11institutetext: Instituto de Radioastronomía Milimétrica, Av. Divina Pastora
7, Núcleo Central E-18012, Granada, Spain 22institutetext: Instituto de
Astrofísica e Ciências do Espaço, Universidade de Lisboa, OAL, Tapada da
Ajuda, 1349-018 Lisbon, Portugal 33institutetext: Departamento de Física,
Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande,
1749-016 Lisbon, Portugal 44institutetext: Department of Physics and
Astronomy, University of Hawai‘i at Mānoa, 2505 Correa Rd., Honolulu, HI,
96822, USA 55institutetext: Institute for Astronomy, University of Hawai‘i,
2680 Woodlawn Dr., Honolulu, HI, 96822, USA 66institutetext: School of
Mathematical and Physical Sciences, 12 Wally’s Walk, Macquarie University, NSW
2109, Australia 77institutetext: Joint ALMA Observatory, Alonso de Córdova
3107, Vitacura 763-0355, Santiago de Chile, Chile 88institutetext: European
Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001,
Santiago de Chile, Chile 99institutetext: International Centre for Radio
Astronomy Research, Curtin University, 1 Turner Avenue, Bentley, WA, 6102,
Australia

# Tracing obscured galaxy build-up at high redshift using deep radio surveys

Stergios Amarantidis<EMAIL_ADDRESS>Jose Afonso 2233 Israel Matute
2233 Duncan Farrah 4455 A. M. Hopkins 66 Hugo Messias 7788 Ciro Pappalardo
2233 N. Seymour 99

(Received March 14, 2023; accepted July 25, 2023)

###### Abstract

Context. A fundamental question of extra-galactic astronomy that is yet to be
fully understood, concerns the evolution of the star formation rate (SFR) and
supermassive black hole (SMBH) activity with cosmic time, as well as their
interplay and how it impacts galaxy evolution. A primary focus that could shed
more light on these questions is the study of merging systems, comprising
highly star-forming galaxies (SFGs) and active galactic nuclei (AGN) at the
earliest stages of galactic formation. However, considering the challenges
associated with identifying these objects, it is essential to explore
complementary selection methods across multiple wavelengths.

Aims. The primary objective of this study is to conduct a comprehensive
analysis of a sample of high-redshift ($z>3$) far-infrared (far-IR) and radio-
emitting galaxies in the highest possible spatial resolution. The aim is to
study the properties of this population, such as their morphological
characteristics, and to explore the interplay of SFR and SMBH activity at this
epoch.

Methods. In order to select the galactic population of our interest, we
employed two selection criteria that have frequently been used as separate
methods in the literature. In more detail, we selected galaxies that present
relatively compact radio morphologies at 1.4 GHz (i.e. an angular size smaller
than 10 arcsec) as well as a far-IR spectrum that peaks in flux at
$\lambda\geq 350\,\mu m$ (i.e. $flux_{350\mu m}>flux_{250\mu m}$). For these
selection criteria, we used the COSMOS and ECDF-S fields, two of the most
extensively observed astronomical fields currently available, which provide
high spectral and spatial resolution at a multi-wavelength scale. By accepting
only galaxies that satisfied these selection criteria, we derived a sample of
eight galaxies that were identified either photometrically or
spectroscopically at $z>3$ from literature studies and by our team.

Results. A thorough investigation of available optical, near-IR, and
millimetre (mm) imaging reveals a possible merging scenario in five out of
eight cases in our sample. Additionally, available multi-wavelength photometry
strongly suggests active star formation at the $10^{3}\,M_{\odot}/yr$ level in
massive systems (stellar masses of $M_{\star}\sim 10^{11}\,M_{\odot}$) co-
hosting an active SMBH.

Conclusions. Comparison of these results with previous studies, suggests that
our selection method preferentially identifies galaxies hosting an active
SMBH, as well as a strong SFG component, resulting in high SFR and IR
luminosity. An additional examination of the efficacy of the radio and far-IR
selection criteria provides further support for their combined application in
selecting co-evolving AGN and star formation activity at high redshift. In
this regard, future use of these selection criteria on radio and far-IR/mm
observations of statistically larger galaxy samples is of high interest.

###### Key Words.:

star formation – radio galaxies – high redshift

##  1 Introduction

One of the main challenges in the field of extra-galactic astronomy is
understanding how the star formation rate density (SFRD -
$M_{\odot}/yr/Mpc^{3}$) evolves with redshift (see Madau & Fragos, 2017; Oesch
et al., 2018; Tacconi et al., 2020, for a detailed review). While we have
accurate estimates of the SFRD at intermediate redshifts ($1<z<3$) using
various selection methods from deep multi-wavelength data (e.g. Gruppioni et
al., 2020; Novak et al., 2017; Ishigaki et al., 2018; Dunlop et al., 2017, for
infrared (IR), radio, ultraviolet-UV and millimetre (mm) criteria,
respectively) that agree with results from hydrodynamical simulations (e.g.
Pillepich et al., 2018; Lagache, 2018), uncertainties regarding higher
redshifts ($z>3$) still remain (e.g. Magnelli et al., 2019; Gruppioni et al.,
2020). For instance, studies of UV-selected galaxy samples (e.g. McLeod et
al., 2016; Ishigaki et al., 2018) estimate a steep decline in the SFRD at
$z>3$, while other works employing far-IR radio selection criteria (e.g. Viero
et al., 2013; Rowan-Robinson et al., 2016; Novak et al., 2017; Malefahlo et
al., 2022) or gamma-ray-burst-selected samples (e.g. Yüksel et al., 2008;
Kistler et al., 2009) suggest a flatter behaviour.

These uncertainties may originate from our current instrumentational
limitations, including low sensitivity, coarse spatial resolution, and
coverage of small parts of the sky, which hinder our ability to properly
characterise distant objects and obtain a statistically unbiased sample at
higher redshifts. For instance, observations based on telescopes such as the
Herschel Space Observatory (Pilbratt et al., 2010) or the Submillimetre
Common-User Bolometer Array (SCUBA; Holland et al., 2013) are bound to limit
the redshift detection to $z_{\rm max}\sim 4$ (e.g. Donevski et al., 2018),
and to uncover only the very bright end of the far-IR luminosity function
(e.g. Gruppioni et al., 2013; Wang et al., 2021). On the other hand, more
sensitive telescopes, such as the Atacama Large Millimeter/submillimeter Array
(ALMA; Wilson et al., 2019) or the Northern Extended Millimeter Array (NOEMA;
Chenu et al., 2016) are limited to smaller areas. Thereby, these telescopes
provide fewer objects (e.g. Dunlop et al., 2017; Faisst et al., 2020; Casey et
al., 2021) and reveal a fainter star-forming galaxy (SFG) population (e.g.
Pantoni et al., 2021). Nevertheless, recent studies using the capabilities of
both single telescopes and interferometers have pushed the limits of our
current understanding regarding the star formation (SF) history, exposing a mm
dusty galaxy population at $z>3$ (see e.g. Donevski et al., 2018; Yamaguchi et
al., 2019; Gruppioni et al., 2020).

Another aspect of our current limitations regarding the accurate estimation of
the SFRD at high redshifts might be hidden in the selection criteria that are
currently used to detect SFGs. Various methods have been employed over the
years in order to detect this galaxy population (e.g. Gruppioni et al., 2013;
Bouwens et al., 2015), however, the bulk of the research is focused on the
detection of Lyman-break galaxies in the rest frame UV and the application of
dust-correction low-redshift empirical relations for the derivation of their
star formation rates (e.g. Bouwens et al., 2015; Oesch et al., 2018). These
relations might not be necessarily valid at $z>3,$ where the SFG population
remains uncertain. Alternatively, recent studies using sensitive far-IR or mm
telescopes (e.g. the ALPINE-ALMA survey; Gruppioni et al., 2020) or radio
surveys (e.g. Talia et al., 2021; Enia et al., 2022) aim to reveal a an SFG
population that has so far remained undetected in the optical/UV bands, which
is estimated to significantly contribute to the SFRD at $z>3$ (e.g. Gruppioni
et al., 2020). In order to obtain accurate SFR estimates, these studies
separate the SFG from the AGN population (e.g. Lanzuisi et al., 2017;
Gruppioni et al., 2020) by removing the AGN contribution from the IR
luminosity. However, this approach can introduce a bias in the sample.
Interestingly, results from the recent surveys of the James Webb Space
Telescope (JWST) (e.g. Harikane et al., 2023; Nakajima et al., 2023; Pérez-
González et al., 2023; Rodighiero et al., 2023) reveal a dusty-galaxy
population like this at even higher redshifts (i.e. $z>6$) that have not been
explored before.

Previous studies, such as the work by Duncan et al. (2019), indicated that the
galaxy pair and merging ratio increases with redshift (from $\sim 5\,\%$ at
$z<1$ to $\sim 40\,\%$ at $z>3$ and for $M_{\star}>10^{10.3}\,M_{\odot}$).
Therefore, in this work, we question the strategy of removing AGN-contaminated
radio sources from selected samples as this approach may overlook merging
systems in which the AGN emission conceals an additional SFG population that
could significantly contribute to the SFRD at high redshifts. Such galaxies
could be detected from the identification of far-IR bright Herschel galaxies
along with a radio emission. Even though these selection criteria might favour
sources with cold gas and negative k-correction, their application to fields
with deep coverage (e.g. the COSMOS field; Scoville et al., 2007) along with
photometric data and high spatial resolution information from telescopes such
as the Hubble Space Telescope (HST) and ALMA could potentially reveal a
population of SFGs that would be omitted by the current selection criteria
because of the AGN contribution of a neighbour galaxy.

In an attempt to explore new techniques to assist radio surveys reaching
$z>3$, we present a complementary study combining the radio selection and far-
IR peak spectrum criteria (e.g. Yan et al., 2022). Furthermore, we suggest
that at $z\sim 3-4$, a substantial amount of star formation occurs that is
hardly revealed from optical or near-IR surveys, and which is consequently
excluded by the current selection methods due to the AGN contribution in the
system. In more detail, we explore a sample of $z>3$ far-IR radio-emitting
galaxies, detected in the COSMOS and ECDF-S fields, that might uncover
interacting systems and merging galaxies with a substantial contribution of an
AGN component, along with the presence of high star formation activity. If
radio emission can help to pinpoint this important ‘missing’ SF component,
upcoming radio surveys, such as the Evolutionary Map of the Universe (EMU)
survey (e.g. Norris, 2009; Norris et al., 2021) as well as novel selection
criteria in the radio (e.g. Talia et al., 2021; Enia et al., 2022) could
become even more important to our understanding of the SFRD at $z>3$.

The paper is structured as follows: in Sect. 2 we present the method and
criteria that were employed in the selection of the original sample and the
subsequent reduction of the sample to the high-redshift far-IR radio
candidates. Section 3 briefly highlights the photometric and spectroscopic
properties of the sample, along with the results obtained from various
properties of the entire sample. In Sect. 4 we compare our work with previous
studies and provide a brief discussion of the results, emphasising the
importance of the merging population for SFRD estimations. Finally, we provide
the conclusions of this study in Sect. 5 and discuss future prospects
regarding our method. Throughout this paper, we assume a $\Lambda$-CDM
cosmology using $H_{\rm 0}=70\,kms^{-1}Mpc^{-1}$, $\Omega_{\rm m}=0.3$, and
$\Omega_{\rm\Lambda}=0.7$.

##  2 Sample selection

Figure 1: Flux density ratios of 350 and 250 $\mu m$ vs the ratios of 500 and
350 $\mu m$ of HerMES in COSMOS (panel a) and ECDF-S (panel b). The sample was
selected by cross-matching the HerMES catalogue with the VLA-COSMOS and VLA-
ECDFS data. The dashed black lines indicate $f_{350}/f_{250}$=1 and
$f_{500}/f_{350}$=1, while the points surpassing the limit of
$f_{350}/f_{250}$¿1 are presented with larger markers. The colour bar
corresponds to the 350 $\mu m$ flux of each source, and the legend shows the
number of sources for each panel and the selected high-redshift candidates
(crosses). The black lines in the subregion of each panel provide information
regarding the shape of the far-IR (FIR) emission. For instance, sources with
$f_{500}/f_{350}$¡1 and $f_{350}/f_{250}$¿1 have a peak in flux between 250
and 500 $\mu m$. The photometric properties of the sources presented in the
legend are provided in Table 1.

For our selection, we examined two of the most deeply explored extra-galactic
fields that were observed under the projects Extended Chandra Deep Field-South
(ECDF-S; Lehmer et al., 2005) and Cosmic Evolution Survey (COSMOS; Scoville et
al., 2007). In past years, these two fields have been extensively observed in
a wide range of electromagnetic frequencies, ranging from X-rays to radio
waves. Radio data were obtained from both fields at 1.4 GHz with the VLA (VLA-
COSMOS; the flux density limit is 7 $\mu Jy/beam,$ and VLA-ECDF-S; the flux
density limit is 7.4 $\mu Jy/beam$; Schinnerer et al., 2010; Smolčić et al.,
2017; Miller et al., 2013) at an angular resolution of $\sim
2.5^{\prime\prime}$.
The far-IR wavelength regime, which is necessary for our selection, is
available through the Herschel Multi-tiered Extragalactic Survey (HerMES;
Oliver et al., 2012). This survey covered a total of 380 $deg^{2}$ using the
two main instruments on board Herschel , namely the Spectral and Photometric
Imaging Receiver (SPIRE; Griffin et al., 2010; Nguyen et al., 2010, ; at 250,
350 and 500 $\mu m$; the confusion noise level is $\sim 5.8$, $\sim 6.3,$ and
$\sim 6.8\,mJy$, respectively) and the Photodetector Array Camera and
Spectrometer (PACS; Poglitsch et al., 2010, ; at 100 and 160 $\mu m$; the
detection limits are $\sim 7.7$ and $\sim 14.5\,mJy$, respectively). The ECDFS
and COSMOS fields are also included in the covered area of HerMES, comprising
a total of $\sim 2\,deg^{2}$. Previous studies using HerMES data have
demonstrated the efficient use of Herschel’s far-IR fluxes for detecting high-
redshift ($z>3$) dusty galaxies. For instance, Dowell et al. (2014) employed
simple criteria for the increasing fluxes for the 250, 350, and 500 $\mu m$
bands, which is indicative of a far-IR peak at longer (redshifted) $>350\,\mu
m$ wavelengths, revealing a dusty SF galaxy population with a mean redshift
distribution of $<z>\sim 4.7$.
To identify potential high-redshift hybrid AGN-SFG systems, we conducted a
cross-match analysis between the VLA COSMOS and ECDF-S 1.4 GHz catalogues with
those of HerMES, applying a maximum separation of 18 arcsec (motivated by the
beam size of Herschel at 250 $\mu m$ of 18.1 arcsec). The resulting cross-
matched sample contained 2292 sources for COSMOS, and for the ECDF-S field is
provided 769 sources. This sample was further reduced by considering robust
radio sources (signal-to-noise ratio; S/N¿5) without an indication for
extended or multicomponent radio emission (according to each catalogue; for
more details, see Lehmer et al., 2005; Scoville et al., 2007). To ensure the
correspondence between radio and far-IR emission, we excluded from the sample
sources that present multiple radio detections within one Herschel beam.
Furthermore, only sources with 250, 350, and 500 $\mu m$ flux errors smaller
than 50 $\%$ were accepted, which ensured that the shape of the far-IR
emission was properly determined. This selection method resulted in 219 and 93
sources for COSMOS and ECDF-S, respectively.
Furthermore, motivated by the efficacy of studies such as Dowell et al. (2014)
and Donevski et al. (2018) in selecting high-$z$ dusty SF galaxies, we adopted
a similar method to identify sources with $f_{350\rm\mu m}/f_{250\rm\mu m}>1$,
which would translate into a potential $z>3$ redshift. This selection method
resulted in a sample of 53 and 30 galaxies (see Fig. 1) for the COSMOS and
ECDF-S fields, respectively. The radio 1.4 GHz and far-IR ($250\,\mu m$)
images of each of these sources were visually inspected in order to secure the
robustness of the matching process and identify potentially uncertain or
confused radio/far-IR detections. We confirmed that all 83 selected sources
display compact morphologies both in the radio and in the FIR, and that the
detected counterparts are robust, with average separations between the radio
and far-IR detections below $\sim 2$ arcsec.

### High-redshift candidates

Application of these far-IR criteria resulted in a sample of $53+30=83$
sources from both fields for which we retrieved possible photometric or
spectroscopic identifications from the literature. For this exploration, we
consulted Simbad (Wenger et al., 2000), the
NED111https://ned.ipac.caltech.eduhttps://ned.ipac.caltech.edu online search
tools, and the Cosmos2015 (Laigle et al., 2016) online
catalogue222https://irsa.ipac.caltech.edu/cgi-bin/Gator/nph-
scan?projshort=COSMOShttps://irsa.ipac.caltech.edu, retrieving redshift values
from sources that are within 2 arcsec of our candidates.
From this analysis, we found 18 sources in the ECDF-S and 19 sources in the
COSMOS field with spectroscopic redshift determination. Five of these radio
sources present a spectroscopic redshift of $z_{\rm spec}>3$, namely
J033140.1-275631 (ECDF-S field; $z_{\rm spec}$=5.14; Suh et al., 2015),
J100028.71+023203.7 (COSMOS field; $z_{\rm spec}$=4.76; Hasinger et al.,
2018), [LBX2017] 214 (ECDF-S field; $z_{\rm spec}$=3.74; Iwasawa et al.,
2012), COSMOSVLA J100256.53+021158.4 (COSMOS field; $z_{\rm spec}$=3.503;
Pâris et al., 2014), and [DSS2017] 554 (ECDF-S field; $z_{\rm spec}$=3.1988;
Danielson et al., 2017). The reported spectroscopic redshifts of the two
highest- $z$ sources, however, are not reliable. For instance, the candidate
with the highest redshift, found in the ECDF-S, reveals contradicting
spectroscopic measurements. More specifically, the $z_{\rm spec}=5.137$ value
(derived from near-IR spectroscopy; Suh et al., 2015) contradicts Coppin et
al. (2012) and Danielson et al. (2017), who estimated a lower spectroscopic
value of $z_{\rm spec}=1.617$ (with spectra from optical and far-IR
wavelengths) for this candidate. Because a high-redshift nature of this source
cannot be excluded, we further explored its available spectra. This
investigation indicated that the emission line responsible for the $z_{\rm
spec}$=5.137 solution can be also interpreted as an emission line at $z_{\rm
spec}$=1.617, which further supports the notion for the intermediate-redshift
solution derived in the literature. Therefore, we considered this source as an
intermediate-redshift galaxy and did not explore it further.

Figure 2: Histograms presenting the number of sources for each redshift bin selected by the criteria for the COSMOS (empty histograms) and ECDF-S (filled histograms). The blue histograms correspond to spectroscopic values derived from the literature, and the orange histograms show the photometric values. The combined sample is depicted with an empty black histogram. The number of sources for each category is presented in the legend. Table 1: Names, abbreviations, equatorial coordinates, flux densities at 1.4 GHz, 250 $\mu m$, 350 $\mu m$ and 500 $\mu m$ (along with their total error - instrumental and confusion), and angular diameter (derived from the 1.4 GHz emission) for our high-redshift ($z_{\rm spec,phot}>3$) radio candidates, ordered by $f_{\rm 1.4GHz}$ (descending order). The second row of the table corresponds to the units of each parameter. Source name | Abbreviation | RA | DEC | $f_{\rm 1.4GHz}$ | $f_{\rm 250}$ | $f_{\rm 350}$ | $f_{\rm 500}$
---|---|---|---|---|---|---|---
– | – | [deg] | [deg] | [mJy] | [mJy] | [mJy] | [mJy]
CVLA J100256.53+021158.4 | CVLA1002 | 150.73553 | 2.19963 | $0.104\pm 0.020$ | $37.0\pm 3.4$ | $39.7\pm 6.6$ | $25.1\pm 5.1$
ALESS J033152.49-280319.1 | ALESS14 | 52.96871 | -28.05531 | $0.103\pm 0.016$ | $13.0\pm 6.2$ | $27.7\pm 3.9$ | $26.7\pm 4.5$
CVLA J100004.81+023045.2 | CVLA1000 | 150.0200 | 2.51255 | $0.096\pm 0.015$ | $25.1\pm 6.3$ | $27.8\pm 6.6$ | $23.1\pm 5.3$
CVLA J095845.94+024329.2 | CVLA958 | 149.69144 | 2.72479 | $0.095\pm 0.018$ | $27.4\pm 6.3$ | $27.7\pm 6.6$ | $28.7\pm 5.4$
CVLA J100233.16+020626.5 | CVLA100 | 150.63817 | 2.10736 | $0.088\pm 0.015$ | $22.3\pm 6.3$ | $24.0\pm 6.6$ | $17.5\pm 5.3$
CVLA J100028.75+023203.5 | CVLA028 | 150.11958 | 2.53438 | $0.057\pm 0.015$ | $28.6\pm 6.3$ | $36.7\pm 6.6$ | $29.3\pm 5.2$
ALESS J033211.34-275211.9 | ALESS33 | 53.04738 | -27.87003 | $0.043\pm 0.012$ | $18.6\pm 6.2$ | $25.6\pm 3.9$ | $25.4\pm 4.7$
$[$DSS2017$]$ 554 | ECDFS544 | 53.42108 | -27.92731 | $0.042\pm 0.014$ | $24.5\pm 3.4$ | $27.0\pm 3.9$ | $23.8\pm 3.9$

Likewise, for J100028.71, inspection of its DEIMOS 10k survey spectrum
(Hasinger et al., 2018) revealed a very low S/N spectrum without an obvious
line detection that could explain the $z=4.76$ indication. We assumed an
erroneous redshift estimate caused by a possible false-positive of the
automated feature recognition from the DEIMOS survey. This source is however
kept in this work as a high-redshift source, prompted by a photometric
redshift determination of $z_{\rm phot}\sim 3.2$ by Miettinen et al. (2015).
Inspection of the remaining three sources with a spectroscopic redshift higher
than 3 revealed that the emission lines used to determine the redshifts of
these sources have either a high S/N (i.e. S/N$>10$) or are denoted with a
secure spectroscopic quality flag. This means that these measurements are
accurate.
Our search identifies 12 sources in the ECDF-S field and 34 sources in the
COSMOS field based on photometric redshifts from the literature. These results
are depicted in Fig. 2, where the histograms for the redshifts of each survey
and the total values are plotted with different colours. This figure reveals
that our selection favours radio sources around $z\sim 2$ with a distribution
that extends to higher redshifts. Four sources of this sample in the COSMOS
field have photometric redshifts (identified in the Cosmos2015 catalogue) of
$z>3$ (i.e. COSMOSVLA J100233.16+020626.5 with $z_{\rm phot}$=3.016, COSMOSVLA
J095845.94+024329.2 with $z_{\rm phot}$=3.172, COSMOSVLA J100028.75+023203.5
with $z_{\rm phot}$=3.175, and COSMOSVLA J100004.81+023045.2 with $z_{\rm
phot}$=3.76). Similarly, in the ECDF-S sample, we identified one source with
$z_{\rm phot}$¿3, namely ALESS J033152.49-280319.1 ($z_{\rm phot}$=3.56;
Wardlow et al., 2011). These five high-redshift candidates, along with the
three sources with a spectroscopic redshift above 3, are presented along with
their 1.4 GHz and far-IR HerMES flux densities in Table 1.
We are particularly interested in sources with the highest redshifts. We
therefore decided to further investigate the five sources for which the
photo-$z$ indicates a $z>3$ nature. Follow-up observations of these sources
aimed to improve the redshift determination (for the photometric redshift
estimates), and a more detailed analysis of their properties was initiated,
using mm observatories from the northern and southern hemispheres (for the
COSMOS and ECDF-S fields, respectively), the first results of which are
described in the following section.
The comparison of our acquired spectroscopic/photometric redshift values for
our sample with the corresponding values from alternative photometric
catalogues in the literature is a valuable exercise. This comparison is
presented in more detail in Appendix A and indicates that the COSMOS and the
ECDF-S fields do not contain additional candidates at $z>3$ that our analysis
could have miss-identified at lower $z$, and the $z>3$ nature of the five
sources with photo-$z$ considered here is confirmed. Therefore, we are more
confident that the sources that are presented in the following sections are
the only ones with $z_{\rm phot,spec}>3$ that could be selected with our
selection criteria from the two fields of study.

##  3 Results

To characterise the morphological and physical properties of the eight high-
redshift galaxies under investigation, a detailed examination of the available
photometric and spectroscopic data from the literature was conducted.
Simultaneously, follow-up observations were initiated, in particular, using
millimetre facilities. The results are described below.

During this analysis, a spectral energy distribution (SED) fitting analysis
was performed for each galaxy in the sample using CIGALE (Noll et al., 2009;
Serra et al., 2011). The resulting fits yield two values for each modelled
physical property of a galaxy (e.g. stellar mass and redshift), one
corresponding to the best-fit solution and another that is derived through a
Bayesian statistical analysis. In the following text, we refer to the best-fit
values when presenting the sample, while the Bayesian-derived estimates along
with their corresponding uncertainties are introduced in Table 3. The images
of the eight sources for four different photometric bands (optical, near-IR,
radio, and mm wavelengths) are presented in Fig. 4. For each wavelength
regime, the observations with the highest available spatial resolution were
selected.

###  3.1 SED fitting results

The SED fitting analysis was conducted using the CIGALE software, employing
consistent input models and parameters for the entire sample. The data fluxes
from the Cosmos2015 catalogue were used in the analysis, while the radio
information as well as any nebular emission component were not taken into
consideration. For the sources that have been identified spectroscopically in
the literature (i.e. CVLA1002, CVLA958, ALESS33, and ECDFS544), the redshift
input was fixed to the corresponding spectroscopic value. From the remaining
galaxies, ALESS14 and CVLA1000, the redshifts were fixed to the photometric
values retrieved from the literature, while for CVLA100 and CVLA028, the best-
fitting SED was selected by introducing $\Delta z=0.1$ intervals. The
resulting fits, along with the model parameters used in this analysis, are
presented in Fig. 3 and Table 2.

To probe the significance of the AGN contribution, we applied the same fitting
strategy using CIGALE without the AGN component. The updated SED fits revealed
that for sources that initially required a lower AGN contribution (e.g.
CVLA1000), the $\chi^{2}$ values changed marginally by approximately $\sim
10\%$. On the other hand, for galaxies such as CVLA100, which presents a
substantial AGN contribution, no accurate SED fit could be provided without an
AGN component ($\Delta\chi^{2}>3$). Consequently, this exercise demonstrates
the necessity of incorporating an AGN component in the SED fit for five out of
eight sources, generally leading to improved fits as compared to those
obtained without an AGN component.

Figure 3: CIGALE SED fitting results for the eight galaxies of our sample. The redshift values for CVLA1002, CVLA958, ALESS33, ECDFS544, ALESS14, and CVLA1000 were fixed to those derived from the literature, and the redshift for CVLA100 and CVLA028 was estimated from a range of values separated by $\Delta z=0.1$. Table 2: Model parameters of CIGALE that were used for the SED fitting. Type | Model name | Reference | Parameters used
---|---|---|---
| Star formation
---
history
sfhdelayed | (Ciesla et al., 2017) |  | age$\\_$main = 1000, 2500, 4500, 6000, 8000
---
tau$\\_$burst = 10000.0
age$\\_$burst = 10, 50, 80, 110
f$\\_$burst = 0.001, 0.01, 0.03, 0.1, 0.2, 0.3
sfr$\\_$A = 10.0,100.0,1000.0,3000.0
Stellar emission | bc03 | (Bruzual & Charlot, 2003) |  | imf = 1(Chabrier)
---
metallicity = 0.0001,0.02
separation$\\_$age = 10
Dust attenuation | dustatt$\\_$2powerlaws | (Charlot & Fall, 2000) |  | Av$\\_$BC = 0.3, 0.8, 1.2, 1.7, 2.3, 2.8, 3.3, 3.8
---
slope$\\_$BC = -0.7
BC$\\_$to$\\_$ISM$\\_$factor = 0.3, 0.5, 0.8, 1.0
slope$\\_$ISM = -0.7
filters = V$\\_$B90, FUV
| Lyman Continuum
---
escape
lyc$\\_$absorption | (Inoue, 2011) |  |  $\rm f_{esc}$ = 0.0,0.5
---
$\rm f_{dust}$ = 0.0,0.5
Dust emission | dl2014 | (Dale et al., 2014) |  | qpah = 0.47, 1.12, 2.5, 3.9
---
umin = 5.0, 10.0, 25.0
alpha = 2.0
gamma = 0.02,0.9
AGN | fritz2006 | (Fritz et al., 2006) |  | r$\\_$ratio=60.0
---
tau=1.0,6.0
beta=-0.5
opening$\\_$angle=100.0
psy=0.001
fracAGNescape=0.0,0.1,0.2,0.6,0.8
Redshift | redshifting | (Meiksin, 2006) | redshift=0.01-7.5 (or fixed)

###  3.2 Sample presentation

#### COSMOSVLA J100256.53+021158.4

The COSMOS field deep multi-wavelength data provide the opportunity of
examining the structure of this source, which appears to be composed of two
galaxies: an optically bright quasar, and a fainter millimetre source (see the
top left panel of Fig. 4). Spectroscopic data from the BOSS survey (Reid et
al., 2016) secured the detection of the Ly-$\alpha$ emission line at a
redshift of $z=3.503\pm 0.0007$. Because the BOSS capabilities cannot resolve
the two sources in study here, this archival spectrum corresponds to both
galaxies. However, considering that the main source (i.e. the quasar) is the
brightest galaxy at optical wavelengths, we assume that the emission line and
subsequently derived redshift correspond to this source.
Furthermore, inspection of the 3 GHz data reveals that the quasar causes most
of the radio flux density emitted from this system with a value of $f_{\rm
3GHz-QSO}=0.052\pm 0.02\,mJy$ ($\sim 80\,\%$ of the total), while the second
source has a flux density of $f_{\rm 3GHz-2^{nd}source}=0.013\pm 0.009\,mJy$.
When we assume a typical radio spectral index of 0.8 for both
sources333$S_{\nu}\propto\nu^{-\alpha}$, where $S_{\nu}$ corresponds to the
flux density, $\nu$ is the frequency, and $\alpha$ is the radio spectral
index, which is 0.8 for pure radio synchrotron emission., the corresponding
radio flux density at 1.4 GHz is equal to $f_{\rm 1.4GHz-QSO}=0.1\pm
0.04\,mJy$ and $f_{\rm 1.4GHz-2^{nd}source}=0.024\pm 0.017\,mJy$. Therefore,
if both galaxies are located at $z\sim 3.5$ and the detected radio emission
was generated solely from SF, the corresponding SFR (Bell, 2003) would be
$SFR_{\rm radio-QSO}=4246\pm 1633\,M_{\odot}/yr$ and $SFR_{\rm
radio-2^{nd}source}=1061\pm 735\,M_{\odot}/yr$ for the quasar and SFG,
respectively ($SFR_{\rm radio-total}=5307\pm 1790\,M_{\odot}/yr$). According
to Lanzuisi et al. (2017), this system of galaxies has a total stellar mass of
$log(M_{\star}/M_{\odot})=11.69\pm 0.01$ and $SFR_{\rm IR-total}$ of 1549
$M_{\odot}/yr$. The latter value is derived from the far-IR flux integrated at
the position of the quasar and is rather close to our SED-fitting, best-
derived SFR estimate of 1721 $M_{\odot}/yr$. Curiously, this value is similar
to the value derived from the radio for the second source in the system (i.e.
$SFR_{\rm radio}-2^{nd}source$). This is compatible with a system with a
powerful quasar and a companion where most of the SF is taking place, and
where no AGN is detected.
It has to be noted that for most bands, we cannot separate the brightness
contribution from each of the two sources because of the sub-arcsec separation
and the insufficient angular resolution of telescopes such as Spitzer. The SED
fit was therefore performed using only total fluxes. A separation of the SED
fitting analysis for the two sources would be non-trivial and would yield high
uncertainties in the fluxes of each source. In the X-ray regime, COSMOS has
been observed both with XMM-Newton (XMM-COSMOS survey; Cappelluti et al.,
2009) and the Chandra X-ray Observatory (Chandra COSMOS Survey and Chandra
COSMOS Legacy Survey; Elvis et al., 2009; Civano et al., 2016), reaching flux
limits of $f_{\rm 0.5-2keV}\sim 1.7\cdot 10^{-15}\,erg/s/cm^{2}$ and $f_{\rm
0.5-2keV}\sim 2.7\cdot 10^{-16}\,erg/s/cm^{2}$, respectively. The source
CVLA1002 has been observed and detected by the Chandra COSMOS survey with a
flux of $f_{\rm 2-10keV}=6.67\cdot 10^{-15}\,erg/s/cm^{2}$ (see Marchesi et
al., 2016).

#### ALESS J033152.49-280319.1

Based on the current photometric data, this source seems to be a single
galaxy, and according to the SED photometric analysis by da Cunha et al.
(2015), it has a photometric redshift of $z\sim 3.38$, an $SFR_{\rm IR}$ of
1202 $M_{\odot}/yr,$ and a stellar mass of $10^{10.96}\,M_{\odot}$. For our
photometric analysis, which takes the possibility of an AGN component into
account, the estimated best SFR value has a similar value of $SFR_{\rm
IR}=1188$ $M_{\odot}/yr$. Assuming that the galaxy is located at $z=3.38$ and
its radio emission is solely due to star formation, the corresponding SFR from
the radio emission would be equal to $SFR_{\rm radio}\sim 4213\pm
757\,M_{\odot}/yr$. A comparison between this value and the value derived from
our SED fitting analysis or by da Cunha et al. (2015) suggests that the source
hosts intense SF, and an accreting SMBH explains the clear radio excess that
is observed. The X-ray energies have been observed in the CDF-S field with
XMM-Newton and the Chandra observatories (for more details, see Ranalli et
al., 2013; Luo et al., 2017) in deep surveys, resulting in sensitivities of
$6.6\cdot 10^{-16}\,erg/s/cm^{2}$ and $2.7\cdot 10^{-17}\,erg/s/cm^{2}$ at
2-10 keV, respectively. However, ALESS14 is not detected in any of these
surveys, supporting the view that if an AGN exists, it contributes mostly to
the radio emission, and at a much lower level at X-ray and optical/IR
wavelengths. This was also suggested by the $\sim 20\%$ AGN contribution to
the IR emission from our CIGALE SED fit.

#### COSMOSVLA J100004.81+023045.2

According to the HELP catalogues (Shirley et al., 2021), this source is
located at a redshift of 3.43 with an $SFR_{\rm IR}$ of 1519 $M_{\odot}/yr$,
while our SED fitting analysis predicts a lower $SFR_{\rm IR}$ of 690
$M_{\odot}/yr$. Based on the SED fitting results, modelling of the IR emission
of CVLA1000 requires an AGN contribution. According to the Cosmos2015 and
Cosmos2020 (Weaver et al., 2022) catalogues, the photometric redshifts are
3.26 and 3.62, respectively. When we accept an average redshift value of $\sim
3.3$ for this source, then according to its radio flux density of $f_{\rm
1.4GHz}=0.096\,mJy,$ the $SFR_{\rm radio}$ should be $\sim
3723\,M_{\odot}/yr$, which in comparison with the previous estimate from the
IR emission strongly suggests an active SMBH at the centre of this galaxy.
Due to its morphology in the near-IR, radio, and mm wavelengths, this source
could be classified as a group of galaxies (see the third row of Fig. 4),
however, catalogues with large samples of galaxies (i.e. Cosmos2015 and
Cosmos2020) identify this galaxy as a single source due to the lack of
resolution. With a radial extent of $\sim 1.146$ arcsec, measured at the 1.6
$\mu m$ \- NICMOS data, and a scaling factor of $7.524\,kpc/^{\prime\prime}$
(at $z=3.3$), the actual size of this galaxy can be estimated as $\sim
8.6\,kpc$. This value is compatible with an ultra-compact dwarf galaxy (e.g.
Rodriguez et al., 2021), however, the images reveal a source that resembles a
disk rather than a spherical object. Inspection of all the available images
cannot exclude the possibility of a merging system, considering that the
NICMOS data depict a structure that could resemble a complex of galaxies. No
detection in the X-ray bands is reported in the literature for this galaxy.

Figure 4: Multi-wavelength view of the eight high-redshift galaxies of our
sample, ordered according to Table 1. From left to right, we present the i- or
r-band (HST, VIMOS, and VOICE data), the Spitzer 3.6$\mu$m/VIDEO Ks-
band/NICMOS 1.6$\mu$m images, the radio VLA 1.4/3 GHz and the FIR data from
ALMA and Herschel, and the red contours correspond to the VLA emission. From
top to bottom, the contour levels of the radio image for each source are
($7.715\cdot 10^{-6}$, $2.093\cdot 10^{-5}$), $2.827\cdot 10^{-5}$,
($-4.710\cdot 10^{-6}$, $1.058\cdot 10^{-5}$), $5.640\cdot 10^{-6}$,
($-4.845\cdot 10^{-6}$, $5.461\cdot 10^{-6}$), ($-5.078\cdot 10^{-6}$,
$1.407\cdot 10^{-5}$), ($-4.633\cdot 10^{-6}$, $1.124\cdot 10^{-5}$,
$2.712\cdot 10^{-5}$), and $2.092\cdot 10^{-5}$ Jy/beam. Each panel has a size
of 7.2x7.2 $arcsec^{2}$ , except for the Herschel image, which for
visualisation purposes has a larger extent. Additionally, the beam size
(shaded light grey area) of each instrument and a 2 arcsec scale bar are added
in each panel.

#### COSMOSVLA J095845.94+024329.2

Source CVLA958 was recently spectroscopically observed by our team with the
Institut de Radioastronomie Millimetrique (IRAM) 30m antenna (project code:
227-19; for more details regarding these observations, see Appendix B).
Unfortunately, its mm faint nature prevented us from acquiring a secure
redshift determination with the IRAM telescope. According to Cosmos2015, the
photometric redshift of this galaxy is $z=3.172$, while our photometric
analysis suggests a minimum of $\chi^{2}$ at 3.21. The latter redshift
solution agrees with the only line that is detected from recent ALMA archival
spectra (project code: 2019.1.01600.S). The detected emission line at $\sim
110.5\,GHz$ has a full width at half maximum of $\sim$0.4 GHz ($\sim
70.4\,km/s$), possibly indicating a merging system or an orderly rotating
spiral galaxy (see Chen et al., 2022, for more details). The recent work by
Birkin et al. (2021) also identified the emission line as due to the CO4-3
rotational transition at band 3 of ALMA, locating this source at $z=3.1755$.
According to our photometric SED fitting analysis, this source has a stellar
mass of $10^{11.07}\,M_{\odot}$, an SFR of 1177 $M_{\odot}/yr,$ and a
significant AGN contribution. Considering the measured flux density at 1.4 GHz
of 90 $\mu Jy$, we estimate that its $SFR_{\rm radio}$ (assuming that its
radio emission comes purely from star formation) should be $\sim
3376\,M_{\odot}/yr$. This value is significantly higher than the value
determined from the SED fitting (see Table 3 for further details), supporting
the presence of an AGN in this source as well. CVLA958 is not detected in any
available dataset in the X-ray wavelengths.

Table 3: Derived and observed properties of the galaxy sample. The SFR values
derived from the IR emission are a product of the SED fitting described in the
text, while the radio-derived SFRs are derived by assuming that the radio
emission is solely due to star formation without contribution by an AGN. The
second row of the table corresponds to the units of each parameter.

Name | Redshift(1) | Stellar Mass | $\rm SFR_{IR}$ | $\rm SFR_{radio}$ | AGN (2) | $\rm L_{1.4GHz}$ |  $\rm L_{IR}$ (3) | $q_{TIR}^{(4)}$
---|---|---|---|---|---|---|---|---
– | – | [$M_{\odot}$] | [$M_{\odot}/yr$] | [$M_{\odot}/yr$] | [$\%$] | [$10^{24}W/Hz$] | [$10^{39}W$] | –
CVLA1002 | $3.503\pm 0.001^{(s)}$ | $11.62\pm 0.44$ | $2496\pm 1266$ | $5307\pm 1790$ | $41\pm 22$ | $8.34\pm 1.60$ | $3.15\pm 0.01$ | $2.00\pm 0.08$
ALESS14 | $3.380\pm 0.25^{(p)}$ | $11.07\pm 0.85$ | $1123\pm 142$ | $4213\pm 757$ | $20\pm 1$ | $7.63\pm 1.37$ | $2.19\pm 0.20$ | $1.88\pm 0.09$
CVLA1000 | $3.430\pm 0.32^{(p)}$ | $12.13\pm 1.30$ | $711\pm 40$ | $4057\pm 792$ | $10\pm 2$ | $7.35\pm 1.44$ | $2.15\pm 0.25$ | $1.89\pm 0.10$
CVLA958 | $3.175\pm 0.001^{(s)}$ | $11.19\pm 0.44$ | $1225\pm 546$ | 3382$\pm 641$ | $20\pm 4$ | $6.13\pm 1.16$ | $0.58\pm 0.02$ | $1.40\pm 0.11$
CVLA100 | $3.290\pm 0.39^{(p)}$ | $10.36\pm 0.52$ | $528\pm 96$ | $3391\pm 781$ | $60\pm 3$ | $6.14\pm 1.41$ | $1.98\pm 0.32$ | $1.93\pm 0.10$
CVLA028 | $3.270\pm 0.24^{(p)}$ | $11.13\pm 0.30$ | $1565\pm 753$ | $2167\pm 602$ | $3\pm 7$ | $3.92\pm 1.09$ | $2.52\pm 0.22$ | $2.23\pm 0.13$
ALESS33 | $3.694\pm 0.001^{(s)}$ | $11.27\pm 0.81$ | $978\pm 113$ | $2140\pm 597$ | $10\pm 1$ | $3.88\pm 1.08$ | $2.20\pm 0.01$ | $2.18\pm 0.12$
ECDFS544 | $3.199\pm 0.001^{(s)}$ | $11.42\pm 0.80$ | $1083\pm 470$ | $1520\pm 507$ | $20\pm 3$ | $2.75\pm 0.92$ | $2.01\pm 0.01$ | $2.29\pm 0.15$

(1)(s): spectroscopic redshift; (p): photometric redshift (accepting the
solution with the lowest $\chi^{2}$ or the best available in the literature),
(2): the AGN percentage (Bayesian values) is estimated by our CIGALE fitting
and corresponds to the same wavelength range in which the dust luminosity of
each source was estimated, (3) this infrared luminosity corresponds to the
wavelength range 42-122 $\mu$m (rest frame), (4) according to equation 5 by
Enia et al. (2022): $\rm q_{TIR}=log(L_{IR}/3.75\cdot 10^{12})-logL_{1.4GHz}$

#### COSMOSVLA J100233.16+020626.5

The source CVLA100 seems to be a single galaxy as well. Our photometric
analysis places this source at a redshift of $z_{\rm phot}\sim 3.3$. This
result agrees with the estimates of other photometric catalogues (see Laigle
et al., 2016; Weaver et al., 2022). ALMA spectroscopic data acquired by our
team (project code: 2017.1.01713.S; see Appendix B for more details) reveal an
emission line at $\sim 107.3\,GHz$, which corresponds to the CO(4-3) emission
line with the galaxy located at $z=3.29$. For the derived physical properties
of this source, our SED fitting analysis indicates a stellar mass of
$10^{10.27}\,M_{\odot}$, an SFR of 566 $M_{\odot}/yr,$ and an AGN fraction of
60 $\%$. Application of equation 6 by Bell (2003) provides us with an estimate
of the $SFR_{\rm radio}$ of 2760 $M_{\odot}/yr$. The SED fitting models for
this redshift suggest an SFR of $\sim 500\,M_{\odot}/yr$, which could mean
that the observed 1.4 GHz radio flux cannot be obtained solely from SF, but by
a combination of star emission and AGN activity (see Table 3 for additional
information). Similarly to CVLA958, this source is not detected at any high-
energy band.

#### COSMOSVLA J100028.75+023203.5

As mentioned in the previous section, this source (which seems to be a single
galaxy) has a spectroscopic redshift of 4.76 (see Hasinger et al., 2018),
however, inspection of the DEIMOS spectrum does not reveal a secure emission
line (i.e. the Ly-$\alpha$ line, according to the authors). No further
spectroscopic data are available in the literature. Our own photometric SED
fitting analysis suggests a redshift of 3.27, which is consistent with the
redshift provided by the HELP database and Miettinen et al. (2015). When we
assume this photo-$z$ to be the true redshift of this galaxy, our SED fit
analysis proposes a stellar mass of $10^{10.74}\,M_{\odot}/yr$, an SFR of 2554
$M_{\odot}/yr,$ and no AGN contribution. The derived SFR from the radio
emission of this galaxy (i.e. 2166 $M_{\odot}/yr$) further supports the
argument that no radio emission is added by SMBH activity. This result
establishes CVLA028 as the only galaxy in our sample without an indication of
an AGN by the SED fit or radio excess. Although this galaxy has been observed
by both XMM-Newton and Chandra, the images at all bands reveal no clear
detection.

#### ALESS J033211.34-275211.9

This source is recognised in the literature as an X-ray AGN with faint radio
emission. Photometric analysis from various deep surveys (e.g. Guo et al.,
2020) reveals a redshift of $z\sim 3.7$, while ALMA spectroscopic data (see
Birkin et al., 2021) show two clear emission lines (namely the CO4-3 and [CI]
lines at $\sim 98.22\,GHz$ and $\sim 104.85\,GHz$, respectively). This secures
the galaxy at $z=3.694$. Assuming that this spectroscopic redshift and that a
radio flux density of $0.043\,mJy$ solely originate from star formation, the
corresponding SFR is $\sim 1884\,M_{\odot}/yr$. However, according to Guo et
al. (2020), following the SED fitting method using CIGALE, this galaxy is a
type II AGN with a stellar mass of $\sim 10^{11.3}\,M_{\odot}$, an SFR of
$\sim 31\,M_{\odot}/yr$ (our CIGALE SED fitting analysis suggests a higher
value of 1194 $M_{\odot}/yr$), and an AGN fraction of 72 $\%$ (our analysis
proposes a lower value of 10 $\%$ ). The source detected in the HST image (see
Fig. 4), which is rather close to the ALMA detection ($\sim 0.5\,arcsec$),
seems to cause the radio and IR emission (see Wardlow et al., 2011).
Therefore, this source could be a complex system of at least two galaxies
possibly interacting, with the presence of a radio-quiet AGN and an SFG. Even
though ALESS33 is not detected in any of the X-ray surveys, another source
lies at a distance of $\sim 2$ arcsec (named $\rm LBX2017-14$; see Ranalli et
al., 2013, for more details) that seems to be an X-ray bright AGN with a flux
of $f_{\rm 2-10keV}=6.26\cdot 10^{-15}erg/s/cm^{2}$ and no counterparts. This
X-ray source was spectroscopically identified by Ranalli et al. (2013) at
$z=3.74\pm 0.06$, meaning that it could be part of a merging system along with
the sub-millimetre and radio galaxy identified with the source ALESS33.

#### [DSS2017] 554

This source is present in the VIMOS spectroscopic survey presented by
Danielson et al. (2017) with a spectroscopic redshift of 3.1988, which was
used in this work as a fixed value for the subsequent SED fitting analysis.
This galaxy is placed at the outskirts of the ECD-S field, meaning that it was
not selected in the photometric analysis by Guo et al. (2020). Inspection of
the VST Optical Imaging of the CDFS and ES1 (VOICE; Vaccari et al., 2016) data
for this source in bands r, g, and i (see the bottom row of Fig. 4) suggests
the possibility of multiple components. These observations might signify a
merging or interacting system. The predicted SFR derived solely from the radio
emission is 1520 $M_{\odot}/yr$, while our SED fitting analysis suggests a
value of 1162 $M_{\odot}/yr$ and an AGN contribution of $20\%$ . No clear
detection has been reported regarding the possible X-ray nature of this
source.

##  4 Discussion

Figure 5: Normalised redshift histograms of all galaxies from the HELP
catalogue (panel a), separated with the criterion $f_{350\rm\mu
m}/f_{250\rm\mu m}=1$. Flux densities were estimated using the Herschel
observational data as provided from the same catalogue. The vertically dotted
coloured lines correspond to the average values of the same histograms,
employing the flux densities from the CIGALE best model fit, and are coloured
accordingly. The text in the panel denotes the percentage of the sources that
present an SFR above $10^{2}\,M_{\odot}/yr$ and are coloured according to the
aforementioned criterion. Panel b illustrates the normalised histograms for
the dust content of the galaxies in both the HELP and Cosmos2015 catalogues.
The blue and orange histograms are derived from galaxies without and with
radio VLA 1.4 GHz emission, respectively. The text of the panel demonstrates
the percentage of the galaxies that present an AGN component ($AGN_{\rm
frac}>40\%$) and are placed at $z>3$.

We have examined a sample of eight high-redshift dusty galaxies with an AGN
component (3 ¡ $z_{\rm spec,phot}$ ¡ 4). The selection process involved using
the 1.4 GHz VLA radio catalogues from the COSMOS and ECDF-S fields in
combination with the far-IR peak criterion ($f_{350\rm\mu m}/f_{250\rm\mu
m}>1$) using the HerMES survey. Moreover, a compact morphology was considered
as an additional criterion. In the following paragraphs, we assess the
efficacy of these criteria in selecting these sources. Furthermore, we provide
a comparison between the derived properties of our sample with those obtained
in previous studies, focusing on similar redshift ranges.

###  4.1 Validity of the selection criteria

While the compactness criterion merely serves as a method for avoiding local
extended galaxies, the far-IR and radio criteria have been employed
extensively in the past in the study of the distant Universe. For instance,
Amblard et al. (2010) demonstrated that the redshift of a source tends to
increase for galaxies that satisfy the criteria $f_{350\rm\mu m}/f_{250\rm\mu
m}>1$ and $f_{500\rm\mu m}/f_{350\rm\mu m}>1$. These results are expected,
considering that the far-IR emission of an SED traces the dust content of a
galaxy, and more specifically, its blackbody radiation, which is shifted to
higher wavelengths with increasing redshift. However, this emission is
affected by the temperature of the dust as well, which consequently introduces
a degeneracy that is non-trivial to break. Another point of attention is the
sensitivity of the Herschel surveys, which are inherently biased towards the
detection of the brightest dust-rich far-IR galaxies.
In order to evaluate this argument and selection method further, we employed
the HELP catalogue, which contains far-IR observational data (i.e. 15747
Herschel detected galaxies) along with SED fitting results from CIGALE (for a
detailed description of the CIGALE parameters used in this catalogue, see
Małek et al., 2018). The galaxies from this catalogue are presented in the
left panel of Fig. 5, in which the blue and orange histograms correspond to
the sources that follow the criterion $f_{350\rm\mu m}/f_{250\rm\mu m}<1$ and
$f_{350\rm\mu m}/f_{250\rm\mu m}>1$, respectively. As anticipated, galaxies
that adhere to the far-IR criterion are situated at higher redshifts on
average. Interestingly, based on the CIGALE fits, a higher percentage (i.e.
$\sim 45\%$) of the galaxies that follow the $f_{350\rm\mu m}/f_{250\rm\mu
m}\geq 1$ criterion exhibit SFRs of $SFR>10^{2}\,M_{\odot}/yr$. In contrast,
for the rest of the sources, a smaller percentage ($\sim 22\%$) demonstrates
high SFR values like this.
A similar exercise was repeated for the radio criterion by estimating the AGN
fraction and dust mass, derived from the CIGALE fits, for each galaxy. Because
the HELP catalogues do not consider the radio emission in their fits, we
cross-matched this catalogue with Cosmos2015, which includes the 1.4 GHz VLA
data. The results of this analysis are displayed in the right panel of Fig. 5,
which shows histograms of the dust mass. The shape and mean values of these
histograms indicate that the radio selection criterion does not guarantee the
preferential selection of highly dusty SFGs. However, upon examining galaxies
at redshifts above 3, it becomes evident that the radio selection method
identifies a higher proportion of sources (i.e. $\sim 22\pm 9\%$) with a
substantial AGN fraction (i.e. $AGN_{\rm frac}>40\%$) compared to cases
without detected radio emission ($\sim 15\pm 4\%$).
While this percentage difference may not be statistically significant, the
combination of the two far-IR and radio selection methods still provides
motivation for their application in identifying high-redshift SFGs with an AGN
component. The detailed analysis of the eight sources in this sample has
reinforced these findings and encourages their application in large-scale
surveys, such as EMU.

###  4.2 Comparison with previous work

Overall, our sample has a mean redshift of $<z>=3.37\pm 0.16$, an average
stellar mass of $<log(M_{\rm\star}/M_{\odot})>=11.27\pm 0.47$, $<L_{\rm
IR,42-122\mu m}>=2.1\pm 0.67\cdot 10^{39}\,W$ (total IR luminosity of $<L_{\rm
IR,8-1000\mu m}>=3.9\pm 0.84\cdot 10^{39}\,W\sim 10^{13}\,L_{\odot}$),
$<L_{1.4GHz}>=5.77\pm 1.9\cdot 10^{24}\,W/Hz,$ and a mean star formation rate
(derived from the $42-122\,\mu m$ rest-frame IR luminosity of each source) of
$<SFR_{\rm IR}>1213\pm 567\,M_{\odot}/yr$. This average SFR is consistent with
previous results that reported an increasing SFR at higher redshift. For
instance, Rowan-Robinson et al. (2016) and Gao et al. (2022) showed that
Herschel-detected samples of hyper-luminous infrared galaxies (HLIRGs; $L\sim
10^{12}-10^{13}\,L_{\odot}$) at $z>3$ have SFR values ranging between $\sim
1000-10000\,M_{\odot}/yr$. Therefore, the inclusion of the 500 $\mu m$
emission in our selection criteria naturally provides us with the high bright
tail of the SFR distribution at these redshifts. Similarly, Wang et al. (2019)
reported that the selection of Herschel galaxies above $z>3$ revealed dusty
SFR galaxies with infrared luminosities of $\sim 10^{12}-10^{13}\,L_{\odot}$,
similar to the derived values of our sample.
Furthermore, for 87.5 $\%$ of the sources in our sample (seven out of eight),
an AGN is indicated either by the radio excess or by the mid-IR signature
obtained from the best SED fit. Although the morphology of these sources still
remains uncertain, five out of eight (63 $\%$) might consist of multiple
galaxies, with significant levels of both SF and AGN activity. In comparison,
Gao et al. (2021) and Wang et al. (2021), who analysed a sample of HLIRGs
selected using data from Herschel and the Deep Fields of the LOFAR Two-metre
Sky Survey (LOTSS; Shimwell et al., 2017). described an AGN contamination of
34-64 $\%$ (using six different models to fit the AGN component) for galaxies
at the redshift range $3<z<4$. These values, although model dependent, seem to
be lower than the AGN contribution of our sample, which can be attributed to
the fact that the LOTSS survey reaches sensitivities of 20-30 $\mu Jy/beam$ at
150 MHz. When we assume a typical radio synchrotron slope, this value
translates into a lower flux density value at 1.4 GHz than the COSMOS-VLA
sensitivity (i.e. 7 $\mu Jy/beam$). Therefore, it seems that by selecting
radio sources from a deeper radio survey, the authors revealed a population of
SF galaxies that receive contributions from AGN less frequently. However, it
is important to emphasise that a comparison of our results with studies
involving larger sample sizes should be approached with caution, considering
the limited statistics that our data provides.

Figure 6: The $q_{TIR}$ parameter over redshift for the sample explored by
Enia et al. (2022) and for our 8 galaxies (panel a). This figure was retrieved
by Enia et al. (2022), published under the CC BY 4.0 international licence and
was modified by adding the results from our work (square points). The dashed
black line and grey shaded region correspond to the best-fit relation of their
total sample and the standard deviation, respectively, while the dashed blue
line shows a similar relation by Delhaize et al. (2017). All points by Enia et
al. (2022) are coloured according to the contribution of AGN to the radio
emission defined as $f_{\rm AGN}=1-10^{q_{\rm tot}-\bar{q_{\rm SF}}}$, with
$q_{\rm tot}$ and $\bar{q_{\rm SF}}$ corresponding to the $q_{\rm TIR}$ and
its evolution with redshift, respectively. The star symbols depict what is
defined by Enia et al. (2022) as H-dark sources (i.e. not detected at the
Hubble-WFC3 H-band). Panel b, presents the SFRs derived from the IR and radio
luminosity, for the 8 sources of our sample, along with the results by Bonzini
et al. (2015) (radio-quiet AGN: blue squares, radio-loud AGN: red triangles,
SF galaxies: green circles) and Talia et al. (2021) (average values of their
sample). All data points with redshift below 3 are coloured with transparency,
while the ones with z¿3 have normal colours.

Additionally, Enia et al. (2022) reported a negligible percentage of AGN for
the entire sample ($0<z<6$), which was selected using deep radio 1.4 GHz
observations from the GOODS-N field (sensitivity of 2.2 $\mu Jy$). This can be
perceived with the parameter $\rm q_{TIR}=log(L_{IR}/3.75\cdot
10^{12})-logL_{1.4GHz}$, which is commonly used as an indicator of the
presence of AGN contribution to the radio emission. Figure 6 depicts the
$q_{TIR}$ values for our sample and those of Enia et al. (2022), demonstrating
the AGN contamination in the radio emission in most of our galaxies. In more
detail, 62.5 $\%$ of the sources in our sample have lower values of $q_{\rm
TIR}$ than the grey region of the plot (which denotes the average redshift-
dependent $q_{\rm TIR}$). These statistics do not fully match the number of
87.5 $\%$ of AGN contamination that was derived from the SED fitting and
described earlier in the text. However, this discrepancy might originate from
the low AGN contribution and uncertainties in the measurements of the radio
and infrared fluxes of the SED of some galaxies. In contrast, the sample by
Enia et al. (2022) for the redshift bin of $3<z<4$ indicates an AGN
contamination of 37.5 $\%$ . This difference can be attributed to the fact
that in their selection criteria, they did not require the far-IR emission (or
better, far-IR peak spectrum) that we employed. As a result, only 38.4 $\%$ of
their total sample has a 500 $\mu m$ Herschel detection.
Another comparison can be conducted with the $q_{TIR}$ relation derived by
Delvecchio et al. (2021) and presents a mass-dependent and almost redshift-
independent equation (i.e. $q_{TIR}(M_{\star},z)=(2.646\pm
0.024)(1+z)^{-0.023\pm 0.008}-(logM_{\star}/M_{\odot}-10)(0.148\pm 0.013)$).
According to this relation, all of the galaxies in our sample satisfy the
criterion, placing them below the mean of the SFR population. However, none of
the sources in our sample follows the relation
$log(L_{1.4GHz}/SFR_{IR})=21.984\times(1+z)^{0.013}$ that was derived earlier
by Delvecchio et al. (2017) and corresponds to the $3\sigma$ deviation from
the peak distribution of the ratio of $L_{\rm 1.4GHz}$ and $SFR_{\rm IR}$ as a
function of redshift.
In the same manner, Novak et al. (2017), which selected sources from the VLA
3GHz survey without a far-IR emission requirement, similarly presented the
trend of a lower contamination of AGN (23 $\%$ and 47 $\%$ of the total sample
and $3<z<4$ redshift bin, respectively) than in our sample. This comparison
favours the inclusion of the criterion of a far-IR rising spectrum to the
radio emission when AGN at high redshifts are to be identified and studied. In
contrast, an interesting comparison can be performed with studies such as that
by Bonzini et al. (2015), in which the authors presented a sample with
galactic properties similar to those of our galaxies. This sample was derived
from the radio 1.4 GHz ECDF-S catalogues (6 $\mu Jy$ sensitivity) with the
addition of the Herschel detection in all sources at the redshift bin of
interest (i.e. $z=3-4$). Fifteen of the 28 galaxies presented in this redshift
range were categorised as radio-quiet (RQ) AGN while one as a radio-loud (RL)
AGN, totalling 16 galaxies, or an AGN contamination of 73 $\%$ . This value is
close to the value derived from our sample, which further supports the
necessity of including the far-IR criterion to detect high-$z$ AGN.
Additionally, the right panel of Fig. 6 shows that our galaxies fall in the RQ
region of their work, even though our estimates indicate that at least two of
our sources should have a significant AGN contribution.
In this sense, our selection method might provide an increased probability for
the detection of dusty galaxies at high-$z$ that present high SFRs with an
additional AGN component, with weak radio emission, as part of a merging
system. Additionally, our galaxies seem to be a more extreme group of
interacting systems, having brighter IR and radio luminosities as well as
stellar masses and SFRs as compared, for instance, to the work by Talia et al.
(2021). This outcome might originate from the fact that their sample, once
again, did not require a far-IR Herschel detection and was originally derived
from the 3 GHz - COSMOS survey, which has a higher detection limit than the
VLA-1.4 GHz-COSMOS survey we employed. Nevertheless, Novak et al. (2017) also
explored the same field at 3 GHz and reached the same flux density sensitivity
(rms of 2.3 $\mu Jy$), from which they achieved more AGN detections. The main
difference lies in their selection criteria, with the latter work requiring an
optical/near-IR counterpart, while Talia et al. (2021) excluded these sources.
The selection of radio sources with a far-IR peak spectrum and an optical
counterpart might therefore reveal merging systems that at high $z$, present
high SFRs ($\sim 10^{3}\,M_{\odot}/yr$) and an AGN counterpart in most cases.
Nevertheless, selecting galaxies that are considered as extreme in terms of
mass and luminosity will naturally limit the study to a rare sample, hence to
a reduced number of selected sources. This infrequency, along with the
complexity of the adopted selection method, will unavoidably provide a less
accurate estimation of the SFRD at $z>3$, which can be only considered as a
lower limit (following the $V_{\rm max}$ method introduced by Avni & Bahcall,
1980), and for our study, this is equal to $SFRD>8\cdot
10^{-3}\,M_{\odot}/yr/Mpc^{3}$. However, these selection criteria might be
quite powerful in the aforementioned identification and in studying the
interplay between AGN and SF activity at high redshift, especially with their
application to the upcoming next-generation radio surveys (e.g. the EMU
survey) and current/future observational data from mm facilities (e.g. Bing et
al., 2023).

##  5 Conclusions

In this work, we present the exploration of a sample of a high-redshift
($3<z<4$) radio galaxy population that was selected with the criterion of a
rising far-IR spectrum (i.e. $f_{350\mu m}>f_{250\mu m}$) along with the
detection of non-extended (¡10 arcsec) radio emission at 1.4 GHz. The aim
behind this selection method was to investigate the potential of combining
these two criteria in identifying high-redshift dusty SFGs that are
contaminated by an AGN and might be ‘hidden’ by other selection methods. A
population like this, due to its extremity and low occurrence, might not
contribute significantly to the SFRD of the Universe, but it can provide us
with further insight into the interplay between SMBHs and SF activity at high
$z$.

In order to explore the morphology of our sample and its physical
characteristics in depth, we took advantage of the multi-wavelength deep data
available in the literature from the COSMOS and ECDF-S fields. These data sets
enabled us to identify that five out of eight galaxies in our sample might
consist of more than one galaxy with possible interaction/merging. It has to
be noted that the low angular extent of these galaxies did not permit
confirmation of this statement due to the spatial resolution of the current
instrumentation. Nevertheless, future observations with telescopes such as the
James Webb Space Telescope (JWST) may provide further insights into this
argument. Furthermore, we determined the presence of an AGN component in seven
out of eight of these sources, which in most cases represents a weak radio
emission, identifying these sources as radio-quiet AGN. An additional highly
star-forming component seems to be present in all cases, with a typical SFR of
$\sim 10^{3}\,M_{\odot}/yr$.

We have demonstrated that the combination of the far-IR and radio emission
criteria is highly efficient in selecting merging systems at $3<z<4$,
typically with the presence of a radio-quiet AGN and a highly SFG. With
upcoming wide radio surveys (e.g. the Evolutionary Map of the Universe, EMU)
and next-generation telescopes such as the SKA, these results motivate further
exploration of alternative selection methods. In order to advance the study of
systems similar to those in our sample, particularly during the epoch of
reionisation (EoR), sources with a mm rise in flux in combination with deep
radio surveys might further be studued. A possible rising mm spectrum profile,
which could be explored for faint individual sources using interferometric
facilities (e.g. ALMA and NOEMA) and single-dish instruments (e.g. the NIKA2
camera of the IRAM 30m antenna, or the ToLTEC of the Large Millimeter
Telescope) targeting a brighter population from larger surveys could assist in
the identification of the redshifted dust component of SFGs at $z>6$.
Similarly, the morphology of these galaxies could be further understood with
infrared facilities, including the JWST.

###### Acknowledgements.

This work was supported by Fundação para a Ciência e a Tecnologia (FCT)
through the research grants PTDC/FIS-AST/29245/2017, 1220UID/FIS/04434/2019,
UIDB/04434/2020, and UIDP/04434/2020. C.P. acknowledges support from DL
57/2016 (P2460) from the ‘Departamento de Física, Faculdade de Ciências da
Universidade de Lisboa’. This paper makes use of ALMA data. ALMA is a
partnership of ESO (representing its member states), NSF (USA), and NINS
(Japan); together with NRC (Canada), MOST, and ASIAA (Taiwan); and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO, and NAOJ. This research made
extensive use of core PYTHON packages for Astronomy as well as TOPCAT. We
thank the anonymous referee for all the valuable comments, which improved the
quality of the paper.

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##  Appendix A Comparison of the retrieved $z$ with photometric catalogues

In this appendix, we present a comparison between the redshift values
estimated by various photometric catalogues and our analysis, as described
above. In more detail, the catalogues consist of the Cosmos2015 (Laigle et al.
2016), the Cosmos2020 (Weaver et al. 2022, employing all possible redshift
estimations, namely the classic and Farmer methods using the SED fitting codes
Lephare and EAZY) and the HELP catalogues (Shirley et al. 2021). The redshift
values derived from these catalogues are compared with the values that our
team acquired from the literature and from the IRAM/ALMA observations of the
aforementioned two targets (i.e. CVLA958 and CVLA100; see Sect. 3). For the
cross match between catalogues, a radius of 1 arcsec was employed in order to
ensure that each source from the catalogues corresponded to the correct source
from our sample. The results of these comparisons are presented in Fig. 7.

In our findings, certain sources exhibit lower redshift values in the
catalogues than in the present study. These differences can be attributed to
the low S/N of the photometric points that were used in the SED fitting.
Furthermore, the authors of these catalogues may not have individually
examined each source due to the extensive volume of data. For example, the
source with coordinates (Ra, Dec)=(150.02005417, 2.51254722) is depicted in
the Cosmos2020 comparison plots (see Fig. 7) with a redshift of $z=2.48$,
derived from the EAZY SED fit ($\chi^{2}$ of $\sim 15$). However, our work
adopts a redshift value of 3.46 obtained from the Lephare SED fit solution for
the same source. This choice was made due to the lower $\chi^{2}$ values
obtained from both the literature Lephare analysis ($\chi^{2}=3.76$) and our
own CIGALE fits ($\chi^{2}=2.24$).

Additionally, this figure shows that according to Cosmos2015, two galaxies
that were identified by our search as $z<3$ sources have a higher photometric
redshift value. In more detail, the sources are identified as UVISTADR1
J100131.33+020325.2 and UVISTADR1 J100114.75+023516.3 and have a $z_{\rm
phot,cosmos2015}$ of 3.152 and 3.007, respectively. However, the more detailed
photometric analysis by Muzzin et al. (2013) proposed lower values for both
sources of $\sim 1.8$ and $\sim 2.7$, respectively. Similarly, for the
cosmos2020-Farmer method, a source named UVISTADR1 J095834.49+015504.9 was
estimated to be at $z_{\rm phot,cosmos2020,EAZY}\sim 3.1$, however, Muzzin et
al. (2013) reported a lower value of $\sim 2.5$. The latter redshift results
can be further supported by the redshift estimate by Lephare cosmos2020 of
$z_{\rm phot,cosmos2020,Lephare}\sim 2.2$. These discrepancies between various
photometric catalogues can mainly be attributed to the faint nature of these
sources and to the subsequent low fit quality of their SED. Finally, according
to the HELP catalogue, no further source could be added to our $z>3$ sample.

Figure 7: Comparison between the Cosmos2015 (a), Cosmos2020 (b and c), and
HELP (d) catalogues and our redshift values for our sample. The blue points in
each plot correspond to the photometric $z$ values from each catalogue, and
the orange points depict the spectroscopic values from the literature.
Notably, the Cosmos2020 catalogue yields two distinct sets of results because
two different SED fitting codes were used: Lephare and EAZY, denoted by
circles and triangles, respectively.

##  Appendix B Spectroscopic follow-up of CVLA958 and CVLA100

Because of the uncertainty of the high-redshift nature of CVLA958 and CVLA100,
these sources were originally included in a follow-up spectroscopic study by
our team with the aim to understand their nature and potentially employ this
knowledge to detect galaxies at higher redshifts. In this appendix, we present
the retrieved spectroscopic data for these two galaxies, bearing in mind the
redshift solution of $z\sim 3$ that the current photometric analysis
indicated. The original idea of the spectra acquired with the Institute Radio
Astronomie Millimétrique (IRAM) 30m antenna and the ALMA telescopes was the
detection of two possible CO emission lines that would constrain the redshift
of each galaxy.

###  B.1 IRAM observations

The IRAM444https://www.iram-institute.orghttps://www.iram-institute.org
institute is composed of two facilities, namely the interferometer NOrthern
Extended Millimeter Array (NOEMA; Chenu et al. 2016) and a single-dish
telescope with a diameter of 30m, located in the Sierra Nevada, Spain. The
adopted strategy to retrieve the millimetre spectrum of our galaxies was to
scan the 3 mm band with the spectral line receiver Eight MIxer Receiver (EMIR;
Carter et al. 2012) of the 30m antenna, along with the Fast Fourier Transform
Spectrometers (FTS200; Klein et al. 2012) and Wideband Line Multiple
Autocorrelator
(WILMA555https://www.iram.fr/IRAMFR/TA/backend/veleta/wilma/index.htmhttps://www.iram.fr/IRAMFR/TA/backend/veleta/wilma/index.htm)
backends (in parallel) at the wavelength range of 73-117 GHz (band E0/E090).
Given the possible AGN nature of the targeted sources, the CO spectral line
energy distribution will likely peak at mid-J transitions (e.g. for the
frequency range of band E0, the CO J=4-3 line could be at $z\sim 2.7-5$ and
the CO J=5-4 line at $z\sim 3.7-6.5$). The expected redshift span for this
candidate is $z\geq 3$, and therefore, the 3 mm spectral range provides an
excellent trade-off of weather versus line detection to search for ideally two
emission lines required to determine the source redshift. Nevertheless, in the
following analysis, only the data from the FTS units were investigated since
they provide a higher spectral resolution than WILMA, which is necessary to
resolve faint emission lines.

The project code is 227-19, and the observations were initially scheduled for
19-24 December 2019. However, considering that the weather conditions were not
appropriate to observe for the full length of our original allocated time,
additional observing time was provided during 21-23 March 2020, 12-14 May
2020, and finally, 14-17 July 2020. Out of a total of 103 hours provided for
this source, only $\sim 8$ hours of observations generated spectra with
acceptable quality to be considered in the assembly of the final spectrum. The
rest of the observing time was lost due to instrumentation malfunctions and
weather conditions (strong wind and high water vapour).

The final spectrum of CVLA958 was constructed using only data from the last
three days of observations (i.e. 21, 22, and 23 December 2019). We further
inspected the histograms of the opacity of the clouds ($\tau$), rms random
noise temperature ($T_{\rm rms}$), and receiver and system temperature
($T_{\rm rec}$ and $T_{\rm sys}$, respectively) from these observations in
order to exclude possible outliers in the scans. After removing these outliers
and correcting for the platforming issue (e.g. Klein et al. 2008, 2009), we
generated the final spectrum for the 3 mm band, depicted in the left panel of
Fig. 8. In this plot, the total spectrum (blue line) and its filtered version
for a Gaussian filter of $50\,km/s$ (orange line) are presented, and the
colours are denoted in the legend. The full applied FTS configuration is also
shown, as is the location of the CO lines, accepting the recent spectroscopic
redshift value of 3.1755 by Birkin et al. (2021) (dashed red lines). The
figure clearly shows that no secure line can be attributed to any feature
because the S/N does not permit this determination.
The high-redshift candidate CVLA100 was also observed by our team with the
IRAM 30m antenna as part of the project 198-16 and during the time period of
23-27 March 2017. In these observations, we also scanned the E0 band of EMIR
(frequency range of 73–117 GHz) using the FTS200 backend. Additional time was
allocated for our source, and observations were conducted remotely on $26$ May
2107, aiming at the 2 mm spectral window. Similarly, with the acquired
spectrum of CVLA958, no secure emission line could be detected for CVLA100. In
order to further test possible lines that are not apparent with visual
inspection, we developed a simple convolution algorithm to search for emission
from CO lines at a given redshift. Essentially, the approach of this code is
to sum the observed spectra at the positions where a CO detection was expected
at a given redshift, and subsequently vary the redshift in steps of 0.01.
Application of this analysis, presents no emission line detection, considering
that the resulting values for specific redshifts ranging from $z=0$ to $z=6$
do not deviate significantly from zero.

Figure 8: Reduced millimetre spectra of CVLA958 and CVLA100 observed by the
IRAM 30m and ALMA telescopes. The IRAM spectrum for CVLA958 is shown, with its
unbinned reduced spectrum presented in blue, and the orange line corresponds
to the Gaussian filtered spectrum using a width of $50\,km/s$ (left panel).
The CO lines for the redshift solution of $z=3.1755$ provided by Birkin et al.
(2021) are depicted with the dashed red lines. The applied FTS setups are
presented at the top of the plot and are coloured according to the adopted
configuration and observing date. The ALMA spectrum for CVLA100 at band 3 is
shown in the right panel. The vertical black line corresponds to the possible
CO(4-3) emission line at $z=3.29$, located at $\nu=107.3\,GHz$. The inset
cutout illustrates the velocity-integrated map of the line emission, and the
Spitzer 3.6 $\mu m$ contour (white isocontour) is overlaid.

###  B.2 ALMA observations

No emission was detected in the IRAM spectrum for CVLA100, our team decided to
secure additional observing time for the same candidate using the ALMA
telescope. More specifically, the source was observed with multiple tunings in
bands 3 (84-116 GHz) and 4 (125-163 GHz) in various configurations (longest
baselines from 300 m to 1400 m), under the project code 2017.1.01713.S. With
acceptable weather conditions (water vapour of $\leq 3\,mm$ at all times; wind
at $\sim 3\,m/s$), our observations achieved a mean angular resolution of
$\sim$1 arcsec and a mean continuum sensitivity of $\sim$0.2 mJy after 4.1
hours at band 3 and 1.4 hours of integration time at band 4. By analysing the
spectrum, extracted using the CASA software (CASA Team et al. 2022), we
detected a feature at 107.3 GHz which, considering the numerous CO emission
lines, assisted in constraining the redshift of the source. One plausible
solution indicates that this feature corresponds to the redshifted CO(4-3)
emission line at $z=3.3$. This proposed redshift possibility closely agrees
with the suggested solution of $z=3.15$ obtained from our photometric analysis
and the referenced literature catalogues (i.e. HELP, Cosmos2015, and
cosmos2020).


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##  Appendix A Figures of the lucky imaging observations

Figure 8:  AstraLux high-spatial resolution image of WASP-103 obtained on 13
January 2021 in the SDSSz bandpass. The image corresponds to the stacking of
the 10% frames with the highest Strehl ratio. We removed a fitted PSF of the
main target as a combination of a Gaussian and a Lorentzian profile. The
location of the previously detected companion by Ngo et al. (2016) is marked
as ’B?’ .  Figure 9:  Sensitivity curve for the AstraLux image of WASP-103.
The 1% contamination level is marked by the horizontal dotted line and the
maximum magnitude of a blended binary to be able to mimic the transit of
WASP-103b is marked by the horizontal dashed line. The location of the
previously detected companion by Ngo et al. (2016) is marked as a star-like
symbol.

##  Appendix B Priors for the limb darkening coefficients

Table 10: Priors for LD coefficients for the quadratic law.

Parameter | Prior
---|---
$LD1$ CHEOPS | $\mathcal{N}(0.5269,0.0218)$
$LD2$ CHEOPS | $\mathcal{N}(0.1279,0.046)$
$LD1$ HST | $\mathcal{N}(0.2346,0.0074)$
$LD2$ HST | $\mathcal{N}(0.1461,0.0266)$
$LD1$ SP1 | $\mathcal{N}(0.1080,0.018)$
$LD2$ SP1 | $\mathcal{N}(0.1220,0.0268)$
$LD1$ SP2 | $\mathcal{N}(0.0920,0.0138)$
$LD2$ SP2 | $\mathcal{N}(0.0976,0.0214)$
111111
$\mathcal{N}(a;b)$ is a normal distribution with mean $a$ and standard
deviation $b$.

Table 11: Priors for LD coefficients for the non-linear law.

Parameter | Prior
---|---
u CHEOPS | $\mathcal{N}(-0.0863,0.0089)$
v CHEOPS | $\mathcal{N}(0.8962,0.0147)$
w CHEOPS | $\mathcal{N}(-0.0438,0.0149)$
z CHEOPS | $\mathcal{N}(-0.1310,0.0065)$
u HST | $\mathcal{N}(-0.1113,0.0084)$
v HST | $\mathcal{N}(1.3060,0.0113)$
w HST | $\mathcal{N}(-1.0551,0.0076)$
z HST | $\mathcal{N}(0.2854,0.0026)$
u SP1 | $\mathcal{N}(-0.0074,0.00258)$
v SP1 | $\mathcal{N}(0.7294,0.00412)$
w SP1 | $\mathcal{N}(-0.7218,0.00332)$
z SP1 | $\mathcal{N}(0.2426,0.0017)$
u SP2 | $\mathcal{N}(0.0078,0.00234)$
v SP2 | $\mathcal{N}(0.5679,0.00418)$
w SP2 | $\mathcal{N}(-0.5661,0.00362)$
z SP2 | $\mathcal{N}(0.1932,0.00162)$

121212
$\mathcal{N}(a;b)$ is a normal distribution with mean $a$ and standard
deviation $b$.