#  Pure Spectroscopic Constraints on UV Luminosity Functions and Cosmic Star
Formation History
From 25 Galaxies at $\mathbf{z_{\mathrm{spec}}=8.61-13.20}$ Confirmed with
JWST/NIRSpec

Yuichi Harikane Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Kimihiko Nakajima National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Masami Ouchi National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka,
Tokyo 181-8588, Japan  Institute for Cosmic Ray Research, The University of
Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan  Kavli Institute for
the Physics and Mathematics of the Universe (WPI), University of Tokyo,
Kashiwa, Chiba 277-8583, Japan  Hiroya Umeda  Institute for Cosmic Ray
Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582,
Japan  Department of Physics, Graduate School of Science, The University of
Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Yuki Isobe Institute for
Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,
Chiba 277-8582, Japan  Department of Physics, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Yoshiaki Ono
Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba 277-8582, Japan  Yi Xu Institute for Cosmic Ray Research, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
Department of Astronomy, Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Yechi Zhang Institute for Cosmic
Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba
277-8582, Japan  Department of Astronomy, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan

###### Abstract

We present pure spectroscopic constraints on the UV luminosity functions and
cosmic star formation rate (SFR) densities from 25 galaxies at
$z_{\mathrm{spec}}=8.61-13.20$. By reducing the JWST/NIRSpec spectra taken in
multiple programs of ERO, ERS, GO, and DDT with our analysis technique, we
independently confirm 16 galaxies at $z_{\mathrm{spec}}=8.61-11.40$ including
new redshift determinations, and a bright interloper at
$z_{\mathrm{spec}}=4.91$ that was claimed as a photometric candidate at $z\sim
16$. In conjunction with nine galaxies at redshifts up to
$z_{\mathrm{spec}}=13.20$ in the literature, we make a sample of 25
spectroscopically-confirmed galaxies in total and carefully derive the best
estimates and lower limits of the UV luminosity functions. These UV luminosity
function constraints are consistent with the previous photometric estimates
within the uncertainties and indicate mild redshift evolution towards $z\sim
12$ showing tensions with some theoretical models of rapid evolution. With
these spectroscopic constraints, we obtain firm lower limits of the cosmic SFR
densities and spectroscopically confirm a high SFR density at $z\sim 12$
beyond the constant star-formation efficiency models, which supports earlier
claims from the photometric studies. While there are no spectroscopically-
confirmed galaxies with very large stellar masses violating the $\Lambda$CDM
model due to the removal of the bright interloper, we confirm star-forming
galaxies at $z_{\mathrm{spec}}=11-13$ with stellar masses much higher than
model predictions. Our results indicate possibilities of high star-formation
efficiency ($>5\%$), hidden AGN, top-heavy initial mass function (possibly
with Pop-III), and large scatter/variance. Having these successful and
unsuccessful spectroscopy results, we suggest observational strategies for
efficiently removing low-redshift interlopers for future JWST programs.

galaxies: formation — galaxies: evolution — galaxies: high-redshift

††journal: ApJ

##  1 Introduction

One of the most important goals in astronomy today is to understand galaxy
formation from their birth stage to the current stage (Stark 2016; Dayal &
Ferrara 2018; Ouchi et al. 2020; Robertson 2021). To accomplish the goal,
observations of present galaxies to first galaxies are key to revealing the
entire process of galaxy formation. Before the operation of the James Webb
Space Telescope (JWST), large telescopes such as the Hubble Space Telescope
(HST) have driven observational studies of galaxy formation with millions of
high redshift galaxies and revealed the evolution of the ultraviolet (UV)
luminosity function and the cosmic star formation rate (SFR) density at
$2\lesssim z\lesssim 10$ (e.g., Madau & Dickinson 2014; Bouwens et al. 2015,
2021; Finkelstein et al. 2015; Ishigaki et al. 2018; Ono et al. 2018; Harikane
et al. 2022b), possibly up to $z\sim 11-13$ (e.g., Coe et al. 2013; Ellis et
al. 2013; Harikane et al. 2022a). Several studies discuss that the evolution
of the cosmic SFR density at high redshifts is well reproduced by models
assuming constant star formation efficiencies (e.g., Bouché et al. 2010; Mason
et al. 2015; Harikane et al. 2018, 2022b; Tacchella et al. 2018; Oesch et al.
2018; Bouwens et al. 2021), which is motivated by the clustering analysis of
galaxies at $z\sim 2-7$ (Harikane et al. 2016, 2018, 2022b) and by the
abundance matching studies (e.g., Behroozi et al. 2013; Mason et al. 2015;
Moster et al. 2018). Such models predict a rapid decline of the cosmic SFR
density at $z>10$ due to the decline of the halo number density (Harikane et
al. 2018, 2022b; Oesch et al. 2018). However, some studies using photometric
galaxy candidates at $z\sim 10-12$ indicate that SFR densities at $z>10$ are
higher than these models predictions (Coe et al. 2013; Ellis et al. 2013;
McLeod et al. 2016). Such high SFR densities at $z>10$ are also suggested by
mature stellar populations in galaxies at $z\sim 6-9$ (Hashimoto et al. 2018;
Mawatari et al. 2020).

JWST started its operation in early 2022 (Rigby et al. 2023), and the first
datasets obtained with NIRCam (Rieke et al. 2003, 2005, 2023; Beichman et al.
2012) and NIRSpec (Jakobsen et al. 2022) were released on July 2022. The early
JWST/NIRCam imaging datasets have allowed us to find a large number of galaxy
candidates at $z\sim 9-20$ (Naidu et al. 2022b; Castellano et al. 2022a, b;
Adams et al. 2022; Atek et al. 2022; Finkelstein et al. 2022b; Yan et al.
2023; Donnan et al. 2023b; Harikane et al. 2023a), including bright galaxy
candidates at $z\sim 16$ (Donnan et al. 2023b; Harikane et al. 2023a).
Subsequent studies have reported more candidates at $z>10$ including sources
found in newly obtained NIRCam images (Finkelstein et al. 2023; Bouwens et al.
2022b, a; Donnan et al. 2023a; Morishita & Stiavelli 2022; Bradley et al.
2022; Pérez-González et al. 2023). These studies suggest that the UV
luminosity function and cosmic SFR density at $z>10$ do not show a rapid
decline, in contrast to the predictions of the constant star formation
efficiency models (e.g., Harikane et al. 2023a; Bouwens et al. 2022a, b).
Several physical interpretations are discussed in Harikane et al. (2023a),
including a high star formation efficiency, AGN activity, and a top-heavy IMF
(see also e.g., Inayoshi et al. 2022). However, these discussions are based on
the photometric galaxy candidates, and there are possibilities that these
candidates are actually low-redshift interlopers (Zavala et al. 2023; Naidu et
al. 2022a; Fujimoto et al. 2022). Since ALMA observations to date have not
yielded conclusive redshifts for these high redshift galaxy candidates (Bakx
et al. 2023; Popping 2023; Yoon et al. 2022; Fujimoto et al. 2022), probably
due to the low metallicity and/or high density (Harikane et al. 2020), JWST
spectroscopy is crucial to obtain spectroscopic redshifts of these galaxy
candidates at $z\gtrsim 10$.

Figure 1:  Absolute UV magnitude as a function of the redshift for galaxies at
$6<z<17$. The red diamonds are spectroscopically-confirmed galaxies at
$z_{\mathrm{spec}}>8.5$ summarized in Table 1. Galaxies at
$z_{\mathrm{spec}}>9.0$ are marked with their names. The red open symbols are
galaxies with photometric redshifts selected with JWST/NIRCam in the
literature (Naidu et al. 2022b; Castellano et al. 2022a, b; Adams et al. 2022;
Atek et al. 2022; Donnan et al. 2023a, b; Finkelstein et al. 2022b, 2023;
Harikane et al. 2023a; Bouwens et al. 2022a, b; Morishita & Stiavelli 2022;
Bradley et al. 2022; Pérez-González et al. 2023). If a photometric candidate
is reported in more than one paper, we represent the candidate with a paper
that reports for the first time. The gray circles denote dropout galaxies
selected with deep HST images (Bouwens et al. 2015).

Recent JWST/NIRSpec spectroscopy has successfully confirmed the redshifts of
galaxies at $z>8$ (Figure 1). Early Director’s Discretionary Time (DDT)
observations obtained spectroscopic redshifts of two galaxies at
$z_{\mathrm{spec}}=9.51$ and $9.76$ (Roberts-Borsani et al. 2022; Williams et
al. 2022). The JWST Advanced Deep Extragalactic Survey (JADES)
spectroscopically confirmed HST-selected galaxy candidates at
$z_{\mathrm{spec}}>10$ (Curtis-Lake et al. 2022), including a bright galaxy at
$z_{\mathrm{spec}}=10.60$, GN-z11 (Bunker et al. 2023, see also Oesch et al.
2016, Jiang et al. 2021). Recently, further NIRSpec spectroscopic observations
have obtained spectroscopic redshifts of JWST-selected candidates at
$z_{\mathrm{spec}}>10$ (Curtis-Lake et al. 2022; Arrabal Haro et al. 2023a,
b), including the highest-redshift galaxy confirmed at
$z_{\mathrm{spec}}=13.20$, GS-z13-0 (Curtis-Lake et al. 2022).

In this study, we present spectroscopic constraints on the UV luminosity
functions and the cosmic SFR densities. Using the NIRSpec datasets obtained in
multiple programs as well as spectroscopically-confirmed galaxies in the
literature, we calculate the UV luminosity functions at $z\sim 9-16$, and
obtain the lower limit of the SFR densities at $z\sim 9-12$. These
spectroscopic constraints allow us to verify the earlier suggestions of the
mild redshift evolution of the luminosity function and the high SFR density at
$z>10$ based on the photometric datasets. We also discuss the physical origin
of the mild redshift evolution at $z>10$ as well as strategies for future JWST
surveys searching for galaxies at $z>10$ based on the spectroscopic results.

This paper is organized as follows. Section 2 describes the JWST/NIRSpec
observational data sets used in this study. In Section 3, we explain the
calculation of the effective volume and present the results of the UV
luminosity functions based on the spectroscopically-confirmed galaxies.
Section 4 presents stellar masses of spectroscopically-confirmed galaxies, and
Section 5 shows our spectroscopic constraints on the SFR densities. We discuss
the physical interpretations of the obtained results and strategies to remove
low redshift interlopers in future JWST observations in Section 6. Section 7
summarizes our findings. Throughout this paper, we use the Planck cosmological
parameter sets of the TT, TE, EE+lowP+lensing+BAO result (Planck Collaboration
et al. 2020): $\Omega_{\mathrm{m}}=0.3111$, $\Omega_{\Lambda}=0.6899$,
$\Omega_{\mathrm{b}}=0.0489$, $h=0.6766$, and $\sigma_{8}=0.8102$. We
basically assume the Salpeter (1955) initial mass function (IMF). All
magnitudes are in the AB system (Oke & Gunn 1983).

##  2 Observational Dataset and Galaxy Sample

###  2.1 ERO, ERS, GO, and DDT NIRSpec Observations

The data sets used in this study were obtained in the Early Release
Observations (EROs; Pontoppidan et al. 2022) targeting the SMACS 0723 lensing
cluster field (ERO-2736, PI: K. Pontoppidan), the Early Release Science (ERS)
observations of GLASS (ERS-1324, PI: T. Treu; Treu et al. 2022) and the Cosmic
Evolution Early Release Science (CEERS; ERS-1345, PI: S. Finkelstein;
Finkelstein et al. 2023, Arrabal Haro et al. 2023a), General Observer (GO)
observations targeting a $z\sim 11$ galaxy candidate (GO-1433, PI: D. Coe),
and the Director’s Discretionary Time (DDT) observations targeting $z\sim
12-16$ galaxy candidates (DD-2750, PI: P. Arrabal Haro; Arrabal Haro et al.
2023b). The ERO data were taken in the medium resolution ($R\sim 1000$)
filter-grating pairs F170LP-G235M and F290LP-G395M covering the wavelength
ranges of $1.7-3.1$ and $2.9-5.1$ $\mu$m, respectively. The total exposure
time of the ERO data is 4.86 hours for each filter-grating pair. The GLASS
data were taken with high resolution ($R\sim 2700$) filter-grating pairs of
F100LP-G140H, F170LP-G235H, and F290LP-G395H covering the wavelength ranges of
$1.0-1.6$, $1.7-3.1$ and $2.9-5.1$ $\mu$m, respectively. The total exposure
time of the GLASS data is 4.9 hours for each filter-grating pair. The CEERS
data were taken with the Prism ($R\sim 100$) that covers $0.6-5.3$ and medium-
resolution filter-grating pairs of F100LP-G140M, F170LP-G235M, and
F290LP-G395M covering the wavelength ranges of $1.0-1.6$, $1.7-3.1$ and
$2.9-5.1$ $\mu$m, respectively. The total exposure time of the CEERS data is
0.86 hours for each filter-grating pair. The GO-1433 and DDT data were
obtained with the Prism and the total exposure times are 3.7 and 5.1 hours,
respectively. These data were reduced with the JWST pipeline version 1.8.5
with the Calibration Reference Data System (CRDS) context file of
jwst_1028.pmap or jwst_1027.pmap with additional processes improving the flux
calibration, noise estimate, and the composition, in the same manner as
Nakajima et al. (2023). Please see Nakajima et al. (2023) for details of the
data reduction.

###  2.2 Obtained Spectra of High Redshift Galaxies

Figure 2:  NIRSpec spectra of Maisie’s Galaxy (CR2-z16-1) at
$z_{\mathrm{spec}}=11.40$, CEERS2_588 at $z_{\mathrm{spec}}=11.04$, and
MACS0647-JD at $z_{\mathrm{spec}}=10.17$. The top panel shows the two-
dimensional spectrum (yellow is positive), and the bottom panel shows the one-
dimensional spectrum. The red dashed line indicates the rest-frame 1215.67
$\mathrm{\AA}$ corresponding to the Ly$\alpha$-break.

Figure 3:  Same as Figure 2 but for CEERS2_7929 at $z_{\mathrm{spec}}=10.10$,
CEERS_99715 at $z_{\mathrm{spec}}=10.01$, and CEERS_35590 at
$z_{\mathrm{spec}}=9.97$.

Figure 4:  Same as Figure 2 but for CEERS2_2324 at $z_{\mathrm{spec}}=9.74$,
Gz9p3 at $z_{\mathrm{spec}}=9.31$, and CEERS-24 at $z_{\mathrm{spec}}=9.00$.
CEERS2_2324 is not used in the constraints in this study because the observed
break is not significant. The spectra of Gz9p3 and CEERS-24 are taken with the
high and medium resolutions, respectively. We plot the smoothed one-
dimensional spectra for them.

Figure 5:  Same as Figure 2 but for CEERS-23 at $z_{\mathrm{spec}}=8.88$,
CEERS1_6059 at $z_{\mathrm{spec}}=8.88$, and CEERS1_3858 at
$z_{\mathrm{spec}}=8.81$. The spectrum of CEERS-23 is taken with the medium
resolution. We plot the smoothed one-dimensional spectrum for CEERS-23.

Figure 6:  Same as Figure 2 but for CEERS_43833 at $z_{\mathrm{spec}}=8.76$,
CEERS-1025 at $z_{\mathrm{spec}}=8.71$, and CEERS_1019 at
$z_{\mathrm{spec}}=8.68$. The spectrum of CEERS-1025 is taken with the medium
resolution. We plot the smoothed one-dimensional spectrum for that source.

Figure 7:  Same as Figure 2 but for CEERS_90671 at $z_{\mathrm{spec}}=8.64$
and EGS_z910_44164 at $z_{\mathrm{spec}}=8.61$.  Figure 8:  NIRSpec spectrum
of 93316 (CR2-z16-1), a $z\sim 16$ galaxy candidate that is found to be
$z_{\mathrm{spec}}=4.912$.  Figure 9:  Spectrum around [Oii]$\lambda$3727 and
[Neiii]$\lambda$3869 emission lines for Maisie’s Galaxy (CR2-z12-1) at
$z_{\mathrm{spec}}=11.40$. The bottom two panels show the stacked two-
dimensional and one-dimensional spectra. The other nine panels are spectra of
nine individual exposures. The [Oii] line is clearly seen in both the stacked
two-dimensional and one-dimensional spectra, and spectra of the individual
exposures are not contaminated by artifacts around the wavelength of [Oii],
suggesting that this emission line feature is real. The red arrow indicates
the position of the [Oii] line.  Figure 10:  Same as Figure 9 but for
CEERS2_588 at $z_{\mathrm{spec}}=11.04$. The [Oii]$\lambda$3727 line is
clearly seen in the stacked two-dimensional and one-dimensional spectra and
some of the individual spectra, and they are not contaminated by artifacts
around the wavelength of [Oii], suggesting that this emission line feature is
real. The red arrows indicate the positions of the [Oii] line.  Table 1: List
of Spectroscopically-Confirmed Galaxies

Name | R.A. | Decl. | $z_{\mathrm{spec}}$ | $M_{\mathrm{UV}}$ | Spec-$z$ Ref. | Phot. Ref.
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
High Redshift Galaxies at $z_{\mathrm{spec}}>8.5$
GS-z13-0 | 03:32:35.97 |  $-$27:46:35.4 | 13.20 (Break) | $-18.5$ | CL22 | Ro22
GS-z12-0 | 03:32:39.92 |  $-$27:49:17.6 | 12.63 (Break) | $-18.8$ | CL22 | Ro22
GS-z11-0 | 03:32:39.54 |  $-$27:46:28.6 | 11.58 (Break) | $-19.3$ | CL22 | Bow11,El13,Ro22
Maisie’s Galaxy (CR2-z12-1) | 14:19:46.36 |  $+$52:56:32.8 | 11.40 (Line)∗ | $-20.1$ | AH23a,This | Fi22b,Har23,Do23,Fi23,Bow22
CEERS2_588 | 14:19:37.59 |  $+$52:56:43.8 | 11.04 (Line) | $-20.4$ | This | Fi23,Do23
GN-z11 | 12:36:25.46 |  $+$62:14:31.4 | 10.60 (Line) | $-21.5$ | Bu23 | Oe18,Tac23
GS-z10-0 | 03:32:38.12 |  $-$27:46:24.6 | 10.38 (Break) | $-18.4$ | CL22 | Oe18,Ro22
MACS0647-JD† | 06:47:55.73 |  $+$70:14:35.8 | 10.17 (Line) | $-20.3$ | Hs-P,This | Co13,Hs22
CEERS2_7929 | 14:19:41.47 |  $+$52:54:41.5 | 10.10 (Break) | $-19.3$ | AH23a,This | Fi23,Do23
CEERS_99715 | 14:19:14.84 |  $+$52:44:13.6 | 10.01 (Break)‡ | $-20.5$ | AH23b,This | Fi23
CEERS_35590 | 14:18:55.81 |  $+$52:45:29.1 | 9.97 (Break)‡ | $-20.1$ | AH23b,This | Fi23
A2744-JD1 | 00:14:22.80 |  $-$30:24:02.6 | 9.76 (Break) | $-17.4$ | RB22 | Zi14,Oe18
CEERS2_2324 | 14:19:26.78 |  $+$52:54:16.6 | 9.74? (Break)¶ | $-20.1$ | AH23a,This | Fi23
11027† | 21:29:41.17 |  $+$00:05:30.1 | 9.51 (Line) | $-17.4$ | Wi22 | Wi22
GS+53.11243-27.77461 | 03:32:26.98 |  $-$27:46:28.6 | 9.437 (Line) | $-20.4$ | Cam23 |  $\cdots$
Gz9p3 | 00:14:28.14 |  $-$30:25:32.0 | 9.313 (Line) | $-21.6$ | Boy23,This | Cas22
MACS1149-JD1† | 11:49:33.58 |  $+$22:24:45.7 | 9.110 (Line) | $-18.5$ | Has18 | Zh12
CEERS-24 | 14:19:35.34 |  $+$52:50:37.9 | 8.998 (Line) | $-19.4$ | Tan23,Fu23,This | Wh23,Fi23
CEERS-23 | 14:19:36.30 |  $+$52:50:49.2 | 8.881 (Line) | $-18.9$ | Tan23,Fu23,This | Fi23
CEERS1_6059 | 14:20:02.81 |  $+$52:59:17.9 | 8.876 (Line) | $-20.8$ | Fu23,Nak23,This | Wh23,Fi23
CEERS1_3858 | 14:19:58.66 |  $+$52:59:21.8 | 8.807 (Line) | $-20.4$ | Fu23,This | Fi23
CEERS_43833 | 14:19:45.27 |  $+$52:54:42.3 | 8.763 (Line) | $-20.4$ | AH23a,This | Fi23
CEERS-1025 | 14:19:52.21 |  $+$52:55:58.6 | 8.715 (Line) | $-21.1$ | Tan23,Nak23,This | Fi22a
CEERS_1019 | 14:20:08.49 |  $+$52:53:26.4 | 8.679 (Line) | $-22.1$ | Zi15,Tan23,Nak23,Sa23,La23,This | Fi22a
CEERS_90671 | 14:19:50.71 |  $+$52:50:32.5 | 8.638 (Line) | $-18.7$ | AH23b,This | Fi23
EGS_z910_44164 | 14:20:52.50 |  $+$53:04:11.5 | 8.612 (Line) | $-21.6$ | La22,Tan23,Nak23,This | Fi22a
A Low Redshift Interloper
93316 (CR2-z16-1) | 14:19:39.49 | $+$52:56:34.9 | 4.912 (Line) |  $\cdots$ | AH23a,This | Do23,Har23,Za22
Nai22,Fi23,Bow22

Note. — (1) Name. Ones in the parentheses are names in Harikane et al.
(2023a). (2) Right ascension. (3) Declination. (4) Spectroscopic redshift. The
spectroscopic feature used to determine the redshift is noted (Break: Lyman
break, Line: emission line). (5) Absolute UV magnitude. (6,7) References for
spectroscopic redshifts and photometry (This: this work, AH23a: Arrabal Haro
et al. (2023b), AH23b: Arrabal Haro et al. (2023a), Bow22: Bouwens et al.
(2022a), Boy23: Boyett et al. (2023), Cam22: Cameron et al. (2023), Cas22:
Castellano et al. (2022a), CL22: Curtis-Lake et al. (2022), Co13: Coe et al.
(2013), El13: Ellis et al. (2013), Do23: Donnan et al. (2023b), Fi22b:
Finkelstein et al. (2022b), Fi23: Finkelstein et al. (2023), Fu23: Fujimoto et
al. (2023), Har22b: Harikane et al. (2022a), Har23: Harikane et al. (2023a),
Has18: Hashimoto et al. (2018), Hs22: Hsiao et al. (2022), Hs-P: Hsiao et al.
in prep., La22: Larson et al. (2022), La23: Larson et al. (2023), Nai22: Naidu
et al. (2022a), Nak23: Nakajima et al. (2023), Oe18: Oesch et al. (2018),
RB22: Roberts-Borsani et al. (2022), Ro22: Robertson et al. (2022), Sa23:
Sanders et al. (2023), Tan23: Tang et al. (2023), Tac23: Tacchella et al.
(2023), Wh23: Whitler et al. (2023), Wi22: Williams et al. (2022), Za22:
Zavala et al. (2023), Zh12: Zheng et al. (2012), Zi14: Zitrin et al. (2014)
Zi15: Zitrin et al. (2015)).

† The estimate of the survey volume for these objects is difficult due to the
lensing effect. We do not use these objects in the constraints on the UV
luminosity functions and the cosmic SFR densities.

∗ The redshift of this object is reported to be
$z_{\mathrm{spec}}=11.44^{+0.09}_{-0.08}$ based on the Lyman break in Arrabal
Haro et al. (2023b). We remeasure the redshift to be $z_{\mathrm{spec}}=11.40$
based on the [Oii] emission line, consistent with their estimate within the
error.

‡ The spectroscopic redshifts presented here are based on our measurements
using the Lyman break. Our redshifts,
$z_{\mathrm{spec}}=10.01_{-0.10}^{+0.04}$ and $9.97_{-0.16}^{+0.01}$ for
CEERS_99715 and CEERS_35590, respectively, are consistent with the
measurements in Arrabal Haro et al. (2023a),
$z_{\mathrm{spec}}=9.77^{+0.37}_{-0.29}$ and $10.01^{+0.14}_{-0.19}$.

¶ The redshift of this object is highly uncertain because the observed break
is not significant. We do not use this object in the constraints on the UV
luminosity function and the cosmic SFR density.

We then investigated the reduced spectra to determine spectroscopic redshifts
of galaxies at $z\gtrsim 9$ by matching the coordinates of the spectroscopic
targets with those of photometric galaxy candidates identified in the
literature (Finkelstein et al. 2022a, b, 2023; Naidu et al. 2022b; Castellano
et al. 2022a, b; Adams et al. 2022; Atek et al. 2022; Donnan et al. 2023b, a;
Harikane et al. 2023a; Bouwens et al. 2022a, b; Morishita & Stiavelli 2022),
as well as by visually inspecting the spectra. We found that a total of 16
galaxies were spectroscopically confirmed at $z_{\mathrm{spec}}>8.5$. Figures
2-8 show spectra of the galaxies, and Table 1 summarizes them. We describe the
spectra of some spectroscopically-confirmed galaxies below.

Maisie’s Galaxy was firstly photometrically identified in Finkelstein et al.
(2022b), and was reported as CR2-z12-1 in Harikane et al. (2023a). As shown in
the top panel of Figure 2, the spectrum of this galaxy shows the Lyman break,
the [Oii]$\lambda$3727 line (5.5$\sigma$), and the possible
[Neiii]$\lambda$3869 line (2.4$\sigma$). The redshift of this galaxy was
reported to be $z_{\mathrm{spec}}=11.44^{+0.09}_{-0.08}$ based on the Lyman
break in Arrabal Haro et al. (2023b). Arrabal Haro et al. (2023b) discuss that
the [Oii] line may not be real due to 1) the presence of image defects in the
individual nods, 2) no clear four negative patterns in the two-dimensional
spectrum, and 3) a low [Neiii]/[Oii] ratio inconsistent with the low
metallicity inferred from the spectral energy distribution (SED) fitting.
However, 1) we do not find any obvious image defects in the spectra of the
individual exposures (Figure 9), 2) it is not unusual that clear four negative
patterns are not seen for the $5.5\sigma$ line, and 3) the metallicity from
the SED fitting is not robustly constrained and the low [Neiii]/[Oii] ratio
([Neiii]/[Oii]$\sim 0.3$) is seen in low-metallicity galaxies (Nakajima et al.
2022). Moreover, the wavelength of the possible [Neiii] line is consistent
with that of [Oii]. Given these facts, we consider this [Oii] line is real. We
determine the redshift to be $z_{\mathrm{spec}}=11.40$ using the [Oii] line,
which is consistent with the previous estimate based on the Lyman break but
slightly lower than that ($z_{\mathrm{spec}}=11.44^{+0.09}_{-0.08}$; Arrabal
Haro et al. 2023b).

The spectrum of CEERS2_588, which was photometrically identified in
Finkelstein et al. (2023) and Donnan et al. (2023b), shows the Lyman break,
the [Oii]$\lambda$3727 line (5.7$\sigma$), and the possible
[Neiii]$\lambda$3869 line (2.8$\sigma$), as shown in the middle panel of
Figure 2. The [Oii] line feature is also seen in some of the individual frames
(Figure 10), and they are not affected by an obvious image defect, indicating
that this [Oii] line is real. The rest-frame equivalent width of the [Oii]
line is $\sim 100\ \mathrm{\AA}$, comparable to that seen in galaxies at
$z\sim 2-3$ (Reddy et al. 2018). We newly determine the spectroscopic redshift
of CEERS2_588 to be $z_{\mathrm{spec}}=11.04$ based on the [Oii] and [Neii]
lines. The wavelength of the Lyman break is consistent with this redshift
estimate. The [Neiii]/[Oii] ratio is low, [Neiii]/[Oii]$\sim 0.6$, and is
comparable to those seen in low-metallicity galaxies in Nakajima et al.
(2022). Given its UV magnitude, $M_{\mathrm{UV}}=-20.4$ mag, this galaxy is
the most luminous galaxy spectroscopically confirmed at
$z_{\mathrm{spec}}>11.0$.

In the bottom panel of Figure 2, we present the spectrum of MACS0647-JD, which
was firstly photometrically reported in Coe et al. (2013). This galaxy is a
triply-lensed galaxy, and a recent study using NIRCam images suggests that
MACS0647-JD is a merger. We show the NIRSpec spectrum of JD1 (MSA ID: 3593) in
the observation ID of 23. The spectrum shows the Lyman break and the
Ciii]$\lambda$1909, [Neiii]$\lambda$3869, and H$\gamma$ emission lines,
suggesting the spectroscopic redshift of $z_{\mathrm{spec}}=10.17$. The data
analysis by the PI team will be presented in Hsiao et al in prep.

There are four galaxies whose spectra show only the Lyman break without clear
emission lines, CEERS2_7929, CEERS_99715, CEERS_35590, and CEERS2_2324 (Figure
3 and the top panel in Figure 4). For these sources, we fit model spectra to
the observed ones at the observed wavelength of $0.6-3.0\ \mu$m including the
break using prospector (Johnson et al. 2021), and obtain the best-fit
spectroscopic redshifts. Model spectra are derived from Flexible Stellar
Population Synthesis (FSPS; Conroy et al. 2009; Conroy & Gunn 2010) package
with the modules for Experiments in Stellar Astrophysics Isochrones and
Stellar Tracks (MIST; Choi et al. 2016). The boost of ionizing flux production
of massive stars is included in the MIST isochrones (Choi et al. 2017). Here
we assume the stellar IMF determined by Chabrier (2003), the Calzetti et al.
(2000) dust extinction law, and the intergalactic medium (IGM) attenuation
model by Madau (1995). The Ly$\alpha$ emission line is also masked considering
the high IGM neutral fraction at these redshifts. We adopt a flexible star
formation history with five bins that are spaced equally in logarithmic times
between 0 Myr and a lookback time that corresponds to $z=30$, where the SFR
within each bin is constant. The parameter settings for the stellar mass, dust
extinction, and metallicity are the same as those in Harikane et al. (2023a).
We search for the best-fit model to the observed photometry with the MCMC
method by using emcee (Foreman-Mackey et al. 2013).

We obtain the best fit redshifts of $z_{\mathrm{spec}}=10.10_{-0.22}^{+0.07}$
for CEERS2_7929, $z_{\mathrm{spec}}=10.01_{-0.10}^{+0.04}$ for CEERS_99715,
$z_{\mathrm{spec}}=9.97_{-0.16}^{+0.01}$ for CEERS_35590, and
$z_{\mathrm{spec}}=9.74_{-0.09}^{+0.03}$ for CEERS2_2324. Our measurements
agree with those in Arrabal Haro et al. (2023b, a),
$z_{\mathrm{spec}}=10.10_{-0.26}^{+0.13}$ for CEERS2_7929,
$z_{\mathrm{spec}}=9.77_{-0.29}^{+0.37}$ for CEERS_99715, and
$z_{\mathrm{spec}}=10.01_{-0.19}^{+0.14}$ for CEERS_35590. Arrabal Haro et al.
(2023b) present the spectrum of CEERS2_2324 (Finkelstein et al. 2023) with a
possible redshift of $z_{\mathrm{spec}}=9.744$, but do not consider this
galaxy confirmed due to the large uncertainty of the redshift estimate. Our
spectrum also shows a continuum detection at $\sim 1.3-1.9\ \mu$m and a
possible break around $\sim 1.3\ \mu$m, suggesting
$z_{\mathrm{spec}}=9.74_{-0.09}^{+0.03}$, consistent with the estimate in
Arrabal Haro et al. (2023b). A possible emission line feature around $\sim
5.15\ \mu$m may not be real because an image defect is seen in one of the
individual nods around this wavelength. Since the significance of the break is
not high, we do not use this galaxy in our analysis.

CEERS_1019 was firstly spectroscopically confirmed with the Ly$\alpha$
emission line in Zitrin et al. (2015). Recently, Larson et al. (2023) report a
broad H$\beta$ emission line in this galaxy, suggesting AGN activity (see also
Harikane et al. 2023b). The NIRSpec spectrum (the bottom panel in Figure 6)
shows multiple emission lines including Niv]$\lambda$1486, Ciii]$\lambda$1909,
[Oii]$\lambda$3727, [Neiii]$\lambda$3869, H$\delta$, H$\gamma$, H$\beta$, and
[Oiii]$\lambda\lambda$4959,5007 emission lines, indicating a spectroscopic
redshift of $z_{\mathrm{spec}}=8.679$. We do not include this galaxy in the
calculation of the cosmic SFR density because of the possible AGN activity.

93316 was firstly photometrically reported in Donnan et al. (2023b) as a
$z\sim 16$ galaxy candidate (CR2-z16-1 in Harikane et al. 2023a), and many
studies discuss the properties of this galaxy including the possible low-
redshift solution based on the available photometric data (Harikane et al.
2023a; Zavala et al. 2023; Naidu et al. 2022a; Finkelstein et al. 2023;
Bouwens et al. 2022a). As shown in Figure 8, the spectrum of 93316 obtained in
Arrabal Haro et al. (2023b) shows two prominent emission lines at $\sim 2.95$
and 3.89 $\mu$m. This indicates that 93316 is a galaxy at
$z_{\mathrm{spec}}=4.91$ whose strong emission lines mimic the photometric
Lyman break at $z\sim 16$, as discussed in Zavala et al. (2023) and Naidu et
al. (2022a). This result indicates that galaxies at $z=4.9$ with strong
emission lines can be interlopers in a $z\sim 16$ galaxy selection, and we
need to carefully remove these interlopers from a sample of high redshift
galaxy candidates. In Section 6.2, we discuss strategies to remove this kind
of low redshift interlopers in the future JWST survey to search for galaxies
at $z\gtrsim 12$.

###  2.3 Spectroscopically-Confirmed Galaxies in the Literature

In addition to the 16 confirmed galaxies at $z_{\mathrm{spec}}>8.5$ described
above, we compile spectroscopically-confirmed galaxies in the literature. The
sample includes four galaxies at $z_{\mathrm{spec}}=10.38-13.20$ reported and
confirmed in Robertson et al. (2022) and Curtis-Lake et al. (2022), GN-z11 at
$z_{\mathrm{spec}}=10.60$ confirmed in Bunker et al. (2023), A2744-JD1 at
$z_{\mathrm{spec}}=9.76$ in Roberts-Borsani et al. (2022), 11027 at
$z_{\mathrm{spec}}=9.51$ in Williams et al. (2022), GS+53.11243-27.77461 at
$z_{\mathrm{spec}}=9.437$ in Cameron et al. (2023), and MACS1149-JD1 at
$z_{\mathrm{spec}}=9.110$ in Hashimoto et al. (2018). Including the galaxies
confirmed in Section 2.2, our sample finally contains a total of 25 galaxies
at $z_{\mathrm{spec}}=8.610-13.20$, which is sufficiently large for the
measurements of the UV luminosity functions and the cosmic SFR densities.
Figure 1 show the absolute UV magnitudes and spectroscopic redshifts of our
final sample of spectroscopically confirmed galaxies at
$z_{\mathrm{spec}}>8.5$, and Table 1 summarizes properties of them.

##  3 UV Luminosity Function

###  3.1 Effective Volume Estimate

We calculate the UV luminosity functions at $z\sim 9-12$ using the
spectroscopically confirmed galaxies at $z_{\mathrm{spec}}>8.5$. We divide our
sample into the three redshift subsample at $z_{\mathrm{spec}}=8.5-9.5$,
$9.5-11.0$, and $11.0-13.5$, and calculate the number densities at $z\sim 9$,
$10$, and $12$. We also calculate an upper limit on the number density at
$z\sim 16$ based on the result of the $z\sim 16$ candidate, 93316, which is
found to be $z_{\mathrm{spec}}=4.912$. Since the galaxies in our spectroscopic
sample are confirmed with NIRSpec/MSA whose target selection and detection
completenesses are not well-known, the estimate of the effective volume for
the luminosity function is not straightforward. We use two methods to estimate
the effective volume, as detailed below.

The first method is using the effective volume estimates published in the
literature. Since sometimes all of the galaxies in a bin of a luminosity
function are spectroscopically confirmed, we can use the effective volume and
the number density estimated in the previous studies. For example, among the
number density bins at $z\sim 10$ in Oesch et al. (2018), the brightest bin
and the two faintest bins are composed of GN-z11 and GS-z10-0 and A2744-JD1,
respectively, all of which are spectroscopically confirmed. Thus we can use
the effective volume estimates in the brightest and two faintest bins in Oesch
et al. (2018) for spectroscopic constraints on the number densities.
Similarly, the brightest bin in Castellano et al. (2022a) is composed of Gz9p3
at $z_{\mathrm{spec}}=9.313$ (DHz1 in Castellano et al. 2022a). We also
constraint the upper limit of the number density at $z\sim 16$ using the
effective volume estimate in Harikane et al. (2023a) without the Stephan’s
Quintet field where one $z\sim 16$ candidate (S5-z16-1) is identified.

The second method is using the field of view of the NIRSpec. Since the other
galaxies in our sample are part of photometric samples that are not completely
observed with spectroscopy, we can place a conservative lower limit on the
number density. The faint galaxies at $z_{\mathrm{spec}}=11.58-13.20$,
GS-z11-0, GS-z12-0, and GS-z13-0, are confirmed in one pointing observation
with NIRSpec in the JADES (Curtis-Lake et al. 2022). Thus we assume the
effective survey area of 9 arcmin2 ($=$1 field of view of NIRSpec) and
calculate the survey volume at the redshift range of $z=11.0-13.5$. Since we
cannot put slits on all of the high redshift galaxy candidates with NIRSpec
MSA due to the overlap of the spectra and the galaxy selection incompleteness,
this survey volume is an upper limit. We also estimate the upper limit of the
survey volume for GS+53.11243-27.77461 at $z_{\mathrm{spec}}=9.437$ similarly
assuming the survey area of 9 arcmin2. For the sources selected with the CEERS
NIRCam images (e.g., Finkelstein et al. 2022b, 2023; Donnan et al. 2023b;
Harikane et al. 2023a; Bouwens et al. 2022a) that were confirmed in the CEERS
and the DDT NIRSpec observations, we assume the survey area of 45 arcmin2,
which is an approximate estimate of the area of the regions where the NIRSpec
pointings are overlapping with the NIRCam images. For the other sources
selected with HST images in the CEERS field (Finkelstein et al. 2022a), we use
the survey area of the EGS field in Finkelstein et al. (2022a), 205 arcmin2.
We do not use some sources in the lensing fields, MACS0647-JD, 11027, and
MACS1149-JD1, because the survey area estimates for these sources are not
straightforward due to the lensing magnification. We discuss that our
constraints are consistent with these spectroscopically-confirmed galaxies in
Section 5.

Based on the effective survey volumes estimated with these two methods, we
obtain the constraints on the number density of galaxies at $z\sim 9$, $10$,
$12$, and $16$. The 1$\sigma$ uncertainty of the number density is calculated
by taking the Poisson confidence limit (Gehrels 1986) and cosmic variance into
account. We estimate the cosmic variance in the number densities following the
procedures in Somerville et al. (2004). To obtain the large-scale bias
parameter needed for the cosmic variance calculation, we estimate the dark
matter halo mass of the galaxies using the simple abundance matching technique
described in Harikane et al. (2016, Equation (66)) assuming a unity duty cycle
and no satellite galaxies. We use the double-power law luminosity functions in
Harikane et al. (2023a) and the halo mass function in Behroozi et al. (2013),
which is a modification of the Tinker et al. (2008) mass function. Then we
calculate the bias parameters from the estimated halo masses using a redshift-
dependent relation between the bias and the halo mass presented in Tinker et
al. (2010).

Figure 11: UV luminosity functions at $z\sim 9$ (upper-left), $z\sim 10$
(upper-right), $z\sim 12$ (lower-left), and $z\sim 16$ (lower-right). The red
diamonds represent the number densities of galaxies with spectroscopic
redshifts derived in this study. The errors include the cosmic variance (see
text). The gray open symbols are estimates based on photometric samples by
previous studies (Harikane et al. 2023a; Pérez-González et al. 2023; Donnan et
al. 2023b; Bouwens et al. 2021, 2022a, 2022b; Leethochawalit et al. 2022;
Stefanon et al. 2019; Bowler et al. 2020; McLeod et al. 2016; Bagley et al.
2022; Castellano et al. 2022a; Morishita et al. 2018; Morishita & Stiavelli
2022; Oesch et al. 2018; Finkelstein et al. 2022a, b; Naidu et al. 2022b). The
gray solid and dashed lines are double power-law and Schechter functions,
respectively, interpolated and extrapolated using the results at $z\sim 9-12$
in Harikane et al. (2023a, Equations (3)-(9)). The brightest bin at $z\sim 9$
(the open red diamond) could be affected by the overdensities (see text).

###  3.2 Results

Figure 11 shows our constraints on the number densities of galaxies at $z\sim
9$, $10$, $12$, and $16$, and Table 2 summarizes them. Our spectroscopic
constraints are consistent with previous estimates of the number densities in
the literature at $z\sim 9-12$ based on the photometric samples, except for
the brightest bin at $z\sim 9$, which is higher than most of the estimates but
consistent with that of Bagley et al. (2022). This bin includes Gz9p3 at
$z_{\mathrm{spec}}=9.313$ in the Abell 2744 field and CEERS_1019 and
EGS_z910_44164 at $z_{\mathrm{spec}}=8.679$ and $8.610$, respectively, in the
CEERS field. As discussed in Larson et al. (2022) and Castellano et al.
(2022a), this high number density may originate from possible galaxy
overdensities at $z\sim 9$ in these fields. Although further spectroscopic
observations in other fields are needed to distinguish whether the high number
density is real or is due to the cosmic variance, we do not use these galaxies
possibly in the overdensities for the cosmic SFR density estimate in Section
5.

In Figure 11, we also plot the double-power law functions,

$\displaystyle\Phi(M_{\rm UV})=\frac{\ln 10}{2.5}\phi^{*}$
$\displaystyle\times\left[10^{0.4(\alpha+1)(M_{\rm UV}-M_{\rm
UV}^{*})}+10^{0.4(\beta+1)(M_{\rm UV}-M_{\rm UV}^{*})}\right]^{-1},$ (1)

and the Schechter functions,

$\displaystyle\Phi(M_{\rm UV})$ $\displaystyle=$ $\displaystyle\frac{\ln
10}{2.5}\phi^{*}10^{-0.4(M_{\rm UV}-M_{\rm UV}^{*})(\alpha+1)}$ (2)
$\displaystyle\times\exp\left(-10^{-0.4(M_{\rm UV}-M_{\rm UV}^{*})}\right).$

The parameters in each function are calculated from the interpolation and
extrapolation of the results at $z\sim 9$ and $12$ in Harikane et al. (2023a),

$\displaystyle M^{*}_{\mathrm{UV}}$ $\displaystyle=-0.09(z-9)-19.33$ (3)
$\displaystyle\mathrm{log}\phi^{*}$ $\displaystyle=-0.28(z-9)-3.50$ (4)
$\displaystyle\alpha$ $\displaystyle=-2.10$ (5) $\displaystyle\beta$
$\displaystyle=0.15(z-9)-3.27$ (6)

for the double-power law function, and

$\displaystyle M^{*}_{\mathrm{UV}}$ $\displaystyle=0.32(z-9)-21.24$ (7)
$\displaystyle\mathrm{log}\phi^{*}$ $\displaystyle=-0.08(z-9)-4.83$ (8)
$\displaystyle\alpha$ $\displaystyle=-2.35$ (9)

for the Schechter function. As shown in Figure 11, these functions are
consistent with our spectroscopic constraints on the number densities within
$1\sigma$ errors.

Table 2: Spectroscopic Constraints on the Luminosity Function at Each Redshift

$M_{\mathrm{UV}}$ | $\Phi$
---|---
(ABmag) | $\mathrm{Mpc^{-3}}\ \mathrm{mag^{-1}}$
$z\sim 9\ (z=8.5-9.5,z_{\mathrm{ave}}=8.93)$
$-22.0$ | $6.6^{+7.1}_{-4.7}\times 10^{-6}$
$-21.0$ | $>5.1^{+7.0}_{-3.8}\times 10^{-6}$
$-20.0$ | $>2.9^{+3.2}_{-2.2}\times 10^{-5}$
$-19.0$ | $>3.5^{+3.7}_{-2.4}\times 10^{-5}$
$z\sim 10\ (z=9.5-11.0,z_{\mathrm{ave}}=10.24)$
$-21.6$ | $1.0^{+2.3}_{-0.9}\times 10^{-6}$
$-20.6$ | $>8.7^{+20.5}_{-8.4}\times 10^{-6}$
$-19.6$ | $>2.6^{+2.8}_{-1.8}\times 10^{-5}$
$-18.6$ | $1.9^{+4.7}_{-1.9}\times 10^{-4}$
$-17.6$ | $6.3^{+15.8}_{-6.3}\times 10^{-4}$
$z\sim 12\ (z=11.0-13.5,z_{\mathrm{ave}}=11.98)$
$-20.1$ | $>8.8^{+9.1}_{-5.5}\times 10^{-6}$
$-18.7$ | $>6.6^{+6.0}_{-4.6}\times 10^{-5}$
$z\sim 16$
$-21.9$ | $<9.8\times 10^{-6}$

Note. — Errors and upper limits are $1\sigma$.

Figure 12:  Comparison of the luminosity functions with theoretical
predictions in the literature at $z\sim 9$ (upper-left), $z\sim 10$ (upper-
right), $z\sim 12$ (lower-left), and $z\sim 16$ (lower-right). The red symbols
show observational results based on the spectroscopically-confirmed galaxies
obtained in this study. The blue lines show the theoretical and empirical
models of Dayal et al. (2014, 2019, solid line), Behroozi et al. (2019, 2020,
dotted-dashed line), Vijayan et al. (2021, double-dotted dashed line at $z\sim
9$), Wilkins et al. (2023, double-dotted dashed line at $z\sim 10-16$), Mason
et al. (2015, 2023, dashed line; their model with dust extinction), and Yung
et al. (2023, dotted line), . At $z\sim 12$, our spectroscopic constraints are
higher than the number densities of some models predicting rapid redshift
evolution.  Figure 13:  Theoretical predictions for the number of bright
galaxies at $z\geq 11.0$ with $M_{\mathrm{UV}}<-19.8$ mag. These numbers are
based on the theoretical models of Dayal et al. (2014, 2019), Behroozi et al.
(2020), Wilkins et al. (2023), Mason et al. (2023), and Yung et al. (2023).
The red horizontal line with the shaded region indicates the number of the
spectroscopically-confirmed galaxies at $z_{\mathrm{spec}}\geq 11.0$ with
$M_{\mathrm{UV}}<-19.8$ mag ($N_{\mathrm{obs,spec}}=2$) and its uncertainty
including both the Poisson error and the cosmic variance. Most of the models
predict a lower number of bright ($M_{\mathrm{UV}}<-19.8$ mag) galaxies at
$z\geq 11.0$ than the observation.

Figure 14:  Comparison of the luminosity functions with models assuming
various star formation efficiencies (the black curves), $f_{\mathrm{SF}}=2\%$,
$5\%$, $15\%$, $40\%$, and $100\%$, which are defined as the ratio of the SFR
to the baryon accretion rate (see also Inayoshi et al. 2022). The red symbols
show observational results based on the spectroscopically-confirmed galaxies.
The number densities at $z\sim 12$ suggest a high star formation efficiency of
$f_{\mathrm{SF}}\gtrsim 5\%$.

###  3.3 Comparison with Model Predictions

In Figure 12, we compare our constraints on the number densities at $z\sim
9-16$ with theoretical model predictions in Dayal et al. (2014, 2019),
Behroozi et al. (2019, 2020), Wilkins et al. (2023), Mason et al. (2023), and
Yung et al. (2023). These model predictions agree with our spectroscopic
constraints at $z\sim 9-10$. However, at $z\sim 12$, the number densities of
Mason et al. (2023) and Yung et al. (2023) are lower than our lower limit
around $M_{\mathrm{UV}}=-20$ mag, implying mild redshift evolution compared to
rapid evolution predicted by these models. Similarly, Figure 13 shows the
predicted number of bright galaxies at $z>11.0$ with $M_{\mathrm{UV}}<-19.8$
mag. Figure 13 indicates that most of the models underpredict the number of
galaxies compared to that of the spectroscopically-confirmed ones in this
study, although the significance is small, and more data are needed to obtain
the conclusion. This difference between the observations and models would
suggest that the feedback effects in the models may be too strong to produce
abundant bright galaxies, lower dust obscuration in these bright galaxies than
the model assumptions, and/or that there exist hidden AGNs that produce
radiation comparable with or more than stellar components of the galaxies,
although there is a possibility that this difference may be caused by other
physical processes, as discussed in Section 6.1. Further spectroscopic
observations will improve the statistics and allow us to distinguish these
models, important to understand star formation and feedback in these early
galaxies.

We also compare our constraints with models assuming different star formation
efficiencies, $f_{\mathrm{SF}}$, which is defined as the ratio of the SFR to
the baryon accretion rate (see also, e.g., Bouché et al. 2010; Behroozi et al.
2013, 2019; Mason et al. 2015; Harikane et al. 2018, 2022b; Tacchella et al.
2018; Moster et al. 2018; Inayoshi et al. 2022). In the models, the SFR is
expressed as,

$\mathrm{SFR}=f_{\mathrm{SF}}\times
f_{\mathrm{b}}\times{\frac{dM_{\mathrm{h}}}{dt}}(M_{\mathrm{h}},z),$ (10)

where $f_{\mathrm{b}}=\Omega_{\mathrm{b}}/\Omega_{\mathrm{n}}=0.157$ is the
cosmic baryon fraction, and ${\frac{dM_{\mathrm{h}}}{dt}}(M_{\mathrm{h}},z)$
is the matter accretion rate in Behroozi & Silk (2015). The SFR is converted
to the UV luminosity using the following equation assuming the Salpeter (1955)
IMF,

$L_{\mathrm{UV}}(\mathrm{ergs^{-1}Hz^{-1}})=\mathrm{SFR}(M_{\odot}\mathrm{yr^{-1}})/(1.15\times
10^{-28}).$ (11)

Then we calculate the UV luminosity function, $\frac{dn}{dM_{\mathrm{UV}}}$,
using the halo mass function, $\frac{dn}{dM_{\mathrm{h}}}$,

$\frac{dn}{dM_{\mathrm{UV}}}=\frac{dn}{dM_{\mathrm{h}}}\left|\frac{dM_{\mathrm{h}}}{dM_{\mathrm{UV}}}\right|,$
(12)

assuming a 0.2 dex scatter in the halo mass,
$\sigma_{\mathrm{log}M_{\mathrm{h}}}=0.2$, following Harikane et al. (2018).

We plot these models in Figure 14 with our spectroscopic constraints. At
$z\sim 9-10$, our constraints on galaxies with $-21\lesssim
M_{\mathrm{UV}}\lesssim-18$ mag are consistent with the models assuming
$f_{\mathrm{SF}}=2\%$, which is the maximum value of the star formation
efficiency inferred from observations at $z\sim 7$ (see Figure 19 in Harikane
et al. 2022b). At $z\sim 12$, our spectroscopic constraints are consistent
with a high star formation efficiency with $f_{\mathrm{SF}}\gtrsim 5\%$ for
galaxies with $-20\lesssim M_{\mathrm{UV}}\lesssim-19$ mag, which is in
contrast to lower redshift results. The physical origin of the high star
formation efficiency is discussed in Section 6.1.

Figure 15:  Stellar masses of spectroscopically-confirmed galaxies. The red-
filled diamonds are stellar masses of bright galaxies at
$z_{\mathrm{spec}}>8.5$ identified in the CEERS field, including the brightest
galaxy at $z_{\mathrm{spec}}>11.0$, CEERS2_588 at $z_{\mathrm{spec}}=11.04$.
The orange-filled squares are stellar masses of the spectroscopically-
confirmed galaxies at $z_{\mathrm{spec}}>10.0$ in the JADES, GS-z13-0,
GS-z12-0, GS-z11-0, and GS-z10-0. The blue open star denotes the stellar mass
of GN-z11 taken from Tacchella et al. (2023). The red and orange shaded
regions indicate the stellar masses that are prohibited by the standard
$\Lambda$CDM cosmology for the CEERS and JADES galaxies, respectively,
calculated from the maximum halo mass that can be observed with the survey
volume in Harikane et al. (2023a) including the CEERS field, and with the
volume of the JADES NIRSpec observation estimated in this study, using the
cosmic baryon fraction, $\Omega_{\mathrm{b}}/\Omega_{\mathrm{m}}=0.157$. The
red and orange solid curves are the stellar masses calculated from the maximum
$M_{*}/M_{\mathrm{h}}$ value in Behroozi et al. (2020, B20) with the maximum
halo mass for the survey volumes of Harikane et al. (2023a, including CEERS)
and the JADES, respectively. While there are no spectroscopically-confirmed
galaxies with very large stellar masses violating the $\Lambda$CDM model, some
galaxies at $z_{\mathrm{spec}}=11-13$ have higher stellar masses than the
model predictions.

##  4 Properties of Galaxies at $z_{\mathrm{spec}}>8.5$

To understand the physical properties of spectroscopically-confirmed galaxies,
we estimate the stellar mass and SFR of the galaxies. We conduct SED fittings
using prospector (Johnson et al. 2021) for galaxies confirmed in the JADES
(Curtis-Lake et al. 2022) and galaxies in the CEERS field, including the two
brightest galaxies, CEERS2_588 and Maisie’s Galaxy at
$z_{\mathrm{spec}}=11.04$ and $z_{\mathrm{spec}}=11.40$, respectively. The
photometric measurements are taken from Finkelstein et al. (2023) and
Robertson et al. (2022). In the SED fitting, we change the optical depth in
the $V$-band, metallicity, star formation history, and total stellar mass as
free parameters while fixing the redshift to the spectroscopically-determined
value. We assume a continuity prior for the star formation history, and flat
priors for other parameters in the range of $0<\tau_{\mathrm{V}}<2$,
$-2.0<\mathrm{log}(Z/Z_{\odot})<0.4$, and $6<\mathrm{log}(M_{*}/M_{\sun})<12$.
For other parameters, we adopt the same assumptions as those in the spectral
fitting in Section 2.2. Table 3 summarizes the results of the SED fittings.

Table 3: SFRs and Stellar Masses of Spectroscopically-Confirmed Galaxies

ID | $z_{\mathrm{spec}}$ | $SFR$ | $M_{*}$
---|---|---|---
|  | ($\mathrm{M_{\odot}\ yr^{-1}}$) | ($\mathrm{M_{\odot}}$)
GS-z13-0 | 13.20 | $0.5_{-0.1}^{+0.7}$ | $(5.1_{-1.2}^{+21.2})\times 10^{7}$
GS-z12-0 | 12.63 | $0.5_{-0.1}^{+1.3}$ | $(4.3_{-2.0}^{+1.8})\times 10^{8}$
GS-z11-0 | 11.58 | $1.8_{-0.5}^{+0.4}$ | $(1.2_{-0.3}^{+0.1})\times 10^{9}$
Maisie’s Galaxy | 11.40 | $0.8_{-0.1}^{+2.1}$ | $(6.1_{-1.4}^{+23.5})\times 10^{7}$
CEERS2_588 | 11.04 | $12.7_{-4.9}^{+9.7}$ | $(9.9_{-4.0}^{+23.9})\times 10^{8}$
GS-z10-0 | 10.38 | $1.6_{-0.2}^{+0.6}$ | $(2.0_{-0.4}^{+0.6})\times 10^{8}$
CEERS2_7929 | 10.10 | $5.9_{-2.6}^{+1.6}$ | $(4.6_{-1.7}^{+5.5})\times 10^{8}$
CEERS-24 | 8.998 | $1.9_{-0.4}^{+1.9}$ | $(9.8_{-0.1}^{+51.2})\times 10^{7}$
CEERS-23 | 8.881 | $1.1_{-0.3}^{+2.9}$ | $(7.6_{-1.5}^{+50.4})\times 10^{7}$
CEERS1_6059 | 8.876 | $8.8_{-2.1}^{+2.4}$ | $(9.6_{-2.1}^{+1.7})\times 10^{8}$
CEERS1_3858 | 8.807 | $3.6_{-0.4}^{+1.9}$ | $(2.3_{-0.3}^{+2.7})\times 10^{8}$

Note. — Errors are $1\sigma$. Assuming the Chabrier (2003) IMF. The SFR is
averaged over the past 50 Myr. See Section 4 for the details of the SED
fitting.

Table 4: Spectroscopic Constraints on Cosmic UV Luminosity Density and SFR
Density

Redshift | $\mathrm{log}\rho_{\mathrm{UV}}$ | $\mathrm{log}\rho_{\mathrm{SFR,UV}}$
---|---|---
| ($\mathrm{erg\ s^{-1}\ Hz^{-1}\ Mpc^{-3}}$) | ($\mathrm{M_{\odot}\ yr^{-1}\ Mpc^{-3}}$)
$z_{\mathrm{ave}}=8.89$ | $>24.81_{-0.42}^{+0.28}$ | $>-3.13_{-0.42}^{+0.28}$
$z_{\mathrm{ave}}=10.04$ | $>24.56_{-0.39}^{+0.38}$ | $>-3.38_{-0.39}^{+0.38}$
$z_{\mathrm{ave}}=11.97$ | $>24.35_{-0.31}^{+0.23}$ | $>-3.59_{-0.31}^{+0.23}$

Note. — Errors are $1\sigma$. $\rho_{\mathrm{SFR,UV}}$ is the SFR density
based on the Salpeter (1955) IMF without dust extinction correction. Galaxies
with possible AGN signatures (GN-z11 and CEERS_1019) and possibly in the
overdensities (Gz9p3 and EGS_z910_44164) are excluded.

Figure 15 shows the stellar mass as a function of the redshift in the same
manner as Harikane et al. (2023a). The SED fittings suggest that the
spectroscopically-confirmed galaxies are massive with stellar masses of
$10^{8}\lesssim M_{*}\lesssim 10^{9}\ M_{\odot}$. These stellar masses are
well below the mass prohibited by the standard $\Lambda$CDM cosmology,
calculated from the maximum halo mass that can be observed with the survey
volume using the cosmic baryon fraction. CEERS2_588 has a large stellar mass
of $M_{*}\sim 10^{9}\ M_{\odot}$ at $z_{\mathrm{spec}}=11.04$, which is higher
than the expected value with the maximum stellar-to-halo mass ratio
($M_{*}/M_{\mathrm{h}}$) in Behroozi et al. (2020). The galaxies identified in
the JADES also have large stellar masses of $M_{*}\sim 10^{8}-10^{9}\
M_{\odot}$ at $z_{\mathrm{spec}}=11.58-13.20$, higher than the expected values
with the maximum $M_{*}/M_{\mathrm{h}}$ ratio. These stellar mass estimates
provide rough lower limits that miss the contribution from old stellar
populations beyond the Balmer break, given high specific SFRs of these galaxy
candidates, $\mathrm{SFR}/M_{*}\sim 10^{-8}\ \mathrm{yr}^{-1}$. These results
indicate that the galaxies at $z_{\mathrm{spec}}\gtrsim 11$ are brighter than
the expectations of the model of Behroozi et al. (2020). Physical
interpretations of the high stellar masses are discussed in Section 6.1.

Figure 16:  Cosmic SFR density evolution. The red diamonds represent the
spectroscopic constraints on the cosmic SFR densities obtained in this study
integrated down to $M_{\mathrm{UV}}=-18.0\ \mathrm{mag}$ (corresponding to
$\mathrm{SFR}_{\mathrm{UV}}=0.8\ M_{\odot}\ \mathrm{yr^{-1}}$, based on the
Salpeter (1955) IMF with a conversion factor of
$\mathrm{SFR}/L_{\mathrm{UV}}=1.15\times 10^{-28}\ M_{\odot}\
\mathrm{yr}^{-1}/(\mathrm{erg\ s^{-1}\ Hz^{-1}})$). These measurements are
firm lower limits because 1) only spectroscopically-confirmed galaxies without
AGN signatures are used, 2) galaxies possibly in the overdensities are
excluded, and 3) the measurements are not corrected for dust extinction. The
error includes both the $1\sigma$ Poisson error and the cosmic variance. The
blue curves are predictions of the constant star formation (SF) efficiency
models of Harikane et al. (2018, 2022b, solid), Mason et al. (2015, 2023,
dashed), and Sun & Furlanetto (2016, dotted). The obtained lower limit of the
SFR density at $z\sim 12$ is higher than the model predictions. Note that the
predictions of Harikane et al. (2018, 2022b) and Mason et al. (2015, 2023) are
integrated down to $M_{\mathrm{UV}}=-18.0\ \mathrm{mag}$, while that of Sun &
Furlanetto (2016) is down to $M_{\mathrm{UV}}=-17.7\ \mathrm{mag}$. The gray
open symbols are estimates of previous studies using photometric samples:
Harikane et al. (2023a, diamond), Donnan et al. (2023b, circle), Pérez-
González et al. (2023, hexagon), Bouwens et al. (2020, left-pointong
triangle), Bouwens et al. (2022b, cross), Bouwens et al. (2022a, square),
Finkelstein et al. (2015, pentagon), Coe et al. (2013, plus), and Ellis et al.
(2013, star). Our spectroscopic constraints are consistent with these
photometric estimates, especially those in Ellis et al. (2013) and Coe et al.
(2013), which are based on the photometric candidates at $z\sim 11-12$ that
are confirmed with JWST, GS-z11-0 and MACS0647-JD.

##  5 Cosmic SFR Density

Using our spectroscopic galaxy sample, we calculate the lower limits of the
cosmic SFR densities at $z\sim 9$, $10$, and $12$, using the effective volume
estimate in Section 3.1. We convert the observed UV luminosity to the SFR
using Equation (11) assuming the Salpeter (1955) IMF. In this calculation, we
do not use CEERS_1019 and GN-z11, because possible AGN activity is suggested
in these galaxies (Larson et al. 2023; Bunker et al. 2023). We also exclude
Gz9p3 and EGS_z910_44164, galaxies possibly in the overdense regions. Since we
only use spectroscopically-confirmed galaxies and do not correct for dust
extinction, these constraints are firm lower limits.

Figure 16 shows our spectroscopic lower limits based on galaxies brighter than
$M_{\mathrm{UV}}=-18.0$ mag, corresponding to the SFR of
$\mathrm{SFR}_{\mathrm{UV}}=0.8\ M_{\odot}\ \mathrm{yr^{-1}}$. We also plot
estimates based on the photometric samples in the literature (Harikane et al.
2023a; Donnan et al. 2023b; Pérez-González et al. 2023; Bouwens et al. 2020,
2022a, 2022b; Finkelstein et al. 2015; Ellis et al. 2013; Coe et al. 2013).
Since some of these studies calculate the SFR densities with different
integration limits from $M_{\mathrm{UV}}=-18.0$ mag, we have corrected their
results based on the difference between the SFR density integrated down to
their limit and that down to $M_{\mathrm{UV}}=-18.0$ mag using their fiducial
luminosity function, in the same manner as Bouwens et al. (2022b). Our lower
limits are consistent with these photometric estimates at $z\sim 9-12$,
especially those in Ellis et al. (2013) and Coe et al. (2013), which are based
on the photometric candidates at $z\sim 11-12$ that are confirmed with JWST,
GS-z11-0 and MACS0647-JD. Our constraint at $z\sim 12$ is $\sim 5$ times
higher than the model predictions assuming the constant star formation
efficiency in Harikane et al. (2018, 2022b), Mason et al. (2015, 2023), and
Sun & Furlanetto (2016) at $\sim 2-3\sigma$ (see also Bouwens et al. 2022a),
supporting earlier suggestions of the slow redshift evolution from $z>10$
based on the photometric samples. This indicates a higher star formation
efficiency in galaxies at $z>12$ or other physical properties different from
galaxies at $z<10$, which will be discussed in Section 6.1.

##  6 Discussion

###  6.1 High Cosmic SFR Density at $z>10$

As presented in Section 5, this study using spectroscopically-confirmed
galaxies suggests that the cosmic SFR density at $z\sim 12$ is $\sim 5$ times
higher than the predictions of the constant star formation efficiency models,
although the models can reproduce the observed SFR densities at $z<10$.
Similarly, the stellar masses of some spectroscopically-confirmed galaxies are
also higher than model predictions (Section 4). Here we discuss the following
five possibilities that explain the observed high SFR densities at $z>10$.

  * (A)

High star formation efficiency. Since the constant star formation efficiency
models predict lower SFR densities than that we have obtained at $z\sim 12$,
one of the natural interpretations is a high star formation efficiency in
galaxies at $z>10$. The comparisons of the UV luminosity functions also
suggest a high star formation efficiency of $f_{\mathrm{SF}}\gtrsim 5\%$ as
shown in Figure 14. Efficient star formation in the early universe can be
achieved with several physical mechanisms, such as no suppression of the UV
background feedback (e.g., Barkana & Loeb 2000; Susa & Umemura 2004), compact
star formation (e.g., Fukushima & Yajima 2021), and the feedback-free
starbursts (Dekel et al. 2023). Regarding the UV background feedback, galaxies
and AGN produce UV radiation by their star formation and nuclear activity and
make strong UV background radiation at the epoch of reionization (EoR; $z\sim
6-10$) and the epoch of post-reionization (post-EoR; $z\lesssim 6$, Ouchi et
al. 2020; Robertson 2021). The UV background radiation heats up Hi gas in low-
mass halos of $M_{\rm h}\lesssim 10^{8}-10^{9}M_{\odot}$ with negligible Hi
self-shielding, suppressing star-formation at the EoR and post-EoR (Barkana &
Loeb 2000; Susa & Umemura 2004; Hoeft et al. 2006; Pawlik & Schaye 2009;
Mesinger et al. 2009; Sawala et al. 2010; Bland-Hawthorn et al. 2015). Before
the EoR ($z\gtrsim 10$), galaxies in the low-mass halos are not expected to be
affected by the UV background feedback, resulting in a high star formation
efficiency at $z\gtrsim 10$ compared to one at $z\lesssim 10$, as discussed in
Harikane et al. (2023a). Also, high redshift galaxies are expected to be
compact and dense. Several simulations predict that such galaxies form stars
efficiently, with star formation efficiencies sometimes higher than 10% (e.g.,
Fukushima & Yajima 2021). Dekel et al. (2023) also discuss that high densities
and low metallicities in galaxies at $z\gtrsim 10$ result in a high star
formation efficiency with feedback-free starbursts.

  * (B)

Presence of AGN activity. Another possibility is that a part of the observed
UV luminosity densities at $z>10$ is produced by AGN, and there are no
excessive SFR densities at $z>10$ beyond the constant star-formation
efficiency model. Although the quasar luminosity function shows a very rapid
decline at $z>4$ compared to that of galaxies (e.g., Harikane et al. 2022b),
recent spectroscopic studies report faint AGNs in galaxy samples at $z>4$
(Kocevski et al. 2023; Übler et al. 2023; Larson et al. 2023; Harikane et al.
2023b). Indeed, two bright galaxies in our spectroscopic sample, CEERS_1019
and GN-z11, could have AGNs (Larson et al. 2023; Bunker et al. 2023), although
they are removed from the sample used for the SFR density calculations.
Harikane et al. (2023b) discuss that the contribution of the AGN to the total
UV light is $\sim 50\%$ on average in these faint AGNs. Given an increasing
AGN fraction from $z\sim 0$ to $z\sim 4$ (Harikane et al. 2023b), the hidden
AGN contribution may ease the tension in the observed vs. predicted SFR
densities at $z>10$.

  * (C)

A top-heavy IMF. In the early universe, the IMF is theoretically expected to
be more top-heavy than that in the lower redshift universe. Low metallicity in
the gas makes the Jeans mass higher, resulting in the formation of many
massive stars, especially for Pop III stellar populations (e.g., Hirano et al.
2014, 2015). Even if the metallicity is moderate due to a possible short
timescale of the metal enrichment, a high CMB temperature makes a top-heavy
IMF whose slope is flatter than that of the Salpeter (1955) IMF (e.g., Omukai
et al. 2005; Chon et al. 2022; Steinhardt et al. 2022). As discussed in
Harikane et al. (2023a), such a top-heavy IMF, especially with Pop-III,
reduces the UV-to-SFR conversion factor by a factor of $3-4$ (Figure 20 in
Harikane et al. 2023a), which will explain the observed high UV luminosity
densities at $z\sim 12$ compared to the constant efficiency models.

  * (D)

A large scatter in the $M_{\mathrm{h}}-SFR$ relation at $z>10$. Some of the
constant star formation efficiency models compared in Section 5 assume the
redshift-independent scatter in the $M_{\mathrm{h}}-SFR$ relation. Harikane et
al. (2018, 2022b) assume the 0.2 dex scatter in the halo mass, corresponding
to the 0.4 dex scatter in the UV luminosity, while Sun & Furlanetto (2016)
assume the 0.2 dex scatter in the UV luminosity. Mason et al. (2023) discuss
that the majority of the JWST-observed galaxies at $z\gtrsim 10$ lie above the
median $M_{\mathrm{h}}-M_{\mathrm{UV}}$ relation due to a large scatter in the
relation. We recalculate the SFR densities using the constant efficiency model
in Harikane et al. (2018, 2022b) using a larger scatter, and find that a 0.4
dex scatter in the halo mass (0.8 dex scatter in the UV luminosity or SFR)
increases an SFR density by a factor of 3 at $z\sim 12$, which makes the model
prediction consistent with the observed lower limit. Thus a large scatter in
the $M_{\mathrm{h}}-\mathrm{SFR}$ relation at $z>10$, probably due to bursty
star formation histories in the early galaxies, may explain the observed high
SFR density at $z\sim 12$.

  * (E)

Cosmic Variance. Since the survey volume of the JWST dataset used in this
study is still not large, the derived cosmic SFR densities suffer from the
cosmic variance. We have evaluated the effects of the cosmic variance using
the large-scale bias estimated with the abundance matching and included them
in the errors of the number densities and the cosmic SFR densities. Thus, our
results indicate that the SFR density at $z\sim 12$ is higher than the model
predictions beyond the uncertainty of the cosmic variance, although a large
spectroscopic survey is needed to conclude this.

###  6.2 Strategies for removing low-redshift interlopers in the future JWST
surveys

As shown in Section 2.1, the previously-claimed $z\sim 16$ galaxy candidate,
93316, firstly identified in Donnan et al. (2023b), is found to be a galaxy at
$z=4.912$ with strong [Oiii]$\lambda\lambda$4959,5007 and H$\alpha$ emission
lines. Many studies independently obtained the best-fit photometric redshift
to be $z_{\mathrm{phot}}\sim 16$ for 93316 based on the red color of
$\mathrm{F200W}-\mathrm{F277}>1.0$ and the flat continuum at
$\lambda_{\mathrm{obs}}>3\ \mu\mathrm{m}$ (Donnan et al. 2023b; Harikane et
al. 2023a; Zavala et al. 2023; Naidu et al. 2022a; Finkelstein et al. 2023;
Bouwens et al. 2022a). However, as shown in Figure 17, the red color and flat
continuum of 93316 are actually mimicked by a red continuum and the strong
[Oiii]$\lambda\lambda$4959,5007 and H$\alpha$ emission lines in the F277W,
F356W, F410M, and F444W bands, as discussed in Zavala et al. (2023) and Naidu
et al. (2022a). The F277W band flux is significantly contributed by the
[Oiii]$\lambda\lambda$4959,5007 lines, while the fluxes in the F356W, F410M,
and F444W bands are boosted by the H$\alpha$ line, which happens in a very
narrow redshift window of $\Delta z\lesssim 0.1$ (Naidu et al. 2022a). Such a
strong-line emitter with the tuned redshift is expected to be rare, but this
result may indicate that a bright $z\sim 16$ galaxy is comparably rare (see
the bottom-right panel in Figure 11).

Figure 17:  SED of 93316 (CR2-z16-1), a $z\sim 16$ galaxy candidate that is
found to be $z_{\mathrm{spec}}=4.912$. The blue circles are magnitudes
calculated from the NIRSpec spectrum multiplied by a factor of 1.4 (the thin
blue line) to correct for the slit loss. The red open diamonds are magnitudes
from the photometry in Harikane et al. (2023a), which are consistent with
those from the spectrum. The upper limits are $3\sigma$. This agreement
indicates that the strong [Oiii]$\lambda\lambda$4959,5007 and H$\alpha$ lines
boost the fluxes in the F277W, F356W, F410M, and F444W bands, and mimic the
Lyman break at $z\sim 16$, as discussed in Zavala et al. (2023) and Naidu et
al. (2022a).

Figure 18:  Importance of the medium-band filter observations in the survey of
$z\gtrsim 12$ galaxies. The top, middle, and bottom panels show SEDs of the
F150W-, F200W-, and F277W-dropouts, respectively. Each dropout has
possibilities of a high redshift solution ($z>10$), in which the observed
break is made by the high redshift Lyman break, and a low redshift solution
($z<8$), where the break is mimicked by strong [Oiii]$\lambda\lambda$4959,5007
and H$\alpha$ emission lines. The red diamonds and blue circles are magnitudes
for the high redshift and low redshift solutions, respectively, calculated
from the spectra for the two cases (the red and blue curve), which are made
from the $z=16.3$ model SED of 93316 and its observed spectrum (Figure 17). We
cannot distinguish the two solutions only with the broad-band magnitudes (the
filled symbols) but can eliminate the low redshift solution with the medium-
band observations with F182M, F250M, F335M, and/or F410M (the open symbols).

Imaging observations with multiple medium-band filters can remove such low-
redshift interlopers with strong emission lines. Figure 18 shows SEDs of the
F150W, F200W, and F277W-dropouts. For example, the F150W-dropout has a high
redshift solution at $z\sim 12$, where the observed red color of
$\mathrm{F150W}-\mathrm{F200W}$ is due to the Lyman break, and a low redshift
solution, where the red color and flat continuum are made by strong
[Oiii]$\lambda\lambda$4959,5007 and H$\alpha$ emission lines, which enter in
the F200W and F277W bands, respectively. Such low redshift interlopers can be
removed with the medium-band filter F182M or F250M images, free from strong
emission lines. Similarly, the F250M and F335M band observations are useful to
distinguish $z\sim 16$ and $z\sim 5$ solutions for the F200W-dropout that
shows a flat continuum with the F277W, F356W, F410M, and F444W bands, like
93316 (Figure 17). For the F277W-dropout that has a possibility of a $z\sim
22$ galaxy, the F335M and F410M images (or possibly F250M) can efficiently
remove a low redshift interloper at $z\sim 6$. Instead of the multiple medium-
band images, short NIRCam Wide-Field Slitlless Spectroscopic observations may
also be useful to eliminate low redshift interlopers by detecting strong
emission lines mimicking the Lyman break at high redshifts. Future JWST
surveys aiming to identify high redshift galaxies, especially bright galaxies
at $z>10$ whose number density is low, should have good strategies such as
using medium-band filters or slitless spectroscopy, to remove low redshift
interlopers including low redshift galaxies with strong emission lines like
93316.

##  7 Summary

In this paper, we present the spectroscopic constraints on the UV luminosity
functions and cosmic SFR densities at $z\sim 9-12$. We have independently
confirmed spectroscopic redshifts of 16 galaxies at $z_{\mathrm{spec}}>8.5$
including new redshift determinations, and a bright interloper at
$z_{\mathrm{spec}}=4.91$ that was claimed as a photometric candidate at $z\sim
16$. Based on the 25 galaxies at $z_{\mathrm{spec}}=8.61-13.20$ (Figure 1,
Table 1), we calculate the UV luminosity functions and the cosmic SFR
densities, which are the firm lower limits based only on the
spectroscopically-confirmed galaxies. Our major findings are summarized below:

  1. 1.

With the conservative treatments of the effective volumes and completeness, we
have obtained the constraints on the UV luminosity functions at $z\sim 9-12$
(Figure 11). Our spectroscopic constraints are consistent with the previous
estimates based on the photometric data. The observed luminosity functions
agree with the theoretical model predictions at $z\sim 9-10$ but are higher
than some models at $z\sim 12$, implying a mild redshift evolution (Figure
12). The lower limits of the number densities at $z\sim 12$ suggest that the
star formation efficiency of galaxies at $z\sim 12$ is high,
$f_{\mathrm{SF}}\gtrsim 5\%$ (Figure 14).

  2. 2.

We have estimated the stellar masses of the spectroscopically-confirmed
galaxies (Figure 15). The estimated stellar masses are $M_{*}\sim
10^{8}-10^{9}\ M_{\odot}$, and some of the galaxies have higher stellar masses
than the model predictions. However, these high stellar masses do not violate
the standard $\Lambda$CDM cosmology.

  3. 3.

We have derived the lower limits of the cosmic SFR densities at $z\sim 9-12$
considering only the spectroscopically-confirmed galaxies without AGN
signatures (Figure 16). The obtained lower limit is higher than the model
predictions assuming the constant star formation efficiencies, supporting
earlier suggestions of the slow redshift evolution at $z>10$ based on the
photometric data. We discuss that the physical origin of the high SFR density
at $z>10$ is a high star formation efficiency, AGN activity, a top-heavy IMF,
a large scatter in the $M_{\mathrm{h}}-\mathrm{SFR}$ relation, and/or the
cosmic variance, although the face value of the SFR density is larger than the
models beyond the uncertainty from the cosmic variance.

  4. 4.

The previous $z\sim 16$ galaxy candidate, 93316, is found to be a galaxy at
$z_{\mathrm{spec}}=4.912$. The strong [Oiii]$\lambda\lambda$4959,5007 and
H$\alpha$ emission lines in 93316 boost the fluxes in the F277W band and the
F356W, F410M, and F444W bands, respectively, mimicking the Lyman break at
$z\sim 16$ (Figure 17). Such a strong line emitter can be significant
contamination in a galaxy selection at $z\sim 16$. We discuss that short
NIRCam spectroscopic observations or multiple medium-band imaging observations
would be useful to remove such low-redshift interlopers with strong lines in
the future JWST survey to search for galaxies at $z\gtrsim 12$ (Figure 18).

This study demonstrates that the JWST/NIRSpec spectroscopic data are useful to
obtain robust constraints on the UV luminosity function and the cosmic SFR
densities at $z\gtrsim 9$. Future large spectroscopic surveys will allow us to
obtain more precise measurements and discover the first-generation galaxies at
$z>15$, crucial to understand the first galaxy formation.

This work is based on observations made with the NASA/ESA/CSA James Webb Space
Telescope. The 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. These observations are associated with programs
ERS-1324 (GLASS), ERS-1345 (CEERS), GO-1433, ERO-2736, and DDT-2750. The
authors acknowledge the ERO, GLASS, CEERS, GO-1433, and DDT-2750 teams led by
Klaus M. Pontoppidan, Tommaso Treu, Steven L. Finkelstein, Dan Coe, and Pablo
Arrabal Haro, respectively, for developing their observing programs with a
zero-exclusive-access period. This publication is based upon work supported by
the World Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan, KAKENHI (20H00180, 21J20785, 21K13953, 21H04467) through Japan
Society for the Promotion of Science, and JSPS Core-to-Core Program (grant
number: JPJSCCA20210003). This work was supported by the joint research
program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo.

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