11institutetext: Centro de Astrobiología, (CAB, CSIC–INTA), Departamento de
Astrofísica, Cra. de Ajalvir Km. 4, 28850 – Torrejón de Ardoz, Madrid, Spain
22institutetext: Kavli Institute for Cosmology, University of Cambridge,
Madingley Road, Cambridge, CB3 0HA, UK 33institutetext: Cavendish Laboratory -
Astrophysics Group, University of Cambridge, 19 JJ Thomson Avenue, Cambridge,
CB3 0HE, UK 44institutetext: Department of Physics, University of Oxford,
Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 55institutetext:
European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748, Garching,
Germany 66institutetext: Scuola Normale Superiore, Piazza dei Cavalieri 7,
I-56126 Pisa, Italy 77institutetext: Sorbonne Université, CNRS, UMR 7095,
Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
88institutetext: Cosmic Dawn Center (DAWN), Copenhagen, Denmark
99institutetext: Niels Bohr Institute, University of Copenhagen, Jagtvej 128,
DK-2200, Copenhagen, Denmark 1010institutetext: National Research Council of
Canada, Herzberg Astronomy & Astrophysics Research Centre, 5071 West Saanich
Road, Victoria, BC V9E 2E7, Canada 1111institutetext: European Space Agency,
c/o STScI, 3700 San Martin Drive, Baltimore, MD 21218, USA 1212institutetext:
European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino
Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain
1313institutetext: Department of Physics and Astronomy, University College
London, Gower Street, London WC1E 6BT, UK 1414institutetext: INAF -
Osservatorio Astrofisco di Arcetri, largo E. Fermi 5, 50127 Firenze, Italy
1515institutetext: AURA for European Space Agency, Space Telescope Science
Institute, 3700 San Martin Drive. Baltimore, MD, 21210, USA

# GA-NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpec

Santiago Arribas e-mail<EMAIL_ADDRESS>The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpec Michele Perna GA-NIFS: The core
of an extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpec Bruno Rodríguez Del Pino GA-
NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpec Isabella
Lamperti GA-NIFS: The core of an extremely massive proto-cluster at the Epoch
of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely
massive proto-cluster at the Epoch of Reionization probed with JWST/NIRSpec
Francesco D’Eugenio GA-NIFS: The core of an extremely massive proto-cluster at
the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpec Pablo G. Pérez-González GA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS:
The core of an extremely massive proto-cluster at the Epoch of Reionization
probed with JWST/NIRSpec Gareth C. Jones GA-NIFS: The core of an extremely
massive proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-
NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpec Alejandro Crespo GA-NIFS: The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpec Mirko Curti GA-NIFS: The core
of an extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpec Andrew J. Bunker GA-NIFS: The
core of an extremely massive proto-cluster at the Epoch of Reionization probed
with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at
the Epoch of Reionization probed with JWST/NIRSpec Stefano Carniani GA-NIFS:
The core of an extremely massive proto-cluster at the Epoch of Reionization
probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-
cluster at the Epoch of Reionization probed with JWST/NIRSpec Stéphane Charlot
GA-NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpec Peter
Jakobsen GA-NIFS: The core of an extremely massive proto-cluster at the Epoch
of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely
massive proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-
NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpec Roberto
Maiolino GA-NIFS: The core of an extremely massive proto-cluster at the Epoch
of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely
massive proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-
NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS:
The core of an extremely massive proto-cluster at the Epoch of Reionization
probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-
cluster at the Epoch of Reionization probed with JWST/NIRSpec Hannah Übler GA-
NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS:
The core of an extremely massive proto-cluster at the Epoch of Reionization
probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-
cluster at the Epoch of Reionization probed with JWST/NIRSpec Chris J. Willott
GA-NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpec Javier
Álvarez-Márquez GA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpec Torsten Böker GA-NIFS: The core of an extremely massive proto-
cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS: The core
of an extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpec Jacopo Chevallard GA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS:
The core of an extremely massive proto-cluster at the Epoch of Reionization
probed with JWST/NIRSpec Chiara Circosta GA-NIFS: The core of an extremely
massive proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-
NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpecGA-NIFS: The core of an extremely massive
proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS:
The core of an extremely massive proto-cluster at the Epoch of Reionization
probed with JWST/NIRSpec Giovanni Cresci GA-NIFS: The core of an extremely
massive proto-cluster at the Epoch of Reionization probed with JWST/NIRSpecGA-
NIFS: The core of an extremely massive proto-cluster at the Epoch of
Reionization probed with JWST/NIRSpec Nimisha Kumari GA-NIFS: The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpec Eleonora Parlanti GA-NIFS: The
core of an extremely massive proto-cluster at the Epoch of Reionization probed
with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at
the Epoch of Reionization probed with JWST/NIRSpec Jan Scholtz GA-NIFS: The
core of an extremely massive proto-cluster at the Epoch of Reionization probed
with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at
the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpec Giacomo Venturi GA-NIFS: The
core of an extremely massive proto-cluster at the Epoch of Reionization probed
with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at
the Epoch of Reionization probed with JWST/NIRSpec Joris Witstok GA-NIFS: The
core of an extremely massive proto-cluster at the Epoch of Reionization probed
with JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at
the Epoch of Reionization probed with JWST/NIRSpecGA-NIFS: The core of an
extremely massive proto-cluster at the Epoch of Reionization probed with
JWST/NIRSpecGA-NIFS: The core of an extremely massive proto-cluster at the
Epoch of Reionization probed with JWST/NIRSpec

(Received TBD; accepted TBD)

###### Abstract

Context.  The SPT0311-58 system resides in a massive dark matter halo at z
$\sim$ 6.9. It hosts two dusty galaxies (E and W) with a combined star
formation rate (SFR) of $\sim$ 3500 M⊙ yr-1, mostly obscured and identified by
their rest-frame IR emission. Its surrounding field exhibits an overdensity of
sub-mm sources, making it a candidate proto-cluster.

Aims. The goal is to characterize the environment and the properties of the
interstellar medium (ISM) within this extraordinary system.

Methods. We use spatially-resolved low ($R$=100) and high ($R$=2700)
resolution spectroscopy provided by the JWST/NIRSpec Integral Field Unit (IFU)
to probe a field of view (FoV) of $\sim$ 17 $\times$ 17 kpc2 around this
object, with a spatial resolution of $\sim$ 0.5 kpc.

Results. These observations have revealed ten new galaxies at z$\sim$ 6.9,
characterised by dynamical masses spanning from $\sim$109 to 1010 M⊙ and a
range in radial velocities of $\sim$ 1500 km s-1, in addition to the already
known E and W galaxies. The implied large number density ($\phi$ $\sim$
10${}^{4}\ $pMpc-3), and the wide spread in velocities, indicate that
SPT0311–58 is at the core of a proto-cluster, immersed in a very massive dark
matter halo of $\sim$ (6 $\pm$ 3) $\times$ 1012 M⊙. We study the dynamical
stage of the system and find that it likely is not fully virialized, although
most of the galaxies are gravitationally bound to the halo. The galaxies
exhibit a great diversity of properties. We derive their ongoing
H$\alpha$-based unobscured SFR, and find that its contribution to the total SF
varies significantly across the galaxies in the system. Their ionization
conditions range from those typical of galaxies at similar redshift recently
studied with JWST to those found in lower redshift objects, with
log([OIII]/H$\beta$) ranging from $\sim$ 0.25 to 1. The metallicity spans more
than 0.8 dex across the FoV, reaching in some cases nearly solar values. The
detailed IFU spectroscopy of the E galaxy reveals that it is actively
assembling its stellar mass, showing inhomogeneities in the ISM properties at
sub-kpc scales, and a metallicity gradient ($\sim$ 0.1 dex/kpc) that can be
explained by accretion of low metallicity gas from the intergalactic medium.
The kinematic maps indicate departures from regular rotation, high turbulence,
and a possible pre-collision minor merger.

Conclusions. These JWST/NIRSpec IFS observations confirm that SPT0311–58 is at
the core of a proto-cluster, embedded in an extremely massive dark matter halo
at the Epoch of Reionization. They also provide insights on the dynamical
status of the system, and unveil the diverse properties and evolutionary
stages of the galaxies residing in this unique environment.

###### Key Words.:

Galaxies: starburst - Galaxies: interactions - - Galaxies: kinematics and
dynamics - Galaxies: clusters - Galaxies: individual (SPT-S J031132-5823.4) -
Techniques: imaging spectroscopy

##  1 Introduction

The major role of dusty, star-forming galaxies (DSFGs) in the assembly of
stellar mass and the evolution of massive galaxies across cosmic time is now
well-established (e.g., Casey et al., 2014). DSFGs are important contributors
to the total star formation rate density (SFRD) in the Universe, accounting
for $\sim$ 20-40 $\%$ at redshifts 2-6 (e.g., Swinbank et al. 2014; Magnelli
et al. 2013; Casey et al. 2021). DSFGs are also thought to be the progenitors
of local massive ellipticals (Simpson et al., 2014, e.g.,) and quiescent
galaxies at intermediate redshifts (z $\sim$ 2-4; e.g., Toft et al., 2014,
Rizzo et al., 2021). Moreover, several studies have highlighted the importance
of DSFGs for identifying overabundances at high-z and studying the proto-
cluster phase, i.e., the stage when the quiescent early-type galaxies we
observe in clusters at low-z were actively forming stars (e.g., Casey et al.,
2014; Casey, 2016; Harikane et al., 2019; Wang et al., 2021a; Calvi et al.,
2023).

DSFGs exhibit infrared luminosities that, in some cases, exceed 1013 L⊙ (e.g.,
Rowan-Robinson, 2000, Riechers et al., 2013, Wang et al., 2021b, Gao et al.,
2021). These luminosities are the result of a massive star formation activity,
which can reach rates of up to several thousand solar masses per year. The
ultraviolet (UV) and optical radiation generated by these enormous star
formation rates is mostly absorbed by the large amount of dust contained in
these objects, and re-emitted as thermal radiation at far-infrared (FIR) and
sub-millimeter (sub-mm) wavelengths. Hence, despite their extreme
luminosities, DSFGs are faint at rest-frame UV and optical wavelengths, and
therefore they have been almost entirely missed by imaging surveys carried out
with the HST and ground-based optical telescopes. DSFGs have been, however,
discovered and characterised in large numbers up to intermediate redshifts
thanks to infrared space satellites and a series of ground-based (sub-)mm
facilities (see Casey et al., 2014 for a review). Moreover, in recent years
high-resolution ALMA observations have enabled the characterisation of the
internal structure of sources at very early cosmic epochs (z $\sim$ 5$-$9,
e.g., Bowler et al., 2018, Hashimoto et al., 2019, Bouwens et al., 2020).

The advent of the James Webb Space Telescope (JWST; Gardner et al., 2023),
with its orders of magnitude improvement in sensitivity and enhanced angular
resolution in near- and mid-infrared wavelengths, is now opening the
opportunity to study the nature of high-z DSFGs in unprecedented detail. In
fact, recent JWST studies based on NIRCam and MIRI imaging are unveiling the
properties of this population and its contribution to the star formation
activity at cosmic dawn (e.g., Pérez-González et al., 2023, Zavala et al.,
2023, Akins et al., 2023, Barrufet et al., 2023). Moreover, JWST enables the
study of the stellar component and ionised gas in these objects by
spectroscopically probing the rest-frame UV-optical spectral range.

Here we present JWST Near Infrared Spectrograph (NIRSpec) Integral Field Unit
(IFU) observations of SPT-S J031132-5823.4 (hereafter SPT0311–58) at z=6.90,
the most distant and massive DSFG known (Strandet et al., 2017). ALMA
observations by Marrone et al. (2018) have revealed that this mildly lensed
system ($\mu$ $\sim$ 1.3-2.2) consists of a pair of massive dusty galaxies
separated by a projected distance of 8 kpc, that form stars at extreme rates
(SFR = 540 and 2900 M⊙ yr-1) and contain enormous amounts of dust (Mdust
$\sim$ 1-3 $\times$ 109 M⊙), and gas (Mgas $\sim$ 1-3 $\times$ 1011 M⊙; see
also Jarugula et al., 2021, Witstok et al., 2023). These galaxies have a
clumpy structure and turbulent kinematics (Spilker et al., 2022; Álvarez-
Márquez et al., 2023), and are undergoing fast metal enrichment (Litke et al.,
2023). Marrone et al. (2018) also show that SPT0311–58 resides in one of the
most massive dark matter halos that are expected to exist at z$\sim$ 7\. Due
to its extreme luminosity at sub-mm wavelengths, SPT0311–58 was also included
in the study by Wang et al. (2021a), who searched for overdensities in the
fields around the brightest high-redshift sources discovered in the SPT 2500
deg2 survey (Vieira et al., 2010, Mocanu et al., 2013, Everett et al., 2020).
This study found that the surroundings of SPT0311–58 (i.e., R $\lesssim$ 1.3
Mpc) have a significant overdensity of sub-mm sources and a compacted central
distribution, suggesting that it could potentially be at the centre of one of
the most distant proto-clusters identified so far. In summary, all these
previous works have established that SPT0311-58 is indeed an extraordinary
system that can be seen as an ideal laboratory to study galaxy evolution in a
massive halo at the Epoch of Reionization (EoR).

The present JWST/NIRSpec IFU observations cover the rest-frame UV wavelengths
and main optical emission lines of SPT0311–58 (i.e., $\sim$ 0.13 – 0.67
$\mu$m). This allows us to characterise the physical and kinematic properties
of the warm ionized gas component of its ISM at sub-kpc scales (i.e., $\sim$
0.5 kpc) over a field of view of $\sim 17\times 17$ kpc2. Being at z=6.9,
SPT0311–58 is at about the highest redshift for which H$\alpha$ can be
observed with the high angular resolution and sensitivity provided by
JWST/NIRSpec.

This paper is organised as follows. In Section 2, the observations and
reductions are described. Section 3 is devoted to the identification and
spectral characterisation of a group of ten newly discovered z $\sim$ 6.9
galaxies around the two already known massive galaxies in SPT0311–58, together
with other lower z sources also found in the field of view (FoV). Section 4 is
focused on the ISM conditions, including the ionization and metallicity
properties. We also obtain and discuss the ongoing H$\alpha$-based
(unobscured) star formation. In Sect. 5, we discuss the kinematic properties
of the system. In light of the results of this study, in Sect. 6, we discuss
the proto-cluster scenario for SPT0311–58. We also discuss the physical,
chemical and kinematic properties of the E galaxy, for which the IFU data give
detailed 2D information. Finally, in Sect. 7, we summarise the main results of
this study. In Appendix A, details on the spectral analysis are included.
Appendix B presents a new gravitational lens model for the SPT0311-58 system,
making use of the new information provided by these IFU data.

In this work we adopt the cosmological parameters from Planck Collaboration et
al. (2016): $H_{0}$ = 67.7 km s-1Mpc-1, $\Omega_{m}$ = 0.307, and
$\Omega_{\Lambda}$ = 0.691, which provide a scale of 5.41
$\mathrm{kpc}/\arcsec$ at $z=$6.90. Unless otherwise stated, we use proper
distances throughout the paper. We refer to the two main galaxies in
SPT0311-58 as W and E, according to their relative position on the sky. We
adopt a Kroupa Initial Mass Function (IMF) (Kroupa, 2001).

##  2 Observations and Reductions

###  2.1 SPT0311-58 observations

The present work is part of the GA-NIFS (Galaxy Assembly with NIRSpec Integral
Field Spectroscopy), a NIRSpec GTO program that aims at studying the internal
structure of a sample of $\sim$ 55 galaxies and active galactic nuclei (AGN)
with redshifts between 2 and 11 by means of spatially-resolved spectroscopy,
obtained with IFS mode (Böker et al., 2022) of the NIRSpec instrument
(Jakobsen et al., 2022). Details on the GA-NIFS program can be found in Perna
(2023). Among other subsamples, the program includes three distant (i.e., z
$>$ 4) infrared-selected dusty star-forming galaxies: HLFS3 (Jones et al.,
2023), GN20 (Übler et al. 2023, in prep.), and SPT0311-58, subject of the
present study.

The NIRSpec-IFU observations of SPT0311-58 were executed in November 20, 2022,
and are specified in PID1264 (PI: L. Colina). These NIRSpec observations were
combined in a single proposal with independent MIRI observations of the same
target (presented in Álvarez-Márquez et al., 2023) with the aim of saving the
telescope slew overhead. The NIRSpec observations include two different
spectral configurations: low-resolution (R100) spectroscopy with the prism
covering the full 0.6–5.3 $\mu$m range, and high-resolution (R2700)
spectroscopy with the grating/filter combination G395H/F290LP which covers
from 2.8 to 5.27 $\mu$m, though due to the gap between the two detectors the
$\sim$ 3.98–4.20 $\mu$m range is not observed at R2700 (Böker et al., 2022).
The FoV is $\sim$ 3′′ $\times$ 3′′ (i.e., $\sim$ 17 $\times$ 17 kpc2 at the
redshift of SPT0311–58), and the native spaxel size 0.1′′ (0.54 kpc).

We constrained the allowed position angle range for the observations to
minimise the presence of bright sources in the MSA FoV that could leak light
and contaminate the IFU spectra. We were primarily concerned about this
potential problem for the R100 spectra, and therefore we protected the region
of the detector occupied by these spectra. For the high-resolution spectra,
MSA-leakage is not so critical as our main goal with this mode is to analyse
emission lines.

We use the first four positions of the medium cycling dither pattern, which
provide offsets large enough to deal with the failed open micro-shutters and
other sources of background, while keeping a large effective FoV with the
complete exposure time. Also, this choice minimizes the number of individual
exposures, and therefore also the (dominant) detector noise.

No dedicated background exposures were obtained, as it was expected that a
substantial number of spaxels were free from galaxy emission and suitable for
deriving the background spectrum.

The detector was configured with the improved reference sampling and
subtraction pattern (IRS2), which significantly reduces the readout noise with
respect to the conventional method (Rauscher et al., 2017). In particular,
NRSIRS2RAPID (60 groups) and NRSIRS2 (25 groups) options were respectively
selected for R100 and R2700 in a single integration. The different temporal
sampling between the two spectral configurations was motivated by data volume
constraints. This gives a total on-source integration time of 3559 and 7352 s
for R100 and R2700, respectively.

###  2.2 NIRSpec IFU data reduction

The data reduction was performed with the JWST calibration pipeline (Bushouse
et al., 2023) version 1.8.2, using the context file jwst_1068.pmap. We follow
the general procedure to correct detector level defects (Stage 1), calibrate
the data (Stage 2), and combine the individual exposures and build the
datacube (Stage 3). However, several modifications to the default reduction
steps were required to enhance data quality. These modifications, described in
detail in Perna et al. (2023), are briefly summarized here.

  * •

The individual count-rate frames were further processed at the end of Stage 1
to correct for the different zero-level in the individual (dither) frames and
subtract the vertical stripes induced by the 1/f noise.

  * •

We masked the regions affected by leakage from failed open microshutters,
including not only the pixels flagged as ‘MSA$\\_$FAILED$\\_$OPEN’, but also
others not correctly associated with such leakage in the context file.

  * •

In Stage 2, we masked detector pixels corresponding to the edges of the IFU
slices with unreliable sflat corrections, as they cause evident artefacts
(stripes) in the final data cube.

  * •

Outliers were flagged and removed on the individual 2D exposures at the end of
Stage 2, using an algorithm similar to lacosmic (van Dokkum, 2001). We
calculated the derivative of the count-rate maps along the (approximate)
dispersion direction, and rejected all measurements above a given percentile
of the resulting distribution (D’Eugenio et al., 2023). This threshold value
was optimised after inspecting the products of the different realisations. We
found that 98th and 99.5th percentiles were appropriate in our case for the
R100 and R2700 data, respectively.

  * •

After experimenting with the different methods and scale options, we generated
the final datacubes with the drizzle weighting method and a spaxel scale of
0.05″in the cube_build step.

Figure 1: Left: Image generated from the NIRSpec R100 datacube, integrating
over the spectral range 1.1-1.4 $\mu$m. The color bar is in units of 10-19 erg
s-1cm-2 per spatial pixel. Right: HST image obtained with WFC3 and F125W
filter resampled at the pixel size used with NIRSpec (i.e., 0.050 ″). The
inset zooms in on the region around the emission peak used to align the
NIRSpec cube (red isophotes) with the image represented with an unsaturated
color code. Here, and in all the spectral maps presented in this work, the
(0,0) corresponds to coordinates $\alpha$ = 03h 11m 33s.248 and $\delta$ =
–58º 23’ 33”.24 (white cross). In the image the peak of the lens galaxy is at
relative coordinates (–0.64 ″, –0.29 ″). The green lines indicate the border
of the NIRSpec IFU FoV. The black and white circles in the bottom-left corners
of the panels represent the FWHW of the respective PSFs.

###  2.3 Astrometric registration

In order to set the reduced data cube coordinates onto the Gaia reference
system, we applied the following two-step procedure. Firstly, we aligned the
IFU data with the HST WFC3 images. With this goal, we generate an image over a
band-pass similar to F125W by collapsing the R100 data-cube in the spectral
range 1.1-1.4 $\mu$m (see Fig. 1). Then we obtained the coordinate offsets to
be applied to align this image with the F125W HST image, taking as reference
the flux peak of the bright foreground galaxy (see inset in Fig. 1). We
estimate the uncertainties associated with this step to be $\lesssim$ 70 mas
on each axis. In this figure we can appreciate the slightly better resolution
provided by NIRSpec IFU with respect to HST at these wavelengths, as expected
from their respective PSFs (NIRSpec: D’Eugenio et al., 2023; WFC3: Anderson et
al. 2023 111https://hst-docs.stsci.edu/wfc3ihb.) Secondly, the absolute
astrometry of the HST/WFC3 F125W image was derived by matching the position of
four stars present in the image FoV with the Gaia DR3 catalogue (Gaia
Collaboration et al., 2022). The absolute sky coordinates (ICRS) of the center
of the FoV (i.e., spaxel [55,55]) are $\alpha$ = 03h 11m 33s.248 and $\delta$
= –58º 23’ 33”.24, and correspond to (0,0) in Fig. 1, and other spectral maps
in the paper. The uncertainty in the astrometric alignment is 17 mas in R.A.
and 30 mas in Dec. The total offsets applied were 0.104 and 0.179 arcsec in
R.A. and Dec., respectively, and the absolute coordinates uncertainty, taking
into account both steps, are estimated to be $\sim$ 80 mas in each direction.

We checked that the images generated from the R100 and R2700 data-cubes were
consistently aligned with each other. In fact, a 2D Gaussian fit to the
brightest region of the lens galaxy in a common wavelength range led to
differences smaller than a fraction of a spaxel (i.e., $<$ 10 mas).

###  2.4 Background

For the R100 datacube, the background spectrum was obtained from the median of
about 1550 spectra free from emission coming from SPT0311–58, the lens galaxy,
or any other object in the FoV. These spaxels were identified by inspecting
the continuum and the line maps generated across the whole R100 wavelength
range (Sect. 3 and Fig. 2) to make sure that neither continuum nor line
emission from the galaxies was included. The area covered by the background
spaxels is shown in Fig. 14.

We report here that the individual R100 spectra presented a strong feature at
1.083 $\mu$m, whose origin is unknown. This feature is found over the whole
FoV of the IFU with a uniform intensity. It is visible in the spaxels used to
obtain the background spectrum (see Fig. 4 bottom panel). It is already
present in the raw data, and also in the reference file
nirspec$\\_$sflat$\\_$0185. As for the present analysis, we note that this
emission is removed in the background subtraction step.

For the R2700 spectra, we do not subtract the sky background. These high-
resolution data are only used for the study of the emission lines and the mean
continuum is removed during the fitting process (see Appendix A).

###  2.5 Ancillary data

In addition to the NIRSpec IFU data presented above, in this study we have
made use of additional data that are briefly described below.

  * •

JWST/NIRSpec IFU: Calibration star: To derive aperture corrections (Appendix
A.2) and the FHWM of the PSF, we used the high-resolution G395H/290LP
observations of the standard star 1808347 (PID:1128, Obs. 9; PI: N.
Luetzgendorf), which were reduced with the same pipeline version and context
file as the data for SPT0311-58.

  * •

HST imaging: HST imaging of SPT0311–58 obtained with the WFC3 (filter F125W)
was used to set the coordinate system of the present NIRSpec data into Gaia
(see Sect. 2.3). The image was directly downloaded from MAST (PID14740; PI: D.
Marrone).

  * •

ALMA: We used ALMA band 6 observations of the [C ii]158$\mu m$ line and the
adjacent continuum from programs 2016.1.01293.S and 2017.1.01423.S (P.I. D.
Marrone). These data have been presented in Marrone et al. (2018) and Spilker
et al. (2022). We requested the calibrated measurement sets (MS) from the EU
ALMA Regional Centre (ARC). Data were calibrated with the standard pipeline
procedure. To analyse the data, we used the ALMA data reduction software CASA
v6.2.1 (CASA Team et al., 2022).

We created a map of the continuum combining two line-free spectral windows,
obtaining a synthesized beam with FWHM of $0.09\times 0.08$ arcsec2 and a
sensitivity of $9.4$ $\mu$Jy beam-1. As for the [C ii] data-cube, the
synthesized beam FWHM is $0.08\times 0.07$ arcsec2 with a sensitivity of $\sim
150$ $\mu$Jy beam-1 in a 20 km s-1channel. The [C ii] moment 0 map (intensity
map) was generated by integrating the continuum subtracted data-cube in the
spectral channels from -1000 km s-1 to 1200 km s-1 with respect to $z=6.900$.
Before producing the moment 1 and moment 2 maps, the spaxels with low signal-
to-noise ratio (S/N¡ 2.3) and those with spurious emission in each 20 km s-1
velocity channel were masked.

Finally, the continuum image and [C ii] data-cube were re-sampled to a spaxel
size of 0.05” to match the size of the NIRSpec spaxels using the CASA task
imregrid.

  * •

JWST/MIRI: The image of STP0311-58 obtained with JWST/MIRI at 10 $\mu$m and
published by Álvarez-Márquez et al. (2023) was kindly provided to us by the
MIRI GTO team. We used this image to compare the morphology observed with
NIRSpec with that at longer wavelengths.

Figure 2: (a): NIRSpec image generated collapsing the R100 datacube in the
3.0-3.8 $\mu$m spectral range (see text). The continuum emitting sources,
whose spectra indicate they are at $z$ $\sim$ 6.9, are identified with yellow
labels (i.e., C1, C2, C3, E). In the E galaxy we distinguish several sub-
regions (ES, ENa,b,c). Sources that according to their spectra are at lower z
(i.e., lz1, lz2, lz3) are labeled in cyan, along with the foreground lens
galaxy. The red iso-contours (in red) correspond to 5, 10, 15, 30, and 100
$\sigma$ values. (b): [OIII] emission image at z $\sim$ 6.9 obtained from the
R100 datacube (see text for specific spectral ranges). Six line-emitting
sources that were not detected in the continuum map (panel a), are identified
in this map as z $\sim$ 6.9 sources and labeled in yellow (L1 to L6). White
iso-lines corresponds to 3, 5, 10, and 15 $\sigma$. Here, and in panels c, d,
and e, the red-contours are the same as in panel a, after removing the ones
for the lens galaxy for clarity. (c): Same as panel (b) for the H$\alpha$
emission. (d): MIRI image at 10 $\mu$m, after subtracting the lens (Álvarez-
Márquez et al., 2023). (e): ALMA continuum at 160 $\mu$m. We mark clumps E1 –
E4 identified by Spilker et al. (2022). (f): ALMA map of the [CII]$\lambda$158
line. The white isolines correspond to the H$\alpha$ emission. Fig. 3 identify
the targets and their apertures ( for details see Appendix A). The green
straight lines in all the pannels indicate the border of the NIRSpec IFU FoV.
Figure 3: Identification of the different targets and regions in the FoV of
SPT0311–58, indicating the apertures used to extract their spectra. The $\sim$
6.9 sources are coded in red (with the sub-regions within E in orange), and
the low-z galaxies in blue. The W galaxy is not detected at the resolution of
the IFU, and is shown with a dashed line. More details about the apertures in
Appendix A.

##  3 Field characterization: discovery of 10 new z$\sim$6.9 galaxies (and 3
low-z objects)

The field around SPT0311-58 is very complex, as it includes a large and bright
lens galaxy, the two previously identified main galaxies (i.e., E and W), ten
additional newly discovered galaxies at z $\sim$ 6.9, as well as other three
lower redshift objects. In this section, we identify all these sources in the
FoV through spectral imaging with the NIRSpec R100 data cube. We then present
their R2700 and R100 spectra, which allow us to confirm their redshifts. We
also briefly compare the NIRSpec images with those obtained with MIRI and ALMA
at longer wavelengths.

###  3.1 NIRSpec R100 spectral imaging: Identification of continuum and line
emitting sources at z$\sim$ 6.9

Fig. 2 (a) shows the image obtained from the R100 data-cube after integrating
over the spectral range 3.0 – 3.8 $\mu$m. For z $\sim$ 6.9 sources, this
spectral range is dominated by the stellar continuum, as it does not include
prominent lines (i.e., rest-frame $\sim$ 0.38 – 0.48 $\mu$m). The image shows
the very bright lens galaxy, the E galaxy, and several continuum-emitting
galaxies labeled as lz1, lz2, lz3, C1, C2, C3. The structure of the E galaxy
is clearly resolved, allowing us to distinguish the south and north regions
(ES and EN, respectively), which are also clearly seen in Fig. 1. We also
define sub-regions within EN with lower case letters (i.e., a, b, c, from
south to north). As we will see below (Sect. 3.3.1), the spectra of the
sources lz1, lz2, and lz3 indicate that they are at lower z than SPT0311–58
(i.e., z ¡ 3, hence the ‘lz’ identification label). Together with the lens
galaxy, all these low-z objects are labeled in cyan in the figure. One of
these low-z objects (lz3) partially overlaps with ES. The spectra of sources
C1, C2, and C3 (Sect. 3.3.2) indicate that they are at z$\sim$ 6.9, and
together with other sources at this redshift (i.e., E and its substructures)
they are labeled in yellow in Fig. 2 (a). To ease the identification of
sources detected in the field we have included in Fig. 3 the apertures used
for extracting their spectra.

Fig. 2 (b) and (c) presents the [OIII]$\lambda$$\lambda$4959,5008 and
H$\alpha$ (+[NII]$\lambda$$\lambda$6548,6584) emission line maps, obtained by
integrating the spectral ranges that cover these lines at a redshift of $\sim$
6.9, and subtracting a continuum measured close to the lines222Specifically
the selected ranges are: 5.00 – 5.15 $\mu$m for H$\alpha$ continuum and 5.17 –
5.23 $\mu$m for the H$\alpha$ line, and 3.72 – 3.82 $\mu$m for the [OIII]
continuum and 3.90 – 4.00 $\mu$m for the [OIII]$\lambda$$\lambda$4959,5008
lines.. Hence, these images allow us to identify line-emitting sources at that
particular redshift. These images reveal a total of nine line-emitters: the
continuum-detected C1, C2, C3 and six new sources L1-L6 (see also Fig. 3). The
detection of several emission lines in the spectra of all these sources
confirmed that they are at a redshift of $\sim$ 6.9 (see below in Sect. 3.3.2
).

We note the morphological differences between the continuum and the line maps.
For instance, some bright line emitting sources like L1, and L5 are barely
detected in the continuum. [OIII] and H$\alpha$ images also show distinct
morphology. The E galaxy is more extended in H$\alpha$ than in [OIII], and
shows a faint diffuse component towards the north-west. The [OIII] shows a
tail towards the south-west, which is not seen in H$\alpha$. Source C3 is much
brighter in H$\alpha$ than in [OIII], and to the north there is also a zone of
faint extended H$\alpha$ emission.

The top panels in Fig. 2 show that the main structure of the W galaxy is not
detected at spaxel level in the NIRSpec images (continuum, H$\alpha$, or
[OIII]). However, as shown in Sect. 3.3.2, the R100 spectrum obtained
integrating over a large aperture enables us to recover its H$\alpha$ flux.

###  3.2 Morphological comparison with MIRI and ALMA imaging

Fig.2(d) shows the 10 $\mu$m image obtained with JWST/MIRI, after subtracting
the lens foreground galaxy (Álvarez-Márquez et al. (2023)). At z=6.9, this
image traces the stellar-continuum at the near-IR rest-frame (i.e., 1.26
$\mu$m). The comparison with the NIRSpec 3.0-3.8 $\mu$m image (Fig.2(a), rest-
frame $\sim$ 0.43 $\mu$m) shows clear differences. Although in the central
regions of the E galaxy the bulk of the NIRSpec emission approximately matches
the MIRI image, in the northern zone the mid-infrared emission seems more
extended to the west. The emission detected with MIRI in the W galaxy has no
NIRSpec counterpart. Conversely, none of the new sources at z $\sim$ 6.9
discovered with NIRSpec were reported by Álvarez-Márquez et al. (2023) as MIRI
emitters (we note, however, that C3 and L4 may have faint low S/N emission at
10 $\mu$m).

The NIRSpec and ALMA continuum images also show very different structures. The
W galaxy, very bright at 160 $\mu$m (Fig. 2(e)), is not seen by NIRSpec in the
3.0-3.8 $\mu$m map (Fig. 2(a)). The NIRSpec region ENa and the ALMA clump E3
are in positional agreement within $\sim$ 0.1 arcsec. None of the other
NIRSpec sources at z $\sim$ 6.9 have ALMA continuum emission.

As for the line emission maps, the main body of E in H$\alpha$ and [OIII]
(Fig. 2(b,c)) approximately matches the [CII]$\lambda$158 $\mu$m emission
(Fig. 2(f)). Regarding the W galaxy, the bright [CII] regions are not seen
with NIRSpec, except at C3. The NIRSpec newly discovered sources at z $\sim$
6.90 are not detected with ALMA at [CII]$\lambda$158 $\mu$m.

###  3.3 Spectral characterization of SPT0311-58

To study the spectral properties in the complex FoV of SPT0311–58, we have
followed different approaches. In this Section, we first present the
integrated spectra extracted from the data-cubes for the sources and regions
identified above. For details on the spectral analysis see Appendix A.1. As we
will see below, these spectra show that three objects (i.e., lz1, lz2, lz3)
are at low redshift, while confirming that nine newly-discovered sources are
at z $\sim$ 6.9.

Later in this section we perform a spatially resolved analysis in the
brightest regions of the E galaxy, using the R2700 data. This analysis reveals
an additional distinct galaxy at z$\sim$ 6.90 overlapping along the line of
sight of E. In summary, we have found ten galaxies at redshift $\sim$ 6.9
(nine sources identified through R100 spectral imaging, and one through the
analysis of the R2700 data), in addition to the already known E and W
galaxies.

####  3.3.1 Spectra of the lens, and other low-z galaxies

The present data have allowed us to determine accurately the redshift of the
foreground lens galaxy (i.e., z= 1.0343 $\pm$ 0.0002) via the identification
of Pa$\alpha$ in the R2700 spectrum, hence improving previous photometric
determinations (zphot=1.43${}_{-0.35}^{+0.42}$, Marrone et al., 2018).
H$\alpha$ and Pa$\alpha$ are also clearly visible in the R100 spectrum shown
in Fig. 4. Using this new redshift, together with a panchromatic image (i.e.,
rest-frame $\sim$ 0.49 - 2.45 $\mu$m) obtained from the R100 datacube, in
Appendix B, we present a new model for the lens galaxy.

Fig. 4 also presents the R100 spectra of lz1, lz2, and lz3. lz1 shows very
prominent emission lines, which are consistent with H$\alpha$,
[OIII]$\lambda$4959,5007 and [OII]$\lambda$3726,3728 for a galaxy at z=2.576
$\pm$ 0.002. lz2 and lz3 have no emission lines, so their redshift
determination is less straightforward. For them we infer the redshifts by
fitting the R100 spectra with EAZY (Brammer et al., 2008), and find zph(lz2)=
2.149${}_{-0.074}^{+0.175}$, and zph(lz3)= 1.678${}_{-0.031}^{+0.032}$ (see
also insets in Fig. 4). Therefore, together with the large foreground galaxy,
lz1, lz2 and lz3 do not form part of the high-z SPT0311-58 system, and they
will not be discussed further in the remainder of the paper.

Figure 4: R100 spectra of three newly identified galaxies with z ¡ 3 found in
the FoV of SPT0311-58, together with the bright foreground (lens) galaxy. The
labels indicate their redshifts (see text). For lz2 and lz3 the insets show
the dependence of fit residuals (reduced $\chi$2) with EAZY (Brammer et al.,
2008) as a function of the redshift. The horizontal blue line in the inset
marks $\chi$${}^{2}_{red}$ = 1. The sky background spectrum inferred from
$\sim$ 1550 spaxels free from galaxy emission is also shown in the bottom
panel.

####  3.3.2 Spectra of the 9 sources at z $\sim$ 6.9 identified with R100
spectral imaging

The R100 and R2700 integrated spectra for the high-z sources are presented in
Fig. 5 and Fig. 15, respectively. These spectra show several emission lines
confirming that these sources are undoubtedly at redshifts around 6.90 (for
the specific redshift values see Table 2). In particular, most of the spectra
include H$\alpha$, [NII]$\lambda$$\lambda$6548,6584,
[OIII]$\lambda$$\lambda$4959,5008, and H$\beta$.
[OII]$\lambda$$\lambda$3726,3728 (unresolved) is also present in the spectra
of the brightest regions of E.

Figure 5: R100 spectra for the different z $\sim$ 6.9 sources and regions
identified in the spectral images presented in Figure 2. The apertures used
are shown in Fig. 3, and further details in Appendix A. The W spectrum is
contaminated by the foreground lens galaxy continuum and Pa$\alpha$ line
(marked in red), which is close to H$\beta$ of W (see the lens spectrum in
Fig. 4).The black vertical dashed lines mark the expected position of the
corresponding lines for a z$\sim$ 6.92.

For the W galaxy, the extraction aperture overlaps with the foreground lens
galaxy (see Appendix A.1), and therefore the resulting integrated spectrum
includes emission from this bright source (i.e., continuum, and Pa$\alpha$ and
H$\alpha$ redshifted for z $\sim$ 1.0343). Still, this spectrum clearly shows
H$\alpha$ at a redshift of $\sim$ 6.90 (Fig. 5). The line profiles for W and E
in the R2700 spectra are largely oversampled due to the line broadening
induced by the wide range in velocities covered by the aperture, and therefore
their S/N is lower than for the R100 spectra.

For the E and W galaxies, we have also obtained the spectra for their external
extended emission using the Eext and Wext apertures defined in Appendix A.

The line fluxes derived from these spectra are given in Table 4. Further
details about the methods followed can be found in Appendix A.

Figure 6: R2700 Spectrum in the H$\beta$-[OIII] (top) and H$\alpha$-[NII]
(bottom) spectral regions showing the presence of double line profiles
corresponding to two kinematically distinct ionised gas components associated
to the E galaxy, and the fainter and redshifted L7 source. The line profiles
are modelled with two Gaussians, distinguished with green and magenta lines
for E and L7, respectively. The total model profile is displayed in red.
Figure 7: Moments 0, 1, and 2 maps for the E galaxy (upper) and the L7 source
(bottom), after the kinematic decomposition (see Sect. 3.3.3). From left to
right, [OIII] flux map (in units of 10-18 erg s-1 cm-2), velocity field (km
s-1), and intrinsic velocity dispersion ($\sigma_{0}$) map (km s-1).

####  3.3.3 Identification of L7 at z $\sim$ 6.9 via 2D kinematic
decomposition

In addition to the nine z $\sim$ 6.9 sources identified with R100 spectral
imaging and confirmed spectroscopically above, we found evidence for a 10th
source overlapping with the main structure of the E galaxy. In fact, some
spectra from individual spaxels close to the central region of the E galaxy
show evidence for a secondary gaseous component on top of the systemic one
(see Fig. 6). Hence we performed a 2-Gaussian model decomposition of these two
kinematically distinct components by comparing the Bayesian Information
Criterion (BIC) of fits performed with one and two components (Perna et al.,
2022). We also considered the possible presence of a third kinematic component
that could be associated with, e.g., an outflow, but the BIC analysis did not
support this possibility. Fig. 7 presents the corresponding moment 0, 1, and 2
maps for the two components for the [OIII] line.

The secondary component (i.e., L7, hereafter) is relatively faint and compact
and it is located in projection about 0.15 arcsec (i.e., 0.8 kpc) to the south
east from the peak of the main systemic component (i.e., E). Its radial
velocity is offset by $\sim$ 300 km s-1with respect to E, hence allowing us a
secure kinematic decomposition between the two sources. The fact the L7 is
redshifted with respect to E excludes the possibility that the detected
emission is due to an outflow. As we will see in Sect. 4.4 and 5.1, the
properties of L7 are similar to those of the other new sources discovered at z
$\sim$ 6.9. The kinematic properties of the E and L7 galaxy are discussed in
Sec. 5.

###  3.4 Summary of sources in the FoV of SPT0311-58

We list below the various sources found in the field of view:

  * •

Lens galaxy (z= 1.0343).

  * •

Three low-z galaxies: lz1 (z=2.576$\pm$0.002), lz2 (zph=2.149), and lz3
(zph=1.678).

  * •

Nine z$\sim$6.9 galaxies identified via R100 spectral imaging (C1, C2, C3, L1,
L2, L3, L4, L5, L6).

  * •

One z$\sim$6.9 galaxy identified through 2D spatially resolved spectroscopy
(L7).

  * •

E galaxy, and its regions ES and EN (a, b, and c).

  * •

W galaxy

In addition, we have obtained the spectra for the extended diffuse emission in
the external regions of E and W. For all these sources we have presented the
integrated spectra in the sections above, except for the one identified via 2D
spectroscopy (L7). For this object, we give the Gaussian model after fitting
two kinematically distinct components to the spectrum (see Fig. 6).

Table 1: ISM properties, star formation, and stellar masses for sources in
SPT0311–58

ID | H$\alpha$ flux | SFR(H$\alpha$) | $A_{V}$ | SFR(H$\alpha$) (dust-corr.) | 12+log(O/H) | $M_{\star}$
---|---|---|---|---|---|---
| (10-18 ergs-1cm-2) | (M⊙ yr-1) | (mag.) | (M⊙ yr-1) |  | ($\times 10^{9}$ M⊙)
(1) | (2) | (3) | (4) | (5) | (6) | (7)
E | 18.8 $\pm$ 0.8 | 62\. $\pm$ 2. | 1.7 $\pm$ 0.3 | 231\. $\pm$ 10. | 8.37 $\pm$ 0.06 | 39.8 $\pm$ 6.9
|  |  | 2.4 $\pm$ 0.31 | 377\. $\pm$ 20.2 |  |
Eext | 11 $\pm$ 2. | 32\. $\pm$ 3. | (0.85) | (61.)3 |  |
W | 4.5 $\pm$ 0.9 | 8\. $\pm$ 1. | 5.8 $\pm$ 0.61 | 603\. $\pm$ 1212 |  | (80. $\pm$24.)5
Wext | 5.3 $\pm$ 1.4 | 9\. $\pm$ 1. | (2.00) | (39.)3 |  |
C1 | 1.5 $\pm$ 0.3 | 5.5 $\pm$ 1. | 0.7 $\pm$ 0.6 | 9.2 $\pm$ 1.8 | 8.37 $\pm$ 0.10 | 1.31 $\pm$ 0.42
C2 | 0.92 $\pm$ 0.25 | 2\. $\pm$ 0.4 | 1.0 $\pm$ 0.9 | 4.2 $\pm$ 1.1 | 7.97 $\pm$ 0.20 | 3.50 $\pm$ 0.41
C33 | 1.3 $\pm$ 0.4 | 6.2 $\pm$ 1.2 | 2.2 $\pm$ 1.6 | 33\. $\pm$ 10. | 8.56 $\pm$ 0.10 | 2.84 $\pm$ 0.67
L1 | 1.2 $\pm$ 0.3 | 2.9 $\pm$ 0.4 | 1.9 $\pm$ 0.8 | 12\. $\pm$ 3. | 7.86 $\pm$ 0.16 | 0.40 $\pm$ 0.06
L2 |  |  |  |  |  | 0.34 $\pm$ 0.04
L33 | 0.47 $\pm$ 0.25 | 1.2 $\pm$ 0.4 | (0.85) | (2.3)3 |  | 1.46 $\pm$ 0.20
L4 | 0.64 $\pm$ 0.22 | 1.1 $\pm$ 0.3 | (0.85) | (2.1)3 |  | 1.15 $\pm$ 0.13
L53 | 1.6 $\pm$ 0.4 | 3.9 $\pm$ 0.6 | 0.8 $\pm$ 0.6 | 7.0 $\pm$ 1.7 | 7.79 $\pm$ 0.35 | 1.13 $\pm$ 0.13
L6 | 0.49 $\pm$ 0.18 | 1.2 $\pm$ 0.3 | (0.85) | (2.0)3 |  | 0.58 $\pm$ 0.09
L74 | 0.85 $\pm$ 0.20 | 1.1 $\pm$ 0.3 | (0.85) | (2.0)3 | (7.58 $\pm$0.46) |

333Column 1: Source identification. Column 2: Observed H$\alpha$ flux obtained
by averaging values from the R2700 and R100 spectra, if S/N $>$ 3\. The R100
fluxes were reduced by 10% to account for a known calibration factor (see
Appendix A). For Eext, W and Wext, only R100 data were used due to the low S/N
of their R2700 spectra. In addition, for Eext the fits could not resolve the
H$\alpha$ and the [NII] lines, and thus an average factor of 0.77 was applied
to account for the [NII] emission when deriving the corresponding H$\alpha$
flux. Further details are provided in Appendix A. Column 3: SFRs obtained with
the relation of Kennicutt & Evans (2012). No dust correction is applied. For
galaxies C4 and L1-L7 values have been corrected by a factor 0.5 to account
for their metallicity (see text). Column 4: Visual attenuation inferred from
the Balmer decrement (H$\alpha$/H$\beta$). For E and W the values inferred
from Pa$\alpha$/H$\alpha$ are also given. For C1 and C2, $A_{V}$ was obtained
from the R100 spectra only, as the H$\beta$ R2700 fluxes have significantly
lower S/N. Column 5: SFR after correction for the visual attenuation in column
4. The SFRs in columns 3 and 5 are corrected for the lensing magnification
factors (see Appendix B) and the finite apertures used (Appendix A). Column 6:
Metallicities based on the Curti et al. (2017) calibration (see Sect. 4.3).
For L7, the metallicity is not well constrained. Column 7: Stellar masses
estimated from the integrated R100 spectra (see Sect. 4.5). For the stellar
masses the aperture correction for an intermediate wavelength ($\sim$ 3.4
$\mu$m) was used. 1 $A_{V}$(Pa$\alpha$/H$\alpha$) using an aperture of 1′′, as
in Álvarez-Márquez et al. (2023). 2 Corrected with
$A_{V}$(Pa$\alpha$/H$\alpha$). 3 These regions are close the edge of the FoV,
and subject to larger calibration uncertainties. 4 For this galaxy the values
are derived from the 2D kinematic decomposition (see text in Sect. 3.3.3). 5
Stellar mass derived by Álvarez-Márquez et al. (2023)

##  4 ISM conditions, and star formation

In this section, we study the ISM properties for the high-z sources identified
above. Details about the methods followed for the spectral analysis (e.g.,
extraction apertures, line fitting) are provided in Appendix A.

###  4.1 Nebular dust attenuation from the recombination lines

The NIRSpec data presented here include the H$\alpha$ and H$\beta$ lines and
therefore they allow us to infer the nebular attenuation due to dust from the
commonly used Balmer decrement (i.e., H$\alpha$/H$\beta$). In addition, for
the E and the W galaxies, the dust attenuation can also be obtained by
combining the H$\alpha$ fluxes measured with NIRSpec with those for Pa$\alpha$
recently obtained with MIRI/MRS (Álvarez-Márquez et al., 2023). To convert
recombination line flux ratios into visual attenuation values (i.e., $A_{V}$)
we take the intrinsic (dust-free) line ratio (H$\alpha$/H$\beta$)int = 2.79
given by Reddy et al. (2023), which is based on photoionization modeling with
CLOUDY version 17.02 (Ferland et al., 2017) for Case B recombination (Te=10000
K, ne= 100 cm-3). This intrinsic line ratio leads to $A_{V}$ values that are
$\sim$ 0.086 mag higher than if (H$\alpha$/H$\beta$)int = 2.86 (Osterbrock,
1989) is adopted. We also use the Calzetti et al. (2000) law as, e.g., in
Domínguez et al. (2013). For comparison, adopting the Cardelli et al. (1989)
reddening law, the derived $A_{V}$ values would be 10% lower (Rodríguez Del
Pino et al., 2023). The $A_{V}$ values for the various sources in SPT0311–58
are presented in Table 1.

For the E galaxy, we find that the value inferred from H$\alpha$/H$\beta$
(i.e., $A_{V}$(H$\alpha$/H$\beta$) = 1.7 $\pm$ 0.3) is lower than the one
obtained from Pa$\alpha$/H$\alpha$ ($A_{V}$(Pa$\alpha$/H$\alpha$) = 2.4 $\pm$
0.3)444Álvarez-Márquez et al. (2023) find a lower limit for
$A_{V}$(Pa$\alpha$/H$\alpha$) ¿ 4.3 mag, which is conflicting with the present
results. This discrepancy can be explained by the inconsistency in the upper
limit for the H$\alpha$ flux reported by Álvarez-Márquez et al. (2023) and the
NIRSpec measurement. . In local dusty objects such as luminous and
ultraluminous infrared galaxies (i.e., LIRGs and ULIRGs, respectively),
several authors have also reported that the visual attenuation derived from
Balmer lines ratios is lower than that obtained from ratios involving infrared
lines (e.g., Piqueras López et al., 2013; Giménez-Arteaga et al., 2022).
Moreover, for intermediate redshifts (i.e., 1 $<$ z $<$ 3.1) Reddy et al.
(2023) have found that ratios involving Paschen lines lead to reddening
measurements that are often larger than those obtained with the Balmer
decrement. These results are generally explained by assuming that part of the
flux in the infrared lines comes from regions that are optically thick for the
Balmer lines. In other words, due to the lower obscuration at longer
wavelengths, H$\beta$, H$\alpha$ and Pa$\alpha$ trace increasingly deeper
dusty regions, which leads to the above-mentioned differences.

For the W galaxy, we cannot reliably infer $A_{V}$(H$\alpha$/H$\beta$), as in
the R100 spectrum H$\beta$ is blended with Pa$\alpha$ from the lens galaxy,
and in the R2700 spectrum the H$\alpha$ and H$\beta$ fluxes are noisy due to
the broadening induced by the large range in velocities covered by the
aperture. However, the H$\alpha$ and Pa$\alpha$ fluxes obtained with NIRSpec
(R100) and MIRI, respectively, imply $A_{V}$(Pa$\alpha$/H$\alpha$) = 5.8 $\pm$
0.6. This value confirms that W is significantly more attenuated by dust than
E, as previous studies (e.g., Marrone et al., 2018) have already shown, based
mainly on far infrared measurements (see ALMA continuum map at 160 $\mu$m in
Fig. 2).

For several of the newly discovered z$\sim$ 6.9 sources (i.e., C1-3, L1, L5)
the $A_{V}$(H$\alpha$/H$\beta$) could be estimated. The obtained values range
from $\sim$ 0.7 to 2.2 mag, and indicate that in general they have lower
attenuation than E and W. We note that the uncertainties for the individual
values are large, and in some cases they are also consistent with low or no
attenuation. The lack of ALMA continuum emission in these sources also
supports a low dust content. The regions in the eastern part (C1, C2, L5)
have, on average, a lower attenuation (¡$A_{V}$¿ $\sim$ 0.85 mag) than the
ones in the western region (C3, L1; ¡$A_{V}$¿ $\sim$ 2 mag). We used these
average values to estimate the attenuation for sources that do not have
reliable H$\beta$ fluxes, including the extended regions Wext and Eext (see
Table 1).

Figure 8: Left: Classical N2-BPT diagram for the z $\sim$ 6.9 sources in
SPT0311-58, colour-coded according to the [OIII]$\lambda$5007/H$\beta$ value.
The line ratios are derived from the integrated R2700 spectra, and include the
subregions within E and the five external z $\sim$ 6.9 sources with reliable
line fluxes. Grey shaded regions show the zones of the diagram covered by 70,
80, and 90 percent of the local SDSS sample (York et al., 2000, Abazajian et
al., 2009). Below the dashed line, local galaxies are ionised by SF (Kauffmann
et al., 2003), while above the dotted line the presence of AGN is required
(Kewley et al., 2001). The region beneath the black solid line corresponds to
LI(N)ERs (Cid Fernandes et al., 2010). The magenta line is the locus of z
$\sim$ 2–3 star-forming galaxies found by Strom et al. (2017). The black
symbols correspond to other galaxies at z ¿ 5 observed with JWST/NIRSpec
(C+23: Cameron et al., 2023; S+23: Sanders et al., 2023; Ü+23: Übler et al.,
2023). Right: Spatial distribution of the different galaxies in the SPT0311-58
system. The apertures used (Appendix 14) are color-coded by their
[OIII]$\lambda$5007/H$\beta$ values, as in the left panel.

###  4.2 Emission line ratio diagnostic diagrams

We now discuss the excitation conditions in the different high-z galaxies
identified in SPT0311–58. In Fig. 8 (left) we present the classical $N2$
$\equiv$ [NII]/H$\alpha$ vs. $R3$ $\equiv$ [OIII]/H$\beta$ BPT diagram
(hereafter N2-BPT; Baldwin et al. 1981). We include the z $\sim$ 6.9 sources
with reliable line flux ratios (i.e., C1, C2, C3, L1, L5, L7), in addition to
the subregions within E. The line fluxes have S/N $>$ 3, except for the [NII]
flux of some sources for which upper limits are provided. Colours are assigned
according to the [OIII]/H$\beta$ value. The figure shows the zones of the
diagram covered by the z $\sim$ 0 SDSS galaxies (Abazajian et al., 2009)
(shaded gray), together with several well-established lines that demarcate the
different excitation regimes at low z. We also include in the plot (magenta
line) the locus of the intermediate redshift (i.e., z $\sim$ 2–3) star-forming
galaxies from the KBSS-MOSFIRE survey (Strom et al., 2017), and results at
redshifts higher than 5 recently obtained with JWST (Cameron et al., 2023,
Sanders et al., 2023, Übler et al., 2023).

Fig. 8 shows that the sources in SPT0311–58 occupy zones of the N2-BPT diagram
barely populated by local galaxies, and are distributed along the SF/AGN
demarcation lines at z=0, covering more than 0.7 dex in $R3$. Although most of
the sources occupy the composite/AGN zones of local galaxies, they do not
exhibit clear evidence of AGN activity (e.g., Balmer broad lines or signs of
strong outflows). However, we do not exclude the presence of AGN, as at high-z
their associated outflows do not appear to be very prominent (Maiolino et al.,
2023, Carniani et al., 2023, Scholtz et al., in prep.). These z $\sim$ 6.9
sources are in a region shifted by 0.1-0.3 dex (in $N2$ or $R3$) with respect
to the star-formation locus at z$\sim$ 2-3, enlarging further the offset with
respect to the z=0 population. We note that none of the galaxies are in the
local LI(N)ER part of the diagram, despite the likely presence of shocks
according to the observed kinematics (see Sect. 5, and Spilker et al., 2022).
The sources in SPT0311–58 with the highest $R3$ (i.e., C2, L1, and L5) are in
similar regions of the N2-BPT diagram as other galaxies at z $>$ 6 recently
observed with JWST (e.g., Cameron et al. 2023, Sanders et al. 2023), and also
have faint or undetected [NII]$\lambda$6583 fluxes.

The spatial distribution of $R3$ reveals interesting trends (Fig. 8, right
panel). First, the sources with the lowest $R3$ (ES, C3) are found within or
close the massive galaxies E and W. Second, E shows a clear variation in $R3$
along its major axis, with increasing values from south to north (i.e., ES,
ENa, ENb, ENc). Note that L7 is seen in projection over the main structure of
E, but it is a separated source (see Sect. 3.3.3). Third, the sources with the
highest $R3$ (C2, L1, L5) are found in the external regions.

We now compare the observed positions in the N2-BPT diagram with the
predictions of the theoretical study by Hirschmann et al. (2022) based on the
coupling of the state-of-the-art cosmological IllustrisTNG simulations
(Springel, 2010) with new-generation nebular-emission models (Hirschmann et
al., 2019). While sources in SPT0311–58 show a wide range in $R3$ (i.e., 0.25
– 1), TNG50 galaxies at z $\sim$ 5-7 have $R3$ ¿ 0.60 (Hirschmann et al. 2022,
their Fig.6). The external sources (i.e., C1, C2, L1, L5) have observed values
within this predicted range. However, for regions within or close the two main
galaxies (i.e., ES, ENa, ENb, ENc, C3) the observed $R3$ are ¡ 0.60, and only
agree with the theoretical predictions at low or intermediate redshifts
(z$\sim$ 0.5–3). Therefore, these galaxies seem to have exceptional properties
for their z, according to theoretical expectations.

Figure 9: R23 – O32 diagnostic diagram for the galaxies in SPT0311–58, as
derived from the R2700 integrated spectra (blue dots). Line flux ratios are
corrected from dust attenuation using the Balmer decrement. Grey shaded
regions show the zones of the diagram covered by 70, 80, and 95 percent of the
local SDSS sample (York et al., 2000, Abazajian et al., 2009). The black
symbols correspond to stacked spectra of other high-z samples observed with
JWST/NIRSpec, as indicated by the labels (C+23: Cameron et al., 2023; S+23:
Sanders et al., 2023).

In Fig. 9, we present the galaxies in the dust-corrected O32-R23 diagram. R23
$\equiv$ log(([OIII]$\lambda$$\lambda$4959,5008 +
[OII]$\lambda$$\lambda$3726,3729)/H$\beta$) is a proxy of the total
excitation, while O32 $\equiv$
log([OIII]$\lambda$5008/[OII]$\lambda$$\lambda$3726,3729) is sensitive to the
ionization parameter and the metallicity (Kewley et al., 2019, Cameron et al.
2023). This diagram reflects again the different ISM conditions of the sources
in SPT0311–58 in comparison with those commonly found in local galaxies. The
large range in O32 indicates a significant variation of the ionization
parameter and/or the metallicity among the sources in the FoV. It also shows
the dichotomy between the external regions (C2, L1, L5) and those associated
with the main structures of the E and W galaxies (C3,ES,ENa,ENb,ENc), with the
latter group having lower O32. Again in this case C3 and regions within E,
have positions in the diagram close to those occupied by local galaxies.

In summary, we find that the ISM conditions in the sources of SPT0311-58 are
very diverse. While some of these z $\sim$ 7 objects have properties similar
to galaxies at these redshifts recently observed with JWST, others have
conditions closer to those found at lower redshifts. Most occupy the region of
SF/AGN composite ionization, and none are in the local LI(N)ER region despite
the likely presence of shocks. The diagnostic diagrams indicate significant
differences in the ionization parameter and/or in the metal content across the
observed sources. In the next section we study in more detail the metal
content in the system.

Figure 10: Metallicity map based on the calibrations of Curti et al. (2017,
2020b) in the 12+(O/H) scale. For the E galaxy, spaxel-by-spaxel information
is provided. For the external regions, the value obtained from the integrated
spectra is assigned to the entire aperture. For E, C1,C2, and L1 the
metallicity is inferred from the ‘R2’ $\equiv$
log($[\mathrm{O}\textsc{ii}]\,\lambda\lambda 3726,29$/H$\beta$), ‘R3’
(log($[\mathrm{O}\textsc{iii}]\,\lambda 5007$/H$\beta$)), ‘R23’, and ‘O32’.
For C3 using ‘R3’ and ‘N2’. For L5, only ‘R3’ was used due to the lack of
reliable [OII] and [NII] line fluxes for these apertures. The region where the
galaxy L7 appears in projection over E, is indicated. The line fluxes were
corrected with the mean $A_{V}$ value for the apertures. The faint gray lines
correspond to two low-level iso-countours of the H$\alpha$ emission (Fig. 2
(c)) (see text).

###  4.3 Metallicity

In Fig. 10, we present the metallicity map of SPT0311-058 according to the
calibration of Curti et al. (2017, 2020b). This calibration is based on local
samples, but it has the advantage of covering well the metallicity range
observed in the system, and it shows relatively low scatter. We also comment
below the results when using the calibration by Sanders et al. (2023), which
is based on samples that cover well the redshift of SPT0311–58, but presents
larger scatter than local ones due to the lower number of sources available
and potential differences in the ISM conditions at high-z. We followed the
procedure developed by Curti et al. (2023), which explores the log(O/H)
parameter space with a Markov Chain Monte Carlo (MCMC) algorithm and finds the
maximum likelihood taking into account the uncertainties of the observed line
ratios and the dispersion of the individual index calibrations. For the E
galaxy, the metallicity is derived for each spaxel (after a 3 x 3 binning)
while for the external sources, and due to their lower S/N, the integrated
spectra are used to derive a global value. For E, C1, C2 and L1 the
metallicity is inferred from the ‘R2’,‘R3’,‘R23’,‘O32’, and R̂ indexes, where
R̂ $\equiv$ $0.47\ \text{R2}+0.88\ \text{R3}$, as defined by Laseter et al.
(2023). For C3, ‘R3’ and ‘N2’ were used, while for L5 and L7 only ‘R3’ was
available due to the lack of reliable line fluxes to derive other indexes. In
Table 3 the individual metallicity values are presented.

Fig. 10 shows that the metallicity spans about 0.8 dex across the FoV. On the
one hand, the southern regions of E and W (i.e., source C3) have the highest
metallicities with values around 8.4-8.6 in the 12+log(O/H) scale. On the
other hand, external galaxies C2, L1, and L5 have low metal content, with
values in the range $\sim$ 7.6-8.1. The calibration by Sanders et al., 2023
leads to qualitatively similar results, although it finds higher metallicity
(by $\sim$ 0.2 dex) in the metal rich regions C1, and E, while leaving
unaltered the values for C2 and L1, and suggesting lower metal content in L5
(by $\sim$ 0.2 dex). Therefore, this calibration indicates an even larger
range in metallicty across the FoV, though it also increases the associated
uncertainties of individual determinations.

Interestingly, the E galaxy shows a prominent metallicity gradient along its
major axis with a total peak-to-peak variation of $\sim$ 0.7 dex, and also
local inhomogeneities at sub-kpc scales. The southern regions are the most
metal-rich (i.e., $\sim$ 8.50), while the lowest metallicity (i.e., $\sim$
7.8) is found to the northwest. This metal-poor region is adjacent to a zone
of diffuse H$\alpha$ (see gray iso-contours in Fig. 10) and [CII] emission
(Fig. 2 c and f), indicating the presence of ionised and neutral gas. This
suggests that the accretion of low metallicity gas into the main structure of
E in the northwest may be the cause for the observed gradient. Gas accretion
has been proposed to explain metal gradients and inhomogeneities at high- and
low-z (e.g., Cresci et al., 2010, Kumari et al., 2017, Curti et al., 2020a,
Rodríguez Del Pino et al., 2023). In Sect. 6.2, we will discuss further this
scenario, as well as other potential alternatives (i.e., merger event,
outflows), also taking into account the kinematic properties of the E galaxy.

Litke et al. (2023) have recently found a nearly solar metallicity (i.e., 0.8
$\pm$ 0.4 Z☉) for the E galaxy based on the observed far-infrared
[OIII]$\lambda$ 88$\mu$m /[NII]$\lambda$ 122$\mu$m emission line ratio, and
the calibration proposed by Pereira-Santaella et al. (2017). To compare with
this result, we have computed the global metallicity over the integrated E
spectrum. Using the ‘R2’,‘R3’,‘R23’ ‘O32’, and R̂indexes and the calibration
by Curti et al. (2017, 2020b), we infer an average value of 0.48 $\pm$ 0.07
Z☉. Although the difference with the Litke at al. result is not significant
taking into account the uncertainties, it could be interpreted as due to the
different regions probed by the FIR and the optical lines. While the FIR
offers access to deeper, dustier, and likely more metal-rich zones, the
optical lines come from the less extincted and likely more metal-poor regions.

In summary, we have found a wide range of metallicities (by 0.7 dex) across
the galaxies in SPT0311–58. The external galaxies have significantly lower
metal content than the regions associated (or close) to the main E and W
galaxies. The E galaxy also shows a metallicity gradient that is likely
evidence for the presence of pristine gas accretion. In Sect. 6.2 we discuss
further this and other alternative scenarios, also considering the kinematic
properties of the system.

###  4.4 H$\alpha$-based star formation

The present observations allow us for the first time to measure H$\alpha$
fluxes in this system, and therefore to infer the unobscured ongoing star
formation using this well-calibrated tracer. We note that SPT0311–58 is at
about the highest z at which H$\alpha$-based SFR can be derived at the
sensitivity and angular resolution provided by NIRSpec (i.e., for z$>$ 7,
H$\alpha$ is redshifted outside the NIRSpec range).

We estimate the nearly instantaneous (i.e., last 3-10 Myrs) star formation
rate (SFR) from the H$\alpha$ luminosity following the commonly used
expression provided by Kennicutt & Evans (2012) (see also Murphy et al., 2011;
Hao et al., 2011): SFR(H$\alpha$)/(M⊙ yr-1) = 5.37 $\times$ 10-42
L(H$\alpha$)/(erg s-1).

This relation presumes constant SFR over the mentioned timescale, solar
metallicity and a Kroupa IMF (Kroupa, 2001) 555Results for a Chabrier IMF
(Chabrier 2003) are nearly identical (c.f. Kennicutt & Evans 2012)., and also
assumes that all ionizing photons actually ionise the ISM. We note, however,
that changing some of these assumptions may have significant effects. For
instance, for subsolar metallicities the SFRs provided by this equation should
be corrected down because metal poor massive stars are more efficient in
producing ionizing photons, which leads to higher L(H$\alpha$) per unit of SFR
as discussed by Reddy et al. (2022) (see also Shapley et al., 2023). To take
into account this effect, and based on the results of the previous section,
the SFRs obtained with the expression quoted above for galaxies C2, L1-L5 have
been corrected accordingly by a factor 0.5. For the more metal-enriched E and
W galaxies and their outskirts (including C1 and C3) we have maintained the
solar abundance assumption. This allows us a more direct comparison with
previous SFR estimates for E and W (Marrone et al., 2018, Álvarez-Márquez et
al., 2023), and with determinations based on general calibrations (e.g., Total
Infrared Luminosity L(TIR)-SFR, Calzetti, 2013) that have used solar
metallicity. The individual H$\alpha$ fluxes and SFRs(H$\alpha$), corrected
and uncorrected for dust attenuation, are presented in Table 1.

As we can see in the table, and also in panel c of Fig. 2, the main body of
the E galaxy is the region in the FoV with the highest H$\alpha$ flux, and
therefore where most of the unobscured (i.e., uncorrected by dust attenuation)
star formation is taking place at a rate of $\sim$ 62 M⊙ yr-1. The dust-
corrected SFR (377 M⊙ yr-1) represents a large fraction (i.,e. $\sim$ 70-80
percent) of the total SFR obtained by Marrone et al. (2018) from a SED
analysis which includes the far-IR (ALMA) dust continuum emission (545 $\pm$
175 M⊙ yr-1), or by using the L(TIR)-SFR calibration by Calzetti (2013)
($\sim$ 466 M⊙ yr-1). In addition to the H$\alpha$ flux associated to E, a low
surface brightness H$\alpha$ emission extends beyond its main structure,
particularly in the direction of source C1. This external H$\alpha$ flux is
obtained integrating within aperture Eext (see Fig. 14), and has an associated
unobscured SFR(H$\alpha$) of $\sim$ 32 M⊙ yr-1($\sim$ 61 M⊙ yr-1dust-
corrected).

For W the inferred unobscured SFR(H$\alpha$) is $\sim$ 8 M⊙ yr-1, i.e., a tiny
fraction of the SFR obtained by Marrone et al., 2018 from a SED analysis
including the FIR emission (i.e., 2900 +/- 1800 M⊙ yr-1), or from the
L(TIR)-SFR relation by Calzetti 2013 (3350 M⊙ yr-1). In this case, the
attenuation correction factor is extremely large, as $A_{V}$ is 5.8
magnitudes. The dust-corrected SFR(H$\alpha$) is $\sim$ 600 M⊙ yr-1, i.e.,
about a fifth of the total SF inferred including the far-infrared dust
emission. This suggests that W hosts a large fraction of star formation that
is totally obscured at rest-frame optical and near-IR wavelengths. We note
that a detailed comparison of the SFR obtained with different tracers should
also take into account the timescale assumed for their calibration and the
Star Formation History (SFH) of the source under study. For example, post-
starburst (PSB) galaxies (where star formation terminated rapidly and
recently) have systematically higher SFR(FIR)s than what is estimated from
recombination lines alone. This is observed both in the local Universe (Baron
et al., 2023) as well as at z $\sim$ 1 (Wu et al., 2023). However, these PSB
galaxies also display lower nebular attenuation than what we infer for the W
galaxy; in our case, because we are dealing with an extremely dusty galaxy, we
favour the star-forming interpretation, and explain the discrepancy between
the SFR(H$\alpha$) and SFR(FIR) as due to heavy dust attenuation. The most
rigorous approach would be to infer the full SFH, but this is beyond the scope
of the present study.

For the newly discovered sources at z $\sim$ 6.9, the unobscured
SFR(H$\alpha$) ranges from 1 to 6 M⊙ yr-1, and totalling 14 M⊙ yr-1. The
sources with $A_{V}$ measurements (i.e., C1, C2, C3, L1, L5), have individual
dust-corrected SFR(H$\alpha$) ranging from $\sim$ 4 to 30 M⊙ yr-1. For the
other sources, the star formation is not well constrained as $A_{V}$ cannot be
reliably derived (see Sect. 4.1). Using for them the average $A_{V}$ derived
for the other sources, the combined dust-corrected SFR(H$\alpha$) for all
these objects is 74 M⊙ yr-1, a small amount compared with the total SFR taking
place in E and W. As the new sources are undetected with ALMA in both the
[CII] 158$\mu$m line and nearby continuum, their dust-corrected SFR(H$\alpha$)
likely represent the total SFR. However, we note that the sensitivity of the
currently available ALMA observations (see Sect. 2.5) imply an (obscured) SFR
detection limit of 10-20 M⊙ yr-1.

In summary, we have found significant unobscured ongoing star formation
activity within and around the E galaxy ($\sim$ 94 M⊙ yr-1). The dust-
corrected SFR(H$\alpha$) ($\sim$ 430 M⊙ yr-1) recovers a large fraction of the
total SFR obtained including the FIR emission. In contrast, for W the current
unobscured SFR(H$\alpha$)s is small (8 M⊙ yr-1), representing a tiny fraction
of the total SFR of this galaxy. An important fraction of the SFR in this
galaxy is totally obscured at optical and near-IR wavelengths, and can not be
recovered with the attenuation correction. As for the newly discovered
external sources, their unobscured SFR(H$\alpha$)s are relatively modest with
individual values $\sim$ 1 - 6 M⊙ yr-1. Correction from dust still leaves the
combined contribution to these sources to the total SFR of the system as minor
(i.e., $\sim$ few percent).

###  4.5 Stellar masses

Here we present stellar masses inferred from the integrated R100 spectra. We
exclude W and L7 from this analysis as their stellar continua are contaminated
by those of the lens and the E galaxies, respectively.

The SED fitting method is described in Pérez-González et al. (2023) and
D’Eugenio et al. (2023). In short, the NIRSpec R100 integrated spectra were
compared to stellar population models from the Bruzual & Charlot (2003)
library, assuming a delayed exponential star formation history characterised
by a timescale $\tau$ (taking values from 1 Myr to 1 Gyr in 0.1 dex steps) and
age $t_{0}$ (ranging from 1 Myr to the age of the Universe at the redshift of
the galaxy). The stellar metallicity $Z$ was left as a free parameter,
allowing us to take all the discrete values provided by the Bruzual & Charlot
(2003) library from 0.02 to 2.5 times solar. Nebular (continuum and line)
emission was taken into account as described in Pérez-González et al. (2003,
2008). The attenuation of the stellar and nebular emission was modelled with a
Calzetti et al. (2000) law, with $A_{\rm V}$ values ranging from 0 to 3
magnitudes in 0.1 mag steps. The stellar mass is obtained by scaling the mass-
normalized stellar model to the spectrum.

The derived values are included in column 6 of Table 3. The stellar mass
obtained for E (4.0$\pm$0.7 $\times$ 1010 M⊙) is in good agreement with the
one derived by Marrone et al. (2018) (3.5$\pm$1.5 $\times$ 1010 M⊙). For the
newly discovered sources, we obtain a median stellar mass of $\sim$ 1.7
$\times$ 109 M⊙, with a range of 0.75 - 4 $\times$ 109 M⊙. This indicates that
these galaxies have already assembled a significant amount of stellar mass.
Further study of the stellar population properties in SPT0311–58 is beyond the
scope of the present work.

##  5 Kinematic properties

This section is devoted to the kinematic properties of the different sources
of SPT0311–58. First, we present the relative motions and velocity dispersion
of the newly discovered sources at z $\sim$ 6.9. We also derive their
dynamical masses, and compare them with the stellar masses. Second, we focus
in more detail on the E and W galaxies, and discuss the kinematics for the
warm and cold gas phases inferred from NIRSpec and ALMA, respectively.

###  5.1 Newly discovered z $\sim$ 6.9 sources: Relative velocities, velocity
dispersions, and dynamical masses

Fig. 11 (upper-right panel) shows the large-scale velocity field in SPT0311–58
derived from the R2700 [OIII] line. In this figure the differences in redshift
are interpreted as relative velocities 666If the observed redshifts were
interpreted as spatial separations only, the proper distance between the two
most distant galaxies (i.e, L3 and C3) would be about 3.6 pMpc. This value is
similar to the extent found by Wang et al. (2021a) for the overdense region
with sub-mm galaxies ( $\sim$ 2.6 pMpc), which suggests that the entire proto-
cluster spans approximately 3 pMpc., taking z = 6.902 the origin for the
velocities (i.e., the redshift of the most massive W galaxy inferred from the
[CII] line, Sect. 5.3). The figure shows that the ten newly discovered
galaxies have a very large range of (projected) velocities: from -595 km s-1
for L3 up to +932 km s-1 for C3 (see Table 2 for the individual redshifts).
These velocities seem randomly distributed across the FoV, with sources with
high positive and negative values relatively close in projection (e.g., L2 and
C4, L3 and C3). This significant spread in velocities suggests that the mass
of the halo must be very large. In Sect. 6 we discuss further the relative
motions of the sources within the system.

The integrated velocity dispersion ($\sigma$, see Table 2) can be used to
estimate the dynamical mass (Mdyn) of the galaxies through the formula:

$M_{\rm dyn}=K\frac{\sigma^{2}R_{e}}{G},$ (1)

where $R_{e}$ is the effective radius, $G$ is the gravitational constant, and
$K$ is a factor that depends on the mass distribution in the galaxy. We have
adopted $K$=5$\pm$0.1, following the best-fitting virial relation calibrated
by Cappellari et al. (2006). We have also applied the correction suggested by
Übler et al. (2023), to account for the differences in the integrated velocity
dispersion of the ionised gas and the stars found by Bezanson et al. (2018).
For the present range of integrated $\sigma$, this $\Delta$ log($\sigma$/(km
s-1)) correction varies from 0. to +0.18. For the determination of the size we
have fit the emission in each source with a 2D Gaussian, and estimate $R_{e}$
from the largest of the two standard deviations provided by the fit, after
deconvolving with the PSF. To derive the properties of the PSF, we used the
data cube of the standard star (Sect. 2.5). We first try to infer the sizes
from a panchromatic image generated from the R100 cube in the 1-5 $\mu$m
spectral range. This led to acceptable results for sources that emit in the
continuum (i.e., C1, C2, C3). For the other objects, which are undetected (or
very faint) at the continuum, we used the [OIII] image instead. The derived
$R_{e}$ values are also given in Table 2.

The dynamical masses for the newly discovered galaxies are in the range $\sim$
109 – 1010 M⊙ (Table 2). Then, although these objects are significantly less
massive than the two main galaxies in SPT0311–58, they contain a substantial
amount of mass for their redshifts. In Figure 12 we compare the dynamical and
stellar masses obtained in Sect. 4.5. The figure shows that on average $M_{\rm
dyn}$ is larger than $M_{\star}$ by a factor 2-3. This result suggests that,
as expected, they may have an important fraction of their mass in gas (Tacconi
et al., 2020). Additionally, or alternatively, they might also have
substantial amounts of dark matter, as suggested by recent NIRSpec
observations of high-z galaxies (de Graaff et al., 2023). Note that in this
figure we have also included the E galaxy, for which we derive a dynamical
mass of 6.0 $\pm$ 2 $\times$1010 M⊙. Marrone et al. (2018) find a total
baryonic (i.e., stars, gas, and dust) mass of 7.5 $\pm$ 3.5 $\times$1010 M⊙
and Álvarez-Márquez et al. (2023) 4-5 $\times$1010 M⊙. Therefore, its Mdyn
determination is consistent with previous results within the estimated
uncertainties.

Figure 11:  Upper panels: large scale velocity fields from the high resolution
NIRSpec IFU spectra in the [OIII]- H$\beta$ complex (left), from the
[CII]$\lambda$158 line observed with ALMA (center), and difference (right).
Lower panels: similar maps for the intrinsic velocity dispersion
($\sigma_{0}$). The NIRSpec maps were obtained after a 3 $\times$ 3 spaxel
binning, except for L2, L3, L4, L6 and L7 for which, due to their low S/N, the
values from the integrated spectrum were assigned to the entire extraction
aperture. The values for L7 were obtained after the kinematic decomposition of
two overlapping systems (Sect. 3.3.3). The relative velocity values are in km
s-1, taking as reference a redshift of 6.902. The NIRSpec velocity dispersions
are corrected for the instrumental broadening according to the dispersion
curves provided by Jakobsen et al. (2022) Table 2: Kinematic properties and
dynamical mass estimates for sources in SPT0311-58

ID | z |  $\Delta$v | $\sigma$ | Re | Mdyn | Dw
---|---|---|---|---|---|---
|  | km s-1 | km s-1 | kpc | 109M⊙ | kpc
C1 | 6.92655 $\pm$ 0.00019 | 932\. $\pm$ 7. | 99\. $\pm$ 8. | 0.64 $\pm$ 0.19 | 7.4${}^{+10.}_{-2.7}$ | 9.4 $\pm$ 0.3
C2 | 6.92419 $\pm$ 0.00010 | 843\. $\pm$ 4. | 57\. $\pm$ 5. | 0.69 $\pm$ 0.21 | 2.6${}^{+3.7}_{-1.0}$ | 10.7 $\pm$ 0.3
C3 | 6.90638 $\pm$ 0.00035 | 166\. $\pm$ 13. | 106\. $\pm$ 15. | 0.75 $\pm$ 0.37 | 10.${}^{+20.}_{-3.7}$ | 3.9 $\pm$ 0.5
L1 | 6.89941 $\pm$ 0.00005 | -98. $\pm$ 2. | 66\. $\pm$ 2. | 0.73 $\pm$ 0.22 | 3.8${}^{+4.7}_{-1.4}$ | 5.1 $\pm$ 0.3
L2 | 6.89256 $\pm$ 0.00025 | -358. $\pm$ 9. | 64\. $\pm$ 12. | 0.51 $\pm$ 0.26 | 2.5${}^{+5.8}_{-0.9}$ | 7.6 $\pm$ 0.5
L3 | 6.88632 $\pm$ 0.00011 | -595. $\pm$ 4. | 36\. $\pm$ 7. | 0.68 $\pm$ 0.20 | 1.0${}^{+2.0}_{-0.4}$ | 15.0 $\pm$ 0.5
L4 | 6.92425 $\pm$ 0.00056 | 845\. $\pm$ 21. | 99\. $\pm$ 24. | 0.70 $\pm$ 0.35 | 7.9${}^{+20.}_{-2.9}$ | 8.2 $\pm$ 0.5
L5 | 6.90940 $\pm$ 0.00004 | 281\. $\pm$ 2. | 36\. $\pm$ 3. | … | … | 14.2 $\pm$ 0.5
L6 | 6.91830 $\pm$ 0.00018 | 619\. $\pm$ 7. | 45\. $\pm$ 10. | 0.58 $\pm$ 0.17 | 1.4${}^{+2.8}_{-0.5}$ | 7.3 $\pm$ 0.5
L71 | 6.92644 $\pm$ 0.00054 | 928\. $\pm$ 20. | 93\. $\pm$ 15. | 0.53 $\pm$ 0.31 | 5.3${}^{+1.8}_{-1.6}$ | 11.0 $\pm$ 0.5
E | 6.91956 $\pm$ 0.00008 | 666\. $\pm$ 3. | 155\. $\pm$ 3. | 2.12 $\pm$ 0.63 | 59.${}^{+70.}_{-21.}$ | 10.7 $\pm$ 0.3
777Column 1: Source identification. Column 2: Redshift from the R2700
[OIII]-H$\beta$ lines. Column 3: Relative velocity with respect to z=6.902, as
inferred for the W galaxy from the ALMA [CII] line (see Sect. 5.3). Column 4:
Integrated velocity dispersion derived from the [OIII]-H$\beta$ lines in the
R2700 integrated spectra, after deconvolving the instrumental profile
(Appendix A). Column 5: Effective radius estimated as the largest sigma of a
2D Gaussian fit to source emission, after deconvolving quadratically the PSF
profile. To derive the properties of the PSF we used the datacube of the
standard star (see Sect. 2.5). For sources with significant continuum emission
(C1, C2, C3) the sizes were inferred from a panchromatic image generated from
the R100 cube in the 1 to 5 $\mu$m spectral range. For the other sources,
which have undetected (or very faint) continuum, the [OIII] image was
preferred. For the E galaxy the combination of continuum plus [OIII] was used.
The Gaussian fits provided unrealistically small errors. Instead,
uncertainties are estimated to be 30%, except for some particularly
problematic cases for which 50% was adopted instead, specifically for C3 (edge
of the FoV), L2 (faint), and L4 (badly defined). For L5, which is very close
to the edge of the FoV, Re could not be estimated. Column 6: Dynamical mass of
the source adopting the calibration by Cappellari et al. (2006) (see Sect.
5.1). Column 7: Distance to spaxel (74,45), which is estimated to be at the
center of gravity of the system. 1 For L7, the kinematic properties were
obtained after the 2D kinematic decomposition of this source and the main
component of the E galaxy (Sect. 3.3.3).

Figure 12: Comparison between the dynamical and the stellar masses for sources
in SPT0311–58 (see Sect. 5.1).

###  5.2 E galaxy

The kinematic maps for the E galaxy as inferred from the [OIII] lines are
presented in the upper panels of Fig. 7. The velocity field shows a
significant gradient of $\sim$ 200 km s-1along the major axis. The iso-
velocity lines clearly deviate from what is expected for a regular rotating
disk, especially to the northwest where they are strongly twisted. For this
spatially extended source, we are able to measure the intrinsic velocity
dispersion ($\sigma_{0}$, moment 2 in Fig. 7). Similarly to the velocity
field, the $\sigma_{0}$ map deviates from the expectation for a rotating disk,
as it shows local changes at sub-kpc scales, and higher values at the edges
than to the center. It also shows a general trend along the major axis with,
on average, higher values to southeast ($\sigma_{0}$ $\sim$ 120-145 km s-1)
than to the northwest ($\sigma_{0}$ $\sim$ 75-100 km s-1). The velocity
dispersion from the integrated spectrum (i.e., $\sigma$), which also includes
the global velocity gradient, is 155 $\pm$ 3 km s-1. The median intrinsic
velocity dispersion ($\sigma_{0}$) over the area covered by Fig. 7 is 113
$\pm$ 19 km s-1. From the fits with a 3 $\times$ 3 binning (Fig. 11), which
cover a larger area but are slightly contaminated by L7, the median
$\sigma_{0}$ = 120 km s-1. These values are above the extrapolation of the
average $\sigma_{0}-z$ evolution found by Übler et al. (2019) (i.e.,
$\sigma_{0}$ $\sim$ 70-80 km s-1at z=6), and lead to a rotational support
factor, v/$\sigma_{0}$, of $\sim$ 0.85. All these characteristics clearly
indicate that the ionised gas disc is not settled yet.

We now compare the kinematic properties of the warm and the cold gas phases,
as inferred from NIRSpec and ALMA, respectively. The upper-right panel of Fig.
11 shows that, excluding the region where the source L7 appears in projection,
the warm and cold gas components in E have similar velocities, though the
latter recedes on average $\sim$ 25 km s-1 faster. This behaviour is not
homogeneous, and in some spaxels towards the south and east the warm gas has
receding velocities of up to $\sim$ +80-100 km s-1 larger. These differences
indicate that the ionised and neutral gas components probed by NIRSpec and
ALMA are not co-spatial. This may happen when they occupy geometrically
distinct regions, and/or as consequence of the dust attenuation. ALMA traces
regions over a large range of dust attenuation, while NIRSpec probes the
ionised gas that is located in less extincted (likely more external) regions.
Hence, relative motions between regions with different dust attenuation could
explain the observed velocity offsets.

As for the velocity dispersion, the cold and warm gas show a significant
difference, particularly towards the south and east (see bottom-right panel of
Fig. 11). Interestingly recent simulations of 4 ¡ z ¡ 9 galaxies by Kohandel
et al. (2023) show that the gas velocity dispersion estimates strongly depend
on the tracer used. According to this study, while [CII] traces the gaseous
disk, H$\alpha$ also includes the contribution from extraplanar ionised gas
beyond the disk. This effect is more significant for massive galaxies (i.e.,
$M_{\star}$¿ 109 M⊙), for which $\sigma$ ([CII]) $>$ 2 $\times$ $\sigma$
(H$\alpha$). These predictions are in general agreement with our observations
in the E galaxy, though we find significant spatial variations. In the north
and in the central regions the velocity dispersion obtained from [CII] and
H$\alpha$ are similar, while to the south and east the H$\alpha$ values are
larger by factors of up to $\sim$ 2-3.

We note that a similarly significant difference between the velocity
dispersion of warm ionized gas and its cold phase counterpart has been
recently reported by Parlanti et al. (2023) in a z= 4.8 DSFG observed with
JWST. However, in that case the difference was found in the central regions of
the galaxy, and explained by the feedback effects of an AGN, a scenario that
can hardly be considered for interpreting the observed behaviour in
SPT0311–58.

In Section 6 we discuss further the kinematics in E in connection with other
ISM properties.

###  5.3 W galaxy

As commented above, the main body of the W galaxy is not detected in a spaxel-
by-spaxel basis, and therefore we cannot infer spatially resolved kinematic
information with NIRSpec for this galaxy. However, the R2700 integrated
spectrum allow us to infer the redshift from the optical emission lines. As
the [OIII] lines are faint in this galaxy, we use the H$\alpha$-[NII] complex
and the H$\beta$ line, and obtain z= 6.907 $\pm$ 0.002. We note that the
uncertainty in this case is relatively large because the spectrum is noisy
(Fig. 15).

We have also derived the redshift from the integrated spectrum of the [CII]
line obtained using the same aperture as above. The [CII] profile in this
spectrum departs significantly from a Gaussian. So, rather than derive the
redshift fitting the line, we obtain the flux-weighted averaged wavelegth, and
obtained z= 6.902 $\pm$ 0.002. This agrees within errors with the Strandet et
al., 2017 determination (z=6.900 $\pm$ 0.002), who used several CO and [CII]
lines and a beam of 2-3″. We note that because the W galaxy is well detected
and resolved with ALMA, and it is also close to the zone of multiple images,
its redshift determination may be affected by the lens, which introduces an
additional uncertainty.

The differences in the redshifts derived from the optical and the far infrared
lines suggest an average shift of $\sim$ +200-300 km s-1 between the warm and
the cold gas phases. Álvarez-Márquez et al. (2023) already reported an offset
of Pa$\alpha$ with respect to the [CII] and [OIII] far-IR lines, though in
this case significantly larger in amplitude (i.e., +700 km s-1). For source
C3, at the south of W, the average relative velocity (cold-warm) is $\sim$ 80
km/s.

As discussed for the E galaxy, these velocity differences may be explained by
the relative motions between the less attenuated regions traced with the
optical lines, and the dustier zones probed at the far infrared. The larger
attenuation in W (see Sect. 4.1) could also qualitatively explain why we find
a significantly larger velocity offset in this galaxy than in E.

##  6 Discussion

Figure 13: Projected Phase Space for the galaxies in SPT0311–58. The x-axis is
the projected distance from the cluster centre in units of R200. The y-axis
shows the absolute value of the relative velocity along the line of sight of
each galaxy with respect to the global recessional velocity, normalized by the
velocity dispersion of the cluster. The red dots correspond to a global
cluster redshift of zsys=6.902 as inferred for the W galaxy from the [CII]
line, and the green dots are for an ad-hoc zsys= 6.912. The blue line
represents the escape velocity normalised by the velocity dispersion of the
cluster for a Navarro, Frenk, and White dark matter halo (Navarro et al.,
1996) of M200 = 6 $\times$ 1012M⊙, with the blue shadow defining the region
for masses between 3 and 9 $\times$ 1012M⊙. The point for the W galaxy at
(0,0) has been slightly shifted to make it more visible.

###  6.1 SPT0311–58: A proto-cluster core in a massive dark matter halo

The possibility that SPT0311–58 forms part of a proto-cluster was proposed by
Wang et al. (2021a), who found a significant overdensity of bright sub-mm
sources in the field surrounding its two main galaxies (r $\lesssim$ 1.3 Mpc).
However, the lack of spectroscopic redshifts prevented Wang and collaborators
to confirm this scenario, and classified SPT0311–58 as a proto-cluster
candidate. As has been detailed in Sect. 5.1, the present NIRSpec data have
revealed ten new spectroscopically confirmed z $\sim$ 6.9 sources in the IFU
FoV (i.e., 17$\times$17 kpc2) in addition to the already known E and W
galaxies, which strongly supports the idea that SPT0311-58 is at the core of
one of the most distant proto-clusters known. Indeed, the number density of
sources ($\phi$ $\sim$ 10${}^{4}\ $Mpc-3) is orders of magnitude above the
expectation for the field ($\phi_{f}$ $\sim$ 5 Mpc-3, for $M_{\star}$¿ 107
M⊙), according to the observed galaxy stellar mass functions (e.g., Conselice
et al., 2016). In addition, we find that the sources in SPT0311–58 exhibit a
relatively large spread in radial velocities ($\Delta$v $\sim$ 1500 km s-1,
Table 2), akin to other high-z proto-clusters (z ¿ 4, e.g., Chanchaiworawit et
al., 2019, Hill et al., 2020, Calvi et al., 2021, Morishita et al., 2023,
Hashimoto et al., 2023, Scholtz et al., 2023). This suggests that the mass of
the halo must be very large. In the following, we estimate the main properties
of the halo, and discuss the dynamical status of the core of this proto-
cluster.

We estimate the mass of the halo from the stellar mass using of the
$M_{\star}$/$M_{h}$ relations found by empirical modeling (UNIVERSEMACHINE,
Behroozi et al., 2019, 2020) and hidrodynamical simulations ( MillenniumTNG,
Kannan et al., 2023). To derive the stellar mass in SPT0311–58 we combine the
results for the E galaxy and the newly discovered sources obtained from the
NIRSpec R100 IFU spectra (Sect. 4.5), together with the one for the W galaxy
determined by Álvarez-Márquez et al. (2023) based on recent JWST/MIRI
observations. The resulting total stellar stellar mass for the system is (1.3
$\pm$ 0.3) $\times$ 1011M⊙. The $M_{\star}$/$M_{h}$ ratio is not well
constrained for the redshift and stellar mass of SPT0311–58, but the relations
by Behroozi et al. (2019, 2020) and Kannan et al. (2023) suggest values of
$\sim$ 0.015–0.04888This $M_{\star}$/$M_{h}$ range is in good agreement with
the one estimated with Flamingos simulations (Schaye et al., 2023, Kugel et
al., 2023) (S. Lim, priv. comm.). Similarly, mock galaxy catalogues generated
by the so called UCHUU-UM approach (Ishiyama et al., 2021; Prada et al., 2023)
lead to a $M_{\star}$/$M_{h}$ ratio range consistent with the one adopted here
(E. Pérez, F. Prada, priv. comm.)., which leads to $M_{h}\sim$ (6$\pm$3)
$\times$ 1012 M⊙.

This result agrees well with the findings of Marrone et al. (2018), who
estimate $M_{h}$ in SPT0311–58 from its gas content (Strandet et al., 2017).
Specifically they found that $M_{h}$ could vary from a conservative lower
limit of 1.7 $\times$ 1012 M⊙ to 7.4 $\times$ 1012 M⊙, depending on the
approach adopted to determine the mass of the gas. However, we note that
recently Jarugula et al. (2021), using non-local thermodynamic equilibrium
radiative transfer models, have found that the actual mass of gas in
SPT0311–58 is larger than previously estimated. This suggests that previous
gas-based Mh determinations should be corrected by factors of 1.5-2.

Having an estimate for the halo mass, we can now derive the expected size,
R200 999R200 is defined as the radius within which the mean enclosed mass
density is 200 times the critical density of the Universe at the corresponding
redshift., for a virialized dark matter (DM) halo at z=6.9. By definition
(e.g., Navarro et al., 1996):

$M_{200}\equiv\frac{4}{3}\pi R_{200}^{3}200\rho_{c}\ ,$ (2)

where M200 $\equiv$ Mh, and $\rho_{c}$ is the critical density that varies as

$\rho_{c}=\frac{3H^{2}(z)}{8\pi G}\ ,$ (3)

with H(z) and G being the Hubble parameter and the gravitational constant,
respectively. From these equations, we derive a size of R200 =
72${}_{-16}^{+10}$ kpc for a halo mass of (6$\pm$3) $\times$ 1012 M⊙.

The center of mass of this proto-cluster should be at the W galaxy. This is
based on previous works (Marrone et al. 2018; Álvarez-Márquez et al. 2023)
that have shown that the baryonic mass of this galaxy is about one order
magnitude larger than its companion, the E galaxy. The newly discovered
galaxies have dynamical masses significantly lower than E (Sect. 5), so they
should not affect significantly the determination of the center of gravity of
the system (CoGsys). We also adopt the redshift of the W galaxy as the mean
redshift for the system. As discussed in Sect. 5.3, the redshift for W is
subject to relatively large uncertainties compared with other sources, with
significant differences among the values inferred from the optical emission
lines (NIRSpec), Pa$\alpha$ (MIRI), and the molecular lines (ALMA), being also
possibly affected by the effects of the lens. We have adopted the
determination based on the [CII] line (i.e., z=6.902 $\pm$ 0.002), because it
is not affected by attenuation, and traces a significantly larger amount of
mass (see Sect. 5.3). Therefore, taking the W galaxy as the center of the
proto-cluster, our NIRSpec observations probe projected cluster-centric
distances of up to 15 kpc (see Table 2), which correspond to about 0.2
$\times$ R200.

Given that we are observing such an inner region of a proto-cluster, one might
inquire whether it has already undergone (partial) virialization. In an
attempt to answer this question, we have analysed whether the radial
velocities of the 12 galaxies may be drawn from a normal distribution, as
expected for a system in equilibrium. With this aim, we performed a
Kolmogorov-Smirnov (K-S) test that led to a p-value of 0.04, and therefore we
conclude that the observed region is not virialized yet. However, we point out
that the result of the K-S test is very dependent on the origin of the
relative velocities. If we allow a global offset to the radial velocities of
380 km s-1, the p-value reaches 0.75, suggesting a normal distribution. This
change would imply that the W galaxy is not at the center of the proto-
cluster, or that its redshift is larger by $\Delta$z $\sim$ 0.01, i.e., z =
6.912. We note that this redshift falls between the one inferred from the
optical lines (z=6.907) and from Pa$\alpha$ (z = 6.918) (Sect. 5.3). Still,
for the reasons mentioned above, we argue that most likely the CoGsys is at W
and zsys = 6.902, and therefore the system does not seem to be at equilibrium.
Nevertheless, for the remainder of the section we will also evaluate how the
results are affected if an ad hoc z = 6.912 is used as reference.

Using the present kinematic information, we can also assess if the system is
at equilibrium, by comparing the velocity dispersion observed in SPT0311–58
with the one expected for a virialized cluster. We derive the velocity
dispersion in our data by bootstrap, and obtain a mean estimate of
$\sigma_{sys}$ = 479 km s-1, and a range from 406 to 551 km s-1with a 68
percent confidence101010The bootstrap estimate is preferred over the standard
deviation (507 km s-1), as it is more robust for small samples (Beers et al.,
1990).. According to the numerical simulations of virialized clusters by
Evrard et al. (2008), the predicted velocity dispersion at equilibrium for a
cluster with a mass of (6 $\pm$ 3) $\times$ 1012 M⊙ is 396${}_{-82}^{+58}$ km
s-1. Since we measure a larger face value of the velocity dispersion, this
comparison suggests that the core in SPT0311–58 is not virialized yet.
However, taking into account the associated uncertainties, the data are also
compatible with being (partially) virialized. Alternatively, using Evrard et
al. (2008) relation and the observed velocity dispersion, the predicted mass
for a virialized proto-cluster is 1.05${}_{-0.41}^{+0.55}$$\times$1013M⊙. The
fact that this value is larger than our halo mass estimate suggests that the
system has not reached equilibrium. However, as mentioned above, the
uncertainties are large and this conclusion lacks robustness.

In order to analyse further the dynamical status of the observed region, in
Fig. 13 we have located the galaxies in the Projected Phase Space diagram
(e.g., Mahajan et al., 2011; Jaffé et al., 2015). The x-axis of the plot is
the projected distance from the cluster centre normalised to R200, while the
y-axis indicates the absolute relative velocity of each galaxy with respect to
the global recessional velocity (v) normalized by the velocity dispersion of
the cluster, i.e., $|$$\Delta$v$|$/$\sigma$sys. In the plot we also include
with a blue line the (projected) escape velocity from a Navarro Frenk and
White (NFW, Navarro et al., 1996) dark matter halo of M200 = 6 $\times$ 1012
M⊙, normalised by $\sigma$sys.The blue shadow region refers to escape
velocities for halo masses from 3 to 9 $\times$ 1012 M⊙. For our primary
reference (i.e., W at the CoGsys, zsys=6.902, red dots), all the galaxies are
below the escape velocity limit, and therefore they are bound gravitationally
to the halo. However, taking into account the uncertainties and the fact that
in the figure we present projected velocities corrected with an average
factor, a few galaxies could also be consistent with being unbound. When the
ad-hoc redshift (i.e., $z$ = 6.912, see above) is adopted (i.e., green
points), one galaxy (L3) appears unbound, though the rest are located deeper
into the potential well, i.e., showing on average lower
$|$$\Delta$v$|$/$\sigma$sys values.

In summary, our data suggest that the observed region in SPT0311–58 is likely
not totally virialized, though it may be not far from reaching equilibrium.
While a few galaxies could be fly-bys, the majority appear to be
gravitationally bound to the halo, for which we estimate a DM mass of $\sim$ 6
$\times$ 1012 M⊙. These results reinforce the findings by Marrone et al.
(2018) who showed that the halo mass for SPT0311–58 is extreme according the
theoretical expectations at z$\sim$ 6.90. This conclusion is strengthened
further by the fact that at this redshift, the most massive proto-clusters in
the IllustrisTNG cosmological simulations discussed by Lim et al. (2021) have
masses lower than SPT0311–58. According to these simulations, proto-clusters
with masses of $\sim$ 1012 M⊙ at z=7 will transform into 1015 M☉ systems at
z=0. These results suggest that SPT0311–58, for which independent approaches
find a DM halo mass higher than 1012 M⊙, could easily be the progenitor of a
Coma-like cluster.

The present study, although admittedly subject to large uncertainties,
represents one of the very first attempts to characterise a proto-cluster at
the epoch of reionization (see also, Endsley & Stark, 2022, Morishita et al.,
2023; Hashimoto et al., 2023, Scholtz et al., 2023), and it shows the power of
JWST for such studies. Further observations expanding the observed area will
enable us to better characterise the field, and constrain the dynamical stage
of this extreme system.

###  6.2 The turbulent early phases of galaxy assembly: accretion, mergers,
and gravitational interactions

As shown in Fig. 10, the E galaxy shows a significant metallicity gradient of
$\sim$ 0.7 dex, with the lowest values (i.e., 12-log(O/H) $\sim$ 7.9) found in
the northwest region. This is close to the zone where the H$\alpha$ and [CII]
emission extends towards the position of the C1 galaxy (see gray iso-contours
in Fig.10, and Fig. 2(c,f)), indicating the presence of significant amounts of
warm and cold gas. In this region, the stellar component, as traced by the
continuum map (left panel in Fig. 2), is not prominent. This indicates that
the outskirts of E in the northwest direction should be dominated by gas.
Moreover, the velocity field in the northwest part of E shows significantly
twisted iso-lines that cannot be explained as part of a global regular
rotation (Sec. 5). All these features can be naturally explained as the result
of the gas accretion of low-metallicity gas from the CGM into the main E
structure.

In lower redshift galaxies (i.e., z$\sim$ 3) metallicity gradients have also
been found, and explained by the accretion of primordial gas (Cresci et al.,
2010) as predicted by theoretical models (Dekel et al., 2009). This gas
accretion could be triggered by the interaction of E with the C1 galaxy, which
is only $\sim$ 100 km s-1 and 2 kpc apart in projection. According to the
velocity field of E and the radial velocity of C1, this possible encounter
would result in a prograde interaction, which is an efficient process at
driving gas inflows (Barnes & Hernquist, 1992, Cox, 2009, Lambas et al.,
2012). Alternatively, E and C1 may not be interacting (i.e., simply appear
close in projection), and the accretion of low-metallicity gas into E could be
the result of smooth streams of gas from the IGM (Sánchez Almeida et al.,
2014). This possibility is consistent with cosmological simulations that
predict that streams of inflowing gas from the cosmic web are the dominant
mechanism for the formation of galaxies in massive halos (e.g., Dekel et al.,
2009, Ceverino et al., 2016). As we have discussed in the previous section,
SPT0311–58 resides in a massive halo and therefore the presence of streams
feeding its central regions with gas from the IGM is naturally expected when
it is actively building its stellar component at rates $>$ 100s M⊙ yr-1.

The observed metallicity gradient could, in principle, also be explained by
the merging of two (or more) galaxies with a significant difference in
metallicity. The continuum maps (Fig. 1, Fig. 2(a)) show two distinct
structures in the E galaxy, i.e., ES and EN, which could be the result of two
separate pre-coalesce merging galaxies. The MIRI image (Fig. 2d) also shows a
structure to the north-west that is not seen in NIRSpec likely due to
obscuration. However, the kinematic properties in the main structure of E do
not show evidence for two kinematically distinct objects, such as peaks in the
velocity dispersion map, or rotation associated to their nuclear regions. The
[OIII] (Fig. 7 upper panels) and [CII] (Fig. 11) velocity fields also show a
smooth behaviour at large scales. The most significant distortion in the
velocity map is found at the northwest edge of the main structure (i.e., not
in an inner region), but the velocity dispersion is relatively low, which does
not suggest gas motions associated to strong interactions. Therefore, while we
cannot rule out completely the merging scenario, the ionised gas, metallicity,
and kinematic maps presented here favour the gas accretion scenario in the E
galaxy.

A third alternative to explain metallicity gradients is the presence of
outflows (e.g., Rodríguez Del Pino et al., 2023). However, we do not see any
clear evidence for outflows in our spectra. Moreover, if an outflow were
present (or had been active in recent times), its most likely orientation
would be along the minor axis, which is perpendicular to the direction of the
metallicity gradient. Therefore, the outflow scenario can hardly explain the
case of SPT.

We now discuss the properties of L7, and the possibility that it will merge
with E. The kinematic maps for these two galaxies are shown in Fig 7. One
clear feature that can be inferred from the figure is that L7 is reshifted by
$\sim$ 250 km s-1with respect to E. This fact has two implications. First,
this significant kinematic decoupling rules out that L7 has been formed by
disc fragmentation in contrast to the clumps identified by Spilker et al.
(2022) at sub-mm wavelengths. Second, the relative velocity between L7 and E
allows us to constrain the geometry. In fact, taking into account the dusty
nature of E, L7 has necessarily to be located between us and the main
structure of E, as otherwise it could not be observed through the dusty disk.
Therefore its receding velocity suggests that it is falling into E. Although
we cannot rule out that L7 has a large velocity component perpendicular to the
line of sight, its already large projected velocity with respect to W (i.e.,
+928 km s-1) makes this possibility unlikely (i.e., otherwise, its actual
velocity with respect to W would be extremely high). So, rather than a
fortuitous alignment of a fly-by source, the data suggest that we are
witnessing the pre-merger phase between L7-E. The mass ratio of both galaxies
is $\lesssim$ 1:8, and thus it should be classified as a minor merger event,
the likes of which are expected to occur frequently according to theoretical
models (e.g., Ceverino et al., 2016).

In Sect. 5 we have shown that, in addition to the irregular velocity field,
the velocity dispersion map of the E galaxy exhibits a non-axisymmetric
structure with variations at sub-kpc scales and a global trend along the
northwest - southeast direction (Fig. 7). The velocity dispersion is high
everywhere, with values ranging from $\sim$ 75 up to 150 km s-1, indicating
high turbulence. The velocity dispersion map does not show the expected
increase due to beam-smeared velocity gradients in the central regions, as
observed in disks formed at later cosmic times (e.g., Übler et al., 2019). The
sparse iso-velocity lines shown in Fig. 7 clearly indicate that the situation
in this galaxy is rather different.

Then, the present 2D kinematic maps (i.e., both the velocity field and the
velocity dispersion map) describe an ionised gas structure that departs from
what is expected for a settled rotating disk. The kinematics of the cold gas
component obtained with high-resolution [CII]$\lambda$158 ALMA observations
(Spilker et al., 2022), also reveal high velocity dispersions associated with
the clumps, indicating large turbulence. The mean radial velocities of the
clumps also follow a global velocity gradient, and thus they could be the
result of disc fragmentation. Hence, the main kinematic properties of the E
galaxy, as inferred from the ionised and neutral gas phases, differ
significantly from the settled and regular behaviour observed in some DSFGs at
lower redshifts (i.e., z $\sim$ 4-5, Fraternali et al., 2021; Rizzo et al.,
2021; Lelli et al., 2021; Parlanti et al., 2023). In fact, the E galaxy is
actively assembling its stellar component undergoing a highly turbulent phase
marked by the presence of gravitational instabilities, gas accretion, and
merger events. All of these processes also account for the irregularities and
sub-kiloparsec variations in the interstellar medium (ISM) conditions, as
evidenced by the ionization and metallicity maps.

Still, despite these tumultuous conditions, the velocity field in E shows a
clear large-scale velocity gradient of about 200 km s-1. If interpreted as
global rotation, the E galaxy will have several complete revolutions ($\sim$ 4
at r = 3 kpc) during the next $\sim$ 0.5 Gyr (i.e. by z $\sim$ 4.5). Together
with other mixing mechanisms, this will contribute to redistributing the gas
and erase the physical and kinematic inhomogeneities produced by present and
recent events. Therefore, it is conceivable that this process will make the
properties of the E galaxy in SPT0311–58 to evolve into those of more settled
DSFGs currently observed at z $\sim$ 4-5 (e.g., Rizzo et al., 2021, Parlanti
et al., 2023), unless additional accretion and/or mergers with nearby objects
continue to generate inhomogeneities in the ISM and disturb its kinematics.

##  7 Summary and Conclusions

We have presented JWST/NIRSpec IFU low (R=100) and high (R=2700) resolution
spectroscopy of SPT0311-58 at z $\sim$ 6.9, the most massive and distant dusty
star-forming system known, and a proto-cluster candidate. The main conclusions
of this paper can be summarised as follows:

i) Our observations have revealed ten new sources at redshift $\sim$ 6.9 (and
three additional foreground galaxies at z $\sim$ 1-3) in the field around the
already-known two main massive galaxies of the system, i.e., the E and W
galaxies. The newly discovered z $\sim$ 6.9 galaxies are characterised by
dynamical masses of $\sim$ 109-1010 M⊙, and a wide range in relative
velocities ($\Delta$v $\approx$ 1500 km s-1). The high density of sources
($\phi$ $\sim$ 10${}^{4}\ $Mpc-3) and their spread in radial velocity confirm
that SPT0311–58 is at the core of a very massive proto-cluster, for which we
estimate a mass of $\sim$ (6 $\pm$ 3) $\times$ 1012 M⊙ from its stellar
content. A basic analysis indicates that the innermost regions of the proto-
cluster (r ¡ 0.2 R200) have likely not reached virial equilibrium yet. We have
also located the galaxies in the Projected Phase Space (PPS) diagram, and find
that most are gravitationally bound to the halo. According to the inferred
mass for the system, IllustrisTNG cosmological simulations (Lim et al., 2021)
predict that this proto-cluster will evolve into a Coma-like system.

ii) The ISM conditions of the sources in SPT0311–58 are very diverse. While
some of these z $\sim$ 6.9 objects have ionization properties similar to
galaxies at these redshifts recently observed with JWST, others have
conditions mostly found at lower redshifts. In the N2-BPT diagnostic diagram,
most sources occupy the region of composite SF/AGN ionization at low-z, but we
do not find clear evidence for AGN. None are in the local LI(N)ER region, but
shocks are likely to be present in a turbulent ISM, which is expected to
experience disc fragmentation, interactions, and minor mergers. Similarly, the
metallicity varies by more than 0.7 dex across the galaxies in the FoV. The E
galaxy shows inhomogeneities at sub-kpc scales and a clear metallicity
gradient of $\sim$ 0.1 dex/kpc along its major axis. The regions with the
highest metallicity regions are associated (or close) to the two main massive
galaxies, with nearly solar values of $\sim$ 8.5-8.6 in the 12+log(O/H) scale.
Three external sources show the lowest metallicities with values of about
7.9-8.1.

iii) Using H$\alpha$ we have characterised the unobscured star-formation
(SFR(H$\alpha$)) in the system, and identified three distinct SF regimes.
First, the most massive W galaxy (Mbar $\sim$ 3 $\times$ 1011 M⊙) has a very
modest SFR(H$\alpha$) ($\sim$ 8 M⊙ yr-1) compared with the enormous obscured
SF detected with ALMA (i.e., SFR(FIR) $\sim$ 3000 M⊙ yr-1). Most of the SF in
this galaxy is totally blocked at optical and near-IR wavelengths, and cannot
be recovered with the dust attenuation correction. Second, the less massive,
though still very massive (Mbar $\sim$ 4 $\times$ 1010 M⊙) E galaxy, has
significant SFR(H$\alpha$) $\sim$ 80 M⊙ yr-1, with a dominant SFR (FIR) of
$\sim$ 500 M⊙ yr-1. The dust-corrected SF (H$\alpha$) recovers most of the
total SF of the system. Finally, the newly discovered sources have
SFR(H$\alpha$) $\sim$ 1-6 M⊙ yr-1, and no obscured SF is detected with ALMA.
Their contribution to the total SF of the system is minor ($\sim$ few
percent).

iv) The 2D physical, chemical and kinematic information of the E galaxy
provided by the present data has allowed us to characterise the current early
phases of its stellar mass assembly. The velocity field and the velocity
dispersion maps show asymmetric and irregular patterns indicating clear
departures from rotation, and high turbulence (intrinsic velocity dispersion,
$\sigma_{0},\approx$ 115 km s-1). We find large-scale motions between the
ionised gas component probed with NIRSpec and the cold gas phase counterpart
traced by ALMA. The observed metallicity gradient is naturally explained by
accretion of low-metallicity gas from the IGM. We also identify a possible
(pre-collision) minor merger event. All these results suggest a scenario in
which the E galaxy is undergoing a tumultuous phase while actively assembling
its stellar component. However, the large global velocity gradient observed
suggests that it could evolve into a regular settled rotating DSFG, similar to
those observed at later cosmic times (i.e., z $\sim$ 4-5).

The present study highlights the role of SPT0311–58 as an ideal astrophysical
laboratory to study galaxy evolution in a very massive dark matter halo at the
epoch of the reionization. It also shows the enormous potential of the
JWST/NIRSpec IFU for detailed spatially-resolved analyses at these very early
cosmic times.

##  Appendix A Spectral characterization: Methods

###  A.1 Definition of extraction apertures for regional integrated spectra

Figure 14: Selected apertures. Left panel: Yellow lines indicate the apertures
for the z $\sim$ 6.9 sources, while with cyan is used for the low-z objects,
including the lens galaxy. For the W galaxy we use a dashed line to indicate
that this aperture is not defined on the basis of the NIRSpec data, but from
the 160 $\mu$m continuum ALMA image. Apertures for the subregions within the E
galaxy are indicated with pink lines. The red region in this panel marks the
region used to obtain the background spectrum. Right panel: Additional
apertures, including the ones with 1 arcsec in diameter (i.e., E1c and W1c, in
orange lines) used by Álvarez-Márquez et al. (2023). In this panel the red
regions represent the spaxels used to probe the diffuse external emission
outside the E and W galaxies (i.e., the Eext and Wext apertures).

We have defined a set of extraction apertures that have been used to obtain
the integrated spectra for the sources in SPT0311–58 (Fig. 14). The definition
of these apertures is, in some cases, difficult because the emission
distribution of the lines and the continuum is different, and also because the
presence of artefacts force us to exclude some spaxels from the aperture that
would have been otherwise included. As a consequence, the absolute fluxes
inferred from the spectra should be taken with caution, also considering that
the calibration of the instrument is not consolidated yet. However, the
velocity determinations, and the flux line ratios should be reliable within
the uncertainties imposed by the S/N. In addition, we have obtained and
applied specific aperture corrections to the individual apertures, which
(partially) correct the derived absolute fluxes (see next section).

Fig. 14 shows the apertures used. On the left panel, yellow lines indicate the
apertures for the z $\sim$ 6.9 sources. For the W galaxy we use a dashed line
to indicate that this aperture is not defined on the basis of the NIRSPec
data, but from the 160 $\mu$m continuum ALMA image. Sub-regions within the E
galaxy are indicated with pink lines. For the low-z objects the apertures are
displayed with cyan lines (see also Fig. 3 for a simplified version of this
figure). The red region in this panel represents the region used to obtain the
background spectrum. In the right panel, we show additional apertures,
including the ones with 1 arcsec in diameter (i.e., E1c and W1c, in orange
lines) used to compare with the Pa$\alpha$ measurements obtained with MIRI
(Álvarez-Márquez et al. (2023)). In this panel the red regions represent the
spaxels used to probe the external diffuse external emission outside the E and
W galaxies (i.e., the Eext and Wext apertures). These extended apertures were
only used to probe the H$\alpha$ emission at the redshift of SPT0311-058, so
the fact that for the W it includes some emission from the lens galaxy is not
critical. Nevertheless, as it can be seen in the figure, we exclude the
spaxels with the brightest emission from the lens galaxy to minimise the
noise.

###  A.2 Aperture corrections

We have obtained the aperture corrections making use of the high-resolution
G395H/290LP observations of the standard star 1808347 (PID:1128, Obs. 9; PI:
N. Luetzgendorf), which were reduced using the same pipeline version and
context as for SPT0311-58. The aperture masks were shifted such that they were
centered with the peak of the star emission. Then we define filters of about
200 spectral pixels around two wavelengths (3.4 aand 5.2 $\mu$m), and the
fractions of light contained within the aperture with respect to the total in
the FoV were computed. The results are summarised in Table 11. Note that for
Eext and Wext, defined for tracing the extended emission, the aperture
correction were not obtained. For L7, characterised with a 2D spectral
decomposition, the aperture correction is not either needed.

Table 3: Aperture positions and corrections

ID | X | Y |  $\Delta$$\alpha$ (”) |  $\Delta$$\delta$ (”) | AC-3.4 | AC-5.2
---|---|---|---|---|---|---
E | 35 | 52 | +1.00 | -0.15 | 0.87 | 0.83
Ecir1 | 34 | 52 | +1.05 | -0.15 | 0.92 | 0.89
ESb | 33 | 43 | +1.10 | -0.60 | 0.72 | 0.65
EN | 36 | 57 | +0.95 | +0.10 | 0.81 | 0.75
ENa | 34 | 52 | +1.05 | -0.15 | 0.50 | 0.44
ENb | 36 | 57 | +0.95 | +0.10 | 0.63 | 0.55
ENc | 36 | 62 | +0.50 | +0.15 | 0.46 | 0.40
W | 73 | 41 | -0.90 | -0.70 | 0.89 | 0.86
Wcir1 | 73 | 41 | -0.90 | -0.70 | 0.92 | 0.89
C1 | 49 | 70 | +0.30 | +0.75 | 0.76 | 0.70
C2 | 55 | 80 | +0.00 | +1.25 | 0.72 | 0.66
C3 | 80 | 32 | -1.25 | -1.15 | 0.51 | 0.45
L1 | 83 | 61 | -1.40 | +0.30 | 0.64 | 0.58
L2 | 67 | 73 | -0.60 | +0.90 | 0.55 | 0.49
L3 | 32 | 80 | +1.15 | +1.25 | 0.60 | 0.54
L4 | 45 | 58 | +0.50 | +0.15 | 0.62 | 0.56
L5 | 24 | 35 | +1.55 | -1.00 | 0.68 | 0.61
L6 | 49 | 36 | +0.30 | -0.95 | 0.59 | 0.56
lz1 | 28 | 32 | +1.35 | -1.15 | 0.34 | 0.27
lz2 | 17 | 39 | +1.90 | -0.80 | 0.54 | 0.48
lz3 | 27 | 42 | +1.40 | -0.65 | 0.61 | 0.54
111111Column 1: Aperture identification, Columns 2-3: X and Y coordinates in
the cube for the central spaxel of the aperture, Columns 4-5: relative sky
coordinates with respect to the center of the FoV (i.e., spaxel [55,55], sky
coordinates: $\alpha$ = 03h 11m 33s.248 and $\delta$ = -58º 23’ 33”.24).
Columns 6-7: Aperture corrections at 3.4 $\mu$m and & 5.2 $\mu$m, which
correspond approximately to the observed wavelength of H$\gamma$ and H$\alpha$
at z=6.90. 1 Circular aperture of 1 arcsec in diameter.

###  A.3 High resolution (R2700) integrated spectra

The integrated R2700 spectra for the sources and regions in SPT0311–58 are
presented in Fig. 15. The R100 spectra were presented in Fig. 5.

Figure 15: R2700 spectra in the region of the
[OII]$\lambda$$\lambda$3726,3728, H$\beta$-[OIII]$\lambda$$\lambda$4959,5007,
and H$\alpha$-[NII]$\lambda$6584 for the sources and regions in SPT0311-58.
The vertical lines mark the central position of these lines at the redshift of
the source.

###  A.4 Line fitting

The emission lines in the R100 and R2700 integrated spectra were fit to single
Gaussian profiles linking their relative fluxes and wavelengths according to
their atomic parameters, and imposing the same line width for all the lines.
The spectral ranges [OII], H$\beta$-[OIII], and H$\alpha$-[NII] were fitted
independently, though for the [OII] and H$\alpha$-[NII], the kinematic
parameters were constrained to the one derived for the H$\beta$-[OIII]
complex.

In the R100 spectra H$\alpha$ and the [NII] lines are not well resolved and
the results of the fit may be degenerated, especially at low S/N. The
[OII]$\lambda\lambda 3726,3728$ doublet is unresolved at R2700 and fit with a
single Gaussian, as in this case we were only interested in the total flux
(i.e., not in the kinematics).

For the regions with the highest S/N (i.e., E galaxy), line fitting at spaxel
level (or with moderate binning) was performed. The R2700 spectra were used to
disentangle two kinematically distinct components following the approach
described in Perna et al. (2022) (see Sect. 3.3.3).

###  A.5 Line fluxes

Table 4 gives the line fluxes and their uncertainties obtained from the
Gaussian fits of the R100 and R2700 spectra for the [OII]$\lambda\lambda
3726,3728$, H$\beta$, [OIII]$\lambda 5008$, H$\alpha$, and [NII]$\lambda 6584$
lines.

As the main lines used in this work appear in both the R100 and R2700
datacubes, this give us the opportunity to compare the derived fluxes with the
two spectral configurations (see Fig. 16). As it is shown in the Figure, the
fluxes span more than two orders of magnitude and indicate in general good
agreement. There is however a systematic trend in the sense that R100 fluxes
are about 10 percent higher than R2700 ones. As the R100 data are subject to
larger uncertainties in the absolute calibration (Charles Profitt, priv.
comm.), we applied this 10 percent correction to the R100 data. After taking
into account this factor fluxes typically agree better than 20 percent, except
few cases. There is not a configuration (i.e., R100 or R2700) that is
generally preferred for obtaining the fluxes. Their spectra have S/N that in
many cases are comparable, though they may differ on other data quality
aspects. For instance, the H$\alpha$ line at the redshift of SPT0311–58 is
very close the edge of the spectra, leaving in the case of R100 few pixels for
the determination of the continuum redwards the line. Moreover, in R100
spectra H$\alpha$ and the [NII] lines are not resolved, which makes the fits
degenerated especially in a low S/N regime. On the other hand, the R2700
datacube is affected in some cases by artefacts that could not be removed by
the pipeline nor by a posterior processing. Table 4 gives further details on
the derived fluxes for the lines.

Figure 16: Comparison of the line fluxes obtained from R100 and R2700
integrated spectra. Fluxes from R100 spectra are reduced by 10 percent (see
text). The dashed lines correspond to the 1:1 and 1:1.1 ratios. Colors
distinguish between lines, and green text labels indicate the
aperture/spectrum.  Table 4: Line fluxes for the sources and regions in the
FoV1

Sp | OII-R100 | OII-R2700 |  H$\beta$-R100 |  H$\beta$-R2700 | OIII-R100 | OIII-R2700 |  H$\alpha$-R100 |  H$\alpha$-R2700 | NII-R100 | NII-R2700
---|---|---|---|---|---|---|---|---|---|---
E |  8.71$\pm$0.48 |  8.52$\pm$0.43 |  4.50$\pm$0.17 |  4.46$\pm$0.24 |  15.49$\pm$0.19 |  14.48$\pm$0.28 |  21.03$\pm$0.33 |  18.59$\pm$0.56 |  5.54$\pm$0.27 |  4.52$\pm$0.48
E1c |  8.08$\pm$0.56 |  8.91$\pm$0.56 |  4.65$\pm$0.19 |  4.77$\pm$0.29 |  15.46$\pm$0.22 |  14.47$\pm$0.34 |  22.91$\pm$0.52 |  18.57$\pm$0.71 |  5.24$\pm$0.43 |  4.12$\pm$0.61
Eext |  |  |  |  |  10.04$\pm$0.56 |  10.77$\pm$0.83 |  15.38$\pm$1.512 |  13.82$\pm$2.34 |  |  0.14$\pm$2.64
ESb |  1.95$\pm$0.18 |  1.51$\pm$0.17 |  0.70$\pm$0.08 |  0.70$\pm$0.08 |  2.00$\pm$0.09 |  1.98$\pm$0.09 |  3.88$\pm$0.11 |  3.09$\pm$0.25 |  0.98$\pm$0.09 |  1.25$\pm$0.23
EN |  4.74$\pm$0.24 |  4.39$\pm$0.25 |  2.81$\pm$0.10 |  2.42$\pm$0.14 |  10.26$\pm$0.11 |  9.14$\pm$0.16 |  11.88$\pm$0.20 |  10.49$\pm$0.32 |  3.33$\pm$0.17 |  1.85$\pm$0.27
ENa |  1.15$\pm$0.07 |  1.41$\pm$0.11 |  0.73$\pm$0.04 |  0.86$\pm$0.07 |  2.93$\pm$0.05 |  2.95$\pm$0.08 |  3.02$\pm$0.05 |  3.34$\pm$0.20 |  0.81$\pm$0.04 |  0.88$\pm$0.19
ENb |  1.86$\pm$0.12 |  1.35$\pm$0.10 |  1.00$\pm$0.05 |  0.87$\pm$0.05 |  3.47$\pm$0.05 |  3.10$\pm$0.06 |  4.06$\pm$0.10 |  3.39$\pm$0.12 |  1.21$\pm$0.08 |  0.56$\pm$0.10
ENc |  0.40$\pm$0.05 |  0.35$\pm$0.07 |  0.20$\pm$0.04 |  0.14$\pm$0.03 |  0.91$\pm$0.04 |  0.69$\pm$0.03 |  0.99$\pm$0.05 |  0.76$\pm$0.06 |  0.22$\pm$0.04 |  0.16$\pm$0.05
W 3 |  |  |  |  1.28$\pm$0.28 |  |  |  4.93$\pm$0.52 |  |  1.08$\pm$0.41 |
W1c |  |  |  |  1.10$\pm$0.24 |  |  |  2.78$\pm$0.32 |  3.02$\pm$0.49 |  1.13$\pm$0.26 |  1.32$\pm$0.41