# GRB host galaxies with strong H2 absorption: CO-dark molecular gas at the
peak of cosmic star formation

K. E. Heintz,1,2,3 G. Björnsson,1 M. Neeleman,4 L. Christensen,2,3 J. P. U.
Fynbo,2,3 P. Jakobsson,1 J.-K. Krogager,5 T. Laskar,6 C. Ledoux,7 G.
Magdis,2,3 P. Møller,3,8 P. Noterdaeme,5 P. Schady,6 A. de Ugarte Postigo,9 F.
Valentino,2,3 & D. Watson,2,3
1Centre for Astrophysics and Cosmology, Science Institute, University of
Iceland, Dunhagi 5, 107 Reykjavík, Iceland
2Cosmic Dawn Center (DAWN), Denmark,
3Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen
N, Denmark
4Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg,
Germany
5Institut d’Astrophysique de Paris, CNRS-SU, UMR 7095, 98bis bd Arago, 75014
Paris, France
6 Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK
7European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla
19001, Santiago, Chile
8European Southern Observatory, Karl-Schwarzschildstrasse 2, D-85748 Garching
bei München, Germany
9 Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía,
s/n, E-18008, Granada, Spain E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a pilot search of CO emission in three H2-absorbing, long-duration
gamma-ray burst (GRB) host galaxies at $z\sim 2-3$. We used the Atacama Large
Millimeter/sub-millimeter Array (ALMA) to target the CO$(3-2)$ emission line
and report non-detections for all three hosts. These are used to place limits
on the host molecular gas masses, assuming a metallicity-dependent CO-to-H2
conversion factor ($\alpha_{\rm CO}$). We find, $M_{\rm mol}<3.5\times
10^{10}\,M_{\odot}$ (GRB 080607), $M_{\rm mol}<4.7\times 10^{11}\,M_{\odot}$
(GRB 120815A), and $M_{\rm mol}<8.9\times 10^{11}\,M_{\odot}$ (GRB 181020A).
The high limits on the molecular gas mass for the latter two cases are a
consequence of their low stellar masses $M_{\star}$ ($M_{\star}\lesssim
10^{8}\,M_{\odot}$) and low gas-phase metallicities ($Z\sim 0.03\,Z_{\odot}$).
The limit on the $M_{\rm mol}/M_{\star}$ ratio derived for GRB 080607,
however, is consistent with the average population of star-forming galaxies at
similar redshifts and stellar masses. We discuss the broader implications for
a metallicity-dependent CO-to-H2 conversion factor, and demonstrate that the
canonical Galactic $\alpha_{\rm CO}$, will severely underestimate the actual
molecular gas mass for all galaxies at $z>1$ with
$M_{\star}<10^{10}\,M_{\odot}$. To better quantify this we develop a simple
approach to estimate the relevant $\alpha_{\rm CO}$ factor based only on the
redshift and stellar mass of individual galaxies. The elevated conversion
factors will make these galaxies appear CO-“dark” and difficult to detect in
emission, as is the case for the majority of GRB hosts. GRB spectroscopy thus
offers a complementary approach to identify low-metallicity, star-forming
galaxies with abundant molecular gas reservoirs at high redshifts that are
otherwise missed by current ALMA surveys.

###### keywords:

gamma-ray burst: general – galaxies: high-redshift, ISM, star formation – ISM:
molecules

††pubyear: 2021††pagerange: GRB host galaxies with strong H2 absorption: CO-
dark molecular gas at the peak of cosmic star formation–GRB host galaxies with
strong H2 absorption: CO-dark molecular gas at the peak of cosmic star
formation

##  1 Introduction

Since long-duration gamma-ray bursts (GRBs) are linked to the death of massive
stars (Hjorth et al., 2003; Stanek et al., 2003; Woosley & Bloom, 2006; Cano
et al., 2017), they are expected to trace star formation through cosmic time
(Wijers et al., 1998; Kistler et al., 2009; Robertson & Ellis, 2012; Greiner
et al., 2015). GRB-selected galaxies therefore probe the underlying population
of star-forming galaxies that are not biased towards the most luminous and
massive galaxies unlike traditional emission-selected galaxy surveys.
Moreover, the short-lived optical afterglows following GRBs are so bright that
the plethora of absorption features that are imprinted from the interstellar
medium (ISM) of the GRB host on the afterglow spectrum can be studied in
detail (e.g., Jakobsson et al., 2004; Prochaska et al., 2007; Vreeswijk et
al., 2007; Fynbo et al., 2009).

Table 1: Sample properties of the H2-bearing GRB host galaxies. GRB | $z_{\rm GRB}$ |  $\log N$(H2) | $\log f_{\rm H_{2}}$ | [X/H] | $A_{V}$ | $\log M_{\star}$ | $L^{\prime}_{\rm CO(3-2)}$ | $\log M_{\rm mol}$
---|---|---|---|---|---|---|---|---
|  | (cm-2) |  |  | (mag) | ($M_{\odot}$) | ($10^{9}$ K km s-1 pc2) | ($M_{\odot}$)
080607 | 3.0363 | $21.20\pm 0.20$ | $-1.23\pm 0.24$ | $>-0.2$ | $2.58\pm 0.45$ | $10.45\pm 0.10$ | $<4.56$ | $<10.54$
120815A | 2.3582 | $20.42\pm 0.08$ | $-1.39\pm 0.09$ | $-1.45\pm 0.03$ | $0.19\pm 0.04$ | $7.90\pm 0.40$ | $<0.55$ | $<11.67$
181020A | 2.9379 | $20.40\pm 0.04$ | $-1.51\pm 0.06$ | $-1.57\pm 0.06$ | $0.27\pm 0.02$ | $7.80\pm 0.40$ | $<0.71$ | $<11.95$

After the first few afterglow spectra were obtained it was clear that GRB-host
absorption systems typically probe sightlines with the highest H i column
densities of the so-called damped Lyman-$\alpha$ absorbers (DLAs; Vreeswijk et
al., 2004; Jakobsson et al., 2006; Fynbo et al., 2009), related to their small
impact parameters. DLAs provide the most effective and detailed probe of
neutral gas in high-redshift galaxies, and contain most of the neutral gas at
high redshift (Noterdaeme et al., 2009). Given their direct link to star
formation and the very high column densities of gas typically detected in GRB
afterglow spectra, the low detection rate of molecular hydrogen H2 (from the
UV Lyman-Werner bands) was initially a puzzle (e.g., Tumlinson et al., 2007;
Ledoux et al., 2009). The first detection of H2 in a GRB absorber was observed
in the remarkable afterglow spectrum of GRB 080607 (Prochaska et al.,
2009)111Though see also the tentative detection of H2 in the afterglow of GRB
060206 reported by Fynbo et al. (2006).. Since then, eight more H2-bearing GRB
absorbers have been detected (Krühler et al., 2013; D’Elia et al., 2014; Friis
et al., 2015; Bolmer et al., 2019; Heintz et al., 2019), largely owing to the
extensive VLT/X-shooter GRB afterglow legacy survey (XS-GRB; Selsing et al.,
2019). This GRB-selected sample of star-forming galaxies provides a unique way
to study the molecular gas properties of high-$z$ galaxies in absorption.

To fully exploit the detailed information of intervening or host galaxy DLAs,
it is important to study the association with their galaxy counterparts in
emission. This has been done extensively for GRB hosts at UV to optical
wavelengths (e.g., Krühler et al., 2015; Arabsalmani et al., 2018a; Corre et
al., 2018). Similarly, surveys targeting GRB hosts at sub-mm wavelengths, in
particular the molecular emission from carbon monoxide (CO) have advanced over
the last few years (Kohno et al., 2005; Endo et al., 2007; Hatsukade et al.,
2007; Hatsukade et al., 2011, 2014; Hatsukade et al., 2019; Hatsukade et al.,
2020; Stanway et al., 2011, 2015; Arabsalmani et al., 2018b; Michałowski et
al., 2016, 2018; de Ugarte Postigo et al., 2020), largely due to the
commissioning of the Atacama Large Millimeter/submillimeter Array (ALMA).
Until now, however, only blind surveys or individual detections of CO emission
in GRB host galaxies have been carried out.

In the pilot study presented here, we target a uniformly selected sample of
GRB hosts, all at $z>2$, and identified solely on the basis of H2 in
absorption. Contrary to the majority of absorption systems in quasar
sightlines (but see Ranjan et al., 2020), we expect the bulk of the absorbing
material to probe the ISM within the host galaxy, close to luminous regions of
star-formation (e.g., Fruchter et al., 2006; Svensson et al., 2010). These
systems will thus allow us to study the molecular gas-phase in the central-
most regions of star-forming galaxies, for which information can be obtained
from the two complementary methods relying on molecular absorption and
emission features.

The present paper is structured as follows. In Sect. 2, we present the sample
criteria of our pilot study and the observational setup. Sect. 3 presents the
results in terms of the inferred molecular gas masses and how they compare to
the overall population of GRB host galaxies. In Sect. 4, we place the GRB
hosts in the context of the underlying population of star-forming galaxies and
discuss how they allow us to probe the elusive high-$z$, low-metallicity
regime. In Sect. 5, we summarize and conclude on our work.

##  2 Sample and observations

###  2.1 GRB hosts with strong H2 absorption

We targeted the three GRB-hosts with strongest H2 absorption known to date,
all at $z>2$. These are: GRB 080607 (Prochaska et al., 2009), GRB 120815A
(Krühler et al., 2013), and GRB 181020A (Heintz et al., 2019). All show H2
column densities above $N{\rm(H_{2})}>10^{20}$ cm-2 and also constitute the
GRB absorbers with the largest molecular-hydrogen fractions, $f_{\rm H_{2}}$
(Bolmer et al., 2019; Heintz et al., 2019). GRB 080607 also shows a high
absorption-derived gas-phase metallicity consistent with Solar ([X/H] $>-0.2$)
and significant dust extinction $A_{V}\sim 3$ mag. On the contrary, GRBs
120815A and 181020A show relatively low metallicities of [X/H] $\approx-1.5$
and modest extinction in the line-of-sight ($A_{V}=0.2-0.3$ mag).

The host-galaxy emission counterpart of GRB 080607 is well detected in several
bands (Chen et al., 2010; Wang et al., 2012). Here, we adopt the stellar
population properties derived by Corre et al. (2018), inferring
$\log(M_{\star}/M_{\odot})=10.45\pm 0.10$. For the other two, more recently
detected GRBs, no host-galaxy counterpart has been identified yet. For these,
we instead rely on previous work connecting DLAs to their emission
counterparts (Neeleman et al., 2013; Møller et al., 2013; Christensen et al.,
2014). Following Arabsalmani et al. (2015), we assign an impact parameter of
2.3 kpc and compute the predicted stellar masses based on the prescription by
Christensen et al. (2014, their eq. 3). This yield stellar masses of
$\log(M_{\star}/M_{\odot})=7.9\pm 0.4$ (GRB 120815A) and
$\log(M_{\star}/M_{\odot})=7.8\pm 0.4$ (GRB 181020A), where the uncertainties
are dominated by the internal scatter in $\log M_{\star}$.

###  2.2 ALMA observations

We observed the fields surrounding GRBs 080607, 120815A, and 181020A,
targeting the CO$(3-2)$ emission line as part of a dedicated ALMA Cycle 7
programme (ALMA Programme ID: 2019.1.00407.S, PI: Heintz). At the redshifts of
the GRB hosts ($z\sim 2-3$), this line falls within the ALMA band 3 receiver.
For each of the GRBs, we tuned one of the 1.875 GHz spectral windows to the
redshifted CO$(3-2)$ emission with a correlator setup yielding 960 channels
with a width of 1.95 MHz. The remaining three bands were used to detect
$\approx\,$95 GHz continuum emission in the fields. Observations were done in
a compact configuration with maximum baselines ranging between 300 and 500 m
depending on the observing block. Total on-source integration times were 3.1,
2.4, and 1.6 hours for GRBs 080607, 120815A, and 181020A, respectively.

The raw data were calibrated using the ALMA Pipeline, which is part of the
Common Astronomy Software Application package (CASA; McMullin et al., 2007).
After the initial calibration, additional manual data editing was performed
using the flagging routines within CASA. Both the continuum image as well as
the spectral cube centered around the redshifted CO$(3-2)$ line were obtained
using the task tclean within CASA by applying natural weighting to maximize
sensitivity to point sources. This resulted in spatial resolutions of
$(4.0"\times 3.6")$, $(2.3"\times 2.1")$, and $(3.8"\times 3.5")$ for GRBs
080607, 120815A, and 181020A, respectively. For all GRBs, we also Hanning-
smoothed the spectral cube to a velocity resolution of 25 and 100 km s-1. The
resultant sensitivities per 100 km s-1 are 94, 85, and $94\,\mu$Jy beam-1 for
the spectral cube and 5.8, 5.7, and $6.3\,\mu$Jy beam-1 at 92.6, 96.0, and
92.9 GHz for the continuum image, for GRBs 080607, 120815A, and 181020A,
respectively.

Figure 1: CO flux density as a function of velocity, where $v_{\rm rel}=0$ km
s-1 corresponds to the redshift of the absorbers, $z_{\rm GRB}$. None of the
spectra show detection of CO$(3-2)$ lines at the redshifts of the GRBs.

##  3 Results

We searched both the 25 and 100 km s-1 channel width ALMA spectral data cubes
for emission originating from CO($3-2$) at the relevant redshifts for the GRBs
in our sample. No line emission is detected at the position of any of the GRBs
(but see, e.g., Neeleman et al., in prep., for a detection of CO$(2-1)$ from
the strong intervening Mg ii absorber towards GRB 120815A). We also did not
detect continuum emission at the positions of any of the three GRBs, but there
appears to be continuum emission from an unrelated galaxy in the field of GRB
120815A.

We extracted 1D spectra from the ALMA spectral data cube centred on the
positions of the GRB afterglows or host galaxies, shown in Fig. 1. We derived
$3\sigma$ upper limits on the velocity-integrated flux densities of $<0.09$ Jy
km s-1 (GRB 080607), $<0.018$ Jy km s-1 (GRB 120815A), and $<0.016$ Jy km s-1
(GRB 181020A), assuming line widths for the emission-line profiles of ${\rm
FWHM}=300\,{\rm km\,s^{-1}}$ (GRB 080607) and ${\rm FWHM}=50\,{\rm
km\,s^{-1}}$ (GRBs 120815A and 181020A), appropriate for galaxies in their
given mass ranges (e.g., Tiley et al., 2016). We then derived the
corresponding CO$(3-2)$ line luminosities (following Eq. 3 from Solomon &
Vanden Bout, 2005) of $L^{\prime}_{\rm CO(3-2)}<4.56\times 10^{9}$ K km s-1
pc2 (GRB 080607), $L^{\prime}_{\rm CO(3-2)}<5.51\times 10^{8}$ K km s-1 pc2
(GRB 120815A), and $L^{\prime}_{\rm CO(3-2)}<7.12\times 10^{8}$ K km s-1 pc2
(GRB 181020A).

We converted the measured CO$(3-2)$ line luminosities into total molecular gas
masses assuming a line ratio of $r_{31}=L^{\prime}_{\rm
CO(3-2)}/L^{\prime}_{\rm CO(1-0)}=0.57$ (which is the observed average for
$z>1$ star-forming galaxies; Dessauges-Zavadsky et al., 2015) and adopting a
metallicity-dependent CO-to-H2 conversion factor:

$\alpha_{\rm CO}(Z)=4.5\times(Z/Z_{\odot})^{-1.40}~{}M_{\odot}\,({\rm
K\,km\,s}^{-1}\,{\rm pc}^{2})^{-1},$ (1)

following Heintz & Watson (2020). This $\alpha_{\rm CO}$-metallicity relation
is the average between the locally-derived (Israel, 1997; Leroy et al., 2011;
Bolatto et al., 2013; Amorín et al., 2016) and high-redshift (Genzel et al.,
2012) inferred relations, and is calibrated to galaxies at $z>1$. This yields
upper limits on the molecular gas masses of $\log(M_{\rm
mol}/M_{\odot})<10.54,11.67,$ and 11.95, for the hosts of GRBs 080607,
120815A, and 181020A, respectively. All the values derived in this section are
summarized in Table 1.

Figure 2: Molecular gas-to-stellar mass ratio for our three GRB host galaxy
sample (star-symbols). Shown are the ratios assuming a MW-like CO-to-H2
conversion factor (blue) and using the $\alpha_{\rm CO}$-metallicity relation
(eq. 1, red). Detections and upper limits for the GRB host sample from
Hatsukade et al. (2020) and quasar DLAs (Kanekar et al., 2020) are also shown.
The dashed curve shows the evolutionary track broadly characterizing
$M_{\star}>10^{10}\,M_{\odot}$ main-sequence star-forming galaxies (Geach et
al., 2011).

In Fig. 2 we show the upper limits of the molecular gas-to-stellar mass ratio
of the three GRB hosts. In each case, the molecular gas mass ratio is
determined by using either the above $\alpha_{\rm CO}$-metallicity relation
and the host absorption-derived metallicities, or by assuming a constant MW-
like conversion factor of $\alpha_{\rm CO,MW}=4.3$
$M_{\odot}\,\mathrm{(K\,km\,s^{-1}\,pc^{2})}^{-1}$ (Bolatto et al., 2013). For
comparison, we overplot the recent compilation of CO observations of GRB host
galaxies from Hatsukade et al. (2020), spanning the redshift range
$z=0.0-2.5$. In addition, we show the track that broadly characterizes the
molecular gas mass evolution of $M_{\star}>10^{10}M_{\odot}$ main-sequence
star-forming galaxies, $M_{\rm mol}/M_{\star}=0.1\times(1+z)^{2}$ (Geach et
al., 2011; Carilli & Walter, 2013).

While the ratio of molecular gas mass to stellar mass of the hosts of GRBs
120815A and 181020A are poorly constrained, the host galaxy of GRB 080607 is
only marginally consistent with that expected for typical star-forming
galaxies at similar redshifts and stellar masses. This is in stark contrast
with the population of DLAs at $z\gtrsim 2$ observed in quasar sightlines,
shown as the green symbols in Fig. 2, which overall show a significant excess
of molecular gas (Neeleman et al., 2018; Kanekar et al., 2020). Assuming a
lower CO excitation of $r_{31}=0.4$ (mostly representative of the $z\lesssim
2$ population; Boogaard et al., 2020), however, results in a consistent limit
on the molecular-to-stellar mass content. The high upper limits on the $M_{\rm
mol}/M_{\star}$ fractions of the hosts of GRBs 120815A and 181020A are mainly
due to their low metallicities requiring high CO-to-H2 conversion factors to
infer their molecular gas content. These host galaxies, however, also have
lower stellar masses than the typical star-forming galaxies probed in CO at
similar redshifts (e.g., Tacconi et al., 2013, 2018). In the next section, we
will explore these particular high-redshift, low-mass, and low-metallicity
hosts in context to the underlying field-galaxy population.

##  4 Discussion

###  4.1 Field environment of GRB hosts

Studies of absorption-selected galaxies in the line-of-sight toward bright
background quasars have revealed that many of these DLAs are found in
environments with other nearby galaxies, both at low- (Kacprzak et al., 2010;
Rahmani et al., 2018) and high-redshifts (Møller & Warren, 1993; Francis &
Hewett, 1993; Fynbo et al., 2003). These DLA counterparts have mostly been
detected based on strong rest-frame optical emission lines, or from
Lyman-$\alpha$ emission from the galaxy counterparts, but are now also being
increasingly detected in CO (Klitsch et al., 2018; Fynbo et al., 2018). With
the data presented here we can examine in an independent way whether GRB-DLAs
also appear to be part of larger galaxy complexes.

Each ALMA cube covers $\approx 45^{\prime\prime}$ subtended on the sky, which
corresponds to 350 kpc and 375 kpc at the target redshifts ($z=3$ and
$z=2.3$), respectively. We do not detect emission from CO$(3-2)$ in galaxies
within the $\approx\pm 1500$ km/s covered by the data cube, corresponding to
$z=\pm 0.02$. We thus do not find evidence for galaxy clustering in this
(albeit small) sample of CO-surveyed GRB hosts. The quasar-DLA bias towards
CO-emitting galaxy groups could partly be explained by the preferential high-
metallicity these systems were selected on. Consequently, absorption-line
features in quasar sightlines could therefore be influenced by these more
populated galaxy environments (e.g., Hamanowicz et al., 2020), rather than
tracing the line-of-sight through a single galactic disk (Fynbo et al., 2018).
GRB sightlines could therefore provide a cleaner probe of absorption-derived
quantities and correlations (e.g., Arabsalmani et al., 2015, 2018a).

###  4.2 Mass and redshift dependence of $\alpha_{\rm CO}$

One of the most promising aspects of studying the CO emission associated to
GRB hosts, is that it allows us to probe the molecular gas content in low-
metallicity galaxies which are otherwise missed by field-selected surveys. As
metallicity decreases both with increasing redshift and decreasing galaxy
mass, we expect $\alpha_{\rm CO}$ to show a strong mass and redshift
dependence. In fact, even a metallicity of $Z/Z_{\odot}\approx 10\%$ would
imply an $\alpha_{\rm CO}$ value more than an order of magnitude higher than
the average Milky Way conversion factor $\alpha_{\rm CO,MW}$. The reason why
this is still largely applied to high-redshift galaxies, is due to the
difficulty in measuring the gas-phase metallicity in low-metallicity, low-
luminosity galaxies. Below, we aim to improve on this and present a simple
relation that conveniently expresses the CO-to-H2 conversion factor, in
addition to the $M_{\rm mol}/M_{\star}$ ratio, as a function of stellar mass
and redshift.

Starting from the metallicity-dependent $\alpha_{\rm CO}$ relation (Eq. 1), we
can connect $\alpha_{\rm CO}$ directly to the galaxy stellar mass at any given
redshift via mass-metallicity relations. For the latter, Savaglio et al.
(2005) provided a convenient fit, valid for a wide mass and redshift ranges,
that has been improved upon by Genzel et al. (2015) who combined metallicity
prescriptions from four different studies. The latter paper, however,
introduces a solar-metallicity cut-off at low redshift, that is too limiting
for our purposes. Instead, we start with the results of Maiolino et al.
(2008), one of the four prescriptions used by Genzel et al. (2015), that
provide at each redshift bin a mass-metallicity relation:

$12+\log({\rm O/H})=-0.0864(\log M_{\star}-\log M_{z})^{2}+K_{0}.$ (2)

Here, $M_{\star}$ is the stellar mass and $\log M_{z}$ and $K_{0}$ are
constants in each bin, their values given in Maiolino et al. (2008, Table 5).
We fit a function of $(1+z)$ to these factors and find that $\log
M_{z}=2.59\times\log(1+z)+11.05$ and $K_{0}=8.9$, provides a good
representation of their redshift evolution. The $K_{0}$-values can also be
well fitted by a $2^{\rm nd}$-order polynomial in $z$, but this significantly
under-predicts the metallicity at $z>3$ compared to other calibrators. With
the above form for $\log M_{z}$ and a constant $K_{0}$, we allow for super-
solar metallicities and find metallicity-values that are bracketed by the
mass-metallicity calibrations of Savaglio et al. (2005) and Genzel et al.
(2015), at all but the lowest masses and lowest redshifts. We include an
uncertainty of 0.2 dex in this mass-metallicity relation, representing the
typical dispersion around this relation.

Figure 3: Metallicity as a function of redshift for four different galaxy
stellar masses. Our fit to the relations in Maiolino et al. (2008) is shown
with solid curves, while Savaglio et al. (2005) is dashed-dotted and Genzel et
al. (2015) is dashed. Note that our fit lies in between the other two at all
redshifts, except for the lowest masses at the lowest redshifts.

Figure 4: The evolution of the molecular gas mass to CO($1-0$) luminosity
ratio, $\alpha_{\rm CO}$, for regular star-forming galaxies typical galaxy
scaling relations. The left panel shows the evolution as a function of
redshift for selected galaxy stellar masses and the right panel shows the
corresponding evolution as a function of stellar mass for the given redshifts.
The shaded error region on each curve represents the combined errors from the
mass-metallicity relation and $\alpha_{\rm CO}$-metallicity relation (see main
text for more details). The average value observed in the Milky Way is shown
with the blue horizontal band in both panels.

Connecting the $\alpha_{\rm CO}$-metallicity relation with the mass-
metallicity calibration derived, we obtain a simple expression for
$\alpha_{\rm CO}$:

$\log\alpha_{\rm CO}=0.121\left(\log M_{\star}-\log M_{z}\right)^{2}+0.359,$
(3)

depending only on the stellar mass, $M_{\star}$, of a galaxy at redshift $z$.
Again, $\log M_{z}=2.59\times\log(1+z)+11.05$, representing the characteristic
turn-over stellar mass at a given redshift. These relations allow us to
estimate directly the appropriate molecular gas mass conversion factors for
main-sequence star-forming galaxies at any given redshift and stellar mass for
which the mass-metallicity calibration is accurate.

Based on this combined relation, we explore how $\alpha_{\rm CO}$ evolves in
typical star-forming galaxies as a function of redshift and stellar mass. We
show these evolutionary tracks in Fig. 4 for a set of representative redshifts
and stellar masses, namely $z=0,1,3,5$ and
$M_{\star}=10^{8},10^{9},10^{10},10^{11}\,M_{\odot}$. The shaded error regions
on each curve represent the combined errors from the typical dispersion around
the mass-metallicity relation ($\sigma_{\rm MZ}=0.2$ dex) and the scatter in
the best-fit $\alpha_{\rm CO}$-metallicity relation. In the figure, we also
compare the evolution of $\alpha_{\rm CO}$ to the average values observed in
the Milky Way and in local galaxies (i.e. equivalent to solar metallicities)
of $\alpha_{\rm CO,MW}=3.5-5.5$
$M_{\odot}\,\mathrm{(K\,km\,s^{-1}\,pc^{2})}^{-1}$ (Bolatto et al., 2013)

As suggested by the mass-metallicity relation (eq. 2), the evolution of
$\alpha_{\rm CO}$ depends strongly on both galaxy mass and redshift. We find
that the Galactic conversion factor $\alpha_{\rm CO,MW}$ is a suitable
approximation for massive galaxies, with $M_{\star}\sim 10^{11}\,M_{\odot}$,
at all redshifts in the range $z=0-5$. Similarly, galaxies at $z\sim 0$ with
stellar masses in the range $M_{\star}=10^{9}-10^{11}\,M_{\odot}$ also show
conversion factors consistent with that observed in the Milky Way within the
uncertainties. However, we do recover an overall increase in $\alpha_{\rm CO}$
at decreasing stellar masses for fixed redshifts as expected. Galaxies with
$M_{\star}<10^{9}\,M_{\odot}$ and also most galaxies at $z>1$ (except the most
massive ones, $M_{\star}\sim 10^{11}\,M_{\odot}$), i.e. the majority of GRB
hosts, show $\alpha_{\rm CO}$ values significantly higher than $\alpha_{\rm
CO,MW}$. For instance, a galaxy with $M_{\star}=10^{8}\,M_{\odot}$ at $z\sim
0$ will have a conversion factor of $\alpha_{\rm CO}=30.5$
$M_{\odot}\,\mathrm{(K\,km\,s^{-1}\,pc^{2})}^{-1}$, almost an order of
magnitude higher than the average value observed in the Milky Way. At higher
redshifts, where CO is now being increasingly detected (e.g., Tacconi et al.,
2010, 2013, 2018; Walter et al., 2011; Decarli et al., 2016; Bothwell et al.,
2017; Aravena et al., 2019; Pavesi et al., 2018; Riechers et al., 2019;
Valentino et al., 2018, 2020), even massive galaxies with $M_{\star}\sim
10^{10}\,M_{\odot}$ at $z=2.5$ will have conversion factors of $\alpha_{\rm
CO}=12.3$ $M_{\odot}\,\mathrm{(K\,km\,s^{-1}\,pc^{2})}^{-1}$, exceeding the
average Galactic value by a factor of $2-3$.

The redshift- and mass-dependent evolution of $\alpha_{\rm CO}$ can now be
used to shed light on the observed $M_{\rm mol}/M_{\star}$-redshift relation.
As discussed in the previous section, it has been shown that this ratio
approximately follows $M_{\rm mol}/M_{\star}\propto(1+z)^{2.5}$, for star-
forming galaxies (Tacconi et al., 2018). In the following analysis we assume a
constant value of $L^{\prime}_{\rm CO(1-0)}=2\times 10^{10}$
$\mathrm{K\,km\,s^{-1}\,pc^{2}}$, which is the average luminosity of galaxies
at $z\sim 1-3$ derived from the PHIBSS sample (corrected by $r_{31}$; Tacconi
et al., 2013, 2018). We caution that the CO luminosity has been found to be
strongly correlated with the stellar mass of each galaxy (e.g. Inami et al.,
2020), and that this analysis should only be treated as a simple model to
explore the evolution of $M_{\rm mol}/M_{\star}$ with redshift.

In Fig. 5, we plot the resulting $M_{\rm mol}/M_{\star}$-curves for a range of
stellar masses, $M_{\star}=10^{8},10^{9},10^{10},10^{11}\,M_{\odot}$, covering
redshifts from $z=0-3$, with the GRB hosts overplotted for reference. For the
hosts of GRBs 120815A and 181020A, the upper limits are weaker than expected
from their host stellar masses, whereas the limit on the molecular gas mass in
GRB 080607 is close to that expected from its host stellar mass. Overall, most
of the CO detections in GRB hosts do fall between the curves with
$M_{\star}=10^{10}-10^{11}\,M_{\odot}$ (Hatsukade et al., 2020), as expected
from their stellar masses, though some hosts appear to show significant
molecular deficits (in particular at $z\lesssim 1$).

Figure 5: Same as Fig. 4, but with curves of constant stellar mass
overplotted. Only the upper-limits using the $\alpha_{\rm CO}$-metallicity
relation (Eq. 1), are shown for the three GRB hosts from this work. All
plotted data points are color-coded according to stellar mass. In addition,
the dotted curve shows the expected evolution for a 0.2 dex metallicity
decrease per unit redshift, starting at $\approx 1.5\times Z_{\odot}$.

Finally, in Fig. 5, we show the expected $M_{\rm mol}/M_{\star}$ evolution
track assuming a 0.2 dex metallicity decrease per unit redshift (as observed
for absorption-selected galaxies, e.g. Rafelski et al., 2012; De Cia et al.,
2018). The trend start at $Z=1.5Z_{\odot}$ at $z=0$, and we again assume
$L^{\prime}_{\rm CO(1-0)}=2\times 10^{10}$ $\mathrm{K\,km\,s^{-1}\,pc^{2}}$
and the mass-metallicity and $\alpha_{\rm CO}$-mass relations described above
(shown as the dotted line in the figure). It is clear that this evolution
track approximately follows the relation, $M_{\rm
mol}/M_{\star}\propto(1+z)^{2.5}$, that has been considered to broadly
describe the evolution of the more massive galaxies,
$M_{\star}>10^{10}M_{\odot}$. While the stellar mass is often thought to be
the main factor regulating the metallicity of galaxies, we here show that the
metallicity evolution of typical massive star-forming galaxies might in fact
be the main driver of the observed molecular gas fraction, $M_{\rm
mol}/M_{\star}$, in galaxies through cosmic time.

##  5 Conclusions

We here presented a pilot survey targeting the CO emission counterparts of the
host galaxies of strong H2-absorbing GRBs at $z\sim 2-3$. We did not detect
the redshifted CO$(3-2)$ line in any of the three GRBs (GRBs 080607, 120815A,
and 181020A). Assuming typical line ratios observed for star-forming galaxies
at similar redshifts and a metallicity-dependent CO-to-H2 conversion factor,
we derived upper limits on the molecular gas masses of $\log(M_{\rm
mol}/M_{\star})<10.54,12.09,$ and 12.47, for the hosts of GRBs 080607,
120815A, and 181020A, respectively.

To place these H2-selected GRB host galaxies into context, we compared them to
the most recent compilation of GRB hosts with CO observations (Hatsukade et
al., 2020, and references therein). First, the systems presented here expand
the redshift range for which CO observations of GRBs have been obtained. Then,
we examined them in terms of their molecular gas and stellar mass contents.
While no strong conclusion can be inferred for the molecular-gas content of
the hosts of GRBs 120815A and 181020A, we demonstrate that the host galaxy of
GRB 080607 is globally deficient in molecules or “CO-dark”. This is surprising
given the high metallicity ([X/H] $>-0.2$) and H2 abundance ($N({\rm
H}_{2})=10^{21.2}$ cm-2) inferred from absorption, and the relatively high
stellar mass inferred from emission ($\log(M_{\star}/M_{\odot})=10.45$).

Motivated by the high inferred limits on the molecular gas masses of the hosts
of GRBs 120815A and 181020A, resulting from their low absorption-derived
metallicities and a metallicity-dependent CO-to-H2 conversion factor
$\alpha_{\rm CO}$, we derived evolutionary tracks for $\alpha_{\rm CO}$ as a
function of redshift and stellar mass. We found that while the Galactic
conversion factor $\alpha_{\rm CO,MW}$ is a suitable for massive galaxies with
$M_{\star}\sim 10^{11}\,M_{\odot}$ at $z\sim 0-5$, lower mass galaxies will
show significantly higher $\alpha_{\rm CO}$ values at all redshifts (by up to
several orders of magnitudes). This will hamper the detection probability of
CO in most star-forming galaxies at $z>1$, since even large molecular gas
reservoirs will show limited CO emission at these redshifts.

We demonstrated in the pilot survey presented here, resulting in non-
detections of CO emission from GRB host galaxies, that the most feasible way
to identify and study the molecular gas reservoirs in high-redshift, low-
metallicity galaxies is through the detection of H2 in absorption. Due to the
metallicity-dependent CO-to-H2 conversion factor $\alpha_{\rm CO}$, these
galaxies that otherwise show strong H2 absorption will be too faint to be
detected in emission using typical molecular gas tracers such as CO.

## Acknowledgements

KEH and PJ acknowledge support by a Project Grant (162948–051) from The
Icelandic Research Fund. MN acknowledges support from ERC Advanced grant
740246 (Cosmic`_`Gas). JPUF thanks the Carlsberg Foundation for support. The
Cosmic DAWN center is funded by the DNRF. PN and JKK acknowledge support from
the French Agence Nationale de la Recherche under grant ANR 17-CE31-0011-01 /
project “HIH2” (PI Noterdaeme).

## Data availability statement

The raw ALMA data is publically available through the ALMA science archive.
The reduced spectral cubes and source codes for the figures and tables
presented in this manuscript are available from the corresponding author upon
reasonable request.

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