# The Gas Accretion Rate of Star-forming Galaxies over the last 4 Gyr

Apurba Bera  International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA 6102, Australia  Inter-University Centre for Astronomy
and Astrophysics, Pune 411007, India  National Centre for Radio Astrophysics,
Tata Institute of Fundamental Research, Pune 411007, India  Nissim Kanekar
National Centre for Radio Astrophysics, Tata Institute of Fundamental
Research, Pune 411007, India  Jayaram N. Chengalur  National Centre for Radio
Astrophysics, Tata Institute of Fundamental Research, Pune 411007, India
Jasjeet S. Bagla  Indian Institute of Science Education and Research Mohali,
Knowledge City, Sector 81, Sahibzada Ajit Singh Nagar, Punjab 140306, India

###### Abstract

Star-forming galaxies are believed to replenish their atomic gas reservoir,
which is consumed in star-formation, through accretion of gas from their
circumgalactic mediums (CGMs). However, there are few observational
constraints today on the gas accretion rate in external galaxies. Here, we use
our recent measurement of the scaling relation between the atomic hydrogen
(Hi) mass $\rm{M_{HI}}$ and the stellar mass $\rm{M_{*}}$ in star-forming
galaxies at $z\approx 0.35$, with the relations between the star-formation
rate (SFR) and $\rm{M_{*}}$, and the molecular gas mass $\rm{M_{Mol}}$ and
$\rm{M_{*}}$, and the assumption that star-forming galaxies evolve along the
main sequence, to determine the evolution of the neutral gas reservoir and the
average net gas accretion rate onto the disks of star-forming galaxies over
the past 4 Gyr. For galaxies with $\rm{M_{*}}\gtrsim 10^{9}\,{M_{\odot}}$
today, we find that both $\rm{M_{*}}$ and $\rm{M_{HI}}$ in the disk have
increased, while $\rm{M_{Mol}}$ has decreased, since $z\approx 0.35$. The
average gas accretion rate onto the disk over the past 4 Gyr is similar to the
average SFR over this period, implying that main-sequence galaxies have
maintained a stable Hi reservoir, despite the consumption of gas in star-
formation. We obtain an average net gas accretion rate (over the past 4 Gyr)
of ${\approx 6\,\rm{M_{\odot}}}$ yr-1 for galaxies with the stellar mass of
the Milky Way. At low redshifts, $z\lesssim 0.4$, the reason for the decline
in the cosmic SFR density thus appears to be the inefficiency in the
conversion of atomic gas to molecular gas, rather than insufficient gas
accretion from the CGM.

Galaxy evolution — Radio spectroscopy — Neutral atomic hydrogen

##  1 Introduction

Neutral gas is the primary constituent of the interstellar medium (ISM) of
star-forming galaxies. It provides the raw material for star-formation, and is
consumed in the process. The gas reservoir is expected to be replenished
through accretion of gas from the circumgalactic medium (CGM) onto the ‘disks’
of galaxies. The accretion may occur either through cooling of the hot
virialized gas in the CGM (the “hot mode”; e.g. Rees & Ostriker, 1977; White &
Rees, 1978) or through gas inflow along cold filaments (the “cold mode”; e.g.
Binney, 1977; Birnboim & Dekel, 2003; Kereš et al., 2005). However,
observational evidence for gas accretion in external galaxies has been scarce,
partly because the inflowing gas is diffuse and difficult to detect, and
partly due to the lack of unambiguous signatures of accretion. Indirect
evidence of gas accretion onto galaxy disks has been found in several recent
studies (e.g. Cheung et al., 2016; Spring & Michałowski, 2017; Kleiner et al.,
2017; Rahmani et al., 2018; Zahedy et al., 2019). Indeed, insufficient gas
accretion to replenish the neutral gas reservoir of galaxies has been proposed
to explain the observed decline in the cosmic star-formation rate (SFR)
density at $z\lesssim 1$ (e.g. Chowdhury et al., 2020, 2022a, 2022b).

Scoville et al. (2017) introduced an interesting approach to determine the gas
accretion rate as a function of redshift, using dust continuum measurements to
infer the ISM masses of galaxies (via an assumed dust-to-gas ratio) and then
fitting for the dependence of the ISM mass on the galaxy redshift, stellar
mass, and offset from the star-forming main sequence (see also Scoville et
al., 2023). They combined the above scaling relation with the assumption of
the continuity of main-sequence evolution to infer the gas accretion rate in
main-sequence galaxies. We note that the continuity of the main sequence is a
standard assumption in the literature (e.g. Renzini, 2009; Peng et al., 2010;
Leitner, 2012a; Speagle et al., 2014; Ciesla et al., 2017), with support from
both observational evidence (e.g. Rodighiero et al., 2011) and hydrodynamical
simulations (e.g. Sparre et al., 2015; Tacchella et al., 2016). However, the
dust-to-gas ratio is known to depend critically on galaxy metallicity; the
calibration of the ISM mass is thus only applicable to massive galaxies, with
near-solar metallicity (stellar mass, $\rm M_{*}\gtrsim 2\times 10^{10}\ \rm
M_{\odot}$ at high redshifts; Scoville et al., 2017). Further, even for
galaxies with near-solar metallicity in the central regions, the dust-to-gas
ratio in the outer disk (which contains a significant fraction of the atomic
phase) would be lower than in the central regions (Draine et al., 2007). As
noted by Scoville et al. (2017, 2023), the inferred ISM masses are applicable
to the inner disks of galaxies, where the assumption of near-solar metallicity
is reasonable. The total ISM mass is hence likely to be under-estimated by
this approach.

The atomic phase (made up of mainly atomic hydrogen, Hi, and helium) is known
to dominate the neutral gas reservoir in main-sequence star-forming galaxies
at $z\approx 0$, accounting for $\gtrsim 85\%$ of the neutral gas mass (e.g.
Saintonge et al., 2017; Catinella et al., 2018). Measurements of the
dependence of the Hi mass of galaxies on the stellar mass and redshift would
thus provide a more reliable way of determining the gas accretion rate,
compared to the estimates of the total ISM mass. In the local Universe, the
dependence of the Hi mass ($\rm M_{H{\textsc{i}}}$) on the stellar mass ($\rm
M_{*}$) has been determined via studies of individual galaxies in the Hi 21 cm
line (e.g. Catinella et al., 2018; Parkash et al., 2018). Unfortunately, the
weakness of this line, the main probe of the Hi mass of galaxies, has meant
that it is very difficult to determine such Hi scaling relations at
cosmological distances via studies of individual galaxies.

We have recently applied the technique of Hi 21 cm stacking (Zwaan, 2000;
Chengalur et al., 2001) to Hi 21 cm data from a deep Giant Metrewave Radio
Telescope (GMRT) survey of the Extended Groth Strip (EGS; Bera et al., 2019,
2022) to determine the $\rm\rm M_{H{\textsc{i}}}-M_{*}$ scaling relation for
star-forming galaxies at $z\approx 0.35$ (Bera et al., 2023, see also
Sinigaglia et al. (2022); Chowdhury et al. (2022c)). In this Letter, we use
this $\rm\rm M_{H{\textsc{i}}}-M_{*}$ relation, with the main-sequence
relation, the scaling relation between molecular gas mass and stellar mass,
and the assumption of the continuity of main-sequence evolution, to study the
evolution of the different baryonic components of star-forming galaxies over
the past 4 Gyr, and to determine the average gas accretion rate over this
period.111Throughout this work, we use a flat $\Lambda$-cold dark matter
($\Lambda$CDM) cosmology, with ($\rm H_{0}$, $\rm\Omega_{m}$,
$\rm\Omega_{\Lambda})=(70$ km s-1 Mpc-1, $0.3,0.7)$. Further, all stellar mass
and SFR estimates assume a Chabrier initial mass function (Chabrier, 2003).

##  2 The evolution of the baryonic content of galaxies

The baryonic mass of a galaxy is made up of stars, atomic gas, molecular gas,
and ionized gas, with a small contribution from interstellar dust. While the
stars and the neutral atomic and molecular gas are predominantly found in the
disk of a galaxy, the ionized gas is found in both the disk and the CGM. The
baryonic content of a galaxy increases due to accretion of gas from the CGM
and the intergalactic medium, and can decrease due to gas outflows driven by
supernovae or stellar winds. The net amount of gas accreted over a given time
is the difference between the amount of gas accreted and the amount of gas
lost in outflows; we will combine these effects to describe the evolution of
the net accreted gas mass, $\rm M_{acc}$. The change in the total baryonic
content in the disk of a galaxy over a given time can then be written as

$\rm\rm M_{acc}=\Delta\rm M_{*}/{\it(1-f_{return})}+\Delta\rm
M_{mol}+\Delta\rm M_{atom}+\Delta\rm M_{ion}$ (1)

where $\Delta\rm M_{*}$, $\Delta\rm M_{atom}$, $\Delta\rm M_{mol}$, and
$\Delta\rm M_{ion}$ are the changes in, respectively, the stellar mass $\rm
M_{*}$, the atomic gas mass $\rm M_{atom}\equiv 1.38\times\rm
M_{H{\textsc{i}}}$, the molecular gas $\rm M_{mol}$, and the ionized gas mass
$\rm M_{ion}$, over this time. $\rm M_{H{\textsc{i}}}$ is the Hi mass and the
factor of 1.38 in $\rm M_{atom}$ accounts for the contribution of helium.
Finally, the factor $(1-f_{return})$ accounts for the fraction of stellar mass
that is returned to the gas phase (see below; Leitner & Kravtsov, 2011;
Scoville et al., 2017).

In the above equation, the stellar mass of a star-forming galaxy is expected
to increase with time, at the expense of the neutral gas mass, along with some
mass loss due to stellar winds and supernovae. The molecular gas mass
increases at the expense of the atomic gas mass, and decreases due to star-
formation. The neutral atomic gas mass decreases via conversion to molecular
gas, but increases through accretion onto the disk. Finally, the ionized gas
mass decreases due to conversion to the neutral atomic phase, but increases
due to stellar and supernova-driven outflows. We will neglect the ionized gas
mass in what follows, as its mass in the disk is expected to be much lower
than the neutral gas mass (e.g. Draine, 2011).

Using the main-sequence relation and its redshift evolution, and the $\rm
M_{H{\textsc{i}}}-\rm M_{*}$ and $\rm M_{mol}-\rm M_{*}$ scaling relations at
any pair of redshifts, we can determine the changes in the stellar mass, the
neutral atomic gas mass, and the neutral molecular gas mass of galaxies over
the redshift range, as a function of their stellar mass. Finally, substituting
for $\Delta\rm M_{*}$, $\Delta\rm M_{mol}$, and $\Delta\rm M_{atom}$ in
Equation 1 would yield the net gas mass accreted by galaxies between the two
redshifts, and thus the average net gas accretion rate. We will apply this
formalism to the redshift range $z\approx 0-0.35$, to determine the net
average gas accretion rate over the last 4 Gyr.

###  2.1 Stellar mass build-up along the main sequence

Star-forming galaxies are known to show a tight correlation between the SFR
and $\rm M_{*}$, known as the main sequence (e.g. Madau & Dickinson, 2014).
The main-sequence relation has been shown to exist out to $z\approx 6$ (e.g.
Popesso et al., 2023), and is known to evolve with redshift; the evolution has
been described using various parametric forms (e.g. Whitaker et al., 2012,
2014; Lee et al., 2015; Leslie et al., 2020; Popesso et al., 2023). Star-
forming field galaxies are thought to evolve along the main sequence, i.e. as
their stellar mass increases, their SFR changes accordingly to keep them on
the main sequence. The knowledge of the redshift evolution of the main-
sequence relation can hence be used to trace the stellar-mass history of
present-day main-sequence galaxies over their lifetimes (see, e.g., Renzini,
2009; Peng et al., 2010; Leitner, 2012b; Speagle et al., 2014; Scoville et
al., 2017, 2023).

Following Scoville et al. (2017, 2023), we will restrict ourselves to main-
sequence galaxies and assume the principle of continuity of main-sequence
evolution. We will ignore both major mergers222The rate of major mergers is
not significant for star-forming galaxies in the stellar mass range considered
in this work (e.g. Rodriguez-Gomez et al., 2015); the stellar mass growth is
dominated by star-formation for these galaxies (Guo & White, 2008). (which can
remove galaxies from the main sequence) and the quenching of star-formation
activity. Assuming that today’s main-sequence galaxies were also on the main
sequence 4 Gyr ago, i.e. at $z\approx 0.35$, we can use the redshift-dependent
main-sequence relation to estimate the net change in their stellar masses from
$z\approx 0.35$ to $z\approx 0$. The increase in the stellar mass of a main-
sequence galaxy from $z\approx 0.35$ to the present epoch is given by

${\rm\Delta\rm M_{*}\equiv M_{*,0}-M_{*,0.35}}=(1-f_{return}){\rm\int_{{\it
z}=0.35}^{0}{\rm SFR}(M_{*},}z)\>dz$ (2)

where $\rm M_{*,0.35}$ and $\rm M_{*,0}$ are the initial (at $z\approx 0.35$)
and final (at $z=0$) stellar masses of the galaxy, respectively, $\rm
SFR(M_{*},z)$ is the redshift-dependent main-sequence relation, and
$f_{return}$ is the fraction of stellar mass that is returned to the gas phase
via stellar winds or supernovae. The value of $f_{return}$ depends on the
initial mass function and typically lies in the range $0.27-0.41$ (see, e.g.,
Madau & Dickinson, 2014). Here, we assume $f_{return}=0.3$, applicable for a
Chabrier initial mass function (Leitner & Kravtsov, 2011)333Our results do not
change significantly if a different value of $f_{return}$, within the range
$0.27-0.41$, is assumed.. We also assume that this processed gas is not
available for further star-formation (e.g. Scoville et al., 2017). We use the
redshift-dependent main-sequence relation of Whitaker et al. (2012)444Errors
associated with the main-sequence relation have been ignored in this work.
Uncertainties in the scaling relations dominate the total errors in our final
results.,

${\rm
log[SFR]}=\alpha(z){\rm\left[log(M_{*}/M_{\odot})-10.5\right]}+\beta(z)\,,$
(3)

where $\alpha(z)=0.70-0.13z$ and $\beta(z)=0.38+1.14z-0.19z^{2}$, to determine
the stellar mass at $z\approx 0.35$ of main-sequence galaxies with present-day
stellar mass $\rm M_{*,0}$. We restrict to galaxies with $\rm M_{*,0}\geq
10^{9}\;M_{\odot}$, for which the local $\rm M_{H{\textsc{i}}}-\rm M_{*}$
scaling relation has been robustly measured today (e.g. Catinella et al.,
2018; Parkash et al., 2018). Using Equation 2, this corresponds to galaxies
with stellar masses $\gtrsim 10^{8.5}\ \rm M_{\odot}$ at $z\approx 0.35$.

Figure 1: Net changes in the average [A] stellar mass ($\rm M_{*}$), [B]
molecular gas mass ($\rm M_{\rm mol}$), and [C] atomic gas mass ($\rm M_{\rm
atom}$) in the disks of main-sequence galaxies from $z\sim 0.35$ to $z\sim 0$
are shown as functions of their present-day stellar mass ($\rm M_{*,0}$,
bottom axis) and their initial stellar mass ($\rm M_{*,0.35}$, top axis). The
shaded regions show the 68% confidence intervals.  Figure 2: Fractional
changes, with respect to their respective present day values, in the average
[A] stellar mass ($\rm\Delta M_{*}/M_{*,0}$), [B] molecular gas mass
($\rm\Delta M_{\rm mol}/M_{\rm mol,0}$), and [C] atomic gas mass ($\rm\Delta
M_{\rm atom}/M_{\rm atom,0}$) in the disks of main sequence galaxies from
$z\sim 0.35$ to $z\sim 0$ are shown as functions of their present-day stellar
mass ($\rm M_{*,0}$, bottom axis) and their initial stellar mass ($\rm
M_{*,0.35}$, top axis). The shaded regions show the 68% confidence intervals.

###  2.2 Molecular gas scaling relations

The evolution of the molecular gas content of main-sequence galaxies has been
quantified through measurements of the redshift-dependent scaling relation
between $\rm M_{mol}$ and $\rm M_{*}$ (e.g. Genzel et al., 2015; Tacconi et
al., 2018). The molecular gas mass of galaxies is typically estimated from the
CO rotational lines, the far-infrared dust continuum emission, or the $\approx
1$ mm dust continuum (see Tacconi et al., 2020, for a review). We used the
redshift-dependent scaling relation555This relation includes the contribution
of helium (Tacconi et al., 2020). connecting the ratio of the molecular gas
mass to the stellar mass, $\rm\mu_{Mol}\equiv\left[\rm M_{mol}/\rm
M_{*}\right]$ of a galaxy to its stellar mass (Tacconi et al., 2020),

${\rm log\left[\mu_{\rm Mol}\right]}=A+B\>[{\rm log}(1+z)-F]^{2}+D[{\rm
log(M_{*}/M_{\odot})}-10.7]$ (4)

where $A=0.06\pm 0.20$, $B=-3.3\pm 0.2$, $D=-0.41\pm 0.03$, and $F=0.65\pm
0.05$, to determine the molecular gas mass of a main-sequence galaxy from its
stellar mass.666Note that we assume that the offset of each galaxy from the
main sequence is zero. For each galaxy with present-day stellar mass $\rm
M_{*,0}$, we can combine Equations 3 and 4 to determine its molecular gas mass
at $z=0$ and $z=0.35$, and thus estimate the change in its molecular gas mass
$\Delta\rm M_{mol}$ between $z=0.35$ and $z=0$ from the relation

$\rm\Delta\rm
M_{mol}=M_{Mol,0}-M_{Mol,0.35}=\mu_{Mol,0}\;M_{*,0}-\mu_{Mol,0.35}\;M_{*,0.35}$
(5)

where $\rm M_{*,0.35}$ and $\rm M_{*,0}$ are again the initial (at $z=0.35$)
and final (at $z=0$) stellar masses, respectively, and $\mu_{{\rm Mol},z}$ can
be inferred from Equation 4.

We note that the galaxies used to measure the scaling relation parameters have
stellar masses in the range $\rm M_{*}=10^{9}-10^{12.2}\ \rm M_{\odot}$
(Tacconi et al., 2020). We assume that the same scaling relation is also
applicable to lower-mass galaxies, with $\rm M_{*}\approx 10^{8.5}\ \rm
M_{\odot}$, which are part of the EGS sample at $z\approx 0.35$ (Bera et al.,
2023).

###  2.3 Atomic gas scaling relations

For neutral atomic gas, the $\rm M_{H{\textsc{i}}}-\rm M_{*}$ scaling relation
is known at $z\approx 0$ from direct Hi 21 cm emission studies of individual
galaxies (e.g. Catinella et al., 2018; Parkash et al., 2018). We will use the
$\rm M_{H{\textsc{i}}}-\rm M_{*}$ relation obtained for blue, star-forming
galaxies of the xGASS sample (Catinella et al., 2018; Bera et al.,
2023)777Note that using the $\rm M_{H{\textsc{i}}}-\rm M_{*}$ scaling relation
of Parkash et al. (2018) does not significantly change our results.

${\rm log}(\rm M_{H{\textsc{i}}}/M_{\odot})=(8.934\pm 0.036)+(0.516\pm
0.030)\left[{\rm log}(M_{*}/M_{\odot})-9.0\right]\,.$ (6)

At present, there are no estimates of the $\rm M_{H{\textsc{i}}}-\rm M_{*}$
relation at cosmological distances, $z\gtrsim 0.1$, based on Hi 21 cm studies
of individual galaxies. However, we have recently used the GMRT to carry out a
deep Hi 21 cm emission survey of the EGS, which yielded an estimate of the
“mean” $\rm M_{H{\textsc{i}}}-\rm M_{*}$ relation in blue, star-forming
galaxies at $z\approx 0.35$ with $\rm M_{*}=10^{8.0}-10^{10.4}\ \rm
M_{\odot}$, based on stacking the Hi 21 cm emission from galaxies in different
stellar-mass bins (Bera et al., 2023). As noted by Bera et al. (2023), this
mean relation may be combined with an assumed lognormal scatter in the $\rm
M_{H{\textsc{i}}}-\rm M_{*}$ relation to infer the “median” $\rm
M_{H{\textsc{i}}}-\rm M_{*}$ scaling relation, which can be directly compared
to the scaling relation obtained from a fit to Hi 21 cm emission measurements
in individual galaxies. Assuming a lognormal scatter in the $\rm
M_{H{\textsc{i}}}-\rm M_{*}$ relation at $z\approx 0.35$ that is equal to that
at $z\approx 0$ in the xGASS sample, Bera et al. (2023) give the following
median $\rm M_{H{\textsc{i}}}-\rm M_{*}$ scaling relation for blue, star-
forming galaxies at $z\approx 0.35$ in the EGS,

${\rm log(\rm M_{H{\textsc{i}}}/\rm M_{\odot})=(8.977\pm 0.069)+(0.183\pm
0.104)\left[log(\rm M_{*}/\rm M_{\odot})-9.0\right]}\,.$ (7)

The net change in the atomic gas mass of a main-sequence galaxy from $z=0.35$
to $z=0$ can then be estimated using the relation

$\rm\Delta\rm M_{atom}=1.38\left[M_{\textrm{H{\sc
i}},0}(M_{*,0})-M_{\textrm{H{\sc i}},0.35}(M_{*,0.35})\right]\,,$ (8)

where $\rm M_{\textrm{H{\sc i}},0.35}$ and $\rm M_{\textrm{H{\sc i}},0}$ are
the initial (at $z=0.35$) and final (at $z=0$) Hi masses, respectively, $\rm
M_{*,0.35}$ and $\rm M_{*,0}$ are the initial and final stellar masses
respectively, and $\textrm{M}_{\textrm{H{\sc i}},z}(\textrm{M}_{*,z})$ is the
$\rm M_{H{\textsc{i}}}-\rm M_{*}$ scaling relation (i.e. Equation 7) at the
redshift $z$.

##  3 The evolution of the baryonic composition of galaxies from $z\approx
0.35$

We have used Equations 2–8 to determine the changes in the stellar mass, the
molecular gas mass, and the atomic gas mass in the disks of present-day main-
sequence galaxies between $z\approx 0.35$ and $z=0$. As noted earlier, we
assume continuity of main-sequence evolution, i.e. that the galaxies evolve
along the main sequence (Scoville et al., 2017, 2023).

Figures 1[A–C] show, respectively, the changes in the stellar mass $\Delta\rm
M_{*}$, the molecular gas mass $\Delta\rm M_{mol}$, and the atomic gas mass
$\Delta\rm M_{atom}$, over the redshift range $z=0.35$ to $z=0$, as a function
of the stellar mass of galaxies today, $\rm M_{*,0}$. Figures 2[A–C] show,
respectively, the fractional changes in the above three quantities (relative
to their present-day values) over the same period as a function of the stellar
mass today. It is clear (see Figures 1[A] and 2[A]) that the stellar masses of
today’s main-sequence galaxies have increased significantly over the past 4
Gyr. Relatively low-mass galaxies, with $\rm M_{*,0}\approx 10^{9}\ \rm
M_{\odot}$, have acquired $\approx 70$% of their current stellar mass since
$z\approx 0.35$, while high-mass galaxies, with $\rm M_{*,0}\approx 10^{10.5}\
\rm M_{\odot}$, have acquired $\approx 30$% of their present stellar mass
during this period.

Conversely, Figure 1[B] shows that the molecular gas content of all galaxies
has declined significantly over the last 4 Gyr. The fractional decline
relative to the present-day molecular gas mass is seen (in Figure 2[B]) to be
the highest for the highest-mass galaxies, with $\Delta\rm M_{mol}/\rm
M_{mol,0}\lesssim-1$ for $\rm M_{*,0}\gtrsim 10^{9.5}\ \rm M_{\odot}$. In
other words, the molecular gas reservoir of present-day main-sequence galaxies
has been steadily consumed by star-formation activity since $z\approx 0.35$.

Finally, the solid blue curves (and blue shaded regions) in Figures 1[C] and
2[C] show the evolution of the atomic gas mass and the fractional atomic gas
mass, relative to today’s atomic gas mass, of galaxies over the redshift range
$z\approx 0.35-0$. We find that $\Delta\rm M_{atom}$ is always positive,
implying a net increase in the atomic gas mass of galaxies over the last 4 Gyr
for all stellar masses. The fractional change in the atomic gas mass (relative
to the atomic gas mass today) is low ($\approx 10$%) for low-stellar-mass
galaxies (with $\rm M_{*,0}\approx 10^{9}\ \rm M_{\odot}$), but substantial
($\approx 70$%) for the highest-stellar-mass galaxies today.

Figure 3: [A] Net change in the average baryonic mass ($\rm\Delta
M_{baryon}\equiv M_{acc}$) in the disks of main-sequence galaxies from $z\sim
0.35$ to $z\sim 0$ as a function of their present-day stellar mass ($\rm
M_{*,0}$, bottom axis) and their initial stellar mass ($\rm M_{*,0.35}$, top
axis). [B] The fractional change in the average baryonic mass, with respect to
the present-day baryonic mass, as a function of the present-day stellar mass
(bottom axis) and the initial stellar mass (top axis). The shaded regions in
the figures show the 68% confidence intervals for the corresponding curves.

Figure 3[A] plots the change in the total baryonic mass $\rm\Delta M_{baryon}$
of main-sequence galaxies from $z\approx 0.35$ to $z=0$ against their stellar
mass today, $\rm M_{*,0}$. We note that $\rm\Delta M_{baryon}\equiv M_{acc}$,
the net average gas mass accreted over the last $\approx 4$ Gyr (i.e. the
difference between the gas mass accreted and the gas mass lost due to winds or
outflows). The figure shows that all galaxies with stellar mass in the range
$\approx 10^{9}-10^{10.6}\,\rm M_{\odot}$ today have increased their total
baryonic mass over the last four Gyr, with the increase being larger for
higher stellar masses. However, the fractional change in the total baryonic
mass, relative to the baryonic mass today (i.e. $\rm M_{acc}/M_{baryon,0}$),
is seen in Fig. 3[B] to be approximately constant, ${\approx 40}$%, across the
above stellar mass range.

For a galaxy of a given stellar mass today, the ratio of $\rm M_{acc}$ to the
elapsed time $\Delta t$ between any two redshifts gives the time-averaged net
gas accretion rate $\rm\dot{\langle M\rangle}_{acc}$ between the two
redshifts. Fig. 4 plots (dashed black curve) the above average net gas
accretion rate from $z\approx 0.35$ to $z\approx 0$ (i.e. over the last
$\approx 4$ Gyr) as a function of the stellar mass of galaxies today, $\rm
M_{*,0}$; the grey shaded region shows the 68% confidence interval. The dashed
red curve shows the average SFR ($\equiv\Delta\rm
M_{*}/\left[(1-f_{return})\Delta t\right]$) over the same period, again as a
function of the stellar mass today, while the dashed blue curve and blue
shaded region show the average net formation rate of molecular hydrogen (i.e.
the difference between the formation rate and the destruction rate). The red
and black curves are in good agreement, within the errors: this indicates that
the average rate of net accretion of gas onto main-sequence galaxies over the
last $\approx 4$ Gyr is sufficient to balance the average SFR in these
galaxies. Star-forming galaxies on the main sequence, with $\rm M_{*}\approx
10^{9}-10^{10.6}\,\rm M_{\odot}$ today, have thus accreted substantial amounts
of gas over the last $\approx 4$ Gyr to replenish their neutral gas reservoir,
maintaining a stable (indeed, slightly increasing) Hi reservoir, despite the
continuous consumption of Hi in the star-formation process. We emphasize that
this is very unlike the situation towards the end of the epoch of galaxy
assembly, $z\approx 1$, where Chowdhury et al. (2022b) find evidence that
insufficient gas accretion is the cause of the decline in the SFR density at
$z<1$.

Figure 4: The time-averaged SFR (dashed red curve), the average net molecular
gas formation rate (dashed blue curve) and the average net gas accretion rate
of the disks of present-day main sequence galaxies (dashed black curve) over
the past 4 Gyr are shown as functions of their present-day stellar mass ($\rm
M_{*,0}$, bottom axis) and their initial stellar mass ($\rm M_{*,0.35}$, top
axis). The dotted curves show the results for the average rates on
extrapolating the Hi scaling relation of Bera et al. (2023) to a stellar mass
of $\rm M_{*}=10^{10.7}\,\rm M_{\odot}$ at $z\approx 0.35$. The star indicates
the average net gas accretion rate for galaxies with stellar masses equal to
that of the Milky Way. The shaded regions show the 68% confidence intervals
for the average net gas accretion rate and the average net molecular gas
formation rate.

Conversely, it is clear from Fig. 4 that the average net rate of H2 formation
over $z\approx 0-0.35$ is significantly lower than both the average SFR and
the average gas accretion rate. Indeed, the net rate of molecular gas
formation is negative, indicating that the atomic-to-molecular gas conversion
does not keep pace with the conversion of molecular gas to stars. This implies
that it is the inefficient conversion of atomic hydrogen to molecular hydrogen
that is likely to be the main cause of the decline in the cosmic SFR density
over the last 4 Gyr.

We note that there could be an environmental dependence to the various scaling
relations (e.g. Cortese et al., 2011; Catinella et al., 2013). The main-
sequence and $\rm M_{mol}-\rm M_{*}$ scaling relations are predominantly based
on field galaxies (e.g. Whitaker et al., 2012; Tacconi et al., 2020). We have
used the Sloan Digital Sky Survey-DR8 group catalog of Tempel et al. (2012) to
find that roughly half of the blue xGASS galaxies used to determine the $\rm
M_{H{\textsc{i}}}-\rm M_{*}$ relation at $z\approx 0$ are field galaxies (see
also Catinella et al., 2013). We note that this estimate of $\approx 50$% of
the xGASS galaxies being field objects is likely to be a lower limit as many
of the group galaxies are in “groups” with only $2-3$ members, and may thus
well be field galaxies (Tempel et al., 2012). Similarly, Gerke et al. (2012)
have used the Voronoi-Delaunay group finder to classify DEEP2 galaxies: of the
260 EGS galaxies used to determine the $\rm M_{H{\textsc{i}}}-\rm M_{*}$
relation at $z\approx 0.35$ and that have been classified by Gerke et al.
(2012), $\approx 70$% are field galaxies (and a significant number of the
remaining systems are in groups with $2-3$ members). It thus appears unlikely
that the environmental dependence of the scaling relations would significantly
affect our results.

The stellar mass of the Milky Way today is $\rm M_{*,0}=(6.08\pm 1.14)\times
10^{10}\,\rm M_{\odot}$ (Licquia & Newman, 2015). This lies beyond the stellar
mass range at $z=0$ ($\rm M_{*,0}\approx 10^{9}-10^{10.6}\,\rm M_{\odot}$)
covered by our results. Assuming that we can extrapolate the Hi scaling
relation at $z\approx 0.35$ to a stellar mass of $\rm M_{*}=10^{10.7}\,\rm
M_{\odot}$, we can estimate the average net gas accretion rate of Milky Way-
like galaxies over the last $\approx 4$ Gyr. The results are shown as the
dotted curve in Fig. 4. We obtain an average net gas accretion rate of
$\rm\approx 6\,\rm M_{\odot}$ yr-1 (indicated by the star in the figure), over
the past 4 Gyr for main-sequence galaxies with the stellar mass of the Milky
Way. This is broadly consistent with estimates of the total gas accretion rate
onto the Milky Way (e.g. Fox et al., 2014; Richter et al., 2017).

In passing, as noted by Bera et al. (2023), we emphasize that the $\rm
M_{H{\textsc{i}}}-\rm M_{*}$ relation used here is based on a relatively small
number of galaxies, and a small cosmic volume, and could hence well be
affected by cosmic variance. The possibility of cosmic variance in this
relation would affect the present results as well. A wide-field determination
of the $\rm M_{H{\textsc{i}}}-\rm M_{*}$ relation at intermediate redshifts
would allow a better estimate of the average gas accretion rate onto the disks
of galaxies, using the approach described here.

##  4 Summary

We present a formalism to determine the evolution of the baryonic composition
of star-forming galaxies between any two redshifts, based on the main-sequence
relation between SFR and stellar mass, the scaling relation between molecular
gas mass and stellar mass, the scaling relation between atomic gas mass and
stellar mass, and the assumption that star-forming galaxies continuously
evolve along the main sequence. We apply this formalism to our recent estimate
of the $\rm M_{H{\textsc{i}}}-\rm M_{*}$ relation at $z\approx 0.35$, to
determine the average changes in the stellar, molecular gas, and atomic gas
contents of the disks of star-forming galaxies from $z\approx 0.35$ to $z=0$,
as a function of the galaxy stellar mass today, for the stellar mass range
$\rm M_{*,0}=10^{9.0}-10^{10.6}\,\rm M_{\odot}$. We find that the stellar and
atomic gas masses of today’s main-sequence galaxies have both increased since
$z\approx 0.35$, while the molecular gas masses of these galaxies have
declined over the same period. The fractional increase in the stellar masses
(relative to the present-day stellar mass) is $\approx 30-70$% , with larger
fractional increases at lower stellar masses, while the fractional increase in
the atomic gas masses (relative to the present-day atomic gas mass) is
$\approx 10-70$%, with a larger fractional increase at high atomic gas masses.

We combine the changes in the stellar mass, the molecular gas mass, and the
atomic gas mass to determine the net change in the baryonic mass of main-
sequence galaxies over the last 4 Gyr. We find that the fractional net
increase in the baryonic mass of these galaxies, relative to the present-day
baryonic mass, is $\approx 40$% for stellar masses today of $\rm\approx
10^{9}-10^{10.6}\,\rm M_{\odot}$. Finally, we determine the average net gas
accretion rate of star-forming galaxies over the last 4 Gyr, finding average
net accretion rates of $\approx 0.2-5\,\rm M_{\odot}$ yr-1, similar to the
average SFR over this period. The average net gas accretion rate for Milky
Way-like galaxies is ${\approx 6\,\rm M_{\odot}}$ yr-1 since $z\approx 0.35$.
We thus find that main-sequence galaxies accrete sufficient amounts of gas
over the last $\approx 4$ Gyr to maintain a stable (and slightly increasing)
Hi reservoir, with the gas accretion compensating for the gas consumption via
star-formation. The observed decline in the cosmic SFR density over the last
$\approx 4$ Gyr thus appears to arise due to the inefficient conversion from
Hi to H2, which does not sufficiently replenish the amount of molecular gas
consumed in the process of star-formation.

## Acknowledgments

We thank the staff of the GMRT who have made these observations possible. The
GMRT is run by the National Centre for Radio Astrophysics of the Tata
Institute of Fundamental Research. AB and NK thank Aditya Chowdhury for many
discussions on Hi 21 cm stacking that have contributed to this paper. We also
thank an anonymous referee whose detailed comments on an earlier version of
the manuscript improved the paper. NK acknowledges support from the Department
of Science and Technology via a Swarnajayanti Fellowship
(DST/SJF/PSA-01/2012-13). AB, NK, $\&$ JNC also acknowledge the Department of
Atomic Energy for funding support, under project 12-R&D-TFR-5.02-0700.

## References

  * Bera et al. (2019) Bera, A., Kanekar, N., Chengalur, J. N., & Bagla, J. S. 2019, The Astrophysical Journal, 882, L7, doi: 10.3847/2041-8213/ab3656
  * Bera et al. (2022) Bera, A., Kanekar, N., Chengalur, J. N., & Bagla, J. S. 2022, ApJ, 940, L10, doi: 10.3847/2041-8213/ac9d32
  * Bera et al. (2023) Bera, A., Kanekar, N., Chengalur, J. N., & Bagla, J. S. 2023, ApJL, in press
  * Binney (1977) Binney, J. 1977, ApJ, 215, 483, doi: 10.1086/155378
  * Birnboim & Dekel (2003) Birnboim, Y., & Dekel, A. 2003, MNRAS, 345, 349, doi: 10.1046/j.1365-8711.2003.06955.x
  * Catinella et al. (2013) Catinella, B., Schiminovich, D., Cortese, L., et al. 2013, MNRAS, 436, 34, doi: 10.1093/mnras/stt1417
  * Catinella et al. (2018) Catinella, B., Saintonge, A., Janowiecki, S., et al. 2018, MNRAS, 476, 875, doi: 10.1093/mnras/sty089
  * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763, doi: 10.1086/376392
  * Chengalur et al. (2001) Chengalur, J. N., Braun, R., & Wieringa, M. 2001, A&A, 372, 768, doi: 10.1051/0004-6361:20010547
  * Cheung et al. (2016) Cheung, E., Stark, D. V., Huang, S., et al. 2016, ApJ, 832, 182, doi: 10.3847/0004-637X/832/2/182
  * Chowdhury et al. (2022a) Chowdhury, A., Kanekar, N., & Chengalur, J. N. 2022a, ApJ, 937, 103, doi: 10.3847/1538-4357/ac7d52
  * Chowdhury et al. (2022b) —. 2022b, ApJ, 931, L34, doi: 10.3847/2041-8213/ac6de7
  * Chowdhury et al. (2022c) —. 2022c, ApJ, 941, L6, doi: 10.3847/2041-8213/ac9d8a
  * Chowdhury et al. (2020) Chowdhury, A., Kanekar, N., Chengalur, J. N., Sethi, S., & Dwarakanath, K. S. 2020, Nature, 586, 369, doi: 10.1038/s41586-020-2794-7
  * Ciesla et al. (2017) Ciesla, L., Elbaz, D., & Fensch, J. 2017, A&A, 608, A41, doi: 10.1051/0004-6361/201731036
  * Cortese et al. (2011) Cortese, L., Catinella, B., Boissier, S., Boselli, A., & Heinis, S. 2011, MNRAS, 415, 1797, doi: 10.1111/j.1365-2966.2011.18822.x
  * Draine (2011) Draine, B. T. 2011, Physics of the Interstellar and Intergalactic Medium
  * Draine et al. (2007) Draine, B. T., Dale, D. A., Bendo, G., et al. 2007, ApJ, 663, 866, doi: 10.1086/518306
  * Fox et al. (2014) Fox, A. J., Wakker, B. P., Barger, K. A., et al. 2014, ApJ, 787, 147, doi: 10.1088/0004-637X/787/2/147
  * Genzel et al. (2015) Genzel, R., Tacconi, L. J., Lutz, D., et al. 2015, ApJ, 800, 20, doi: 10.1088/0004-637X/800/1/20
  * Gerke et al. (2012) Gerke, B. F., Newman, J. A., Davis, M., et al. 2012, ApJ, 751, 50, doi: 10.1088/0004-637X/751/1/50
  * Guo & White (2008) Guo, Q., & White, S. D. M. 2008, MNRAS, 384, 2, doi: 10.1111/j.1365-2966.2007.12619.x
  * Kereš et al. (2005) Kereš, D., Katz, N., Weinberg, D. H., & Davé, R. 2005, Monthly Notices of the Royal Astronomical Society, 363, 2, doi: 10.1111/j.1365-2966.2005.09451.x
  * Kleiner et al. (2017) Kleiner, D., Pimbblet, K. A., Jones, D. H., Koribalski, B. S., & Serra, P. 2017, MNRAS, 466, 4692, doi: 10.1093/mnras/stw3328
  * Lee et al. (2015) Lee, N., Sanders, D. B., Casey, C. M., et al. 2015, ApJ, 801, 80, doi: 10.1088/0004-637X/801/2/80
  * Leitner (2012a) Leitner, S. N. 2012a, ApJ, 745, 149, doi: 10.1088/0004-637X/745/2/149
  * Leitner (2012b) —. 2012b, ApJ, 745, 149, doi: 10.1088/0004-637X/745/2/149
  * Leitner & Kravtsov (2011) Leitner, S. N., & Kravtsov, A. V. 2011, ApJ, 734, 48, doi: 10.1088/0004-637X/734/1/48
  * Leslie et al. (2020) Leslie, S. K., Schinnerer, E., Liu, D., et al. 2020, ApJ, 899, 58, doi: 10.3847/1538-4357/aba044
  * Licquia & Newman (2015) Licquia, T. C., & Newman, J. A. 2015, ApJ, 806, 96, doi: 10.1088/0004-637X/806/1/96
  * Madau & Dickinson (2014) Madau, P., & Dickinson, M. 2014, ARA&A, 52, 415, doi: 10.1146/annurev-astro-081811-125615
  * Parkash et al. (2018) Parkash, V., Brown, M. J. I., Jarrett, T. H., & Bonne, N. J. 2018, ApJ, 864, 40, doi: 10.3847/1538-4357/aad3b9
  * Peng et al. (2010) Peng, Y.-j., Lilly, S. J., Kovač, K., et al. 2010, ApJ, 721, 193, doi: 10.1088/0004-637X/721/1/193
  * Popesso et al. (2023) Popesso, P., Concas, A., Cresci, G., et al. 2023, MNRAS, 519, 1526, doi: 10.1093/mnras/stac3214
  * Rahmani et al. (2018) Rahmani, H., Péroux, C., Augustin, R., et al. 2018, MNRAS, 474, 254, doi: 10.1093/mnras/stx2726
  * Rees & Ostriker (1977) Rees, M. J., & Ostriker, J. P. 1977, MNRAS, 179, 541, doi: 10.1093/mnras/179.4.541
  * Renzini (2009) Renzini, A. 2009, MNRAS, 398, L58, doi: 10.1111/j.1745-3933.2009.00710.x
  * Richter et al. (2017) Richter, P., Nuza, S. E., Fox, A. J., et al. 2017, A&A, 607, A48, doi: 10.1051/0004-6361/201630081
  * Rodighiero et al. (2011) Rodighiero, G., Daddi, E., Baronchelli, I., et al. 2011, ApJ, 739, L40, doi: 10.1088/2041-8205/739/2/L40
  * Rodriguez-Gomez et al. (2015) Rodriguez-Gomez, V., Genel, S., Vogelsberger, M., et al. 2015, MNRAS, 449, 49, doi: 10.1093/mnras/stv264
  * Saintonge et al. (2017) Saintonge, A., Catinella, B., Tacconi, L. J., et al. 2017, ApJS, 233, 22, doi: 10.3847/1538-4365/aa97e0
  * Scoville et al. (2017) Scoville, N., Lee, N., Vanden Bout, P., et al. 2017, ApJ, 837, 150, doi: 10.3847/1538-4357/aa61a0
  * Scoville et al. (2023) Scoville, N., Faisst, A., Weaver, J., et al. 2023, ApJ, 943, 82, doi: 10.3847/1538-4357/aca1bc
  * Sinigaglia et al. (2022) Sinigaglia, F., Rodighiero, G., Elson, E., et al. 2022, ApJ, 935, L13, doi: 10.3847/2041-8213/ac85ae
  * Sparre et al. (2015) Sparre, M., Hayward, C. C., Springel, V., et al. 2015, MNRAS, 447, 3548, doi: 10.1093/mnras/stu2713
  * Speagle et al. (2014) Speagle, J. S., Steinhardt, C. L., Capak, P. L., & Silverman, J. D. 2014, ApJS, 214, 15, doi: 10.1088/0067-0049/214/2/15
  * Spring & Michałowski (2017) Spring, E. F., & Michałowski, M. J. 2017, MNRAS, 471, L101, doi: 10.1093/mnrasl/slx108
  * Tacchella et al. (2016) Tacchella, S., Dekel, A., Carollo, C. M., et al. 2016, MNRAS, 457, 2790, doi: 10.1093/mnras/stw131
  * Tacconi et al. (2020) Tacconi, L. J., Genzel, R., & Sternberg, A. 2020, ARA&A, 58, 157, doi: 10.1146/annurev-astro-082812-141034
  * Tacconi et al. (2018) Tacconi, L. J., Genzel, R., Saintonge, A., et al. 2018, ApJ, 853, 179, doi: 10.3847/1538-4357/aaa4b4
  * Tempel et al. (2012) Tempel, E., Tago, E., & Liivamägi, L. J. 2012, A&A, 540, A106, doi: 10.1051/0004-6361/201118687
  * Whitaker et al. (2012) Whitaker, K. E., van Dokkum, P. G., Brammer, G., & Franx, M. 2012, ApJ, 754, L29, doi: 10.1088/2041-8205/754/2/L29
  * Whitaker et al. (2014) Whitaker, K. E., Franx, M., Leja, J., et al. 2014, ApJ, 795, 104, doi: 10.1088/0004-637X/795/2/104
  * White & Rees (1978) White, S. D. M., & Rees, M. J. 1978, MNRAS, 183, 341, doi: 10.1093/mnras/183.3.341
  * Zahedy et al. (2019) Zahedy, F. S., Rauch, M., Chen, H.-W., et al. 2019, MNRAS, 486, 1392, doi: 10.1093/mnras/stz861
  * Zwaan (2000) Zwaan, M. A. 2000, PhD thesis, University of Groningen, The Netherlands