# Birds of a Feather: Resolving Stellar Mass Assembly With JWST/NIRCam in a
Pair of Kindred $\boldsymbol{z\sim 2}$ Dusty Star-forming Galaxies Lensed by
the PLCK G165.7+67.0 Cluster

Patrick S. Kamieneski School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Brenda L. Frye Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009  Rogier A. Windhorst School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Kevin C. Harrington European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001, Santiago de Chile, Chile  Min S. Yun Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA  Allison Noble School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Massimo Pascale Department of Astronomy, University of California, 501 Campbell Hall #3411, Berkeley, CA 94720, USA  Nicholas Foo School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Seth H. Cohen School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Rolf A. Jansen School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Timothy Carleton School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Anton M. Koekemoer Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA  Christopher N. A. Willmer Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009  Jake S. Summers School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Nikhil Garuda Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009  Reagen Leimbach Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ, 85721-0009  Benne W. Holwerda Department of Physics and Astronomy, University of Louisville, Louisville KY 40292, USA  Justin D. R. Pierel Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA  Eric F. Jiménez-Andrade Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Antigua Carretera a Pátzcuaro # 8701, Ex-Hda. San José de la Huerta, Morelia, Michoacán, C.P. 58089, México  S. P. Willner Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA  Belén Alcalde Pampliega European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001, Santiago de Chile, Chile  Amit Vishwas Cornell Center for Astrophysics and Planetary Science, Cornell University, Space Sciences Building, Ithaca, NY 14853, USA  William C. Keel Department of Physics and Astronomy, University of Alabama, Box 870324, Tuscaloosa, AL 35404, USA  Q. Daniel Wang Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA  Cheng Cheng Chinese Academy of Sciences South America Center for Astronomy, National Astronomical Observatories, CAS, Beijing, 100101, China  Dan Coe Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA  Association of Universities for Research in Astronomy (AURA) for the European Space Agency (ESA), STScI, Baltimore, MD 21218, USA  Center for Astrophysical Sciences, Department of Physics and Astronomy, The Johns Hopkins University, 3400 N Charles St. Baltimore, MD 21218, USA  Christopher J. Conselice Jodrell Bank Centre for Astrophysics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK  Jordan C. J. D’Silva International Centre for Radio Astronomy Research (ICRAR) and the International Space Centre (ISC), The University of Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009, Australia  ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia  Simon P. Driver International Centre for Radio Astronomy Research (ICRAR) and the International Space Centre (ISC), The University of Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009, Australia  Norman A. Grogin Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA  Tyler Hinrichs School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  James D. Lowenthal Smith College, Northampton, MA 01063, USA  Madeline A. Marshall National Research Council of Canada, Herzberg Astronomy & Astrophysics Research Centre, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada  ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia  Mario Nonino INAF-Osservatorio Astronomico di Trieste, Via Bazzoni 2, I-34124 Trieste, Italy  Rafael Ortiz III School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Alex Pigarelli School of Earth and Space Exploration, Arizona State University, PO Box 871404, Tempe, AZ 85287-1404, USA  Nor Pirzkal Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA  Maria del Carmen Polletta INAF – Istituto di Astrofisica Spaziale e Fisica cosmica (IASF) Milano, Via A. Corti 12, 20133 Milan, Italy  Aaron S. G. Robotham International Centre for Radio Astronomy Research (ICRAR) and the International Space Centre (ISC), The University of Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009, Australia  Russell E. Ryan, Jr. Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA  Haojing Yan Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, USA

###### Abstract

We present a new parametric lens model for the G165.7+67.0 galaxy cluster,
which was discovered with Planck through its bright submillimeter flux,
originating from a pair of extraordinary dusty star-forming galaxies (DSFGs)
at $z\approx 2.2$. Using JWST and interferometric mm/radio observations, we
characterize the intrinsic physical properties of the DSFGs, which are
separated by only $\sim 1\arcsec$ (8 kpc) and a velocity difference $\Delta
V\lesssim 600~{}{\rm km}~{}{\rm s}^{-1}$ in the source plane, and thus likely
undergoing a major merger. Boasting intrinsic star formation rates ${\rm
SFR}_{\rm IR}=320\pm 70$ and $400\pm 80~{}M_{\odot}~{}{\rm yr}^{-1}$, stellar
masses ${\rm log}[M_{\star}/M_{\odot}]=10.2\pm 0.1$ and $10.3\pm 0.1$, and
dust attenuations $A_{V}=1.5\pm 0.3$ and $1.2\pm 0.3$, they are remarkably
similar objects. We perform spatially-resolved pixel-by-pixel SED fitting
using rest-frame near-UV to near-IR imaging from JWST/NIRCam, resolving some
stellar structures down to 100 pc scales. Based on their resolved specific
SFRs and $UVJ$ colors, both DSFGs are experiencing significant galaxy-scale
star formation events. If they are indeed interacting gravitationally, this
strong starburst could be the hallmark of gas that has been disrupted by an
initial close passage. In contrast, the host galaxy of the recently discovered
triply-imaged SN H0pe has a much lower SFR than the DSFGs, and we present
evidence for the onset of inside-out quenching and large column densities of
dust even in regions of low specific SFR. Based on the intrinsic SFRs of the
DSFGs inferred from UV through FIR SED modeling, this pair of objects alone is
predicted to yield an observable $1.1\pm 0.2~{}{\rm CCSNe~{}yr}^{-1}$, making
this cluster field ripe for continued monitoring.

Strong gravitational lensing(1643) — Starburst galaxies(1570) — James Webb
Space Telescope(2291)

††facilities: JWST(NIRCam), ALMA, VLA††software: APLpy (Robitaille & Bressert
2012; Robitaille 2019), astropy (Astropy Collaboration et al. 2013), bagpipes
(Carnall et al. 2018), blobcat (Hales et al. 2012), casa (McMullin et al.
2007), glue (Beaumont et al. 2015; Robitaille et al. 2017), lenstool (Kneib et
al. 1993, 1996; Jullo et al. 2007; Jullo & Kneib 2009), Ned Wright’s Cosmology
Calculator (Wright 2006), piXedfit (Abdurro’uf et al. 2021, 2022a, 2022b,
2022c), photutils (Bradley et al. 2022)

##  1 Introduction

Out to at least $z\sim 3$, the fraction of obscured star formation in galaxies
appears to increase nearly monotonically with increasing stellar mass (Bourne
et al. 2017; Whitaker et al. 2017; McLure et al. 2018), a relation that has
also been recovered by simulations at $z\lesssim 2$ (e.g., Zimmerman et al.
2024). Recent results from the Atacama Large Millimeter/submillimeter Array
(ALMA) have suggested even the presence of a population of galaxies that are
both massive and dust-rich as early as $z\approx 7$ (Algera et al. 2024). The
dust-obscured mode of star formation has been found to dominate the cosmic
star formation rate density (SFRD) for 12 billion years, to at least $z\sim 4$
(e.g., Bouwens et al. 2020; Zavala et al. 2021). Long et al. (2023) predicted
that, at the peak of the Universe’s star formation history at $z\sim 2$
(“Cosmic Noon"), $25-60\%$ of the stellar mass density of the Universe may be
contributed by dusty star-forming galaxies111There is no widely-adopted strict
definition for DSFGs, but they are generally considered to have large SFRs
$\gtrsim 100~{}M_{\odot}~{}{\rm yr}^{-1}$ and significant quantities of dust
capable of reprocessing a sizable fraction of the ultraviolet (UV) emission
from young stars into the infrared. A subset of DSFGs, hyperluminous infrared
galaxies (HyLIRGs) are more clearly defined, such that they exceed $L_{\rm
IR}>10^{13}~{}L_{\odot}$. (DSFGs) and their quiescent descendants. Together,
this implies that our understanding of the growth and evolution of stellar
mass in the Universe from rest-frame UV/optical studies alone would be
horribly incomplete.

Conversely, our knowledge of the typical stellar masses and properties of
older, extant stellar populations of DSFGs is still woefully lacking, due in
large part to intense dust attenuation at crucial rest-frame UV/optical
wavelengths. Now, the sensitivity and resolution of JWST in this regime is
beginning to facilitate novel interpretations of this frequently perplexing
population—often also referred to as submillimeter galaxies (SMGs; Blain et
al. 2002), although these may more accurately be said to comprise a subset of
DSFGs (Casey et al. 2014a). They are widely believed to be the progenitors of
massive quiescent, early-type galaxies seen at lower redshifts (Lilly et al.
1999; Farrah et al. 2006; Toft et al. 2014), and given their association with
galaxy overdensities and protoclusters (Geach et al. 2006; Chapman et al.
2009; Clements et al. 2014; Dannerbauer et al. 2014; Casey et al. 2015;
Umehata et al. 2015; Casey 2016; Clements et al. 2016; Hung et al. 2016;
Miller et al. 2018; Long et al. 2020; Caputi et al. 2021; Calvi et al. 2023),
it is presumed that many will evolve into the ultra-massive population of
brightest cluster galaxies (BCGs).

DSFGs at Cosmic Noon are typically considered to be dominated by compact
nuclear starburst events, occurring within a thick shroud of dust. These have
been postulated to signal the inside-out growth of galactic bulges that will
persist in the massive descendents (e.g., Gullberg et al. 2019; Nelson et al.
2019). This activity may be driven by major mergers (Hopkins et al. 2008), or
more secularly through in situ star formation in gravitationally-unstable,
gas-rich disks, followed by inward clump migration (e.g. Bournaud et al. 2007,
2008; Elmegreen et al. 2008; Ceverino et al. 2010; Elmegreen 2011; Wuyts et
al. 2012). However, for DSFGs, this picture often lacks the context of how the
extant stellar mass (or even ongoing unobscured star formation) is
distributed. Many studies have found a factor of $>2$ larger effective radii
in the rest-frame UV/optical vs. FIR continuum for DSFGs (including Ikarashi
et al. 2015; Simpson et al. 2015; Barro et al. 2016; Hodge et al. 2016;
Pantoni et al. 2021). However, these half-light sizes are subject to gradients
in the mass-to-light (M/L) ratios, which may be the result of spatial
variation in dust attenuation (e.g., Byun et al. 1994; Jansen et al. 1994;
Tuffs et al. 2004; Möllenhoff et al. 2006; Pastrav et al. 2013; Popping et al.
2022), stellar population ages (e.g., Carrasco et al. 2010; van Dokkum et al.
2010; Tacchella et al. 2015), and metallicities (Franx & Illingworth 1990;
Mollá et al. 1997; Carrera et al. 2008; Cresci et al. 2010; Kewley et al.
2010; Pilkington et al. 2012)—or realistically, some combination of all three
(e.g., Keel & White 2001; Miller et al. 2022).

Figure 1:  North-aligned RGB image of the G165 cluster, with (R, G, B) =
(F444W + F356W, F277W + F200W, F150W + F090W). Colored curves show the best-
fit model-derived lensing caustics at the source-plane redshifts of Arc system
2abc (the host of SN H0pe; $z=1.78$), Arc 1abc ($z=2.24$), Arc 5abc
($z=3.95$), and at $z=6$; see Fig. 2 of F24 for the full set of arcs. Violet
contours show JVLA 6 GHz continuum, blue contours show ALMA 3 mm continuum,
and green contours show ALMA 2 mm continuum. Their respective synthesized
beams are shown in the lower left. Spurious features in the ALMA contours
towards the edge of the field of view are due to lower sensitivity near the
edge of the primary beam. Arcs 1a, 1bc, and 3c are strongly detected at 2 mm,
3 mm, and 6 GHz. Arc 3ab yields only a marginal detection at 6 GHz, which is
due to the lack of dust/radio continuum in the region of the galaxy that is
lensed into this merging pair of arcs (see Fig. 2, for example). Dashed white
boxes show the source-plane locations of DSFG-1 and DSFG-3 (smaller northwest
and larger southeast boxes, respectively), and match the fields of view shown
in Fig. 3.

In the era of JWST, great progress is being made in directly capturing the
rest-frame near-IR structure of galaxies at Cosmic Noon and earlier, offering
a significantly less biased view of the actual distribution of stellar mass
(Lang et al. 2019; Chen et al. 2022; Cheng et al. 2022; Suess et al. 2022;
Abdurro’uf et al. 2023; Cheng et al. 2023; Colina et al. 2023; Kamieneski et
al. 2023; Kokorev et al. 2023; Le Bail et al. 2023; Liu et al. 2023;
Rujopakarn et al. 2023; Smail et al. 2023; Amvrosiadis et al. 2024; Kalita et
al. 2024; Sun et al. 2024), including their clumpiness, compactness, and
lopsidedness. This pixel-by-pixel comparison of stellar mass and star
formation provides a key snapshot of the structural history of a galaxy,
especially when supplemented with mapping of the molecular gas kinematics and
dust continuum at comparable resolutions with facilities like ALMA. For
example, an examination of how the star-forming main sequence of galaxies
translates on resolved scales at $z>1$ has now been made feasible (e.g.,
Abdurro’uf & Akiyama 2018; hereafter A18), along with its connection to
fundamental scaling relations between the surface density of gas mass vs. star
formation rate and between gas mass surface density vs. stellar mass surface
density. While a full systematic study has not yet been carried out, results
for a small number of objects can offer preliminary insights.

Figure 2:  Source-plane reconstruction of the DSFG system at $z=2.24$. The
background RGB image shows JWST filters F444W, F200W, and F115W
(respectively). The gold/white dotted line shows the lens model-derived
caustic curve at $z=2.24/z=2.12$ (the latter of which is a secondary redshift
solution for DSFG-3, as discussed in Appendix B). The portions of the source-
plane interior to the caustics are triply-imaged; structures above and below
the caustics in this figure are only singly-imaged (but still magnified by
$\mu\gtrsim 5$, primarily in the direction perpendicular to the caustic). The
constituent arcs (Arcs 1, 3, 4, and 6), as labeled by Frye et al. (2024), are
indicated with circles, in addition to the NS_969 and NS_46 components that
are associated with DSFG-1 (but at slightly different redshifts, indicating
perhaps the presence of two orbiting clumps seen in projection).

In this work, we focus on two DSFGs and a massive red galaxy at Cosmic Noon
that have been strongly lensed by a massive galaxy cluster at $z=0.35$. PLCK
G165.7+67.0 (hereafter G165) was first identified by Cañameras et al. (2015)
and Harrington et al. (2016)222The cluster has also been denoted in literature
as PJ112714.5., hereafter H16, as an exceptionally IR-bright source in Planck
and Herschel surveys, but was first examined in-depth by Frye et al. (2019)
and Pascale et al. (2022). Multi-wavelength follow-up imaging at higher
resolution revealed that the substantial submillimeter flux is driven by a
highly-magnified DSFG (Table 1) at $z=2.236$ (H16), lensed into a giant arc by
a galaxy cluster that appears to be undergoing a merger of two primary mass
components (Fig. 1, and Pascale et al. 2022). Hereafter, we refer to this
known DSFG, PLCK_G165.6295+67.0026333Galactic coordinates are used to remain
consistent with the name of the cluster, and the location of Arc 1a is chosen
as it contains the entire object, in contrast with the fold image 1bc., as
G165-DSFG-1 (or just DSFG-1), given its appearance as Arcs 1a and 1bc, per the
labeling introduced by Frye et al. (2019). With this work, we identify another
strongly-lensed submillimeter source, which is believed to lie at the same
redshift. Our current photometric redshift and lens model-predicted geometric
redshift are consistent with $z\approx 2.23$, making it likely that the two
are gravitationally interacting (see Fig. 2). For this second object,
G165.6354+67.0087, we use the nickname G165-DSFG-3 (or DSFG-3), in accordance
with its appearance in Arcs 3c and 3ab. Through its inferred properties
(including $L_{\rm IR}>10^{12.6}$), it is clearly also safely classified as a
DSFG. Frye et al. (2024), hereafter F24, revealed substantial candidate galaxy
overdensities at $z\approx 1.8$ and at $z\approx 2.2$ using a combination of
spectroscopic redshift identifications with JWST NIRSpec and photometric
redshifts from 8-filter NIRCam photometry. The number of candidates identified
for these structures ($N>150$) rivaled even the number of objects with
photometric redshifts consistent with the G165 cluster itself.

As first reported by Frye et al. (2023) and Polletta et al. (2023), initial
inspection of the NIRCam observations immediately revealed a conspicuous
multiply-imaged supernova in a set of arcs on the opposite side of the cluster
from the DSFGs. Named SN H0pe, this serendipitous discovery was quickly
followed-up with additional NIRCam imaging and NIRSpec spectroscopy to track
the transient’s light curve and classify its type. At $z=1.78$, it was
confirmed to be one of the most distant known Type Ia supernovae. This
discovery may not be purely coincidence: the fortuitous alignment of a rich
collection of star-forming $z\sim 2$ objects with a massive and highly
elliptical foreground cluster ($M_{\rm 600kpc}=(2.6\pm 0.3)\times
10^{14}~{}M_{\odot}$; F24) along the line of sight provides ideal conditions
for the detection of Type Ia and core-collapse supernovae. As the area of
time-delay cosmography has become increasingly fruitful in recent years (Treu
et al. 2022), the G165 cluster is likely to endure as a subject of intense
interest and legacy value for years to come.

This paper is organized as follows: in Section 2, we outline the near-IR,
radio, and mm-wave imaging used in this study. In Section 3, we discuss our
approach to lens modeling and performing pixel-by-pixel SED fitting. The
results of these analyses are outlined in Section 4, and their implications
are discussed in Section 5 and summarized in Section 6.

In this work, we adopt a $\Lambda$CDM cosmological model of $\Omega_{m}=0.3$,
$\Omega_{\Lambda}=0.7$, and $H_{0}=70~{}{\rm km}~{}{\rm s}^{-1}~{}{\rm
Mpc}^{-1}$, and a Kroupa (2001) initial mass function (IMF). At the redshift
of the cluster ($z=0.348$), the angular-to-physical size conversion is
$1\arcsec=4.921$ kpc, and at the redshift of the DSFGs ($z=2.236$),
$1\arcsec=8.243$ kpc (Wright 2006). To comment on our notation: throughout, we
frequently use DSFG-1 and DSFG-3 as shorthand for PLCK_G165.6295+67.0026 and
G165.6354+67.0087, respectively. These names themselves are introduced to
avoid confusion when referring to the actual objects in the background of the
cluster. To clarify, DSFG-1 is visible in its entirety in Arc 1a, while only a
fraction of the intrinsic source is visible in the merging pair Arc 1bc.
Likewise, DSFG-3 is visible entirely in Arc 3c, with a small portion of the
source visible in the merging pair Arc 3ab.

Figure 3:  Source-plane reconstruction of Arcs 1abc at $z=2.236$ (left) and
3abc at the lens model-preferred redshift of $z=2.12$ (right), with 8-filter
RGB images scaled according to the Trilogy prescription. The locations of
these reconstructions are indicated in Fig. 1. The filters are assigned to the
red/green/blue channels as (R,G,B) = (F356W + F410M + F444W, F200W+F277W,
F090W + F115W + F150W). Caustics for their respective redshifts are shown as
white dotted curves, while contours show the reconstruction of JVLA 6 GHz
(violet), ALMA 3 mm (light blue), and ALMA 2 mm (lime green), as in Fig. 1.
Representative source-plane elliptical PSF beams (§3.1.2) for the radio/mm
images are shown in the lower left, and for the finest/coarsest NIRCam filters
in the lower right (teal = F090W, orange = F444W). Only the regions of the
source inside the caustic curves (southeastern end of G165-DSFG1, northwestern
end of G165-DSFG3) are included in their respective highly-magnified fold
images, 1bc and 3ab (Fig. 18). However, by reconstructing together with the
much lower-magnification images 1a and 3c, they are effectively convolved to
lower resolution.

##  2 Data

###  2.1 JWST NIRCam

A more in-depth description of the JWST NIRCam data is given in Windhorst et
al. (2023) and Frye et al. (2024), but we provide relevant details here. G165
was observed with NIRCam in three epochs: first through the Prime
Extragalactic Areas for Reionization and Lensing Science (PEARLS) program (PID
1176; PI: Windhorst) on 2023 March 30 (Epoch 1), followed by two later epochs
on 2023 April 22 (Epoch 2) and 2023 May 9 (Epoch 3) through a DDT program (PID
4446, PI: Frye), which were designed to track the light curve of SN H0pe
following its discovery in Epoch 1. As the exposure times of Epoch 1 were
longer (see Table 1 of F24), and 8 filters were used instead of 6, for
simplicity we performed most of our analysis in this work on data from Epoch 1
only, in part to minimize variance in the PSF which could impact the
spatially-resolved SED analysis. Moreover, the primary targets here are
detected with very high significance, and do not benefit enormously from
stacking additional exposures. However, the image families used in
constraining the lens model (which are often faint) benefit greatly from this
additional effective exposure time, and so the stacked images are employed for
that purpose. Limiting magnitudes are $m_{\rm AB}\sim 28.4-28.5$ for the 4
short wavelength (SW) filters (F090W / F115W / F150W / F200W) and $m_{\rm
AB}\sim 28.6-28.9$ for the 4 long wavelength (LW) filters (F277W / F356W /
F410M / F444W); see Figure 6 of F24. Data were reduced in line with Windhorst
et al. (2023), using the STScI JWST
Pipeline444https://github.com/spacetelescope/jwst version 1.11.2 (Bushouse et
al. 2023) and context file pmap_1100, but with additional improvements in
“wisp" removal as described by Robotham et al. (2023). Images in this work
were drizzled on to 30 milliarcsecond square pixels.

###  2.2 ALMA Bands 3 and 4

G165 was observed with ALMA band 3 (representative wavelength 2.7 mm, 111 GHz;
hereafter referred to as 3 mm) and band 4 (2.2 mm, 138 GHz; hereafter referred
to as 2 mm) as part of program 2021.1.00607.S (PI: R. Cañameras). These data
were retrieved from the public ALMA
archive555https://almascience.nrao.edu/aq/.

Continuum multi-frequency synthesis images and spectral cubes were created
using the Common Astronomy Software Applications (casa) version 6.2.1
(McMullin et al. 2007), with robust$=$0.5 Briggs weighting (Briggs 1995). The
unweighted baselines ranged from $0.03-0.3$ km (5th to 80th percentile). This
results in synthesized beams of $2\farcs 0\times 0\farcs 9$ at
PA$=-2.7^{\circ}$ for 3 mm (maximum recoverable scale of $18\farcs 4$) and
$1\farcs 6\times 0\farcs 7$ at PA$=-18.4^{\circ}$ for 2 mm (maximum
recoverable scale of $15\farcs 1$)666Imaging with $\texttt{robust}=-1$ (i.e.,
closer to uniform weighting) was tested in an attempt to improve angular
resolution, but S/N was insufficient to yield useful results.. The continuum
noise rms levels achieved are 22 $\mu$Jy/beam (3 mm) and 20 $\mu$Jy/beam (2
mm), both over a 3.3 GHz bandwidth. Weather conditions were favorable, with
mean precipitable water vapor PWV $=0.9$ mm for the 3 mm observations
(executed on 2022 May 26) and PWV$=1.4$ mm for the 2mm observations (executed
on 2022 May 28).

As the lensed objects of interest are spatially resolved, photometry was
performed using the flood-filling algorithm blobcat (Hales et al. 2012). In
Table 1, we compare the measured 2 mm and 3 mm fluxes against the 1 mm flux
from the Large Millimeter Telescope (LMT) AzTEC, reported by H16, under
assumption of a modified blackbody SED, and find relative consistency.

###  2.3 VLA C Band

G165 was observed with the Karl G. Jansky Very Large Array in C band (4 - 8
GHz) in full polarization during the most extended A configuration through our
program 18A-399 (PI: P. Kamieneski), which targeted 26 members of the Planck
All-Sky Survey to Analyze Gravitationally-lensed Extreme Starbursts (PASSAGES)
sample; see Kamieneski et al. (2024) for additional details. The target was
observed on 2018 April 6 during a shared track (switching with another nearby
object) lasting a total of 3.0 hrs, with an effective total on-source
integration time of 1.1 hrs. Weather conditions were favorable during the
observations: the phase RMS measured by the Atmospheric Phase Interferometer
at the VLA site was $\approx 2^{\circ}$. Antenna baselines ranged from
$0.8-36.6$ km, yielding a maximum recoverable scale of $9\arcsec$. The
sensitivity reached was $2.7~{}\mu{\rm Jy}~{}{\rm beam}^{-1}$. Natural
weighting achieved a synthesized beam of $0\farcs 65\times 0\farcs 37$ at
PA$=72.2^{\circ}$.

The data were reduced using casa v6.1.0. Initial flagging and calibration was
carried out through the VLA Calibration pipeline v2018.1. The Högbom (1974)
CLEAN algorithm was used to deconvolve the image down to a $2\sigma$
threshold. As with the ALMA images, photometry was performed using blobcat
(Hales et al. 2012), also reported in Table 2. As discussed by Pascale et al.
(2022), these data also reveal a number of point sources and narrow angle
tails (e.g., Rudnick & Owen 1977; Venkatesan et al. 1994), which offer some
insight into the intracluster medium and dynamical state of the galaxy
cluster.

##  3 Methodology

###  3.1 Lens modeling

We constructed a parametric lens model for the G165 cluster using
lenstool777https://projets.lam.fr/projects/lenstool/wiki (Kneib et al. 1993,
1996; Jullo et al. 2007; Jullo & Kneib 2009), which has been tested
extensively for galaxy clusters (e.g., Johnson et al. 2014; Meneghetti et al.
2017). The model is constrained by 41 families of multiply-imaged features—of
which 14 have spectroscopic redshift information (Frye et al. 2019; Pascale et
al. 2022; Frye et al. 2024), covering 4 distinct source planes. Some details
of this model and the cluster members are presented in the appendix of Pascale
et al. (2024), but we include additional information here. For this iteration
of the model, only the image-plane positions were considered, which are
reported in F24. To estimate the astrometric uncertainty in these positions,
multiple team members independently marked the positions for the known image
family members, and found a median offset of $0\farcs 03$, which is consistent
with the pixel scale used for identifying images. For the mass profiles, we
used two cluster halo-scale profiles for the merging cluster components, in
addition to perturbations to this large-scale potential in the form of the 165
identified cluster member galaxies at $z\approx 0.35$. Each of these profiles
was chosen to be a pseudo-isothermal elliptical mass distribution, or PIEMD
(Kassiola & Kovner 1993).

To reduce the number of free parameters, the position of each cluster member
was held fixed to the centroids of their luminous components. Likewise, the
ellipticities and orientations were held fixed to match their morphologies in
the F200W filter. The masses of the cluster members were scaled all together
in a power law, according to their F200W flux, per Limousin et al. (2005). The
power law was normalized such that a characteristic $L_{\star}$ galaxy at the
redshift of the cluster ($m_{\rm F200W}=17.0$ AB mag) has a velocity
dispersion of $\sigma_{0}^{\star}=120~{}{\rm km~{}s}^{-1}$, a core radius of
$r_{\rm core}^{\star}=0.15$ kpc, and a cut radius of $r_{\rm cut}^{\star}=30$
kpc. All galaxies were scaled in mass based on their flux relative to
$L_{\star}$. We settled on these choices after some trial and error, as it was
determined that the current set of lensing evidence does not sufficiently
constrain the masses of perturbing cluster galaxies, so they were instead held
fixed. The choices of $\sigma_{0}^{\star}$, $r_{\rm core}^{\star}$, and
$r_{\rm cut}^{\star}$ match those used in several other works that use
lenstool, including Limousin et al. (2008) and Johnson et al. (2014). However,
one foreground galaxy near Arc 3ab, located at $(\alpha,\delta)=$
(11h27m14$\fs$304, $+42^{d}$28m32$\fs$32), was optimized independently from
the other cluster members, given the large effect it has on the arc. Its
velocity dispersion $\sigma$ was kept as a free parameter, assuming a uniform
prior of 0 to 250 km s-1.

As for the two cluster-scale halos, their free parameters include centroid
position, ellipticity, orientation, velocity dispersion, and core radius. As
their cut/truncation radii are poorly constrained, these were fixed to values
of 1000 kpc. Uniform priors were used for these parameters, including a wide
range of $\pm 5\arcsec$ in $x$ and $y$ position from the approximate center of
both cluster cores, a very wide range in ellipticity from 0 to 0.85, a maximum
velocity dispersion of 800 km s-1, and a maximum core radius of
$20\arcsec\approx 100$ kpc. We also left as free parameters the redshifts for
16 background lensed image families for which no spectroscopic information is
currently available. To simplify the parameter space, we placed relatively
tight priors on these, however, based on their photometric redshifts. This was
done iteratively, where we widened or adjusted the redshift range if prior
iterations of the model optimization resulted in any parameters clearly being
overly limited by the prior boundaries. Still, the uncertainties in these
“geometric" redshifts (often as small as $\Delta z=\pm 0.01$) are likely
drastic underestimates. We present the full set of best-fit parameter values,
and the corresponding convergence map, in Appendix A. In this work, we adopt
the median of each parameter’s posterior distribution, which is only
appropriate if the posteriors are well-behaved and approximately normally
distributed (and especially only if the distributions are not multi-modal). We
also provide the minimum-$\chi^{2}$ solutions, which are nearly always in
close agreement with the median solution.

To determine magnifications, we used slightly different approaches for JWST
vs. mm/radio images. For the ALMA and VLA images, which are not contaminated
by nearby foreground sources888While several of the cluster members are
detected in 6 GHz continuum, they are sufficiently far from the lensed arcs of
interest to have negligible impact., we measured magnifications simply by
taking the ratio of image-plane to source-plane area. For the JWST images, we
measured representative magnifications using the F200W filter, and compute an
average of the magnification map weighted by source-plane flux. In both cases,
to include the relevant statistical uncertainties, we resampled 300 iterations
uniformly from the MCMC-derived posterior distribution, resulting in an
approximately Gaussian distribution of magnifications. The results for the Arc
1abc and Arc 3abc systems are summarized in Table 1. Any slight discrepancies
beyond statistical uncertainty between the magnifications for rest-frame near-
IR, far-IR, and radio continuum are likely due in part to very different
observing parameters (e.g. PSFs), but also due to differences in source-plane
distribution (as seen in Fig. 3).

####  3.1.1 Redshift of DSFG-3

Since DSFG-3 lacks a secure spectroscopic redshift, we left this as a free
parameter in our lens model, and allowed the redshift to vary from $z=2.0-2.2$
in the final model. This range was chosen based on previous iterations of the
model with wider redshift priors. This is also in line with the photometric
redshift of $z_{\rm phot}=2.23\pm 0.09$ and with previous iterations of the
lens model using a wider prior range, which suggested a geometric redshift of
$z_{\rm geom}\approx 2.1$. The MCMC posterior distribution yields a quite
precise value of $z_{\rm geom}=2.11\pm 0.01$ (although the uncertainty is
likely underestimated). This narrow range is probably a result of the
constraint offered by DSFG-1, which is lensed by a similar portion of the
foreground cluster and has a known redshift from the LMT Redshift Search
Receiver (RSR) and other single-dish sub-mm observations (H16; Harrington et
al. 2021) and from NIRSpec (F24).

There is no clear evidence of a second line detected in the LMT/RSR spectrum
(H16), which means either that the CO(3–2) line from DSFG-3 is blended with
the line from DSFG-1 at $z=2.236$, or is undetected. Indeed, this target has
the widest linewidth ($\Delta V=650\pm 10$ km s-1) of the sample presented by
H16, and a possible secondary component at $\sim-400$ km s-1, giving weight to
the hypothesis that the lines are blended. Since DSFG-3 is inferred to have a
similar stellar mass, size, and star formation rate as DSFG-1 (as we discuss
in §4), it is reasonable to expect comparable CO flux (or only $\sim 4$ times
greater flux for DSFG-1 given its larger magnification, Table 1). A non-
detection by virtue of a faint line flux is thus unlikely. However, the
pointing center of the LMT observations coincides with the peak of the sub-mm
flux, which is dominated by the high-magnification Arc 1bc. At an angular
separation of $<11\arcsec$, Arc 3c would be near the edge of the primary beam
($\sim 13\farcs 1$ at 106.86 GHz), so it is also somewhat possible that any
lines were simply missed by the observations.

We adopt the redshift of DSFG-3 to be $z=2.23$ in this work ($\Delta V\sim
600~{}{\rm km~{}s}^{-1}$ from DSFG-1), but we briefly discuss the possibility
and implications of the secondary redshift solution, $z\approx 2.12$, in
Appendix B. Also, we do not hold fixed the redshift of DSFG-3 in our lens
model given this lingering uncertainty. For the source-plane reconstructions
shown in Fig. 2 and 3, we ray trace back to $z=2.12$ for the highest-fidelity
view of the source, but this has no impact on our interpretation of results.

Table 1: Model-derived magnifications and observed mm/radio fluxes.

Arc | $\mu_{2\mu{\rm m,JWST}}$ | $\mu_{2{\rm mm}}$ | $\mu_{3{\rm mm}}$ | $\mu_{6{\rm GHz}}$ | $S_{\rm 2mm}$ | $S_{\rm 3mm}$ | $S_{\rm 6GHz}$
---|---|---|---|---|---|---|---
|  |  |  |  | [$\mu$Jy] | [$\mu$Jy] | [$\mu$Jy]
Arc 1a | $5.5\pm 0.1$ | $7.6\pm 0.2$ | $8.1\pm 0.2$ | $6.6\pm 0.2$ | $260\pm 30$ | $110\pm 30$ | $190\pm 10$
Arc 1bc | $42\pm 1$ | $44\pm 3$ | $41\pm 4$ | $46\pm 5$ | $1620\pm 90$ | $860\pm 60$ | $960\pm 50$
Arc 1abc | — | $28.4\pm 2.4$ | $29.1\pm 1.7$ | $24^{+6}_{-3}$ | $1900\pm 100^{\dagger}$ | $980\pm 60^{\dagger}$ | $1150\pm 50$
Arc 3ab | $40\pm 2$ | — | — | $29\pm 5$ | $<80$ | $<90$ | $39\pm 5$
Arc 3c | $5.9\pm 0.1$ | $6.8\pm 0.2$ | $7.5\pm 0.3$ | $5.9\pm 0.2$ | $620\pm 40$ | $310\pm 30$ | $132\pm 8$
Arc 3abc | — | $7.3\pm 0.7$ | $7.4\pm 0.6$ | $6.8^{+1.1}_{-0.4}$ | $620\pm 40^{\dagger}$ | $310\pm 30^{\dagger}$ | $171\pm 9$

^†^†footnotetext: The combined flux of of Arcs 1abc and 3abc at 2 mm
($S\approx 2.5$ mJy) is consistent with that expected from a modified
blackbody (dust emissivity index $\beta=1.8$) with $S_{\rm 1mm}=24\pm 2.0$ mJy
(H16), which is $S_{\rm 2mm}\approx 2.5$ mJy. The observed flux at 3 mm (1.3
mJy) exceeds the modified blackbody prediction (0.5 mJy), possibly owing to
contamination from synchrotron emission.

Note. — Magnifications are measured for the F200W filter (as representative
for all of the NIRCam filters) by taking the source-plane flux-weighed average
over the magnification map. For the mm and radio images, magnifications are
measured by simply taking the ratio of image-plane to source-plane areas.

####  3.1.2 Source-plane reconstruction

The reconstruction of all images in this work in the source plane does not
impose regularization or parameterization of the intrinsic structure of the
source. Instead, the observed images are ray-traced to the source plane pixel-
by-pixel through the cleanlens function of lenstool, with the image-plane
oversampled by a factor of 4 and the source-plane by a factor of 2. Further
details of our specific procedure may be found in Section 3.3 of Kamieneski et
al. (2024). Fig. 3 reconstructs the full structure of G165-DSFG-1 and
G165-DSFG-3 by including both low- (1a and 3c) and high-magnification (1bc and
3ab) arcs. Effectively, this simple approach convolves the high-magnification
arcs with the PSF of the low-magnification counterparts (e.g., Sharma et al.
2018). We use this version to estimate the source-plane size of the dust
continuum, where we use the casa imfit task to fit the emission with 2-d
Gaussian profiles (and adopt a minimum 10% fractional uncertainty). Despite
the smaller beam size of the 3 mm observations, these dust sizes are measured
for the 2 mm images given their higher S/N, as reported in Table 2.
Additionally, the possible contamination from synchrotron at 3 mm is expected
to result in slightly larger sizes than from thermal dust alone. In Appendix
D, we show the reconstructions from including only the high-magnification
images, which are at essentially the finest achievable resolution. There is
general agreement with the structure seen in Fig. 3, adding confidence to this
reconstruction.

RGB images of the 8 NIRCam filters are constructed according to the
Trilogy999https://www.stsci.edu/$\sim$dcoe/trilogy prescription (Coe et al.
2012). In all cases, we also show representative source-plane PSFs to
illustrate the approximate resolution of the reconstructions. These are
created by placing a simulated PSF or beam at a location in the image plane
and ray-tracing it to the source-plane in the same manner as the image itself
(then refitting with a 2-dimensional Gaussian). The distorted source-plane
PSFs arise because the image-plane PSFs are relatively circularly symmetric to
first order, but this symmetry is lost through the ray-tracing process.
However, as there is a gradient in magnification over each of the arcs (which
is much more extreme for 1bc and 3ab), the source-plane PSF is finer in
regions closer to the caustic curve than those further away. While the source-
plane reconstructions assist in our interpretation of these objects, most of
this work computes properties in the image plane and applies a magnification
correction (or relies on magnification-invariant properties). This approach is
less subject to any pitfalls of the intricacies of source-plane PSFs.

The approximate source-plane major axes of the ALMA 3 mm and 2 mm beams
(roughly oriented along the major axis of the galaxies) are $1\farcs 0\approx
8.2$ kpc and $0\farcs 8\approx 6.6$ kpc for G165-DSFG-1, and $0\farcs 7\approx
5.8$ kpc and $0\farcs 6\approx 5.0$ kpc for G165-DSFG-3. For VLA 6 GHz, the
source-plane beam minor axes are approximately oriented along the galaxies’
major axes, which are $0\farcs 2\approx 1.6$ kpc for G165-DSFG-1, and $0\farcs
1\approx 0.83$ kpc for G165-DSFG-3. These are indicated in Fig. 3.

###  3.2 Pixel-by-pixel SED fitting with piXedfit and bagpipes

As the cluster lensing provides an areal magnification and a flux
amplification for the DSFGs, it becomes easier to ascertain the spatially-
resolved properties of their stellar populations through so-called pixel-by-
pixel SED fitting. To accomplish this, we used the publicly-available package
piXedfit101010https://pixedfit.readthedocs.io/en/latest/ (Abdurro’uf et al.
2021; also Abdurro’uf et al. 2022a, b, 2023) to bin pixels such that each
satisfies a S/N condition in all filters, while preserving similarity in SED
shape between pixels within each bin. In brief, starting with the brightest
unbinned pixel, each bin is required to be larger than the F444W PSF FWHM
($0\farcs 14$). If a S/N threshold for each filter is met (a conservative
S/N$=20$, in our case), the bin is established; otherwise, the radius of the
bin is increased. Before adding the new pixels from the larger radius to the
bin, their SED shape is confirmed to be consistent through a $\chi^{2}$
calculation. This process then repeats, starting with the brightest pixel not
already in a bin, and terminates once no additional bins can be constructed.
Full details are provided in Section 3.3 of Abdurro’uf et al. 2021.

With the bin-wise photometry collected for the 8 NIRCam bands
($0.9-4.4~{}\mu$m), all convolved to the resolution of F444W, we fitted the
SED of each bin with bagpipes (Bayesian Analysis of Galaxies for Physical
Inference and Parameter EStimation; Carnall et al. 2018). The star formation
history (SFH) was parameterized as an exponentially declining or “tau" model,
where star formation promptly begins at time $T_{0}$ and decays relative to a
timescale $\tau$. The stellar population age for each bin could vary from 0.1
Gyr to the age of the Universe at the given redshift, and $\tau$ was allowed
to vary from $0.3-10$ Gyr. We consider this simple smooth (i.e. not bursty)
parameterization of the SFH to be appropriate given the small physical scale
of each bin. In other words, given the high spatial resolution, these sub-kpc
regions are perhaps best described with low time resolution (to avoid
overfitting), and thus a less stochastic star formation history than a galaxy
in aggregate. The total mass formed was allowed to vary widely, from ${\rm
log}_{10}(M/M_{\odot})=1-15$, and metallicity could vary from 0 to 3.5
$Z_{\odot}$. The standard nebular component in the bagpipes fits was accounted
for through the ionization parameter, log($\cal{U}$), which was allowed to
vary from $-4$ to $-2$, pre-computed from cloudy photoionization models
(Ferland et al. 2017). While higher values (up to $-1$) can be found for
individually resolved H ii regions (e.g., Snijders et al. 2007), an upper
limit of $-2$ is typically sufficient for starbursts resolved on kpc scales
(e.g., for M82’s inner 500 pc; Thornley et al. 2000; Förster Schreiber et al.
2001). A Calzetti et al. (2000) dust attenuation model was imposed, with
attenuation allowed to vary from $A_{V}=0-3$, which we found to be a
sufficient range111111To test this, we examined the 84th percentile of the
derived $A_{V}$ posteriors for the DSFG-1 and DSFG-3 bins. Only 12% and 8%
(respectively) exceeded $A_{V}>2.5$..

Figure 4:  Spatially resolved properties of DSFG-1 (Arc 1a), including image-
plane properties $\mu{\rm SFR}$ and $\mu M_{\star}$, and their magnification-
corrected intrinsic values, SFR and $M_{\star}$. As bin size varies, these
properties are best assessed through their respective surface densities,
$\Sigma_{\rm SFR}$ and $\Sigma_{\star}$, which are also unaffected by
magnification, as are the specific star formation rates, sSFR. Dust
attenuation $A_{V}$, ionization parameter log($\cal{U}$), stellar metallicity
$Z$, and mass-weighted ages are likewise assumed to not be affected by lensing
magnification. These properties are inferred through SED fitting with bagpipes
at a fixed redshift, $z=2.236$. The bin index map shows the order in which the
piXedfit bins are allocated. The $R_{\rm equiv}$ panel shows the radius of a
circle with equivalent area to each bin corrected for areal magnification
factor $\mu$, such that $R_{\rm equiv}\equiv(A\mu^{-1}/\pi)^{0.5}$. The field
of view of each panel is $120\times 120$ pix, or $3\farcs 6\times 3\farcs 6$.
Figure 5:  Spatially resolved image-plane properties for DSFG-3 (Arc 3c), as
with Fig. 4. Redshift is held fixed at an assumed $z=2.23$. The field of view
is $140\times 140$ pix, or $4\farcs 2\times 4\farcs 2$.

##  4 Results

###  4.1 The complex, galaxy-wide assembly of stellar mass in two DSFGs at
Cosmic Noon

One of the fundamental questions we have sought to address through our
resolved pixel-by-pixel SED modeling of G165-DSFG-1 and G165-DSFG-3 is how
their significant ongoing star formation is spatially distributed. If DSFGs
are primarily powered by violent major mergers of massive galaxies, as for
local ultra-luminous infrared galaxies (ULIRGs; Armus et al. 1987; Soifer et
al. 1987; Sanders et al. 1988; Sanders & Mirabel 1996; Lonsdale et al. 2006),
is it reasonable to expect that their burst of star formation might be
confined only to nuclear regions, with shocks and tidal forces driving gas
inwards (e.g., Hernquist 1989; Stockton 1990; Barnes & Hernquist 1991, 1992)?
Or if other modes of extreme star formation are prevalent, such as
gravitational instabilities within massive gas-rich disks—fueled continuously
by cold gas streams (e.g., Kereš et al. 2005; Dekel et al. 2009; Davé et al.
2010)—would we expect to find a greater number of global star formation
events, reaching well beyond the nucleus?

Numerous recent works have demonstrated that the effective radii of DSFGs at
rest-frame far-IR are compact ($\sim 1-2$ kpc) and smaller than at rest-frame
UV/optical (e.g., Simpson et al. 2015; Ikarashi et al. 2015; Barro et al.
2016; Hodge et al. 2016; Tadaki et al. 2017). On the other hand, some of the
most luminous DSFGs—so-called hyperluminous infrared galaxies (HyLIRGs), which
exceed $L_{\rm IR}>10^{13}~{}L_{\odot}$—may be more extended, in part
explaining their ability to sustain SFRs above $1000~{}M_{\odot}~{}{\rm
yr}^{-1}$ without significantly breaching the Eddington limit (e.g., Bussmann
et al. 2013; Kamieneski et al. 2024). In a large sample of Herschel-selected
candidate lensed DSFGs, Borsato et al. (2024) found rest-optical sizes to be
$\approx 3$ times smaller than far-IR sizes, determined by Enia et al. 2018 to
be $\sim 2-3$ kpc. However, these isophotal rest-optical sizes are SNR-
dependent, and the far-IR sizes are derived from lower-resolution
Submillimeter Array imaging, both of which may bias results towards smaller
optical-to-FIR size ratios.

In this work, we find 2-mm dust continuum effective radii of $R_{e,2{\rm
mm}}=1.6\pm 0.2$ kpc and $2.1\pm 0.2$ kpc for DSFG-1 and DSFG-3, respectively
(Table 2).121212At 3 mm, we find $\sim 1.5\times$ larger sizes of $R_{e,3{\rm
mm}}=2.6\pm 0.3$ for both DSFGs, consistent with contribution from synchrotron
(Thomson et al. 2019). These values are quite consistent with other members of
PASSAGES (Kamieneski et al. 2024) and other lensed DSFGs (e.g., Bussmann et
al. 2013), and in line with what might be expected from the size-luminosity
relation derived by Fujimoto et al. (2017) and their intrinsic luminosities of
${\rm log}[L_{\rm IR}/L_{\odot}]\approx 12.5$. Given their apparent elongated
morphology suggestive of high-inclination disks (Fig. 2), this seems to
suggest a large region of ongoing obscured-mode star formation. However,
higher-resolution dust continuum imaging is required to comment on this more
robustly, as Arcs 1a and 3c are only marginally resolved at 2 mm.

The UV-NIR photometry captures primarily the unobscured mode of star
formation, which is likely to be a small fraction of the total (obscured plus
unobscured) SFRUV+IR for such massive, highly star-forming galaxies (Whitaker
et al. 2017). Regardless, it is still informative to investigate the spatial
scales over which unobscured star formation is occurring, as an insight into
the driving mechanisms and present evolutionary stages of the galaxies. For
example, galaxies with a higher central specific SFR (sSFR$\equiv{\rm
SFR}/M_{\star}$) are likely undergoing inside-out growth, by which a
spheroidal bulge component is built up in situ through gas compaction (e.g.,
Dekel & Burkert 2014; Zolotov et al. 2015; Tacchella et al. 2016a, 2018). In
later stages, sSFR may be elevated in outer regions of the galaxy, pointing to
inside-out quenching. This may be driven by morphological quenching (Martig et
al. 2009; Genzel et al. 2014; Tacchella et al. 2015), where the central
spheroidal component renders the disk stable against gravitational
fragmentation of the molecular gas, even without requiring energetic feedback
from a central starburst or active galactic nucleus (AGN).

Figs. 4 and 5 show the spatially-resolved bin-wise properties derived through
SED fitting with piXedfit and bagpipes, including corrections for
magnification from our lens model. Given the variation in bin size, we
primarily examine the surface densities of SFR and $M_{\star}$, or
$\Sigma_{\rm SFR}$ and $\Sigma_{\star}$. As gravitational lensing conserves
surface brightness, these properties do not need to be corrected for
magnification. We also perform the same analysis for the high-magnification
Arcs 1bc and 3ab as a consistency check in Appendix D. These arcs arise only
from a portion of the galaxy that crosses inside the caustics (see Fig. 2 and
the red demarcation line in left panels of Figs. 6 and 7). For that reason, we
focus here primarily on Arcs 1a and 3c.

For G165-DSFG-1 (Arc 1a), there appears to be a single central peak in
$\Sigma_{\rm SFR}\approx 15~{}M_{\odot}~{}{\rm yr}^{-1}~{}{\rm kpc}^{-2}$, and
a roughly axially symmetric radial decrease (except for perhaps lower
$\Sigma_{\rm SFR}$ in the northwest region). The stellar mass surface density
$\Sigma_{\star}$ also peaks in the center of the
arc—log$[\Sigma_{\star}/(M_{\odot}~{}{\rm kpc}^{-2})]\approx 9.5$—although the
peak is more consistent with an elongated ellipse parallel to the major axis
of the arc. In fact, there appear to be two peaks in $\Sigma_{\star}$, with a
slight dip between them corresponding to the peak in $\Sigma_{\rm SFR}$. Of
possible relevance is the discovery by F24 that Arc 1a is actually composed of
components at slightly different redshifts seen in projection, as indicated in
the source-plane reconstruction in Fig. 2. The redder, southeasternmost
portion (NIRSpec aperture NS_969) is found to be at $z=2.2355\pm 0.0003$,
while the bluer region $0\farcs 2$ (1.6 kpc in projection) more to the
northwest (NS_46) is at $z=2.2401\pm 0.0002$ (i.e., a rest-frame redshift
difference of $\approx$ 400 km s-1 relative to the southeast clump). Despite
this, we still refer to the system as G165-DSFG-1, as it is not possible to
segregate them with the current data, and as it seems likely that they will
ultimately merge into one object.

There is uncertainty in the spectroscopic redshift of G165-DSFG-3, although it
shares a similar color gradient as DSFG-1 (Fig. 2), and an even clumpier
distribution of $\Sigma_{\star}$ (peaking at $\Sigma_{\star}\approx
10^{9.5}~{}M_{\odot}~{}{\rm kpc}^{-2}$), with as many as five distinct peaks
along the major axis. This may be suggestive even of spiral arm structures
within the plane of a disk, a history of recent minor mergers, or the result
of gravitational fragmentation of molecular clouds throughout the galaxy.
Without kinematic information, our interpretation is necessarily limited. The
distribution of $\Sigma_{\rm SFR}$ (peaking at $\approx 8~{}M_{\odot}~{}{\rm
yr}^{-1}~{}{\rm kpc}^{-2}$) is elongated along the major axis and decreasing
radially, not dissimilar to G165-DSFG-1.

Combining the SFR and $M_{\star}$ information yields a map of sSFR (or
alternatively the mass doubling time, sSFR-1) on sub-kpc scales. This provides
a direct measure of how and where the galaxies are actively growing (in tandem
with the distribution of mass-weighted ages, also shown in Figs. 4 and 5). In
§4.3, we discuss the relation of these sSFR values to the resolved star-
forming main sequence, but first we examine the relative values of sSFR as a
function of projected galaxy radius. For G165-DSFG-1, there is a clear
bimodality in sSFR, with more starbursting regions in the southeast vs. the
northwest (by nearly 1 dex). This may also support the picture of DSFG-1
comprising multiple orbiting clumps seen in projection, with one both forming
stars more actively and containing greater dust content, based on the $A_{V}$
map. The map of mass-weighted stellar ages also supports this picture, with
the southeast component more or less entirely younger than $\sim 100$ Myr.
Alternatively, if the orbiting clumps are not responsible for this effect,
then there must be some other mechanism by which gas is driven preferentially
to one side of the galaxy to form stars. While it remains to be confirmed that
DSFG-3 lies at the same redshift, one might speculate that the asymmetry is
driven by a recent gravitational interaction between the two DSFGs, with
molecular gas driven towards the center of mass of the system to form stars.

If a similar morphology were seen in DSFG-3, this might support an
interpretation of tidal interaction between DSFG-1 and DSFG-3. There is no
such obvious asymmetry in DSFG-3, however. In fact, the scenario for DSFG-3 is
rather different: here there is an extensive central starbursting region with
sSFR$>10~{}{\rm Gyr}^{-1}$, with patchy regions of more mild star formation in
the outskirts of the disk. However, the very central region (where
$\Sigma_{\star}$ peaks) has a slight depression in sSFR, which we speculate
might reveal the very early stages of inside-out quenching. In a similar
resolved SED-fitting analysis, Giménez-Arteaga et al. (2023) suggested that
dramatic spatial variations in sSFR could signal the important role of bursty
star formation on small (kpc-sized) sub-galactic scales. This stochastic star
formation at fine spatial and temporal resolutions also makes it difficult to
determine if lower sSFR is due to quenching or merely a period of inactivity
between bursts.

To condense the overwhelming amount of information contained in Figs. 4 and 5,
we also extract profiles of $\Sigma_{\rm SFR}$, $\Sigma_{\star}$, sSFR, and
$A_{V}$ by taking linear profiles along the major axes of Arcs 1a and 3c,
which are shown in Figs. 6 and 7. This approach is largely independent of
inclination (Devour & Bell 2017, 2019). In these plots, we take the piXedfit
bins from Figs. 4 and 5 and compile these into binned slices of various
properties along the major axis. We do this in two ways: for the “weighted
median" curves, we do not account for overcounting pixels. That is, the
piXedfit-binned properties consist of regions of multiple pixels that are all
assigned the same sSFR, $A_{V}$, etc. By not removing duplicate pixels, the
result is effectively weighted by the area of each piXedfit bin contained
within the major axis slices. For the “median" curve, these duplicates are
removed, which gives all piXedfit bins within the major axis slice equal
weight. The inner 68% confidence intervals of pixel values are also shown as
shaded curves.

Here, the skew in sSFR and $A_{V}$ is even more readily observable for DSFG-1,
with both transitioning between higher and lower values at a similar offset of
$\approx+0\farcs 5$. It is also clear that the $\Sigma_{\rm SFR}$ peak is
skewed relative to the broad $\Sigma_{\star}$ plateau131313The eastern NIRSpec
slit from F24 was consistent with a redshift of $z=2.2355\pm 0.0003$, while
the western slit yields $z=2.2401\pm 0.0002$. This redshift difference is
insufficient to affect the derived properties, but it is worth recalling that
both components are contained within the offset range of $-1\farcs 0$ to
$0\farcs 0$, and it is not clear yet how to visually separate them in these
images.. For DSFG-3, the sSFR and $A_{V}$ profiles are less pronounced, but
are ultimately greater in the center than in the outskirts. They appear
largely symmetric, thus more consistent with a simpler picture of a more
centrally-concentrated dusty starburst.

Figure 6:  Linear profiles (right) of $\Sigma_{\rm SFR}$,
log$(\Sigma_{\star})$, log(sSFR), $A_{V}$, and magnification $\mu$ created by
taking bins along the major axis, shown over the RGB image-plane image of Arc
1a (left). Dashed white lines perpendicular to the major axis correspond to
those in the right panel (for the center and at an offset of $\pm 1\arcsec$,
where positive is northwest). The relative source-plane offsets in physical
units (kpc) corresponding to the image-plane offsets of $\pm 1\arcsec$ and
$\pm 2\arcsec$ are shown at the top, which are asymmetric due to the gradient
in magnification. The F444W PSF is shown as a red beam in the left panel. The
gray contour in the left panel shows the border of the bins used to make the
profiles (as seen also in Fig. 4). In each bin along the major axis, we show
the weighted median (dashed curve) for each property, which does not account
for overcounting pixels within each bin, so they are effectively weighted by
the area of each SED fit bin within the major axis bin. The median (solid
curve) counts each SED bin only once. The inner 68% interval is shown as
shaded curves. A red line denotes where the source crosses the caustic curve
according to our lens model; in this case, regions south of the line is
triply-imaged, and north are singly-imaged. The locations of the NIRSpec MSA
slits from F24 are shown in cyan and labeled for reference. As a caveat,
properties at an offset $>+1\farcs 0$ are less reliable, given possible
blending with a nearby object of unknown redshift.

Figure 7:  As with Fig. 6, but for Arc 3c. In this case, the portion of the
object northwest of the red line is triply-imaged, and the rest is singly-
imaged.

The ionization parameter is found to generally peak at ${\rm
log}(\mathcal{U})\approx-2.3$, consistent with the inner 500 pc core of the
starburst M82 (Förster Schreiber et al. 2001; see also Engelbracht et al.
1998; Thornley et al. 2000). This is also slightly below the upper limit of
${\rm log}(\mathcal{U})=-2$ used for our SED fitting. For DSFG-1, the inner
68% confidence interval ranges from ${\rm log}(\mathcal{U})=-3.6$ to $-2.7$,
while for DSFG-3 the range is ${\rm log}(\mathcal{U})=-3.3$ to $-2.3$ (median
uncertainty $0.3-0.4$ dex), but these are likely poor estimates from SED
fitting alone without spectroscopy. This is likewise the case for the
estimation of stellar metallicity. While robust super-solar gas-phase
metallicity has been found in the nucleus of DSFGs even at $z\sim 4$ (e.g.,
Birkin et al. 2023; Peng et al. 2023), we do not place much confidence in the
peak of $Z/Z_{\odot}\approx 3$ for G165-DSFG-1 and G165-DSFG-3 (apart from
perhaps a believable radial decrease in both cases). However, we also find
similar results of $Z/Z_{\odot}=3.1\pm 0.3$ for DSFG-1 and $Z/Z_{\odot}=3.2\pm
0.3$ from the global galaxy-wide SED fits described in §4.6, but we consider
the unaccounted-for systematic uncertainties too great to further interpret
these results.

###  4.2 A favorable dust geometry for detection of supernovae?

We also remark that the widespread star formation and starbursting
regions—along with some regions of decreased dust attenuation—have
implications for the observable rate of supernovae, which we discuss in detail
in §5.3. If the formation of massive stars (i.e. progenitors of core-collapse
supernovae, or CCSNe) is not limited just to highly dust-obscured nuclear
starbursts (Mattila & Meikle 2001), but is apportioned more evenly throughout
the disk, then the turbulence in the ISM may result in a patchy, porous, and
generally complex star vs. dust geometry that would facilitate easier
detection of SNe than a scenario of a uniform dust screen (Casey et al. 2014b;
Narayanan et al. 2018). Likewise, star formation in the outskirts of the
galaxy might also result in a “frosting" of unobscured UV-bright massive stars
(e.g., Safarzadeh et al. 2017). This has not yet been tested in practice, but
such DSFGs with evidence for galaxy-wide star formation events and
inhomogeneous distributions of dust may thus be ideal candidates for continued
monitoring for supernovae.

Figure 8:  Cumulative distribution function of SFRUV-NIR below a given
threshold of dust attenuation, $A_{V}<a_{V}$, as determined through piXedfit
binning. For both the DSFGs and Arc 2a, 50% of the star formation is confined
to regions with $A_{V}<2$.

To get a sense of what supernova rates would be observed given different
observing depths, we sum the resolved SFRs for only bins with attenuations
below a certain value, shown in Fig. 8. For example, all bins of DSFG-1 are
consistent with $A_{V}<3$, so the summed (magnification-uncorrected) $\mu{\rm
SFR}_{A_{V}<3}\approx 380~{}M_{\odot}~{}{\rm yr}^{-1}$ (see Table 3), while
below $A_{V}<2$ yields $\mu{\rm SFR}_{A_{V}<2}\approx 130~{}M_{\odot}~{}{\rm
yr}^{-1}$, or 34% of the total. Likewise, for DSFG-3, the total SFR satisfies
$\mu{\rm SFR}_{A_{V}<3}\approx 340~{}M_{\odot}~{}{\rm yr}^{-1}$ (Table 3),
while $\mu{\rm SFR}_{A_{V}<2}\approx 190~{}M_{\odot}~{}{\rm yr}^{-1}$ (56% of
total). Essentially, this implies that a shallower point-source sensitivity
accounting for only up to 2 mag dust attenuation for supernovae would still
detect them at a rate of $30-60$% the intrinsic rate. This contrasts with the
estimated $\lesssim 20$% recovered rate of supernovae at optical wavelengths
for some local ULIRGs (e.g., Arp 220 and Arp 299; Leaman et al. 2011; Mattila
et al. 2012; also Richmond et al. 1998). Mattila et al. (2012) and Dahlen et
al. (2012) recognized that the increasing fraction of the cosmic star
formation rate occurring within luminous infrared galaxies (LIRGs;
$10^{11}<L_{\rm IR}/L_{\odot}<10^{12}$) and ULIRGs (Magnelli et al. 2011)
might result in dire, prohibitively large fractions of missed CCSNe at $z>1$.
However, the concrete differences in the distribution and driving mechanisms
of star formation in distant vs. local ULIRGs that have become apparent in
recent years may work immensely in our favor, as we contend here.

Moreover, a more dispersed distribution of CCSNe is favorable for lensed
objects like G165-DSFG-1 and G165-DSFG-3, as it improves the odds that any
given supernova will occur within a multiply-imaged region of the galaxy. If
all SNe were confined to the galaxy nucleus, this would effectively require
that this region falls by chance within the caustics in order to ever observe
a multiply-imaged supernova.

Figure 9:  Individual piXedfit bins of G165 DSFGs (Arc 1, left panel; Arc 3,
right panel) placed on the resolved star formation main sequence, or
$\Sigma_{\rm SFR}$ vs. $\Sigma_{\star}$. The color scale indicates sSFR.
Median uncertainties are indicated in the center left. These properties are
not corrected for lensing, as they are related to surface brightness and thus
independent of magnification. However, we filter bins to only include those
with areas equivalent to $0.15-1.5$ kpc in radius, where we assume a fiducial
magnification factor of $\mu=6$ for Arc 1 and $\mu=7$ for Arc 3 (i.e., image-
plane areas are divided by this factor before filtering). A number of
measurements of the rSFMS in the local Universe (Cano-Díaz et al. 2016;
Abdurro’uf & Akiyama 2017; Ellison et al. 2023), at low-$z$ (Jafariyazani et
al. 2019; Yao et al. 2022), and at $z\sim 1$ (Wuyts et al. 2013; Magdis et al.
2016; A18) are shown as dashed, dash-dotted, and solid lines, respectively.
Legends in the two panels indicate the redshift ranges included for each. To
make an approximate comparison at $z\sim 2$, we extrapolate the redshift
evolution model of $\Sigma_{\rm SFR}(r)$ and $\Sigma_{\star}(r)$ from A18
backwards to $z=2.24$, which is shown as black stars. The vast majority of
pixels in each galaxy appear to be on or above the rSFMS, as 99% (89%) of the
bins in Arc 1 (Arc 3) lie above this backward-evolved curve. In terms of areal
fractions, these translate to $\sim 100\%$ (Arc 1) and $\sim 85\%$ (Arc 3).

###  4.3 Resolved star-forming main sequence in DSFG-1 and DSFG-3

Remarkably, most star-forming galaxies (SFGs) fall on a tight, nearly linear
scaling relation (that evolves with redshift) between SFR and $M_{\star}$,
known commonly as the star-forming main sequence, as has been borne out
extensively (including Brinchmann et al. 2004; Daddi et al. 2007; Elbaz et al.
2007; Noeske et al. 2007a, b; Salim et al. 2007; Whitaker et al. 2012; Speagle
et al. 2014; Renzini & Peng 2015). While the meaning behind this result is
still somewhat subject to debate, it is commonly interpreted as an indication
that most SFGs grow steadily when considered over $>100$ Myr timescales
(although individual objects are likely to oscillate around the main sequence;
e.g., Tacchella et al. 2016b). What remains highly uncertain is if and how
this relation holds for $\sim$kpc-scale regions within galaxies. In parallel
with the integrated properties of SFR and $M_{\star}$, their respective
surface densities $\Sigma_{\rm SFR}$ and $\Sigma_{\star}$ are correlated
through what is referred to as the spatially resolved star formation main
sequence (rSFMS), first explored by Sánchez et al. (2013) and Cano-Díaz et al.
(2016) locally and at higher redshift by Wuyts et al. (2013). A number of
works since have supported the existence of the correlation, suggesting that
it is physically motivated at sub-galactic scales (Hemmati et al. 2014;
González Delgado et al. 2016; Magdis et al. 2016; Abdurro’uf & Akiyama 2017;
Hsieh et al. 2017; Maragkoudakis et al. 2017; Abdurro’uf & Akiyama 2018;
Ellison et al. 2018; Hall et al. 2018; Liu et al. 2018; Medling et al. 2018;
Cano-Díaz et al. 2019; Vulcani et al. 2019; Bluck et al. 2020a, b; Enia et al.
2020; Morselli et al. 2020; Pessa et al. 2021, 2022; Ellison et al. 2023).

In particular, Erroz-Ferrer et al. (2019) demonstrated that the relation
between $\Sigma_{\rm SFR}$ and $\Sigma_{\star}$ may be more fundamental; the
global SFR$-M_{\star}$ relation could be in part a consequence of the
generally small variation for $\Sigma_{\rm SFR}$ and $\Sigma_{\star}$
throughout a galaxy, combined with the size-mass relation (Shen et al. 2003;
van der Wel et al. 2014) that scales both up and yields a linear relation
between the integrated properties SFR and $M_{\star}$ (Vulcani et al. 2019).
The rSFMS has also been observed through simulations (e.g. Trayford & Schaye
2019; Hani et al. 2020; McDonough et al. 2023). However, we caution that the
relation is likely non-universal, with non-trivial variations in the rSFMS
from galaxy to galaxy (e.g., Vulcani et al. 2019; Ellison et al. 2021), and
the slope of the sequence may be influenced by the minimum size scales that
are probed (e.g., Hani et al. 2020).

Fig. 9 compares the distribution of individual piXedfit bins in Arcs 1 and 3
in the $\Sigma_{\rm SFR}-\Sigma_{\star}$ plane. To facilitate comparison with
work at lower-$z$, we include only the bins with an area approximately
equivalent to a radius of $150-1500$ pc. Additionally, we expect that SED
fitting at $\lesssim 100$ pc physical scales (comparable to individual giant
molecular clouds) becomes much less physically meaningful. As Vulcani et al.
(2019) point out, it is highly possible for star clusters to migrate somewhat
from the natal molecular cloud where they formed, which perturbs the one-to-
one relationship between $\Sigma_{\rm SFR}$ and $\Sigma_{\rm gas}$ (and by
extension, between $\Sigma_{\rm SFR}$ and $\Sigma_{\star}$). Since our method
is applied to the magnified image-plane arcs, we first divide the bin areas by
a fiducial magnification ($\mu=6$ for Arcs 1 and 3, and $\mu=8$ for Arc 2)
before filtering to include only intrinsic sizes from $\approx 150-1500$ pc.

So far, there are rather few studies that have examined the rSFMS beyond the
local Universe. Jafariyazani et al. (2019) used integral-field spectroscopy to
place a sample covering $0.1<z<0.4$ on the rSFMS, while Yao et al. (2022)
likewise covered $0.2<z<0.4$. Further afield, our comparison is primarily
limited to Wuyts et al. (2013) at $0.7<z<1.5$, Magdis et al. (2016) at
$0.8<z<1.0$, A18 at $0.8<z<1.8$, and Nelson et al. (2021) at $0.8<z<1.5$. Fig.
9 shows the best-fit rSFMS relations from these works (over approximately the
pertinent range in $\Sigma_{\star}$ that each study dealt with), in addition
to local relations from Cano-Díaz et al. (2016), Abdurro’uf & Akiyama (2017),
and Ellison et al. (2023). Abdurro’uf & Akiyama (2017) and A18 first examined
how the radial profiles of $\Sigma_{\rm SFR}(r)$, $\Sigma_{\star}(r)$, and
${\rm sSFR}(r)$ generally evolve with redshift, and by extension, how the
rSFMS evolves. We can use this to predict where the pixels of the G165 DSFGs
lie relative to the rSFMS—with the caveat that the rSFMS is not universal from
one galaxy to another.

For now, we employ the redshift evolutionary model of A18, which tracked
progenitors from $z\approx 1.06$ ($t(z)\approx 5.5$ Gyr in our assumed
cosmology) to $z\approx 0.02$ ($t(z)\approx 13.2$ Gyr), and extend it
backwards in time to $z=2.24$ ($t(z)\approx 2.9$ Gyr). The result is shown as
a dotted curve of black stars in Fig. 9 (and see also Fig. 15 of A18 for $z=0$
to 1.1 in steps of $\Delta z=0.11$). Relative to this curve, we find that the
overwhelming majority ($\approx 99\%$) of the bins in Arc 1 lie above the
model rSFMS at $z\approx 2.2$ (accounting for nearly $\approx 100\%$ of the
total area), whereas $\approx 89\%$ of the bins in Arc 3 lie above (here
accounting for $85\%$ of the area)141414While the main sequence has large
intrinsic scatter, this binary calculation of the areal fraction “above” vs.
“below” the main sequence treats it as an exact discrete curve, in part to
draw clear contrast with Arc 2 in §4.4. In this instance, “above the main
sequence” does not equate to “starbursting.”. This suggests that a clear
majority of the bins shown in Figs. 4 and 5 are actively star-forming (and
some of which are starbursting). For these objects, at least, it appears to be
the case that star formation is not confined purely to a central starburst
region, but is widely distributed (often in clumps) throughout the plane of
the disk. This has been demonstrated to be the case in recent years for a
number of high-$z$ DSFGs, especially in contrast with local ULIRGs (e.g.,
Chapman et al. 2004; Carilli et al. 2010; Ivison et al. 2010, 2011; Rujopakarn
et al. 2011; Hodge et al. 2012; Hatsukade et al. 2015; Miettinen et al. 2015;
Simpson et al. 2015; Swinbank et al. 2015; Hodge et al. 2016; Sharda et al.
2018; Tadaki et al. 2018; Hodge et al. 2019; Kamieneski et al. 2024). Here, we
have demonstrated that the unobscured mode of star formation inferred from the
UV-NIR SED is similarly widespread for DSFG-1 and DSFG-3, matching the
widespread obscured star formation suggested by our finding of $R_{e}\sim 3$
kpc for the dust continuum. This is a result that we revisit in §4.5, using
$U-V$ and $V-J$ colors alone for a largely independent and non-parametric
characterization.

The evolutionary model we employ for the rSFMS was only designed for $z\sim 1$
to $z=0$, and it is not clear how the evolution might extend to our objects at
$z\approx 2.2$. However, the $z\sim 1$ sample from A18 includes objects up to
$z\approx 1.8$, only $500-700$ Myr after the G165 DSFGs. Their sample of high-
mass objects ($M_{\star}>10^{10.5}~{}M_{\odot}$, SFR $\lesssim
300~{}M_{\odot}~{}{\rm yr}^{-1}$) is also selected to focus on face-on (low-
ellipticity) disk-like objects, so it is subject to different systematic
effects than our work. Nevertheless, properly quantifying the spatial extent
of starbursting regions in G165 and other $z\sim 2$ galaxies will not be
possible until a more detailed study of the rSFMS at higher redshifts has been
undertaken. With JWST covering an ever-increasing number of strongly lensed
objects, this is becoming more feasible, but beyond the scope of our current
study.

####  4.3.1 A mode of elevated star formation efficiency in the resolved star-
forming main sequence

We also observe for the first time what appears to be a secondary “starburst"
sequence $\sim 1$ dex above the “star-forming" main sequence (Fig. 9). This
elevated mode of star formation is not dissimilar to the higher star formation
efficiency (SFE) mode of the Schmidt-Kennicutt relation between molecular
hydrogen mass $M_{H_{2}}$ and SFR on resolved scales (Sharon et al. 2019).
Recent works (e.g., Baker et al. 2022b) have demonstrated that the
(integrated) SFMS is not a fundamental correlation, but rather the result of
two more fundamental scaling relations with less scatter: the Schmidt-
Kennicutt relation, and the Molecular Gas Main Sequence (MGMS) between
$M_{H_{2}}$ and $M_{\star}$. Lin et al. (2019) found likewise (and since
supported by Ellison et al. 2021 and Baker et al. 2022a) that the correlation
between $\Sigma_{\rm SFR}$ and $\Sigma_{\star}$ is merely a byproduct of the
correlations between $\Sigma_{\rm SFR}$ and $\Sigma_{H_{2}}$ (which is the
strongest correlation) and between $\Sigma_{H_{2}}$ and $\Sigma_{\star}$.
Together, this suggests a picture where the rate of star formation is
generally driven (or regulated) by the availability of molecular gas fuel
(although this is not necessarily a universal slope, e.g., Daddi et al. 2010),
or neutral Hi in the case of low-density environments at the outskirts of
galaxies (Leroy et al. 2008). The correlation between $\Sigma_{H_{2}}$ vs.
$\Sigma_{\star}$ is suggestive of gas being preferentially accreted in regions
of deeper gravitational potential, as traced by the distribution of stellar
mass (Lin et al. 2019).

The bimodalities in the rSFMS and the Schmidt-Kennicutt relation may in fact
be directly related if, for example, the MGMS that is hypothesized to link the
two is not itself bimodal. In other words, regions of elevated SFE could also
be detected merely through $\Sigma_{\rm SFR}$ vs. $\Sigma_{\star}$, under the
assumption that each galaxy satisfies a unimodal
$\Sigma_{H_{2}}$-$\Sigma_{\star}$ relation (without large scatter). Without
information on the molecular gas (or dust) distribution of the G165 DSFGs on
similar spatial scales, or the gas excitation conditions through multiple
spectral lines to accurately infer gas mass (e.g., Harrington et al. 2021), we
are missing part of the picture in characterizing what drives stellar mass
assembly across the galaxy. However, the rSFMS is still helpful in
characterizing the widespread distribution of star formation and the rapid
consumption of molecular gas.

Figure 10:  RGB source-plane reconstruction of NIRCam imaging of Arc 2b at
$z=1.78$, as with Fig. 3. Caustics are shown as sparse white dotted curves,
and F090W/F444W source-plane PSFs are shown in the lower right as teal/orange
beams, respectively. Only Arc 2b is shown in this reconstruction, as small but
visually detectable offsets ($\sim 0\farcs 05$) in the source-plane location
of Arcs 2a, 2b, and 2c are introduced from the lens model.  Figure 11:  As
with Fig. 4, but for Arc 2a, which hosts SN H0pe, with a fixed redshift of
$z=1.78$. A small black circle indicates the location of the supernova. We
choose this image for SED fitting rather than 2b or 2c as it is the one in
which the SN is faintest (and thus contaminates the least).

###  4.4 Resolved properties of the massive host galaxy of SN H0pe

As discussed by Polletta et al. (2023) and F24, the rest-frame optical
spectrum of Arc system 2abc, host to SN H0pe, shows evidence for a mostly
intermediate-age stellar population at $\sim 2$ Gyr (e.g., a strong NaD line
and larger D(4000) value). The discovery of a Type Ia supernova itself helps
support this, as these require delay times for white dwarf progenitors ($\sim
3-8~{}M_{\odot}$) to evolve off of the main sequence and reach the end of
their lifetimes. This delay time distribution appears to peak at $\sim 1-3$
Gyr (Maoz & Mannucci 2012, and references therein). While the galaxy has
evidence of some ongoing star formation, $10^{10.9}~{}M_{\odot}$ of stellar
mass has already been assembled by the time it is observed at $z=1.78$ (F24).
We fit its UV-NIR SED with bagpipes in Appendix E, and remark on possible
degeneracies with this narrow wavelength range which affect our
interpretations..

Fig. 11 shows the results of spatially-resolved SED fitting for Arc 2a, which
is host to SN H0pe. As this image is understood to arrive first (followed by
2c then 2b), the supernova is faintest in this image, having already faded
substantially in its light curve. We thus choose this arc for SED fitting, as
it is subject to the least contamination from the SN. Moreover, it is also
less contaminated by intracluster light than image 2b (which coincides with
the axis of the merging cluster), and more isolated than 2c (which is nearby
another very red arc, of currently-unknown redshift). While the properties of
this massive, mildly star-forming galaxy are naturally of interest in relation
to SN H0pe, it also serves as a useful control for comparison with the more
extreme DSFGs in this work.

Fig. 12 shows the relation of Arc 2a to the rSFMS, which shows clear contrasts
with Fig. 9. While 99% and 89% of bins in Arcs 1a and 3c (respectively) lie
above the center of the rSFMS evolved to their respective redshifts (or areal
fractions of $\sim$100% and 85%), merely 44% of bins in Arc 2a are above the
rSFMS at $z=1.78$ (or $\approx 57\%$ in terms of galaxy area). This is
strongly influenced by the presence of a low-$\Sigma_{\rm SFR}$ (and low-sSFR)
sequence at the center of the galaxy—with high stellar mass surface density,
log$(\Sigma_{\star}[M_{\odot}~{}{\rm yr}^{-1}])>9$. This locus is entirely
absent for the DSFGs, although the sSFRs are still above the nominal “green
valley" transition at log(sSFR)$<-10.8$ yr-1 (Salim 2014). It also appears to
be the case that many bins are clustered around the main sequence, adding some
further confidence in the validity of backward-evolving the rSFMS model from
A18, although it is true that this arc actually falls within the redshift
range used for the model. There are a select number of regions that appear to
be starbursting, perhaps in line with the conclusion by F24 that the galaxy is
still star-forming (despite its very large stellar mass), as evidenced by the
detection of H$\alpha$ emission (alongside spectral features that confirm the
presence of a much older stellar population).

Figure 12:  The $\Sigma_{\rm SFR}-\Sigma_{\star}$ plane for Arc 2, as with
Fig. 9. A fiducial magnification of $\mu=8$ is chosen to filter intrinsic bin
sizes. Note that, given the larger stellar mass for this object, we shift the
horizontal axis by 1 dex relative to Fig. 9. Here, for a fiducial
magnification of $\mu=8$ and filtering to only include bins with intrinsic
equivalent radii of $0.15-1.5$ kpc, we find that $\approx 44\%$ of bins lie
above the redshift-evolved rSFMS to $z=1.78$ (accounting for $\approx 57\%$ of
the galaxy area).

The large stellar mass and non-negligible ongoing star formation may in part
help explain the favorable conditions leading to the discovery of SN H0pe, as
the SN Ia rate is hypothesized to depend on an instantaneous term proportional
to the SFR and on an extended term proportional to the stellar mass (e.g.,
Scannapieco & Bildsten 2005; Smith et al. 2012). Similarly, the Type Ia SN
Requiem (Rodney et al. 2021) was discovered in a low-sSFR host, classified as
a Massive/Magnified Red Galaxy (MRG; Newman et al. 2018; Akhshik et al. 2020).
Since then, an additional SN Ia (“Encore") has recently been found in the same
host galaxy (Pierel et al. 2024a). In §5.3, we discuss the prospects for
discovering core-collapse SNe behind G165 given the high rate of star
formation. However, given the large stellar mass of the host of SN H0pe
(Polletta et al. 2023; Frye et al. 2024), we also consider that the G165 field
may be a source for the discovery of additional Type Ia SNe, as well (e.g.,
Shu et al. 2018; Holwerda et al. 2021).

Figure 13:  Spatially-resolved image-plane $UVJ$ diagrams for DSFG-1 (Arc 1a),
DSFG-3 (Arc 3c), and Arc 2a (left panels), and bin maps of $UVJ$
classifications (right panels); these are Quiescent (Q), Star-forming (SF),
and Dusty Star-forming (DSF). The fields of view match those in Figs. 4, 5,
and 11. The selection of quiescent galaxies is from Muzzin et al. (2013),
while the distinction of SF vs. DSF is from Miller et al. (2022). Colors of
points in the $UVJ$ diagrams show sSFRs derived from the full observed-frame
near-IR SED fit. A transparent gray square shows the location of the global
$UVJ$ colors for each arc. The black vector in the bottom panel indicates the
effect of increasing $A_{V}$ by 1 mag, per the Calzetti et al. (2000)
attenuation curve.

###  4.5 Resolved $UVJ$ diagrams

The plane of $U-V$ and $V-J$ colors (or the well-known $UVJ$ diagram) is an
efficient method to discern between stellar populations that are red due to
age vs. those that are reddened by dust (e.g., Labbé et al. 2005; Wuyts et al.
2007; Williams et al. 2009; Brammer et al. 2011; Whitaker et al. 2011). For
objects at $z\approx 2$, it was not practically feasible before JWST to reach
the same resolution for rest-frame $J$-band (near-IR) imaging as for rest-
frame (optical) $U$ and $V$. With the advent of NIRCam, an essentially direct
conversion from F115W, F200W, and F356W photometry to rest-frame $U$, $V$, and
$J$ (respectively) at $z\approx 2$ is now possible. For the range of
$1.7<z<2.3$, Miller et al. (2022) calibrated a linear relation for rest-frame
$U-V$ and $V-J$ colors as a function of redshift and photometry in the NIRCam
filters. While the classification of stellar populations with the $UVJ$
diagram is imperfect, and made more uncertain by the assumptions of the
conversion to rest-frame colors, it benefits from a reliance on very few
pieces of information. No SED templates are imposed directly, and only 3
photometric measurements are employed (with which a robust SED fit would
hardly be practical). Fortunately, the objects of interest in this work are
all understood to lie within the redshift range for which the Miller et al.
(2022) calibration is valid, so we use this semi-independent method alongside
the measurement of sSFR from the SED fits (§4.1 and §4.4).

The left panels of Fig. 13 plot the rest-frame $U-V$ and $V-J$ colors of the
bins in the DSFG-1 and DSFG-3 alongside arc 2a, all still shown in the image
plane. As differential magnification is negligible within the small bin size
(i.e., $U$, $V$, and $J$ are all magnified equally in each bin), there is no
need to correct the $UVJ$ colors. The marker colors indicate sSFR as derived
from the SED. Before introducing the selection criteria, we first remark on
the approximate similarity in the locus for Arcs 1a and 3c. Arc 2a occupies a
clearly different region, in particular towards redder $U-V$ colors. These
reddest $U-V$ bins include substantially lower sSFRs, about 3 orders of
magnitude lower than the lowest-sSFR bins in DSFG-1 and DSFG-3. Referring to
Fig. 11, we find unsurprisingly that these are concentrated at the nucleus
(central kpc) of the galaxy. Given the high stellar mass density here,
$\Sigma_{\star}>10^{10.5}~{}M_{\odot}~{}{\rm kpc}^{-2}$, it seems likely that
this galaxy has an early formation redshift (Estrada-Carpenter et al. 2020)
and has already built up a dense stellar bulge and started quenching inside-
out (e.g., Fang et al. 2013; Ji & Giavalisco 2022a, b; Ji et al. 2023). This
is consistent with (but not necessarily implicitly due to) the gravitational
stabilization of gas by an increasing dominance of a spheroidal bulge
component, known as morphological quenching (Martig et al. 2009).

We use the higher-redshift selection box for quiescent galaxies defined by
Muzzin et al. (2013). While the distinction between star-forming vs. dusty
star-forming populations is more arbitrary and less pronounced, we also show
the $V-J>1.25$ cut from Miller et al. (2022). Differential magnification may
indeed impact the global colors for each of the three arcs (shown as gray
squares in Fig. 13; see discussion of global properties in §4.6). We find that
DSFG-1 and DSFG-3 lie in very similar locations, safely in the star-
forming/dusty star-forming region, but essentially close to the dividing line
between the two. Arc 2a, on the other hand, lies on the cusp of the division
between quiescent and dusty star-forming. The right-column panels of Fig. 13
show the classification of bins from the $UVJ$ diagram as quiescent (Q), star-
forming (SF), and dusty star-forming (DSF). Curiously, while there is a
greater presence of quiescent regions in the massive galaxy seen in Arc 2a
(relative to the DSFG arcs 1a and 3c), a clear majority of the galaxy’s area
consists of stellar populations consistent with the dusty star-forming region
of the $UVJ$ diagram, with only its outskirts being classified as less-
obscured star-forming. Extending the quiescent selection box slightly to
redder $V-J$ colors would change the classification of a large number of bins
from dusty star-forming to quiescent. Yet, the overall classification of this
object as quiescent would likely be erroneous, despite its low sSFR ($\sim
0.08~{}{\rm Gyr}^{-1}$; F24). Less-obscured star formation is found in the
outskirts of the galaxy, and the galaxy center (with peak $\Sigma_{\star}$) is
coincident with the quiescent bins, perhaps signifying inside-out quenching.
In §4.4 and Fig. 12, we found that 57% of the galaxy’s area was consistent
with being above the center of the rSFMS evolved to $z=1.78$. In other words,
$<43\%$ was quiescent based on sSFR, or at least on the lower end of the
resolved main sequence; now, using the independent $UVJ$ selection, we find
that $\sim 20\%$ of the area is classified as quiescent.

For the DSFGs, Arcs 1a and 3c, there are remarkably few regions that fall into
the quiescent selection. For Arc 1a, there is a clear separation with the
southeast portion dominated by dusty star-forming populations and a northwest
portion of regular star-forming populations. This could further support the
orbiting clumps hypothesis (§4.1), with both components actively forming
stars, but one hosting greater dust content (or at least, somehow more
obscured due to projection effects, although this region is actually at a
lower redshift). As for Arc 3c, the inner regions of the galaxy are DSF with
outskirts that are classified as SF, with isolated embedded quiescent regions.
These seem consistent with the observation of multiple peaks in
$\Sigma_{\star}$, perhaps as a result of a late-stage merger or the large-
scale fragmentation of gas within the disk. In both DSFG-1 and DSFG-3, the
most active star formation (highest sSFR) is clearly concentrated in the DSF
box, which is to be expected given the connection between starbursts and dust
obscuration (e.g., Whitaker et al. 2017).

Figure 14: Best-fit near-IR SED and properties from bagpipes for Arc 1a (top)
and Arc 3c (bottom). The top subpanels show the measured photometry as blue
data points, with the best-fit SED as an orange curve. The expected photometry
from the SED is shown as orange points. The bottom subpanels of each show the
posterior distribution for apparent SFR (before correction for magnification),
the mass-weighted age, the apparent (pre-lensing correction) stellar mass, the
sSFR, and the dust attenuation $A_{V}$. Dashed vertical lines show the 16th,
50th, and 84th percentiles. These are distinct objects (not multiple images)
but are very similar in these given properties.

###  4.6 A comparison of integrated vs. resolved properties

In addition to the spatially-resolved SED fitting, we also use Bagpipes to
measure global properties of G165-DSFG-1 and G165-DSFG-3. For simplicity, we
measure the photometry of the counterimage arcs 1a and 3c, as these contain
the entire object (in contrast with the high-magnification arcs 1bc and 3ab,
which arise from only a small fraction of their respective sources). Moreover,
the magnification gradient for these counterimages, while not entirely
negligible, is much smaller. After convolving all filters with a kernel to
match the PSF of the coarsest-resolution filter (F444W), we use photutils
(Bradley et al. 2022) to capture Kron aperture fluxes (Kron 1980).

We fit the 8-filter photometry with bagpipes in an identical manner as for the
pixel-by-pixel fluxes, described in §3.2, using the same set of priors. While
we measure photometry at 2 mm, 3 mm, and 6 GHz (Table 1), these are only
included in a combined SED fit for the sum of Arc 1a, 1bc, 3ab, and 3c. The
resolution of these long-wavelength observations ($\sim 1\farcs 1$, $\sim
1\farcs 3$, and $\sim 0\farcs 5$, respectively) are much coarser than what is
enabled by JWST (see Fig. 3). Moreover, the synthesized beam of the ALMA
observations is highly elongated, owing to this being a high-declination
($+42^{\circ}$) target for ALMA’s southern latitude. Unfortunately, this
elongation is essentially aligned with the orientation of greatest
magnification, so lensing does not offer an easy remedy. With these parameters
of the currently-available data, it is therefore not worthwhile to attempt
spatially-resolved SED fitting with both ALMA and JWST, as only a few
resolving elements cover the extent of the arcs.

Table 2: Properties of DSFG-1 and DSFG-3 from total IR dust SED, and
corresponding SN rates.

Object | $\mu_{\rm 2mm}{\rm SFR}_{\rm IR}^{\dagger}$ | SFRIR | log[$L_{\rm IR}/L_{\odot}$] | $R_{\rm CC}^{\ddagger}$ | $R_{\rm CC}(1+z)^{-1}$ | $R_{e,2{\rm mm}}$ | $R_{e,2{\rm mm}}$
---|---|---|---|---|---|---|---
| [$M_{\odot}~{}{\rm yr}^{-1}$] | [$M_{\odot}~{}{\rm yr}^{-1}$] |  | [yr-1] | [yr-1] | [arcsec] | [kpc]
G165-DSFG-1 | $9100\pm 1900$ | $320\pm 70$ | $12.5\pm 0.1$ | $3.3\pm 0.7$ | $1.0\pm 0.2$ | $0.20\pm 0.02$ | $1.6\pm 0.2$
G165-DSFG-3 | $2900\pm 500$ | $400\pm 80$ | $12.6\pm 0.1$ | $4.2\pm 0.8$ | $1.3\pm 0.2$ | $0.25\pm 0.03$ | $2.1\pm 0.2$

^†^†footnotetext: The total apparent (i.e. lensing-uncorrected) IR-derived SFR
of $12000\pm 2000~{}M_{\odot}{\rm yr}^{-1}$ from H16 is divided between Arcs
1abc and 3abc according to their ratio of millimeter flux, $S_{\rm
1abc}/S_{\rm 3abc}=3.1\pm 0.3$ (averaged for 2 mm and 3 mm).

^‡^‡footnotetext: Expected rate of core-collapse supernovae tabulated as
$R_{\rm CC}=k_{\rm CC}\cdot{\rm SFR}$; $k_{\rm CC}=0.0104~{}M_{\odot}^{-1}$
(e.g., Strolger et al. 2015). These are intrinsic SN rates; observed SN rates
will be dilated by a factor of $(1+z)^{-1}$, but also increased for any SNe
occurring in regions of the galaxy that are multiply imaged.

Note. — We derive the SFR of the two DSFGs independently using the far-IR
photometry (H16) vs. the rest-frame UV-NIR photometry from NIRCam (which is
shown in Table 3). As the far-IR photometry largely does not resolve Arcs 1abc
and 3abc, we make a preliminary estimate by apportioning the total ${\rm
SFR}_{\rm IR}$ according to their share of flux at 2 mm and 3 mm (combining
Arcs 1a and 1bc, and Arcs 3ab and 3c), as both arc systems are understood to
be at similar enough redshifts. Then, these apportioned SFRs are corrected for
the total 2 mm magnifications of Arc 1abc and Arc 3abc, respectively. Based on
these various SFRs, we estimate the rate of occurrence of core-collapse
supernovae (for the rest-frame of each galaxy); the observable rate is likely
bracketed by the SED- vs. IR-inferred SFRs, given that SNe in some very dusty
environments may be missed (or, alternatively, the extreme star formation may
lead to a porosity in the dust geometry that lends to a more favorable chance
for discovery).

However, we can make a comparison of SED-fitting with vs. without the
inclusion of far-IR rest-frame photometry for the unresolved, combined SED of
both DSFG-1 and DSFG-3 together. Flux measurements that better constrain the
peak of the dust SED at rest-frame 100 $\mu$m are available (Cañameras et al.
2015; H16), but these all have resolutions of $8\farcs 5$—in the case of the
LMT AzTEC—or much larger. In Appendix F, we sum the NIR photometry of the four
arcs to revisit the SED fit of Harrington et al. (2016). As shown by Battisti
et al. (2019), the stellar mass of DSFGs can be underestimated by 0.3 dex with
only UV to near-IR coverage (a discrepancy that is greatest for objects with
$M_{\star}>10^{10}~{}M_{\odot}$). More broadly, for galaxies simulated with
the Evolution and Assembly of GaLaxies and their Environments (EAGLE; Schaye
et al. 2015) project and the SKIRT dust radiative transfer code (Baes et al.
2011), Dudzevičiūtė et al. (2020) found that even with UV-to-radio coverage,
SED-modeling resulted in a systematic under-prediction of stellar mass by
$0.5\pm 0.1$ dex. It also appears to be the case that inclusion of far-IR
photometry in an SED fit tends to reduce the inferred SFR (Pacifici et al.
2023), although Wise et al. (2023) found that SFR can be under- or over-
estimated (up to 1 dex) from UV to near-IR alone. For galaxies with lower
SFR${}_{\rm UV-NIR}\lesssim 300~{}M_{\odot}~{}{\rm yr}^{-1}$, they found that
adding FIR data boosted the estimated SFR; for high SFRUV-NIR galaxies, FIR
data lowered the estimated SFR. Indeed, in our case, DSFG-1 and DSFG-3 have a
combined SFR${}_{\rm UV-NIR}\approx 90~{}M_{\odot}~{}{\rm yr}^{-1}$, but the
unresolved UV-FIR SED yields SFR${}_{\rm UV-FIR}=350\pm 70~{}M_{\odot}~{}{\rm
yr}^{-1}$, a $\sim 0.6$ dex increase (Table 3). As is also emphasized by Smail
et al. (2023), spatially resolved “pixel-by-pixel" SED modeling is made
significantly more robust by the inclusion of long-wavelength photometry at
similar angular resolution, highlighting the particular need for submillimeter
interferometry in this analysis.

Table 3: Properties of DSFG-1 and DSFG-3 from rest-frame UV through near-IR
SED fitting, and corresponding SN rates.

Object | $\mu_{\rm 2\mu m}{\rm SFR}$ | ${\rm SFR}$ | $R_{\rm CC}^{\ddagger}$ | log[$\frac{\mu_{\rm 2\mu m}M_{\star}}{M_{\odot}}$] | log[$\frac{M_{\star}}{M_{\odot}}$] | log[$\frac{\rm sSFR}{{\rm yr}^{-1}}$] | $A_{V}$
---|---|---|---|---|---|---|---
| [$M_{\odot}~{}{\rm yr}^{-1}$] | [$M_{\odot}~{}{\rm yr}^{-1}$] | [yr-1] |  |  |  | [mag]
Arc 1a (SED) | $280\pm 130$ | $50\pm 20$ | $0.5\pm 0.2$ | $10.9\pm 0.1$ | $10.2\pm 0.1$ | $-8.5\pm 0.3$ | $1.5\pm 0.3$
Arc 1a (H$\alpha$) | $320\pm 140^{\dagger}$ | $60\pm 30^{\dagger}$ | $0.6\pm 0.3$ | $10.5\pm 0.1^{\dagger}$ | $9.7\pm 0.1^{\dagger}$ | $-8.4\pm 0.3^{\dagger}$ | $1.5\pm 0.4^{\dagger\dagger}$
Arc 1a (bin-wise sum) | $380\pm 10$ | $69\pm 3$ | $0.72\pm 0.03$ | $10.91\pm 0.01$ | $10.17\pm 0.02$ | $-8.34\pm 0.02$ | $\approx 1.0\pm 0.8^{\dagger\dagger}$
Arc 3c (SED) | $260\pm 110$ | $40\pm 20$ | $0.4\pm 0.2$ | $11.1\pm 0.1$ | $10.3\pm 0.1$ | $-8.7\pm 0.3$ | $1.2\pm 0.2$
Arc 3c (bin-wise sum) | $440\pm 10$ | $75\pm 3$ | $0.78\pm 0.03$ | $11.06\pm 0.01$ | $10.29\pm 0.02$ | $-8.42\pm 0.02$ | $1.2\pm 0.7$
Arcs 1abc+3abc (UV-FIR)⋆ | $5560\pm 280$ | $350\pm 70$ | $3.6\pm 0.7$ | $11.68\pm 0.03$ | $10.48\pm 0.09$ | $-7.93\pm 0.03$ | $2.54\pm 0.04$

^‡^‡footnotetext: Expected rate of core-collapse supernovae tabulated as in
Table 2.

^†^†footnotetext: The H$\alpha$-derived SFR and stellar masses from F24 are
the summed values from NIRSpec apertures NS_46 and NS_969. The intrinsic
(magnification-corrected) properties use the 2 $\mu$m magnifications from
Table 1. The sSFR is determined from this SFRHα and the stellar mass, which is
determined from an SED model that incorporated H$\alpha$ flux as a constraint.

^††^††footnotetext: As described in §4.6, the dust attenuation $A_{V}$ is
estimated from the Balmer-decrement derived $E(B-V)_{\rm gas}=0.85\pm 0.25$
for NS_969 (F24) and a Calzetti et al. (2000) reddening law,
$R_{V}=A_{V}/E(B-V)=4.05$. We lastly impose a conversion to a stellar
continuum color excess through $E(B-V)_{\rm stellar}=0.44E(B-V)_{\rm gas}$,
per Calzetti (1997a). For the bin-wise dust attenuation, we provide the inner
68% interval for the bins’ values.

^⋆^⋆footnotetext: Here, we use the total flux-weighted magnification for both
DSFG-1 and DSFG-3 combined, $\mu=16\pm 3$, to derive intrinsic values.

Note. — We compute the UV-NIR SED-derived SFRs for Arcs 1a and 3c (see Fig.
14), and correct these for respective magnifications at $2\mu$m. These values
are lower than the SFRIR estimates, but this is expected given their large
dust attenuations. In Appendix F, we fit the summed SED of both DSFGs so that
the unresolved far-IR photometry can be incorporated. The intrinsic, combined
SFRUV-FIR is $\approx 4$ times the sum of SFRUV-NIR for the two DSFGs, while
the summed stellar masses are more consistent with the integrated value. We
also compare with the properties derived through the H$\alpha$ measurement by
F24, incorporating the two NIRSpec apertures that covered this object (but
applying our 2 $\mu$m magnifications, from Table 1). In all cases, we find
quite close agreement with our independently-derived values.

Zibetti et al. (2009) first presented an important bias to consider when
inferring the stellar mass of a galaxy: mass estimates based on global
(galaxy-integrated) fluxes were 40% (0.2 dex) lower than those determined by
summing over the map of stellar mass derived from spatially-resolved SED
fitting. This effect is the direct result of the stellar mass contribution
from dust-obscured regions being underestimated through the unresolved
photometry (although Sorba & Sawicki 2015, 2018 point out that an “outshining"
bias from the youngest stellar populations may be responsible, as well). Song
et al. (2023) found recently that without coverage of rest-frame near-IR in
SED fitting, $M_{\star}$ can be overestimated by 0.2 dex, owing directly to an
overestimate of both dust attenuation and stellar ages from only rest-frame UV
and optical photometry. As shown in Table 3, there is excellent agreement in
the stellar masses determined through integrated photometry vs. a sum of
stellar masses derived through resolved bin-wise SED fitting with piXedfit,
which may indeed be the result of photometric coverage out to rest-frame 1.4
$\mu$m.

Table 3 also compares with the H$\alpha$-inferred SFR and dust attenuation
properties presented by F24 for G165-DSFG-1. These are the result of combining
measurements from NIRSpec slits NS_46 and NS_969, the latter of which is
labeled “Arc 1a" in their Table 4. We find strong agreement for SFR,
especially within uncertainties: global $\mu{\rm SFR}_{\rm NIR}=280\pm
130~{}M_{\odot}~{}{\rm yr}^{-1}$ vs. spatially-integrated $\mu{\rm SFR}_{\rm
NIR}=380\pm 10~{}M_{\odot}~{}{\rm yr}^{-1}$ vs. $\mu{\rm SFR}_{{\rm
H}\alpha}=320\pm 140~{}M_{\odot}~{}{\rm yr}^{-1}$. The stellar mass derived by
F24 is not completely independent of our result, as they used a joint
photometric and spectroscopic SED fit with FAST$++$ (Kriek et al. 2009;
Schreiber et al. 2018). Whether due to the different SED fitting algorithms or
to the exclusion of spectroscopic information, the stellar mass we
estimate—log$[\mu M_{\star}/M_{\odot}]=10.9\pm 0.1$—is 0.4 dex larger than the
value quoted in F24. Our assumption of a “single" SFH in the form of an
exponentially declining “tau" model may underestimate the stellar mass by
$0.1-0.3$ dex relative to a multi-component SFH (Michałowski et al. 2014;
Carnall et al. 2019; Jain et al. 2023). Given the typical uncertainty of 0.1
dex in stellar mass for SED modeling at $z>1$ (e.g., Pacifici et al. 2023), or
even slightly elevated uncertainty of $\sim 0.3$ dex for DSFGs (Michałowski et
al. 2014), it is perhaps most likely that the statistically significant
discrepancy is explained by the information provided by the near-IR
spectroscopy (or perhaps the extrapolation from limited slit coverage of the
source).

Finally, we compare the estimate of dust attenuation that we find from
modeling the UV-NIR SED against that derived from the Balmer decrement in
H$\alpha$ vs. H$\beta$ from F24. They measured $E(B-V)=0.85\pm 0.25$ for
NS_969 (with which we choose to compare here, given its larger uncertainty).
For a Calzetti et al. (2000) reddening law, or $R_{V}=A_{V}/E(B-V)=4.05$, this
yields $A_{V}=3.0\pm 1.0$. However, this discrepancy with our smaller value of
$A_{V}=1.5\pm 0.3$ can be explained by the typically larger color excess for
ionized gas relative to stellar continuum. Calzetti (1997b) quantify this as
$E(B-V)_{\rm stellar}=0.44\times E(B-V)_{\rm gas}$ (cf. Shivaei et al. 2015).
Applying this correction to the value from F24, we recover $A_{V}=1.5\pm 0.4$,
in superb agreement with our finding. For the diffuse ISM in the outskirts of
galaxies, the ratio of $E(B-V)_{\rm stars}/E(B-V)_{{\rm H}\textsc{ii}}$ is
larger than for stellar birth clouds in star-forming regions, and even
approaches unity (e.g., Robertson et al. 2024). Our finding that a ratio of
0.44 is appropriate for DSFG-1 may actually be in line with our interpretation
of widespread star-forming regions, but this claim is limited for now by the
current spectroscopic coverage of just a narrow region of the arc.

##  5 Discussion

###  5.1 How $UVJ$ selection breaks down on resolved scales, and possible
evidence for radial dust transport within the disk

There are at least a dozen bins for Arc 2a in Fig. 13 with very low sSFRs
(log$[{\rm sSFR}/{\rm yr}^{-1}]<-11.5$) that are still safely outside of the
classical quiescent selection region, and would instead be classified as dusty
star-forming populations. The quiescent sequence has been found to evolve with
age—Belli et al. 2019 found a best-fit relation for the median stellar age of
${\rm log}(t_{\rm 50}~{}[{\rm yr}])=7.03+0.84(V-J)+0.74(U-V)$. These redder
colors in resolved $U-V$ and $V-J$ can thus partly be the result of older
stellar populations ($\sim 3$ Gyr), but this model still severely
underpredicts the finding of $V-J\approx 1.7$ and $U-V\approx 2.3$.
Ultimately, the most likely explanation seems to be that these quiescent
regions are more dust-attenuated than typically expected. The reddening vector
of $\Delta A_{V}$ shown in Fig. 13 implies that several of these low-sSFR bins
have at least $A_{V}>0.5$ mag. This may also be borne out by the finding of
$A_{V}\gtrsim 1$ for nearly the entirety of Arc 2a in Fig. 11. As some added
confirmation, SN H0pe is located nearby this dusty quiescent region, and
estimated via light-curve fitting to be attenuated by $A_{V}=1.21\pm 0.05$
(Pierel et al. 2024b). While dust in quiescent galaxies is still very much an
area of active research, owing to the difficulty this presents for
observations (e.g., Gomez et al. 2010; Rowlands et al. 2012; Magdis et al.
2021; Donevski et al. 2023; and references therein), they are often assumed to
be quite dust-poor. Revisions to the original $UVJ$ selection frequently
exclude the $V-J<1.5$ cut to accommodate older and dustier quiescent galaxies
(van der Wel et al. 2014; Whitaker et al. 2015; Belli et al. 2019). For Arc
2a, this correction would appropriately enclose the low-sSFR bins within the
quiescent regime, but would also erroneously include regions with sSFRs that
are over 2 dex larger (sSFR $\gtrsim 0.5$ Gyr-1).

This might indicate a shortcoming of our interpretation of the classical $UVJ$
selection when applied to resolved scales. In a similar way that the resolved
star formation main sequence breaks down at $\sim 100$ pc and smaller, owing
to stars migrating from their parent star-forming clouds, the $UVJ$ diagram
presumably also breaks down when significant amounts of dust and metals formed
in regions of active star formation are transported to adjacent, more
quiescent regions. This can occur over short distances within the plane of the
disk through turbulent motions in the ISM (Cho et al. 2003), but Vorobyov &
Shchekinov (2006) showed that spiral stellar density waves could help shepherd
dust several kpc in the radial direction over the course of $\sim 1$ Gyr (see
also Mishurov et al. 2014).

The extra reddening seen may also be due to extraplanar dust that has been
transported a few kpc into the circumgalactic medium by the likes of galactic-
scale winds driven by supernovae and winds from massive stars (Mac Low et al.
1989; Norman & Ikeuchi 1989; Heckman et al. 1990; Heckman & Thompson 2017;
Richie et al. 2024) or simply through momentum injection by radiation pressure
(e.g., Ferrara et al. 1991; Murray et al. 2005; Thompson et al. 2015; Barnes
et al. 2020; Kannan et al. 2021). However, it is not clear if extraplanar dust
(or even a thick disk of dust) can constitute a large enough fraction of the
total dust mass, as it is estimated to be $<5\%$ (Howk & Savage 2000; Popescu
et al. 2000; Bianchi & Xilouris 2011; Seon et al. 2014). The vertical
stability of the disk also plays a role, as diffuse distributions of dust
appear to be more predominant in low-mass, slow-rotating disks (Dalcanton et
al. 2004; Holwerda et al. 2012). This extraplanar dust is also typically
thought to be highly filamentary in nature, associated with chimney vents and
extending a few kpc from the midplane (Wang et al. 2001; Stein et al. 2020;
Mosenkov et al. 2022; see also review by Veilleux et al. 2020). It is unlikely
to provide the necessary column densities to reproduce the observed $A_{V}$
over large scales (Thompson et al. 2004), especially when considering the
effect of grain destruction by sputtering (e.g., Spitzer 1978). Admittedly,
widespread large attenuations do not necessitate uniform, thick screens of
dust; they merely need large quantities of dust to be spatially correlated
with regions of visible starlight. We thus consider it most likely that the
bulk of the dust responsible for the attenuation is still largely contained
(and well mixed) within the disk.

Over larger, galaxy-wide scales, this effect is likely to be less of a
concern, as $UVJ$ colors will be biased in favor of the unobscured sightlines.
For globally quiescent objects, this means that the more dust-free regions
with older stellar populations will dominate. This appears to be the case for
the global colors of Arc 2a, as the object just barely falls within the
$UVJ$-quiescent regime. Relatively few studies so far have examined spatially-
resolved $UVJ$ diagrams in the era of JWST (including Miller et al. 2022;
Kokorev et al. 2023; Suess et al. 2023), so our speculation on the effect of
dust transport is necessarily limited. Yet, we can expect a better
understanding of these systematics (and in what other circumstances the
selection criteria break down; e.g., Leja et al. 2019) with time.

###  5.2 Starbursts or not? The proximity of DSFG-1 and DSFG-3 to the star-
forming main sequence

In §4.3 we considered the location of spatially-resolved regions relative to
the resolved star formation main sequence, but now we examine the position of
DSFG-1 and DSFG-3 as a whole. For both DSFG-1 at $z=2.236~{}(t=2.88~{}{\rm
Gyr})$ and DSFG-3 at $z=2.1~{}(t=3.07~{}{\rm Gyr})$, the Speagle et al. (2014)
model for the evolution of the main sequence151515The main sequence follows
${\rm log}[{\rm SFR}(M_{\star},t)]=(0.84\pm 0.02-0.026\pm 0.003\times
t)~{}{\rm log}M_{\star}-(6.51\pm 0.24-0.11\pm 0.03\times t)$, for the age of
the Universe $t$ in Gyr. This model uses a Kroupa (2001) IMF, consistent with
our SFRs. yields ${\rm log}[{\rm SFR}(M_{\star},t)]\approx 0.8~{}{\rm
log}M_{\star}-6.2$. Using our derived stellar mass of DSFG-1 of ${\rm
log}M_{\star}=10.2$, we find ${\rm SFR}_{\rm MS}\approx 90~{}M_{\odot}~{}{\rm
yr}^{-1}$. The IR-inferred SFR${}_{\rm IR}\approx 320~{}M_{\odot}~{}{\rm
yr}^{-1}$ is therefore approximately 3.6 times that of the main sequence at
this redshift. As for DSFG-3, with stellar mass ${\rm log}M_{\star}=10.3$,
${\rm SFR}_{\rm MS}\approx 110~{}M_{\odot}~{}{\rm yr}^{-1}$, and the inferred
SFR${}_{\rm IR}\approx 400~{}M_{\odot}~{}{\rm yr}^{-1}$ is likewise
approximately 3.6 times that of the main sequence at this redshift. With
distance to the main sequence defined as $\Delta_{\rm MS}\equiv{\rm
log}_{10}({\rm SFR}/{\rm SFR}_{\rm MS}$, both cases yield $\Delta_{\rm
MS}\approx 0.6$ dex. Adopting a $\Delta_{\rm MS}\geq 0.6$ dex threshold (e.g.,
Rodighiero et al. 2011) for starbursts above the main sequence—chosen to be a
factor of 2 above the typical scatter of $\approx 0.3$ dex (Brinchmann et al.
2004; Daddi et al. 2007; Elbaz et al. 2007; Noeske et al. 2007a; Salim et al.
2007; Whitaker et al. 2012; Speagle et al. 2014; Tacchella et al. 2016b)—this
means that DSFG-1 and DSFG-3 are only just marginally classified as
starbursts.

While DSFGs with such high SFRs are often incorrectly presumed to lie
universally within the starburst regime, Rodighiero et al. (2011) found that
only $20\%\pm 4\%$ of galaxies with ${\rm SFR}>100~{}M_{\odot}~{}{\rm
yr}^{-1}$ satisfied the $+0.6$ dex criterion. Similarly, Barrufet et al.
(2020) found that approximately 60% of DSFGs (in a sample of 185) have SFRs
consistent with the star-forming main sequence. Indeed, many claims have been
made that DSFGs simply occupy the high-$M_{\star}$ end of the SFMS (e.g.,
Finlator et al. 2006; Dunlop 2011; Magnelli et al. 2012; Michałowski et al.
2012; Sargent et al. 2012; Targett et al. 2013; Koprowski et al. 2016; Dunlop
et al. 2017; Elbaz et al. 2018; Drew et al. 2020; Pantoni et al. 2021;
contrast with Hainline et al. 2011, for example). Furthermore, for a set of 15
other lensed DSFGs in the PASSAGES sample (not including G165), Kamieneski et
al. (2024) found that proxies for $\Delta_{\rm MS}$ (namely, molecular gas
depletion time and $\Sigma_{\rm SFR}$) predicted broad consistency with the
upper end of the star-forming main sequence, but not necessarily
classification as starbursts. In this work, we have also found that the radial
profiles of sSFR (Figs. 6 and 7) are broadly consistent with those expected
for starbursts from Nelson et al. (2021).

The finding that the DSFGs in the G165 field can be classified as starbursts
may have some connection to the finding that they reside within a highly
active star-forming environment (and possible galaxy overdensity) by
F24—although Drew et al. 2020 contend that distance to the SFMS may be a poor
indicator of recent merger activity. Regardless, the locus of high-$z$ DSFGs
on the SFR$-M_{\star}$ plane helps to quantify the role of gas supply-
dependent, secular evolution (Davé et al. 2012). A more systematic examination
of the stellar masses of DSFGs—especially now, with JWST helping to circumvent
their prohibitive degree of dust obscuration—will ultimately shed more light
on their connection to the star-forming main sequence. For now, the
inconsistency of DSFGs and SMGs lying on or above the main sequence (Hodge &
da Cunha 2020) may ultimately arise from the apparent heterogeneity of the
overall population (e.g., Hayward et al. 2011, 2012; Magnelli et al. 2012;
Hayward et al. 2013).

###  5.3 Prediction of observer-frame supernova rates from G165-DSFG-1 and
G165-DSFG-3, and prospects for future discovery

The connection of core-collapse supernova rates to SFRs is straightforward, as
CCSN progenitors are short-lived massive stars, believed to range in mass from
$8-50~{}M_{\odot}$ (Nomoto 1984; Tsujimoto et al. 1997). The supernova rate
can thus be tabulated as $R_{\rm CC}=k_{\rm CC}\cdot{\rm SFR}$, where the
scale factor is derived from the IMF $\phi(M)$ as

$k_{\rm
CC}=\frac{\int_{8~{}M_{\odot}}^{50~{}M_{\odot}}\phi(M)dM}{\int_{0.1~{}M_{\odot}}^{125~{}M_{\odot}}M\phi(M)dM}$
(1)

and essentially represents the fraction of the forming stellar populations
that consists of CCSN progenitors. These lower and upper mass limits and a
Kroupa (2001) IMF yield $k_{\rm CC}=0.0104~{}M_{\odot}^{-1}$ (e.g. Strolger et
al. 2015), which we adopt for this work to remain consistent with the SED
fitting. A Salpeter (1955) IMF for the same $8-50~{}M_{\odot}$ progenitor
range would yield $k_{\rm CC}=0.0070~{}M_{\odot}^{-1}$ (e.g., Mattila & Meikle
2001; Young et al. 2008), but this difference is essentially canceled out by
the corresponding effect on the SFR (e.g., Madau & Dickinson 2014; Ziegler et
al. 2022).

With this in mind, the predicted rest-frame supernova rate for G165-DSFG-1 is
$3.3\pm 0.7~{}{\rm yr}^{-1}$ using the FIR-derived SFR, or $0.5\pm 0.2~{}{\rm
yr}^{-1}$ using the lower, UV-NIR SED-derived SFR (see Tables 2 and 3). For
G165-DSFG-3, these rates are comparable: $4.2\pm 0.8~{}{\rm yr}^{-1}$ (FIR)
and $0.4\pm 0.2~{}{\rm yr}^{-1}$ (UV-NIR SED). Before taking into account the
multiplexing provided by multiple images of any SNe, these rates translate to
observed-frame values161616Cosmological time dilation decreases the apparent
rate of occurrence for SNe, $R_{\rm CC,obs}=R_{\rm CC}\cdot(1+z)^{-1}$, but it
also stretches out the time for which a given supernova’s light curve is
detectable. for G165-DSFG-1 of $R_{\rm CC,obs}\approx 0.2$ yr-1 (SED) to
$\approx 1.0$ yr-1 (FIR), and for G165-DSFG-3 of $R_{\rm CC,obs}\approx 0.1$
yr-1 (SED) to $\approx 1.3$ yr-1 (FIR).

However, for the most reliable rate estimate, we use the SFR from the best-fit
UV-through-FIR SED for DSFG-1 and DSFG-3 combined, ${\rm SFR}_{\rm UV-
FIR}=350\pm 70~{}M_{\odot}~{}{\rm yr}^{-1}$. This results in a total expected
rest-frame rate of $R_{\rm CC}=3.6\pm 0.7~{}{\rm yr}^{-1}$, or a remarkable
observer-frame rate of $1.1\pm 0.2~{}{\rm yr}^{-1}$. This very high rate is
derived only from the pair of DSFGs; however, as discussed by F24, the
H$\alpha$-derived SFR for the objects associated with Arcs 1abc and 2abc
groups surpasses 500 $M_{\odot}~{}{\rm yr}^{-1}$, which comes from only the
unobscured mode of star formation. Given that these groups appear to signal
galaxy overdensities at $z\sim 2$, it may well be the case that $\gtrsim 1-2$
CCSNe yr-1 may be observable for the G165 cluster. This makes the G165 field
an extremely attractive target for follow-up monitoring campaigns with JWST,
as even the six Frontier Field clusters (Lotz et al. 2017) all together would
provide only 0.9 CCSNe per year, with a cadence of four 1-hour visits with
F150W within a year (Petrushevska et al. 2018). Given the best-fit
$A_{V}\approx 2.5$ from the UV-FIR SED, it is entirely feasible to detect the
descendent supernovae with only modest integration with JWST. An appropriate
cadence would be designed to span a baseline of at least several months, so
that enough time elapses for a clearly observable light curve evolution of any
SNe (which is subject to a factor of $1+z$ in cosmological time dilation).

This last point is crucial, and helps to answer the inevitable question of why
no CCSNe have yet been discovered in G165, given that two follow-up visits
were undertaken with JWST on 22 April 2023 and 9 May 2023 after the discovery
of the Type Ia SN H0pe in imaging on 30 March 2023. At only $+23$ d and $+40$
d from Earth’s perspective, this is a mere $+8$ d and $+13$ d in the rest-
frame at $z\sim 2$, which is hardly sufficient to guarantee a CCSN detection.
The cadence of observations for SN H0pe was designed for monitoring of a
discovered event; a cadence for discovering new SNe will simply need to extend
over a longer period of time.

##  6 Summary and Conclusions

With this work, we have examined in detail the magnified, massive galaxies at
$z\sim 2$ behind the G165 galaxy cluster, including two luminous dusty star-
forming galaxies (log$(M_{\star}/M_{\odot})\approx 10.2$; SFR${}_{\rm
IR}\approx 300-400~{}M_{\odot}~{}{\rm yr}^{-1}$) and a massive galaxy that
appears to be exiting a phase of dusty star formation
(log$(M_{\star}/M_{\odot})\approx 10.8-11.0$ and SFR${}_{{\rm H}\alpha}\approx
5-10~{}M_{\odot}~{}{\rm yr}^{-1}$; F24). This latter object has been the
subject of intense interest as the host of a recently-discovered triply-lensed
Type Ia supernova, SN H0pe. Yet, the lensed DSFGs are responsible for the
intense submillimeter flux by which the cluster was first discovered
(Cañameras et al. 2015; Harrington et al. 2016; Frye et al. 2019; Pascale et
al. 2022), and we have sought to further elucidate their properties through
recent imaging in the near-IR with JWST and in the millimeter-wave through
ALMA. We also present a new parametric lens model for the G165 cluster,
developed with lenstool. This model was constructed predominantly for the
purpose of predicting lensing time delays for SN H0pe, in order to compare
with values measured independently through the photometric and spectroscopic
light curve, and thus infer the Hubble constant $H_{0}$. However, this model
is also invaluable for reconstructing the intrinsic (i.e. magnification-
corrected) properties of the lensed DSFGs.

Coincidentally, what was initially thought to be a single DSFG strongly lensed
by the foreground merging galaxy cluster has turned out to be two DSFGs with a
peculiar (almost uncanny) similarity. The original known DSFG, with a
detection of CO(3–2) at $z=2.236$ (H16; also Harrington et al. 2021) was
identified as a merging image pair Arc 1bc, and the counterimage Arc 1a by
Frye et al. (2019) and Pascale et al. (2022). With new NIRCam and ALMA
continuum imaging, it has become clear that the object identified as Arc 3ab
(merging pair) and Arc 3c (counterimage) on the same side of the lensing
cluster also harbors considerable dust and star formation. In fact, while the
Arc 1abc system (which we denote G165-DSFG-1) is considerably brighter in dust
continuum (primarily due to the contribution from the high-magnification Arc
1bc) and likewise presumably CO emission, the intrinsic properties are very
similar to the Arc 3abc system (G165-DSFG-3), which may even have a larger
SFRIR ($400\pm 80~{}M_{\odot}~{}{\rm yr}^{-1}$ vs. $320\pm
70~{}M_{\odot}~{}{\rm yr}^{-1}$).

The redshift of DSFG-3 is still somewhat uncertain, as the current molecular
line detections are all spatially unresolved and cannot distinguish between
the two. It is most likely that it also lies at $z\approx 2.236$ with DSFG-1,
so that their CO lines are blended at the 100 km s-1 spectral resolution of
LMT/RSR. If they do in fact lie at the same redshift, then this proximity
($\approx 1\arcsec$, $<10$ kpc) would certainly imply that the objects are in
the early stages of a spectacular gas-rich major merger (with a nearly 1:1
mass ratio, each with log$(M_{\star}/M_{\odot})\approx 10.2$). The color
gradients that are clearly apparent in Fig. 2 may even be the result of tidal
disruption of molecular gas after a first passage, driving an asymmetry of
dust-obscured star formation towards the center of mass of the system (which
is most readily observable in DSFG-1; Fig. 6). Moreover, the fainter Arc 3, 4,
and 6 clumps that lie in between DSFG-1 and DSFG-3 (and predicted by the lens
model to lie at the same redshift) could be additional evidence for such a
significant gravitational interaction. Future spectroscopic follow-up and
kinematic information for the molecular gas is ultimately necessary to help
resolve this open question.

Regardless of their distances along the line-of-sight, the presence of two
strongly star-forming DSFGs only $1\arcsec$ apart in the sky is a remarkable
finding. In fact, one could perhaps even make the case that this conjunction
would be more impressive in the sky without the magnification and distortion
contributed by the lensing galaxy cluster! More quantitatively, they share
similar IR-inferred SFRs, as well as UV-NIR SED-derived SFRs ($50\pm
20~{}M_{\odot}~{}{\rm yr}^{-1}$ for DSFG-1, $40\pm 20~{}M_{\odot}~{}{\rm
yr}^{-1}$ for DSFG-3), stellar masses (${\rm log}(M_{\star}/M_{\odot})=10.2\pm
0.1$ vs. $10.3\pm 0.1$, respectively), and dust attenuation ($A_{V}=1.5\pm
0.3$ vs. $A_{V}=1.2\pm 0.2$). Their intrinsic (source-plane) dust continuum
sizes are $1.6\pm 0.2$ kpc and $2.1\pm 0.2$ kpc. Unsurprisingly, given the
well-known correlation of radio with the far-IR (e.g., van der Kruit 1971; de
Jong et al. 1985; Helou et al. 1985; Condon 1992; Lisenfeld et al. 1996; Yun
et al. 2001; Bell 2003; Murphy et al. 2006a, b), they also have similar radio
continuum fluxes ($S_{\rm 6GHz}/\mu\approx 30~{}\mu{\rm Jy}$ for Arc 1a and
$\approx 20~{}\mu{\rm Jy}$ for Arc 3c). Their position in the $UVJ$ diagram is
functionally identical ($U-V\approx 1.1$, $V-J\approx 1.2$), so their
similarities cannot be attributed to limited flexibility in SED fitting.

Given the diversity and heterogeneity of DSFGs as a population—see Smail et
al. (2023) for another recent example of two DSFGs nearby to each other behind
a lensing cluster, but with markedly different properties—this chance ultra-
close alignment of two remarkably similar DSFGs is very peculiar. While we do
not wish to fully speculate on whether or not this is purely coincidence,
especially given the uncertainty in exact redshift for one of the objects, it
is maybe not entirely outside the realm of possibility that these objects
could be associated with filamentary nodes of large-scale structure in the
Universe. As discussed in our introduction, DSFGs are widely believed to be
associated with galaxy overdensities and protoclusters (see the theoretical
work by Chiang et al. 2013, and recent review by Alberts & Noble 2022 and
references therein). This is similar to the association of protoclusters with
high-redshift radio galaxies (HzRGs; e.g. Venemans et al. 2007; Miley & De
Breuck 2008) or Ly$\alpha$ emitters (e.g., Keel et al. 1999; Umehata et al.
2015; Harikane et al. 2019) and extended Ly$\alpha$ nebulae, also known as
Lyman Alpha Blobs (e.g. Steidel et al. 2000; Hennawi et al. 2015; Arrigoni
Battaia et al. 2018; Ramakrishnan et al. 2023; review by Overzier 2016). These
signposts, and their connections to each other, are still very much an active
area of research. And indeed, Frye et al. (2024) found evidence of a
substantial overdensity of $z\approx 2$ galaxies behind G165 through
photometric redshifts (rivaling even the number of identified members of the
foreground cluster at $z=0.35$). The authors also identified several groups of
objects found serendipitously to lie at the same redshift (one association of
six objects at $z=1.8$ and another of several galaxies at $z=2.2$) with the
JWST NIRSpec Micro-Shutter Array (MSA). Given the limitations of how MSA slits
could be arranged, there are almost certainly other members of these
associations that are not yet uncovered. With this rich set of background
objects, all strongly lensed by a massive merging galaxy cluster, the G165
field is sure to be the source of a wealth of future astrophysical
discoveries.

We summarize the primary scientific findings of our work:

  * •

Through resolved SED fitting of DSFG-1 and DSFG-3, we find complex
distributions of $\Sigma_{\rm SFR}$, $\Sigma_{\star}$, and $A_{V}$ (which are
of primary concern for this work). For DSFG-1, there is a noticeable
asymmetry, with star formation and dust attenuation skewed higher towards the
southeast, with the peak in $\Sigma_{\star}$ essentially occupying a highly
elliptical central region rather than a single peak. These likely are related
to the presence of multiple orbiting components at slightly different
redshifts in this arc ($z=2.2355$ vs. $z=2.2401$).

  * •

In the case of DSFG-3, there is not this clear asymmetry. Here, $\Sigma_{\rm
SFR}$ is centrally peaked but again in a rather elliptical nuclear region.
Stellar mass density $\Sigma_{\star}$, on the other hand, shows multiple
distinct peaks throughout the disk plane, possibly signaling a history of
(major or minor) merger activity. Dust attenuation $A_{V}$ is patchy
throughout the object, with two distinct peaks, neither of which seem to be at
the galaxy center.

  * •

We examine the position of the spatially-resolved regions (on scales of
$150-1500$ pc) on the resolved star formation main sequence, which extends the
global relation down to sub-galactic scales. There is a clear distinction
between DSFG-1 and DSFG-3 vs. the more quiescent (or at least inside-out
quenching) host of SN H0pe, examined through Arc 2a. For the DSFGs, we even
observe a possible sequence at an elevated mode of star formation efficiency.

  * •

By extrapolating the model from Abdurro’uf & Akiyama (2018) for the evolution
of the rSFMS between $z\approx 1.8$ to 0 backwards to $z\approx 2$, we gain an
initial sense of what areal fractions of the two DSFGs and Arc 2a are
consistent with being above the rSFMS. For the DSFGs, these star-
forming/starburst regions account for an overwhelming portion of the area
($80-100\%$), in contrast with only $\sim 60\%$ for Arc 2a. This supports the
picture of many luminous DSFGs sustaining large-scale star-forming events,
which are difficult to fully explain through major mergers.

  * •

The two DSFGs occupy similar locations in the $UVJ$ plane, as do the loci of
their resolved, pixel-by-pixel colors. While not all regions are classified as
dusty star-forming, only a small minority would be classified as
$UVJ$-quiescent, further indicating the presence of large-scale galaxy-wide
star formation events in tandem with the rSFMS results.

  * •

For the more quiescent galaxy seen in Arc 2a (according to its global $UVJ$
color), which is possibly undergoing inside-out quenching, we find a higher
fraction of regions within the quiescent regime. Curiously, this object falls
primarily within the dusty star-forming region of the $UVJ$-plane, with only a
small area in the outskirts of the disk falling within the unobscured star-
forming regime.

  * •

Comparing $UVJ$ colors and SED-derived sSFRs indicates a possible breakdown of
the classical $UVJ$ selection on resolved scales. Moreover, we find evidence
for a substantial degree of dust attenuation for the low-sSFR regions,
including the region where SN H0pe is located. We suggest that this could be
the result of efficient radial dust transport within the galaxy.

  * •

In comparing the properties we derive for DSFG-1 and DSFG-3 by using global
galaxy-wide photometry vs. SED-fitting the spatially-resolved photometry and
summing the relevant properties, we find approximate agreement for both SFRs
and $M_{\star}$. As Arc 1a also has an H$\alpha$ detection reported by F24, we
also compare SFRHα with ${\rm SFR}_{\rm SED}$, finding very close agreement.
Comparing the attenuation $A_{V}$ found through SED fitting vs. through the
Balmer decrement, we reach excellent agreement for $A_{V}=1.5$, if we make the
assumption that $E(B-V)_{\rm stellar}=0.44\cdot E(B-V)_{\rm gas}$ (Calzetti
1997b; Calzetti et al. 2000).

  * •

Using the global properties of DSFG-1 and DSFG-3, we find that they lie
directly on the $\Delta_{\rm MS}=+0.6$ dex threshold above the star-forming
main sequence to be classified as starbursts. The location of DSFGs on the
SFMS is still subject to debate, but it is obvious that they cannot simply be
uniformly considered starbursts or high-$M_{\star}$ end main sequence
galaxies. In this case, we consider it perhaps notable that these DSFGs are
starbursting despite being largely disk-like and without particularly obvious
signs of merger activity, which is often responsible for a temporary elevation
in sSFR.

  * •

Employing our lens model for the purpose of estimating the magnifications and
time delays of the three images of SN H0pe, we find $\mu_{a}=6.7\pm 0.1$,
$\mu_{b}=9.8\pm 0.3$, and $\mu_{c}=8.9\pm 0.3$, which are in reasonable