# The Mass-Metallicity Relation of Dwarf Galaxies at Cosmic Noon from JWST
Observations

Mingyu Li<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Astronomy, Tsinghua University, Beijing 100084, China  Zheng Cai
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Astronomy,
Tsinghua University, Beijing 100084, China  Fuyan Bian<EMAIL_ADDRESS>European Southern Observatory, Alonso de Córdova 3107, Casilla 19001,
Vitacura, Santiago 19, Chile  Xiaojing Lin<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Astronomy, Tsinghua University,
Beijing 100084, China  Zihao Li<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Astronomy, Tsinghua University, Beijing 100084, China  Yunjing
Wu<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Astronomy,
Tsinghua University, Beijing 100084, China  Steward Observatory, University of
Arizona, 933 N Cherry Ave, Tucson, AZ 85721, USA  Fengwu Sun Steward
Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ 85721, USA
Shiwu Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Astronomy, Tsinghua University, Beijing 100084, China  Emmet Golden-Marx
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Astronomy,
Tsinghua University, Beijing 100084, China  Zechang Sun<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Astronomy, Tsinghua University,
Beijing 100084, China  Siwei Zou<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Astronomy, Tsinghua University,
Beijing 100084, China  Xiaohui Fan Steward Observatory, University of Arizona,
933 N Cherry Ave, Tucson, AZ 85721, USA  Eiichi Egami Steward Observatory,
University of Arizona, 933 N Cherry Ave, Tucson, AZ 85721, USA  Stephane
Charlot Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris,
F-75014, Paris, France  Gustavo Bruzual Institute of Radio Astronomy and
Astrophysics, National Autonomous University of Mexico, San José de la Huerta
58089 Morelia, Michoacán, México  Jacopo Chevallard Department of Physics,
University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK

(Received 2022 Nov.2; Revised 2023 Aug.22; Accepted 2023 Aug.28)

###### Abstract

We present a study of the mass-metallicity relation (MZR) of 51 dwarf galaxies
($M_{\star}\approx 10^{6.5}-10^{9.5}\mathrm{M}_{\odot}$) at $z=2-3$ from the
Abell 2744 and SMACS J0723-3732 galaxy cluster fields. These dwarf galaxies
are identified and confirmed by deep JWST/NIRISS imaging and slitless grism
spectroscopic observations. By taking advantage of the superior performance of
JWST and the gravitational lensing effect, we extend the previous MZR relation
at $z=2-3$ to a much lower mass regime down by $\approx$ 2.5 orders of
magnitude as compared with previous studies. We find that the MZR has a
shallower slope at the low-mass end
($M_{\star}<10^{9}~{}\rm\mathrm{M}_{\odot}$), with a slope turnover point of
$\approx$ $10^{9}~{}\rm\mathrm{M}_{\odot}$. This implies that the dominating
feedback processes in dwarf galaxies may be different from that in massive
galaxies. From $z=3$ to $z=2$, the metallicity of the dwarf galaxies is
enhanced by $\approx 0.09$ dex for a given stellar mass, consistent with the
mild evolution found in galaxies with higher mass. Furthermore, we confirm the
existence of a fundamental metallicity relation (FMR) between the gas-phase
metallicity, stellar mass, and star formation rate in dwarf galaxies at
$z=2-3$. Our derived FMR, which has no significant redshift evolution, can be
used as a benchmark to understand the origin of the anti-correlation between
the SFR and metallicity of dwarf galaxies in the high-$z$ Universe.

Galaxies: abundances – Galaxies: ISM – ISM: abundances – Galaxies: evolution –
Galaxies: formation – Galaxies: high redshift

††journal: ApJL††facilities: JWST(NIRISS and NIRCam), HST(ACS and
WFC3)††software: Astropy (Astropy Collaboration et al. 2022), Source Extractor
(Bertin & Arnouts 1996), Grizli (Brammer et al. 2022), BEAGLE (Chevallard &
Charlot 2016)

##  1 Introduction

The mass-metallicity relation (MZR) is one of the most fundamental relations
for quantifying the chemical enrichment and gas properties of star-forming
galaxies and has been studied across cosmic time (e.g., Tremonti et al., 2004;
Erb et al., 2006; Maier et al., 2014; Steidel et al., 2014; Sanders et al.,
2015; Guo et al., 2016; Sanders et al., 2021; Papovich et al., 2022; Li et
al., 2022). In this scaling relation, the metallicity increases with stellar
mass, and galaxies at high redshifts tend to have lower metallicity than their
low-redshift counterparts for a given stellar mass. The normalization, slope,
and scatter of the MZR provide insight into the physical mechanisms regulating
galaxy evolution, including how metals have been produced and driven out of
galaxies and how pristine/metal-populated gas has been accreted into galaxies
(e.g., Peeples & Shankar, 2011; Lilly et al., 2013; Torrey et al., 2019).

Dwarf galaxies are the most numerous type of observed galaxies. They are
believed to dominate the ionizing photon budget and play the most important
role in the metal enrichment of the circum-/intergalacitic medium (CGM/IGM)
(e.g., Robertson et al., 2010; Lin et al., 2022; Wu et al., 2021), giving them
great cosmological importance. Understanding the gas properties of dwarf
galaxies and studying the evolution of gas-phase metallicity of dwarf galaxies
at a given stellar mass can provide insight into the baryonic processes that
regulate star formation and galaxy growth throughout cosmic history.
Nevertheless, to date, most studies focus on the MZR of high to intermediate
mass galaxies ($>$ 109 $\mathrm{M}_{\odot}$). At these mass ranges, there have
been multiple examples of the MZR at different redshifts haing similar slopes
(e.g., Zahid et al., 2014; Sanders et al., 2021).

At low redshift, Zahid et al. (2012) studied the MZR $z\sim 0$ using a sample
of local dwarf galaxies. They found that the scatter of MZR increases with
decreasing stellar mass. Similarly, Guo et al. (2016) identified an increase
in MZR scatter in a sample of dwarf galaxies at $0.5<z<0.7$ and tentatively
suggested a shallower MZR slope when the stellar mass is lower than
$10^{9}~{}\mathrm{M_{\odot}}$. However, the MZR has been probed in very few
high-$z$ ($z>2$), low-mass galaxies ($<$ $10^{9}\mathrm{M}_{\odot}$).
Therefore, it is unclear whether the MZR has the same slope at the low mass
end and how the MZR for dwarf galaxies evolves with cosmic time beyond $z=2$.

In this paper, we probe the MZR in a sample of 51 dwarf galaxies
($M_{\star}\approx 10^{6.5}-10^{9.5}\ \mathrm{M}_{\odot}$) at $z=2-3$ using
Near-Infrared Imager and Slitless Spectrograph (NIRISS) grism spectroscopy
mounted on James Webb Space Telescope (JWST) in the Abell 2744 and SMACS
J0723-3732 fields.

This paper is organized as follows. We briefly describe the observations,
data, and sample selection in Section 2. In Section 3 we measure and derive
the physical properties of our sample and we report the results of MZR for
dwarf galaxies at $z=2-3$. We investigate the physical interpretation of the
shape and evolution of MZR and discuss the existence of fundamental
metallicity relation in Section 4. Throughout this paper, we use AB magnitudes
(e.g., Oke & Gunn, 1983). We adopt a flat $\mathrm{\Lambda CDM}$ cosmology
with $\Omega_{\mathrm{m}}=0.3$ and
$H_{0}=70\mathrm{~{}km}\mathrm{~{}s}^{-1}\mathrm{~{}Mpc}^{-1}$ and a Chabrier
(2003) Initial Mass Function (IMF).

##  2 Data and Sample

Figure 1(a): The $1.5\arcsec\times 1.5\arcsec$ RGB (red for JWST/NIRISS F200W,
green for F150W, and blue for F115W) cutouts, extracted 1D spectra, and
corresponding 2D spectra for six galaxies in our sample. Their IDs, redshifts,
and stellar masses are labeled. The flux density of 1D spectra, $f_{\lambda}$,
is in unit of $10^{-19}\mathrm{\,erg\,s^{-1}\,cm^{-2}\,\text{\AA}^{-1}}$.
Figure 1(b): One-dimensional spectral stacks binned by redshift and stellar
mass, with each individual continuum-subtracted spectrum normalized by a de-
reddening flux of [O III] (see Section 3.3 for more details). The solid lines
show the weighted average stacked spectra, and the shaded regions indicate the
corresponding 1$\sigma$ uncertainties. The low S/N wavelength ranges located
at the edge of filter bands are masked. Emission lines are labeled as black
dashed lines. The [O III]/[O II] line ratio evolves prominently with the
stellar mass, demonstrating that galaxies with lower stellar mass tend to be
more metal-poor.

We create our galaxy sample in two cluster fields, Abell 2744 (A2744) field
and SMACS J0723-3732 (SMACS 0723). These fields are extensively covered by
JWST/NIRISS grism spectroscopy and imaging data from HST and JWST. The
wavelength range (0.9 - 2.3 $\mu$m) of the JWST/NIRISS spectroscopy of these
galaxies covers the metallicity diagnostics emission lines, including [O II],
[O III], H$\beta$ at $z=2-3$. The high signal-to-noise (S/N) of JWST/NIRISS
grism spectroscopy plus gravitational lensing allows us to detect the emission
lines of galaxies with extraordinarily low stellar masses ($\approx$
$10^{6.5}\ \mathrm{M}_{\odot}$ at $z=2-3$). The complete coverage of the
optical and IR imaging in SMACS 0723 (e.g., Pontoppidan et al., 2022; Coe et
al., 2019) and A2744 fields (e.g., Treu et al., 2022; Lotz et al., 2017) also
allows us to have an excellent estimate of the stellar mass down to $10^{6}\
\mathrm{M}_{\odot}$ at $z=2-3$, allowing for our measure of the MZR using a
sample of dwarf galaxies at $z=2-3$.

###  2.1 JWST/NIRISS Grism Spectroscopic Observations

NIRISS observations of the A2744 field were taken by the GLASS JWST ERS
program (ERS-1324: Treu et al., 2022). The integration time is
$2835\mathrm{~{}s}$ ($\simeq 47\mathrm{\,minutes})$ for pre-imaging and
$2\times 5196.5\mathrm{~{}s}~{}(\simeq 2.9\mathrm{\,hr})$ for GR150 row and
column (R+C) grism with three filters (F115W, F150W, F200W). This observation
setup provides almost continuous wavelength coverage from $1$ to
$2.2\mathrm{~{}\mu m}$ in the A2744 field.

SMACS 0723 field was targeted by the JWST early-release observation program
(ERO 2736: Pontoppidan et al., 2022). The JWST/NIRISS grism spectroscopy was
obtained in F115W (1–1.3 µm) and F200W (1.7–2.2 µm) filters. The exposure time
of the NIRISS grism observations of SMACS 0723 is $2\times
2834.5\mathrm{~{}s}~{}(\simeq 1.6\mathrm{~{}hr})$ for each filter,
respectively.

The JWST data were reduced and calibrated following the methodology of Wu et
al. (2022). The data were reduced using the standard JWST
pipeline111https://github.com/spacetelescope/jwst v1.9.4 with calibration
reference files “jwst_1041.pmap”. First, we modeled the 1/f noise (see
Schlawin et al. 2020) and removed it using the tshirt/roeba
code222https://github.com/eas342/tshirt. Then, we identified “snowball”
artifacts from cosmic rays (Rigby et al., 2022a) and masked them. We then
registered the world coordinate system of mosaicked images using the Pan-
STARRS DR1 catalog (Flewelling et al., 2020) (Gaia DR2 catalog, Gaia
Collaboration et al., 2018) for A2744 (SMACS 0723) field. Lastly, the pixel
scale of final mosaicked images was resampled to 0.03″ with pixfrac $=$ 0.8.
We reduced NIRISS grism data using the Grism Redshift & Line Analysis tool
(Grizli333https://github.com/gbrammer/grizli; Brammer et al. 2022). The 2D
grism spectra were drizzled with a pixel scale of 0.065″.

The JWST/NIRISS grism observations cover the metallicity-senstive lines [O
II]$~{}\lambda\lambda 3727,3729$, H$\beta$, and [O III]$~{}\lambda\lambda
4959,5007$, allowing for our gas-phase oxygen abundance estimate. The spectral
resolution ($R\equiv\lambda/\Delta\lambda$) of JWST/NIRISS observations is
$\approx$150, enabling the separation of the [O III]$~{}\lambda\lambda
4959,5007$ doublet.

We used Grizli to perform target detection, contamination modeling, and
spectrum extraction. First, we constructed pre-imaging mosaics, which were
used to generate catalogs of sources and associated segmentation maps. Then,
we built contamination models for all sources by forward modeling, which
eliminates both zeroth and higher-order spectra from nearby sources. Finally,
we extracted contamination-removed spectra for detected sources.

We determined galaxy redshifts by spectral template synthesis (SSP) fitting.
From the extracted sources, we generated a catalog of galaxies with robust
redshift solutions, which required only one significant solution
($\log(p(z)/\chi^{2})>1$) for redshift template fitting. Specifically, at
$z\simeq 2-3$, we expect multiple strong emission lines within the NIRISS
wavelength coverage, e.g., [O II], H$\beta$, [O III], and H$\alpha$. The
redshifts can be well fitted by the resolved [O III] doublets and the
secondary lines (e.g., [O II], H$\beta$, and H$\alpha$) allow us to perform a
more robust redshift fitting.

###  2.2 HST and JWST Photometric Imaging

Because A2744 cluster is one of the Hubble Space Telescope Frontier Fields
(HFF) program444https://archive.stsci.edu/prepds/frontier/ (e.g., Lotz et al.,
2017) and the SMAC 0723 field is part of the Reionization Lensing Cluster
Survey: RELICS555https://archive.stsci.edu/prepds/relics/ (Coe et al., 2019),
each cluster has extensive archival photometric imaging across a range of
wavelengths. Specifically, these include HST/ACS in F435W, F606W, F814W
filters, and WFC3/IR in F105W, F125W, F140W, F160W filters. In this study, we
use the high-level science products released by Space Telescope Science
Institute (STScI) with a pixel size of 30 mas.

Both A2744 and SMACS 0723 have grism pre-imaging from JWST/NIRISS as
references to extract the grism spectra (see Section 2.1). Additionally, SMACS
0723 field has deep JWST/Near Infrared Camera (NIRCam) imaging data in the
F090W, F150W, F200W, F277W, F356W, and F444W bands.

For HST imaging in the A2744 field, we used photometric results from the HFF-
DeepSpace multiple wavelength catalogs (e.g., Shipley et al., 2018; Nedkova et
al., 2021). For HST imaging in the SMACS 0723 field and JWST imaging data in
both fields, we use SExtractor (Bertin & Arnouts, 1996) to measure Kron-like
AUTO photometry of the galaxies on our custom calibrated JWST images and
public HST data product. We performed individual measurements and subsequently
crossmatched them to obtain a photometric catalog using the detection of
NIRISS F200W as the reference. Finally, we built the combined catalog with
photometry in F435W, F606W, F814W, F105W, F115W, F125W, F140W, F150W, F160W,
and F200W bands for both fields. For the SMACS 0723 field, we also include the
F090W, F277W, F356W, and F444W photometry. A 5% photometry relative
uncertainty floor is set for JWST bands to involve the systematic zero-point
error. Based on this photometric catalog, we perform spectral energy
distribution (SED) fitting to derive the physical properties of each galaxy,
as explained in Section 3.2.

###  2.3 Sample Selection

We selected star-forming galaxies based on the JWST/NIRISS wide field slitless
spectroscopy. We used the following criteria to select galaxies from the
source catalog as established in Section 2.1.

  * •

With the redshifts determined by NIRISS grism, we select galaxies at
$1.8<z<2.3$ and $2.65<z<3.4$. For the SMACS 0723 field, the absence of grism
observations in the F150W band limited us to only select galaxies at
$2.65<z<3.4$. These redshift ranges ensure that NIRISS spectra completely
cover the [O II], H$\beta$, and [O III] emission lines in the A2744 field, and
H$\beta$ and [O III] emission lines in the SMACS 0723 field. Our redshift
selection criteria also ensure these emission lines are located in the high
response wavelength interval ($>90\%$ response at filter center), which avoids
the impact of large flux calibration uncertainty. Based on this redshift
criterion, we selected 76 galaxies to be the primary sample to apply the
following criteria.

  * •

The signal-to-noise ratio of the [O III] line should be greater than ten. This
criterion was established by empirical inspection. We did not require the
signal-to-noise of secondary lines, which could lead to the Eddington
bias(Eddington, 1913). The galaxies that have secondary line detection (e.g.,
H$\beta$ or [O II]) tended to have higher metallicity (e.g., Henry et al.,
2021). To minimize this selection effect, only [O III] detection significance
was required instead. 72 galaxies satisfied this criterion.

  * •

To remove AGN contamination, we excluded galaxies that fall in the “AGN
regime” of the mass–excitation (MEx) diagram ([O III]$~{}\lambda
5007$/H$\beta$ versus $M_{\star}$) defined by Juneau et al. (2011) and
modified by Coil et al. (2015). In all galaxies, we found H$\beta$ to be at
least marginally detected ($>1.5\sigma$), so there are very few sources with
ambiguous [O III]/H$\beta$ on the MEx diagram. Eight galaxies were classified
in the AGN-like category and were removed from our sample. Because we detected
no other AGN diagnostic lines (e.g., [S II] and [N II]), the MEx diagnostic
was the best method available to remove AGN candidates from our sample.

  * •

We performed visual inspections on each individual galaxy and removed heavily
contaminated sources. Due to the challenges of extracting the spectrum of
Multi-images and arclets, we removed these sources following Mahler et al.
(2018); Bergamini et al. (2022).

Ultimately, we selected 51 galaxies, where 44 are in A2744 and seven in SMACS
0723. We show six examples of the JWST/NIRISS color composite stamps and
spectra in Figure 1(a). Among all galaxies in the sample, 27 galaxies are at
redshift $1.8<z<2.3$ ($z_{\mathrm{med}}=2.01$), and 24 galaxies are at
redshift $2.65<z<3.4$ ($z_{\mathrm{med}}=2.97$). The redshifts of 42 galaxies
are spectroscopically determined by NIRISS observations for the first time,
while the remaining nine galaxies have been previously confirmed by HST/WFC3
or VLT/MUSE (e.g., Wang et al., 2020; Mahler et al., 2018; Boyett et al.,
2022). Among the nine galaxies, the NIRISS redshifts of two galaxies (ID 1692
and ID 2225) are inconsistent with those in previous studies. For these two
galaxies, we re-checked their spectra (shown in Figure1(a)) and confirm the
reliability of the NIRISS redshifts based on the appearance of multiple
emission lines with S/N $>$ 3.

##  3 Analysis and Results

Figure 2: The distribution of galaxies in our sample on the SFR–$M_{\star}$
plane. The red squares show the $z\simeq 2$ sample and the red hexagons show
the $z\simeq 3$ sample. The MOSDEF stacked results are denoted in black
triangles. The gray shaded regions shows the (extrapolated) galaxy main
sequence at $z=2-3$ based on the results from Speagle et al. (2014), Schreiber
et al. (2015), Santini et al. (2017), Pearson et al. (2018), and Leja et al.
(2022). The origin results from these works are denoted as gray dashed lines.
The top and right panels show the stellar mass and SFR distributions
respectively, with the $z\simeq 2$ sample in red and $z\simeq 3$ sample in
blue.

Based on the spectral and photometric data, we measure and derive the galaxy
properties from the emission line intensities and the SED fitting using
multiple-band photometry. In this section, we present the methodology for the
measurement and derivation of galaxy properties and show the results of MZR.

###  3.1 Emission-line Flux

To measure the intensity of galaxy emission lines, we use the Gaussian profile
models to fit the lines in continuum-subtracted grism spectra. The line
wavelengths are fixed by the galaxy’s redshift, which is determined by the SSP
fitting as presented in Sec 2.1. For the H$\alpha$ and [O II] lines, we fit
the spectra using a single 1D Gaussian function. For the H$\beta$+[O
III]$\lambda\lambda 4959,5007$ lines, we fit the spectra using three Gaussian
functions. We fix the line ratio of the [O III] doublet to [O III]$\lambda
4959$/[O III] $\lambda 5007=1/3$ and require the line width of [O III] doublet
to be equal. We further verify the accuracy of our derived flux by comparing
our measurements to the modeled flux solution fitted by Grizli and find the
two to be consistent. A thorough summary of the best-fit observed emission
line fluxes can be found in Appendix A.

In the early era of JWST observations, data calibration cannot be completely
accurate (e.g., Rigby et al., 2022b). To minimize calibration issues, we
employed reference calibration files that have been corrected using space-
based observations. To quantify the effect of any remaining systematic errors
on our data, we compared the NIRISS-extracted spectra with previous HST/WFC3
grism observations. We found that the estimated relative flux calibration
error between different filter bands was limited to $\lesssim 5\%$, which is
consistent with values reported in previous literature (e.g., Wang et al.,
2022a).

###  3.2 Stellar Mass and Star Formation Rate

Galaxy stellar masses are derived by SED fitting to the photometry shown in
Sec 2.2. Note that both A2744 and SMACS 0723 clusters significantly magnify
the background sources, including galaxies in this work due to the
gravitational lensing. In order to calculate the lensing magnification of each
galaxy at the appropriate redshift, we adopt lensing models from Mahler et al.
(2018) and Golubchik et al. (2022), respectively (see Appendix A). To do our
analysis, we first correct the magnitudes in all bands by the lensing
magnification factor. Note that we do not introduce the systematic error of
lensing models. To simultaneously measure the stellar mass ($M_{\star}$), the
stellar dust attenuation ($\rm E(B-V)_{star}$), and the best-fit model of the
stellar continuum with emission lines, we fit the SED using the Bayesian code
BEAGLE (Chevallard & Charlot, 2016). For our SED fitting, we fix the galaxy
redshift, assume a constant star formation history, and model the dust
attenuation curve following Calzetti et al. (2000). We set the dust
attenuation in the $\mathrm{V}$ band $A_{\mathrm{V}}$ to vary from $0-8$ and
use the Chabrier (2003) initial mass function (IMF) with an upper limit of
$300~{}\mathrm{M}_{\odot}$. The metallicity of the interstellar medium (ISM)
is assumed to be the same as that of the stellar populations. The nebular line
and continuum emission are included in our fitting based on the CLOUDY
photoionization code (Ferland et al., 2017). To model the metallicity, we use
a flat prior with values ranging from $-2<\log(Z/\mathrm{Z}_{\odot})<0.4$. For
our model, we bound the ionization parameter in log-space between $-4<\log
U<-1$.

The derivation of star formation rate (SFR) and gaseous metallicity is based
on the attenuation-corrected emission line intensities. We correct the
reddening effect of dust attenuation for all measured nebular lines using the
$\rm E(B-V)_{star}$ which is derived from the SED fitting, assuming the dust
reddening for the nebular and stellar components are equal. Indeed, several
works (e.g., Shivaei et al., 2020) pointed out that $\rm E(B-V)_{star}$ could
equal to $\sim 2\rm E(B-V)_{gas}$ for galaxies at $z=2$, we test using this
assumption and found out that the SFR could be increased by 0.1 dex.
Furthermore, the relative changing line ratios (e.g., $\rm O_{32}$) are at an
average level of 0.03 dex, which will lead to consistent conclusions in this
paper. Therefore, in the rest of the paper, we adopt the assumption of $\rm
E(B-V)_{star}=\rm E(B-V)_{gas}$.

Note that the nebular dust attenuation can be also derived from
H$\alpha$/H$\beta$ Balmer decrement (e.g., H$\alpha$/H$\beta$ or
H$\beta$/H$\gamma$) for a few galaxies in our sample. But half of our galaxy
sample does not have H$\alpha$ coverage. The absence of H$\alpha$ line
coverage and lack of significant H$\gamma$ line detection makes it impossible
to measure dust attenuation via the Balmer decrement. To maintain consistency
among our entire sample, we adopt the dust attenuation from the SED fitting to
perform the de-reddening correction for emission lines.

We measure the SFR using the attenuation-corrected H$\alpha$ luminosity. For
galaxies without H$\alpha$ coverage, we derive the H$\alpha$ luminosity from
the attenuation-corrected H$\beta$ flux by adopting a Balmer decrement of
H$\alpha$/H$\beta$=2.86. In order to estimate the SFR we assume the Kennicutt
(1998) SFR calibration and the Chabrier (2003) initial mass function. The SFR
is modeled as

$\mathrm{SFR}=4.6\times 10^{-42}\frac{L_{\mathrm{H\alpha}}}{\rm
erg~{}s^{-1}}[\mathrm{M}_{\odot}~{}\mathrm{yr}^{-1}],$ (1)

where $L_{\mathrm{H\alpha}}$ is the H$\alpha$ luminosity. Throughout this
paper, SFR corresponds to the $\rm SFR_{H\alpha}$, which is derived from the
attenuation-corrected H$\alpha$ luminosity. And for those galaxies without
H$\alpha$ coverage, we get H$\alpha$ luminosity using the attenuation-
corrected H$\beta$ assuming a Balmer decrement of H$\alpha$/H$\beta$=2.86.

Within our sample, it is worth mentioning that the H$\alpha$ line is blended
with [N II] lines due to the low spectral resolution, which biases our
estimate of H$\alpha$ towards a higher value. However, the contribution of [N
II] decreases with metallicity and typically the [N II]/H$\alpha$ is less than
0.07 for star-forming galaxies with the stellar mass of
$M_{\star}<10^{9}\mathrm{M}_{\odot}$ at $z=2-3$ (e.g., Bian et al., 2018).
Therefore, the [N II] contamination should be negligible to our SFR
measurement, especially when compared with the uncertainty of dust attenuation
correction. Besides, Balmer line stellar absorption can bias the intrinsic
line luminosity to a lower value. Nevertheless, for the young star-forming
galaxies in our sample, the Balmer line stellar absorption is comparatively
weak and marginalized.

Using the measurements from SED fitting and SFR calibration, we plot Log(SFR)
as a function of the Log(M⋆), in Figure 2. We find that the relation between
SFR and $M_{\star}$ displays a positive correlation across the complete range
of stellar masses and redshift. We also find this slope agrees with the more
massive galaxies measured in the MOSDEF sample Sanders et al. (2021) and the
extrapolated galaxy star-forming main sequence. Despite a large degree of
scatter, the SFR at fixed $M_{\star}$ increases with the redshift (Figure 2).
However, when compared with previous studies of star-forming main sequence
galaxies at redshifts of $2<z<3$, our sample of galaxies has a slightly higher
mean SFR.

Table 1: Properties of stacked spectra in bins of $M_{\star}$ for the $z\simeq
2.0$ and $z\simeq 3.0$ samples.

log$(M_{\star}/\mathrm{M}_{\odot})$ | N | ($\rm 12+log(O/H)$)stack | ($\rm 12+log(O/H)$)med | $\rm O_{32}$ | $\rm O_{3}$ | SFR [$\mathrm{M_{\odot}~{}yr^{-1}}$]
---|---|---|---|---|---|---
$1.8<z_{\mathrm{grism}}<2.3$
$6.64\pm{0.12}$ | 8 | $8.03\pm{0.05}$ | $8.00\pm{0.10}$ | $0.86\pm{0.08}$ | $0.59\pm{0.04}$ | $0.61\pm{0.12}$
$7.56\pm{0.09}$ | 9 | $8.11\pm{0.03}$ | $8.05\pm{0.06}$ | $0.73\pm{0.06}$ | $0.79\pm{0.04}$ | $1.09\pm{0.25}$
$8.50\pm{0.09}$ | 6 | $8.23\pm{0.03}$ | $8.26\pm{0.06}$ | $0.52\pm{0.05}$ | $0.74\pm{0.04}$ | $2.61\pm{0.95}$
$9.25\pm{0.20}$ | 4 | $8.35\pm{0.02}$ | $8.38\pm{0.07}$ | $0.32\pm{0.03}$ | $0.55\pm{0.03}$ | $2.83\pm{0.57}$
$2.65<z_{\mathrm{grism}}<3.4$
$7.48\pm{0.09}$ | 7 | $8.00\pm{0.04}$ | $7.98\pm{0.08}$ | $0.91\pm{0.06}$ | $0.75\pm{0.03}$ | $5.02\pm{1.77}$
$8.39\pm{0.10}$ | 6 | $8.14\pm{0.03}$ | $8.10\pm{0.05}$ | $0.67\pm{0.04}$ | $0.78\pm{0.03}$ | $5.60\pm{2.35}$
$9.62\pm{0.06}$ | 4 | $8.34\pm{0.02}$ | $8.40\pm{0.09}$ | $0.34\pm{0.03}$ | $1.01\pm{0.10}$ | $10.64\pm{6.97}$

Note. — Column 1 is the median stellar mass for each bin; Column 2 is the
number of galaxies in each bin; Columns 3 and 4 are gas phase metallicity
measured from the stacked spectra and statistical median of individual
galaxies, respectively; Column 5 and 6 are the line ratios O${}_{32}=$ log([O
III]$\lambda\lambda 4959,5007$/[O II]$\lambda\lambda 3727,3729$) and
$\mathrm{O}_{3}$=log([O III]$\lambda\lambda 4959,5007$/H$\beta$) derived from
the stacked spectra; Column 7 is the median star formation rate.

###  3.3 Gas-phase Metallicity

Bian et al. (2018) established empirical relations between the strong
metallicity diagnostic line ratios and the direct-$\mathrm{T_{e}}$ metallicity
using a sample of local analogs of high-redshift galaxies. These calibrations
have been demonstrated as one of the most reliable metallicity calibrations at
$z>1$ (Sanders et al., 2020).

We derive the gas-phase oxygen abundance by adopting the Bian et al. (2018)
calibration and measuring the strong metallicity diagnostic line ratios
O${}_{32}=$ log([O III]$\lambda\lambda 4959,5007$/[O II]$\lambda\lambda
3727,3729$) and $\mathrm{O}_{3}$=log([O III]$\lambda\lambda
4959,5007$/H$\beta$). This calibration allows us to minimize the systematic
uncertainty of the metallicity measurements when comparing our MZR results
with those from Sanders et al. (2021). For the galaxies with both [O III] and
[O II] coverage (44/51), we use the $\mathrm{O_{32}}$ to measure the gas-phase
oxygen abundance as follows

$12+\mathrm{log(O/H)}=8.54-0.59\times\mathrm{O_{32}}.$ (2)

As shown above, we use the Bian et al. (2018) calibration, which is suitable
in the range of $0.3<\mathrm{O_{32}}<1.2$ to measure the oxygen abundance in
our galaxies. For the galaxies (11/48) where $\rm O_{32}$ is out of this
range, we extrapolate this relation linearly to estimate their oxygen
abundance. The linearity of the O32 calibration allows us to extrapolate with
minimal systematic error. For the same reason, we try not to use the $\rm
R_{23}$ and $\rm O_{3}$ to further constrain the metallicity, which will cause
more systematic uncertainties, especially for low-mass and low-metallicity
galaxies.

For galaxies without [O II] coverage (7/51), we calculate the metallicity by
using line ratio $\mathrm{O_{3}}\equiv$ [O
III]$\lambda\lambda$4959,5007/H$\beta$, and use the following equation:

$\mathrm{O_{3}}=43.9836-21.6211x+3.4277x^{2}-0.1747x^{3},$ (3)

where $x=12+\mathrm{log(O/H)}$.

For the above $\mathrm{O_{3}}-Z$ relation, there exist two oxygen abundance
solutions for a given $\mathrm{O_{3}}$ value. We adopt the solution within the
range of $7.8<12+\log(\mathrm{O/H})<8.4$, which is consistent with the oxygen
abundance range derived from $\mathrm{O_{32}}$ indicator in a similar mass
range. Our derived metallicities are presented in Appendix A.

Additionally, we stack each individual spectrum to increase the signal-to-
noise and measure the average metallicity in different stellar mass and
redshift bins. We perform the spectrum stacking in two redshift bins
($1.8<z<2.3$ and $2.65<z<3.4$). For our stacking analysis, we only use the
galaxies in the A2744 field, because those in the SMACS 0723 field do not have
[O II] coverage. We divide the entire galaxy sample into four (three) stellar
mass bins within the redshift range $1.8<z<2.3$ ($2.65<z<3.4$). Each stellar
mass bin contains between four and nine galaxies, which is determined by the
bootstrap simulation to minimize the uncertainty of derived metallicities of
stacked spectra. Moreover, we ensure uniformity of the median stellar mass of
each mass bin when setting the galaxy number in each bin.

For our stacking analysis, we first shift each spectrum into the rest frame
according to the grism redshift. We then apply dust-corrections following the
E(B-V) values from our SED fitting and the Calzetti et al. (2000) attenuation
curve. We then subtract off the continuum for each individual spectrum before
normalization and stacking. We normalized each spectrum based on their [O III]
flux and resampled it onto a uniform wavelength grid. Lastly, the spectra in
each mass bin are combined by taking the mean of the 3-$\sigma$ clipping of
each wavelength grid. In Figure 1(b), we present the final stacked spectra in
different stellar mass and redshift bins. We measure the relative emission
line flux in the stacked spectra using the same method as for the individual
galaxies. We derive gas-phase metallicities using the $\mathrm{O}_{32}$
indicator. The measurement uncertainties are propagated from both the stacking
and line flux fitting. The median stellar mass, measured line ratios, median
SFR, and the derived metallicity of stacked spectra are presented in Table 1.

It’s important to note that performing the dust correction either before or
after stacking during the stacking procedure could potentially introduce bias
to the measured metallicity(e.g., Henry et al., 2021). We performed a test by
changing the sequence of dust corrections. Consequently, we find that the
difference in derived metallicity 12+log(O/H) is 0.01 - 0.03 typically, which
is within $1\sigma$ error of metallicity $\simeq 0.02-0.04$. Therefore, our
results are not sensitive to the dust correction sequence when stacking. We
also measure the dust attenuation using the Balmer decrement on the stacked
spectra that are not dust-corrected.

Figure 3: The mass-metallicity relation (MZR) for dwarf galaxies at $z\simeq
2$ (blue) and $z\simeq 3$ (red). The unfilled squares and hexagons denote
individual galaxies at $z\simeq 2$ and $\simeq 3$ respectively. The large blue
squares and red hexagons mark the results based on stacked spectra. The error
bar in the lower left corner displays the median uncertainty of the individual
galaxies. The best-fit results are denoted by the colored dashed lines, with
their $1\sigma$ uncertainties in the shaded region.The MOSDEF stacked results
(Sanders et al., 2021) are marked as black triangles and Henry et al. (2021)
stacked results are marked as black crosses. The MOSDEF best-fit result and
the extrapolation to the low-mass end are in solid and dashed lines. Figure 4:
Projection of the fundamental metallicity relation (FMR) as O/H vs.
$\mu_{\alpha_{\mathrm{best}}}=\operatorname{log}(M/\mathrm{M_{\odot}})-\alpha_{\mathrm{best}}\times\operatorname{log}(\mathrm{SFR/M_{\odot}~{}yr^{-1}})$.
The coefficient $\alpha_{\mathrm{best}}=0.60$, minimizes the scatter of the
O/H for our sample. The solid black line shows the best linear fit from
Equation 4 and the grey-shaded region shows the $1\sigma$ uncertainty. The
unfilled squares and hexagons represent individual galaxies at $z\simeq 2$ and
$z\simeq 3$ respectively, and the large colored ones show the stacked results.
The black triangles denote the MOSDEF stacked results from Sanders et al.
(2021) and the black dotted-dashed line marks their best-fit. The error bar in
the lower right corner displays the median uncertainty of the individual
galaxies.

###  3.4 The Mass–Metallicity Relation

We find a clear correlation between gas-phase metallicity and stellar mass for
our measurements of both individual galaxies and composite spectra. This is
the first-ever effort to derive the MZR down to galaxies of
$M_{\star}=10^{6.5}\mathrm{M}_{\odot}$ at $z=2-3$ using grism slitless
spectroscopy. Within both the $z\simeq 2$ and $z\simeq 3$ sample, we find
Spearman correlation coefficients of 0.70 and 0.68, respectively, with the
statistical-significant p-value $<10^{-4}$. This means a positive correlation
with strong evidence to reject the null hypothesis.

Beyond the strong trends of the MZR within each sample, we also look for signs
of redshift evolution. Using the stacked spectra for the two samples, we find
that the MZRs at $z\simeq 2.0$ and $z\simeq 3.0$ show a monotonic evolution
toward lower metallicity with increasing redshift. As seen in Figure 3, the
scale of this evolution is similar to that of MOSDEF from $z\simeq 2.3$ to
$z\simeq 3.3$.

To quantify the MZR and derive slope and normalization, we fit it using a
power law of the form:

$\mathrm{12+log(O/H)}=\beta\times\mathrm{log}\left(\frac{M_{\star}}{10^{8}~{}\mathrm{M}_{\odot}}\right)+Z8,$
(4)

where $Z8$ is the metallicity at the stellar mass of
$10^{8}~{}\mathrm{M}_{\odot}$. Using all galaxies in the sample, we get the
best-fit relations with $1\sigma$ uncertainties are:

$\displaystyle\mathrm{12+log(O/H)}$ $\displaystyle=$ (5)
$\displaystyle(0.14\pm 0.04)\times$
$\displaystyle\mathrm{log}\left(\frac{M_{\star}}{10^{8}~{}\mathrm{M}_{\odot}}\right)+(8.14\pm
0.03).$

We also fit the stacked spectra in the $z\simeq 2.0$ and $z\simeq 3.0$. The
best-fit parameters with $1\sigma$ uncertainties are as follows: for $z\simeq
2.0$, $\beta_{z\simeq 2}=0.13\pm 0.04$, $Z8_{z\simeq 2}=8.18\pm 0.02$; and for
$z\simeq 3.0$, $\beta_{z\simeq 3}=0.16\pm 0.03$, $Z8_{z\simeq 3}=8.08\pm
0.03$. These two best-fit relations and the M-Z values of each individual
galaxy are shown in Figure 3.

Note that the slope of best-fit MZR for all stacked spectroscopy is $0.14\pm
0.04$, consistent with both $z\simeq 2$ and $z\simeq 3$ best-fit slopes,
implying that there is no significant evolution in the slope between $z=2-3$.
In contrast to the lack of evolution in the slope, we do find moderate
evolution in the MZR normalization with redshift between $z=2-3$.
Specifically, we find that the difference in the normalization at $z\simeq 2$
and $z\simeq 3$ is $\mathrm{\Delta log(O/H)}/\mathrm{\Delta}z=-0.09\pm 0.06$.
Beyond redshift evolution, we also probe for differences in the MZR as a
function of stellar mass. We find that at the low-mass end, we measure a much
shallower slope than at the high-mass end mass where $\beta\simeq 0.30$
(Sanders et al., 2021). This suggests a transition in the slope of the MZR at
$M_{\star}\simeq 10^{9}\mathrm{M}_{\odot}$.

Although our sample is not large enough to allow a statistically robust study
of the scatter of the MZR at the low-mass end, we find a tentative increase of
the MZR scatter at the stellar mass less than $10^{8}~{}\mathrm{M}_{\odot}$.
By defining the intrinsic scatter as
$\sigma_{\mathrm{ins}}=\sqrt{\sigma_{\mathrm{obs}}^{2}-\sigma_{\mathrm{meas}}^{2}}$,
where $\sigma_{\mathrm{obs}}$ is the observation scatter and
$\sigma_{\mathrm{meas}}$ is the measurement uncertainties, we measure the
intrinsic scatter to be $0.16-0.18$ dex at
$M_{\star}=10^{8}-10^{9}\mathrm{M}_{\odot}$. However, we see this value
increase to $\approx 0.23$ dex at $M_{\star}=10^{7}\mathrm{M}_{\odot}$ bins
implying a tentative evolution in the scatter.

##  4 Discussion

Using imaging from HST/ACS, HST/WFC3, JWST/NIRISS, and JWST/NIRCAM and grism
spectroscopy from JWST/NIRISS, we present a study of the MZR in high-$z$
($z=2-3$), low-mass galaxies
($10^{6.5}~{}\mathrm{M}_{\odot}<M_{\star}<10^{9.5}\mathrm{M}_{\odot}$), which
probes approximately two orders of magnitude lower than previous studies. The
best-fit slope of the MZR in dwarf galaxies at $z=2-3$ is $0.14\pm 0.04$,
shallower than that of more massive galaxies ($M_{\star}>10^{9}\
\mathrm{M}_{\odot}$) at $z\approx 2-3$. Additionally, we measure moderate
evolution in the MZR between the $z\simeq 2$ and $z\simeq 3$. The primary
difference between our two redshift samples is the normalization, which
differs by $\mathrm{\Delta log(O/H)}/\mathrm{\Delta}z=-0.09\pm 0.06$. In the
following discussion, we provide a detailed physical interpretation of these
results.

###  4.1 The MZR slope at low-mass end

Deep JWST/NIRISS spectroscopy of a sample of dwarf galaxies allows us, for the
first time, to make a robust measurement of the slope of MZR at $z=2-3$ using
low mass galaxies ($M_{\star}\approx 10^{6.5-9.5}\ \mathrm{M}_{\odot}$). At
$z=2-3,$ we find a shallower slope in MZR at the low-mass end, in contrast to
Sanders et al. (2021), which examines a sample of galaxies with
$M_{\star}=10^{9}-10^{11}\mathrm{M}_{\odot}$. This difference indicates a
transition in MZR slope around $M_{\star}\simeq 10^{9}\mathrm{M}_{\odot}$.

The differences in our measurements of the MZR slope may be indicative of
differences in the feedback mechanisms that regulate it (e.g., Wang et al.,
2022b; Strom et al., 2022). Typically, stellar feedback drives the gas and
metal out of galaxies, which will result in differences in the gas-phase
metallicity of the ISM. The feedback of stellar winds can be described by the
wind mass loading factor ($\eta\equiv\dot{M}_{\mathrm{out}}/\mathrm{SFR}$),
which presents the fraction of gas ejected in winds. The wind mass loading
factor $\eta$ is related to the wind velocity ($v_{w}$), with $\eta\propto
v_{w}^{-2}$ for the energy-driven wind model, $\eta\propto v_{w}^{-1}$ for the
momentum-driven wind model, and $\eta\equiv\mathrm{const}$ for the constant
wind model. We argue that the shallower MZR slope at the low-mass end is
caused by that galaxies with different masses could be dominated by different
feedback mechanisms.

For example, similar to our analysis, Torrey et al. (2019) used IllustrisTNG
simulations and similarly found that the MZR has a shallower slope at the low-
mass end. They argue that their results are indicative that feedback in low-
mass galaxies may be dominated by constant wind ($\eta=\rm constant$), while
feedback in high-mass galaxies may be dominated by energy-driven wind
($\eta\propto v_{w}^{2}$). Similarly, Pillepich et al. (2018) presents the
necessity to introduce a constant wind model to match the low mass end of the
galaxy stellar mass function. Specifically, the constant wind velocity reduces
the wind mass loading factor. The larger-than-expected wind velocity and lower
mass loading factor further result in longer wind recycling times and less
direct metal ejection, which manifests as somewhat higher metallicities in the
ISM and shallower MZR slope for low-mass galaxies, in agreement with our
results.

Our best fit MZR slope is $\beta=0.14\pm 0.04$, which is consistent with the
prediction of the momentum-driven wind model (e.g., Finlator & Davé, 2008; Guo
et al., 2016). Such a slope suggests that momentum-driven wind is the dominant
feedback in dwarf galaxies, in contrast to the constant wind models in Torrey
et al. (2019). One possible interpretation is that the measured MZR slope in
our sample is the average effect of a transition between the energy-driven
wind model and constant wind models (the slope of the MZR in momentum-driven
wind models is between the other two models). In this interpretation, the
momentum-driven wind model ($\eta\propto v_{w}^{-1}$) acts as an intermediate
state between the energy-driven wind ($\eta\propto v_{w}^{-2}$) and constant
wind ($\eta\equiv\mathrm{const}$). Thus, a more general power-law wind model
could be written as $\eta\propto v_{w}^{-\alpha_{\mathrm{fb}}}$ where
$\alpha_{\mathrm{fb}}$ is the power index parameter of the feedback. Our
results indicate that there could exist a continuous transition of wind model
power index from $\alpha_{\mathrm{fb}}=2$ to $\alpha_{\mathrm{fb}}=0$ in low-
mass galaxies with a transition stellar mass of $M_{\star}\simeq
10^{9}~{}\mathrm{M}_{\odot}$.

The slope of the MZR at the low-mass end remains not well constrained and is
under debate. For example, Henry et al. (2021) found that the MZR slope for
galaxies with stellar masses $M_{\star}>10^{8}~{}M_{\odot}$ at $z\simeq 1.9$
could remain steep, similar to that of higher-mass galaxies. However, the low-
mass MZR slope we find is shallower. The discrepancy with Henry et al. (2021)
could arise from several factors, including the different metallicity
calibration methods used. As Sanders et al. (2020) suggested, we employ strong
line calibration from Bian et al. (2018) and use $\rm O_{32}$ as the
metallicity indicator. In contrast, Henry et al. (2021) derived metallicities
based on Curti et al. (2017) using joint fitting of multiple line ratios. The
choice of strong lines and calibration method could lead to a systematic
discrepancy in MZR slope and normalization. To better compare our results with
Henry et al. (2021), we recalculated the metallicities from their stacked
measurements (overlaid in Figure 3), using our method based on the Bian et al.
(2018) calibration. We find that the stacked results at
$M_{\star}<10^{9}\mathrm{~{}M_{\odot}}$ show a tendency to follow the
shallower-sloped MZR compared to the higher-mass points. To better constrain
the validity of the slope transition hypothesis, we require further
observations of the low-mass MZR and a more robust comparison to simulations.

###  4.2 The Evolution of MZR from z=3 to z=2

For fixed stellar mass $M_{\star}$, we find that the MZR normalization
moderately evolves from lower O/H to higher O/H from $z=3$ to $z=2$. Moreover,
we also aim to determine any evolution with stellar mass by comparing with
Sanders et al. (2021). For the high-mass end ($M_{\star}\simeq
10^{10}\mathrm{~{}M_{\odot}}$), the normalization evolution is $\mathrm{\Delta
log(O/H)}/\mathrm{\Delta}z=-0.11\pm 0.02$, while for the result in this work,
we found $\mathrm{\Delta log(O/H)}/\mathrm{\Delta}z=-0.09\pm 0.06$
($M_{\star}\simeq 10^{8}\mathrm{~{}M_{\odot}}$). It concludes that the
evolution of the MZR is independent of galaxy stellar mass, as indicated by an
unchanged MZR shape from $z=3$ to $z=2$ in a wide stellar mass range of
$10^{6}-10^{10}~{}\mathrm{M}_{\odot}$.

The evolution of MZR is the consequence of a complicated interplay between
star formation efficiency, galactic feedback, gas fraction, gas accretion, and
recycling (e.g., Lilly et al., 2013). Thus, studies of MZR evolution can be
used to constrain how the above processes evolve with cosmic time. Along these
lines, Onodera et al. (2016) explain the evolution of the MZR normalization by
suggesting that star-formation efficiency evolves weakly with redshift.
However, Sanders et al. (2021) suggest an increasing star-formation efficiency
with redshift to match their mild MZR evolution ($\mathrm{\Delta
log(O/H)}/\mathrm{\Delta}z\simeq-0.1$). An increase in star-formation
efficiency with redshift is consistent with recent studies on the evolution of
star-formation efficiency (e.g., Liu et al., 2019). In this work, we also find
a similar mild MZR evolution ($\mathrm{\Delta
log(O/H)}/\mathrm{\Delta}z=-0.09\pm 0.06$) at the low-mass end, which suggests
that star-formation efficiency increases with redshift across a wide stellar
mass range.

The gas fraction of galaxies is another crucial factor for regulating the
metallicity of galaxies. Observations have found that the gas fraction
increases significantly with redshift at fixed $M_{\star}$ and has been
quantified by $\mu_{\mathrm{gas}}\equiv
M_{\mathrm{gas}}/M_{\star}\propto(1+z)^{2.5}$ (e.g., Tacconi et al., 2018).
These large gas fractions can dilute the ISM of galaxies making lower gas-
phase metallicity for higher redshift galaxies. Theoretically, the evolution
of the gas fraction is considered the driving force of MZR evolution in some
cosmological simulations (e.g., Ma et al., 2016; Torrey et al., 2019).
However, the gas fraction is also predicted to increase with decreasing
stellar mass at fixed redshift, which is quantified as
$\mu_{\mathrm{gas}}\propto M_{\star}^{-0.4}$ for $z=2$ (e.g., Torrey et al.,
2019). In contrast to previous studies, our results indicate that the
evolution rate of MZR appears to be the same in the low-mass and high-mass
end, implying that the high gas fraction of dwarf galaxies at $z=2-3$ may have
a weak effect on the evolution of MZR. In the future, direct measurements of
the gas fraction in dwarf galaxies (e.g., using the Square Kilometre Array,
SKA) are crucial to understanding the effect and contribution of gas
reservoirs on galaxy metal enrichment.

###  4.3 Existence of Fundamental Metallicity Relation in Dwarf Galaxies at
$z=2-3$

Previous works (e.g., Ellison et al., 2008; Mannucci et al., 2010; Andrews &
Martini, 2013) have suggested that a fundamental relation exists between
metallicity and other galaxy properties. This fundamental metallicity relation
(FMR), correlates the metallicity with both stellar mass and SFR. For a given
stellar mass, the metallicity is enhanced with the decreasing SFR. Several
works probing large samples of galaxies from $z=3$ to $z=0$ have found little
redshift evolution in the FMR in galaxies above 108 M⊙ (e.g., Henry et al.,
2013; Sanders et al., 2015, 2018; Curti et al., 2020; Henry et al., 2021).
These results emphasize the dominance of stellar mass and SFR on the gas-phase
metallicity of galaxies, regardless of the cosmic redshift.

The FMR relation can be explicitly parameterized using a two-dimensional
projection with the relation between O/H and is written as

$\mu_{\alpha}\equiv\operatorname{log}\left(\frac{M_{\star}}{\mathrm{M}_{\odot}}\right)-\alpha\times\operatorname{log}\left(\frac{\mathrm{SFR}}{\mathrm{M}_{\odot}~{}\mathrm{yr}^{-1}}\right).$
(6)

In the above equation, $\alpha$ is introduced as the parameter to minimize the
scatter in O/H. For the dwarf galaxies at $z=2-3$ in our sample, the scatter
of O/H is minimized at $\alpha_{\mathrm{best}}=0.60\pm 0.13$, suggesting the
existence of FMR for $z=2-3$ dwarf galaxies.

Based on our fitting, we construct an FMR with the linear function form of

$\mathrm{12+log(O/H)}=(0.20\pm 0.02)\mu_{\alpha_{\mathrm{best}}}+(6.59\pm
0.15),$ (7)

where
$\mu_{\alpha_{\mathrm{best}}}=\operatorname{log}\left(M_{\star}/\mathrm{M}_{\odot}\right)-0.60\operatorname{log}\left(\mathrm{SFR}/{\mathrm{M}_{\odot}~{}\mathrm{yr}^{-1}}\right)$.
We show the projection of the FMR, the O/H vs. $\mu_{\alpha_{\mathrm{best}}}$
in Figure 4. As seen in Figure 4, we find that the FMR exists within a large
stellar mass range ($M_{\star}=10^{6}-10^{10}\mathrm{M}_{\odot}$) at $z=2-3$.

The existence of the FMR in the low stellar mass end demonstrates that the SFR
and stellar mass act as the main physical properties for determining the gas-
phase metallicity in dwarf galaxies at $z=2-3$. However, it is important to
note that the exact measurement of parameter $\alpha$ and interpretation of
the FMR can vary depending on the method used to measure it. For example, some
works that have measured alpha for the FMR have found quite a range, with a
systematic difference depending on whether the FMR was measured using a strong
line calibration or direct metallicities in individual galaxies or stacks
(e.g., Andrews & Martini, 2013; Henry et al., 2021). In the context of these
studies, the results presented in this paper suggest that the FMR for dwarf
galaxies at $z=2-3$ follows a consistent $\alpha$ value with previous
measurements ($\alpha=0.55-0.7$; Andrews & Martini, 2013; Sanders et al.,
2017, 2021; Curti et al., 2020). These measurements are over cosmic time
($z=0-3$), suggesting a slight evolution of FMR. However, we note that the
slope of FMR in the low-mass end is much shallower than previous measurements,
and further studies will be needed to fully understand the evolution of the
FMR over cosmic time.

In the future, deeper imaging and spectroscopic observations over larger
survey area will allow us to probe the MZR and FMR relations out to even
higher redshifts and down to even lower metallicity. This will allow us to
better understand the physical processes driving chemical enrichment at
different epochs in the history of the Universe.

## Acknowledgments

Our team thanks Feige Wang and Zuyi Chen for very helpful discussions. Z.C.,
M.L, X.L., Z.L., Y.W., & S.Z. are supported by the National Key R&D Program of
China (grant no. 2018YFA0404503), the National Science Foundation of China
(grant no. 12073014), the Tsinghua University Initiative Scientific Research
Program (No.20223080023), and the science research grants from the China
Manned Space Project with No. CMS-CSST2021-A05. F.S. acknowledges support from
the NRAO Student Observing Support (SOS) award SOSPA7-022. F.S. and E.E.
acknowledge funding from JWST/NIRCam contract to the University of Arizona,
NAS5-02105.

This work is based on observations made with the NASA/ESA Hubble Space
Telescope and NASA/ESA/CSA James Webb Space Telescope. HST and JWST data were
obtained from the Mikulski Archive for Space Telescopes at the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST and NAS
5–26555 for HST. The JWST observations are associated with programs ERS-1324
and ERO-2736. The HST observations are associated with the Hubble Space
Telescope Frontier Fields program and the Reionization Lensing Cluster Survey.
The authors acknowledge the GLASS team for developing their observing program
with a zero-exclusive-access period. This work is based on data and catalog
products from HFF-DeepSpace, funded by the National Science Foundation and
Space Telescope Science Institute (operated by the Association of Universities
for Research in Astronomy, Inc., under NASA contract NAS5-26555). This work is
based on observations taken by the RELICS Treasury Program (GO 14096) with the
NASA/ESA HST, which is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5-26555.

Some of the data presented in this paper were obtained from the Mikulski
Archive for Space Telescopes (MAST) at the Space Telescope Science Institute.
The JWST observations analyzed can be accessed via
https://doi.org/10.17909/91zv-yg35 (catalog doi:10.17909/91zv-yg35) and
https://doi.org/10.17909/12rr-2a67 (catalog doi:10.17909/12rr-2a67). The HFF
data can be accessed via https://doi.org/10.17909/T9KK5N (catalog
doi:10.17909/T9KK5N) and the RELICS data via https://doi.org/10.17909/T9SP45
(catalog doi:10.17909/T9SP45).

##  Appendix A Physical parameters for our sample

In Table A1, we show the measured and derived physical parameters of all of
the galaxies in our sample, including galaxy ID defined by the Grizli spectrum
extraction procedure, galaxy coordinates (R.A. and Decl.), galaxy redshift
($z_{\mathrm{grism}}$), stellar mass
($\mathrm{log}(M_{\star}/\mathrm{M}_{\odot})$), dust attenuation ($A_{V}$),
star formation rate (SFR) derived from Balmer emission lines, line ratios
($\mathrm{O_{32}}=\operatorname{log}$([O III]/[O II])) and
($\mathrm{O_{3}}=\operatorname{log}$([O III]/H$\beta$)), and gas-phase
“metallicity (12+log(O/H)). In Table A2, we present the observed line fluxes
of [O II], H$\beta$, [O III], and H$\alpha$. Note that these fluxes have not
been corrected by dust attenuation and gravitational lensing. The sample is
divided into two redshift bins and presented for two fields A2744 and SMACS
0723.

Table A1: Measured and Derived Properties of Individual Galaxies

ID | R.A. | Decl. | $z_{\rm grism}$ | log$(M_{\star}/\mathrm{M}_{\odot})$ | $A_{V}$ | SFR | $\mathrm{O}_{32}$ | $\mathrm{O}_{3}$ | $\rm 12+log(O/H)$ | $\mu$
---|---|---|---|---|---|---|---|---|---|---
| deg. | deg. |  |  |  | $\rm M_{\odot}~{}yr^{-1}$ |  |  |  |
A2744 Field
$1.8<z_{\mathrm{grism}}<2.3$
34 | 3.591921 | -30.415740 | 2.07 | $7.72^{+0.19}_{-0.14}$ | $0.33^{+0.05}_{-0.04}$ | $2.82\pm{0.15}$ | $0.87\pm{0.03}$ | $0.85\pm{0.02}$ | $8.03\pm{0.02}$ | 3.04
245 | 3.599965 | -30.409570 | 1.81 | $7.78^{+0.19}_{-0.17}$ | $0.39^{+0.06}_{-0.08}$ | $0.47\pm{0.05}$ | $1.60\pm{0.89}$ | $1.02\pm{0.11}$ | $7.60\pm{0.52}$ | 6.29
282 | 3.577672 | -30.408856 | 2.26 | $7.05^{+0.09}_{-0.09}$ | $0.01^{+0.01}_{-0.01}$ | $0.45\pm{0.06}$ | $0.69\pm{0.05}$ | $0.48\pm{0.03}$ | $8.13\pm{0.03}$ | 2.31
614 | 3.618857 | -30.403802 | 2.20 | $7.11^{+0.08}_{-0.08}$ | $0.19^{+0.04}_{-0.04}$ | $1.24\pm{0.25}$ | $0.86\pm{0.13}$ | $1.15\pm{0.28}$ | $8.03\pm{0.08}$ | 1.00
882 | 3.602338 | -30.400736 | 1.81 | $8.58^{+0.16}_{-0.15}$ | $0.20^{+0.08}_{-0.04}$ | $0.64\pm{0.08}$ | $0.04\pm{0.08}$ | $0.63\pm{0.08}$ | $8.52\pm{0.05}$ | 2.23
1018 | 3.606238 | -30.397984 | 2.28 | $6.89^{+0.24}_{-0.17}$ | $0.12^{+0.05}_{-0.03}$ | $1.05\pm{0.10}$ | $1.68\pm{0.61}$ | $0.80\pm{0.07}$ | $7.55\pm{0.36}$ | 1.78
1085 | 3.604176 | -30.397170 | 2.07 | $9.37^{+0.15}_{-0.20}$ | $0.38^{+0.05}_{-0.04}$ | $3.93\pm{0.26}$ | $0.16\pm{0.03}$ | $0.75\pm{0.09}$ | $8.44\pm{0.02}$ | 1.85
1333 | 3.606056 | -30.393533 | 2.18 | $8.42^{+0.18}_{-0.25}$ | $0.30^{+0.09}_{-0.07}$ | $1.48\pm{0.16}$ | $0.80\pm{0.06}$ | $1.13\pm{0.11}$ | $8.07\pm{0.03}$ | 1.68
1407 | 3.611638 | -30.392479 | 2.28 | $7.45^{+0.30}_{-0.30}$ | $0.13^{+0.04}_{-0.03}$ | $0.86\pm{0.08}$ | $1.03\pm{0.10}$ | $1.05\pm{0.09}$ | $7.93\pm{0.06}$ | 1.53
1409 | 3.599706 | -30.392378 | 1.87 | $7.85^{+0.15}_{-0.14}$ | $0.15^{+0.04}_{-0.04}$ | $0.71\pm{0.06}$ | $0.58\pm{0.24}$ | $0.63\pm{0.11}$ | $8.20\pm{0.14}$ | 1.95
1445 | 3.607556 | -30.391864 | 2.06 | $6.34^{+0.06}_{-0.04}$ | $0.16^{+0.06}_{-0.05}$ | $0.44\pm{0.13}$ | $1.85\pm{1.04}$ | $0.87\pm{0.08}$ | $7.45\pm{0.61}$ | 1.62
1461 | 3.604243 | -30.391652 | 1.88 | $8.06^{+0.14}_{-0.14}$ | $0.29^{+0.06}_{-0.07}$ | $0.85\pm{0.09}$ | $0.50\pm{0.08}$ | $0.92\pm{0.12}$ | $8.25\pm{0.05}$ | 1.71
1500 | 3.607616 | -30.391073 | 2.06 | $7.26^{+0.12}_{-0.12}$ | $0.47^{+0.06}_{-0.07}$ | $1.00\pm{0.13}$ | $0.71\pm{0.15}$ | $0.95\pm{0.14}$ | $8.12\pm{0.09}$ | 1.62
1503 | 3.603143 | -30.391055 | 2.18 | $7.57^{+0.15}_{-0.13}$ | $0.13^{+0.03}_{-0.03}$ | $1.72\pm{0.09}$ | $1.01\pm{0.05}$ | $0.87\pm{0.03}$ | $7.94\pm{0.03}$ | 1.80
1504 | 3.613264 | -30.391095 | 1.88 | $8.66^{+0.19}_{-0.23}$ | $0.42^{+0.06}_{-0.06}$ | $4.50\pm{0.32}$ | $0.48\pm{0.04}$ | $0.87\pm{0.04}$ | $8.26\pm{0.02}$ | 1.48
1573 | 3.609447 | -30.389859 | 2.21 | $8.96^{+0.12}_{-0.15}$ | $0.25^{+0.05}_{-0.05}$ | $2.09\pm{0.15}$ | $0.52\pm{0.05}$ | $0.66\pm{0.06}$ | $8.23\pm{0.03}$ | 1.57
1596 | 3.610005 | -30.389482 | 2.18 | $8.63^{+0.24}_{-0.31}$ | $0.32^{+0.07}_{-0.07}$ | $2.92\pm{0.24}$ | $0.52\pm{0.03}$ | $0.90\pm{0.05}$ | $8.23\pm{0.02}$ | 1.55
1685 | 3.603072 | -30.387831 | 1.88 | $6.23^{+0.10}_{-0.07}$ | $0.28^{+0.10}_{-0.08}$ | $0.29\pm{0.05}$ | $1.29\pm{0.72}$ | $1.07\pm{0.26}$ | $7.78\pm{0.43}$ | 1.73
1823 | 3.579059 | -30.385947 | 2.20 | $7.53^{+0.19}_{-0.16}$ | $0.14^{+0.03}_{-0.04}$ | $1.06\pm{0.06}$ | $0.81\pm{0.05}$ | $0.82\pm{0.04}$ | $8.06\pm{0.03}$ | 3.42
2032 | 3.592031 | -30.382505 | 1.87 | $6.87^{+0.11}_{-0.12}$ | $0.16^{+0.04}_{-0.03}$ | $0.56\pm{0.05}$ | $0.56\pm{0.06}$ | $0.77\pm{0.06}$ | $8.21\pm{0.04}$ | 2.43
2084 | 3.591101 | -30.381687 | 1.89 | $8.38^{+0.22}_{-0.13}$ | $0.27^{+0.02}_{-0.03}$ | $7.38\pm{0.22}$ | $0.70\pm{0.03}$ | $1.00\pm{0.02}$ | $8.13\pm{0.02}$ | 2.48
2152 | 3.585912 | -30.380660 | 1.87 | $7.32^{+0.32}_{-0.15}$ | $0.29^{+0.03}_{-0.03}$ | $2.07\pm{0.08}$ | $0.77\pm{0.04}$ | $0.86\pm{0.02}$ | $8.08\pm{0.02}$ | 2.82
2186 | 3.610360 | -30.380189 | 1.88 | $8.92^{+0.15}_{-0.15}$ | $0.48^{+0.05}_{-0.05}$ | $5.72\pm{0.33}$ | $0.05\pm{0.04}$ | $0.63\pm{0.08}$ | $8.51\pm{0.03}$ | 1.63
2303 | 3.607664 | -30.378280 | 1.85 | $6.38^{+0.21}_{-0.15}$ | $0.07^{+0.04}_{-0.03}$ | $0.19\pm{0.03}$ | $0.77\pm{0.30}$ | $0.43\pm{0.09}$ | $8.09\pm{0.18}$ | 1.76
2313 | 3.598602 | -30.378502 | 1.91 | $8.88^{+0.20}_{-0.25}$ | $0.27^{+0.07}_{-0.05}$ | $2.05\pm{0.17}$ | $0.45\pm{0.09}$ | $0.88\pm{0.11}$ | $8.27\pm{0.05}$ | 2.04
2330 | 3.599784 | -30.377873 | 1.89 | $7.96^{+0.22}_{-0.29}$ | $0.10^{+0.07}_{-0.05}$ | $0.51\pm{0.05}$ | $0.99\pm{0.19}$ | $0.74\pm{0.05}$ | $7.96\pm{0.11}$ | 2.03
2421 | 3.599383 | -30.375941 | 1.91 | $7.30^{+0.13}_{-0.12}$ | $0.06^{+0.02}_{-0.02}$ | $0.36\pm{0.03}$ | $1.05\pm{0.50}$ | $0.63\pm{0.10}$ | $7.92\pm{0.30}$ | 2.11
SMACS 0723 Field
$2.65<z_{\mathrm{grism}}<3.4$
50 | 110.821398 | -73.474602 | 2.98 | $8.78^{+0.07}_{-0.09}$ | $0.38^{+0.02}_{-0.04}$ | $8.66\pm{1.42}$ | - | $0.73\pm{0.07}$ | $8.35\pm{0.08}$ | 1.44
1347 | 110.823498 | -73.444440 | 3.37 | $8.22^{+0.06}_{-0.07}$ | $0.17^{+0.02}_{-0.01}$ | $14.00\pm{1.28}$ | - | $0.89\pm{0.04}$ | $8.12\pm{0.09}$ | 1.73
1361 | 110.859711 | -73.444213 | 2.73 | $9.01^{+0.08}_{-0.07}$ | $0.12^{+0.03}_{-0.03}$ | $6.27\pm{0.99}$ | - | $0.97\pm{0.07}$ | $7.87\pm{0.15}$ | 1.79
1424 | 110.828726 | -73.442593 | 3.10 | $9.59^{+0.04}_{-0.04}$ | $0.31^{+0.01}_{-0.01}$ | $47.29\pm{1.63}$ | - | $0.77\pm{0.01}$ | $8.30\pm{0.02}$ | 1.63
1475 | 110.863244 | -73.441489 | 2.73 | $8.33^{+0.09}_{-0.12}$ | $0.24^{+0.04}_{-0.07}$ | $2.90\pm{0.51}$ | - | $0.90\pm{0.07}$ | $8.10\pm{0.16}$ | 1.75
1491 | 110.859274 | -73.441230 | 2.73 | $9.83^{+0.02}_{-0.05}$ | $0.30^{+0.04}_{-0.02}$ | $9.45\pm{1.18}$ | - | $0.65\pm{0.05}$ | $8.43\pm{0.05}$ | 1.94
1520 | 110.850517 | -73.440461 | 3.04 | $7.71^{+0.10}_{-0.12}$ | $0.18^{+0.07}_{-0.05}$ | $6.77\pm{0.90}$ | - | $0.63\pm{0.05}$ | $8.44\pm{0.04}$ | 1.69

Note. — To be continued.

Table A1: (Continued)

ID | R.A. | Decl. | $z_{\rm grism}$ | log$(M_{\star}/\mathrm{M}_{\odot})$ | $A_{V}$ | SFR | $\mathrm{O}_{32}$ | $\mathrm{O}_{3}$ | $\rm 12+log(O/H)$ | $\mu$
---|---|---|---|---|---|---|---|---|---|---
| deg. | deg. |  |  |  | $\rm\mathrm{M}_{\odot}~{}yr^{-1}$ |  |  |  |
A2744 Field
$2.65<z_{\mathrm{grism}}<3.4$
404 | 3.613460 | -30.406860 | 2.85 | $6.84^{+0.09}_{-0.05}$ | $0.06^{+0.03}_{-0.04}$ | $2.56\pm{0.35}$ | $0.90\pm{0.08}$ | $0.69\pm{0.06}$ | $8.01\pm{0.05}$ | 1.69
434 | 3.607437 | -30.406471 | 3.20 | $9.25^{+0.24}_{-0.42}$ | $0.35^{+0.05}_{-0.09}$ | $6.79\pm{1.03}$ | $0.52\pm{0.04}$ | $0.80\pm{0.05}$ | $8.23\pm{0.02}$ | 2.18
538 | 3.612998 | -30.405067 | 3.04 | $9.35^{+0.24}_{-0.43}$ | $0.43^{+0.04}_{-0.09}$ | $36.97\pm{3.36}$ | $0.53\pm{0.03}$ | $0.46\pm{0.02}$ | $8.23\pm{0.01}$ | 1.69
903 | 3.609749 | -30.400342 | 2.97 | $9.68^{+0.13}_{-0.19}$ | $0.41^{+0.06}_{-0.04}$ | $5.58\pm{1.85}$ | $-0.06\pm{0.04}$ | $0.77\pm{0.14}$ | $8.57\pm{0.02}$ | 1.73
988 | 3.613656 | -30.398646 | 2.85 | $9.29^{+0.28}_{-0.27}$ | $0.21^{+0.09}_{-0.04}$ | $1.31\pm{1.02}$ | $0.02\pm{0.07}$ | $0.87\pm{0.34}$ | $8.53\pm{0.04}$ | 1.56
1016 | 3.607800 | -30.398054 | 3.20 | $7.28^{+0.07}_{-0.06}$ | $0.06^{+0.02}_{-0.02}$ | $3.10\pm{0.50}$ | $0.83\pm{0.05}$ | $0.97\pm{0.07}$ | $8.05\pm{0.03}$ | 1.77
1042 | 3.610340 | -30.397893 | 3.36 | $7.79^{+0.03}_{-0.03}$ | $0.02^{+0.01}_{-0.01}$ | $14.97\pm{0.55}$ | $0.96\pm{0.02}$ | $0.82\pm{0.02}$ | $7.98\pm{0.01}$ | 1.67
1206 | 3.607114 | -30.395622 | 2.98 | $8.27^{+0.31}_{-0.29}$ | $0.15^{+0.03}_{-0.03}$ | $7.25\pm{0.46}$ | $1.17\pm{0.05}$ | $0.85\pm{0.02}$ | $7.85\pm{0.03}$ | 1.72
1275 | 3.572769 | -30.394578 | 2.94 | $8.16^{+0.33}_{-0.37}$ | $0.24^{+0.04}_{-0.05}$ | $1.58\pm{0.35}$ | $0.67\pm{0.06}$ | $0.84\pm{0.09}$ | $8.15\pm{0.03}$ | 3.25
1394 | 3.611255 | -30.392440 | 2.98 | $7.13^{+0.16}_{-0.07}$ | $0.11^{+0.03}_{-0.03}$ | $3.69\pm{0.35}$ | $1.71\pm{0.62}$ | $0.44\pm{0.04}$ | $7.53\pm{0.37}$ | 1.57
1535 | 3.604198 | -30.390338 | 2.71 | $8.40^{+0.10}_{-0.16}$ | $0.09^{+0.06}_{-0.04}$ | $1.07\pm{0.30}$ | $0.67\pm{0.16}$ | $0.49\pm{0.13}$ | $8.15\pm{0.10}$ | 1.79
1692 | 3.604118 | -30.387855 | 2.84 | $7.52^{+0.25}_{-0.07}$ | $0.11^{+0.02}_{-0.02}$ | $5.42\pm{0.35}$ | $1.09\pm{0.04}$ | $0.92\pm{0.03}$ | $7.90\pm{0.02}$ | 1.76
2029 | 3.596739 | -30.382570 | 3.21 | $7.43^{+0.08}_{-0.06}$ | $0.20^{+0.03}_{-0.03}$ | $5.89\pm{0.66}$ | $1.30\pm{0.09}$ | $0.85\pm{0.05}$ | $7.77\pm{0.06}$ | 2.16
2060 | 3.585941 | -30.382101 | 3.06 | $8.52^{+0.32}_{-0.49}$ | $0.17^{+0.04}_{-0.04}$ | $4.72\pm{0.35}$ | $0.83\pm{0.02}$ | $0.90\pm{0.03}$ | $8.05\pm{0.01}$ | 3.33
2225 | 3.591399 | -30.379771 | 2.73 | $8.55^{+0.19}_{-0.15}$ | $0.70^{+0.04}_{-0.05}$ | $21.43\pm{1.34}$ | $0.75\pm{0.01}$ | $0.91\pm{0.01}$ | $8.10\pm{0.01}$ | 2.50
2237 | 3.604111 | -30.379262 | 2.83 | $7.67^{+0.38}_{-0.39}$ | $0.22^{+0.06}_{-0.06}$ | $1.42\pm{0.36}$ | $0.79\pm{0.11}$ | $0.78\pm{0.11}$ | $8.08\pm{0.07}$ | 1.95
2287 | 3.607475 | -30.378498 | 2.69 | $8.34^{+0.11}_{-0.20}$ | $0.06^{+0.04}_{-0.03}$ | $1.99\pm{0.25}$ | $0.63\pm{0.08}$ | $0.46\pm{0.06}$ | $8.17\pm{0.05}$ | 1.83

Note. — Column 1 is the source ID defined by our source detection procedure;
Columns 2 and 3 are the equatorial coordinates right ascension (R.A.) and
declination (Decl.) in equinox with an epoch of J2000; Column 4 is the
redshift determined by NIRISS grism spectra; Column 5 is the stellar mass;
Column 6 is the dust attenuation ($A_{V}$); Column 7 is the star formation
rate derived from Balmer emission lines; Columns 8 and 9 are the line ratios
O${}_{32}=$ log([O III]$\lambda\lambda 4959,5007$/[O II]$\lambda\lambda
3727,3729$) and $\mathrm{O}_{3}$=log([O III]$\lambda\lambda
4959,5007$/H$\beta$); Column 10 is gas phase metallicity represented by oxygen
abundance; Column 11 is the magnification of gravitational lensing effect.

Table A2: Measured emission line fluxes of Individual Galaxies

ID | $f_{\rm[OII]}$ | $f_{\rm H\beta}$ | $f_{\rm[OIII]}$ | $f_{\rm H\alpha}$
---|---|---|---|---
| $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$ | $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$ | $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$ | $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$
A2744 Field
$1.8<z_{\mathrm{grism}}<2.3$
34 | $1.47\pm{0.10}$ | $1.54\pm{0.07}$ | $10.82\pm{0.10}$ | $4.12\pm{0.11}$
245 | $0.08\pm{0.17}$ | $0.32\pm{0.08}$ | $3.32\pm{0.09}$ | $1.84\pm{0.12}$
282 | $0.42\pm{0.05}$ | $0.68\pm{0.04}$ | $2.06\pm{0.06}$ | $0.58\pm{0.08}$
614 | $0.23\pm{0.07}$ | $0.12\pm{0.07}$ | $1.64\pm{0.09}$ | $0.60\pm{0.12}$
882 | $2.64\pm{0.45}$ | $0.68\pm{0.12}$ | $2.88\pm{0.12}$ | $1.10\pm{0.11}$
1018 | $0.04\pm{0.05}$ | $0.29\pm{0.05}$ | $1.85\pm{0.06}$ | $0.89\pm{0.07}$
1085 | $2.42\pm{0.15}$ | $0.63\pm{0.13}$ | $3.53\pm{0.15}$ | $3.29\pm{0.13}$
1333 | $0.49\pm{0.06}$ | $0.23\pm{0.06}$ | $3.10\pm{0.07}$ | $1.09\pm{0.06}$
1407 | $0.23\pm{0.05}$ | $0.22\pm{0.05}$ | $2.51\pm{0.06}$ | $0.62\pm{0.05}$
1409 | $0.36\pm{0.20}$ | $0.32\pm{0.08}$ | $1.38\pm{0.08}$ | $1.05\pm{0.08}$
1445 | $0.02\pm{0.05}$ | $0.19\pm{0.04}$ | $1.43\pm{0.05}$ | $0.41\pm{0.12}$
1461 | $0.53\pm{0.09}$ | $0.20\pm{0.05}$ | $1.65\pm{0.07}$ | $0.92\pm{0.08}$
1500 | $0.29\pm{0.10}$ | $0.17\pm{0.05}$ | $1.47\pm{0.07}$ | $0.67\pm{0.08}$
1503 | $0.49\pm{0.05}$ | $0.68\pm{0.05}$ | $5.00\pm{0.07}$ | $1.63\pm{0.06}$
1504 | $1.68\pm{0.14}$ | $0.68\pm{0.06}$ | $5.04\pm{0.08}$ | $3.69\pm{0.07}$
1573 | $0.68\pm{0.08}$ | $0.49\pm{0.07}$ | $2.25\pm{0.08}$ | $1.47\pm{0.07}$
1596 | $1.25\pm{0.07}$ | $0.53\pm{0.06}$ | $4.13\pm{0.09}$ | $1.95\pm{0.07}$
1685 | $0.04\pm{0.06}$ | $0.06\pm{0.04}$ | $0.73\pm{0.05}$ | $0.33\pm{0.04}$
1823 | $0.75\pm{0.08}$ | $0.74\pm{0.07}$ | $4.82\pm{0.09}$ | $1.86\pm{0.08}$
2032 | $0.69\pm{0.10}$ | $0.42\pm{0.05}$ | $2.50\pm{0.07}$ | $1.01\pm{0.09}$
2084 | $5.90\pm{0.43}$ | $2.93\pm{0.14}$ | $29.62\pm{0.17}$ | $11.71\pm{0.14}$
2152 | $1.85\pm{0.16}$ | $1.51\pm{0.06}$ | $11.00\pm{0.10}$ | $3.77\pm{0.08}$
2186 | $3.21\pm{0.29}$ | $0.85\pm{0.15}$ | $3.63\pm{0.17}$ | $4.86\pm{0.14}$
2303 | $0.11\pm{0.08}$ | $0.24\pm{0.04}$ | $0.65\pm{0.05}$ | $0.27\pm{0.04}$
2313 | $1.51\pm{0.31}$ | $0.57\pm{0.14}$ | $4.29\pm{0.16}$ | $2.61\pm{0.14}$
2330 | $0.22\pm{0.10}$ | $0.39\pm{0.05}$ | $2.13\pm{0.06}$ | $0.80\pm{0.05}$
2421 | $0.10\pm{0.11}$ | $0.26\pm{0.06}$ | $1.10\pm{0.07}$ | $0.60\pm{0.05}$
SMACS 0723 Field
$2.65<z_{\mathrm{grism}}<3.4$
50 | - | $0.77\pm{0.12}$ | $4.12\pm{0.18}$ | -
1347 | - | $1.43\pm{0.13}$ | $11.19\pm{0.20}$ | -
1361 | - | $1.18\pm{0.18}$ | $11.05\pm{0.21}$ | -
1424 | - | $4.72\pm{0.14}$ | $27.79\pm{0.21}$ | -
1475 | - | $0.46\pm{0.08}$ | $3.65\pm{0.11}$ | -
1491 | - | $1.54\pm{0.18}$ | $6.86\pm{0.22}$ | -
1520 | - | $0.86\pm{0.09}$ | $3.68\pm{0.13}$ | -

Note. — To be continued.

Table A2: (Continued)

ID | $f_{\rm[OII]}$ | $f_{\rm H\beta}$ | $f_{\rm[OIII]}$ | $f_{\rm H\alpha}$
---|---|---|---|---
| $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$ | $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$ | $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$ | $10^{-17}~{}\rm erg~{}s^{-1}~{}cm^{-2}$
A2744 Field
$2.65<z_{\mathrm{grism}}<3.4$
404 | $0.27\pm{0.05}$ | $0.44\pm{0.06}$ | $2.18\pm{0.08}$ | -
434 | $1.51\pm{0.13}$ | $0.80\pm{0.10}$ | $5.03\pm{0.11}$ | -
538 | $2.91\pm{0.16}$ | $3.44\pm{0.14}$ | $9.83\pm{0.18}$ | -
903 | $3.89\pm{0.29}$ | $0.58\pm{0.19}$ | $3.41\pm{0.20}$ | -
988 | $1.22\pm{0.14}$ | $0.17\pm{0.13}$ | $1.28\pm{0.13}$ | -
1016 | $0.59\pm{0.06}$ | $0.43\pm{0.07}$ | $3.97\pm{0.08}$ | -
1042 | $1.34\pm{0.06}$ | $1.82\pm{0.07}$ | $12.14\pm{0.11}$ | -
1206 | $0.49\pm{0.05}$ | $1.03\pm{0.05}$ | $7.24\pm{0.08}$ | -
1275 | $0.58\pm{0.07}$ | $0.39\pm{0.08}$ | $2.68\pm{0.11}$ | -
1394 | $0.03\pm{0.04}$ | $0.50\pm{0.04}$ | $1.38\pm{0.06}$ | -
1535 | $0.14\pm{0.05}$ | $0.21\pm{0.06}$ | $0.65\pm{0.07}$ | -
1692 | $0.62\pm{0.06}$ | $0.93\pm{0.05}$ | $7.65\pm{0.08}$ | -
2029 | $0.29\pm{0.06}$ | $0.82\pm{0.09}$ | $5.80\pm{0.10}$ | -
2060 | $1.37\pm{0.07}$ | $1.19\pm{0.07}$ | $9.38\pm{0.10}$ | -
2225 | $3.97\pm{0.08}$ | $2.74\pm{0.08}$ | $22.15\pm{0.13}$ | -
2237 | $0.23\pm{0.06}$ | $0.24\pm{0.06}$ | $1.42\pm{0.07}$ | -
2287 | $0.30\pm{0.05}$ | $0.43\pm{0.05}$ | $1.25\pm{0.06}$ | -

Note. — Column 1 is the source ID defined by our source detection procedure;
Columns 2-5 present the observed emission line fluxes of [O II], H$\beta$, [O
III], and H$\alpha$. Note that the “-” value means that the grism observations
do not cover the corresponding wavelength.

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