# Atomic Gas Scaling Relations of Star-forming Galaxies at $z\approx 1$

Aditya Chowdhury  National Centre for Radio Astrophysics, Tata Institute of
Fundamental Research, Pune, India.  Nissim Kanekar  National Centre for Radio
Astrophysics, Tata Institute of Fundamental Research, Pune, India.  Jayaram N.
Chengalur  National Centre for Radio Astrophysics, Tata Institute of
Fundamental Research, Pune, India.

###### Abstract

We use the Giant Metrewave Radio Telescope (GMRT) Cold-Hi AT $z\approx 1$
(CAT$z1$) survey, a 510 hr Hi 21cm emission survey of galaxies at
$z=0.74-1.45$, to report the first measurements of atomic hydrogen (Hi)
scaling relations at $z\approx 1$. We divide our sample of 11,419 blue star-
forming galaxies at $z\approx 1$ into three stellar mass ($\textrm{M}_{*}$)
subsamples and obtain detections (at $\geq 4\sigma$ significance) of the
stacked Hi 21cm emission signal from galaxies in all three subsamples. We fit
a power-law relation to the measurements of the average Hi mass
($\textrm{M}_{\rm H{\textsc{i}}}$) in the three stellar-mass subsamples to
find that the slope of the $\textrm{M}_{\rm H{\textsc{i}}}-\textrm{M}_{*}$
relation at $z\approx 1$ is consistent with that at $z\approx 0$. However, we
find that the $\textrm{M}_{\rm H{\textsc{i}}}-\textrm{M}_{*}$ relation has
shifted downwards from $z\approx 1$ to $z\approx 0$, by a factor of $3.54\pm
0.48$. Further, we find that the Hi depletion timescales (${\rm
t_{dep,H{\textsc{i}}}}$) of galaxies in the three stellar-mass subsamples are
systematically lower than those at $z\approx 0$, by factors of $\approx 2-4$.
We divide the sample galaxies into three specific star-formation rate (sSFR)
subsamples, again obtaining $\geq 4\sigma$ detections of the stacked Hi 21cm
emission signal in all three subsamples. We find that the relation between the
ratio of Hi mass to stellar mass and the sSFR evolves between $z\approx 1$ and
$z\approx 0$. Unlike the efficiency of conversion of molecular gas to stars,
which does not evolve significantly with redshift, we find that the efficiency
with which Hi is converted to stars is much higher for star-forming galaxies
at $z\approx 1$ than those at $z\approx 0$.

Galaxy evolution — Neutral hydrogen clouds — High-$z$ galaxies

††software:  astropy (Astropy Collaboration et al., 2013)

##  1 Introduction

Measurements of the neutral atomic hydrogen (Hi) properties of galaxies as a
function of their redshift, environment, and stellar properties are important
to obtain a complete picture of galaxy evolution. In the local Universe, the
Hi properties of galaxies are known to depend on their global stellar
properties, e.g. the stellar mass ($\textrm{M}_{*}$), the star-formation rate
(SFR), etc. (see Saintonge & Catinella, 2022, for a review). Such “Hi scaling
relations” at $z\approx 0$ serve as critical benchmarks for numerical
simulations and semi-analytical models of galaxy formation and evolution (e.g.
Lagos et al., 2018; Diemer et al., 2018; Davé et al., 2019).

Unfortunately, the faintness of the Hi 21 cm line has severely hindered the
use of Hi 21 cm emission studies to probe the Hi properties of galaxies at
cosmological distances. Even very deep integrations with today’s best radio
telescopes (e.g. Jaffé et al., 2013; Catinella & Cortese, 2015; Gogate et al.,
2020) have yielded detections of Hi 21 cm emission from individual galaxies
out to only $z\approx 0.376$ (Fernández et al., 2016). Thus, until very
recently, nothing was known about the Hi properties of high-$z$ galaxies and
how the Hi properties depend on the stellar mass, the SFR, or other galaxy
properties.

The above lack of information about Hi scaling relations at high redshifts has
meant that simulations of galaxy evolution are not well constrained with
regard to gas properties beyond the local Universe. Specifically, while a
number of simulations broadly reproduce the Hi scaling relations at $z\approx
0$ (e.g. Lagos et al., 2018; Diemer et al., 2018; Davé et al., 2019), the
predictions for the evolution of these relations differ significantly (e.g.
Davé et al., 2020). Measurements of Hi scaling relations at $z\gtrsim 1$,
along with similar relations for the molecular component (e.g. Tacconi et al.,
2020), would hence provide a crucial benchmark for simulations of galaxy
evolution. Further, such Hi scaling relations at $z\approx 1$ would be useful
in estimating the individual Hi masses of galaxies at these redshifts, and the
sensitivity of upcoming Hi 21 cm surveys to both individual and stacked Hi 21
cm emission from galaxies at high redshifts (e.g. Blyth et al., 2016).

The Hi 21 cm stacking approach (Zwaan, 2000; Chengalur et al., 2001), in which
the Hi 21 cm emission signals from a large number of galaxies with accurate
spectroscopic redshifts are co-added to measure the average Hi mass of a
galaxy sample, can be used to overcome the intrinsic weakness of the Hi 21 cm
line (e.g. Lah et al., 2007; Delhaize et al., 2013; Rhee et al., 2016; Kanekar
et al., 2016; Bera et al., 2019; Sinigaglia et al., 2022). This approach has
been used to measure the global Hi properties of local Universe galaxies as a
function of their global stellar properties (e.g. Fabello et al., 2011; Brown
et al., 2015; Guo et al., 2021). The Hi scaling relations obtained from these
stacking analyses have been shown to be consistent with those derived from
individual Hi 21 cm detections (e.g. Saintonge & Catinella, 2022). It should
thus be possible to use the Hi 21 cm stacking approach to determine the Hi
scaling relations at cosmological distances (e.g. Sinigaglia et al., 2022).

Hi 21 cm stacking experiments with the Giant Metrewave Radio Telescope (GMRT)
have recently been used to measure the average Hi properties of blue star-
forming galaxies at $z\gtrsim 1$ (Chowdhury et al., 2020, 2021). These studies
have shown that star-forming galaxies at $z\approx 1$ have large Hi masses but
that the Hi reservoirs can sustain the high SFRs of the galaxies for a short
period of only $1-2$ Gyr. More recently, Chowdhury et al. (2022a) used the
GMRT Cold-Hi AT $z\approx 1$ (GMRT-CAT$z1$; Chowdhury et al., 2022b) survey, a
510 hr GMRT Hi 21 cm emission survey of the DEEP2 fields (Newman et al.,
2013), to find that the average Hi mass of star-forming galaxies declines
steeply by a factor of $\approx 3.2$ from $z\approx 1.3$ to $z\approx 1.0$,
over a period of $\approx 1$ Gyr. This is direct evidence that the the rate of
accretion of gas from the circumgalactic medium (CGM) on to galaxies at
$z\approx 1$ was insufficient to replenish their Hi reservoirs, causing a
decline in the star-formation activity of the Universe at $z\lesssim 1$.
Subsequently, Chowdhury et al. (2022c) used the GMRT-CAT$z1$ measurements of
the average Hi mass of galaxies at $z\approx 1.0$ and $z\approx 1.3$ to show
that Hi dominates the baryonic content of high-$z$ galaxies.

In this _Letter_ , we use the GMRT-CAT$z1$ survey to report, for the first
time, measurements of Hi scaling relations at $z\approx 1$, at the end of the
epoch of peak cosmic star-formation activity in the Universe.

##  2 Observations and Data Analysis

###  2.1 The GMRT-CAT$z1$ Survey

The GMRT-CAT$z1$ survey (Chowdhury et al., 2022b) used $\approx$510 hrs with
the upgraded GMRT $550-850$ MHz receivers to carry out a deep Hi 21 cm
emission survey of galaxies at $z=0.74-1.45$, in three sky fields covered by
the DEEP2 Galaxy Survey (Newman et al., 2013). The three DEEP2 fields covered
by the CAT$z1$ survey contain seven sub-fields of size $\approx
52^{\prime}\times 28^{\prime}$, each of which was covered using a single GMRT
pointing. The design, the observations, the data analysis, and the main sample
of galaxies of the GMRT-CAT$z1$ survey are described in detail in Chowdhury et
al. (2022b). We provide here a summary of the information directly relevant to
this paper.

The observations for the GMRT-CAT$z1$ survey were obtained over three GMRT
observing cycles. The data of each subfield from each observing cycle were
analysed separately. This was done to prevent systematic effects present in
the data of one cycle (e.g. low-level RFI, deconvolution errors, etc), from
affecting the quality of the data from the other cycles (see Chowdhury et al.,
2022b, for a detailed discussion). The analysis resulted in $2-3$ spectral
cubes for each of the seven DEEP2 fields. The cubes have channel widths of
$48.8$ kHz, corresponding to a velocity resolution of $18$ km s-1$-25$ km s-1,
over the redshift range $z=0.74-1.45$. The FWHM of the synthesized beams of
the spectral cubes are $4\farcs 0-7\farcs 5$ over the frequency range
$580-830$ MHz, corresponding to spatial resolutions in the range $29$ kpc$-63$
kpc111Throughout this work, we use a flat “737” Lambda-cold dark matter
cosmology, with $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and $H_{0}=70$ km
s-1 Mpc-1. for galaxies at $z=0.74-1.45$.

The GMRT-CAT$z1$ survey covers the Hi 21 cm line for 16,250 DEEP2 galaxies
with accurate spectroscopic redshifts (velocity errors $\lesssim 62$ km s-1;
Newman et al., 2013) at $z=0.74-1.45$. We excluded (i) red galaxies,
identified using a cut in the $\rm(U-B)$ vs ${\rm M_{B}}$ colour-magnitude
diagram (Willmer et al., 2006; Chowdhury et al., 2022b), (ii) radio-bright
AGNs, detected in our radio-continuum images at $>4\sigma$ significance with
rest-frame 1.4 GHz luminosities $\textrm{L}_{1.4\textrm{GHz}}\geq 2\times
10^{23}$ W Hz-1 (Condon et al., 2002), (iii) galaxies with stellar masses
$\textrm{M}_{*}<10^{9}~{}\textrm{M}_{\odot}$, and (iv) galaxies whose Hi 21 cm
subcubes were affected by discernible systematic effects (Chowdhury et al.,
2022b). This yielded a total of 11,419 blue star-forming galaxies with
$\textrm{M}_{*}\geq 10^{9}~{}\textrm{M}_{\odot}$ at $z=0.74-1.45$, the main
sample of the GMRT-CAT$z1$ survey. The survey provides a total of 28,993 Hi 21
cm subcubes for the 11,419 galaxies. The subcube of each galaxy covers a
region of $\pm 500$ kpc around the galaxy location, with a uniform spatial
resolution of 90 kpc, and a velocity range of $\pm 1500$ km s-1 around its
redshifted Hi 21 cm frequency, with a channel width of 30 km s-1. The median
spectral RMS noise on the 28,993 Hi 21 cm subcubes is $297\ \mu$Jy per 30 km
s-1 velocity channel, at a spatial resolution of 90 kpc.

We note that the average Hi 21 cm emission signal from the sample of 11,419
galaxies is consistent with being unresolved at a spatial resolution of 90 kpc
(Chowdhury et al., 2022b). Further, the compact resolution of 90 kpc ensures
that the average Hi 21 cm emission signal does not include a significant
contribution from companion galaxies in the vicinity of the target galaxies
(Chowdhury et al., 2022b).

The stellar masses of the individual DEEP2 galaxies were obtained using a
relation between the stellar mass222All stellar masses and SFRs in this work
assume a Chabrier initial mass function (IMF). Estimates in the literature
that assume a Salpeter IMF were converted a Chabrier IMF by subtracting 0.2
dex (e.g. Madau & Dickinson, 2014). and the absolute rest-frame B-band
magnitude (${\rm M_{B}}$), the rest-frame (U$-$B) colour, and the rest-frame
(B$-$V) colour (Weiner et al., 2009). The relation was calibrated using a
subset of the DEEP2 galaxies with K-band estimates of the stellar masses
(Weiner et al., 2009). The SFRs of the individual galaxies were inferred from
their ${\rm M_{B}}$ values and rest-frame (U$-$B) colours, via the SFR
calibration of Mostek et al. (2012); these authors used the SFRs of galaxies
in the Extended Groth Strip (obtained via spectral-energy distribution (SED)
fits to the rest-frame ultraviolet, optical, and near-IR photometry; Salim et
al., 2009) to derive the SFR calibration for the DEEP2 galaxies. Mostek et al.
(2012) found that the scatter between the SED SFRs of Salim et al. (2009) and
the SFRs obtained via the calibration based on the ${\rm M_{B}}$ and (U$-$B)
values is $\approx 0.2$ dex333We note that we used the SFR calibration of
Mostek et al. (2012) that relates the SFR of a galaxy to its ${\rm M_{B}}$,
(U$-$B), and (U$-$B)2 values. We divided our sample of galaxies into multiple
${\rm M_{B}}$ and (U$-$B) subsamples and, for each subsample, compared the
average SFR obtained from the Mostek et al. (2012) calibration with that
obtained from the stack of the rest-frame 1.4 GHz continuum luminosity
density. We find that the difference in SFRs from the two approaches (as a
function of colour and ${\rm M_{B}}$) is consistent with the SFR scatter of
0.2 dex obtained by Mostek et al. (2012)..

###  2.2 The Stacking Analysis

We estimate the average Hi mass and the average SFR of subsamples of galaxies
by stacking, respectively, the Hi 21 cm line luminosities and the rest-frame
1.4 GHz continuum luminosities. The procedures used in stacking the Hi 21 cm
emission signals and the rest-frame 1.4 GHz continuum emission signals are
described in detail in Chowdhury et al. (2022a, b). We provide here, for
completeness, a brief review of the procedures.

The stacked Hi 21 cm spectral cube of a given subsample of galaxies was
computed by taking a weighted-average of the individual Hi 21 cm subcubes, in
luminosity-density units, of the galaxies in the subsample. During the
stacking analysis, each Hi 21 cm subcube is treated as arising from a separate
“object”. The weights were chosen to ensure that the redshift distributions of
the different subsamples are identical; the specific choices of weights for
the different stacks are discussed in Section 3.1 and Section 3.3. For each
subsample, we then fitted a second-order polynomial to the spectra at each
spatial pixel of the stacked Hi 21 cm cube, and subtracted this out to obtain
a residual cube; the polynomial fit was performed after excluding spectral
channels in the velocity range $\pm 250$ km s-1. For each subsample, the RMS
noise at each spatial and velocity pixel of the stacked Hi 21 cm cube was
obtained via Monte Carlo simulations (Chowdhury et al., 2022b). Finally, for
each subsample, the average Hi mass was obtained from the measured velocity-
integrated Hi 21 cm line luminosity.444 Note that the quoted average Hi masses
of the different subsamples in this Letter do not include the mass
contribution of Helium. The velocity integral was carried out over a
contiguous range of central velocity channels containing emission at $\geq
1.5\sigma$ significance, after smoothing the stacked Hi 21 cm subcubes to a
velocity resolution of 90 km s-1.

The average SFR of each subsample was computed by stacking the rest-frame 1.4
GHz luminosity density of the galaxies in the subsample (e.g. White et al.,
2007; Chowdhury et al., 2022a). We used the GMRT 655 MHz radio-continuum
images of the DEEP2 subfields to extract subimages around each of the 11,419
galaxies of the full sample. We convolved all subimages to an uniform spatial
resolution of 40 kpc, regridded them to a uniform grid with $5.2$ kpc pixels
spanning $\pm 260$ kpc, and converted the flux-density values (in Jy) to rest-
frame 1.4 GHz luminosity density values (in W Hz-1), assuming a spectral index
of $\alpha=-0.8$ (Condon, 1992), with $S_{\nu}\propto\nu^{\alpha}$. The
stacked rest-frame 1.4 GHz luminosity density of a subsample of galaxies was
computed by taking a weighted-median of the individual subimages, with the
weights being the same as those used during the Hi 21 cm stacking of the
subsample. Finally, the stacked rest-frame 1.4 GHz continuum luminosity
density of a subsample of galaxies is converted to an estimate of the average
SFR of the subsample, using the relation SFR
$(\textrm{M}_{\odot}/\textrm{yr})=3.7\times 10^{-22}\times{\rm L_{1.4GHz}\
(W~{}Hz^{-1})}$ (Yun et al., 2001). The errors on our measurements of the
average SFRs include both the statistical uncertainty and a 10$\%$ flux-scale
uncertainty (Chowdhury et al., 2022b).

##  3 Results and Discussion

###  3.1 Hi Mass as a Function of Stellar Mass

We divide our sample of 11,419 galaxies (28,993 Hi 21 cm subcubes) into three
stellar-mass subsamples with $1.0\times
10^{9}~{}\textrm{M}_{\odot}<\textrm{M}_{*}\leq 6.0\times 10^{9}\
\textrm{M}_{\odot}$ (“Low”), $6.0\times 10^{9}\
\textrm{M}_{\odot}<\textrm{M}_{*}\leq 1.3\times 10^{10}\ \textrm{M}_{\odot}$
(“Intermediate”), and $\textrm{M}_{*}>1.3\times 10^{10}\ \textrm{M}_{\odot}$
(“High”)555 The stellar-mass ranges of the three subsamples were chosen such
that a clear ($\geq 4\sigma$) detection of the stacked Hi 21 cm emission
signal is obtained for each subsample. However, we emphasise that the
conclusions of this Letter do not depend on the exact choice of the stellar-
mass bins. The number of galaxies and Hi 21 cm subcubes in each subsample are
provided in Table 1. The redshift distributions of the three stellar-mass
subsamples are different (see Figure 1). We correct for this difference by
assigning weights to each Hi 21 cm subcube such that the redshift distribution
of each stellar-mass subsample is effectively identical. Specifically, the
weights ensure that the effective redshift distributions of the intermediate-
and high-stellar-mass subsamples are identical to that of the low-stellar-mass
subsample; the mean redshift of the final redshift distribution is $\langle
z\rangle=1.01$. We use these weights while computing all average quantities
for the three stellar-mass subsamples.

We separately stacked the Hi 21 cm emission and the rest-frame 1.4 GHz
continuum emission of the galaxies in the three stellar-mass subsamples,
following the procedures of Sections 2.2. Figure 2 shows the stacked Hi 21 cm
emission images, the stacked Hi 21 cm spectra, and the stacked rest-frame 1.4
GHz continuum images of the three subsamples. We obtain clear detections of
the average Hi 21 cm emission signal in all three cases, at $4.2-4.9\sigma$
statistical significance. We also detect the stacked rest-frame 1.4 GHz
continuum emission at high significance ($>28\sigma$) in all three subsamples.
The average Hi masses and the average SFRs of galaxies in the three subsamples
are listed in Table 1. We find that the average SFR and the average stellar
mass of the galaxies in the three subsamples are in excellent agreement with
the star-forming main sequence at $z\approx 1$ (see Table 1; Whitaker et al.,
2014; Chowdhury et al., 2022a).

Figure 1:  The redshift distributions of the three stellar-mass subsamples. The blue histograms show, for each stellar-mass subsample, the number of Hi 21 cm subcubes in different redshift intervals (N), obtained after normalising by the total number of subcubes in the corresponding subsample. The Hi 21 cm subcubes of each stellar-mass subsample were assigned weights such that each effective redshift distribution is identical to the redshift distribution of the low stellar-mass subsample (orange lines). The number of galaxies in the subsample is indicated in each panel, with the number of Hi 21 cm subcubes shown in parentheses. Figure 2: The average Hi 21 cm emission signal and the average rest-frame 1.4 GHz continuum emission from star-forming galaxies in the three stellar-mass subsamples. Panels [A] show the average Hi 21 cm emission images of galaxies of the three stellar-mass subsamples. The Hi 21 cm subcubes of each subsample were assigned weights such that their effective redshift distributions are identical. The circle on the bottom left of each panel indicates the 90-kpc spatial resolution of the images. The contour levels are at $-3.0\sigma$ (dashed), $+3.0\sigma$, and $+4.0\sigma$ significance. Panels [B] show the average Hi 21 cm emission spectra of the three stellar-mass subsamples. The $\pm 1\sigma$ errors on the stacked Hi 21 cm spectra are indicated by the dashed black curves. We clearly detect the stacked Hi 21 cm emission signals in all three subsamples. Panels [C] show the average rest-frame 1.4 GHz luminosity density of the galaxies in the three stellar-mass subsamples. The contour levels are at $5\sigma,\ 10\sigma,\ 20\sigma,\ 40\sigma,\ {\rm and}\ 80\sigma$ statistical significance. The circle at the bottom left of each panel indicates the 40 kpc resolution of the images. | Low | Intermediate | High
---|---|---|---
Stellar Mass Range ($\times 10^{9}\ \textrm{M}_{\odot}$) | $1.0-6.0$ | $6.0-13$ | $13-240$
Number of Hi 21 cm Subcubes | 13,954 | 8,635 | 6,404
Number of Galaxies | 5,455 | 3,422 | 2,542
Average Redshift | 1.01 | 1.01 | 1.01
Average Stellar Mass ($\times 10^{9}\ \textrm{M}_{\odot}$) | $3.3$ | $8.9$ | $25.9$
Average Hi Mass ($\times 10^{9}\ \textrm{M}_{\odot}$) | $9.5\pm 2.2$ | $20.3\pm 4.1$ | $18.2\pm 4.3$
Average SFR ($\textrm{M}_{\odot}\textrm{yr}^{-1}$) | $4.04\pm 0.43$ | $8.95\pm 0.92$ | $21.1\pm 2.1$
Main-sequence SFR ($\textrm{M}_{\odot}\textrm{yr}^{-1}$) | 3.8 | 8.9 | 17.3
Hi depletion timescale (Gyr) | $2.35\pm 0.61$ | $2.27\pm 0.52$ | $0.86\pm 0.22$
Table 1: Average properties of galaxies in the three stellar-mass subsamples. For each subsample, the rows are (1) the range of stellar masses, in units of $10^{9}\ \textrm{M}_{\odot}$, (2) the number of Hi 21 cm subcubes, (3) the number of galaxies, (4) the average redshift of the galaxies, (5) the average stellar mass of the galaxies, (6) the average Hi mass of the galaxies, measured from the stacked Hi 21 cm emission spectra of Figure 2[B], (7) the average SFR of the galaxies, measured using the stacked rest-frame 1.4 GHz luminosity densities of Figure 2[C], (8) the expected SFR at this average stellar mass, for the star-forming main sequence at $z\approx 1$ (Whitaker et al., 2014), and (9) the characteristic Hi depletion timescale, $\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle{\rm SFR}\rangle$. Note that all quantities are weighted averages, with weights such that the redshift distributions of the three stellar-mass subsamples are identical. | Low | Intermediate | High
---|---|---|---
Stellar Mass Range ($\times 10^{9}\ \textrm{M}_{\odot}$) | $1.0-6.0$ | $6.0-13$ | $13-240$
Average Stellar Mass ($\times 10^{9}\ \textrm{M}_{\odot}$) | $3.3$ | $8.9$ | $25.9$
Average Hi Mass ($\times 10^{9}\ \textrm{M}_{\odot}$) |  |  |
$z\approx 0$ | $2.7\pm 0.2$ | $4.5\pm 0.4$ | $5.9\pm 0.4$
$z\approx 1$ | $9.5\pm 2.2$ | $20.3\pm 4.1$ | $18.2\pm 4.3$
Average SFR ($\textrm{M}_{\odot}\textrm{yr}^{-1}$) |  |  |
$z\approx 0$ | $0.44\pm 0.03$ | $0.88\pm 0.07$ | $1.83\pm 0.15$
$z\approx 1$ | $4.07\pm 0.43$ | $8.93\pm 0.86$ | $21.1\pm 2.1$
Hi depletion timescale (Gyr) |  |  |
$z\approx 0$ | $6.11\pm 0.48$ | $5.12\pm 0.42$ | $3.23\pm 0.29$
$z\approx 1$ | $2.33\pm 0.60$ | $2.27\pm 0.51$ | $0.86\pm 0.26$
Table 2: A comparison of the average Hi properties of blue star-forming
galaxies at $z\approx 1$ with those of blue galaxies in the local Universe.
For galaxies in each of the three stellar-mass subsamples at both $z\approx 0$
and $z\approx 1$, the rows are (1) the range of stellar masses, in units of
$10^{9}\ \textrm{M}_{\odot}$, (2) the average stellar mass, (3) the average Hi
mass, (4) the average SFR, (5) the characteristic Hi depletion timescale,
$\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle{\rm SFR}\rangle$. The Hi and
stellar properties of the $z\approx 0$ subsamples are derived from the xGASS
survey (Catinella et al., 2018), using appropriate weights (see main text for
details). The errors on the local Universe measurements were derived using
bootstrap resampling with replacement.

We use the extended GALEX Arecibo SDSS survey (xGASS; Catinella et al., 2018)
to compare our measurements of the Hi properties of star-forming galaxies at
$z\approx 1$ to those of galaxies in the local Universe. The xGASS Survey used
the Arecibo telescope to measure the Hi masses of a stellar-mass-selected
sample of galaxies with $\textrm{M}_{*}>10^{9}\ \textrm{M}_{\odot}$ at
$z=0.01-0.05$. The stellar masses and SFRs of the xGASS galaxies used in this
work were obtained from the publicly available catalogue of the “xGASS
representative sample”. The stellar masses in this catalogue are from
Kauffmann et al. (2003) and Brinchmann et al. (2004), while the SFRs were
computed using a combination of Galex near-ultraviolet (NUV) and WISE mid-
infrared (MIR) data or via spectral energy distribution fits for galaxies for
which MIR data were not available (Catinella et al., 2018).

The main sample of the GMRT-CAT$z1$ survey consists of blue, star-forming
galaxies at $z=0.74-1.45$. In order to ensure a fair comparison between the Hi
properties of the GMRT-CAT$z1$ galaxies and those of the xGASS galaxies, we
restrict to blue galaxies, with NUV$-$r$<4$, in the xGASS sample. We divide
the xGASS galaxies into three stellar-mass subsamples, using the same “Low”,
“Intermediate”, and “High” stellar-mass ranges as for the DEEP2 galaxies.
Further, for each xGASS subsample, we use weights to ensure that the stellar-
mass distribution within the subsample is effectively identical to that of the
corresponding (Low, Intermediate, or High) subsample at $z\approx 1$. In
passing, we note that the average Hi mass of xGASS galaxies in the three
stellar-mass subsamples obtained using a cut in the SFR-$\textrm{M}_{*}$ plane
to select main-sequence galaxies is consistent with the values obtained by
selecting blue galaxies with NUV$-$r$<4$.

The average Hi masses of the blue xGASS galaxies in the three stellar-mass
sub-samples are listed in Table 2; the errors on the averages were computed
using bootstrap resampling with replacement. The table also lists, for
comparison, the GMRT-CAT$z$1 measurements of the average Hi masses of blue
galaxies in the same stellar-mass subsamples at $z\approx 1$. We find that,
across the stellar-mass range $10^{9}-2.4\times 10^{11}~{}\textrm{M}_{\odot}$,
the average Hi mass of the $z\approx 1$ galaxies is higher than that of local
Universe galaxies, by a factor of $\approx 3.1-4.5$.

We determined the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at
$z\approx 1$ by fitting a power-law relation to our measurements of the
average Hi mass of blue star-forming galaxies in the three stellar-mass
subsamples at $z\approx 1$, following the procedures in Appendix A. We find
that the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation for main-sequence
galaxies at $z\approx 1$ is

$\log\left[\rm{M_{H{\textsc{i}}}}/\textrm{M}_{\odot}\right]=(0.32\pm
0.13)\log\left[{\textrm{M}_{*,10}}\right]+(10.183\pm 0.056)\;,$ (1)

where ${\textrm{M}}_{*,10}=\textrm{M}_{*}/10^{10}~{}\textrm{M}_{\odot}$. In
order to compare the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation of blue
star-forming galaxies at $z\approx 1$ to that of blue star-forming galaxies at
$z\approx 0$, we also fitted a power-law relation, using the procedures of
Appendix A, to the measurements of $\langle\rm{M_{H{\textsc{i}}}}\rangle$ in
blue xGASS galaxies in the three stellar-mass subsamples of Table 2, with
stellar-mass distributions identical to those of the subsamples of galaxies at
$z\approx 1$. We find that the best-fit
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation for blue galaxies at
$z\approx 0$ is
$\log\left[\rm{M_{H{\textsc{i}}}}/\textrm{M}_{\odot}\right]=(0.38\pm
0.05)\log\left[{\textrm{M}_{*}}_{,10}\right]+(9.634\pm 0.019)$.666We note that
the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation for blue xGASS galaxies
obtained by fitting to the $\langle\rm{M_{H{\textsc{i}}}}\rangle$ values in
the three stellar-mass subsamples is consistent with that obtained by fitting
to $\langle\rm{M_{H{\textsc{i}}}}\rangle$ values in small $\textrm{M}_{*}$
bins, separated by 0.1 dex.

Figure 3[A] shows the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relations for
blue star-forming galaxies at $z\approx 1$ and $z\approx 0$. We find no
statistically significant evidence for an evolution in the slope of the
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation from $z\approx 1$ to
$z\approx 0$. However, we find clear evidence that the relation has shifted
downwards from $z\approx 1$ to $z\approx 0$. Specifically, our measurements
show that the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation of blue star-
forming galaxies at $z\approx 1$ lies a factor of $3.54\pm 0.48$ above the
local Universe relation.

In passing, we emphasize that the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$
relations of this Letter, at both $z\approx 0$ and $z\approx 1$, were obtained
by fitting a relation to measurements of
$\langle\rm{M_{H{\textsc{i}}}}\rangle$ in three stellar-mass subsamples. This
approach is different from that typically followed for galaxies at $z\approx
0$, where the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation is obtained by
fitting to estimates of $\langle\log\rm{M_{H{\textsc{i}}}}\rangle$ in multiple
stellar-mass subsamples (e.g. Saintonge & Catinella, 2022). The difference
arises from the fact that the averaging in a stacking analysis is carried out
on the Hi masses themselves, rather than on the logarithm of the Hi masses; in
general, the logarithm of the average value of a given quantity is not the
same as the average of the individual logarithms (e.g. Brown et al., 2015).
Care must hence be taken when comparing scaling relations obtained from
simulations with those obtained from stacking analyses such as the present
work, or when comparing the scaling relations from stacking analyses with
those based on direct measurements of $\rm{M_{H{\textsc{i}}}}$, and hence on
estimates of $\langle\log\rm{M_{H{\textsc{i}}}}\rangle$. Specifically, the
scaling relations obtained from the stacking analysis yield the mean Hi mass
at a given stellar mass. Conversely, for a log-normal distribution of Hi
masses, the scaling relations obtained from direct measurements yield the
median HI mass at a given stellar mass. Further, again for a log-normal
distribution of Hi masses with scatter $\sigma$,
$\langle\log\rm{M_{H{\textsc{i}}}}\rangle$ =
$\log\langle\rm{M_{H{\textsc{i}}}}\rangle-\frac{\ln 10}{2}\sigma^{2}$.
Assuming that the scatter of the scaling relation at $z\approx 1$ is
independent of $\textrm{M}_{*}$ and that it is equal to the scatter of $0.4$
dex measured at $z=0$ (Catinella et al., 2018), the “direct”
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation would be offset downward from
Equation 1 by 0.184 dex.

Figure 3: The Hi properties of star-forming galaxies at $z\approx 1$, as a
function of their stellar masses. The red circles in panels [A] and [B] show,
respectively, our measurements of the average HI mass,
$\langle\rm{M_{H{\textsc{i}}}}\rangle$, and the characteristic Hi depletion
timescale, $\langle{\rm
t_{dep,H{\textsc{i}}}}\rangle=\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle\textrm{SFR}\rangle$,
for star-forming galaxies at $z\approx 1$ in the three stellar-mass subsamples
of Figure 2. The blue squares indicate the same quantities for the blue xGASS
galaxies in three $\textrm{M}_{*}$ subsamples with stellar-mass distributions
identical to those of the three subsamples at $z\approx 1$. The
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at $z\approx 1$, derived by
fitting a power-law relation to our measurements of the average Hi mass in the
three stellar-mass subsamples, is shown as the green line in Panel [A], with
the green shaded region showing the $1\sigma$ error on the relation. Panel [A]
also shows the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation for blue
galaxies at $z\approx 0$ (orange line), obtained by fitting a power-law
relation to the average Hi mass of xGASS galaxies in the three
$\textrm{M}_{*}$ subsamples. In Panel [B], the green curve shows the ${\rm
t_{dep,H{\textsc{i}}}}-\textrm{M}_{*}$ relation at $z\approx 1$, derived by
combining our estimate of the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation
at $z\approx 1$ with the equation describing the star-forming main sequence at
$z\approx 1$ (Whitaker et al., 2014); the green shaded region shows the
$1\sigma$ error on the relation. The orange line in panel [B] shows an
estimate of the ${\rm t_{dep,H{\textsc{i}}}}-\textrm{M}_{*}$ relation at
$z\approx 0$ derived in a similar manner, by combining the
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at $z\approx 0$ with the
equation describing the star-forming main sequence at $z\approx 0$ (Whitaker
et al., 2012). The figure shows that blue star-forming galaxies at $z\approx
1$, with stellar masses in the range $\textrm{M}_{*}\approx
10^{9}-10^{11}~{}\textrm{M}_{\odot}$, have $\approx 3-4$ times more Hi than
blue galaxies at $z\approx 0$, but have far lower characteristic depletion
timescales, by a factor of $\approx 2-4$.

###  3.2 The Hi Depletion Timescale as a Function of Stellar Mass

The availability of cold gas regulates the star-formation activity in a
galaxy. The Hi depletion timescale (${\rm t_{dep,H{\textsc{i}}}}$), defined as
the ratio of the Hi mass of the galaxy to its SFR, quantifies the approximate
timescale for which the galaxy can sustain its current SFR, in the absence of
accretion of fresh Hi from the CGM. In other words, accretion of gas from the
CGM on a timescale of $\approx{\rm t_{dep,H{\textsc{i}}}}$ is required to
sustain the current star-formation activity of the galaxy.

We define the “characteristic” Hi depletion timescale of a sample of galaxies
as $\langle{\rm
t_{dep,H{\textsc{i}}}}\rangle\equiv\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle{\rm
SFR}\rangle$. We combined the average SFRs of galaxies in the three subsamples
with their average Hi masses to estimate the characteristic Hi depletion
timescale, $\langle{\rm
t_{dep,H{\textsc{i}}}}\rangle\equiv\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle{\rm
SFR}\rangle$, of galaxies at $z\approx 1$, as a function of their average
stellar masses. Table 1 lists the $\langle{\rm t_{dep,H{\textsc{i}}}}\rangle$
values of the galaxies in the three stellar-mass subsamples at $z\approx 1$,
while the estimates of $\langle{\rm t_{dep,H{\textsc{i}}}}\rangle$ are plotted
against the average stellar mass in Figure 3[B]. For comparison, the figure
also shows the characteristic Hi depletion timescale of the xGASS galaxies in
the same three stellar-mass subsamples, while Table 2 compares the values of
$\langle{\rm t_{dep,H{\textsc{i}}}}\rangle$ for the galaxies at $z\approx 0$
and $z\approx 1$. We find that the characteristic Hi depletion timescale of
blue star-forming galaxies at $z\approx 1$ is $\approx 2-4$ times lower than
that of similar galaxies with the same stellar mass distribution at $z\approx
0$.

In passing, we note that the “characteristic” Hi depletion timescale,
$\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle{\rm SFR}\rangle$, for a sample
of galaxies may be different from the average of the depletion timescales of
the individual galaxies, $\langle\rm{M_{H{\textsc{i}}}}/SFR\rangle$. Indeed,
for the xGASS galaxies, we find that the $\langle\rm{M_{H{\textsc{i}}}}/{\rm
SFR}\rangle$ values in the three stellar-mass subsamples are higher than the
corresponding $\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle{\rm SFR}\rangle$
values by factors of $\approx 1.2-1.6$. However, this does not affect the
results of this _Letter_ because we consistently compare the characteristic
depletion timescales of the different galaxy subsamples, at both $z\approx 1$
and $z\approx 0$.

We obtained the ${\rm t_{dep,H{\textsc{i}}}}-\textrm{M}_{*}$ relation at
$z\approx 1$ by combining our estimate of the
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at $z\approx 1$ (Equation 1)
with a relation for the star-forming main sequence at $z\approx 1$ from
Whitaker et al. (2014). These authors provide best-fitting relations to the
star-forming main sequence for the redshift ranges $z=0.5-1.0$ and
$z=1.0-1.5$; we interpolated the best-fit parameters between the two redshift
intervals to find that the main-sequence relation at $z\approx 1$ is
$\log\left[{\rm SFR}/(\textrm{M}_{\odot}{\rm
yr}^{-1})\right]=0.976+0.720\log\left[{\textrm{M}_{*}}_{,10}\right]-0.205\log\left[{\textrm{M}_{*}}_{,10}\right]^{2}$.
Combining this relation with the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$
relation of Equation 1, we find that the ${\rm
t_{dep,H{\textsc{i}}}}-\textrm{M}_{*}$ relation for main-sequence galaxies at
$z\approx 1$ is777We note that the uncertainties on the ${\rm
t_{dep,H{\textsc{i}}}}-\textrm{M}_{*}$ relation of Equation 2 are dominated by
the uncertainties on the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at
$z\approx 1$, with relatively little contribution from the uncertainties in
the main-sequence relation of Whitaker et al. (2014). The errors on the
parameters in Equation 2 were hence obtained by ignoring the uncertainties in
the main-sequence relation.:

$\log\left[{\rm t_{dep,H{\textsc{i}}}}/{\rm Gyr}\right]=(0.207\pm
0.056)+(-0.40\pm
0.13)\log\left[{\textrm{M}_{*}}_{,10}\right]+0.205\log\left[{\textrm{M}_{*}}_{,10}\right]^{2}$
(2)

We emphasise that Equation 2 was not obtained by fitting a relation to our
measurements of the characteristic Hi depletion timescale in the three
$\textrm{M}_{*}$ subsamples. However, Figure 3[B] shows that our measurements
of $\langle{\rm t_{dep,H{\textsc{i}}}}\rangle$ in the three subsamples are
consistent with the ${\rm t_{dep,H{\textsc{i}}}}-\textrm{M}_{*}$ relation of
Equation 2.

Overall, we find that blue star-forming galaxies at $z\approx 1$, with stellar
masses in the range $\textrm{M}_{*}\approx 10^{9}-2.4\times
10^{11}~{}\textrm{M}_{\odot}$ have larger Hi reservoirs than those of blue
galaxies at $z\approx 0$, by a factor of $3.54\pm 0.48$. However, the
evolution of the star-forming main-sequence by a factor of $\approx 10$ from
$z\approx 0$ to $z\approx 1$ (e.g. Whitaker et al., 2014) implies that the
characteristic Hi depletion timescales of blue star-forming galaxies at
$z\approx 1$ are lower, by factors of $\approx 2-4$, than those of local
galaxies. The results of this _Letter_ thus extend the findings of the earlier
GMRT Hi 21 cm stacking studies (Chowdhury et al., 2020, 2021, 2022a, 2022b)
that blue star-forming galaxies at $z\approx 1$ have a large average Hi mass
but a short characteristic Hi depletion timescale to the entire stellar mass
range $\textrm{M}_{*}\approx 10^{9}-2.4\times 10^{11}~{}\textrm{M}_{\odot}$.

###  3.3 The Hi Fraction as a Function of the Specific SFR

The Hi fractions (${\rm f_{\rm
H{\textsc{i}}}}\equiv\rm{M_{H{\textsc{i}}}}/\textrm{M}_{*}$) of galaxies in
the local Universe and their specific SFRs
($\textrm{sSFR}\equiv\textrm{SFR}/\textrm{M}_{*}$) are known to be correlated,
with a scatter of $\approx 0.5$ dex (Catinella et al., 2018); this is one of
the tightest atomic gas scaling relations at $z\approx 0$ (Catinella et al.,
2018). The locations of galaxies in the ${\rm f_{\rm H{\textsc{i}}}}-$sSFR
plane are indicative of the efficiency with which their Hi is being converted
to stars. In this section, we investigate the redshift evolution of the
relation between ${\rm f_{\rm H{\textsc{i}}}}$ and sSFR, for blue star-forming
galaxies, from $z\approx 1$ to $z\approx 0$.

We divide our sample of 11,419 galaxies into three sSFR subsamples with sSFR
$\leq 0.8~{}\textrm{Gyr}^{-1}$ (“Low”),
$0.8~{}\textrm{Gyr}^{-1}<~{}$sSFR$~{}\leq 1.3~{}\textrm{Gyr}^{-1}$
(“Intermediate”), and sSFR$~{}>1.3~{}\textrm{Gyr}^{-1}$ (“High”)888 The sSFR
ranges of the three subsamples were chosen such that a clear ($\geq 4\sigma$)
detection of the stacked Hi 21 cm emission signal is obtained for each
subsample. However, we emphasise that the conclusions of this section do not
depend on the exact choice of the sSFR bins.. The numbers of galaxies and Hi
21 cm subcubes in each subsample are listed in Table 3, while the redshift
distributions of the three sSFR subsamples are shown in Figure 4. The high-
sSFR subsample contains a significantly larger number of galaxies at higher
redshifts than the other two subsamples; this is primarily due to the redshift
evolution of the star-forming main sequence within our redshift coverage,
$z=0.74-1.45$ (e.g. Whitaker et al., 2014). We corrected for this difference
in the redshift distributions of the subsamples by using weights such that the
effective redshift distributions of the intermediate- and high-sSFR subsamples
are identical to that of the low-sSFR subsample. We separately stacked the Hi
21 cm subcubes of the galaxies in the three subsamples, following the
procedures of Section 2.2, using the above weights to ensure that the redshift
distributions of the three subsamples are identical.

Figure 3 shows the stacked Hi 21 cm emission images and the stacked Hi 21 cm
spectra of galaxies in the three sSFR subsamples. We obtain clear detections,
with $\approx 4.3-4.4\sigma$ statistical significance, of the average Hi 21 cm
emission signals from galaxies in the three subsamples. The average Hi mass
and the “characteristic” Hi fraction, $\langle{\rm f_{\rm
H{\textsc{i}}}}\rangle\equiv\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle\textrm{M}_{*}\rangle$,
of the galaxies in each subsample are listed in Table 3.

Figure 4:  The redshift distributions of the three sSFR subsamples. The blue histograms show, for each sSFR subsample, the number (N) of Hi 21 cm subcubes normalized by the total number of subcubes in the corresponding subsample, for the different redshift intervals. The Hi 21 cm subcubes of each sSFR subsample were assigned weights such that each effective redshift distribution is identical to the redshift distribution of the low-sSFR subsample (orange lines). The total number of galaxies in the subsample is indicated in each panel, with the number of Hi 21 cm subcubes shown in parentheses. | Low | Intermediate | High
---|---|---|---
sSFR Range ($\textrm{Gyr}^{-1}$) | $0.1-0.8$ | $0.8-1.3$ | $1.3-4.2$
Number of Hi 21 cm Subcubes | 6,975 | 6,049 | 15,969
Number of Galaxies | 2,793 | 2,417 | 6,209
Average Redshift | 0.97 | 0.97 | 0.97
Average sSFR ($\textrm{Gyr}^{-1}$) | $0.5$ | $1.1$ | $1.9$
Average Stellar Mass ($\times 10^{9}\ \textrm{M}_{\odot}$) | $20.6$ | $9.4$ | $4.5$
Average Hi Mass ($\times 10^{9}\ \textrm{M}_{\odot}$) | $15.1\pm 3.4$ | $16.4\pm 3.7$ | $9.1\pm 2.1$
Characteristic Hi Fraction | $0.73\pm 0.17$ | $1.75\pm 0.39$ | $2.02\pm 0.47$
Table 3: Average properties of galaxies in the three sSFR subsamples. For each
sSFR subsample, the rows are (1) the range of sSFR values, in units of
$\textrm{Gyr}^{-1}$, (2) the number of Hi 21 cm subcubes, (3) the number of
galaxies, (4) the average redshift, (5) the average sSFR, (6) the average
stellar mass, (7) the average Hi mass, measured from the stacked Hi 21 cm
emission spectra of Figure 5, and (8) the characteristic Hi fraction, ${\rm
f_{\rm
H{\textsc{i}}}}\equiv\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle\textrm{M}_{*}\rangle$.
Note that all quantities are weighted averages, with weights such that the
redshift distributions of the three sSFR subsamples are identical. Figure 5:
The average Hi 21 cm emission signals from star-forming galaxies in the three
sSFR subsamples. Panels [A] show the average Hi 21 cm emission images of the
three sSFR mass subsamples. The circle on the bottom left of each panel
indicates the 90-kpc spatial resolution of the images. The contour levels are
at $-3.0\sigma$ (dashed), $+3.0\sigma$, and $+4.0\sigma$ significance. Panels
[B] show the average Hi 21 cm emission spectra of the same galaxies in the
three sSFR subsamples. The $\pm 1\sigma$ errors on the stacked Hi 21 cm
spectra are indicated with dashed black curves. We clearly detect the stacked
Hi 21 cm emission signals in all three subsamples. The Hi 21 cm subcubes of
each subsample were assigned weights such that their effective redshift
distributions are identical.

Our measurements of the characteristic Hi fraction of star-forming galaxies in
the three sSFR subsamples at $z\approx 1$ are shown in Figure 6; also shown
for comparison are the characteristic Hi fractions of blue xGASS galaxies at
$z\approx 0$ (Catinella et al., 2018). Note that the average sSFR of the DEEP2
galaxies in the low-sSFR subsample is $0.5~{}\textrm{Gyr}^{-1}$, while there
are only 3 galaxies in the xGASS survey with sSFR $>0.5~{}\textrm{Gyr}^{-1}$.
This is because the main sequence evolves between $z\approx 1$ and $z\approx
0$, with the sSFR of galaxies at a fixed stellar mass being $\approx 10$ times
higher at $z\approx 1$ than at $z\approx 0$ (e.g. Whitaker et al., 2014).

The straight lines in Figure 6 are the loci of constant depletion timescales
on the ${\rm f_{\rm H{\textsc{i}}}}-\textrm{M}_{*}$ plane. The characteristic
Hi depletion timescale of main-sequence galaxies in the local Universe is
$\approx 4.5$ Gyr, with a large scatter around the mean (Saintonge et al.,
2017). Figure 6 shows that the characteristic Hi fractions and the average
sSFRs of blue xGASS galaxies at $z\approx 0$ are consistent with the
$\langle{\rm t_{dep,H{\textsc{i}}}}\rangle=4.5$ Gyr line. However, it is clear
from Figure 6 that star-forming galaxies at $z\approx 1$ do not follow the
${\rm f_{\rm H{\textsc{i}}}}-$sSFR relation of local Universe galaxies. This
is consistent with our earlier results (e.g. Chowdhury et al., 2020, 2021)
that blue star-forming galaxies at $z\approx 1$ have a low characteristic Hi
depletion timescale of $\approx 1-2$ Gyr. This evolution of the ${\rm f_{\rm
H{\textsc{i}}}}-$sSFR relation from $z\approx 1$ to $z\approx 0$ is different
from the behaviour of the molecular component: the molecular gas depletion
timescales in main-sequence galaxies are typically $\approx 0.5-0.7$ Gyr at
$z\approx 0-1.5$, with no significant evidence for redshift evolution over
$z\approx 0-1.5$ (e.g. Saintonge et al., 2017; Genzel et al., 2015).

The short Hi depletion timescale of galaxies at $z\approx 1$ (or,
equivalently, the high Hi star-forming efficiency) is indicative of a very
efficient conversion of Hi to ${\rm H_{2}}$, which then directly fuels the
high star-formation activity. The difference between local Universe galaxies
(with massive Hi reservoirs but low star-forming efficiency) and star-forming
galaxies at $z\approx 1$ may lie in the typical Hi surface densities in the
galaxies; a high Hi surface density is likely to be a requirement for
efficient conversion of Hi to ${\rm H_{2}}$ (e.g. Leroy et al., 2008). In
other words, it appears that the efficiency of conversion of Hi to stars is
different at $z\approx 1$, towards the end of the epoch of peak star-formation
activity in the Universe, from that at $z\approx 0$, with the Hi in galaxies
at $z\approx 1$ being able to fuel star-formation far more efficiently than at
$z\approx 0$. Measurements of the average Hi surface density profiles of the
GMRT-CAT$z1$ galaxies would allow one to test this hypothesis.

Figure 6: The characteristic Hi fractions of star-forming galaxies at
$z\approx 1$, as a function of their specific star-formation rates. The red
circles show our measurements of the characteristic Hi fraction,
$\langle\rm{M_{H{\textsc{i}}}}\rangle$/$\langle\textrm{M}_{*}\rangle$, of
star-forming galaxies at $z\approx 1$ in the three sSFR subsamples of Figure
5. The black squares indicate the characteristic Hi fractions of blue xGASS
galaxies at $z\approx 0$ in multiple sSFR bins. The dashed lines show the loci
of constant gas depletion timescales. The relation between the Hi fraction and
the sSFR shows clear evolution from $z\approx 1$ to $z\approx 0$, with blue
galaxies at $z\approx 0$ having a characteristic Hi depletion timescale of
$\approx 4.5$ Gyr (see also Saintonge et al., 2017) but those at $z\approx 1$
having an Hi depletion timescale of just $\approx 1.5$ Gyr.

##  4 Summary

In this _Letter_ , we report the first determinations of Hi scaling relations
of galaxies at $z\approx 1$, measuring the Hi properties of blue star-forming
galaxies at $z=0.74-1.45$ as a function of stellar mass and sSFR, based on
data from the GMRT-CAT$z$1 survey. We divided our main sample of 11,419 blue
star-forming galaxies at $z\approx 1$ into three stellar-mass subsamples and
detected the stacked Hi 21 cm emission signals from all three subsamples at
$4.3-4.9\sigma$ significance. We fitted a power-law relation for the
dependence of the average Hi mass on the average stellar mass, to obtain
$\log\left[\rm{M_{H{\textsc{i}}}}/\textrm{M}_{\odot}\right]=(0.32\pm
0.13)\log\left[{\textrm{M}_{*}}_{,10}\right]+(10.183\pm 0.056)$. We compared
the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at $z\approx 1$ to that
for blue galaxies at $z\approx 0$ to find that the slope of the
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at $z\approx 1$ is consistent
with that at $z\approx 0$. However, we find that the
$\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at $z\approx 1$ has shifted
upwards from the relation at $z\approx 0$, by a factor of $3.54\pm 0.48$. We
combined our measurements of the average Hi mass in the three stellar-mass
subsamples with measurements of their average SFRs, obtained by stacking the
rest-frame 1.4 GHz continuum emission, to obtain the characteristic Hi
depletion timescale,
$\langle\rm{M_{H{\textsc{i}}}}\rangle/\langle\textrm{SFR}\rangle$, of the
three subsamples. We find that the characteristic Hi depletion timescale of
blue star-forming galaxies at $z\approx 1$, over the stellar mass range
$\textrm{M}_{*}\approx 10^{9}-2.4\times 10^{11}~{}\textrm{M}_{\odot}$, is
$\approx 2-4$ times lower than that at $z\approx 0$, for blue galaxies with
similar stellar masses. We also divided the galaxies into three sSFR
subsamples, obtaining detections of the stacked Hi 21 cm emission signals in
all three subsamples, at $\approx 4.3-4.4\sigma$ significance. We find that
the ${\rm f_{\rm H{\textsc{i}}}}-$sSFR relation shows evidence for redshift
evolution, with galaxies at $z\approx 1$ having a lower characteristic Hi
fraction, by a factor of $\approx 3$, than what is expected from the
extrapolation of the relation at $z\approx 0$ to higher sSFR values. We thus
find that star-forming galaxies at $z\approx 1$ are able to convert their Hi
reservoirs into stars with much higher efficiency than galaxies at $z\approx
0$. This is unlike the situation for molecular gas, where the efficiency of
conversion of molecular gas to stars in main-sequence galaxies shows no
significant evolution over $z\approx 0-1.5$.

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. We thank an anonymous referee for
suggestions that improved this manuscript. NK acknowledges support from the
Department of Science and Technology via a Swarnajayanti Fellowship
(DST/SJF/PSA-01/2012-13). AC, NK, $\&$ JNC also acknowledge the Department of
Atomic Energy for funding support, under project 12-R&D-TFR-5.02-0700.

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##  Appendix A Fitting Power-law Relations to Stacked Measurements

We fitted a power-law relation of the form in Equation A1 to our measurements
of the average Hi mass in the three stellar-mass subsamples to determine the
dependence of the Hi mass of star-forming galaxies at $z\approx 1$ on their
stellar mass.

$\log\left[\rm{M_{H{\textsc{i}}}}(\alpha,\beta)/\textrm{M}_{\odot}\right]=\alpha+\beta\log\left[{\textrm{M}_{*}}_{,10}\right]\;.$
(A1)

The fitting was done via a $\chi^{2}$ minimization, taking into account the
stellar-mass distribution of the galaxies in each of the three subsamples.
Specifically, for given trial values of $\alpha$ and $\beta$, we use the
stellar masses of the 11,419 galaxies of our sample in Equation A1 to estimate
their individual Hi masses, ${\rm{M_{H{\textsc{i}}}}}(\alpha,\beta)$. Next, we
use these individual Hi masses to compute the weighted-average Hi mass of the
$i$’th subsample, $\langle{\rm{M_{H{\textsc{i}}}}}(\alpha,\beta)\rangle^{i}$,
with the weights being the same as those used to stack the Hi 21 cm emission
signals of the subsample. Through this procedure, we effectively obtain the
average Hi masses of the three subsamples as a function of $\alpha$ and
$\beta$, assuming that the $\rm{M_{H{\textsc{i}}}}-\textrm{M}_{*}$ relation at
$z\approx 1$ can be described by Equation A1. The parameters $\alpha$ and
$\beta$ are finally obtained by minimising, using a standard steepest-descent
approach999The optimization was carried out using an implementation of the
Levenberg-Marquardt algorithm in the scipy package (Virtanen et al., 2020).,
the $\chi^{2}$ given by

$\chi^{2}(\alpha,\beta)=\sum_{i=1}^{3}\left(\frac{\langle{\rm{M_{H{\textsc{i}}}}}\rangle^{i}-\langle{\rm{M_{H{\textsc{i}}}}}(\alpha,\beta)\rangle^{i}}{\sigma^{i}_{\rm{M_{H{\textsc{i}}}}}}\right)^{2}$
(A2)

In the above equation, ${\langle\rm{M_{H{\textsc{i}}}}\rangle}^{i}$ and
$\sigma^{i}_{\rm{M_{H{\textsc{i}}}}}$ are the measurement of the average Hi
mass in the $i$’th subsample and the uncertainty on the measurement,
respectively.