# Resolved CO(1$-$0) emission and gas properties in luminous dusty star
forming galaxies at $z=2-4$

F. Stanley Sorbonne Université, UPMC Université Paris 6 & CNRS, UMR 7095,
Institut d’Astrophysique de Paris, 98b Boulevard Arago, 75014 Paris, France
Institut de Radioastronomie Millimétrique (IRAM), 300 Rue de la Piscine, 38400
Saint-Martin-d’Hères, France  B. M. Jones I. Physikalisches Institut,
Universität zu Köln, Zülpicher Strasse 77, D-50937 Köln, Germany  Jodrell Bank
Centre for Astrophysics, Department of Physics and Astronomy, School of
Natural Sciences, The University of Manchester, Manchester, M13 9PL, UK  D. A.
Riechers I. Physikalisches Institut, Universität zu Köln, Zülpicher Strasse
77, D-50937 Köln, Germany  C. Yang Department of Space, Earth and Environment,
Chalmers University of Technology, Onsala Space Observatory, SE-439 92 Onsala,
Sweden  S. Berta Institut de Radioastronomie Millimétrique (IRAM), 300 Rue de
la Piscine, 38400 Saint-Martin-d’Hères, France  P. Cox Sorbonne Université,
UPMC Université Paris 6 & CNRS, UMR 7095, Institut d’Astrophysique de Paris,
98b Boulevard Arago, 75014 Paris, France  T. J. L. C. Bakx Division of
Particle and Astrophysical Science, Graduate School of Science, Nagoya
University, Aichi 464-8602, Japan  National Astronomical Observatory of Japan,
2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan  A. Cooray  Department of Physics
and Astronomy, University of California, Irvine, CA92697, USA  H. Dannerbauer
Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain
Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife,
Spain  S. Dye School of Physics and Astronomy, University of Nottingham,
University Park, Nottingham, NG7 2RD, UK  D. H. Hughes  Instituto Nacional de
Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, Santa María
Tonantzintla, Puebla 72840, Mexico  R. J. Ivison European Southern
Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany  S. Jin
Cosmic Dawn Center (DAWN)  DTU Space, Technical University of Denmark,
Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark  M. Lehnert Centre de Recherche
Astrophysique de Lyon - CRAL, CNRS UMR 5574, UCBL1, ENSLyon, 9 avenue Charles
André, F-69230 Saint-Genis-Laval  R. Neri Institut de Radioastronomie
Millimétrique (IRAM), 300 Rue de la Piscine, 38400 Saint-Martin-d’Hères,
France  A. Omont Sorbonne Université, UPMC Université Paris 6 & CNRS, UMR
7095, Institut d’Astrophysique de Paris, 98b Boulevard Arago, 75014 Paris,
France  P. van der Werf Leiden Observatory, Leiden University, P.O. Box 9513,
2300 RA Leiden, The Netherlands  A. Weiß Max-Planck-Institut für
Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany

###### Abstract

We present the results of a survey of CO(1$-$0) emission in 14 infrared
luminous dusty star forming galaxies (DSFGs) at $2<z<4$ with the NSF’s Karl G.
Jansky Very Large Array. All sources are detected in 12CO(1$-$0), with an
$\sim 1\arcsec$ angular resolution. Seven sources show extended and complex
structure. We measure 12CO luminosities of $(\mu)L^{\prime}_{\rm
CO(1-0)}=0.4-2.9\times 10^{11}$ K km s-1 pc2, and molecular gas masses of
$(\mu)M_{\rm H_{2}}=1.3-8.6\times 10^{11}$ M⊙, where $(\mu)$ is the
magnification factor. The derived molecular gas depletion times of $t_{\rm
dep}=40-460$ Myr, cover the expected range of both normal star forming
galaxies and starbursts. Comparing to the higher$-J$ 12CO transitions
previously observed for the same sources, we find CO temperature brightness
ratios of $r_{32/10}=0.4-1.4$, $r_{43/10}=0.4-1.7$, and $r_{54/10}=0.3-1.3$.
We find a wide range of CO spectral line energy distributions (SLEDs), in
agreement with other high-$z$ DSFGs, with the exception of three sources that
are most comparable to the Cloverleaf and APM08279$+$5255\. Based on radiative
transfer modelling of the 12CO SLEDs we determine densities of $\rm
n_{H_{2}}=0.3-8.5\times 10^{3}$ cm-3 and temperatures of $\rm
T_{K}=100-200\,K$. Lastly, four sources are detected in the continuum, three
have radio emission consistent with their infrared derived star formation
rates, while HerBS-70E requires an additional synchrotron radiation component
from an active galactic nucleus. Overall, we find that even though the sample
is similarly luminous in the infrared, by tracing the CO(1$-$0) emission a
diversity of galaxy and excitation properties are revealed, demonstrating the
importance of CO(1$-$0) observations in combination to higher-$J$ transitions.

##  1 Introduction

Dusty star-forming galaxies (DSFGs) have played an important role in galaxy
growth at high-$z$, as they dominate the cosmic star formation rate density
(SFRD) up to redshifts of $z\sim 4$ (e.g., Magnelli et al., 2013; Bourne et
al., 2017; Bouwens et al., 2016, 2020; Hatsukade et al., 2018; Zavala et al.,
2021), including the peak of the SFRD at $z=1-3$ (see Madau & Dickinson,
2014). The most luminous DSFGs in the infrared (IR) can have luminosities at
rest-frame $8-1000\,\mu$m of $L_{\rm IR}>10^{12}-10^{13}$ L⊙, corresponding to
some of the most intense episodes of star formation activity, with star
formation rates (SFRs) that can exceed 1000 M⊙ yr-1 (see reviews by Blain et
al., 2002; Casey et al., 2014; Hodge & da Cunha, 2020).

The number of detected DSFGs at high-$z$ has been steadily increasing over the
last decades, with the advent of large area surveys, including the all-sky
Planck-HFI (Planck Collaboration et al., 2015; Cañameras et al., 2015), and
the South Pole Telescope (SPT; Carlstrom et al., 2011) surveys (Vieira et al.,
2010, 2013), amongst others. The large field surveys on the Herschel Space
Observatory (Herschel; Pilbratt et al., 2010) covering $\rm>1000\,deg^{2}$ of
the extragalactic sky at $\rm 70-500\,\mu m$, have significantly contributed
to the increased number of known DSFGs. Surveys such as the Herschel
Astrophysical Terahertz Large Area Survey (H-ATLAS; Eales et al., 2010), the
Herschel Multi-tiered Extragalactic Survey (HerMES; Oliver et al., 2012), and
the Herschel Stripe 82 Survey (HerS; Viero et al., 2014), have led to the
detection of $>10^{5}$ DSFGs, including $>200$ luminous DSFGs with $S_{\rm
500\,\mu m}=80-900\,\rm mJy$ (e.g., Nayyeri et al., 2016; Bakx et al., 2018,
and references therein).

Dedicated follow-up studies of the luminous DSFGs identified with Herschel,
have found that these sources cover a wide redshift range of $1<z<6$ (e.g.,
Nayyeri et al., 2016; Bakx et al., 2018, and references therein), include
numerous gravitationally amplified galaxies (e.g., Negrello et al., 2010,
2017; Conley et al., 2011; Riechers et al., 2011b; Cox et al., 2011; Wardlow
et al., 2013; Bussmann et al., 2013; Nayyeri et al., 2016; Bakx et al., 2020a,
b), rare galaxies with $L_{\rm IR}>10^{13}\,{\hbox{$L_{\odot}$}}$ classified
as hyper-luminous infrared galaxies (HyLIRGs) (e.g., Ivison et al., 2013,
2019; Fu et al., 2013; Oteo et al., 2016; Riechers et al., 2013, 2017), and
over-densities of DSFGs that are blended due to the large Herschel beam (e.g.,
Bussmann et al., 2015; Oteo et al., 2018; Gómez-Guijarro et al., 2019; Ivison
et al., 2019). The diversity and size of the Herschel selected DSFG population
makes it ideal for dedicated follow-up surveys of large statistical samples to
determine the galaxy properties and nature of bright DSFGs at high-$z$.

Tracing the molecular gas in the interstellar medium (ISM) of these galaxies
is of particular interest, as it is the fuel for star formation. Direct
observations of the 12CO(1$-$0) transition are necessary to measure total
molecular gas masses, and depletion timescales, probe the cold gas
distribution and morphology, and anchor the modeling of the 12CO spectral line
energy distributions (SLEDs) from which the physical properties of the cold
gas (e.g., kinetic temperature, and density) can be derived (e.g., Ivison et
al., 2011; Riechers et al., 2011a, b, c; Harris et al., 2012; Sharon et al.,
2016; Aravena et al., 2016).

A first necessary step towards this direction is the robust determination of
the redshifts of these sources. Ongoing efforts have lead to the measurement
of spectroscopic redshifts for over $\sim 300$ of the brightest high-$z$ DSFGs
(e.g., Weiß et al., 2009; Walter et al., 2012; Harris et al., 2012; Lupu et
al., 2012; Weiß et al., 2013; Strandet et al., 2016; Fudamoto et al., 2017;
Danielson et al., 2017; Reuter et al., 2020; Urquhart et al., 2022), with
$z$-GAL (Neri et al., 2020, Cox et al. in prep) being the largest redshift
survey among them.

The first series of $z$-GAL sources for which reliable spectroscopic redshifts
were determined, was presented in Neri et al. (2020). In total 14 individual
high-$z$ luminous DSFGs were detected with NOEMA in 11 target fields that were
Herschel selected based on their 500$\mu$m fluxes. The sources were detected
either in multiple 12CO lines or in a combination of 12CO and other molecular
or atomic species. However, despite the multi-line detections, the conclusions
of this study on the molecular gas properties of the sample remained limited
due to the lack of information on the lowest-$J$ 12CO lines. This motivated
targeted 12CO(1$-$0) follow-up of these galaxies with the NSF’s Karl G. Jansky
Very Large Array (VLA).

In this paper, we present the survey of 12CO(1$-$0) emission and the
underlying radio continuum of 14 luminous DSFGs from the $z$-GAL Pilot Program
in the redshift range of $2<z<4$. In Section 2, we describe the sample, the
observations and the data reduction. In Section 3, we report the main results
from the VLA observations both for the 12CO(1$-$0) emission line and the
continuum. Section 4 outlines the implications of these results; particularly,
addressing the total gas molecular gas mass, the excitation conditions and the
nature of these sources. Finally, Section 5 summarizes the main findings of
this paper. Throughout this paper, we adopt a spatially flat $\Lambda$CDM
cosmology with $H_{0}=67.4\,{\rm km\,s^{-1}\,Mpc^{-1}}$ and
$\Omega_{\mathrm{M}}=0.315$ (Planck Collaboration et al., 2020).

Table 1: The $z$-GAL Pilot Program sources

Source | RA | Dec. | $z_{\rm spec}$ | $\rm\Delta V$ | $\mu_{L}L_{\rm IR}$ | $\mu_{L}M_{\rm dust}$
---|---|---|---|---|---|---
| (J2000) |  | $\rm(km\,s^{-1})$ | $10^{12}\,L_{\odot}$ | $10^{10}\,M_{\odot}$
HerBS-34 | 13:34:13.9 | 26:04:57.5 | 2.663 | 330$\pm$10 | 40.6$\pm$2.4 | 1.16$\pm$0.07
HerBS-43a | 13:24:18.8 | 32:07:54.4 | 3.212 | 1070$\pm$90 | 33.4$\pm$4.1 | 0.69$\pm$0.04
HerBS-43b | 13:24:19.2 | 32:07:49.2 | 4.054 | 800$\pm$50 | 15.0$\pm$2.0 | 0.45$\pm$0.04
HerBS-44 | 13:32:55.8 | 34:22:08.4 | 2.927 | 520$\pm$50 | 108.1$\pm$7.8 | 0.7$\pm$0.04
HerBS-54 | 13:15:40.7 | 26:23:19.6 | 2.442 | 1020$\pm$190 | 23.6$\pm$1.6 | 1.25$\pm$0.12
HerBS-58 | 13:03:33.2 | 24:46:42.3 | 2.084 | 970$\pm$50 | 17.7$\pm$1.3 | 0.83$\pm$0.06
HerBS-70E | 13:01:40.3 | 29:29:16.2 | 2.308 | 770$\pm$50 | 40.5$\pm$5.2 | 0.37$\pm$0.02
HerBS-70W | 13:01:39.3 | 29:29:25.2 | 2.311 | 140$\pm$20 | 8.7$\pm$4.2 | 0.09$\pm$0.02
HerBS-79 | 13:14:34.1 | 33:52:20.1 | 2.078 | 870$\pm$70 | 18.1$\pm$1.2 | 0.83$\pm$0.18
HerBS-89a | 13:16:11.5 | 28:12:17.7 | 2.949 | 1080$\pm$60 | 28.9$\pm$2.5 | 1.30$\pm$0.07
HerBS-95E | 13:43:42.7 | 26:39:18.0 | 2.972 | 870$\pm$50 | 11.4$\pm$1.9 | 0.50$\pm$0.05
HerBS-95W | 13:43:41.5 | 26:39:22.7 | 2.973 | 540$\pm$30 | 16.1$\pm$1.8 | 0.77$\pm$0.04
HerBS-113 | 13:12:11.3 | 32:38:37.8 | 2.787 | 900$\pm$200 | 29.6$\pm$2.8 | 0.72$\pm$0.06
HerBS-154 | 13:22:58.1 | 32:50:51.7 | 3.707 | 310$\pm$40 | 81.3$\pm$7.3 | 0.46$\pm$0.03

††footnotetext: Notes. The coordinates and properties of the sources are taken
from Neri et al. (2020), with the exception of HerBS-89a for which we include
the revised values reported in Berta et al. (2021). None of the properties in
this table have been corrected for gravitational magnification ($\mu_{L}$ is
the magnification factor, assuming no differential lensing between the CO and
dust emission). The $\rm\Delta V$ corresponds to the mean FWHM of the CO
transitions observed. The $\rm 8-1000\,\mu m$ rest-frame infrared luminosities
($L_{\rm IR}$) and dust masses ($M_{\rm dust}$) are those derived using the
Draine & Li (2007) approach - see Neri et al. (2020) and Berta et al. (2021)
for further details.

##  2 Sample, Observations & Analysis

###  2.1 Sample

The sources of the Pilot Program (Neri et al., 2020) were selected from the
Herschel Bright Sources (HerBS) sample Bakx et al. (2018); they are located
close to the North Galactic pole (NGP) and have flux densities in the range
$\rm 80\,mJy\lesssim S_{500\,\mu m}\lesssim 130\,mJy$. From the 13 Herschel
targets observed with NOEMA for the Pilot Program, three were resolved into
multiple sources. HerBS-70 and HerBS-95 were resolved into binary systems, and
HerBS-43 was resolved into two galaxies at different redshifts. This resulted
in a total of 16 individual galaxies that were presented in Neri et al.
(2020), 14 of which have a reliable redshift in the range of $2.08<z<4.05$ and
were selected for follow-up VLA observations. These sources are spatially
unresolved or barely resolved with the current NOEMA data with angular
resolutions between $1\farcs 2$ and $6\arcsec$ (Neri et al., 2020).

Table 2: Observed properties of the CO(1$-$0) data

Source | Moment-0 map properties |  | CO(1$-$0) line properties
---|---|---|---
| rmsa | beamb |  | spec-rmsc | $S_{\rm peak}^{d}$ | $\rm FWHM^{e}$ | $I_{\rm CO(1-0)}^{f}$ | Extent of emissiong
| [Jy km s-1 beam-1] | [arcsec2] |  | [mJy] | [mJy] | [km s-1] | [Jy km s-1] | [arcsec2]
HerBS-34 | 0.055 | $0.8\times 0.7$ |  | $0.19$ | $0.7\pm 0.12$ | $593\pm 112$ | $0.44\pm 0.11$ | $(1.7\pm 0.4)\times(0.9\pm 0.2)$
HerBS-43a | 0.048 | $1.0\times 0.9$ |  | $0.13$ | $0.3\pm 0.06$ | $1166\pm 249$ | $0.37\pm 0.10$ | $(1.2\pm 0.2)\times(0.9\pm 0.3)$
HerBS-43b | 0.016 | $1.3\times 1.0$ |  | $0.05$ | $0.08\pm 0.03$ | $744\pm 290$ | $0.064\pm 0.033$ | $(1.1\pm 0.3)\times(0.3\pm 0.6)$
HerBS-44 | 0.047 | $0.9\times 0.8$ |  | $0.18$ | $0.95\pm 0.14$ | $377\pm 63$ | $0.38\pm 0.08$ | $(1.1\pm 0.3)\times(0.5\pm 0.2)$
HerBS-54 | 0.055 | $0.8\times 0.7$ |  | $0.25$ | $0.8\pm 0.11$ | $1087\pm 176$ | $0.92\pm 0.19$ | $\lesssim 2.2\times 2.1$
HerBS-58 | 0.046 | $0.9\times 0.6$ |  | $0.57$ | $2.12\pm 0.32$ | $363\pm 64$ | $0.82\pm 0.19$ | $(2.1\pm 0.4)\times(1.6\pm 0.3)$
HerBS-70e | 0.03 | $0.9\times 0.7$ |  | $0.15$ | $0.49\pm 0.09$ | $622\pm 130$ | $0.32\pm 0.09$ | $(1.6\pm 0.3)\times(1.2\pm 0.3)$
HerBS-70w | 0.019 | $0.9\times 0.7$ |  | $0.15$ | $0.83\pm 0.15$ | $197\pm 39$ | $0.17\pm 0.04$ | $(2.1\pm 0.5)\times(1.1\pm 0.3)$
HerBS-79 | 0.043 | $0.8\times 0.6$ |  | $0.34$ | $1.24\pm 0.17$ | $787\pm 125$ | $1.04\pm 0.22$ | $(2.9\pm 0.5)\times(1.0\pm 0.2)$
HerBS-89a | 0.07 | 1.2 $\times$ 0.8 |  | $0.24$ | $0.64\pm 0.09$ | $1586\pm 247$ | $1.08\pm 0.22$ | $(2.1\pm 0.5)\times(1.3\pm 0.4)$
HerBS-95e | 0.02 | $1.0\times 0.8$ |  | $0.06$ | $0.2\pm 0.04$ | $658\pm 137$ | $0.14\pm 0.04$ | $(1.5\pm 0.3)\times(0.8\pm 0.2)$
HerBS-95w | 0.024 | $1.0\times 0.8$ |  | $0.19$ | $0.95\pm 0.12$ | $522\pm 76$ | $0.52\pm 0.10$ | $(2.5\pm 0.4)\times(1.9\pm 0.3)$
HerBS-113 | 0.035 | $1.0\times 0.7$ |  | $0.32$ | $1.46\pm 0.21$ | $497\pm 81$ | $0.77\pm 0.16$ | $\lesssim 3\times 2.8$
HerBS-154 | 0.035 | $1.2\times 1.0$ |  | $0.16$ | $0.95\pm 0.11$ | $384\pm 54$ | $0.39\pm 0.07$ | $(2.7\pm 0.4)\times(1.7\pm 0.3)$

††footnotetext: Notes. (a) The rms of the moment-0 maps; (b) the beam sizes of
the moment-0 maps; (c) the rms per 100 km/s channels of the extracted spectra,
based on the line free channels; (d) the fitted flux peak of the CO(1$-$0)
emission line; (e) the fitted full width half-max (FWHM) of the CO(1$-$0)
emission line from single Gaussian fits; (f) the integrated line flux derived
from the fit to the line; (g) estimates using imfit on the size of the
CO(1$-$0) emission are based on the moment-0 maps, deconvolved from the beam,
with the exception of HerBS-54 and HerBS-113 for which we were not able to use
the single gaussian fit of imfit due to their complexity. For HerBS-54 and
HerBS-113 we instead provide a rough upper limit of the extent of the
emission.

###  2.2 VLA observations & data reduction

We observed the 12CO(1$-$0) - hereafter CO(1$-$0) - emission line for the 14
luminous DSFGs with reliable redshifts from the $z$-GAL Pilot Program (Table
1). Observations were carried out using the NSF’s Karl G. Jamsky Very Large
Array (VLA), in the C configuration (program I.D.:VLA/20A-083 - P.I.: D.
Riechers). We used the 0.9 cm Ka and K band, with the correlator configured to
an 8-bit sampling mode, achieving a spectral resolution of 2 MHz (i.e., $\rm
16-21\,km\,s^{-1}$), over a total bandwidth of 2 GHz in dual polarization. The
observations were set up so that one of the two side-bands was centred on the
expected frequency of the redshifted CO(1$-$0) emission line ($\nu_{\rm
rest}=115.271\,\rm GHz$) for each source. The second sideband was either
positioned alongside the first to provide contiguous frequency coverage and
maximise continuum sensitivity or re-positioned to a significantly different
frequency to obtain data with which to measure the continuum spectral index
and cover faint spectral lines. Three target fields were observed with the
former setting, and nine with the latter. However, the integration times were
chosen to detect the CO(1$-$0) emission, not the underlying continuum. Note
that HerBS-43a was also in the field of view for the observations of
HerBS-43b, and was observed with two different frequency tunings. The data
were acquired in January-May 2020 under stable atmospheric conditions and each
source was observed for between $1-7.6$ hours with $0.6-5.4$ hours on-source
integration. The 2 GHz bandwidth setup was used to maximize the potential for
stacking of faint lines, while at the same time retaining sufficient spectral
resolution (2 MHz) to finely sample the CO emission line for each source. The
8-bit samplers were selected to maximize sensitivity. The gain, bandpass, and
flux calibrators used were 3C286, J1327$+$2210, J1310$+$3220, and
J1310$+$3230.

The data were reduced, calibrated and imaged using CASA v5.6.2 (Common
Astronomy Software Application111https://casa.nrao.edu; McMullin et al. 2007).
Manual data flagging during reduction was necessary for most sources. The
accuracy of the flux density calibration is within 10–15%, when comparing the
measured fluxes for the calibrators from our observations to the calibrator
models, with a median deviation of 3.33%.

Spectral cubes, continuum, and CO(1$-$0) moment-0 maps were created and
cleaned using the tclean function of casa with a natural weighting. The
spectral channel width in the cubes was set to 100 km s-1 and continuum maps
were created using the line-free channels. The resulting moment-0 maps have
spatial resolutions varying from $0\farcs 9\times 0\farcs 6$ to $1\farcs
2\times 1\farcs 0$ and rms noise levels between 16 and $\rm 55\,\mu
Jy\,km\,s^{-1}$ beam-1 (Table 2). For those sources where the underlying
continuum is detected, we performed continuum subtraction for the spectral
cubes using the line-free channels. Two continuum images were produced for the
sources, with the exception of the three sources with continuous frequency
coverage between the two sidebands for which only a single continuum image is
produced.

###  2.3 Spectral analysis

In order to obtain the best quality CO(1$-$0) spectra, we follow an iterative
process for defining the extraction region used. Starting with a $1\arcsec$
circular aperture centered on the source position, we extract initial spectra,
from which we define the line channels to be used for the moment-0 maps. Once
the moment-0 map is created, we define new extraction regions based on the
2$\sigma$ contours of the map. We repeat the steps of selecting the line
channels to be used for the moment-0 maps and defining the 2$\sigma$ contour
extraction region, until the 2$\sigma$ contour converges. To achieve a higher
SNR and retrieve the full information on the extent of the emission as best as
possible, the total line intensity maps were created directly from the uv data
and cleaned using tclean in casa. In Figure 1, we present the extracted
spectra and moment-0 maps for each source. The properties of the extracted
CO(1$-$0) lines are listed in Table 2 together with the rms noise levels and
the spatial resolution of the moment-0 maps. For sources where a clear double
peak is seen in the line profile, we fit with both single and double Gaussian
components; however, we find that the integrated flux does not change
significantly between the two. As the more complex fit is not required by the
data and will only increase the uncertainties due to the larger number of
fitting parameters, we chose to only use the results of the simpler, single
Gaussian fit.

###  2.4 Continuum analysis

As the aim of the observations was to detect the CO(1$-$0) emission, and the
observations were not designed to detect the continuum, the majority of our
sample is not detected in the continuum, with the exception of four sources,
namely HerBS-43a, HerBS-43b, HerBS-54 and HerBS-70E. For these sources, we
used the 2D Gaussian fitting tool imfit of casa to retrieve the total
continuum flux density. In Table 3, we give the continuum flux densities, the
rms values, and the corresponding frequencies. The radio continuum emission
for the four sources detected are displayed as white contours on the moment-0
maps of Fig. 1. For the sources where no continuum emission was detected,
Table 3 lists the rms noise level of each map.

Within the field of HerBS-44, a serendipitous source is detected in the radio
continuum (at 29 and 38 GHz) at a distance of $\sim 34\arcsec$ from the main
source. The position of the source is at RA: 13:32:57.8, Dec: +34:22:27.0
(J2000.0), and the continuum fluxes are reported in Table 3 under the source
name HerBS-44s. A likely counterpart to this source is FIRST J133257.7+342227
with a separation of 0.36$\arcsec$, or NVSS J133257+342228 at a separation of
1$\arcsec$.

Figure 1: CO(1$-$0) spectra (left), moment-0 (right) for each source in our
sample. The source names are indicated on the top of the left panels. Each
CO(1$-$0) emission line is plotted in yellow and centered at the zero velocity
corresponding to the redshift of the source (see Table 1), and the Gaussian
fits to the lines are shown with red curves. For comparison, we plot the next
lowest-$J$ CO line from Neri et al. (2020) in grey, normalized to 1.5$\times$
the peak of the CO(1$-$0) line to show the respective line profiles with more
clarity. The comparison line used is indicated on the top right of each plot.
The moment-0 map contours are plotted starting at $2\sigma$ and increase in
steps of $1\sigma$, where the $1\sigma$ noise levels for each source are
listed in Table 2. The symmetric negative contours are also shown, with dashed
curves. The synthesized beam is shown in the lower left corner of each
moment-0 map. The maps are centered on the positions listed in Table 1. For
the sources detected in continuum (HerBS-43a, 43b, 54, and 70E), we overplot
in white contours the continuum emission on the moment-0 maps, and the
corresponding synthesized beam on the lower right corner. The continuum
contours start at $2\sigma$ and increase in steps of $1\sigma$, with the
exception of HerBS-70E for which we plot contours at 2, 5, 10, 30, and
40$\sigma$ (see Table 3). The red polygon on the moment-0 maps corresponds to
the region from which we extracted the spectra.

.

Figure 1 (continued)

Figure 1 (continued)

Figure 1 (continued)

Figure 2: A comparison of the detected emission lines of HerBS-58. The lines
plotted are indicated on the upper left of each panel. Both the ${\rm
CO}(3-2)$ and [CI] spectra have additional emission at negative velocities,
not seen in the CO(1$-$0) spectra.

Figure 3: Observed spectral energy distributions for the four sources detected
in the radio continuum, HerBS-43a, HerBS-43b, HerBS-54, and HerBS-70E (from
top left to bottom right panel). The data include SPIRE and SCUBA-2 flux
densities (black and blue dots; Bakx et al., 2018) and NOEMA continuum flux
densities at 2 and 3 mm (red dots; Neri et al., 2020); the VLA measurements at
22–38 GHz (13–7.8 mm) are shown as purple dots and the 3 $\sigma$ upper limits
as purple arrows. The fits are based on modified black body (MBB) dust models
[green and red lines] including corrections for the effects of the CMB (see
text for details). The light-blue continuous lines represent the synchrotron
emission due to star formation, with a fixed spectral index $\alpha=-0.8$,
normalised on the basis of two radio/far-infrared correlations derived by
Delhaize et al. (2017) (lower line), and Magnelli et al. (2015) (upper line).
In the case of HerBS-70E, we also plot the synchrotron emission due to the AGN
(purple dotted line), and the sum of all components (purple dashed line), and
additional radio data from the FIRST survey at 1.4 GHz and LOFAR at 126-173
MHz, for comparison.  Table 3: Continuum measurements

Source | $\nu_{\rm cen}$ | rms | Scont
---|---|---|---
| [GHz] | [mJy/beam] | [mJy]
| LSB | USB | LSB | USB | LSB | USB
HerBS-34 | 32.02 | 0.011 | –
HerBS-43a | 27.3 | 37.45 | 0.014 | 0.018 | – | –
HerBS-43a | 22.24 | 0.004 | 0.024$\pm$0.007
HerBS-43b | 22.24 | 0.004 | 0.019$\pm$0.003
HerBS-44 | 29.16 | 38.51 | 0.016 | 0.023 | – | –
HerBS-44s | 29.16 | 38.51 | 0.037 | 0.045 | 0.77$\pm$0.04 | 0.65$\pm$0.08
HerBS-54 | 33.06 | 0.009 | 0.072$\pm$0.013
HerBS-58 | 28.80 | 37.31 | 0.014 | 0.02 | – | –
HerBS-70E | 27.11 | 34.78 | 0.01 | 0.01 | 0.60$\pm$0.02 | 0.47$\pm$0.02
HerBS-70W | 27.11 | 34.78 | 0.01 | 0.01 | – | –
HerBS-79 | 28.73 | 37.51 | 0.01 | 0.015 | – | –
HerBS-89a | 29.18 | 38.49 | 0.02 | 0.05 | – | –
HerBS-95E | 28.95 | 38.5 | 0.008 | 0.01 | – | –
HerBS-95W | 28.95 | 38.5 | 0.008 | 0.01 | – | –
HerBS-113 | 30.37 | 38.5 | 0.016 | 0.022 | – | –
HerBS-154 | 19.15 | 24.42 | 0.022 | 0.01 | – | –

††footnotetext: Notes. The source HerBS-43a was observed with two different
tunings. The results for HerBS-89a are from Berta et al. (2021). HerBS-44s is
a serendipitous source detected in the radio continuum located $\sim
34\arcsec$ to the north-east of HerBS-44 at RA: 13:32:57.8, Dec: +34:22:27.0
(J2000.0).

##  3 Results

In this section we present the initial results on the observed CO(1$-$0) line
emission (Section 3.1), and the radio continuum measurements (Section 3.2) of
our sample.

###  3.1 Observed emission line properties

All sources are detected in CO(1$-$0) with at least $\rm 5\sigma$ significance
for the integrated line flux. Here we provide an overview of the observed
properties of our sample, with more detailed descriptions for the individual
sources reported in Appendix A.

In Fig. 1 we compare the CO(1$-$0) line to the higher-$J$ CO lines detected in
2 and 3 mm with NOEMA (Neri et al., 2020). Specifically we compare the FWHM
from the Gaussian fit to the CO(1$-$0), with the average of the FWHM from the
respective Gaussian fits to the higher-$J$ transitions (expressed as $\Delta$V
in Neri et al., 2020). To quantify the difference, we use the relative
difference of the two, expressed as $\rm|FWHM-\Delta V|/\Delta V$. We find
that the line profiles are similar for most sources in our sample, with 12/14
sources having a CO(1$-$0) FWHM consistent within 50% of the $\Delta$V from
Neri et al. (2020) for the higher $J$ transitions. The two sources with a
significant difference in line-widths are HerBS-34 with a ratio of
FWHM/$\Delta$V$=1.8\pm 0.3$, and HerBS-58 with a ratio of
FWHM/$\Delta$V$=0.4\pm 0.1$. The large ratio of HerBS-34 is most likely due to
the low signal to noise ratio of the CO(1$-$0) observations. The case of
HerBS-58 is more complex. When comparing to the previously observed ${\rm
CO}(3-2)$ and $\rm[CI](^{3}P_{1}-^{3}P_{0})$ in Fig. 2 it becomes apparent
that we are only detecting part of the line emission in CO(1$-$0).
Specifically, we are only detecting the red component of the line.
Consequently, as we are only fitting the red component, this also leads to an
underestimation of the error on the FWHM/$\Delta$V. When comparing the
CO(1$-$0) moment-0 map to the resolved velocity map of
$\rm[CI](^{3}P_{1}-^{3}P_{0})$ in Neri et al. (2020), we find that the
CO(1$-$0) is located in the red-shifted western region, with no emission
covering the blue-shifted eastern part of the [CI] emission, consistent with
the lack of emission seen for the expected blue component of the line. A more
detailed analysis of the CO and [CI] emission in HerBS-58, will be provided in
Ismail et al. (in prep.) based on new high-angular resolution NOEMA data.

The resolution of $\sim 1\arcsec$ achieved with the VLA has allowed for all
sources to be at least partially resolved (see Table 2, and Fig. 1). Half of
the sample shows compact to somewhat extended emission over 9–18 kpc scales.
For the other 7/14 sources extended and complex structure is revealed, over
scales of 18–25 kpc, with HerBS-54, 58, 79, 89a, 113, and 154 showing multiple
peaks in their emission, while HerBS-95W has a single peak and extends over 20
kpc. The complex, multi-peak structure observed for these sources is a
possible indicator for lensing, as shown in Berta et al. (2021) for HerBS-89a.
However, we are not able to perform a detailed lensing analysis in order to
confirm this with the current data. A brief discussion on this point is
provided in Appendix B.

###  3.2 Radio continuum

As reported in Sect. 2, due to the limited sensitivity of the VLA data, the
majority of the sample is not detected in continuum, with the exception of
four sources. The typical rms levels are between $\sim 4-45$ $\mu$Jy beam-1 in
the observed frequency range of 20 to 38 GHz. In Table 3 we give the continuum
properties of the full sample. The four sources detected in the high-frequency
radio continuum are: HerBS-43a, HerBS-43b, HerBS-54, and HerBS-70E; the
continuum emission is displayed as white contours in the respective panels in
Fig. 1.

Sources HerBS-43a and HerBS-43b are similarly compact in the 22.2 GHz (rest-
frame 93.5 GHz and 112.2 GHz, respectively) continuum than in the CO(1$-$0)
emission, but we observe a small offset of $\sim 0\farcs 5$ between the
continuum and CO(1$-$0) emission for both sources. Similarly, the continuum of
HerBS-70E, detected at 27.1 and 34.8 GHz (rest-frame 89.6 GHz and 115.1 GHz)
is also somewhat offset with respect to the CO(1$-$0) emission, and displays a
more compact morphology. Interestingly, HerBS-54, detected at 33.0 GHz (rest-
frame 113.6 GHz) has continuum emission that is aligned with only the south
peak of the CO(1$-$0) emission, although there is some extension towards the
north. This could be a result of the north component being fainter in the
continuum and therefore not detected in these observations.

For the sources detected in the radio continuum we define simple SED models,
in which we use the modified black body (MBB) fits done to the NOEMA continuum
data in Neri et al. (2020), with correction for the cosmic microwave
background (CMB) following da Cunha et al. (2013), and simply add a power-law
spectrum of $S_{\nu}\propto\nu^{-0.8}$ representing the synchrotron emission
due to star formation (SF synchrotron), normalized on the basis of the
radio/far-infrared correlation taking into account the redshift evolution
described in Delhaize et al. (2017) and the infrared emission derived using
the DL07 model (Draine & Li, 2007) as reported in (Neri et al., 2020) (see
Table 1). We show these SEDs in Fig. 3.

In the case of HerBS-43a, the continuum at 22.2 GHz lies on the expected SF
synchrotron power-law, while the upper limits at 27.2 and 37.4 GHz are also
consistent. Although they lie above the expected emission based on the
Delhaize et al. (2017) relation, HerBS-43b and HerBS-54 have flux densities
that are compatible with the synchrotron emission due to star formation as
they are in agreement with expectations based on the Magnelli et al. (2015)
relation. However, we note that measurements at lower frequencies (e.g., 4-8
GHz) would be useful to further assess the slope and derive the properties of
the radio emission in these SMGs and, in particular, to estimate the
contribution of the free-free (Bremsstrahlung) emission with a power-law radio
spectrum of $S_{\nu}\propto\nu^{-0.1}$ that contributes significantly at
higher frequencies (e.g., Condon, 1992; Thomson et al., 2014). All the sources
with only upper limits in the radio continuum are compatible with the
synchrotron emission levels derived from the radio/far-infrared relationship.

In the case of HerBS-70E, the flux densities are well above the expected level
of the synchrotron emission due to star formation. This clearly indicates the
presence of a radio luminous AGN in this source (see Fig. 3). HerBS-70E was
also detected in the FIRST survey (Becker et al., 1995) at 1.4 GHz and with
LOFAR at $\rm 126-173\,MHz$ (Hardcastle et al., 2016), tracing the increase of
the radio flux density at these lower frequencies (corresponding to 4.63 GHz
and 416-572 MHz in the rest-frame), including the expected turn-over of the
radio AGN emission. Based on the 1.4 GHz data we estimate a radio luminosity
of $L_{\rm 1.4GHz}=2.4\times 10^{26}$ W Hz-1, placing it among typical radio
luminous AGN at these redshifts. A more detailed analysis of this source will
be presented in a separate paper.

##  4 Discussion

In this discussion, we use the observed CO(1$-$0) line properties to determine
the molecular gas properties of our sample of luminous DSFGs, namely: the CO
luminosities and masses (Sect. 4.1), the gas properties, including the gas
depletion time, that are compared to other samples previously observed in
CO(1$-$0) (Sect. 4.2), and the gas to dust mass ratios (Sect. 4.3). In Sect.
4.4, we estimate the CO line ratios by combining the CO(1$-$0) measurements
with the previously observed higher-$J$ CO transitions from Neri et al.
(2020), and, in Sect. 4.5, we analyse the spectral line energy distributions
(SLEDs) of our sample with radiative transfer modeling. An important caveat
that must be noted, and could affect most if not all of the points in this
discussion, is the likelihood of our observations missing extended CO(1$-$0)
emission. This would increase the line fluxes and luminosities, but it is not
clear by how much, and how significant that difference could be.

###  4.1 CO luminosities, molecular gas masses, and the choice of
$\alpha_{CO}$

The direct measure of the CO(1$-$0) emission allows for the estimation of the
line luminosity, $L^{\prime}_{\rm CO(1-0)}$, and molecular gas masses, $\rm
M_{\rm H_{2}}$, without the uncertainty of conversion from higher-$J$ CO
transitions. We estimate these values following the standard equations given
below (e.g., Carilli & Walter, 2013):

$L^{\prime}_{\rm CO}=3.25\times 10^{7}\times\frac{S_{\rm
CO}\,\Delta\upsilon_{CO}\,D_{\rm
L}^{2}}{\nu_{CO,rest}^{2}\times(1+z)}\,\rm[K\,km\,s^{-1}\,pc^{-2}]$

$M_{\rm H_{2}}=\alpha_{CO}\times L^{\prime}_{\rm CO(1-0)}\,[M_{\sun}]$

where $S_{\rm CO}\,\Delta v_{CO}$ is the measured flux of the line in Jy km
s-1, $D_{\rm L}$ is the luminosity distance in Mpc, $\nu_{CO,rest}$ is the
rest frequency of the line in GHz, and $\rm\alpha_{CO}$ is the $\rm
CO(1-0)-H_{2}$ conversion factor.

The estimation of $M_{\rm H_{2}}$ remains uncertain even with the direct
measure of the CO(1$-$0) emission, due to the dependence on $\rm\alpha_{CO}$,
which itself is dependent on other properties of the galaxies, such as
metallicity, that are difficult to determine over a large sample of galaxies,
especially at high redshifts (see e.g. Dunne et al., 2021; Bolatto et al.,
2013). Historically, studies of classical sub-mm galaxies (SMGs) have assumed
an $\rm\alpha_{CO}=0.8\,\rm M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$ (see
review by Carilli & Walter, 2013), in agreement with the dynamical constraints
on the $\rm\alpha_{CO}$ in local starburst galaxies placing it within the
range of $0.8-1.5\,\rm M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$ (e.g.,
Downes & Solomon, 1998; Genzel et al., 2010). Normal star-forming galaxies
seem to have an $\rm\alpha_{CO}\sim 4\,\rm
M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$ (e.g. Tacconi et al., 2013, 2018;
Genzel et al., 2015; Carilli & Walter, 2013) consistent with Galactic studies
(e.g. Bolatto et al., 2013). More recently a study on Herschel selected
galaxies using multiple gas tracers to calibrate the gas mass, found an
average $\rm\alpha_{CO}=3\,\rm M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$
(Dunne et al., 2021). For the purposes of this discussion, we chose to use the
results of Dunne et al. (2021), and assume an $\rm\alpha_{CO}=3\,\rm
M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$, but in addition we examine the
range of results possible when assuming the two extremes of
$\rm\alpha_{CO}=0.8-4.3\,\rm M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$. We
note that during the final stages of this study a more updated analysis on the
$\rm\alpha_{CO}$ was presented in Dunne et al. (2022), using a much larger
sample of 407 galaxies spanning up to $z\sim 6$, and found $\rm\alpha_{CO}$
values consistent with those of Dunne et al. (2021), assumed in our analysis.
We also note that in a detailed high-resolution study of a high-$z$ luminous
Herschel selected galaxy, Dye et al. (2022) estimated the gas mass of the
galaxy following different tracers and methods - including from CO(1$-$0)
assuming $\rm\alpha_{CO}=3\,\rm M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$
\- and found a surprising agreement between the resulting gas masses.

The estimated CO(1$-$0) luminosities and gas masses for our sample are given
in Table 4. We find molecular gas masses of $1.2-13.2\times 10^{11}$M⊙, with
no corrections made for the possible magnification due to gravitational
lensing. The sources with the largest values ($\geq 5.4\times 10^{11}$) are
all identified to have extended and complex structure and are likely
magnified. For HerBS-89a, correction for gravitational lensing ($\mu=5$)
reduces the molecular gas mass from $(13.2\pm 2.7)\times 10^{11}$ to $(2.6\pm
0.5)\times 10^{11}$ M⊙ (see also Berta et al., 2021). The large molecular gas
masses measured are consistent with the necessary conditions to support the
SFRs of such galaxies, and are in agreement with literature results for
luminous DSFGs (e.g., Tacconi et al., 2008; Ivison et al., 2011; Bothwell et
al., 2013; Aravena et al., 2016; Harrington et al., 2021).

Table 4: Derived properties of the molecular gas

Source | ($\mu$)$L^{\prime}_{\rm CO}$ | ($\mu$)M${}_{\rm H_{2}}$ | tdep
---|---|---|---
| [1011 K km s-1 pc2] | [1011 M⊙] | [102 Myr]
HerBS-34 | $1.5\pm 0.4$ | $4.5\pm 1.2$ | $1.1\pm 0.3$
HerBS-43a | $1.7\pm 0.5$ | $5.1\pm 1.5$ | $1.5\pm 0.5$
HerBS-43b | $0.4\pm 0.2$ | $1.2\pm 0.6$ | $0.8\pm 0.4$
HerBS-44 | $1.5\pm 0.3$ | $4.5\pm 0.9$ | $0.4\pm 0.1$
HerBS-54 | $2.7\pm 0.6$ | $8.1\pm 1.8$ | $3.4\pm 0.8$
HerBS-58 | $1.8\pm 0.4$ | $5.4\pm 1.2$ | $3.1\pm 0.7$
HerBS-70E | $0.9\pm 0.2$ | $2.7\pm 0.6$ | $0.7\pm 0.2$
HerBS-70W | $0.5\pm 0.1$ | $1.5\pm 0.3$ | $1.7\pm 0.9$
HerBS-79 | $2.3\pm 0.5$ | $6.9\pm 1.5$ | $3.8\pm 0.9$
HerBS-89a | $4.4\pm 0.9$ | $13.2\pm 2.7$ | $4.6\pm 1.0$
HerBS-95E | $0.6\pm 0.2$ | $1.8\pm 0.6$ | $1.6\pm 0.6$
HerBS-95W | $2.2\pm 0.4$ | $6.6\pm 1.2$ | $4.1\pm 0.9$
HerBS-113 | $2.9\pm 0.6$ | $8.7\pm 1.8$ | $2.9\pm 0.7$
HerBS-154 | $2.3\pm 0.4$ | $6.9\pm 1.2$ | $0.9\pm 0.2$

††footnotetext: Note. Where $\mu$ is the magnification factor due to
gravitational lensing, $L^{\prime}_{\rm CO}$ is the CO(1$-$0) luminosity,
M${}_{\rm H_{2}}$ is the molecular gas mass, and tdep is the depletion time.

Figure 4: $\it(left)$ Infrared luminosity (LIR; 8–1000 $\mu$m) as a function
of L${}^{\prime}_{\mathrm{CO(1-0)}}$ for the DSFGs reported in this paper
(shown as red squares and identified by their names). For comparison we also
plot low and high redshift DSFGs from the literature, categorized as sub-mm
galaxies (SMGs), Herschel selected galaxies (HSGs), galaxies from the SPT
survey, cluster galaxies (CGs), galaxies on the main sequence (MS), DSFGs
hosting AGN (AGN) and ULIRGs (see section 4.2 for a detailed list of
references). Only sources with reported CO(1$-$0) emission lines have been
included to avoid uncertainties in estimating the CO(1$-$0) luminosities from
higher-$J$ transitions. Trends of constant SFE values are plotted with grey
dashed lines. $\it(right)$ The L${}^{\prime}_{\mathrm{CO(1-0)}}$/LIR ratio as
a function of ($1+z$). In both plots, corrections for amplification were not
applied to the infrared and CO luminosities for the gravitationally amplified
galaxies.

###  4.2 Molecular Gas Depletion Times & Star Formation Efficiencies

Two parameters of interest when investigating the molecular gas properties of
galaxies, are the star formation efficiency (SFE) and its inverse, the gas
depletion time (tdep). The SFE is a measure of the efficiency of star
formation given the molecular gas reservoir available, and can be expressed as
$\rm SFE=SFR/{\it M_{\rm gas}}$, where SFR[ M⊙ yr-1]$=1.09\times
10^{-10}\,L_{\rm IR}$[L⊙], and $L_{\rm IR}$ is the infrared luminosity at
rest-frame 8–1000$\mu m$ (Kennicutt, 1989, corrected for a Chabrier 2003
initial mass function). The tdep is a measure of the time it would take for
the molecular gas to be depleted by the current SFR, assuming that the SFR
will remain constant throughout this period and that no other processes (such
as gas accretion or outflows) affect the gas reservoir.

To compare the line emission properties of our sample to previously observed
DSFGs at low and high redshifts, we plot $L_{\rm IR}$ as a function of
$L^{\prime}_{\rm CO(1-0)}$ and the ratio of $L^{\prime}_{\rm CO(1-0)}/L_{\rm
IR}$ in Figure 4. For this comparison we only include DSFGs with measured
CO(1$-$0) and we split the sources based on redshift and in the following
categories: classical SMGs selected at 850 $\mu$m and/or 1200 $\mu$m (see
compilation in Carilli & Walter, 2013), with CO(1$-$0) measurements from
Riechers et al. (2011c, a); Swinbank et al. (2011); Dannerbauer et al. (2019);
Lestrade et al. (2011); Harris et al. (2010); Carilli et al. (2010); Ivison et
al. (2011); Thomson et al. (2012); Sharon et al. (2016); galaxies from the
south pole telescope (SPT) survey (e.g. Weiß et al., 2013; Spilker et al.,
2014), and with CO(1$-$0) measurements presented in Aravena et al. (2013,
2016); Herschel selected galaxies (HSGs) from surveys such as Herschel-ATLAS
(e.g. Maddox et al., 2018; Valiante et al., 2016) and the HerBS catalogue
(Bakx et al., 2018), with CO(1$-$0) measurements from Harris et al. (2012);
George et al. (2013); Bakx et al. (2020b); galaxies from cluster studies
(Rudnick et al., 2017; Wang et al., 2018); galaxies on the main sequence (MS)
(Aravena et al., 2014; Villanueva et al., 2017); high-$z$ DSFGs hosting AGN
(from the compilations of Harris et al., 2012; Carilli & Walter, 2013; Sharon
et al., 2016; Penney et al., 2020); and ULIRGs (Solomon et al., 1997; Chung et
al., 2009; Combes et al., 2011).

In Figure 4 we can see that our sample falls within the scatter of high
redshift DSFGs, without much distinction between the different selection
methods. Furthermore, although there is an apparent $L_{\rm IR}$–
$L^{\prime}_{\rm CO(1-0)}$ correlation for most sources, it is worth noting
that the scatter corresponds to a factor of $\sim$10 in SFEs. When taking the
ratio of $L^{\prime}_{\rm CO(1-0)}/L_{\rm IR}$ with redshift the comparison
amongst samples becomes somewhat clearer as the effects of magnification are
removed. The luminosity ratios of high redshift SMGs, HSGs, AGN, and SPT
galaxies, cover the same range as lower redshift ULIRGs, while there is quite
some overlap between these high redshift luminous DSFGs and MS galaxies at
those redshifts.

Although, the ratios of $L^{\prime}_{\rm CO(1-0)}/L_{\rm IR}$ (or reverse) can
be interpreted as a proxy to $t_{\rm dep}$ (or SFE) allowing for comparisons
without the interference of $\rm\alpha_{CO}$ assumptions, such comparisons,
especially between different populations, remain limited precisely due to the
fact that the $\rm\alpha_{CO}$ could vary significantly among the range of the
sources examined.

Figure 5: Depletion time (tdep) as a function of redshift. Our sample is
plotted with red squares and the individual sources are labelled. We compare
to sources from the literature, specifically the high redshift DSFGs detected
in CO(1$-$0) (see relevant references in section 4.2). We estimate the tdep of
the literature sources making the same assumptions as for our sample, with an
$\alpha_{\rm CO}=3\,\rm M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$. However,
the error bars plotted, include the range of tdep that would be estimated
using $\alpha_{\rm CO}$ from 0.8 to 4.3$\,\rm
M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$

To further examine the tdep of our sample we plot in Figure 5 tdep as a
function of redshift, comparing to the high redshift luminous DSFGs for which
it is more likely that a similar value of $\alpha_{\rm CO}$ could apply. We
also plot the range of tdep covered by MS galaxies from the relation presented
in the Tacconi et al. (2020) review. To examine the effect that the assumed
$\alpha_{\rm CO}$ can have on the tdep values, we include in the plotted
errorbars the range of tdep that results from assuming $\alpha_{\rm
CO}=0.8\,\rm and\,4.3$.

The range in tdep that we find for our sample is consistent with what is seen
for the luminous DSFGs at the same redshifts. Furthermore, we observed the
decreasing trend of tdep with redshift that has been previously observed for
the general population of DSFGs (see review by Tacconi et al., 2020).

Overall, we find a tdep of 40–460 Myr for our sources (see Table 4). These
values place our sample both on and below the MS values for the redshift range
covered, demonstrating that luminous DSFGs from Herchel selected samples,
include both normal and star-bursting galaxies (see also Berta et al., 2021).

###  4.3 Gas to Dust ratios

We use the dust masses (Mdust) derived for our sample by Neri et al. (2020)
using Draine & Li (2007) templates (see Table 1), to estimate the gas to dust
ratio ($\rm M_{\rm H_{2}}$/Mdust). We find a wide range of values, $\rm M_{\rm
H_{2}}/M_{\rm dust}=27-167$, with a mean of $\sim 82$ (see solid histogram in
Figure 6). For comparison, $z\sim 0.3$ Herschel selected galaxies have ratios
of 64–261, with a mean of $128^{+54}_{-35}$ (Dunne et al., 2021), these values
remain consistent with the larger sample of DSFGs of Dunne et al. (2022) at
$z<6$. For nearby star-forming galaxies the ratio is $\sim 70$ (Sandstrom et
al., 2013). Although, the majority of our sample is in agreement with previous
measurements of the gas to dust ratio, HerBS-43b, HerBS-95E, and HerBS-34 have
surprisingly low ratios of 27, 36, and 39, respectively. These low ratios
could be a result of variation of $\alpha_{CO}$ amongst the sample, or due to
the presence of an AGN contaminating the SED and artificially elevating the
consequent Mdust estimates. Indeed in the study of Sharon et al. (2016) where
the properties and line ratios of AGN and SMGs were compared, they found that
although there is significant overlap amongst the two samples the lowest
values were those of AGN.

In Figure 6 we also plot the distribution of the ratio when assuming
$\alpha_{\rm CO}=0.8\,\rm and\,4.3$ with hollow histograms. An assumption of
$\alpha_{\rm CO}=0.8$ results in the whole sample having ratios below 40,
while an $\alpha_{\rm CO}=4.3$ does not change the distribution significantly.

Figure 6: Histogram of the gas to dust ratio (Mgas/Mdust) of our sample, for
an $\alpha_{\rm CO}=3\,\rm M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$, is
plotted in solid. We also plot the histograms that result from assumptions of
$\alpha_{\rm CO}=0.8$ and 4.3$\,\rm
M_{\sun}\,{\rm(K\,km\,s^{-1}\,pc^{2})^{-1}}$ for comparison.

###  4.4 CO line ratios & SLEDs

Figure 7: Histograms of the CO brightness temperature ratios, for different
transitions: (top) r${}_{32/10}=\rm
L^{\prime}_{CO(3-2)}/L^{\prime}_{CO(1-0)}$, (middle) r${}_{43/10}=\rm
L^{\prime}_{CO(4-3)}/L^{\prime}_{CO(1-0)}$, (bottom) r${}_{54/10}=\rm
L^{\prime}_{CO(5-4)}/L^{\prime}_{CO(1-0)}$. The solid line corresponds to the
mean of the distribution, while the dot-dashed corresponds to the median. The
highlighted area corresponds to the range of average ratios found in the
literature for the corresponding transitions.

Figure 8: CO SLEDs, normalized to CO(1$-$0) for our sample. The sample has
been split in three sub-samples for easier distinction of the sources in the
figures. The normalized integrated CO fluxes measured for each source are
plotted as circles. For comparison we also plot SLEDs of other sources from
the literature in dot-dashed curves (see section 4.5 for references). Note
that for HerBS-89a we include the CO(9$-$8) transition and use the best fit
SLED model originally presented in Berta et al. (2021).

As our sample has been observed in multiple CO transitions in addition to
CO(1$-$0), we can directly measure the CO brightness temperature ratios, $r$,
by taking the ratio of the L${}^{\prime}_{CO}$ of the different transitions
over CO(1$-$0) (e.g., $r_{32/10}=\rm
L^{\prime}_{CO(3-2)}/L^{\prime}_{CO(1-0)}$). For HerBS-58 that only has the
red component of the higher-$J$ transitions detected in CO(1$-$0) we calculate
an upper limit on the total flux and luminosity of the CO(1$-$0) line over the
full linewidth observed for the higher-$J$ lines and report the corresponding
lower limit on the $r$ values of this source. We find that the line ratios
have a similar range of values across the different CO transitions, between
0.3–1.7 (see Table 5). In Figure 7 we plot the observed distributions for
$r_{32/10}=\rm L^{\prime}_{CO(3-2)}/L^{\prime}_{CO(1-0)}$, r${}_{43/10}=\rm
L^{\prime}_{CO(4-3)}/L^{\prime}_{CO(1-0)}$, r${}_{54/10}=\rm
L^{\prime}_{CO(5-4)}/L^{\prime}_{CO(1-0)}$, with the typical range of values
found for SMGs shown with a yellow filled region. Typical values of $r$ for
SMGs are $r_{32/10}=0.4-0.9$, $r_{43/10}=0.4-0.7$, $r_{54/10}=0.4-0.7$, and
$r_{65/10}=0.2-0.5$ (e.g., Ivison et al., 2011; Bothwell et al., 2013; Spilker
et al., 2014; Sharon et al., 2016; Harrington et al., 2021; Carilli & Walter,
2013, and references therein). The majority of our sample have ratios
consistent with the typical SMG values. However, it is interesting to note
that the majority of the r54/10 ratios are toward the higher end of the
typical SMG values, although the small sample size limits the significance of
this. HerBS-43b, and HerBS-44 have ratios more in line with starburst-quasar
systems such as APM08279$+$5255 (e.g., Riechers et al., 2009), but are still
within the range of possible values for SMGs (see Sharon et al., 2016). We
note that the $r_{65/10}$ of HerBS-43b is quite large, with a value that is
nearly twice that of APM08279$+$5255; however, there is a large uncertainty on
the CO(6$-$5) flux detected by Neri et al. (2020), which is reflected in the
large error on this ratio.

Table 5: CO brightness temperature ratios Source | r32/10 | r43/10 | r54/10 | r65/10
---|---|---|---|---
HerBS-34 | $0.7\pm 0.2$ | $-$ | $0.7\pm 0.2$ | $-$
HerBS-43a | $-$ | $0.9\pm 0.3$ | $0.7\pm 0.2$ | $-$
HerBS-43b | $-$ | $1.7\pm 0.9$ | $0.9\pm 0.5$ | ($2.1\pm 1.1$)
HerBS-44 | $1.4\pm 0.4$ | $-$ | $1.3\pm 0.3$ | $-$
HerBS-54 | $0.5\pm 0.1$ | $0.6\pm 0.1$ | $-$ | $-$
HerBS-58 | $>0.4$ | $>0.2$ | $-$ | $-$
HerBS-70E | $0.6\pm 0.2$ | $0.6\pm 0.2$ | $-$ | $-$
HerBS-70W | $1.1\pm 0.4$ | $0.7\pm 0.2$ | $-$ | $-$
HerBS-79 | $0.4\pm 0.1$ | $0.3\pm 0.1$ | $-$ | $-$
HerBS-89a | $0.4\pm 0.1$ | $-$ | $0.31\pm 0.07$ | $-$
HerBS-95E | $0.8\pm 0.2$ | $-$ | $1.0\pm 0.3$ | $-$
HerBS-95W | $0.5\pm 0.1$ | $-$ | $0.3\pm 0.1$ | $-$
HerBS-113 | $0.9\pm 0.3$ | $-$ | $0.7\pm 0.2$ | $-$
HerBS-154 | $-$ | $-$ | $-$ | $0.5\pm 0.1$

In Figure 8, we plot the integrated CO fluxes of our sources, normalized to
the CO(1$-$0). We have divided the sources into three different plots for
easier comparison with the SLEDs of other DSFG samples from Bothwell et al.
(2013) (32 SMGs at $1.2<z<4.1$), and Spilker et al. (2014) (22 SPT-selected
SMGs at $2.0<z<5.7$), and the mean SLED of “main sequence” star-forming
galaxies from Valentino et al. (2020). We also make a comparison with
individual sources: Cosmic Eyelash (Swinbank et al., 2011; Danielson et al.,
2011); GN20 (Carilli et al., 2010; Cortzen et al., 2020); the starburst-quasar
galaxies Cloverleaf (Barvainis et al., 1997; Weiß et al., 2003; Bradford et
al., 2009; Riechers et al., 2011d), and APM08279$+$5255 (Papadopoulos et al.,
2001; Riechers et al., 2006; Weiß et al., 2007; Riechers et al., 2009); and M
82 (Weiß et al., 2005). All SLEDs used for comparison are also normalised to
the CO(1$-$0).

As can be seen in Figure 8, when comparing the normalised integrated flux
values and respective curves, sources HerBS-54,-58,-79,-89a,-95W are most
comparable to other high-$z$ SMGs, such as the Cosmic Eyelash, GN20, and the
SMG sample of Bothwell et al. (2013). Sources
HerBS-34,-43a,-70E,-70W,-113,-154 are most comparable to the SPT SMGs of
Spilker et al. (2014), the Cloverleaf galaxy, and the M82 center. Finally,
sources HerBS-43b,-44,-95E all lie between the two quasar-starburst systems
Cloverleaf and APM08279$+$5255, which could indicate that these sources likely
host an AGN. However, CO transitions at $J_{up}>6$ would be necessary to
determine the presence of AGN excitation. Overall, the wide range of CO
excitation we find for these sources demonstrates the importance of observing
the CO(1$-$0) emission in galaxies when studying their molecular gas
properties.

Table 6: Properties derived from the radiative transfer analysis, where both the median (med) and the maximum likelihood (max) results for each property are given (with the exception of $P_{\rm thermal}$) Source | log($n_{\rm H2}$) | log($T_{\rm K}$) | log($N_{\rm CO}$/dV) | log($P_{\rm thermal})$
---|---|---|---|---
| [cm-3] | [K] | [cm-2 km-1 s] | [K cm-3]
| med | max | med | max | med | max | med |
HerBS-34 | 3.69${}^{+0.95}_{-0.79}$ | 4.61 | 2.06${}^{+0.42}_{-0.40}$ | 1.93 | 18.09${}^{+0.28}_{-0.42}$ | 18.34 | 5.80${}^{+0.71}_{-0.65}$ |
HerBS-43a | 2.97${}^{+0.66}_{-0.57}$ | 2.66 | 2.22${}^{+0.49}_{-0.49}$ | 2.02 | 17.58${}^{+0.52}_{-0.65}$ | 17.87 | 5.29 ${}^{+0.45}_{-0.62}$ |
HerBS-43b | 3.92${}^{+0.73}_{-0.81}$ | 3.95 | 2.24${}^{+0.39}_{-0.45}$ | 2.36 | 17.82${}^{+0.47}_{-0.68}$ | 18.01 | 6.19${}^{+0.54}_{-0.72}$ |
HerBS-44 | 3.76${}^{+0.73}_{-0.59}$ | 3.84 | 2.27${}^{+0.34}_{-0.41}$ | 2.33 | 17.65${}^{+0.57}_{-0.53}$ | 17.91 | 6.08${}^{+0.47}_{-0.43}$ |
HerBS-54 | 3.18${}^{+1.00}_{-0.69}$ | 2.84 | 2.09${}^{+0.44}_{-0.43}$ | 2.50 | 17.89${}^{+0.41}_{-0.41}$ | 17.93 | 5.32${}^{+0.75}_{-0.49}$ |
HerBS-58 | 2.83${}^{+0.72}_{-0.51}$ | 2.96 | 2.00${}^{+0.61}_{-0.62}$ | 1.81 | 16.74${}^{+0.68}_{-0.72}$ | 16.68 | 4.96${}^{+0.41}_{-0.48}$ |
HerBS-70E | 2.86${}^{+0.72}_{-0.52}$ | 3.38 | 2.25${}^{+0.45}_{-0.44}$ | 2.39 | 17.57${}^{+0.49}_{-0.60}$ | 17.37 | 5.19${}^{+0.51}_{-0.52}$ |
HerBS-70W | 3.06${}^{+0.74}_{-0.57}$ | 3.35 | 2.05${}^{+0.56}_{-0.51}$ | 1.88 | 17.27${}^{+0.67}_{-0.85}$ | 16.55 | 5.26${}^{+0.48}_{-0.68}$ |
HerBS-79 | 2.49${}^{+0.42}_{-0.33}$ | 2.18 | 2.29${}^{+0.45}_{-0.48}$ | 2.40 | 16.89${}^{+0.42}_{-0.51}$ | 17.01 | 4.82${}^{+0.36}_{-0.37}$ |
HerBS-95E | 3.93${}^{+0.66}_{-0.74}$ | 3.44 | 2.25${}^{+0.33}_{-0.39}$ | 2.38 | 18.03${}^{+0.33}_{-0.88}$ | 18.31 | 6.19${}^{+0.56}_{-0.72}$ |
HerBS-95W | 2.55${}^{+0.66}_{-0.74}$ | 2.21 | 2.31${}^{+0.44}_{-0.50}$ | 2.65 | 17.21${}^{+0.35}_{-0.37}$ | 17.23 | 4.93${}^{+0.34}_{-0.38}$ |
HerBS-113 | 3.07${}^{+0.83}_{-0.63}$ | 2.62 | 2.10${}^{+0.51}_{-0.43}$ | 2.51 | 17.78${}^{+0.44}_{-0.53}$ | 17.66 | 5.28${}^{+0.59}_{-0.57}$ |
HerBS-154 | 3.09${}^{+0.63}_{-0.64}$ | 3.16 | 2.24${}^{+0.48}_{-0.47}$ | 1.87 | 17.51${}^{+0.57}_{-0.92}$ | 17.78 | 5.44${}^{+0.41}_{-0.64}$ |

###  4.5 Radiative Transfer Modelling

Using the large velocity gradient (LVG) statistical equilibrium method (e.g.,
Sobolev, 1960), we modeled the molecular gas excitation conditions through the
observed integrated fluxes of the multi-$J$ CO lines. We adopt the one-
dimensional (1D) non-LTE radiative transfer code RADEX (van der Tak et al.,
2007), with an escape probability of $\beta=(1-e^{-\tau})/\tau$ derived from
an expanding sphere geometry. We used the CO collisional data from the LAMDA
database (Schöier et al., 2005). With an MCMC approach following Yang et al.
(2017), we explored the parameter space consisting of the kinetic temperature
of the molecular gas (TK), the volume density (n${}_{\mathrm{H_{2}}}$), the
column density of CO per unit velocity gradient (NCO/dV), and the solid angle
($\Omega_{\mathrm{app}}$) of the source. The overall shape of the CO SLEDs
only depends on TK, n${}_{\mathrm{H_{2}}}$ and NCO/dV, and scales with
$\Omega_{\mathrm{app}}$ (the magnification factors are also included in this
factor). Therefore, we only focus on the parameters TK, n${}_{\mathrm{H_{2}}}$
and NCO/dV hereafter. As noted in the previous section, the CO(6$-$5) flux of
HerBS-43b is quite uncertain and is an outlier when compared to the other
transitions, therefore we chose to exclude it from our SLED modelling
analysis. We note that a detailed analysis for the SLED of HerBS-89a was
presented in Berta et al. (2021), where additional higher-$J$ transitions were
included. Here we present the analysis of the rest of the sample.

As most of the transitions of the CO lines observed are below $J_{up}=$6,
there will be insufficient data to constrain the highly-excited molecular gas
component, assuming the HerBS galaxies are similar to those high-$z$ SMGs, in
which the CO excitation is dominated by two components peaking around
$J_{up}=$6 and $J_{up}=$8 respectively (Yang et al., 2017; Cañameras et al.,
2018). To better constrain the posteriors, we have given slightly tighter
boundaries for the flat priors of n${}_{\mathrm{H_{2}}}$ and NCO/dV than those
used in Yang et al. (2017) (while other priors stay the same). Taking the
values of the parameters from statistically studied SMG samples (Yang et al.,
2017; Cañameras et al., 2018), we have chosen flat priors of $\rm
log(n_{\mathrm{H_{2}}}/cm^{-2})=2.0-5.5$ and $\rm
log(N_{\mathrm{CO}}/dV/cm^{-2}(km/s)^{-1})=15.5-18.5$. Similarly, we also
limited the range of the thermal pressure Pthermal (defined by
P${}_{\mathrm{thermal}}=n_{\mathrm{H_{2}}}\times T_{\mathrm{K}}$) to be within
104 and 107 K cm-3.

Two hundred walkers have been deployed with five hundred iterations after the
one hundred burn-in ones. Thus, a total of 100,000 points of the solutions
have been explored in parameter space.

The results of the radiative transfer analysis are reported in Table 6,
indicating the $\pm 1\sigma$ values and the median of the posteriors. The
maximum likelihood values are also listed in the table. The lack of
measurements at $J_{up}>$6 for the majority of our sample means that the CO
SLED models within $\pm 1\sigma$ of the best fit are significantly different
at high-$J$, leading to a relatively flat posterior of TK, and therefore a
poor constraint on the maximum likelihood values (max). Therefore, we use the
median (med) values to describe the fits. The CO data can be described with
models of dense warm gas of $n_{\rm H_{2}}=0.31-8.5\times 10^{3}$ cm-3, and
$T_{\rm K}=100-200$ K. These values are in general agreement with what has
been found previously in high-$z$ luminous DSFGs and SMGs (e.g. Combes et al.,
2012; Danielson et al., 2013; Riechers et al., 2013; Spilker et al., 2014;
Yang et al., 2017; Harrington et al., 2021).

A relation between $\rm P_{\mathrm{thermal}}$ and SFE has been discussed in
theoretical work (e.g. Elmegreen & Efremov, 1997; Wong & Blitz, 2002),
suggesting that the main driver of the SFE in collapsing molecular clouds is
the thermal gas pressure, or alternatively that higher pressures lead to more
molecular clouds and a higher molecular gas fraction. Yang et al. (2017)
examined whether there is evidence for this through the relationship between
$\rm P_{\mathrm{thermal}}$ and LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$ (as a
proxy for SFE), in a sample of $Herschel$ selected luminous lensed DSFGs. A
statistically significant correlation was found between $\rm
log_{10}(P_{thermal})$ and $\rm log_{10}(L_{\rm
IR}/L^{\prime}_{\mathrm{CO(1-0)}})$, spanning the range of environments from
nearby galaxies to high-$z$ luminous DSFGs, with no evidence of it being
driven by the $n_{\rm H_{2}}$ or $T_{\mathrm{K}}$ values. Following Yang et
al. (2017), in Figure 9 we plot the median values of $\rm
P_{\mathrm{thermal}}$ from the radiative transfer analysis as a function of
LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$. In agreement with what was reported in
Yang et al. (2017) we find that the data are following a correlation. We fit a
line through the data in log-space and find a strong correlation of $\rm
log_{10}(P_{thermal})\varpropto 0.97\times log_{10}(L_{\rm
IR}/L^{\prime}_{\mathrm{CO(1-0)}})$, with Pearson coefficient and p-value of
0.69 and 0.006, respectively. In addition, we color the datapoints based on
the $\rm log_{10}(n_{H_{2}})$ value, which allows us to simultaneously examine
the presence of a dependence of $\rm n_{H_{2}}$ with
LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$. We observe a trend in the $\rm
n_{H_{2}}$ values showing an increase with higher
LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$. To examine wether this positive trend
of $\rm n_{H_{2}}$ values could be the driver of the observed correlation
between $\rm P_{\mathrm{thermal}}$ and LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$,
we calculate the Pearson coefficient and p-value for a correlation between
$\rm n_{H_{2}}$ and LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$. Indeed, we find
evidence for a correlation of $\rm n_{H_{2}}$ with
LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$, with a Pearson coefficient and p-value
of 0.63 and 0.016, respectively. No evidence of a correlation between
$T_{\mathrm{K}}$ and LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$ is found. This
suggests that the $\rm n_{H_{2}}$ trend could be significantly contributing to
the observed relation, but is not likely the driver as the correlation of $\rm
n_{H_{2}}$ is less significant than that of $\rm P_{\mathrm{thermal}}$.
However, we remind the reader that our radiative transfer analysis is limited
by the lack of CO transitions at $J>6$.

Figure 9: Median gas pressure (Pthermal) from the median of the fitted models
of the SLEDs, as a function of the LIR/L${}^{\prime}_{\mathrm{CO(1-0)}}$. The
sources are plotted with gradient colour that corresponds to the median volume
densities ($n_{\mathrm{H_{2}}}$).

##  5 Conclusions

In this paper, we have presented CO(1$-$0) observations using the VLA for 14
luminous DSFGs, including two binary systems, in the redshift range $2<z<4$.
These sources are part of the pilot project sample for the large NOEMA project
$z$-GAL (Cox et al. in prep.), and were originally presented in Neri et al.
(2020). Our main findings are the following:

  * •

We successfully detected the CO(1$-$0) emission in the entirety of our sample,
resolving the molecular emission into extended and complex morphologies in
nearly half the sample (7/14 sources). For the majority of our sample, the
CO(1$-$0) line profiles agree with those of the higher-$J$ transitions.

  * •

Four of the sources were detected in the underlying radio continuum at
observed frequencies of 20-38 GHz. Simple SED fitting to the far-infrared and
radio emission reveals that HerBS-43a, -43b, and -54 have radio emission
consistent with their SFRs, while HerBS-70E has additional synchrotron
radiation from an AGN.

  * •

We find that our sources have CO luminosities of
$(\mu)L^{\prime}_{CO}=0.4-2.9\times 10^{11}$ K km s-1 pc2. From these
CO(1$-$0) luminosities, we estimated molecular gas masses of $(\mu)M_{\rm
H_{2}}=1.3-8.6\times 10^{11}$ M⊙. By combining our CO(1$-$0) luminosities with
the total infrared (8–1000$\mu$m) luminosities from Neri et al. (2020), we
compare the molecular gas depletion times ( $t_{\rm dep}$) and star formation
efficiencies (SFEs) to other galaxy samples. We find that our sources have
SFEs consistent with high-$z$ DSFGs and low-$z$ ULIRGs. Similarly, the
depletion times of our sources cover a wide range and are consistent with both
the main sequence and the starburst phase, with values of $t_{\rm dep}=40-460$
Myr.

  * •

We find a wide range of gas to dust ratios, $\rm M_{\rm H_{2}}/M_{\rm
dust}=27-167$, in agreement with previous measurements for the majority of our
sample. Three sources have surprisingly low ratios of $<40$, which could be a
result of variation of $\alpha_{CO}$ amongst the sample, or due to the
presence of an AGN.

  * •

We find CO temperature brightness ratios of $r_{32/10}=0.4-1.4$,
$r_{43/10}=0.4-1.7$, and $r_{54/10}=0.3-1.3$, with median values of 0.7, 0.6,
and 0.7 respectively. The ratios are relatively consistent for the different
transitions, with similar distributions. The range of values observed are in
agreement with previous observations of high-$z$ DSFGs and AGN (e.g. Sharon et
al., 2016).

  * •

We find a wide range in the shapes of the CO SLEDs of our sample, highlighting
the importance of CO(1$-$0) in revealing the range of excitation in such
galaxies. We compare the SLEDs of our sources to those of main sequence,
typical SMGs, and starburst-quasar systems. The majority of our sample is
consistent with high-$z$ SMGs, with the exception of HerBS-43b, HerBS-44, and
HerBS-95E that are more comparable with the quasar-starburst systems,
Cloverleaf and APM08279$+$5255.

  * •

Finally, we perform radiative transfer modeling of the SLEDs following Yang et
al. (2017). We find that our sample can be described with models of $\rm
n_{H_{2}}=0.3-8.5\times 10^{3}$ cm-3 and temperatures of $\rm
T_{K}=100-200\,K$. However, the results of the modelling are limited due to
the lack of CO observations at the highest transitions. We examine the
possible relation of the thermal gas pressure with SFE, by using the
approximation of the $\rm L_{IR}/L^{\prime}_{CO(1-0)}$. We find strong
evidence of a correlation, in agreement with the theoretical idea of thermal
gas pressure having a direct role in the star formation process of these
galaxies.

The results of this study emphasize the importance of anchoring the CO SLED to
the ground state in order to determine the properties of the dense gas.
Furthermore, by probing the underlying radio continuum we can trace the
contribution to the radio emission from AGN and/or intense star formation in
high-$z$ DSFGs. Building on the previous studies of CO(1$-$0) in high-$z$
DSFGs, our successful detection of the full sample with the high angular
resolution possible with the VLA, revealed a wide range of galaxy and
excitation properties for a sample that would otherwise be classified as
similarly luminous highly star-forming systems. The demonstrated success of
this Pilot study highlights the progress we will be able to achieve by
measuring CO(1-0) in the full $z$-GAL sample of 126 Herschel selected sources,
in understanding the range of physical conditions that lead to the formation
and evolution of luminous high-$z$ DSFGs.

We thank the anonymous referee for their comments that helped to improve the
clarity of various aspects of the paper. This work is based on observations
carried out under program VLA/20A-083 using the National Radio Astronomy
Observatory’s (NRAO) Karl G. Jansky Very Large Array (VLA). The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. This
work benefited from the support of the project Z-GAL ANR-AAPG2019 of the
French National Research Agency (ANR). B.M.J. acknowledges the support of a
studentship grant from the UK Science and Technology Facilities Council (STFC)
and funding from the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) –Project SCHI 561/3-1. C.Y. acknowledges support from an ESO
Fellowship. T.B. acknowledges funding from NAOJ ALMA Scientific Research Grant
2018-09B and JSPS KAKENHI No. 17H06130. HD acknowledges financial support from
the Agencia Estatal de Investigación del Ministerio de Ciencia e Innovación
(AEI-MCINN) under grant (La evolución de los cíumulos de galaxias desde el
amanecer hasta el mediodía cósmico) with reference
(PID2019-105776GB-I00/DOI:10.13039/501100011033) and acknowledge support from
the ACIISI, Consejería de Economía, Conocimiento y Empleo del Gobierno de
Canarias and the European Regional Development Fund (ERDF) under grant with
reference PROID2020010107. S.J. acknowledges the financial support from the
European Union’s Horizon research and innovation program under the Marie
Skłodowska-Curie grant agreement No. 101060888.

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##  Appendix A Individual sources

In this section, we provide details on the CO(1$-$0) emission line profiles
and the integrated-velocity maps for each source and compare them with the
NOEMA 3 and 2-mm results described in Neri et al. (2020).

HerBS-34 is well detected in the CO(1$-$0) emission line displaying a profile
that is comparable to the CO(3$-$2) line, although the ratio of CO(1$-$0) FWHM
over the $\Delta$V of higher-$J$ transitions (FWHM/$\Delta$V) is 1.8$\pm$0.3.
The large ratio is likely due to the lower SNR of the CO(1$-$0) line compared
to the higher-$J$ transitions. The CO emission remains fairly compact even
when resolved, with an estimated size (deconvolved from the beam) of $1\farcs
7\times 0\farcs 9$ ($14\times 7$ kpc).

HerBS-43a displays a similarly broad CO(1$-$0) emission line as the higher-$J$
CO lines, with a ratio of FWHM/$\Delta$V$=1.1\pm 0.2$. The source is partially
resolved, with an estimated size (deconvolved from the beam) of $1\farcs
2\times 0\farcs 9$ ($9\times 7$ kpc).

HerBS-43b,is the highest redshift source in our sample, and therefore has the
weakest CO(1$-$0) emission line of the entire sample with a line flux of $\rm
0.064\pm 0.033\,Jy\,kms^{-1}$. The line profile of the CO(1$-$0) is nearly
identical to that of the CO(4$-$3) line, showing the same double peaked
profile. The ratio of FWHM/$\Delta$V$=0.9\pm 0.4$. The source is partially
resolved, with an estimated size (deconvolved from the beam) of $1\farcs
1\times 0\farcs 3$ ($8\times 2$ kpc).

HerBS-44 has a strong CO(1$-$0) emission line with a single peaked profile and
a $\rm FWHM=377\pm 64\,km\,s^{-1}$ that is somewhat narrower than the high-$J$
transitions, with a ratio of FWHM/$\Delta$V$=0.7\pm 0.1$. The source is
elongated in the east-west direction with an estimated size (deconvolved from
the beam) of $1\farcs 1\times 0\farcs 5$ ($9\times 4$ kpc).

HerBS-54 has a very broad CO(1$-$0) line with FWHM$=1087\pm 176\,\rm
km\,s^{-1}$ comparable to the high-$J$ CO lines, with a ratio of
FWHM/$\Delta$V$=1.1\pm 0.3$. With the new VLA observations, the source is
resolved into a complex morphology showing two peaks within extended weaker
emission. The peaks are separated by $\sim 1\farcs 0$ and the CO(1$-$0)
emission extends over a region of $\sim 2\farcs 2$ (18 kpc).

HerBS-58 shows a complex morphology with two peaks along an arc-like feature
to the west and a well-defined peak to the east. The overall extent of the CO
emission is estimated to be $2\farcs 1\times 1\farcs 6$ ($18\times 14$ kpc).
The CO(1$-$0) line profile, which is best described by a single and relatively
narrow gaussian of $\rm FWHM=363\pm 64\,km\,s^{-1}$, is red-shifted at
$\rm+300\,km\,s^{-1}$ with respect to the center velocity corresponding to a
$z_{spec}=2.0842$ as reported in Neri et al. (2020). The spectroscopic
redshift for HerBS-58 was derived from the double-peaked
$\rm[CI](^{3}P_{1}-^{3}P_{0})$ emission line where both peaks have similar
line fluxes. It is interesting to note that the $\rm CO$(3$-$2) emission line
shows a strong red-shifted peak (at $\rm\sim 300\,km\,s^{-1}$), corresponding
to the CO(1$-$0) emission line, whereas the blue-shifted component is at least
$2\times$ weaker. There is no clear evidence for a blue-shifted CO component
in the VLA data. A more detailed study of this source based on higher angular
resolution NOEMA data will be presented in Ismael et al. (in prep.)

HerBS-70 is a binary system which was originally resolved in Neri et al.
(2020), including the sources HerBS-70E and HerBS-70W that are separated by
$16\farcs 5$ corresponding to a projected distance of $\rm\sim 140\,kpc$. Both
are well detected and resolved in the CO(1$-$0) emission. HerBS-70E (to the
east) has a wide emission line with $\rm FWHM=622\pm 130$ km s-1 comparable to
the higher-$J$ transitions, with a ratio of FWHM/$\Delta$V$=0.8\pm 0.2$. In
contrast, HerBS-70W has narrow line of $\rm FWHM=197\pm 39$ km s-1, similar to
the one seen in the higher-$J$ CO lines with a ratio of FWHM/$\Delta$V$=1.4\pm
0.3$. Both sources display extended and somewhat resolved CO(1$-$0) emission
with sizes of $1\farcs 6\times 1\farcs 2$ ($13\times 10$ kpc), and $2\farcs
1\times 1\farcs 1$ ($18\times 9$ kpc), for HerBS-70E and HerBS-70W,
respectively.

HerBS-79 displays a broad CO(1$-$0) emission line with $\rm FWHM=787\pm 125$
km s-1 that is comparable to the CO(3$-$2) and CO(4$-$3) profiles, with a
ratio of FWHM/$\Delta$V$=0.9\pm 0.2$. The moment-0 map shows an extended arc-
like morphology with a size of $2\farcs 9\times 1\farcs 0$ ($25\times 8$ kpc)
in the east-west direction.

The VLA data of HerBS-89a were originally published separately in Berta et al.
(2021), a dedicated study of the source that combined the CO(1$-$0)
observations with high angular resolution NOEMA observations of the $\rm
CO(9-8)$ emission line and additional emission or absorption lines of
molecular tracers (including $\rm H_{2}O$ and $\rm OH^{+}$). We have
reanalysed the CO(1$-$0) emission to be consistent with the analysis of the
whole sample. We find a similar morphology and line profile as reported in
Berta et al. (2021). However, the CO(1$-$0) from our extraction is slightly
wider, with a $\rm FWHM=1586\pm 244$ km s-1 (versus $1433\pm 293$ km s-1) and
the integrated line flux is significantly larger with $I_{\rm CO(1-0)}=1.08\pm
0.22$Jy km s-1 (versus $0.64\pm 0.13$Jy km s-1). The difference between the
two results is due to the larger extraction region used in our analysis. The
morphology of the CO(1$-$0) emission is arc-like, which was clearly resolved
in the higher angular resolution NOEMA observations into a partial $1\farcs 0$
diameter Einstein ring in the dust emission and the molecular emission lines
of $\rm CO(9-8)$ and $\rm H_{2}O(2_{02}-1_{11})$. Based on a lensing model,
the magnification of HerBS-89a was estimated to be $\mu\sim 5$ (see Berta et
al., 2021).

HerBS-95 is the second binary system resolved by Neri et al. (2020), including
the sources HerBS-95E and HerBS-95W, which are separated by $16\farcs 4$
corresponding to a projected distance of $\rm\sim 130\,kpc$. HerBS-95E
displays a CO(1$-$0) emission line that is somewhat blue-shifted with respect
to the expected rest-frame zero velocity, with a ratio of
FWHM/$\Delta$V$=0.8\pm 0.2$. The emission is slightly extended with a size of
$1\farcs 5\times 0\farcs 8$ ($12\times 6$ kpc). HerBS-95W shows a double-
peaked line profile with a $\rm FWHM=522$ km s-1, in agreement with the
higher-$J$ transitions, with a ratio of FWHM/$\Delta$V$=1.0\pm 0.2$. The
CO(1$-$0) emission has a centrally-peaked but extended structure with a size
of $2\farcs 5\times 1\farcs 9$ ($20\times 15$ kpc).

HerBS-113 has a CO(1$-$0) profile somewhat narrower than the higher-$J$
transitions with a $\rm FWHM=497$ km s-1, with a ratio of
FWHM/$\Delta$V$=0.6\pm 0.2$. The morphology of CO(1$-$0) emission is complex
with three distinct peaks, distributed along an elongated ring-like structure
and extending over $\sim 3\arcsec$ (24 kpc). This morphology has been
confirmed by higher resolution CO observations that will be presented in
Ismail et al. (in prep.).

HerBS-154 displays a strong single-peaked CO(1$-$0) emission line with a $\rm
FWHM=384\pm 54$ km s-1, comparable to the higher-$J$ CO and
$\rm[CI](^{3}P_{1}-^{3}P_{0})$ lines, with a ratio of FWHM/$\Delta$V$=1.2\pm
0.2$. The CO(1$-$0) emission has an extended arc-like morphology with a size
of $2\farcs 7\times 1\farcs 7$ ($20\times 13$ kpc).

##  Appendix B Lensing

The large sub-millimeter surveys made with telescopes such as Herschel or the
South Pole Telescope, have revealed a significant population of bright high-
redshift SMGs that are strongly gravitionally lensed, that becomes dominant
for high flux densities ($S_{\rm 500\mu}>100\rm mJy$) (e.g., Negrello et al.,
2010; Wardlow et al., 2013; Mocanu et al., 2013). We therefore expect that
some sources in our sample will be lensed, as discussed in Neri et al. (2020).
The new CO(1$-$0) data have revealed that sources HerBS-54, HerBS-58,
HerBS-79, HerBS-89a, HerBS-113 and HerBS-154, have complex morphologies such
as arc-like structures, filaments or multiple components, suggesting that
these sources could be gravitationally lensed (see Fig. 1). In the case of
HerBS-89a higher resolution followup observations confirmed that it is indeed
lensed (see Berta et al., 2021). However, for the rest of the sources, a
derivation of the magnification based on the available data remains uncertain
and deeper observations with higher angular resolution are needed to make a
detailed lensing model.

In past studies, the relationship between $L^{\prime}_{\rm CO(1-0)}$ and the
full width half maximum (FWHM) of the CO(1$-$0) emission line has been
discussed as a tool to estimate the magnification factor of high-redshift
galaxies (e.g., Harris et al., 2012; Bothwell et al., 2013; Aravena et al.,
2016; Neri et al., 2020). As $L^{\prime}_{\rm CO(1-0)}$ is a measure of
$M_{\rm H_{2}}$ and the CO(1$-$0) FWHM is related to the dynamical mass, this
relationship is seen as a proxy to the Tully-Fisher relation (e.g., Isbell et
al., 2018). Therefore, any sources lying above the expected correlation would
be lensed. However, as shown in Aravena et al. (2016), the use of the
$L^{\prime}_{\rm CO(1-0)}$ versus FWHM relationship for measuring
magnifications is highly unreliable, and can be typically off by factors of
$\sim 2$ or more.

In Figure 10, we plot $L^{\prime}_{\rm CO(1-0)}$ as a function of the
CO(1$-$0) FWHM for the sources of our sample together with sources previously
observed in CO(1$-$0). We limit our comparison to high-redshift sources that
are unlensed and sources for which the gravitational magnification has been
reliably estimated based on lensing model analysis. Therefore, we remove
uncertainties that can accompany conversions from higher-$J$ CO transitions or
sources which were defined as magnified using this same relation. The known
lensed sources (identified in the figure) were not corrected for
magnification. The dotted line shows the best-fitting relationship from
Bothwell et al. (2013) and the dashed line represents the derived
parametrization for $L^{\prime}_{\rm CO}$ by Bothwell et al. (2013) for a disk
galaxy, that can be described as:

${L^{\prime}_{\rm CO}}=C\,\left(\frac{\Delta
V}{2.35}\right)^{2}\frac{R}{\alpha\,G},$ (B1)

where $\Delta V$ is the FWHM of the CO in $\rm km\,s^{-1}$, $R=7\,\rm kpc$ is
the radius of the CO emitting region in parsecs, $G$ is gravitational
constant, $\alpha$ is the CO luminosity to gas mass conversion factor, and
$C=2.1$ is the constant parametrizing the galaxy’s disk morphology.

We use the compilation of CO(1$-$0) detected galaxies and fit all sources that
are unlensed and the lensed sources after correcting for magnification, as
well as only those at $z>1$. Both relationships show a less steep correlation
than previously found. We note that we plot the lensed sources without the
magnification correction, even though we correct them for magnification for
the fit, in order to better visualise the parameter space covered by the
lensed sources, in relation to the unlensed sources and the relations.

Overall, we find a large scatter for both the lensed and unlensed galaxies,
consistent with what has been reported in previous studies (e.g., Aravena et
al., 2016). Sources with magnifications of $<10$ mostly cover a similar region
of the parameter space as the unlensed sources, with only highly lensed
galaxies ($\mu>10$) showing clear offsets. This strengthens the conclusions of
Aravena et al. (2016) on the unreliability of this relation as a precise
measure of magnification (see also discussions in, e.g., Dannerbauer et al.,
2017; Aravena et al., 2019; Jin et al., 2021). It is interesting to note the
specific example of HerBS-89a, which based on the $L^{\prime}_{\rm
CO(1-0)}$–FWHM relationship would be considered to be an unlensed source when
comparing to the Bothwell et al. (2013) relation, and therefore a HyLIRG (see
Neri et al., 2020), was found to be lensed with a magnification of $\sim 5$
based on high-angular resolution observations and a lensing model (Berta et
al., 2021). For the above reasons, we do not attempt to determine the
magnifications of our sample using this method.

Figure 10: L${}^{\prime}_{\mathrm{CO(1-0)}}$ versus the full width half max
(FWHM) of the $\rm CO(1-0)$ emission line. The sample of sources presented in
this paper are shown with large red squares (and identified by their name),
and are compared to other high-$z$ lensed and unlensed luminous DSFGs (Ivison
et al., 2011; Harris et al., 2012; Aravena et al., 2016; Bakx et al., 2020b;
Carilli & Walter, 2013, and references therein), $z>1$ main-sequence (MS)
galaxies (Aravena et al., 2014) and cluster galaxies (Rudnick et al., 2017;
Wang et al., 2018), as well as low-$z$ ultra-luminous infrared galaxies
(ULIRGs), mild-starburst (mild-SBs) and MS galaxies (Solomon et al., 1997;
Chung et al., 2009; Combes et al., 2011; Geach et al., 2011; Villanueva et
al., 2017). Only sources with reported $\rm CO(1-0)$ emission lines have been
included to avoid uncertainties in estimating
L${}^{\prime}_{\mathrm{CO(1-0)}}$ from higher-$J$ transitions. No correction
for magnification was applied to the CO luminosities. Sources with
magnifications $>10$ are classified as Highly-lensed SMGs. The dotted and
dashed lines represent the relationships described in Bothwell et al. (2013)
(see text and Eq. B1); the solid and dash-dotted lines display fits to all the
$z>1$ galaxies and the entire sample of sources, respectively, that are
unlensed and have been corrected for magnification.