{MS}})$ to highlight this
effect. We find positive median values of $\Delta\log\rm{sSFR}$ for the SMG
sample at all redshifts. We applied the Mann-Whitney test to check if the
medians of SMGs could be derived from the same parent distribution as those of
SFGs, and we find the differences to be robust at $1<z<4$, once we consider
the uncertainties on redshift estimations through a bootstrap: $p\sim
0.009-7\times 10^{-10}$ at different redshift bins (see Table 7 in the
Appendix).

We observe at $2.5<z<3$ the highest fraction of SMGs above the main sequence:
$\sim$89 per cent of SMGs (8/9) have sSFRs above $1\sigma$ of the main
sequence. This is also the redshift bin where the $850~{}\mu\rm m$-selected
SMG population and the dust-obscured SFR density peak ($z\sim 2-2.5$, e.g.
Zavala et al., 2021). At higher redshift bins, $3<z<4$, the faint SMGs do not
have larger SFRs than the SFGs and we see a larger dispersion of faint SMGs
across the main sequence.

Elbaz et al. (2011) initially used $R_{\rm SB}=\rm sSFR/sSFR_{MS}\geq 2$ as
the threshold for defining a starburst. Nowadays a factor of 3 is more
commonly used for this starburstiness parameter (Franco et al., 2020). If we
consider this threshold, we would classify 23 (40 per cent) of the faint SMGs
as starburst galaxies.

We applied the same colour and dust extinction selection as in section 4.1 to
define a sub-sample of SFGs with the same dust extinction properties as SMGs,
to check if this could bias our results. The fraction of starbursts and
location of SMGs with respect to the main sequence of SFGs remains unchanged.

We hence conclude that the faint SMG population is mostly located above the
main sequence of star formation (56 per cent above $1\sigma$), particularly at
$z>2$ where the population peaks (65 per cent). There is also a significant
fraction of starbursts (40 per cent).

####  4.2.3 Impact of different SFR estimates

Figure 5:  (Top panel) Comparison of IR-derived SFRs: $\rm{SFR_{FIR}}$ by the
S2CLS team (Zavala et al., 2018), using the 450/850 $\mu$m SCUBA-2 and
Herschel deconvolved photometry at the NIR or radio positions of the
counterparts and $\rm{SFR_{IR}}$ by the CANDELS team (Barro et al., 2019),
using either a fit to the Herschel photometry extracted with a PSF or,
whenever that was not available, extrapolating the Spitzer/MIPS 24 $\mu$m
photometry with templates that extend to the FIR. The arrows indicate the
upper limits derived from Herschel data for the CANDELS catalog whenever a
detection at 24 $\mu$m or at longer wavelengths was not available. (Bottom
panel) SFR ratio versus redshift, dust extinction (color gradient) and stellar
mass (sizes). The color in the symbols indicate the dust extinction $A_{V}$
and the sizes are proportional to the stellar mass $M_{\star}$ of each galaxy.
We use the total SFRs coadding the IR SFR and raw UV-based SFR considering
both FIR and IR-derived SFR estimations. When the SFR is estimated without any
IR data it uses the UV SFR corrected for dust extinction, which in the cases
of great discrepancy shows small values.

We explore how different estimations of the total SFRs for faint SMGs impact
their location with respect to the main sequence. The top panel of Figure 5
compares the SFRs based on IR observations derived by the CANDELS
($\rm{SFR_{IR}},$ Barro et al., 2019) and the S2CLS teams ($\rm{SFR_{FIR}}$,
Zavala et al., 2018).

There are six SMGs that did not have Herschel nor Spitzer detections at the
time the CANDELS catalog was produced (marked with blue arrows in Figure 5).
These galaxies have higher values of $\rm{SFR_{FIR}}$ than the upper limits of
$\rm{SFR_{IR}}$ derived from Herschel non-detections. In these cases, the best
SFR estimate uses the UV SFR corrected for dust extinction, but this dust
correction fails to recover the total SFR from the UV continuum solely. These
are also galaxies with $A_{V}$ values unusually small (tipically $A_{V}<1$),
compared to the obscuration values $A_{V}>2$ of the bulk of the faint SMG
optical counterparts. This can be appreciated in the bottom panel of Figure 5,
where the SMGs with larger discrepancies between the SFR estimations also have
low dust extinction values. This effect of small derived obscuration but high
$\rm SFR_{FIR}$ could be explained in a patchy dust distribution scenario,
where most of the UV light would come from areas of less obscuration, hence
rendering an underestimation of dust obscuration for the full galaxy. The
S2CLS $\rm{SFR_{FIR}}$ is in these cases higher than the upper values derived
from Herschel photometry, as they are based on the detections at 450/850
$\mu$m at a higher spatial resolution and the deconvolved Herschel fluxes at
the position of the IR and radio counterparts, providing a better
characterization of the dust-enshrouded star formation rate.

For most other galaxies $\rm{SFR_{FIR}}$ are similar or smaller than
$\rm{SFR_{IR}}$. We note, however, that $\rm{SFR_{FIR}}$ was estimated by
deconvolving the Herschel flux densities at the position of the radio or mid-
IR counterpart sources (Zavala et al., 2018). Meanwhile, $\rm{SFR_{IR}}$ is
based on Herschel flux densities extracted with a PSF model, without
deconvolution (Barro et al., 2019). Hence $L\rm{{}_{IR}}$ is likely boosted by
the crowding and merging of fainter sources into the main SMG extracted flux
density, and $\rm{SFR_{IR}}$ could be overestimated.

Table 3: Median SFRs of the faint SMGs in our sample, considering various sets of data and estimation methods. The medians of $\rm{SFR_{FIR}}$ and $\rm{SFR_{IR}}$ agree with each other within the errors.  $\rm{SFR_{FIR}}$ | $132\pm 28~{}\rm{M_{\odot}~{}yr^{-1}}$
---|---
$\rm{SFR_{IR}}$ | $151\pm 33~{}\rm{M_{\odot}~{}yr^{-1}}$
$\rm{SFR_{UV}}$ | $2.8\pm 0.5~{}\rm{M_{\odot}~{}yr^{-1}}$
$\rm{SFR_{FIR}+SFR_{UV}}$ | $133\pm 27~{}\rm{M_{\odot}~{}yr^{-1}}$
$\rm{SFR_{UV+IR}}$ | $152\pm 30~{}\rm{M_{\odot}~{}yr^{-1}}$

In Table 3 we present the median SFRs for the faint SMG sample based on the
estimates presented in section 3.3. We also present a total SFR for faint SMGs
by coadding the $\rm{SFR_{FIR}}$ calculated by Zavala et al. (2018) and the
raw UV-based SFR not corrected for dust obscuration of Barro et al. (2019):
$\rm{SFR_{FIR}+SFR_{UV}}$. This estimate is also represented in Figure 3. The
median values of $\rm{SFR_{FIR}+SFR_{UV}}$ and $\rm{SFR_{UV+IR}}$ are in
agreement with each other within the errors. The ratio of median SFRs is
$\langle\rm{SFR_{FIR}+SFR_{UV}}\rangle/\rm{\langle SFR_{UV+IR}}\rangle=0.9\pm
0.2~{}\rm M_{\odot}~{}yr^{-1}$ and the median of the ratios
$\langle\rm{SFR_{FIR}+SFR_{UV}}/\rm{SFR_{UV+IR}}\rangle=0.59\pm 0.04$,
highlighting the overall tendency for
$\rm{SFR_{FIR}+SFR_{UV}}<\rm{SFR_{UV+IR}}$ in our sample. The SMGs that do not
follow this tendency are mainly those with $\rm SFR_{UV+IR}$ estimations
derived from UV-based SFRs corrected by dust extinction. These discrepant SMGs
also show low values of $A_{V}$ (see Fig. 5).

Adopting the $\rm{SFR_{FIR}+SFR_{UV}}$ estimations for SMGs, the location of
faint SMGs in the sSFR vs $M_{\star}$ diagram is such that 43 galaxies (75 per
cent) are above the main sequence of SFGs, 29 SMGs (51 per cent) are located
above $1\sigma$, 7 SMGs (12 per cent) are above $2\sigma$, and only 1 (2 per
cent) is located above $3\sigma$. On the other hand, there are 4 SMGs (7 per
cent) below $-1\sigma$ and one galaxy below $-2\sigma$ of the main sequence.
Hence the number of starbursts is slightly reduced from those adopting the
CANDELS SFR estimates, but they still are indicative of higher SFRs than the
optically selected population of SFGs of the same stellar mass. We confirm
this statement with a Mann-Whitney test, finding that at $1<z<4$ the sSFR of
SMGs is larger than that of SFGs ($p=10^{-3}-10^{-9}$).

###  4.3 H-band Morphology

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:

  * •

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
$n=1.23$.

  * •

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 $n=0.68$.

  * •

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.

  * •

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.

####  4.3.1 Effective radii

Figure 6: Normalized distribution of the effective radius $R_{\rm e}$ of SMGs (orange) and SFGs (grey) with $\log(M_{\star}/\rm M_{\odot})>10$, using $R_{\rm e}$ values estimated by Van der Wel et al. (2014) and García-Rivero (2018). The median effective radii $R_{\rm e}$ of SMGs and SFGs are represented by vertical lines. The median values of the effective radii of SMGs are 50 per cent larger than those of SFGs at $z<3$. Table 4: Median values of structural parameters for SFGs and SMGs with $\log(M_{\star}/\rm M_{\odot})>10$. The columns present: (1) redshift range; (2) median effective radii of SFGs along the semi-major axis; (3) median effective radii of SMGs along the semi-major axis; (4) median radius difference to the size-mass relation of SMGs; (5) median Sérsic index of SFGs; (6) median Sérsic index of SMGs; (7) median axis ratio of SFGs and (8) median axis ratio of SMGs.  Redshift | $R_{\rm{e,SFGs}}$ | $R_{\rm{e,SMGs}}$ | $\Delta\log R_{\rm e,SMGs}$ | $n_{\rm{SFGs}}$ | $n_{\rm{SMGs}}$ | $q_{\rm SFGs}$ | $q_{\rm SMGs}$
---|---|---|---|---|---|---|---
$z$ | [kpc] | [kpc] |  |  |  |  |
0.2–1.0 | $3.62\pm 0.13$ | $4.8\pm 0.9$ | $0.09\pm 0.09$ | $1.16\pm 0.04$ | $1.7\pm 0.4$ | $0.52\pm 0.02$ | $0.64\pm 0.17$
1.0–2.0 | $3.65\pm 0.10$ | $5.2\pm 0.4$ | $0.006\pm 0.03$ | $1.03\pm 0.02$ | $1.2\pm 0.3$ | $0.55\pm 0.01$ | $0.67\pm 0.06$
2.0–3.0 | $2.76\pm 0.09$ | $4.7\pm 0.6$ | $0.22\pm 0.05$ | $1.19\pm 0.08$ | $1.2\pm 0.22$ | $0.55\pm 0.02$ | $0.47\pm 0.07$
3.0–4.0 | $2.59\pm 0.19$ | $2.5\pm 1.6$ | $-0.05\pm 0.07$ | $1.23\pm 0.11$ | $2.4\pm 1.0$ | $0.51\pm 0.03$ | $0.68\pm 0.11$
Median | $3.33\pm 0.05$ | $4.8\pm 0.4$ | $0.11\pm 0.07$ | $1.11\pm 0.02$ | $1.23\pm 0.22$ | $0.550\pm 0.008$ | $0.62\pm 0.05$

The optical counterparts of faint SMGs have a median effective radius along
the semi-major axis $R_{\rm e,SMG}=4.8\pm 0.4~{}\rm kpc$, larger than the
median effective radius of the SFG sample $R_{\rm e,SFG}=2.71\pm 0.03~{}\rm
kpc$. Since we had found that SMGs are more massive than SFGs (section 4.1),
this result is not entirely surprising.

In Figure 6 we show the normalized distributions of sizes for galaxies within
four redshift bins. The corresponding medians for galaxies with
$\log(M_{\star}/\rm M_{\odot})>10$ are listed in Table 4. The median $R_{\rm
e,SFG}$ is slightly smaller at redshifts $z>2$ than at $z<2$. The same effect
is seen in the SMG population, where the median effective radius of SMGs at
high redshift ($z>3$) is smaller than that at $z<3$. The sizes of both
populations are only comparable at the highest redshift bin. The median
$R_{\rm e,SMGs}$ are significantly larger than the median effective radii of
SFGs at $1<z<3$ ($p<0.05$, see Table 9).

Figure 7: Radii differences vs. stellar mass of the SFGs (black dots) and SMGs
(squares and triangles) to the $\log R_{e}-\log M_{\star}$ relation traced by
SFGs (cyan dashed line). We represent with green squares the SMGs with $R_{\rm
e}$ derived by Van der Wel et al. (2014). The empty squares connected to red
triangles are those radii replaced with $R_{\rm e}$ derived by García-Rivero
(2018). The redshift bin where the median size of SMGs is significantly larger
than that of SFGs of the same stellar mass is $2<z<3$.

In order to further explore the sizes of SMGs with respect to the typical
sizes of SFGs with the same stellar mass, we fitted a linear $\log R_{\rm
e}-\log~{}M_{\star}$ relation, following a similar procedure to that employed
to derive the main sequence of star-forming galaxies (section 4.2.1). We then
calculate the residual radii, $\Delta\log R_{\rm e}$, as the difference
between the galaxy effective radius and the mean effective radius of the SFG
sample at the galaxy’s stellar mass. In Figure 7 we present the residual radii
versus stellar mass for SFGs and SMGs and the median $\Delta\log R_{\rm
e,SMG}$ values are listed in Table 4.

The median effective radii of SMGs are overall significantly larger than the
median radii of SFGs of the same mass ($p=0.001$, see Table 9). The
differences are mainly carried by the population of SMGs at $2<z<3$:
probability $p=0.0004$ that the median radii for the same stellar mass could
originate from a common distribution. A KS test with $p<0.05$ also confirms
that the residual distributions of SMGs and SFGs are different at this
redshift bin. At $0.2<z<2$, however, we cannot reject the null hypothesis.

Hence, SMGs are in general larger than the SFG population of the same stellar
mass. This difference is more significant at $2<z<3$, where the fraction of
starbursts is also larger.

####  4.3.2 Sérsic index

Figure 8: Sérsic index $n$ vs. $M_{\star}$ for SFGs (black dots) and SMGs
(green squares and red triangles), following the same symbols as in Figure 7.
When we select $\log(M_{\star}/\rm M_{\odot})>10$ galaxies, the indices of
SMGs seem to be derived from the same distribution as those of SFGs.

We present the Sérsic index vs stellar mass for SFGs and SMGs in Figure 8.
Figure 9 shows the normalized distribution of the Sérsic index of both
populations with masses $\log(M_{\star}/\rm M_{\odot})>10$ separated in four
redshift bins and their corresponding median values are presented in Table 4.
We find no differences between the distributions of $n$ values (Table 9).

Zavala et al. (2018) claimed an evolution in the Sérsic index of SMGs between
redshift bins $0.2<z<1.4$ and $1.4<z<3$, such that $n$ increases towards lower
redshifts, which we also confirm. Once we split the sample in four redshift
bins, however, the evolution is not as clear. The Mann-Whitney test indicates
that the median Sérsic indices of SMGs and SFGs in this mass range are
compatible with a common parent distribution ($p=0.3$) at $1.4<z<3$. At
$0.2<z<1.4$, however, the null hypothesis of identity is rejected and the
median Sérsic indices are not compatible with a common parent distribution
($p=0.009$): the median Sérsic index of SMGs at $0.2<z<1.4$ is larger than
that of SFGs of the same stellar mass. We will test this result further in
section 4.3.4.

Figure 9: Distribution of the Sérsic index ($n$) of SMGs (orange) and SFGs
(grey) with $\log(M_{\star}/M_{\sun})>10$. Both distributions are normalized
by the number of galaxies in each sample. The Mann-Whitney and KS test show
that the medians are not significantly different at any redshift.

####  4.3.3 Axis ratio

The projected axis ratio $q=b/a$, describes the roundness or elongation of the
galaxy, and varies from 0 (very elongated projected shape) to 1 (circular).
The median values of axis ratios for SMGs and SFGs in each redshift bin are
listed in Table 4. The median axis ratio for SMGs is $q_{\rm SMGs}=0.62\pm
0.05$ and for the optically-selected SFGs in the comparison sample is $q_{\rm
SFGs}=0.550\pm 0.008$. This implies that the projected shape of SMGs is
slightly rounder and the Mann-Whitney test shows this difference is
statistically significant ($p=0.04$, see Table 9).

We present the normalized distributions of axis ratios in four redshift bins
in Figure 10. The median $q_{\rm SMG}$ are slightly larger than $q_{\rm SFG}$
at all redshifts, except $2<z<3$, where they have similar values. We applied
the Mann-Whitney test to evaluate if the medians stem from the same parent
distribution, and find the difference could be significant only at $1<z<2$
($p=0.01$).

Figure 10: Normalized distribution of the axis ratio ($q=b/a$) for SMGs
(orange) and SFGs (grey) with $\log(M_{\star}/{\rm M}_{\sun})>10$. The median
$q$ (vertical lines) of SMGs are larger at all redshifts, except $2<z<3$. The
Mann-Whitney test indicates that this difference is significant only at
$1<z<2$.

####  4.3.4 CAS parameters

The morphology of galaxies can also be described with non-parametric indices
like concentration (C), asymmetry (A) and clumpiness (S) (Conselice et al.,
2003; Lotz et al., 2004). Since these indices do not assume a specific
function of the light distribution, they are specially useful as we move to
higher redshifts or explore irregular and merger populations. We calculated
these indices for the SFG and faint SMG samples using the same procedure as in
Lotz et al. (2004), taking special care of the calculations for the SMG
counterparts in the procedure, as a good fraction of these are faint and do
not have available CAS indices in the CANDELS catalogs.

In Figure 11 we present the CAS indices versus redshift for SFGs and SMGs with
$\log(M_{\star}/\rm M_{\odot})>10$. At low $z$ the C and A indices show higher
values for both samples, implying an apparent higher concentration and
asymmetry of the galaxies in H-band.

We explore the correlation between the indices and redshift with the Kendall
rank correlation index. We find a strong correlation for the SFGs between the
C index and $z$ ($p=5.5\times 10^{-7}$) and A and $z$ ($p=9\times 10^{-7}$).
The SMGs show a correlation as well ($p=6\times 10^{-7}$ for C vs. $z$ and
$p=4\times 10^{-4}$ for A vs. $z$). Both SMGs and SFGs show no trend between
the clumpiness index of the galaxies and redshift ($p\sim 0.5$).

The median values of the CAS indices can be found in Table 5. The differences
in the medians of the concentration parameter C are significant globally (see
Table 10). The difference is most significant at $2<z<3$ ($p=9\times 10^{-4}$)
and marginal, when uncertainties in redshift are taken into account, at
$0.2<z<1$ ($p\sim 0.05$). The differences in A and S between both populations
are not significant.

We hence find a larger concentration for both SMGs and SFGs with decreasing
redshift, and the growth in concentration to be larger for SMGs than for SFGs
at later times, confirming the results we found for the Sérsic $n$ index in
section 4.3.2.

Table 5: Median values of the non-parametric indices for SFGs and SMGs with $\log(M_{\star}/\rm M_{\odot})>10$. The columns present: (1) redshift range; (2) median concentration index of SFGs; (3) median concentration index of SMGs; (4) median asymmetry of SFGs; (5) median asymmetry of SMGs; (6) median clumpiness index of SFGs and (7) median clumpiness index of SMGs. Redshift | CSFGs | CSMGs | ASFGs | ASMGs | SSFGs | SSMGs
---|---|---|---|---|---|---
0.2-1.0 | $2.73\pm 0.03$ | $3.11_{-0.003}^{+0.04}$ | $1.23_{-0.11}^{+0.14}$ | $1.62\pm 0.04$ | $0.41\pm 0.04$ | $0.29\pm 0.11$
1.0-2.0 | $2.47\pm 0.03$ | $2.60\pm 0.18$ | $1.04\pm 0.03$ | $0.98\pm 0.05$ | $0.38\pm 0.01$ | $0.35\pm 0.03$
2.0-3.0 | $2.39\pm 0.03$ | $2.13\pm 0.09$ | $0.92\pm 0.02$ | $0.84\pm 0.07$ | $0.38\pm 0.02$ | $0.39\pm 0.06$
3.0-4.0 | $2.43\pm 0.25$ | $1.98^{+0.28}_{-0.24}$ | $1.03\pm 0.19$ | $0.91^{+0.13}_{-0.10}$ | $0.37\pm 0.02$ | $0.35\pm 0.06$
$0.2-4.0$ | $2.49\pm 0.02$ | $2.34\pm 0.11$ | $1.03\pm 0.03$ | $0.98\pm 0.03$ | $0.38\pm 0.01$ | $0.36\pm 0.02$
Figure 11: Non-parametric morphology indices for SFGs (black dots) and faint
SMGs (red dots) with $\log(M_{\star}/\rm M_{\odot})>10$: (Top) Concentration
(C), (Middle) asymmetry (A) and (Bottom) clumpiness (S). The mean of the SFG
indices are marked by blue squares in four redshift bins, and the mean of the
SMG indices with orange diamonds. We find a correlation between redshift and
the concentration and asymmetry for SFGs and SMGs, and differences in median
concentrations, such that SMGs have higher concentrations at lower redshift.

####  4.3.5 Machine-learned classification

In order to further check possible discrepancies between the SFG and faint SMG
populations, we use the morphological classes presented by Huertas-Company et
al. (2015): pure disks, pure spheroids, disks+spheroids, irregular disks and
irregular/mergers. We applied their criteria to both SMGs and SFGs with
$\log(M_{\star}/\rm M_{\odot})>10$, finding classifications for 44 SMGs and
975 SFGs. In Figure 12 we present the fraction of SMGs and SFGs classified by
morphological type at each redshift bin.

We find that the predominant morphological type for faint SMGs is disk-like
galaxies at $z<2$ (pure disks, irregular disks and disks+spheroids), while at
$z>2$ the fraction of mergers is roughly the same as that of irregular disks.
All SMGs and half of SFGs (51 per cent) at the highest redshift bin $3<z<4$
are classified as mergers. However, there are only 3 SMGs with classifications
in the highest redshift bin.

Irregular disks and mergers in the SMG population decrease to give rise to
pure disks and disk+spheroids. At the lowest redshift bin 67 per cent of SMGs
are classified as pure disks, with equal fractions ($\sim 17$ per cent) of
irregular disks and disks+spheroids. SFGs show similar fractions of pure disks
and irregular disks in this redshift bin with a $\sim 15$ per cent of
disk+spheroids. Both galaxy populations show an increment of the disk+spheroid
and pure disk fraction towards lower redshifts.

There are no SMGs classified as pure spheroids and there are $\leq 10~{}$per
cent of spheroidal SFGs at any redshift. Overall the main morphology of SMGs
is disks, and an evolution in both faint SMG and SFG populations can be seen,
where the merger fraction decreases with redshift and, irregular disks
dominate at intermedate redshifts ($1<z<3$), while pure disks and
disk+spheroids rise at the lowest redshift bin. There are no clear differences
in the evolution of both populations.

The selection criteria we use in this work is that of Huertas-Company et al.
(2015), which require probabilities $>2/3$ for disks and spheroids
classification, and produces a different result to that applied by Zavala et
al. (2018), where probabilities $>1/3$ were used. This difference results in a
smaller number of galaxies classified as pure spheroids and disk+spheroids.

In section 4.2.2 we found that 82 per cent of SMGs are located above the main
sequence of star formation, and 40 per cent can be classified as starbursts.
Among starbursts galaxies we find 27 per cent to have merger morphologies and
73 per cent to have disk-like morphologies: 61 per cent irregular disks, 6 per
cent disk+spheroids and 6 per cent pure disks. On the other hand, the
morphologies of the faint SMGs that lie within $\pm 1\sigma$ of the main
sequence are classified as 24 per cent mergers, 35 per cent irregular disks,
35 per cent pure disks and 6 per cent disk+spheroids. Hence there are slightly
more starburst SMGs classified as mergers and irregular galaxies than among
main-sequence faint SMGs, but the differences are within the poisson errors of
the samples. However, at the redshift bin where we find significantly larger
sizes for SMGs than for SFGs and a larger number of starbursts ($2<z<3$,
section 4.3.1), the galaxies above the $\log R_{\rm e}-\log M_{\star}$
sequence are classified as mergers (43 per cent) and irregular disks (57 per
cent). This could be a trace of recent interactions.

Figure 12: Fraction of SMGs (circles) and SFGs (squares) with
$\log(M_{\star}/M_{\sun})>10$ that are classified as pure disks, pure
spheroids, disks+spheroids, irregular disks and irregulars/mergers, from the
machine learning visual classification catalog.

###  4.4 The impact of possible misidentifications of counterparts of faint
SMGs

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.

##  5 Discussion

###  5.1 Stellar Mass

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.

###  5.2 Starbursts among faint SMGs?

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.

###  5.3 Star formation rate indicators of faint SMGs and patchy obscuration

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.

###  5.4 H-band Morphology of Faint SMGs

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.

##  6 Conclusions

We studied the physical and morphological properties of 57 faint submillimeter
galaxies detected at 450 and $850~{}\mu\rm m$ in the S2CLS Extended Groth
Strip field, with flux densities in the ranges $S_{850\mu\rm m}=0.7-6~{}\rm
mJy$ and $S_{450\mu\rm m}=3-17~{}\rm mJy$, in the flux density regime below
classical submillimeter galaxies ($S_{850\mu\rm m}\lesssim 6~{}\rm mJy$). We
compare them to a sample of optically-selected star-forming galaxies of
similar mass extracted from the same field and analyzed with the same
techniques in order to detect if there are differences between the
populations. Our main conclusions are:

  * •

Adopting our best estimates of total SFR, faint SMGs are on average located
within the $3\sigma$ scatter of the main sequence of star formation, defined
within the same field and with the same methodology. About 82 per cent of
faint SMGs are located above the main sequence and only 4 per cent above the
$3\sigma$ intrinsic scatter of the main sequence, but even within the main
sequence, 40 per cent of our faint SMG sample has starburst characteristics,
with $R_{\rm SB}(\rm{SFR/SFR_{MS}})>3$. Faint SMGs have significantly larger
sSFRs at $z>1$ than optically-selected star-forming galaxies of the same mass.

  * •

The sizes of SMGs and SFGs are significantly different at $2<z<3$. We find
that, for galaxies of the same stellar mass ($\log(M_{\star}/\rm
M_{\odot})>10$), SMGs are 50 per cent larger than SFGs. In this redshift bin
we also find the largest starburst fraction in the SMG sample. Hence we could
be witnessing merging processes, also supported by a high fraction of SMG
counterparts with morphological classifications falling in the merger (43 per
cent) or irregular disk (57 per cent) classes at this redshift bin.

  * •

We find an evolution in the morphology of SMGs from mergers and irregular
disks at high redshift to pure disks and disk+spheroids at low redshift. A
similar evolution is seen in high-mass SFGs, although the fraction of mergers
in SMGs is slightly larger.

  * •

We find an evolution of the Sérsic index of the optical counterparts of SMGs
from $n\sim 1.0\pm 0.2$ at $1.4<z<3$ to $n\sim 1.8\pm 0.4$ at $0.2<z<1.4$,
consistent with the findings of Zavala et al. (2018). When we compare this
evolution with that of SFGs of the same stellar mass, $\log(M_{\star}/\rm
M_{\odot})>10$, we find that SMGs are significantly more concentrated at
$z<1.4$. The same redshift evolution can be inferred from the non-parametric
concentration index C. The concentration of SMGs is significantly smaller than
that of SFGs at $2<z<3$ and marginally larger at $0.2<z<1$. No differences can
be found with the asymmetry A and clumpiness indices between both populations.

  * •

We find that the $\rm SFRs$ estimated without the use of FIR data are
underestimated by factors of 3–570 with respect to the FIR-based SFR
measurements. The intense obscuration in these systems tends to underestimate
the SFR when using only optical-to-near IR data. These systems have the lowest
$A_{V}$ values from the whole SMG sample. This is likely an effect of
differential obscuration and patchiness at rest-frame UV-to-optical
wavelengths. This implies that the optical dust extinction parameter is unable
to correct the absorption by dust in heavily obscured systems and,
consequently, the total SFR from optical-to-near IR data alone is
underestimated. The UV-optical emission could correspond to less obscured
regions, implying patchiness or core concentration in the dust distribution,
which could be further studied with higher resolution FIR imaging.

  * •

We find a discrepancy between the FIR-based SFRs derived from different teams,
which we trace to the different methodologies to extract flux densities from
the Herschel data. The median of the ratios is
$\langle\rm{SFR_{FIR}+SFR_{UV}}/\rm{SFR_{UV+IR}}\rangle=0.59\pm 0.04$.
Although in principle, this discrepancy coud have consequences for the number
of faint SMGs above the main sequence and the starbursts in the sample, these
numbers are very similar for both SFR estimations.

The initial understanding of SMGs located them as high redshift analogues of
ULIRGs (Sanders, 2003), although detail morphological analysis supported the
view that they were mainly massive isolated disks experiencing large star
formation rates (Targett et al., 2013). SMGs have also been linked as possible
precursors to QSOs and elliptical galaxies (Granato et al., 2001). The faint
SMGs in our sample are also hosted in massive disk-like galaxies, while we
have shown that they grow spheroidal components more prominently at later
times than optically selected SFGs of the same mass. We also find larger
effective radii at $2<z<3$, where more starbursts are present. This is also
the redshift bin where irregular disks and mergers are dominant morphological
classes, possibly reflecting close interactions that increase star-formation,
in excess of those found among the optically-selected SFGs. Therefore, this
population of SMGs might be the link between the extreme starbursts and main
sequence massive star-forming galaxies, represented by the SFG behaviour.

The future exploration of wider and deeper submillimeter surveys with larger
single dish and larger samples of interferometric resolved follow-up will
allow better statistical comparisons between galaxy populations. Furthermore,
the exploration of the gas content of SMGs will indicate whether these are
post-starburst or highly efficient star-forming galaxies.

## Acknowledgements

We thank the referee for useful comments that significantly improved the
sistematics in the conclusions of the paper. This work has been funded by
CONACYT grants FDC2016-1848 and CB2016-81948. A.M. thanks support from CONACYT
grant A1-S-45680. This work has made use of the Rainbow Cosmological Surveys
Database, which is operated by the Centro de Astrobiología (CAB/INTA),
partnered with the University of California Observatories at Santa Cruz
(UCO/Lick,UCSC).

## Data availability

The S2CLS catalogue of submillimeter galaxies used in this article is
available at https://doi.org/10.1093/mnras/sty217. The catalogues of the
CANDELS program can be found at
https://archive.stsci.edu/missions/hlsp/candels/egs/catalogs/v1/. The
morphological parameters catalogue by Van der Wel et al. (2014) is available
at https://www2.mpia-hd.mpg.de/homes/vdwel/3dhstcandels.html. Multiple
catalogues used here can be explored in the Rainbow navigator, at
https://arcoirix.cab.inta-csic.es/Rainbow_navigator_public/.

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##  Appendix A Color selection for all stellar mass and redshift bins

In order to select the star forming galaxies in the EGS field we use the $U-V$
vs $V-J$ color diagram. In Figure 13 we plot the $UVJ$ color diagram for the
five stellar mass and four redshift bins considered, including all CANDELS
galaxies with well-constrained parameters (as described in section 2.2). We
apply the selection criteria by Williams et al. (2009) at the corresponding
redshifts. At $2.5<z<3$ the galaxies with $\log(M_{\star}/\rm M_{\odot})>10.5$
start to populate the quiescent region. At lower redshifts the quiescent
region is more populated by galaxies with lower stellar mass.

Figure 13: UVJ color diagram of EGS CANDELS galaxies, highlighting the optical
counterparts of submillimeter galaxies. The columns separate the galaxies
according to their stellar mass, and the rows according to their redshift. The
color gradient shows the $\log\rm sSFR$ derived from optical to min-IR data.
The empty triangles are the optical counterparts of SMGs that comply with the
selection criteria and the empty squares are the SMGs that do not comply with
the selection criteria. The region selected by the dashed lines marks the
location of quiescent galaxies.

##  Appendix B Mann-Whitney test results

This appendix provides the results of the different Mann-Whitney tests
performed on the physical properties and morphological parameter distributions
presented in this work: stellar mass (Table 6), residual SFR to main sequence
(Tables 7 and 8), parametric morphology (Table 9) and non-parametric
morphology (Table 10).

Table 6: Median stellar mass of the optically-selected star-forming galaxy (SFG) and faint submillimeter galaxy (SMG) samples. The columns are (1) redshift range; (2) median stellar mass of SFGs; (3) median stellar mass of faint SMGs; (4) probability ($p$) of the Mann-Whitney test that characterizes if the median stellar masses of faint SMGs and SFGs could correspond to a common parent distribution, with the upward arrows marking that the median mass of SMGs is statistically larger than that of SFGs; (5) 68-per cent confidence limits of $p$, taking into account redshift uncertainties. We assign to each galaxy a new redshift under its redshift uncertainty distribution and recalculate the test 5000 times in order to produce this confidence limit. We highlight probabilities $<0.05$, which we adopt as a threshold to reject the null hypothesis of a common parent distribution for both galaxy samples. The errors of the median values of stellar mass were calculated with a bootstrap. $z$ | $\log M_{\star,\rm{SFGs}}$ | $\log M_{\star,\rm{SMGs}}$ | $p$ | $p$
---|---|---|---|---
| [$\rm M_{\odot}$] | [$\rm M_{\odot}$] |  | 68% CL
0.2–1.0 | $9.46\pm 0.01$ | $10.3\pm 0.6$ |  $\uparrow~{}$6x10-5 | 2x10-4 – 3x10-4
1.0–2.0 | $9.70\pm 0.01$ | $10.9\pm 0.1$ |  $\uparrow~{}$1x10-10 | 6x10-13 – 1x10-10
2.0–3.0 | $9.86\pm 0.02$ | $10.8\pm 0.2$ |  $\uparrow~{}$3x10-10 | 3x10-13 – 2x10-8
3.0–4.0 | $10.28\pm 0.05$ | $10.5\pm 0.1$ |  $\uparrow~{}$0.03 | 2x10-6 – 5x10-3
all $z$ | $9.68\pm 0.01$ | $10.8\pm 0.1$ |  $\uparrow~{}$3x10-26 | 2 – 8 x10-26
Table 7: Median difference of specific star formation rate of SMGs $\rm{SFR_{IR+UV}}$, $\Delta\log\rm{sSFR_{SMG}}$, to the star-formation main sequence. The columns present: (1) redshift range; (2) median $\Delta\log\rm sSFR$ of SMGs to main sequence; (3) probabilities ($p$) of the Mann-Whitney test for the sSFR difference to the main sequence of SFGs and SMGs, the upward arrows marking that the median values of SMGs are statistically larger than those of SFGs; (4) bootstrapped 68 per cent confidence interval of $p$, considering redshift uncertainties. We highlight probabilities $<0.05$, adopted as a threshold to reject the null hypothesis of a common parent distribution for both galaxy samples.  $z$ | $\Delta\log\rm{sSFR_{SMGs}}$ | $p$ | $p$
---|---|---|---
| [$\rm yr^{-1}$] |  | 68% CL
0.2–1.0 | $0.19\pm 0.21$ | 0.08 | 0.55 – 0.63
1.0–2.0 | $0.38\pm 0.13$ |  $\uparrow~{}$ 3.8x10-5 | 7x10-6 – 5x10-5
2.0–3.0 | $0.40\pm 0.04$ |  $\uparrow~{}$ 7.6x10-6 | 7x10-10 – 3x10-6
3.0–4.0 | $0.51\pm 0.14$ |  $\uparrow~{}$ 0.07 | 2x10-6 – 9x10-3
all $z$ | $0.39\pm 0.4$ |  $\uparrow~{}$ 2.4x10-11 | 2x10-11 – 5x10-10
Table 8: Median difference of specific star formation rate of SMGs $\rm{SFR_{FIR+UV}}$, $\Delta\log\rm{sSFR_{SMG}}$, to the star-formation main sequence. The columns present: (1) redshift range; (2) median $\Delta\log\rm sSFR$ of SMGs to main sequence; (3) probabilities ($p$) of the Mann-Whitney test for the sSFR difference to the main sequence of SFGs and SMGs, the upward arrows marking that the median values of SMGs are statistically larger than those of SFGs; (4) bootstrapped 68 per cent confidence interval of $p$, considering redshift uncertainties. We highlight probabilities $<0.05$, adopted as a threshold to reject the null hypothesis of a common parent distribution for both galaxy samples. $z$ | $\Delta\log\rm{sSFR_{SMGs}}$ | $p$ | $p$
---|---|---|---
| [$\rm yr^{-1}$] |  | 68% CL
0.2–1.0 | $-0.06\pm 0.3$ | 0.9 | 0.32-0.37
1.0–2.0 | $0.21\pm 0.12$ |  $\uparrow$7x10-3 | 1x10-3 – 3x10-3
2.0–3.0 | $0.44\pm 0.11$ |  $\uparrow~{}$ 1x10-4 | 9x10-10 – 5x10-7
3.0–4.0 | $0.40\pm 0.05$ |  $\uparrow~{}$ 2x10-4 | 2x10-7 – 7x10-5
all $z$ | $0.37\pm 0.06$ |  $\uparrow~{}$ 3x10-8 | 3x10-8 – 1x10-7
Table 9: Probabilities of the Mann-Whitney tests applied to the distributions of structural parameters of SMGs and SFGs with $\log(M_{\star}/\rm M_{\odot})>10$. The top row shows the redshift bins considered in the analysis. Rows 2 and 3 give the number of SMGs and SFGs at each redshift bin. Rows 4-11 give the probabilities for the null hypothesis of identity of the medians of structural parameters to be true. We state the values for the best redshifts estimated for the SMG and SFG samples, and also for the bootstrap when we randomly assign a redshift for each galaxy within its uncertainty distribution. We highlight in bold the values that reject the null hypothesis of a common parent distribution for SMGs and SFGs taking into account best and randomized redshifts within their uncertainties, and the upward arrows mark when the median values of SMGs are statistically larger than those of SFGs. Redshift bin | all $z$ | $0.2<z<1.0$ | $1.0<z<2.0$ | $2.0<z<3.0$ | $3.0<z<4.0$
---|---|---|---|---|---
Num. submillimeter galaxies | 55 | 6 | 20 | 20 | 9
Num. star-forming galaxies | 1349 | 201 | 693 | 367 | 88
Effective radius ($R_{\rm{e,SMA}}$,best $z$) |  $\uparrow~{}$7x10-6 | 0.08 |  $\uparrow~{}$1x10-3 |  $\uparrow~{}$2x10-4 | 0.07
Effective radius ($R_{\rm{e,SMA}}$) | 3x10-7 – 7 x10-7 | 7x10-3 – 0.02 | 2x10-5 – 1x10-3 | 7x10-6 – 9x10-3 | 0.02 – 0.2
Residual effective radius ($\Delta\log R_{\rm{e,SMA}}$,best $z$) |  $\uparrow~{}$1x10-3 | 0.13 | 0.14 |  $\uparrow~{}$4x10-3 | 0.16
Residual effective radius ($\Delta\log R_{\rm{e,SMA}}$) | 1x10-4 – 8x10-4 | 6x10-3 – 0.01 | 0.06 – 0.1 | 4x10-4 – 0.04 | 0.06 – 0.2
Sérsic index ($n$,best $z$) | 0.11 | 0.13 | 0.17 | 0.29 | 0.15
Sérsic index ($n$) | 0.1 - 0.12 | 0.08 – 0.11 | 0.12 – 0.23 | 0.32 – 0.47 | 0.08 – 0.35
Axis ratio ($q$,best $z$) |  $\uparrow~{}$ 0.04 | 0.3 |  $\uparrow~{}$ 0.01 | 0.3 | 0.1
Axis ratio ($q$) | 0.03 – 0.05 | 0.3 – 0.4 | 7x10-3 – 0.02 | 0.3 – 0.5 | 0.06– 0.25
Table 10: Probabilities of the Mann-Whitney test that measure if the medians of the SFG and SMG indices can be derived from the same parent distribution. The top row shows the redshift bins considered in the analysis. Rows 2 and 3 give the number of SMGs and SFGs at each redshift bin. Rows 4-9 give the probabilities for the null hypothesis of identity of the medians of non-parametric indices to be true for both the best-$z$ and the 68 percent confidence intervals of $p$ considering redshift uncertainties. We highlight in bold the values that reject the null hypothesis of a common parent distribution for SMGs and SFGs. Upward arrows mark the bins where the median values of SMGs are statistically larger than those of SFGs, and downward arrows where the median values of SMGs are statistically smaller than those of SFGs.  Redshift | $0.2<z<4$ | $0.2<z<1.0$ | $1.0<z<2.0$ | $2.0<z<3.0$ | $3.0<z<4.0$
---|---|---|---|---|---
Num. SMGs | 54 | 6 | 20 | 19 | 9
Num. SFGs | 361 | 51 | 203 | 96 | 11
C (best $z$) |  $\downarrow~{}$0.03 |  $\uparrow~{}$0.05 | 0.19 |  $\downarrow~{}$9x10-4 | 0.06
C | 0.03–0.046 | 0.05–0.06 | 0.17 –0.27 | 3x10-5 – 6x10-4 | 0.03–0.15
A (best $z$) | 0.13 | 0.09 | 0.48 | 0.04 | 0.13
A | 0.11–0.15 | 0.10–0.14 | 0.42–0.50 | 0.04–0.14 | 0.02–0.11
S (best $z$) | 0.22 | 0.19 | 0.22 | 0.44 | 0.47
S | 0.20–0.23 | 0.18–0.21 | 0.13–0.28 | 0.32–0.48 | 0.3–0.5