# Main Sequence to Starburst Transitioning Galaxies:
Gamma-ray Burst Hosts at $z\sim 2$

Jakub Nadolny Michał Jerzy Michałowski Astronomical Observatory Institute,
Faculty of Physics, Adam Mickiewicz University, ul. Słoneczna 36, 60-286
Poznań, Poland  J. Ricardo Rizzo ISDEFE, Beatriz de Bobadilla 3, 28040 Madrid,
Spain  Centro de Astrobiología (CAB), CSIC-INTA, Ctra. M-108, km. 4, 28850
Torrejón de Ardoz, Spain  Agata Karska Institute of Astronomy, Faculty of
Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziadzka 5, 87-100 Toruń, Poland  Max-Planck-Institut für Radioastronomie,
Auf dem Hügel 69, 53121, Bonn, Germany  Jesper Rasmussen Technical University
of Denmark, Department of Physics, Fysikvej 309, DK-2800 Lyngby, Denmark
Jesper Sollerman Department of Astronomy, The Oskar Klein Centre, Stockholm
University AlbaNova, SE-106 91 Stockholm, Sweden  Jens Hjorth DARK, Niels Bohr
Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark
Andrea Rossi INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via
Piero Gobetti 93/3, 40129 Bologna, Italy  Martín Solar Radosław Wróblewski
Aleksandra Leśniewska Astronomical Observatory Institute, Faculty of Physics,
Adam Mickiewicz University, ul. Słoneczna 36, 60-286 Poznań, Poland

(Received -; Revised -; Accepted -)

###### Abstract

Star-forming galaxies populate a main sequence (MS), a well-defined relation
between stellar mass ($M_{*}$) and star-formation rate (SFR). Starburst (SB)
galaxies lie significantly above the relation whereas quenched galaxies lie
below the sequence. In order to study the evolution of galaxies on the
SFR-$M_{*}$ plane and its connection to the gas content, we use the fact that
recent episodes of star formation can be pinpointed by the existence of gamma-
ray bursts (GRBs). Here we present sensitive [C I]-nondetections of $z\sim\,2$
ultra luminous infrared (ULIRG) GRB host galaxies. We find that our GRB hosts
have similar molecular masses to those of other ULIRGs. However, unlike other
ULIRGs, the GRB hosts are located at the MS or only a factor of a few above
it. Hence, our GRB hosts are caught in the transition toward the SB phase.
This is further supported by the estimated depletion times, which are similar
to those of other transitioning galaxies. The GRB hosts are [C I]-dark
galaxies, defined as having a [C I]/CO temperature brightness ratio of $<$0.1.
Such a low [C I]/CO ratio has been found in high-density environments ($n_{\rm
H}>10^{4}\,{\rm cm}^{-3}$) where CO is shielded from photodissociation,
leading to under-abundances of [C I]. This is consistent with the merger
process that is indeed suggested for our GRB hosts by their morphologies.

Galaxies (573), Galaxy evolution (594), Gamma-ray bursts (629), Molecular gas
(1073), Starburst galaxies (1570)

††journal: ApJ††software: astropy (Astropy Collaboration et al., 2013, 2018),
GILDAS (Pety, 2005)

##  1 Introduction

Star formation occurs in molecular gas clouds (Wong & Blitz 2002; Gao &
Solomon 2004a; Bigiel et al. 2008; but see Glover & Clark 2012; Krumholz 2012;
Michałowski et al. 2015). The molecular gas fraction and its availability for
star formation are key ingredients that shape the evolution of galaxies (see
review by Saintonge & Catinella, 2022) and determine the place where a galaxy
is found in the star-formation rate (SFR)–stellar mass ($M_{*}$) plane. Normal
star-forming galaxies (SFGs) form a well-defined ‘main sequence’ (MS) on this
plane with a scatter of about 0.2 dex (Brinchmann et al., 2004; Noeske et al.,
2007; Speagle et al., 2014).

The so-called starburst galaxies are found above the MS with extremely high
SFRs for a given stellar mass (Combes et al., 2011; Rodighiero et al., 2011;
Larson et al., 2016). It is not clear whether an increase in star formation
efficiency (SFE; Cheng et al., 2018; Hogan et al., 2022) or gas mass fraction
(Lee et al., 2017; Valentino et al., 2020a) drives the departure from the MS.
As pointed out by Gao & Solomon (2004b), the global SFR depends mainly on the
amount of dense molecular gas, which can be traced for example by a hydrogen
cyanide (HCN) line. This dependence remains nearly the same (with a slope of
about 1) for normal and starburst galaxies, including ultra-luminous galaxies
(ULIRGs). The HCN observations however are limited to the local universe due
to the weakness of this line. Thus, until deep HCN (or other dense gas
tracers) observations are available, we need to rely on other approximations.

Major and minor mergers have been invoked as possible causes for triggering
the starburst behavior (Combes et al., 2011; Rodighiero et al., 2011; Larson
et al., 2016; Saintonge & Catinella, 2022). Eventually, some galaxies will
terminate their star formation. These galaxies tend to have red colors,
compact and spheroidal morphologies (Schawinski et al., 2014; Nadolny et al.,
2021), with relatively low gas and dust content.

Most of the information on molecular gas in galaxies comes from observations
of the carbon monoxide (CO) lines (Bolatto et al., 2013; Carilli & Walter,
2013). The notion that the neutral carbon line ([C I]) traces the bulk of the
molecular gas mass has been investigated for more than four decades now
(Phillips & Huggins, 1981; Papadopoulos et al., 2004; Jiao et al., 2017;
Valentino et al., 2018). On the other hand, in UV-intense and metal-poor
environments, the use of [C I] as a molecular gas tracer is limited due to the
increased ionisation of carbon. The limited usefulness of [C I] has also been
shown in dense conditions, i.e in the collisional fronts of mergers (Michiyama
et al., 2021). Bisbas et al. (2017) showed however, that even if [C I] is
limited in such cases, it is still a more reliable tracer of global molecular
mass than CO. This is due to the high sensitivity of the [C/CO] abundance to
even small changes in the cosmic ray ionization rate, especially when the
average gas densities are low ($\leq 10^{3}{\rm cm}^{-3}$). It is in such
densities that the bulk of the H2 reservoirs in galaxies is often found.

Recent episodes of star formation in galaxies can be pinpointed by the
existence of gamma-ray bursts (GRBs) which are explosions of short-lived
massive stars (Hjorth et al., 2003; Stanek et al., 2003). In this paper we
take advantage of this feature to study the evolution of galaxies on the
SFR-$M_{*}$ plane and its connection to the gas content by analyzing high-
sensitivity observations of [C I] line emission towards selected GRB hosts. In
particular, we want to shed light on the cause of weak [C I] lines of
galaxies, i.e. if they are due to gas properties (e.g., density) or due to
true low molecular gas content.

This paper is organized as follows. In Section 2 we describe the sample
selection, observations, and reduction process. We also define the comparison
sample from the literature and describe the methods used to derive fluxes,
luminosities, and molecular masses. In Section 3 we describe the results of
our analysis. Section 4 discusses our interpretation of the observables
together with alternative scenarios. Finally, in Section 5 we present the
conclusions of our work. Throughout this paper we use a cosmological model
with $H_{0}$ = 70 km s-1Mpc-1, $\Omega_{\Lambda}=0.7$, and $\Omega_{m}=0.3$.

##  2 Data

###  2.1 Sample Selection

The GRB host sample observed in [C I](${}^{3}P_{1}-^{3}P_{0}$) (hereafter [C
I]) was selected based on the availability of infrared or radio detections
(Hunt et al., 2014; Perley et al., 2015; Michałowski et al., 2015), allowing
precise estimates of SFRs. We selected hosts with spectroscopic redshifts so
that their [C I] lines were expected to be observed away from atmospheric
water lines. This resulted in seven potential targets (the hosts of GRB
051006, 051022, 060814, 061121, 080207 100316D, 111005A). Depending on their
declinations, these sources were observed with the Atacama Pathfinder
Experiment (APEX) or the IRAM 30m radio telescope. The hosts of GRB 111005A,
051006, 051022 were not observed, because they are at the lower redshift range
proposed for a given telescope, resulting in a high observing frequency at
which the weather requirements were challenging.

Low-redshift targets (GRB 061121 and 100316D) were only suitable for the APEX
telescope because for them the observing frequency is high and requires very
stable weather conditions and also a very low amount of precipitable water
vapor. These stringent conditions are often attained at Chajnantor (the APEX
site), but not at Pico Veleta (the IRAM 30m telescope site).

###  2.2 Observations and Data Reduction

Figure 1: The [C I] spectra of GRB 060814 and 080207. Vertical dotted lines
indicate the velocity range used for the flux integration (from $-100$ to
$100$ km s-1). The upper x-axis labels the observed frequencies. Inset panels
show the whole obtained spectra, while shaded areas indicate ranges used to
estimate the noise.

We observed the hosts of GRB 060814 and 080207 with the IRAM 30m telescope
(proposal 172-16; PI: M.J.M.), equipped with the Eight MIxer Receiver111
www.iram.es/IRAMES/mainWiki/EmirforAstronomers (Carter et al., 2012). We
implemented the wobbler-switching mode and the Fourier Transform Spectrometers
200 (FTS-200) providing 195 kHz spectral resolution and 16 GHz bandwidth in
each linear polarisation. The observations for both targets were executed
between 2017-Feb-01 and 2017-May-22 and lasted in total 13.1 hr on-source for
GRB 060814 and 17.8 hr for GRB 080207. The observations were divided into 6
min scans, each consisting of 12 subscans 30 s long. Pointing was checked and
corrected every 1–2 hr. Each spectrum was calibrated, and corrected for
baseline shape. The spectra were aligned in frequency and noise-weight
averaged. Some well-known platforming, due to the fact that the instantaneous
bandwidth of 4 GHz is sampled by three different FTS units, was corrected off-
line by a dedicated procedure within the Continuum and Line Analysis Single
Dish Software (Class). In all cases, the [C I] line is far away from the step
of the platforming.

The hosts of GRB 061121 and 100316D were observed with the APEX telescope
(Güsten et al. 2006; proposals 098.F-9300 and 098.D-0243; PI: M.J.M.),
equipped with the Swedish Heterodyne Facility Instrument (SHeFI; Vassilev et
al. 2008; Belitsky et al. 2006). However the upper limits were not
sufficiently constraining to allow robust conclusions about the molecular
content of these sources, hence we do not report the results for these
galaxies.

All data were reduced and analyzed using the Class package within the Grenoble
Image and Line Data Analysis Software: Gildas222www.iram.fr/IRAMFR/GILDAS
(Pety, 2005).

The obtained [C I] spectra of our two GRB (060814 and 080207) hosts are shown
in Figure 1.

###  2.3 Comparision Sample

In order to place our two GRB hosts into a general perspective we compiled a
sample of galaxies spanning several orders of magnitude in stellar mass, SFR
and gas mass. In particular, this compilation contains normal SFG (Bourne et
al., 2019; Dunne et al., 2021; Valentino et al., 2020b), (Ultra) Luminous IR
Galaxies [(U)LIRGs] (Valentino et al., 2020b; Michiyama et al., 2021; Lu et
al., 2017), intermediate-$z$ isolated LIRGs (Lee et al., 2017) and merging
starburst ULIRGs (Combes et al., 2011), high-$z$ starbursts (Shi et al.,
2018), and transitioning isolated and merging galaxies (Cheng et al., 2018;
Hogan et al., 2022). For all sources with [C I] data, $M_{\rm mol}$ has here
been estimated in a consistent manner (see Sec. 2.4). We used the total far-
infrared luminosity to estimate the SFR (Kennicutt, 1998), as

$SFR=L_{\rm IR}[L_{\odot}]/(9.86\times 10^{9})\rm{M}_{\odot}\rm{yr}^{-1}$

(converted to the Chabrier 2003 IMF), and the fundamental plane (Lara-López et
al., 2010) to estimate the oxygen metallicity (Valentino et al., 2018; Bourne
et al., 2019; Dunne et al., 2021), for galaxies without these estimates in the
literature. The latter is needed to estimate the metallicity-dependent
conversion factor $\alpha_{\rm[CI]}$ introduced below.

###  2.4 Flux, Luminosity and Mass Measurements

Our deep observations of [C I] in GRB hosts show a lack of significant
emission. We estimated $2\sigma$ upper limits of the line flux, luminosity,
and molecular gas masses by integration within a velocity range of -100 to 100
km s-1 around the expected velocity of the [C I] line at the redshift of each
source. For our sample, as well as for the data from other works, we estimated
the [C I] luminosity $L^{\prime}_{\rm CI}$ using the same prescription –
equation 3 in Solomon et al. (1997).

We employed two methods to estimate the molecular gas mass from the luminosity
of the [C I] line. The first method is based on a theoretical analysis of the
[C I] emission from the ISM assuming local thermal equilibrium given by
Papadopoulos et al. (2004). Using their Equation 11, evaluating all the
constants we derived an expression to estimate the molecular gas mass from the
[C I](1-0) line flux as:

$\begin{split}\frac{M_{\rm mol}}{\rm M_{\odot}}=1.3747\times
10^{-9}\frac{D_{\rm L}^{2}}{1+z}\frac{I_{\rm CI}}{(X_{\rm CI}A_{\rm
10}Q_{10})},\end{split}$ (1)

where the abundance ratio is $X_{\rm CI}=3\times 10^{-5}$ (Valentino et al.,
2018; Jiao et al., 2017), Einstein coefficient for this transition is
$A_{10}=7.93\times 10^{-8}$s-1, $Q_{10}$ is given by Equation A15 from
Papadopoulos et al. (2004) with assumed $T_{\rm kin}=40\rm{\ K}$, $D_{\rm L}$
is the luminosity distance given in Mpc, and $I_{\rm CI}$ is the velocity
integrated [C I] line flux in units of Jy km s-1. We refer to this method as
P04 (see Weiß et al. 2003 for a similar method). The second method is taken
from Heintz & Watson (2020), and is based on the conversion factor
$\alpha_{\rm[CI]}$ between $M_{\rm mol}$ and $L^{\prime}_{\rm CI}$, estimated
from observations of the [C I] absorption line in spectra of GRB afterglows
and QSOs. In this case $\alpha_{\rm[CI]}$ is metallicity dependent. We refer
to this method as H20. Note that in both methods a factor $\mu=1.36$ that
corrects for the contribution from helium and heavier elements is included.

##  3 Results

Table 1: Results from our [C I] observations. The [C I] fluxes are given with their 2$\sigma$ errors, while luminosities and molecular masses are 2$\sigma$ upper limits. The errors were obtained by randomly perturbing 1000 times the fluxes of each channel in the spectra within their errors and assessing the 95% confidence interval of the obtained integrated fluxes. The measurements of $M_{\rm mol}$ for all the methods include the $\mu=1.36$ factor accounting for the contribution from helium and heavier elements. The last column (M18) reports CO-based molecular mass from Michałowski et al. (2018). GRB | $I_{\rm[CI]}$ |  $\log($$L^{\prime}_{\rm CI}$$)$ | $\log(M_{\rm mol}/M_{\odot})$
---|---|---|---
| (Jy km s-1) | (K km s-1 pc-2) | P04(a) | H20(a) | M18
060814 | -0.064 $\pm 0.113$ | 9.178 | 9.852 | 10.859 | 10.92
080207 | 0.016 $\pm 0.139$ | 9.504 | 10.17 | 10.777 | 11.3
Table 2: GRB host properties from the literature.  GRB | $z$ | SFR (M⊙/yr) |  $\log$(M∗/M⊙)
---|---|---|---
060814 | 1.92 (H12) | 256.0 (P15) | 10.2 (P15)
080207 | 2.09 (H12) | 170.0 (H12) | 11.17 (H14)

All the measurements obtained for our GRB hosts are given in Table 1. The
upper limits are 2$\sigma$. The SFRs, stellar masses, and spectroscopic
redshifts from the literature are given in Table 2.

###  3.1 [C I] Luminosity

Figure 2: Star-formation rate as a function of [CI] emission line luminosity
$L^{\prime}_{\rm CI}$. The dashed line and shaded region mark our linear fit
(Eq. 2) and its $2\sigma$ scatter, respectively, based on data (gray open
markers) from the literature (Bourne et al., 2019; Dunne et al., 2021; Lu et
al., 2017; Valentino et al., 2020b). The right panel shows the residuals
relative to the best-fit relation. The red X symbol shows the [C I] upper
limit of [C I]-dark galaxy found in Michiyama et al. (2021). See main text.

In Figure 2 we show the SFRs as a function of [C I] line luminosity
$L^{\prime}_{\rm CI}$ $[{\rm K\,km\,s}^{-1}{\rm pc}^{-2}]$ for our sample and
for data from the literature for which the [C I] fluxes were available (Bourne
et al., 2019; Lu et al., 2017; Valentino et al., 2020b; Dunne et al., 2021;
Michiyama et al., 2021). Using only data for SFG from the literature (gray
empty markers), we find the best-fit relation between $L^{\prime}_{\rm CI}$
and SFR as:

$\log(\mbox{SFR})[{\rm M_{\odot}}/{\rm
yr}]=0.952\times\log({\rm\mbox{$L^{\prime}_{\rm CI}$}})-7.165$ (2)

which holds over a wide redshift range ($0\,<\,z\,<\,3$) with $2\sigma$
scatter of 0.27 dex. As shown in Figure 2, the hosts of GRBs 060814 and 080207
have lower $L^{\prime}_{\rm CI}$ (by $>$0.8 and $>$0.3 dex, respectively) than
expected for their SFRs based on our best-fit. It has been shown for GRB
hosts, in general, that, we can rule out the possible contamination of the
emission by active galactic nuclei at the wavelengths used to estimate SFRs
(Perley et al., 2015), so we consider these SFRs to be robust.

In Figure 2 we also show the local (U)LIRG merging galaxies (Michiyama et al.,
2021), which together with our low-$L^{\prime}_{\rm CI}$ GRB hosts are found
above the SFG population in terms of their SFR. This is also clearly visible
on the right-hand panel in the same Figure where the scatter around the best-
fit relation is shown. In particular, the red X symbol shows NGC 7679, the [C
I]-dark AGN host found by Michiyama et al. (2021). The [C I]/CO ratio of 0.07
of this galaxy is below those of other ULIRGs (Jiao et al., 2017). Using the
CO data from Michałowski et al. (2018) we estimated the [C I]/CO ratio of our
GRB hosts to be of the order of $\leq$0.1. This is lower than normal ULIRGs
but not as low as [C I]-dark galaxies like NGC 7679 or NGC 6052 studied in
Michiyama et al. (2020, 2021). On average (U)LIRGs have 0.6 dex lower
$L^{\prime}_{\rm CI}$ while the [C I]-dark NGC 7679 has 1.2 dex lower
$L^{\prime}_{\rm CI}$ than the expectations from the $L^{\prime}_{\rm CI}$-SFR
fit.

###  3.2 Molecular Gas Mass

Figure 3: Star formation rate as a function of molecular gas mass, $M_{\rm
mol}$, for the two estimation methods used (labeled in each panel). Magenta
arrows show the upper-limit estimation of $M_{\rm mol}$ of GRB hosts from this
work. Gray open markers represent the data from the literature used in the
fitting (Bourne et al., 2019; Valentino et al., 2020b; Dunne et al., 2021).
Black filled markers represent intermediate- and high-$z$ starbursts ULIRGs
(Combes et al., 2011; Shi et al., 2018); magenta X, diamond, cross symbols
represent merging (U)LIRGs (Michiyama et al., 2021; Shangguan et al., 2019),
transitioning intermediate-$z$ galaxies (Hogan et al., 2022), transitioning
local merging systems (Cheng et al., 2018), respectively; blue diamonds and
squares represent intermediate-$z$ isolated LIRGs (Lee et al., 2017), and
local isolated transitioning galaxies (Cheng et al., 2018); red star and cross
reprsenent local LIRG NGC 1674 (König et al., 2013) and the molecule-deficient
GRB 051022 host galaxy (Hatsukade et al., 2014). For these galaxies, we use
the same $M_{\rm mol}$ in both panels (mainly due to a lack of
$L^{\prime}_{\rm CI}$ estimates), and these are only for comparison. The
results of the fitting are given in each panel.

In Figure 3 we show the estimated molecular gas masses of our GRB hosts based
on $L^{\prime}_{\rm CI}$ using our two different methods (see section 2.4 for
details). We include the same data as in Figure 2,333Not all the galaxies from
Fig. 2 had metallicity estimates available, nor was it possible to estimate
these using the fundamental metallicity relation (Lara-López et al., 2010).
These galaxies could not be included in the method where the conversion factor
is metallicity-dependent, i.e. for Method H20. together with local LIRG NGC
1674 (König et al., 2013), intermediate-$z$ isolated LIRGs (Lee et al., 2017),
intermediate-$z$ merging starburst ULIRGs (Combes et al., 2011), and high-$z$
starburst ULIRGs (Shi et al., 2018), and low- and intermediate-$z$
transitioning isolated and merging galaxies (Cheng et al., 2018; Hogan et al.,
2022) with the molecular mass estimated from CO emission (i.e. these have the
same $M_{\rm mol}$ in both panels, and are shown for comparison only). Using
the SFG sample from the literature (gray empty markers) we find the best-fit
relation between SFR and [C I]-based $M_{\rm mol}$ for each of the methods.
The results of the fitting are given in each panel. The $M_{\rm mol}$ for the
GRB 060814 and 080207 host galaxies are found to be lower by $>$0.8 and $>$0.2
dex from our best-fit for the P04 method. At least in the case of the host
galaxy of GRB 060814 the offset is in agreement with the distance from the
best-fit estimated for (U)LIRGs ($\sim$0.8 dex), high-$z$ starburst
($\sim$0.7), NGC 1674 ($\sim$0.9 dex), and for the transitioning SFG mergers
($\sim$0.4 dex).

The range of the molecular gas mass in isolated SFG (blue markers), merging
transitioning systems (magenta), and in starburst (black) is similar (between
$10^{9.5}$ and $10^{11}$$\,\rm M_{\odot}$), but their SFRs vary significantly
from tens to thousands solar masses per year, which translates to different
depletion times (or SFEs; see Sec. 3.3). Considering the second method used
(H20), which is metallicity-dependent, we can see that the majority of the
ULIRGs, including our GRB hosts, have $M_{\rm mol}$ above, but within the
scatter of, the best-fit relation.

###  3.3 Main Sequence and Depletion Times

To establish whether the position of GRB hosts in Figures 2 and 3 is due to
low $L^{\prime}_{\rm CI}$ or elevated SFRs, we investigate their location
relative to the MS. Figure 4 shows the stellar mass as a function of SFR, and
the MS for SFGs at different redshifts (Speagle et al., 2014). While
intermediate-$z$ LIRGs are found on their MS, the GRB hosts, (U)LIRGs, and
transitioning SFG are found to lie above the MS for their redshifts. To better
quantify this, the distance from the MS is estimated as the ratio of the
observed SFR to the SFR of a galaxy on the MS with the same stellar mass and
redshift (SFR/SFRMS).

On the right panel of Figure 4, we show the molecular gas depletion timescale
($M_{\rm mol}$/SFR using P04 method, or CO-based molecular gas masses from the
literature when necessary), as a function of the distance to the MS. We find
that GRBs 060814 is about seven times above the MS, while GRB 080207 lies on
the corresponding MS within the scatter. The intermediate-$z$ isolated LIRGs
lie close to the MS, while isolated transitioning galaxies are found one order
of magnitude above it. The transitioning merging galaxies and starburst
merging ULIRGs are found at even greater distances (with an average offset of
a factor 20 and 77, respectively).

We obtained relatively short gas depletion times of $<$64 and $<$238 Myr
(2$\sigma$ upper limits) for the hosts of GRB 060814, and 080207,
respectively. These values are similar to those of high-$z$ starbursts for
which we obtain a mean depletion time of 84 Myr and of transitioning mergers
with an average of 205 Myr. The shortest depletion times (15 Myr on average)
are found in intermediate-$z$ merging ULIRGs galaxies. As expected, the
isolated LIRGs and isolated transitioning galaxies show longer depletion times
of $\sim$1 Gyr on average.

In the right panel of Figure 4 we can see that with the gradual departure from
the MS, the depletion time decreases. Gao & Solomon (2004b) showed that the
fraction of the dense gas is the primary predictor of the SFR, and that the
relation between these quantities is similar for different galaxy types. Here
we can see that the merging process plays an important role in decreasing
depletion time (or increasing SFE). Indeed, for the dense environment, it has
been shown that the gas is converted quicker to stars, in particular in gas-
rich mergers (Genzel et al., 2010). Perhaps this is to be expected given that
in mergers there are much more turbulent molecular gas reservoirs, whose
higher Mach numbers will place more gas at high densities ($\geq 10^{4}\,{\rm
cm}^{-3}$).

Figure 4: Star-formation rate as a function of stellar mass (left panel), and
depletion time as a function of distance from the MS SFR/SFRMS (right panel).
All the symbols are the same as in Figure 3. The MS for the corresponding
redshift has been estimated using results from Speagle et al. (2014). The
shaded region represents 0.2 dex scatter. Black symbols correspond to galaxies
that are in the starburst regime or will likely reach it thanks to their high
gas masses.

##  4 Discussion

A common feature for both GRB hosts studied in this work is a sensitive [C I]
nondetection implying relatively low [C I] emission. The disturbed multi-
component morphology in the Hubble Space Telescope imaging of both galaxies
suggests an ongoing merger process (Svensson et al., 2012; Blanchard et al.,
2016; Chrimes et al., 2019; Schneider et al., 2022). In what follow we put
forward two non-exclusive scenarios to explain this feature. The first one is
that the GRB hosts are caught at their transition from the main sequence
toward the starburst phase (Michiyama et al., 2021; Hogan et al., 2022; Cheng
et al., 2018). The second possibility is that they are [C I]-dark galaxies,
which may explain their low [C I]/CO brightness temperature ratio (Jiao et
al., 2017; Michiyama et al., 2020).

###  4.1 Transition To Starburst

Our GRB hosts exhibit lower molecular masses than the best fit to normal
galaxies (Fig. 3), similar to merging ULIRGs and high-$z$ starbursts. Moderate
SFRs (as for the redshift of our GRB hosts) result in distances from their MS
in between that measured for intermediate-$z$ isolated LIRGs and that for
intermediate- and high-$z$ starbursts. Estimated depletion times ($M_{\rm
mol}$/SFR) are similar to that of transitioning merger galaxies, while being
shorter than intermediate-$z$ isolated LIRGs, and isolated transitioning
galaxies, and longer (by about an order of magnitude) than the merging
intermediate- and high-$z$ starburst galaxies (Fig. 4, right panel). The
moderate distances from the MS and short depletion times suggest that our GRB
hosts are observed in their transition toward the starburst phase. Given the
morphology, this increase in SFR is likely caused by mergers. The existence of
GRB events indicates that this may be the beginning of such a transition,
because the progenitors of GRBs are short-lived stars.

We note that estimated depletion times assume no feedback effects (e.g.
energetic winds from massive stars, radiative pressure). These effects have
been found to be a possible cause of the enlargement of the time over which
starburst galaxies consume their available cold gas reservoirs. In this work
we treat all the galaxies in the same manner, i.e. depletion times are
estimated without such feedback effects (Semenov et al., 2017; Díaz-García &
Knapen, 2020), thus we consider this comparison as valid. Moreover, the
depletion times are not used to draw any conclusions about the timescale of
running out of gas.

###  4.2 [C I]-dark Galaxies

The [C I] nondetection may be caused by an underabundance of carbon, rather
than low molecular hydrogen mass. This would result in a [C I]-dark object.
The [C I]-dark galaxies are characterised by a low [C I]/CO temperature
brightness ratio, below 0.1 (Michiyama et al., 2020, 2021), while regular
ULIRGs have this ratio not lower than 0.2 (Jiao et al., 2017). These [C
I]-dark galaxies have high hydrogen densities between $10^{5}$ and $10^{6}$
cm-3, as shown using photodissociation region (PDR) models (Valentino et al.,
2020b; Michiyama et al., 2020, 2021).

In standard models of photodissociation regions, CO molecules are efficiently
shielded from ultraviolet radiation at the dust extinction $A_{V}$ above a few
mag, which could lead to low abundances of [C I] (Tielens & Hollenbach, 1985).
Recent models show, however, that the abundance ratio of [C I] over CO is
linked more strongly with the cosmic rays ionization rate rather than the UV
radiation field (Bisbas et al., 2017). Nevertheless, [C I] can be considered
as a good tracer of H2 gas in mergers, except in the highest-density medium
where carbon is locked in CO. Such high densities are indeed commonly detected
toward [C I]-dark objects which have undergone a merger event (Michiyama et
al., 2020, 2021).

The proposed explanations of transitioning-to-starburst galaxies and [C
I]-dark galaxies may be related. The low [C I]/CO line ratio in [C I]-dark
galaxies could be an effect of mergers yielding high average density H2 gas
reservoirs but with a yet-to-be-fully-ignited starburst, producing lower
average cosmic ray energy densities. The combination of high $n_{H}$ and low
cosmic ray energy density can then naturally produce [C I]-dark galaxies,
albeit only for short cosmic time intervals. In that regard, it would be
interesting to examine whether [C I]-dark galaxies of this type deviate from
the far infrared-radio correlation, i.e., with lower synchrotron emission for
a given far-infrared luminosity.

The estimated upper limit [C I]/CO temperature brightness ratio of $<0.1$ for
the GRB 080207 hosts places it in the regime of [C I]-dark galaxies. Very high
gas density between $10^{5}$ and $10^{6}$ cm-3 have also been inferred for
this galaxy using a [C II] marginal detection (Hashimoto et al., 2019), which
is consistent with the proposed [C I]-dark nature of the host. Both our GRB
hosts have also low [C I] luminosities for their SFRs. In the case of the GRB
060814 host, the [C I] and CO were not detected, so we can not confirm or rule
out this interpretation for this galaxy. High gas densities were also claimed
for other GRB hosts (Christensen et al., 2008; Michałowski et al., 2014, 2015,
2016; Arabsalmani et al., 2015, 2019, 2020, 2022; de Ugarte Postigo et al.,
2020).

Thus, until additional observations (e.g. of HCN to trace the dense gas phase)
are available we conclude that both GRB hosts are candidates for [C I]-dark
galaxies that are transitioning toward the starburst phase with the merger
event being the cause.

###  4.3 Ruled Out Mechanism: Post-starbursts

In principle, our galaxies might be observed in the post-starburst phase, on
the way down from the starburst regime. This would explain their SFRs and low
gas content. However, this possibility is inconsistent with the presence of
optical emission lines, indicating recent star formation. Furthermore, Balmer
absorption features are not detected in the spectra of these GRB hosts
(Krühler et al., 2015), unlike for post-starburst galaxies. Post-starburst
galaxies have low SFRs and high metallicities, so while it is not impossible
(e.g. Rossi et al., 2014; Levan et al., 2023), they are less likely to host a
GRB, as opposed to an early starburst phase.

##  5 Conclusions

Our targets display spectra indicative of a young stellar population and have
slightly increased SFRs relative to their MS (Perley et al., 2013; Krühler et
al., 2015). Based on our [C I] emission line measurements, we have inferred a
molecular gas content similar to that of local ULIRGs but lower than would be
expected from the best fit to the normal star-forming galaxies (see Figs. 2
and 3). We propose that these are high-$z$ merger systems caught at the
transition from the main sequence toward the starburst phase. These are high
redshift analogs to the intermediate-$z$ transitioning galaxies (Cheng et al.,
2018; Hogan et al., 2022). Furthermore, their low [C I]/CO ratio point to a
high-density environment observed in the collisional fronts of mergers
(Valentino et al., 2020b; Michiyama et al., 2020, 2021). Indeed, the merger
signatures have been observed for our GRB hosts (Svensson et al., 2012;
Blanchard et al., 2016; Chrimes et al., 2019; Schneider et al., 2022). While
such merger-driven starbursts play a lesser role in the overall star-formation
rate density at $z=2$ (Rodighiero et al., 2011), these systems may play a
crucial role in the quenching and morphological transformation.

To test whether these galaxies indeed contain dense molecular star-forming
clouds it would be essential to observe their emission of the high dipole-
moment molecule HCN (Gao & Solomon, 2004a). This, however, will be challenging
because of the weakness of the HCN line. Additionally, the deep, high-
resolution imaging of [C I], and CO (e.g., with the NOEMA interferometer)
would be helpful to study the relative location of [C I] and CO, which again
is not an easy task considering that both GRB hosts are at $z\sim 2$.

##  6 Acknowledgments

We thank the anonymous reviewer for all the comments that helped to improve
our work. J.N., M.J.M., M.S. and A.L. acknowledge the support of the National
Science Centre, Poland through the SONATA BIS grant 2018/30/E/ST9/00208. This
research was funded in whole or in part by National Science Centre, Poland
(grant number: 2021/41/N/ST9/02662). J.R.R. acknowledges support by grant
PID2019-105552RB-C41 funded by MCIN/AEI/10.13039/501100011033. A.K.
acknowledges support from the First TEAM grant of the Foundation for Polish
Science No. POIR.04.04.00-00-5D21/18-00 and the Polish National Agency for
Academic Exchange grant No. BPN/BEK/2021/1/00319/DEC/1. This article has been
supported by the Polish National Agency for Academic Exchange under grant No.
PPI/APM/2018/1/00036/U/001. A.R. acknowledges support from the INAF project
Premiale Supporto Arizona & Italia. J.H. was supported by a VILLUM FONDEN
Investigator grant (project number 16599). Based on observations collected at
the European Organisation for Astronomical Research in the Southern Hemisphere
under ESO programmes 098.F-9300(A) and 098.D-024 (A). Based on observations
with the Atacama Pathfinder EXperiment (APEX) telescope. APEX is a
collaboration between the Max Planck Institute for Radio Astronomy, the
European Southern Observatory, and the Onsala Space Observatory. Swedish
observations on APEX are supported through Swedish Research Council grant No
2017-00648. This work is based on observations carried out under project
number 172-16 with the IRAM 30m telescope. IRAM is supported by INSU/CNRS
(France), MPG (Germany) and IGN (Spain). The research leading to these results
has received funding from the European Union’s Horizon 2020 research and
innovation program under grant agreement No 730562 [RadioNet]. This research
has made use of the GHostS database (www.GRBhosts.org), which is partly funded
by Spitzer/NASA grant RSA Agreement No. 1287913; and the NASA’s Astrophysics
Data System Bibliographic Services.

APEX (Güsten et al., 2006), IRAM-30m (Carter et al., 2012)

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