11institutetext: Centro de Astrobiología (CAB), CSIC-INTA, Ctra. de Ajalvir km
4, Torrejón de Ardoz, E-28850, Madrid, Spain; 11email<EMAIL_ADDRESS>csic.es 22institutetext: Departamento de Física Teórica, Universidad Autónoma
de Madrid, E-28049, Cantoblanco (Madrid), Spain 33institutetext: Université
Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette,
France 44institutetext: Departamento de Física de la Tierra y Astrofísica,
Fac. CC. Físicas, Universidad Complutense de Madrid, Plaza de las Ciencias 1,
Madrid, E-28040, Spain 55institutetext: IPARCOS (Instituto de Física de
Partículas y del Cosmos), Facultad de Ciencias Físicas, Ciudad Universitaria,
Plaza de las Ciencias, 1, Madrid, E-28040, Spain 66institutetext: The
University of Texas at Austin, 2515 Speedway Blvd Stop C1400, Austin, TX
78712, USA 77institutetext: Cosmic Dawn Center (DAWN), Jagtvej 128, DK2200
Copenhagen N, Denmark 88institutetext: DTU-Space, Technical University of
Denmark, Elektrovej 327, DK2800 Kgs. Lyngby, Denmark 99institutetext: Niels
Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen N,
Denmark 1010institutetext: Department of Astronomy, University of Geneva,
Chemin Pegasi 51, 1290 Versoix, Switzerland

# Measuring the gas reservoirs in $10^{9}<$ M${}_{\star}<10^{11}$ M⊙ galaxies
at $1\leq z\leq 3$

Rosa M. Mérida 1122 Carlos Gómez-Guijarro 33 Pablo G. Pérez-González 11
Patricia Sánchez-Blázquez 4455 David Elbaz 33 Maximilien Franco 66 Lucas Leroy
33 Georgios E. Magdis 778899 Benjamin Magnelli 33 Mengyuan Xiao 1010

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. Understanding the gas content in galaxies, its consumption and
replenishment, remains pivotal in our comprehension of the evolution of the
Universe. Numerous studies have addressed this, utilizing various
observational tools and analytical methods. These include examining low-
transition 12CO millimeter rotational lines and exploring the far-infrared and
the (sub-)millimeter emission of galaxies. With the capabilities of present-
day facilities, much of this research has been centered on relatively bright
galaxies.

Aims. We aim at exploring the gas reservoirs of a more general type of galaxy
population at $1.0<z<3.0$, not restricted to bright (sub-)millimeter objects.
We strive to obtain a measurement that will help to constrain our knowledge of
the gas content at $10^{10-11}$ M⊙, and an upper limit at lower stellar
masses, $\sim 10^{9-10}$ M⊙.

Methods. We stack ALMA 1.1 mm data to measure the gas content of a mass-
complete sample of galaxies down to $\sim 10^{8.6}$ M⊙ at $z=1$ ($\sim
10^{9.2}$ M⊙ at $z=3$), extracted from the HST/CANDELS sample in GOODS-S. The
selected sample is composed of 5,530 on average blue ($<b-i>\sim 0.12$ mag,
$<i-H>\sim 0.81$ mag), star-forming main sequence objects ($\Delta$MS=log SFR
- log SFR${}_{\mathrm{MS}}\sim-0.03$).

Results. We report measurements at $10^{10-11}$ M⊙ and upper limits for the
gas fractions obtained from ALMA stacked data at $10^{8-10}$ M⊙. At
$10^{10-11}$ M⊙, our gas fractions (fgas=Mgas/(Mgas+M⋆)), ranging from 0.32 to
0.48 at these redshifts, agree well with other studies based on mass-complete
samples down to $10^{10}$ M⊙, and are lower than expected according to other
works more biased to individual detections. At $10^{9-10}$ M⊙, we obtain
3$\sigma$ upper limits for fgas which range from 0.69 to 0.77. These upper
limits are on the level of the extrapolations of scaling relations based on
mass-complete samples and below those based on individual detections. As such,
it suggests that the gas content of low-mass galaxies is at most what is
extrapolated from literature scaling relations based on mass-complete samples
down to $10^{10}$ M⊙. Overall, the comparison of our results with previous
literature reflects how the inclusion of bluer, less obscured, and more MS-
like objects progressively pushes the level of gas to lower values.

###### Key Words.:

Submillimeter: galaxies – Galaxies: high-redshift – star formation – evolution

##  1 Introduction

Cold molecular gas is the material that fuels the galaxy machinery that works
to form stars. Knowing the amount of available gas in galaxies, how
efficiently it is converted into stars, as well as how it is replenished is
crucial to understanding their evolutionary pathways. The cosmic history of
the gas mass density resembles that of the star formation rate density
(Decarli et al. 2019, Riechers et al. 2019, Magnelli et al. 2020, Walter et
al. 2020), peaking at $z\sim 2$ and steadily decreasing until now. The gas
mass (Mgas) content in galaxies at a fixed stellar mass (M⋆) increases with
redshift at least at $0<z<3$. At a fixed redshift, the gas fraction (fgas =
Mgas/(M⋆+Mgas)) decreases with M⋆ (Genzel et al. 2010, Béthermin et al. 2015,
Morokuma-Matsui & Baba 2015, Dessauges-Zavadsky et al. 2020, Tacconi et al.
2020, Magnelli et al. 2020, Wang et al. 2022).

The relation between Mgas and M⋆ at different redshifts has been quantified by
a variety of studies (e.g., Scoville et al. 2016, Tacconi et al. 2018, Liu et
al. 2019b, Tacconi et al. 2020, Kokorev et al. 2021, Wang et al. 2022),
covering $0<z<6$. It is typically parameterized according to cosmic time or
redshift, and the distance from the galaxies to the main sequence (MS) of
star-forming galaxies (SFGs). The term MS refers to the tight correlation that
exists between the SFR and the M⋆ (e.g., Noeske et al. 2007, Elbaz et al.
2011, Whitaker et al. 2012, Speagle et al. 2014, Tomczak et al. 2016, Santini
et al. 2017, Pearson et al. 2018, Barro et al. 2019), which is seen to be
present at least at $0<z<6$.

The cold molecular gas can be studied directly using rotational lines of
molecular hydrogen, H2. However, the transition probabilities are very small,
line emission is weak, and transitions are sufficiently excited only in
radiation or shock-warmed molecular gas-like photodissociation regions and
outflows (Parmar et al. 1991, Richter et al. 1995). Common alternatives to
study the gas content in distant galaxies include the use of the low-
transition 12CO millimeter rotational lines and dust continuum measurements.

For the first approach, it is typically assumed that the CO lower rotational
lines are optically thick and the CO line luminosity is proportional to the
total molecular gas mass (M${}_{\mathrm{H_{2}}}$), using an empirical
conversion factor (Dickman et al. 1986, Solomon et al. 1987, Bolatto et al.
2013).

For the second approach, the Mgas can be derived based on the dust content,
converting the dust mass (M${}_{\textrm{dust}}$) obtained by fitting the
infrared (IR) spectral energy distribution (SED) (Draine & Li, 2007) to Mgas,
for which it is typically assumed a metallicity-dependent gas-to-dust ratio
($\delta_{\mathrm{GDR}}$; e.g. Magdis et al. 2012, Genzel et al. 2015). One
can also use the photometry measured in the Rayleigh–Jeans (RJ) tail of the
SED (e.g. Scoville et al. 2016, Hughes et al. 2017). The Scoville et al.
(2016) method (S16 hereafter) works similarly to the previous one, assuming a
constant $\delta_{\mathrm{GDR}}$ with a mass-weighted dust temperature (Tdust)
of 25 K. These approaches assume that zero-point calibrations based on $z=0$
measurements are also valid at higher redshifts.

These methods have been previously used in other works to study the gas
content in the local and distant universe (e.g. Saintonge et al. 2017, Decarli
et al. 2019, Freundlich et al. 2019, Sanders et al. 2023 derived Mgas using
the 12CO rotational lines; Schinnerer et al. 2016, Wiklind et al. 2019,
Kokorev et al. 2021 used the dust emission; and Liu et al. 2019b, Aravena et
al. 2020, and Birkin et al. 2021 used both methods). However, despite the
increasing number of studies in the field, most of the efforts so far focus on
individual (sub-)millimeter detections of massive objects ($>10^{10-11}$ M⊙).
For instance, the Schinnerer et al. 2016 sample is made up of ALMA detections
at 240 GHz, with M${}_{\star}>10^{10.7}$ M⊙ at $z\sim 3.2$. In Freundlich et
al. (2019), they include CO emitters with 1010-11.8 M⊙ within $0.5<z<3$. The
Liu et al. 2019b sample contains galaxies at $0.3<z<6$ that show high-
confidence ALMA detections, with median M${}_{\star}=10^{10.7}$M⊙.

Alternatively, other studies sought to extend this analysis to fainter
galaxies and improve the completeness of the data-sets by stacking the
emission of similar sources, not imposing a flux criterion on the
(sub-)millimeter emission of the sources. In Tacconi et al. (2020), part of
their sample is based on this strategy, made up of stacks of $Herschel$ far
infrared (FIR) spectra. Their data-set also includes individual CO emitters
though. In Magnelli et al. (2020), they measure the cosmic density of dust and
gas by stacking $H$-band selected galaxies above a certain M⋆. In Garratt et
al. (2021), they study the evolution of the H2 mass density back to $z\approx
2.5$ measuring the average observed 850 $\mu$m flux density of near-infrared
selected galaxies. In Wang et al. (2022), they employ stacking to derive the
mean mass and extent of the molecular gas of a mass-complete sample down to
1010 M⊙. In the latter study, they obtain molecular gas masses which are
generally lower than previous estimates, based on individual detections.

In this work, we use the emission at 1.1 mm measured with observations
obtained by the Atacama Large Millimeter/submillimeter Array (ALMA) to infer
the content of gas present in a mass-complete sample of galaxies at
$1.0<z<3.0$, analyzing stacked ALMA images for subsamples in different
redshift ranges and M⋆ bins. Taking advantage of the galaxy catalog provided
by the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey
(CANDELS; Grogin et al. 2011, Koekemoer et al. 2011) in The Great
Observatories Origins Deep Survey (GOODS; Dickinson et al. 2003, Giavalisco et
al. 2004), specifically in GOODS-S (Guo et al. 2013, G13 hereafter), we probe
the $10^{10-11}$ M⊙ stellar mass regime with a complete sample whose 80%
completeness level reaches down to 108.6 M⊙ at $z=1$ (109.2 M⊙ at $z=3.0$)
(Barro et al., 2019). Our analysis aims at removing potential biases at the
high-mass end when based on detections of individual galaxies. We aspire to
check whether faint sources in ALMA follow the same scaling relations derived
from brighter sources or, on the contrary, present a distinct molecular gas
content than that prescribed for their stellar masses. Moreover, this sample
gives us the chance to explore the gas reservoirs of less massive galaxies,
$\sim 10^{9-10}$ M⊙, for which previous scaling relations are still not well
calibrated.

The structure of the paper is as follows. In Section 2 we present the data and
sample selection. We then describe the physical properties of the sample and
compare them with other catalogs in Section 3. In Section 4 we present our
stacking and flux measurement methodology applied to the ALMA data. In
Sections 5 and 6, we present and discuss our results regarding the gas
reservoirs of our sample, comparing them with previous scaling relations. The
conclusions are summarized in Section 7.

Throughout the paper we assume a flat cosmology with
$\Omega_{\mathrm{M}}=0.3$, $\Omega_{\lambda}=0.7$ and a Hubble constant
H${}_{0}=70$ km s-1 Mpc-1. We use AB magnitudes (Oke & Gunn, 1983). All M⋆ and
SFR estimations refer to a Chabrier (2003) initial mass function (IMF).

##  2 Data and sample

###  2.1 Data

We base this work on the images provided by the GOODS-ALMA 1.1 mm galaxy
survey (Franco et al. 2018, Gómez-Guijarro et al. 2022a) in GOODS-S, carried
out in ALMA Band 6. This survey extends over a continuous area of 72.42
arcmin2 with a homogeneous average sensitivity. It is the result of two
different array configurations. Cycle 3 observations (program 2015.1.00543.S;
PI: D. Elbaz) were based on a more extended array configuration that provided
a high angular resolution data-set (Franco et al. 2018). Cycle 5 observations
(program 2017.1.00755.S; PI: D. Elbaz) were based on a more compact array
configuration, which resulted in a lower angular resolution data-set (Gómez-
Guijarro et al., 2022a). In this work, we use the low-resolution data-set,
which has a sensitivity of 95.2 $\mu$Jy beam-1 and an angular resolution of
$1\aas@@fstack{\prime\prime}330\times\leavevmode\nobreak\
0\aas@@fstack{\prime\prime}935$. This choice is motivated by our interest in
detections and flux measurements, as opposed to resolving the extent of the
sources.

###  2.2 Sample selection

In this research, we use the source catalog provided by CANDELS in GOODS-S
(G13). Following Mérida et al. (2023) (M23 hereafter), we select galaxies in
the redshift range $1.0\leq z\leq 3.0$, where G13 cataloged 18,459 galaxies
(out of the full sample of 34,930 galaxies). We focus on galaxies with
M${}_{\star}>10^{8}$ M⊙, taking into account the G13 mass completeness limits,
which were computed based on the idea that the maximal mass permitted at a
given redshift for a galaxy is the mass of a source with a magnitude equal to
the faint limit of the sample (Fontana et al. 2004, Grazian et al. 2015). The
limits we report were obtained considering a 90% completeness limit in flux of
$H=26$ mag, which corresponds to the limit computed for the shallow part of
GOODS-S. Additionally, we only keep in our sample those sources with
M${}_{\star}<10^{11}$ M⊙, given that the number density of the G13 sample
sharply decreases towards $>$1011 M⊙.

We, moreover, restrict the sample to SFGs as indicated by the $UVJ$ diagram
(Whitaker et al., 2011), which allows us to classify galaxies in quiescent or
star-forming according to their rest-frame colors. This $UVJ$ selection
guarantees that the Mgas we derive, based on stacking galaxies, is not biased
to lower values because of the contribution of quiescent galaxies. All these
criteria leave us with a sample of 15,236 sources.

Finally, we discard any source lying outside the GOODS-ALMA map coverage. Our
final sample is thus composed of 5,530 star-forming objects located at
$1.0\leq z\leq 3.0$, with stellar masses ranging 108-11 M⊙.

Within this sample, we looked for the ALMA counterparts of our galaxies, as
well as for galaxies in the vicinity of sources showing an ALMA counterpart,
using the source catalog listed in Gómez-Guijarro et al. (2022a), GG22
hereafter. We select those galaxies closer than 5″ to any object included in
GG22. This 5″ radius is chosen considering the growth curve of the low-
resolution ALMA map point spread function (PSF) and the trade-off between the
number of objects in the sample and the possible contamination of ALMA
detected galaxies. This condition affects just $\sim$3% of the objects in the
sample. The effect of the exclusion of these individually detected sources and
their neighbors in the photometry and further fgas is discussed in Sec. 4 and
5.1. We will refer to the subsample obtained when excluding these galaxies as
the undetected data-set hereafter.

Additionally, we also looked for counterparts of these sources in other ALMA-
based catalogs, namely the ALMA twenty-six arcmin2 survey of GOODS-S at one
millimeter (ASAGAO; Hatsukade et al. 2018) and the ALMA Hubble Ultra Deep
Field (ALMA-HUDF; Dunlop et al. 2017, Hill et al. 2023). In Yamaguchi et al.
(2020), they include a list of those ASAGAO sources that have an optical
counterpart in the Fourstar Galaxy Evolution Survey (ZFOURGE; Straatman et al.
2016), and in Hill et al. (2023) they include the optical counterparts in G13
of the sources from the ALMA-HUDF catalog. Only 4 sources from the undetected
data-set coincide with objects from ASAGAO and another 4 with sources from
ALMA-HUDF.

We also investigated if any of these sources belonging to the undetected data-
set shows significant emission at (sub-)millimeter wavelengths. We measure the
photometry of each of these objects using the aperture photometry method
provided by photutils from Python, selecting an aperture radius
$r=0\aas@@fstack{\prime\prime}8$. Only 18 out of the undetected sources show a
signal-to-noise ratio SNR$>$3\. Only 3 out of these 18 galaxies show an
SNR$>$3.5. When considering the SNR at the peak, only 1% of these galaxies
show an SNR${}_{\mathrm{peak}}>3.5$ (including the latter 18 sources). Based
on the analysis carried out in GG22, none of the sources is massive enough for
the ALMA emission excess to be regarded as real and, therefore, they are
indistinguishable from random noise fluctuations (see Gómez-Guijarro et al.
2022a for more details). Given the low SNR of our sources, in order to look
into the gas reservoirs of these galaxies we need to analyze stacked data.

##  3 Properties of the sample and comparison with other catalogs

In this section, we compare the properties of the galaxies from our sample
with those from other catalogs that were used by previous studies that also
aimed at inferring the gas content of galaxies. In particular, we will refer
to (i) the sources from the ”super-deblended” catalogs, performed in GOODS-N
and in the Cosmic Evolution Survey (COSMOS; Scoville et al. 2007), that were
used to derive the Kokorev et al. (2021) scaling relation; (ii) the galaxies
from the Automated ALMA Archive mining in the COSMOS field (A3COSMOS), used to
obtain the Liu et al. (2019b) scaling relation; (iii) the objects from Tacconi
et al. (2020), based on a compilation of individually detected galaxies from
different surveys and also stacks, and used to derive their scaling relation;
(iv) the galaxies from the COSMOS2020 catalog that lie in the A3COSMOS
footprint (COSMOS2020∗ hereafter), which is the sample the Wang et al. (2022)
scaling relation is based on and, finally, (v) the sources from GG22. All
these comparison samples are cut to only include galaxies within the same M⋆
and redshift intervals that we are considering in this work (see Sec. 2).
Later, in Sec. 5.2, we will refer to these same data-sets in the context of
the scaling relations that were derived based on them. In Fig. 1, we show
different diagrams that highlight the properties of the listed samples
together with our data-set, based on G13. In Table 1, we summarize the
information contained in Fig. 1.

Given that some of these works do not report the values of the rest-frame
colors of their sources, the $i-H$ and $b-i$ colors allow us to build a
diagram that works similarly to an $UVJ$, but using apparent magnitudes. In
Fig. 1, for the panel showing this color vs. color diagram, we use the
photometry measured within the $F435W$ ($b$), the $F775W$ ($i$), and the
$F160W$ ($H$) bands from HST for our sample and for GG22. For A3COSMOS and
COSMOS2020∗, we use the photometry measured within the Subaru Prime Focus
Camera (Suprime-Cam) $b$ band, the Hyper Suprime-Cam (HSC) $i$ band, and the
UltraVISTA $H$ band. For the super-deblended catalogs and the Tacconi et al.
(2020) data-set we also use the HST photometry, together with the Canada-
France-Hawaii Telescope (CFHT) and Subaru observations in the absence of HST
data.

Fig. 1 also compares the position of the samples in the SFR vs M⋆ plane. In
that panel, we include the M23 fit, defined for $1.5<z<2.0$, up to $10^{10}$
M⊙, and the Barro et al. (2019) MS fit, B19 hereafter, above $10^{10}$ M⊙. The
distance of each point to the MS is re-scaled to its corresponding redshift.
In the fourth panel, we show the difference with respect to the MS for each
galaxy in these samples, $\Delta$MS ($\Delta$MS = log SFR$-$log SFRMS, with
$\Delta$MS = 0 being equivalent to $\delta$MS = SFR/SFRMS = 1). We use M23 to
calculate the $\Delta$MS of galaxies with M⋆/M${}_{\odot}<10^{10}$ and B19 for
sources with higher stellar masses, given that M23 focuses on the low-mass end
of the MS.

It is important to mention that for our G13-based sample, Tacconi et al.
(2020), and COSMOS2020∗, the SFR were computed following the ladder technique
(Wuyts et al., 2011), which combines SFR indicators at UV, mid-infrared (MIR)
and FIR. For the galaxies from the super-deblended catalogs, the SFR was
computed from the integrated IR luminosity (LIR) using the Daddi et al. (2010)
relation. The SFRs for A3COSMOS were computed from the IR luminosity using the
Kennicutt (1998) calibration. For the GG22 galaxies, the SFRs were calculated
as the sum of the SFRIR (using the Kennicutt 1998 calibration) and the SFRUV
(using the Daddi et al. 2004 calibration). We check that only 219 of our
galaxies (4%, with only 10 galaxies with M${}_{\star}>10^{10}$ M⊙) are
detected in the MIR and/or FIR using Spitzer MIPS and Herschel PACS and SPIRE.
This means that the SFRs of our sample come mostly from the UV emission.

Figure 1: From left to right and up and down we show: stellar mass vs.
redshift, a color vs. color diagram based on $i-H$ and $b-i$, the star
formation rate vs. stellar mass, and histograms showing the distance to the
main sequence in log scale, $\Delta$MS, calculated using the Mérida et al.
(2023) and Barro et al. (2019) fits to the main sequence. We cut the
comparison samples to only include those galaxies within $1.0<z<3.0$ and
having $10^{8-11}$ M⊙. In blue, we represent our sample. The GG22 galaxies are
identified in yellow. The Tacconi et al. (2020) and A3COSMOS samples are shown
in gray and green, respectively. The COSMOS2020∗ sample is represented in
magenta and the galaxies from the super-deblended catalogs are displayed in
maroon. In the stellar mass vs. redshift plot, the blue contours showing our
sample enclose roughly 20%, 50%, 60%, 70%, 80%, and 90% of the data. In the
color-color diagram, the contours roughly enclose the 10%, 20%, 40%, 60%, 80%,
and 90% of the data. In these two panels, histograms of the quantities there
represented are also included, following the same color code. Quartiles are
represented as horizontal segments in all the histograms. The Mérida et al.
(2023) main sequence up to $10^{10}$ M⊙ and the Barro et al. (2019) main
sequence above $10^{10}$ M⊙ are shown in the third panel as a solid black
line. The dashed lines in the third and fourth panels show the area enclosed
within 3$\sigma$ with respect to the main sequence, based on the typical
scatter reported in Speagle et al. (2014) ($\sim$0.30 dex).

###  3.1 This work: the G13-based data-set

The sample evenly populates the redshift range considered in this work (first
panel of the first row in Fig. 1), with median and quartiles
$z=1.9_{1.5}^{2.4}$. The low-mass coverage of G13 allows us to reach down to
$10^{8-9}$ M⊙ (log (M⋆/M⊙)=$8.6_{8.3}^{9.1}$). In terms of the optical colors
(second panel of the first row), our sample shows typical values of
0.81${}_{0.29}^{1.31}$ mag for $i-H$ and 0.12${}_{-0.04}^{0.35}$ mag for
$b-i$.

The position of our galaxies in the SFR vs M⋆ plane (first panel of the second
row) is compatible with the MS, with only a minor population of galaxies above
or below three times the typical scatter ($\sim 0.04$% and $\sim 1$% of the
galaxies above/below 3$\sigma$, respectively, with $\sigma$ being $\sim\,$0.3
dex according to Speagle et al. 2014). The median $\Delta$MS (second panel of
the second row) of our galaxies is $\Delta$MS $=-0.03_{-0.25}^{0.17}$ dex.

Table 1: Summary of the properties of our data-set and the comparison samples shown in Fig. 1, and described along Sec. 3. The values here shown correspond to the median and first and third quartiles. The names between brackets refer to the scaling relations that are based on each data-set (see Sec. 5.2). W22 refers to the Wang et al. (2022) relation, K21 to the relation from Kokorev et al. (2021), and L19 to that of Liu et al. (2019b). The T20 data-set was used in that same work to obtain their scaling relation. | z |  log M⋆ | $i-H$ | $b-i$ |  $\Delta$MS
---|---|---|---|---|---
|  |  (M⊙) | (mag) | (mag) | (dex)
Mass-complete samples
This work |  1.9${}_{1.5}^{2.4}$ |  8.6${}_{8.3}^{9.1}$ |  0.81${}_{0.29}^{1.31}$ |  0.12${}_{-0.04}^{0.35}$ | $-0.03_{-0.25}^{0.17}$
COSMOS2020∗ (W22) |  1.8${}_{1.3}^{2.2}$ |  9.0${}_{8.6}^{9.6}$ |  0.85${}_{0.51}^{1.21}$ |  0.59${}_{0.37}^{0.89}$ | $-0.06_{-0.39}^{0.28}$
Individual (sub-)mm detections + stacks
T20 |  1.4${}_{1.1}^{2.0}$ |  10.6${}_{10.4}^{10.8}$ |  1.87${}_{1.45}^{2.36}$ |  0.95${}_{0.76}^{1.40}$ | $0.33_{0.02}^{0.67}$
Individual (sub-)mm detections
Super-deblended (K21) |  1.6${}_{1.1}^{2.0}$ |  10.4${}_{10.1}^{10.7}$ |  1.85${}_{1.35}^{2.34}$ |  1.24${}_{0.88}^{1.71}$ | $0.34_{0.14}^{0.55}$
A3COSMOS (L19) |  2.1${}_{1.8}^{2.6}$ |  10.7${}_{10.5}^{10.9}$ |  1.89${}_{1.46}^{2.27}$ |  1.24${}_{0.86}^{1.61}$ | $0.58_{0.36}^{0.76}$
GG22 |  2.2${}_{1.9}^{2.7}$ |  10.5${}_{10.3}^{10.7}$ |  1.98${}_{1.30}^{2.54}$ |  0.82${}_{0.41}^{1.36}$ | $0.46_{0.22}^{0.83}$

###  3.2 Comparison data-sets

  * •

”Super-deblended” catalogs

The ”super-deblended” catalogs (Liu et al. 2018, Jin et al. 2018), performed
in GOODS-N and COSMOS and constructed using FIR and sub-millimeter images, use
the prior positions of sources from deep Spitzer/IRAC and Very Large Array
(VLA) 20 cm observations to obtain the photometry of blended FIR/sub-
millimeter sources. They also employ the SED information from shorter
wavelength photometry as a prior to subtract lower redshift objects. In the
case of the COSMOS super-deblended catalog, the authors additionally select a
highly complete sample of priors in the $Ks$-band from the UltraVista
catalogs. Apart from selecting those galaxies satisfying our redshift and M⋆
cuts, we only keep those galaxies showing an SNR $>3$ in at least 3 FIR to
sub-millimeter bands from 100 $\mu$m to 1.2 mm, following Kokorev et al.
(2021). The optical photometry of these galaxies is obtained by looking for
possible optical counterparts in the CANDELS catalog performed in GOODS-N
(B19) and in the COSMOS2020 catalog (Weaver et al., 2022).

The median redshift and quartiles of the galaxies satisfying our selection
criteria are $z=1.6_{1.1}^{2.0}$, with a higher concentration of lower
redshift galaxies compared to our sample, and in line with T20. This data-set
is biased towards more massive galaxies than our sample (log M⋆/M⊙ =
10.4${}_{10.1}^{10.7}$), $\sim 1.8$ dex more massive than our galaxies. Their
optical $i-H$ and $b-i$ colors are redder than the ones traced by our sources
($i-H=1.85_{1.35}^{2.34}$ mag and $b-i=1.24_{0.88}^{1.71}$ mag, respectively),
$\sim 1$ mag redder in both colors.

In terms of the position of these galaxies with respect to the MS, these
sources show $\Delta$MS values compatible with being MS galaxies, showing
$\Delta$MS = $0.34_{0.14}^{0.55}$ dex. This value corresponds though to a more
star-forming data-set, more compatible with the upper envelope of the MS,
given the typical scatter. 6% of the galaxies show values $>$3$\sigma$.

  * •

A3COSMOS

The A3COSMOS dataset (Liu et al., 2019a) contains $\sim$700 galaxies
($0.3<z<6$) with high-confidence ALMA detections in the (sub-)millimeter
continuum. It consists of a blind extraction, imposing a
S/N${}_{\mathrm{peak}}>5.40$, and on a prior-based extraction, using the known
positions of sources in the COSMOS field, cutting the final sample to
S/N${}_{\mathrm{peak}}>4.35$. We extract the photometry of these sources from
the COSMOS2020 catalog.

The A3COSMOS galaxies with redshifts and M⋆ in common with this work are
mainly located at higher redshifts ($z=2.11_{1.75}^{2.64}$) compared to our
galaxies, and are also biased towards more massive objects (log (M⋆/M⊙) =
$10.7_{10.5}^{10.9}$), $\sim 2$ dex more massive than our sources in this
case. They also display redder optical colors, with values of
1.89${}_{1.46}^{2.27}$ mag for $i-H$ and 1.24${}_{0.86}^{1.61}$ mag for $b-i$,
$\sim$1 mag redder in both colors than our sample.

According to their position in the SFR vs. M⋆ plane, these objects are also
compatible with the MS, but, as well as the galaxies from the super-deblended
catalogs, T20, and GG22, they are located in the upper envelope, showing
values nearly 2 times the typical scatter ($\Delta$MS = 0.58${}_{0.36}^{0.76}$
dex, with 13% of the galaxies above 3$\sigma$).

  * •

Tacconi et al. (2020)

The Tacconi et al. (2020) sample (T20 hereafter) is based on the existing
literature and ALMA archive detections for individual galaxies and stacks. It
consists of 2,052 SFGs. 858 of the measurements are based on CO detections,
724 on FIR dust measurements, and 470 on $\sim$1 mm dust measurements. We
extract their photometry looking for the counterparts of the individual
objects in the CANDELS catalogs performed in the different cosmological
fields, using the catalogs already specified together with the Stefanon et al.
2017 catalog for the Extended Groth Strip (EGS; Davis et al. 2007), and
Galametz et al. 2013 for the Ultra Deep Survey (UDS; Lawrence et al. 2007,
Cirasuolo et al. 2007). It is however true that, since part of their sample is
based on stacking, our results regarding the colors will only reflect the
nature of the individual detections that make up the sample.

We see that the Tacconi et al. (2020) galaxies meeting our redshift and M⋆
criteria are centered at $z=1.4_{1.1}^{2.0}$, in line with the super-deblended
sample. In terms of M⋆, this data-set is made up mostly of massive objects
(log (M⋆/M⊙) = $10.6_{10.4}^{10.8}$), 2 dex more massive than our sample.
According to the optical colors, this sample traces redder values of $i-H$ and
$b-i$, typically $1.87_{1.45}^{2.36}$ mag and $0.95_{0.76}^{1.40}$ mag for
each of these colors. This is more than 1 mag redder in $i-H$ and $\sim 0.8$
mag redder in $b-i$.

These galaxies are more star-forming than our sources, showing $\Delta$MS =
0.33${}_{0.02}^{0.67}$ dex, which is compatible with them being in the upper
envelope of the MS (13% of the galaxies are located above 3$\sigma$).

  * •

COSMOS2020∗

The COSMOS2020 catalog comprises 1.7 million sources across the 2 deg2 of the
COSMOS field, $\sim$966,000 of them measured with all available broad-band
data. Compared to COSMOS2015 (Laigle et al., 2016), it reaches the same
photometric redshift precision at almost one magnitude deeper. It goes down to
$10^{8.43}$ M⊙ at $z=1$ with 70% completeness ($10^{9.03}$ M⊙ at $z=3$). We
keep those galaxies that lie within the A3COSMOS footprint, which we will call
COSMOS2020∗, consisting of 207,129 objects. This sample is not biased towards
ALMA-detected galaxies, CO emitters, or high-mass systems, which makes it more
similar to our data sample.

The median redshift of the galaxies within our redshift and M⋆ intervals is
$z=1.77_{1.31}^{2.24}$, comparable to the values we retrieve for our sample.
The COSMOS2020∗ data-set shows a typical M⋆ of log (M⋆/M⊙) =
$9.0_{8.6}^{9.6}$, $\sim$0.5 dex more massive than our sample. According to
the optical colors, these sources show similar $i-H$ colors
($0.85_{0.51}^{1.21}$ mag) to our galaxies, and around $\sim 0.50$ mag redder
colors of $b-i$ ($0.59_{0.37}^{0.89}$ mag).

These objects are located well within the MS typical scatter, with a
$\Delta$MS very similar to the one we obtain for our data-set ($\Delta$MS =
$-0.06_{-0.39}^{0.28}$ dex, with 4% of the galaxies above 3$\sigma$ and 7% of
the galaxies below 3$\sigma$).

  * •

GG22

GG22 presented an ALMA blind survey at 1.1 mm and built a bona fide sample of
88 sources, comprising mostly massive dusty star-forming galaxies. Half of
them are detected with a purity of 100% with a S/N${}_{\mathrm{peak}}>5$ and
half of them with 3.5 $\leq$ S/N${}_{\mathrm{peak}}\leq$ 5, aided by the
Spitzer/IRAC and the VLA prior positions. We retrieve the optical fluxes of
the GG22 ALMA-selected galaxies from ZFOURGE.

The GG22 sources compatible with our redshifts and M⋆ cuts are also biased
towards high redshifts compared to our sample, similarly to A3COSMOS
($z=2.15_{1.91}^{2.67}$). As well as the super-deblended data-set, T20, and
A3COSMOS, GG22 is mainly made up of massive galaxies, $\sim$2 dex more massive
than our objects (log (M⋆/M⊙) = $10.5_{10.3}^{10.7}$). Their optical colors
are also redder than the ones showed by our sample, with median and quartiles
being $1.98_{1.30}^{2.54}$ mag for $i-H$ ($\sim$1 mag redder) and
$0.82_{0.41}^{1.36}$ mag for $b-i$ (0.7 mag redder).

These galaxies are also MS objects but, as well as the sources from the super-
deblended data-set, T20, and A3COSMOS, they are more star-forming than our
sources ($\Delta$MS = 0.46${}_{0.22}^{0.83}$ dex), located above the typical
scatter of the MS (20% of the galaxies above 3$\sigma$).

###  3.3 Comparison remarks

The main differences between our data-set and the comparison samples, with the
exception of COSMOS2020∗, are the blue optical colors of our galaxies, their
low M⋆ coverage, and their closer proximity to the MS. However, the latter
results concerning the blue optical colors of our galaxies can be a
consequence of mixing different redshifts and M⋆ when producing the color-
color diagram. We thus decided to cut it in 0.5 redshift bins and select those
galaxies with $>10^{10}$ M⊙, which allows a direct comparison with the other
catalogs. In Fig. 2, we show the color-color diagram included in Fig. 1
divided into different redshift bins. We only show the super-deblended,
A3COSMOS and COSMOS2020∗ galaxies as comparison data-sets since the number of
objects in each redshift bin included in these catalogs still provides the
means to obtain meaningful number statistics to compare with.

Figure 2: Color vs. color diagram based on $i-H$ and $b-i$ in different
redshift bins. We only include galaxies with M${}_{\star}\geq 10^{10}$ M⊙
within our data-set, the super-deblended catalogs, A3COSMOS, and COSMOS2020∗.
See Fig. 1 for the description of the color codes and markers here shown.

When restricting our sample to galaxies with $>10^{10}$ M⊙, the difference in
$i-H$ diminishes and we retrieve similar values to those obtained for the
comparison samples. We get $2.23_{1.71}^{2.66}$ mag at $1.0\leq z<1.5$, and
$2.25_{1.51}^{2.84}$ mag at $2.5\leq z\leq 3.0$ for our data-set.

For the $b-i$ color, we trace bluer values than the super-deblended catalogs
and A3COSMOS while getting similar results to COSMOS2020∗. The difference
between the set of our sample and COSMOS2020∗, and the super-deblended
catalogs and A3COSMOS increases with redshift, and the color gets bluer as
well. We obtain $1.07_{0.82}^{1.32}$ mag at $1.0\leq z<1.5$
($0.25_{0.12}^{0.42}$ mag at $2.5\leq z\leq 3.0$) according to our data-set,
compared to $1.56_{1.19}^{1.98}$ mag at $1.0\leq z<1.5$ ($1.20_{0.99}^{1.48}$
mag at $2.5\leq z\leq 3.0$) for the super-deblended catalogs, and
$1.54_{1.02}^{1.76}$ mag at $1.0\leq z<1.5$ ($1.36_{1.01}^{1.61}$ mag at
$2.5\leq z\leq 3.0$) for A3COSMOS. The affinity with COSMOS2020∗ and the
discrepancy with the super-deblended catalogs and A3COSMOS in this color are
expected. The COSMOS2020∗ includes all the galaxies at these stellar masses,
regardless of their flux at (sub-)millimeter wavelengths, hence being mass-
complete at $10^{10}$ M⊙, similarly to our sample. On the contrary, the super-
deblended catalogs use prior positions from deep Spitzer/IRAC and VLA
observations, and the A3COSMOS only considers sources with high-confidence
ALMA detections, which is translated to redder colors of $b-i$ and higher dust
obscuration. Our galaxies show median optical extinctions, A(V), ranging
1.03-1.71 mag, smaller as we increase in redshift, whereas these numbers are
2.08-2.28 mag for A3COSMOS.

##  4 Stacking analysis and flux measurements

In order to study the gas content of our galaxies, we stack the emission of
objects similar to each other. We group galaxies according to (1) redshift and
(2) log M⋆. We distinguish:

  * •

4 redshift bins: $1.0\leq z<1.5$, $1.5\leq z<2.0$, $2.0\leq z<2.5$, and
$2.5\leq z\leq 3.0$

  * •

3 M⋆ bins, 8$\leq$log M${}_{\star}/$M${}_{\odot}<$9, 9$\leq$log
M${}_{\star}/$M${}_{\odot}<$10, 10$\leq$log M${}_{\star}/$M${}_{\odot}\leq$11

These divisions in redshift and stellar mass are chosen as a result of an
estimation used to evaluate and maximize the probability of obtaining
detections according to different combinations of redshift and stellar mass
intervals. The estimation is based on the depth of the observations and the
previous knowledge about the gas reservoirs in galaxies as given by the
scaling relations derived in other works (see Sec. 5.2). If we consider the
expected gas fractions provided by these relations and use the
$\delta_{\mathrm{GDR}}$ approach (see Sec. 1 and 5.1), we can calculate the
typical flux density that corresponds to those gas fractions and we can
roughly infer the number of objects necessary to obtain a measurement with SNR
$>3$. For this, we quantify the relation $\sigma$ $\;\propto 1/\sqrt{N}$ (with
$\sigma$ being the noise and $N$ the number of objects) considering different
combinations of redshift and mass bins and obtain that, for getting a
measurement (SNR $>3$) at 10 $\leq$ log M${}_{\star}/$M${}_{\odot}\leq$ 11,
just a few objects ($<$ 10) are required. For the 9 $\leq$ log
M${}_{\star}/$M${}_{\odot}<$ 10 bin, we require a number of objects of the
order of hundreds. Finally, for the 8 $\leq$ log M${}_{\star}/$M${}_{\odot}<$
9 bin, we would need tens of thousands of objects to reach the necessary depth
according to our current knowledge of the gas reservoirs in galaxies. We check
that the adopted redshift division guarantees these numbers for the 9 $\leq$
log M${}_{\star}/$M${}_{\odot}<$ 10 and 10 $\leq$ log
M${}_{\star}/$M${}_{\odot}\leq$ 11 mass bins, while for the 8 $\leq$ log
M${}_{\star}/$M${}_{\odot}<$ 9 bin, we lack objects, regardless of how we
divide in redshift, what already warns that the probability to obtain a
measurement in this mass bin is very low. This estimation does not, however,
ensure that we are obtaining measurements for the two remaining bins, given
that scaling relations are not calibrated for the kind of objects we are
considering in this work, but still can be used as a starting point.

After defining the bins, we stack 50$\times$50 arcsec2 cutouts within the low-
resolution ALMA mosaic, centered at each source and using the coordinates of
the centroids provided by G13. Before the stacking, we corrected these
centroids for a known offset between the HST and ALMA data, reported in
different studies (e.g. Dunlop et al. 2017, Franco et al. 2018). We apply the
correction from Whitaker et al. (2019), that corresponds to $\delta$R.A.(deg)
= (0.011$\pm$0.08)/3600 and $\delta$decl.(deg) = ($-0.26\pm 0.10)/3600$. We
opted for median stacking galaxies instead of mean stacking them. We obtain
higher SNR in the median stacks (10% more SNR) while getting similar
photometric fluxes. We also check that the centroids computed using the
stacked emission in ALMA are compatible with those provided by G13, based on
the HST imaging, within $0\aas@@fstack{\prime\prime}06$.

The photometry is calculated within an aperture of
$r=0\aas@@fstack{\prime\prime}8$. We then apply the corresponding aperture
correction by dividing this flux density by that enclosed within the
synthesized dirty beam (normalized to its maximum value) using the same
aperture radius (see Gómez-Guijarro et al. 2022a for more details). This
aperture correction is $\sim$1.67 for $r=0\aas@@fstack{\prime\prime}8$.

When the SNR $<3$, we calculate an upper limit for the flux density based on
the surrounding sky emission by placing 10,000
$r=0\aas@@fstack{\prime\prime}8$ apertures at random positions across a
20$\times$20 arcsec2 cutout centered at the source. We measure the photometry
within each of these apertures and produce a histogram with all the values,
fitting the resulting Gaussian distribution leftwards to the peak, to skip the
possible emission of the source. We compute the upper limit as 3 times the
standard deviation of the fit.

If SNR $>3$ within the aperture we repeat the measurement using an aperture
radius $r=1\arcsec$. We check that this larger radius allows us to optimize
the flux/SNR gain/loss, recovering $\sim$7$\%$ more flux. The aperture
correction for $r=1\arcsec$ is $\sim$1.28. The uncertainty associated with the
measurements is calculated by placing 10,000 $r=1\arcsec$ apertures at random
positions across the 50$\times$50 arcsec2 cutout. We measure the photometry
within each aperture and fit the histogram leftwards to the peak, as done in
the calculation of the upper limits in the previous case. The standard
deviation provided by this fit is taken as the uncertainty of the measurement.

In Fig. 3 we show an image of all the stacks. In Table 2 we list the flux
densities we measure, together with the derived uncertainties. In both, Fig. 3
and Table 2, we also include the results for the undetected data-set, defined
in Sec. 2. In Sec. 5.1, we discuss the effects of the inclusion or exclusion
of the GG22 sources and their neighbors in the fgas. For both, the whole
sample and the undetected data-set, we obtain SNR $>3$ flux density
measurements for $10^{10-11}$ M⊙ (high-mass bin) at all redshifts. The flux
density enclosed within this mass bin increases towards higher redshifts. For
the intermediate-mass ($10^{9-10}$ M⊙) and the low-mass bins ($10^{8-9}$ M⊙),
we provide 3$\sigma$ upper limits. In the case of the intermediate-mass one,
we obtain a signal close to our SNR threshold at $1.5\leq z<2.0$, with an SNR
= 2.6 (SNR = 2.5 for the undetected data-set), and again at $2.5\leq z\leq
3.0$, with an SNR = 2.2 (SNR = 2.1 for the undetected data-set).

Figure 3: Cutouts of 7$\times$7 arcsec2 of the ALMA low-resolution map showing the median stacked galaxies in each redshift and mass bin. For each bin, we include cutouts that correspond to the whole sample and the undetected data-set, as defined in Sec. 2. The apertures used to measure the photometry are displayed in green. The size of the beam is shown in the first panel of the figure in gray. The flux densities and the corresponding uncertainties for each stacked galaxy are included in Table 2. Table 2: Summary of the photometry derived in this work. We show the flux densities for our sample, and a subsample excluding the Gómez-Guijarro et al. (2022a) sources and those from Guo et al. (2013) which are closer than 5 ″ to them, as explained in Sec. 2. The absence of uncertainty denotes an upper limit at the 3$\sigma$ level. z bin | log M⋆ bin | Nobj | Flux density | erFlux
---|---|---|---|---
| (M⊙) |  | ($\mu$Jy) | ($\mu$Jy)
| Whole | sample |  |
|  $8\leq$ log M${}_{\star}<$ 9 | 910 |  $<$ 17.31 | -
1.0 $\leq z<$ 1.5 |  $9\leq$ log M${}_{\star}<$ 10 | 299 |  $<$ 25.88 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 92 | 132.10 | 20.36
|  $8\leq$ log M${}_{\star}<$ 9 | 1423 |  $<$ 12.70 | -
1.5 $\leq z<$ 2.0 |  $9\leq$log M${}_{\star}<$10 | 353 |  $<$ 26.13 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 96 | 148.31 | 18.67
|  $8\leq$ log M${}_{\star}<$ 9 | 881 |  $<$ 14.99 | -
2.0 $<z<$ 2.5 |  $9\leq$ log M${}_{\star}<$ 10 | 274 |  $<$ 25.94 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 54 | 219.30 | 24.01
|  $8\leq$ log M${}_{\star}<$ 9 | 751 |  $<$ 15.71 | -
2.5 $\leq z\leq$ 3.0 |  $9\leq$log M${}_{\star}<$10 | 340 |  $<$ 26.38 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 57 | 211.26 | 24.79
| Undetected | sources |  |
|  $8\leq$ log M${}_{\star}<$ 9 | 890 |  $<$ 16.83 | -
1.0 $\leq z<$ 1.5 |  $9\leq$ log M${}_{\star}<$ 10 | 285 |  $<$ 25.71 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 85 | 115.97 | 20.91
|  $8\leq$ log M${}_{\star}<$ 9 | 1387 |  $<$ 11.75 | -
1.5 $\leq z<$ 2.0 |  $9\leq$ log M${}_{\star}<$ 10 | 338 |  $<$ 25.57 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 84 | 110.66 | 19.72
|  $8\leq$ log M${}_{\star}<$ 9 | 854 |  $<$ 14.64 | -
2.0 $<z<$ 2.5 |  $9\leq$ log M${}_{\star}<$ 10 | 265 |  $<$ 30.55 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 46 | 165.87 | 24.83
|  $8\leq$ log M${}_{\star}<$ 9 | 732 |  $<$ 14.86 | -
2.5 $\leq z\leq$ 3.0 |  $9\leq$ log M${}_{\star}<$ 10 | 325 |  $<$ 27.91 | -
|  $10\leq$ log M${}_{\star}\leq$ 11 | 47 | 165.93 | 27.05

The use of a certain aperture radius in our measurements, in this case,
$r=1.0$ ″, involves some flux loss. Departure from a point-like source may
involve an additional flux correction based on the galaxy morphology (see
Blánquez-Sesé et al. 2023). We consider two size estimations: the size of the
dust component as prescribed by GG22 and the size of the stellar component as
measured and reported in G13, based on $H$-band data.

As pointed out in several studies, the dust component is usually more
concentrated than the stellar one (e.g. Kaasinen et al. 2020, Tadaki et al.
2020, Gómez-Guijarro et al. 2022a, Liu et al. 2023). However, it is currently
uncertain if our stacks, based on a mass-complete sample including faint
objects, follow the latter statement, given that previous size estimations of
the dust component rely on individual detections of bright objects at
(sub-)millimeter wavelengths. Due to this, we also include a size estimation
based on the stellar component.

According to GG22, the effective radius Reff (the radius that contains half of
the total light) of the dust component of a source with $z=1.9$ and
M${}_{\star}=10^{10.5}\;$M⊙ is 0$\aas@@fstack{\prime\prime}$10\. At HST
$H$-band resolution our galaxies are fitted by a Sérsic profile characterized
by a median Sérsic index n = 1.36 and a median effective radius
R${}_{\mathrm{eff}}=0\aas@@fstack{\prime\prime}36$. In Fig. 4, we show the
flux correction factor one should take for our measurements (i.e., at 1010-11
M⊙) versus the Reff. Focusing on the size estimation provided by GG22, the
flux correction associated with our measurements is negligible. According to
the size of the stellar component, for a Sérsic index n = 1.0-1.5, this
correction ranges from 1.17-1.22. If the size of the dust component resembled
that of the stellar component, this $\sim$20% correction would translate to
0.08 dex larger M${}_{\textrm{gas}}$ than those reported in Table 3 and Fig.
5.

Figure 4: Flux ratio between a point source emission and a modeled galaxy
profile versus the effective radius. The blue line shows the profile for a
Sérsic index, n = 0.5, the orange one for n = 1, the green one for n = 1.5,
and the red one for n=4. The vertical solid line shows the typical size of the
dust component according to Gómez-Guijarro et al. (2022a) for a $z=1.9$, log
M⋆/M${}_{\odot}\sim$ 10.5 galaxy. The vertical dashed line shows the typical
size according to the median Sérsic parameters extracted from G13 for the
M${}_{\star}\geq 10^{10}\;$M⊙ galaxies in our sample.

##  5 Gas reservoirs

###  5.1 Observed evolution of the gas reservoir of our sample

We calculate the gas content of our sample following two approaches. The first
is based on the computation of a $\delta_{\mathrm{GDR}}$ using a mass
metallicity relation (MZR), and the second on the RJ dust continuum emission
(see Sec. 1).

For the first one, we produce synthetic spectra of the dust emission of our
galaxies, according to their median redshift and $\Delta$MS, using the
Schreiber et al. (2018) IR SED template
library111http://cschreib.github.io/s17-irlib/. This library contains 300
templates, divided into two classes: 150 dust continuum templates due to the
effect of big dust grains, and 150 templates that include the MIR features due
to polycyclic aromatic hydrocarbon molecules (PAHs). These templates, which
can be co-added, correspond to the luminosity that is emitted by a dust cloud
with a mass equal to 1 M⊙. After scaling each template to the measured flux
density of the stacked galaxy at 1.1 mm, we obtain the LIR by integrating the
rest-frame template flux between 8 and 1000$\mu$m. This luminosity is then
translated to Mdust by multiplying the intrinsic Mdust/LIR of the template by
the LIR that corresponds to the measured flux density. Schreiber et al. (2018)
models use different dust grain composition and emissivity yielding lower
Mdust by a factor of two on average when compared to the more widely used
Draine & Li (2007) models. Therefore, in order to have comparable Mdust with
the literature studies and prescriptions needed to convert them into Mgas, we
re-scale the results based on the Schreiber et al. (2018) by an appropriate
factor at each source redshift (Leroy et al. in prep). Mgas is then obtained
through the dust emission using the $\delta_{\mathrm{GDR}}$-Z relation derived
by Magdis et al. (2012), assuming the MZR from Genzel et al. (2015), using the
median M⋆ and $z$ of the corresponding bin. The Mgas that we get using this
approach corresponds to the total gas budget of the galaxies, including the
molecular and atomic phases. As explained in Gómez-Guijarro et al. (2022b) and
references therein, the molecular gas dominates over the atomic one within the
physical scales probed by the dust continuum observations at this wavelength.

Let us note that this approach assumes that the emissivity index ($\beta$)
adopted in the Schreiber et al. (2018) templates ($\sim$1.5, the average value
for local dwarf galaxies (Lyu et al., 2016)) is accurate for our galaxies
since we do not have FIR data to better constrain it. Leroy et al. (in prep)
perform stacking using ALMA and $Herschel$ data to obtain the SED of typical
MS galaxies. They obtain this $\beta$ by SED fitting, getting values that are
compatible with the $\beta=1.5$ assumed in the Schreiber et al. (2018) models.
In Shivaei et al. (2022), they use stacks of Spitzer, $Herschel$, and ALMA
photometry to examine the IR SED of high-z subsolar metallicity ($\sim$0.5
Z$\odot$) luminous IR galaxies (LIRGs), adopting $\beta=1.5$ for their sample.
In this paper, they also discuss other possible values of this parameter, but
still, they perform the analysis using $\beta=1.5$.

For the second approach, we follow S16, using the corrected version of
equation 16 from that paper. In this paper, they affirm that the luminosity-
to-mass ratio at 850 $\mathrm{\mu}$m is relatively constant under a wide range
of conditions in normal star-forming and starburst galaxies, at low and high
redshifts. We can thus use the measurements of the RJ flux density, derive the
luminosity, and estimate Mgas. They note that this approach is equivalent to a
constant $\delta_{\mathrm{GDR}}$ for high stellar mass galaxies. They insist
that the calibration samples that they use are intentionally restricted to
objects with high stellar mass (M${}_{\star}>5\times
10^{10}\leavevmode\nobreak\ $M⊙), hence not probing lower metallicity systems.
As a consequence, we only use this approach for the calculation of the Mgas in
the high-mass bin. We discuss the effect of this and other prescriptions in
the calculation of the gas content in lower mass galaxies in Sec. 6. An offset
between both approaches is expected though at high stellar masses, as reported
in Gómez-Guijarro et al. (2022b), where they find a median relative difference
(M${}_{\mathrm{gas}}^{\mathrm{RJ}}$-M${}_{\mathrm{gas}}^{\delta_{\mathrm{GDR}}}$)/M${}_{\mathrm{gas}}^{\delta_{\mathrm{GDR}}}$=$\leavevmode\nobreak\
0.23\pm 0.84$ between both measurements.

The errors are estimated through 10,000 Monte-Carlo simulations perturbing the
photometry randomly within the uncertainties and assuming an error of 0.20 dex
for the metallicity (Magdis et al., 2012). All these values are included in
Table 3.

To fully understand and check the consistency of our measurements, we include
two additional calculations of the gas fractions for the high-mass bin: one
for the undetected data-set, and another considering only the GG22 galaxies.
This is to both compare the measurement of the gas fractions that we obtain
when considering the full mass-complete sample with the results excluding
individually detected sources and to check what is the contribution of these
bright individually-detected galaxies in the stacked measurements. All these
values are also included in Table 3.

In Fig. 5, we show the evolution with stellar mass of the gas fractions
derived for each redshift bin. As mentioned in Sec. 4, we provide 3$\sigma$
upper limits for the low- and intermediate-mass bins and measurements for the
high-mass bin. For the low-mass bin we get f${}_{\mathrm{gas}}<0.97-0.98$
along $1\leq z\leq 3$. These numbers are f${}_{\mathrm{gas}}<0.69-0.77$ for
the intermediate-mass bin. For the high-mass bin, focusing first on the
$\delta_{\mathrm{GDR}}$ results, we obtain f${}_{\mathrm{gas}}=0.32-0.48$.
Looking at the two additional cases, we check that removing the GG22 galaxies
and their neighbors from our sample drops the measurements, but the effect is
not very significant (f${}_{\mathrm{gas}}=0.30-0.45$). The number of detected
objects is much lower compared to the contribution of the undetected galaxies
(see Table 3), which dominate the emission from the stack. If we only consider
the detected galaxies from GG22 and stack them following the procedure
described along Sec. 4, we get, as expected, much higher gas fractions
(f${}_{\mathrm{gas}}=0.46-0.66$). Taking the individual gas fractions of the
GG22 objects, provided in Gómez-Guijarro et al. (2022b), and calculating the
mean for each redshift bin, we get values of f${}_{\mathrm{gas}}=0.53-0.69$.
If we turn to the gas fractions as derived using the S16 relation, we see that
the latter provides higher values of the gas fractions but both results are
compatible within the uncertainties.

Figure 5: Gas fractions versus stellar masses derived for each redshift range.
Squares represent the gas fractions obtained for our sample, circles show the
gas fractions derived for the undetected data-set. Diamonds represent the gas
fractions calculated for the GG22 galaxies only. The differences between
samples highlight the effect of the exclusion and inclusion of individually-
detected galaxies in the gas fractions. Gas fractions from the colored filled
markers are computed following the gas-to-dust ratio prescription whereas
colored empty markers with a cross within represent the gas fractions as
calculated using the S16 method. Both values are included in Table 3.
3$\sigma$ upper limits at lower stellar masses are shown with smaller squares
and vertical arrows. Uncertainties are also included for the measurements. We
also show the Liu et al. (2019b) (L19), Tacconi et al. (2020) (T20), Kokorev
et al. (2021) (K21), and Wang et al. (2022) (W22) scaling relations in green,
gray, pink, and black, respectively. For each line, there is a dashed and a
solid part. The solid part represents the mass range for which these relations
are calibrated, whereas the dashed one shows an extrapolation of these
relations for lower stellar masses. For the T20 relation, we also include the
uncertainty as a shaded gray region. The scaling relations are computed using
the median redshift of each bin and a $\Delta$MS = 0. The distance from each
of the points to these scaling relations is re-scaled to their corresponding
redshifts and $\Delta$MS using T20.

###  5.2 Scaling relations framework and comparison with our sample

Based on galaxy samples such as the ones described in Sec. 3, several works
provide us with different scaling relations that allow us to obtain the gas
content of galaxies given the redshift, the $\Delta$MS, and the M⋆. Some of
these are Liu et al. (2019b) (L19), T20, Kokorev et al. (2021) (K21), and Wang
et al. (2022) (W22).

The L19 relation is based on the A3COSMOS project, already introduced in Sec.
3, together with $\sim$1,000 CO-observed galaxies at $0<z<4$ (75% of them at
$z<0.1$). The galaxies from the A3COSMOS project probe the log
M${}_{\star}/$M${}_{\odot}\sim 11-12$ MS. Complementary sources (most of them
belonging to Tacconi et al. 2018 and Kaasinen et al. 2019) sample the log
M${}_{\star}/$M${}_{\odot}\sim 10-11$ MS at $z>1$. For $z<0.03$, the
complementary sample also covers the log M${}_{\star}/$M${}_{\odot}\sim 9-10$
MS, but they insist that the metallicity-dependent CO-to-H2 conversion factor
$\alpha_{\mathrm{CO}}$ might be more uncertain, and so the estimated gas mass.

T20 is based on individually detected objects plus stacks of fainter galaxies,
as pointed out in Sec. 3. This relation is an expansion of the results
obtained in Tacconi et al. (2018).

K21 uses $\sim$5,000 SFGs at $z<4.5$, drawn from the super-deblended catalogs
introduced in Sec. 3. The median redshift of the sample K21 is based on is
$z\sim 0.90$, with a median M${}_{\star}\sim$ 4.07$\times$1010 M⊙. The low-
mass and low-redshift part of their sample is restricted to galaxies that lie
above the MS. Nevertheless, 69% of the galaxies qualify as MS, 26% are
classified as starbursts and 5% qualify as passive galaxies.

W22 is based on the COSMOS2015 galaxy catalog (in Sec. 3 we referred to
COSMOS2020, which is an updated version of COSMOS2015). They select star-
forming MS galaxies with ALMA band 6 or 7 coverage in the A3COSMOS database
and well within the ALMA primary beam, obtaining a final sample of 3,037
sources. They stack galaxies, binning in redshift and stellar mass, in the
$uv$ domain, covering the mass range 1010-12 M⊙. They do not select galaxies
according to a certain SNR threshold at (sub-) millimeter wavelengths, so the
sample includes both, detected and undetected ALMA sources.

In Fig. 5, the values of the scaling relations there represented are derived
using the median redshift of the bin and a $\Delta$MS = 0.

If we compare our results with the previous scaling relations, for the high-
mass bin, we see that the measurements for our sample are more compatible with
the W22 scaling relation than with any of the latter relations. These
measurements are also within the uncertainties defined by T20 but below L19
and K21. In the case of the intermediate-mass bin, the upper limits lie on the
level established by the W22 and T20 extrapolations, and well below L19 and
K21. In the low-mass bin, upper limits are poorly constraining and lie above
most of the scaling relations.

##  6 Discussion

Pushing the limit to bluer, less dusty, more MS-like, and more mass-complete
samples yields lower levels of gas than those prescribed by literature scaling
relations based on redder and less complete data-sets. The super-deblended
catalogs and the A3COSMOS samples are highly dust-obscured, showing red
optical colors, and are more star-forming than our galaxies. As a consequence,
the L19 and K21 scaling relations yield f${}_{\mathrm{gas}}-$M⋆ relations with
a higher normalization. Going to mass-complete samples, like the one used in
the W22 relation, leads to the inclusion of blue objects with low obscurations
and SFRs compatible with being right on top of the MS. As a result, the W22
relation exhibits a lower normalization, better matching our results. T20 lies
between the two regimes, presumably because it is based on a combination of
individually-detected red, dust-obscured objects, complemented by stacks of
bluer and fainter galaxies.

Regarding our results for the high-mass bin, our low values of fgas could
still be contaminated to a certain extent by the presence of post-starburst
galaxies in their way to quiescence. These galaxies can have passed the $UVJ$
screening due to their blue $U-V$ color and can be pulling the fgas to lower
values. It is true, however, that this effect gains importance at $z>3$
(D’Eugenio et al. 2020, Forrest et al. 2020, Valentino et al. 2020), out of
the redshift range considered in this work. In Antwi-Danso et al. (2023), they
test the performance of the $UVJ$ diagram selecting quiescent galaxies,
including post-starbursts. According to their results based on the
Prospector-$\alpha$ SED modeling, the $UVJ$ selection reaches the $\sim$90%
completeness at $z\leq 4$. They define this completeness as the number of
quiescent galaxies that are selected divided by the total number of quiescent
galaxies in the sample, with quiescence being defined as a specific SFR below
the threshold of the green valley.

To quantify the effect of the possible contamination due to these post-
starburst objects, we produce mock sources, based on sky positions where no
galaxies have been cataloged, and introduce them in the stack, checking their
imprint on the resulting fgas. Considering the $UVJ$ selection to be 90%
complete at these redshifts, we see that after introducing these mock sources
we obtain between 5%-7% less fgas. This difference is smaller than the
uncertainty we derive for this parameter.

Additionally, the fact that we are comparing our values of fgas, obtained
using a certain method, with the results provided by scaling relations whose
measurements of fgas come from different conversion prescriptions, might be
another source of discrepancy.

The L19 and W22 scaling relations rely on the RJ-tail continuum method of
Hughes et al. (2017). Using this prescription to compute the fgas of our
sample, adopting $\alpha_{\textrm{CO}}$=6.5 (K km s-1 pc${}^{2})^{-1}$, we get
7% larger values, similarly to what we get using S16. T20 uses the Leroy et
al. (2011) $\delta_{\mathrm{GDR}}$ together with the Genzel et al. (2015) MZR.
Using this prescription, we obtain 3% larger values of fgas. K21 is based on
the Magdis et al. (2012) $\delta_{\mathrm{GDR}}$ prescription together with
the Mannucci et al. (2010) fundamental metallicity relation (FMR) calibrated
for Kewley & Dopita (2002) that they convert to the Pettini & Pagel (2004)
(PP04 N2) scale following Kewley & Ellison (2008). Using this method, we get
5% larger values of fgas.

The discrepancy between some of the scaling relations and our data is
therefore not a consequence of the methodology or other factors that might be
artificially pulling down our values, but simply the end result of considering
a mass-complete sample that includes bluer less-dusty objects in comparison
with other samples progressively more and more biased to redder dustier
galaxies.

Concerning our findings for the intermediate-mass bin, it is important to take
into account that at low stellar masses, the link between metallicity and the
$\alpha_{\mathrm{CO}}$ or the $\delta_{\mathrm{GDR}}$ is still not well
constrained and can lead to a bad estimation of the gas content. Up to date,
there is very little information about the gas content of galaxies with $\sim
10^{9}$ M⊙ at high redshifts. Most efforts so far focused on galaxies at
$z\sim 0$ (e.g., Jiang et al. 2015, Cao et al. 2017, Saintonge et al. 2017,
Madden et al. 2020, Leroy et al. 2021). According to T20 and references
therein, it is hard or impossible to detect low-mass galaxies with
substantially subsolar metallicity and to determine their gas content
quantitatively. They suggest that there might be an interstellar medium
component that might be missed or overlooked with the current techniques, such
as gas/dust at very low temperatures. Deeper observations would be required to
provide a better constraint on the fgas of these systems.

We also test the effect of using different prescriptions to compute the fgas
in the intermediate-mass bin. Using RJ continuum methods such as S16 or Hughes
et al. (2017) yields $\sim 10$% lower values than the ones we report.
Discrepancies are expected since these methods are calibrated for more massive
galaxies. S16 relies on a sample of 0.2$-$4$\times 10^{11}$ M⊙ galaxies
whereas the Hughes et al. (2017) sample comprises stellar masses ranging from
$6-11\times 10^{10}$ M⊙. On the other hand, the Leroy et al. (2011)
prescription provides values which are compatible with our results (they
differ in less than 1%), whereas the use of the Magdis et al. (2012)
prescription using the Mannucci et al. (2010) FMR yields similar fgas at $z<2$
but starts differing at higher redshifts, where this approach reports 8% less
fgas. This difference is compatible with the uncertainties but still could
reflect that the metallicity of low-mass galaxies at $z>2$ deviates from that
observed for local galaxies, contrary to what is seen in higher-mass systems,
whose metallicity does not evolve with redshift until $z>2.5$ (Mannucci et
al., 2010). This highlights the need to re-calibrate these relations for less
massive objects compatible with being MS galaxies. In most cases, the low-mass
sample of this kind of studies are mainly made up of galaxies showing very
high SFRs.

##  7 Summary and conclusions

Taking advantage of the CANDELS mass-complete catalog performed in GOODS-S
(Guo et al., 2013), we are able to explore the gas content of galaxies in
ALMA, using Band-6 observations at 1.1 mm (Gómez-Guijarro et al., 2022a). Our
sample is composed of 5,530 star forming blue ($<b-i>\sim 0.12$ mag,
$<i-H>\sim$0.81 mag) galaxies at $1.0\leq z\leq 3.0$, located in the main
sequence. It allows us to explore the gas content of $10^{10-11}$ M⊙ star-
forming galaxies regardless of their emission at (sub-)millimeter wavelengths.
Additionally, and thanks to the mass coverage and completeness of the sample,
we can provide an upper limit of the gas content of lower mass galaxies at
$\sim 10^{9-10}$ M⊙. We report measurements at $10^{10-11}$ M⊙ and 3$\sigma$
upper limits for the gas fraction at $10^{8-10}$ M⊙.

At $10^{10-11}$ M⊙, we are tracing lower gas fractions,
f${}_{\mathrm{gas}}=0.32-0.48$, than those derived from other scaling
relations that use samples of redder and dustier objects on average, being
biased towards individually-detected sources at (sub-)millimeter wavelengths,
more subject to higher attenuations and also more star-forming than our
galaxies. Relations based on more general mass-complete samples show more
compatible values to the ones we report.

At $10^{8-9}$ M⊙, the values we retrieve lie well above the scaling relations
extrapolation, whereas at $10^{9-10}$ M⊙ the upper limits, ranging from 0.69
to 0.77, are located well within the region defined by the Wang et al. (2022)
and Tacconi et al. (2020) scaling relations. The position of the upper limits
at these intermediate masses supports the idea that the extrapolation derived
from these scaling relations is representative of the upper bound of the
underlying f${}_{\mathrm{gas}}-$M⋆ relation as traced by the bulk of star-
forming galaxies.

Table 3: Summary of the physical properties derived in this work for our sample, the undetected data-set, and GG22. The Tdust are those corresponding to the Schreiber et al. (2018) templates derived for the galaxies. The subindex GDR denotes that the quantity has been calculated using the gas-to-dust ratio approach. The subindex S16 means that the quantity is derived following Scoville et al. (2016). The lack of uncertainty in the gas fractions, gas, and dust masses, and luminosities denotes an upper limit (3$\sigma$ level). Let us recall that for the undetected data-set we also remove the G13 sources in the neighborhood of the GG22 galaxies. We have 3 of these neighbor objects at $1<z<1.5$ and another 3 at $1.5<z<2.0$ in the high-mass bin; 2 at $2.0<z<2.5$, and none at $2.5<z<3.0$. z bin | log M⋆ bin | N${}_{\textrm{obj}}$ | zmedian | log M∗,median |  $\Delta$MSmedian | log LIR | Tdust | log Mdust | log Mgas,GDR | log Mgas,S16 | fgas,GDR | fgas,S16
---|---|---|---|---|---|---|---|---|---|---|---|---
| (M⊙) |  |  | (M⊙) | (dex) | (L⊙) | (K) | (M⊙) | (M⊙) | (M⊙) |  |
|  |  |  |  | Our sample |  |  |  |  |  |  |
|  $8\leq$ log M${}_{\star}<$ 9 | 910 | 1.3 | 8.41 | $-0.02$ | 10.26 | 29.36 | 7.15 | 10.00 |  |  $<$ 0.98 |
1.0 $\leq z<$ 1.5 |  $9\leq$ log M${}_{\star}<$ 10 | 299 | 1.2 | 9.37 | 0.00 | 10.43 | 29.36 | 7.35 | 9.72 |  |  $<$ 0.69 |
|  $10\leq$ log M${}_{\star}<$ 11 | 92 | 1.2 | 10.35 | 0.07 | 11.18$\pm$0.07 | 30.04 | 7.97$\pm$0.07 | 10.02$\pm$0.27 | 10.20$\pm$0.07 | 0.32$\pm$0.13 | 0.42$\pm$0.04
|  $8\leq$ log M${}_{\star}<$ 9 | 1423 | 1.7 | 8.41 | $-0.08$ | 10.21 | 30.84 | 6.97 | 9.97 |  |  $<$ 0.97 |
1.5 $\leq z<$ 2.0 |  $9\leq$ log M${}_{\star}<$ 10 | 353 | 1.8 | 9.31 | 0.13 | 10.66 | 33.16 | 7.28 | 9.80 |  |  $<$ 0.75 |
|  $10\leq$ log M${}_{\star}<$ 11 | 96 | 1.8 | 10.30 | 0.17 | 11.44$\pm$0.05 | 33.49 | 8.00$\pm$0.05 | 10.14$\pm$0.25 | 10.27$\pm$0.05 | 0.41$\pm$0.14 | 0.48$\pm$0.03
|  $8\leq$ log M${}_{\star}<$ 9 | 881 | 2.3 | 8.50 | $-0.15$ | 10.38 | 32.59 | 7.04 | 10.12 |  |  $<$ 0.98 |
2.0 $\leq z<$ 2.5 |  $9\leq$log M${}_{\star}<$10 | 274 | 2.3 | 9.31 | 0.04 | 10.74 | 34.70 | 7.17 | 9.80 |  |  $<$ 0.75 |
|  $10\leq$ log M${}_{\star}<$ 11 | 54 | 2.2 | 10.38 | $-0.03$ | 11.61$\pm$0.05 | 33.59 | 8.16$\pm$0.05 | 10.34$\pm$0.25 | 10.43$\pm$0.05 | 0.47$\pm$0.14 | 0.53$\pm$0.03
|  $8\leq$ log M${}_{\star}<$ 9 | 751 | 2.7 | 8.56 | 0.00 | 10.58 | 36.21 | 6.90 | 10.03 |  |  $<$ 0.97 |
2.5 $\leq z\leq$ 3.0 |  $9\leq$ log M${}_{\star}<$ 10 | 340 | 2.8 | 9.29 | 0.06 | 10.84 | 37.00 | 7.12 | 9.82 |  |  $<$ 0.77 |
|  $10\leq$ log M${}_{\star}<$ 11 | 57 | 2.7 | 10.32 | $-0.02$ | 11.61$\pm$0.05 | 36.03 | 8.16$\pm$0.05 | 10.34$\pm$0.25 | 10.43$\pm$0.05 | 0.48$\pm$0.14 | 0.54$\pm$0.03
|  |  |  |  | Undetected data-set |  |  |  |  |  |  |
1.0 $\leq z<$ 1.5 |  $10\leq$ log M${}_{\star}<$ 11 | 85 | 1.2 | 10.34 | 0.06 | 11.12$\pm$0.08 | 29.89 | 7.91$\pm$0.08 | 9.97$\pm$0.28 | 10.15$\pm$0.08 | 0.30$\pm$0.13 | 0.39$\pm$0.04
1.5 $\leq z<$ 2.0 |  $10\leq$ log M${}_{\star}<$ 11 | 84 | 1.8 | 10.29 | 0.12 | 11.28$\pm$0.08 | 33.01 | 7.90$\pm$0.08 | 10.05$\pm$0.28 | 10.14$\pm$0.08 | 0.36$\pm$0.15 | 0.42$\pm$0.04
2.0 $<z\leq$ 2.5 |  $10\leq$ log M${}_{\star}<$ 11 | 46 | 2.2 | 10.37 | $-0.03$ | 11.49$\pm$0.06 | 33.64 | 8.04$\pm$0.06 | 10.22$\pm$0.26 | 10.30$\pm$0.06 | 0.42$\pm$0.15 | 0.46$\pm$0.04
2.5 $\leq z\leq$ 3.0 |  $10\leq$ log M${}_{\star}<$ 11 | 47 | 2.7 | 10.27 | 0.00 | 11.60$\pm$0.07 | 36.17 | 7.92$\pm$0.07 | 10.19$\pm$0.27 | 10.29$\pm$0.07 | 0.45$\pm$0.15 | 0.51$\pm$0.04
|  |  |  |  | G22 |  |  |  |  |  |  |
1.0 $\leq z<$ 1.5 |  $10\leq$ log M${}_{\star}<$ 11 | 4 | 1.2 | 10.63 | 0.10 | 11.77$\pm$0.08 | 30.33 | 8.56$\pm$0.08 | 10.55$\pm$0.28 | 10.79$\pm$0.08 | 0.46$\pm$0.16 | 0.59$\pm$0.05
1.5 $\leq z<$ 2.0 |  $10\leq$ log M${}_{\star}<$ 11 | 9 | 1.9 | 10.53 | 0.34 | 12.31$\pm$0.03 | 36.02 | 8.65$\pm$0.03 | 10.74$\pm$0.23 | 10.99$\pm$0.03 | 0.62$\pm$0.13 | 0.74$\pm$0.01
2.0 $<z\leq$ 2.5 |  $10\leq$ log M${}_{\star}<$ 11 | 6 | 2.2 | 10.65 | 0.09 | 12.40$\pm$0.04 | 34.77 | 8.84$\pm$0.04 | 10.94$\pm$0.24 | 11.02$\pm$0.04 | 0.66$\pm$0.12 | 0.70$\pm$0.02
2.5 $\leq z\leq$ 3.0 |  $10\leq$ log M${}_{\star}<$ 11 | 10 | 2.8 | 10.49 | 0.11 | 12.26$\pm$0.04 | 37.43 | 8.51$\pm$0.04 | 10.71$\pm$0.23 | 10.89$\pm$0.04 | 0.62$\pm$0.13 | 0.72$\pm$0.02

###### Acknowledgements.

RMM acknowledges support from Spanish Ministerio de Ciencia e Innovación
MCIN/AEI/10.13039/501100011033 through grant PGC2018-093499-B-I00, as well as
from MDM-2017-0737 Unidad de Excelencia “Maria de Maeztu” - Centro de
Astrobiología (CAB), CSIC-INTA, “ERDF A way of making Europe”, and INTA
SHARDSJWST project through the PRE-SHARDSJWST/2020 PhD fellowship. CGG
acknowledges support from CNES. PGP-G acknowledges support from grants
PGC2018-093499-B-I00 and PID2022-139567NB-I00 funded by Spanish Ministerio de
Ciencia e Innovación MCIN/AEI/10.13039/501100011033, FEDER, UE. PSB
acknowledges support from Spanish Ministerio de Ciencia e Innovación
MCIN/AEI/10.13039/501100011033 through the research projects with references
PID2019-107427GB-C31 and PID2022-138855NB-C31. MF acknowledges NSF grant
AST-2009577 and NASA JWST GO Program 1727. GEM acknowledges the Villum Fonden
research grant 13160 “Gas to stars, stars to dust: tracing star formation
across cosmic time,” grant 37440, “The Hidden Cosmos,” and the Cosmic Dawn
Center of Excellence funded by the Danish National Research Foundation under
the grant No. 140. This work has made use of the Rainbow Cosmological Surveys
Database, which is operated by the Centro de Astrobiología (CAB/INTA),
partnered with the University of California Observatories at Santa Cruz
(UCO/Lick, UCSC).

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