# Searching for Emission Lines at $z>11$: The Role of Damped Lyman-$\alpha$
and Hints About the Escape of Ionizing Photons

Kevin N. Hainline Steward Observatory, University of Arizona, 933 N. Cherry
Ave, Tucson, AZ 85721, USA  Francesco D’Eugenio Kavli Institute for Cosmology,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK  Cavendish
Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE,
UK  Peter Jakobsen Cosmic Dawn Center (DAWN), Copenhagen, Denmark  Niels Bohr
Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen, Denmark
Jacopo Chevallard Department of Physics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK  Stefano Carniani Scuola Normale
Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy  Joris Witstok Kavli
Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK  Cavendish Laboratory, University of Cambridge, 19 JJ Thomson
Avenue, Cambridge CB3 0HE, UK  Zhiyuan Ji Steward Observatory, University of
Arizona, 933 N. Cherry Ave, Tucson, AZ 85721, USA  Emma Curtis-Lake Centre for
Astrophysics Research, Department of Physics, Astronomy and Mathematics,
University of Hertfordshire, Hatfield AL10 9AB, UK  Benjamin D. Johnson Center
for Astrophysics $|$ Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138
USA  Brant Robertson Department of Astronomy and Astrophysics, University of
California, Santa Cruz, 1156 High Street, Santa Cruz CA 96054, USA  Sandro
Tacchella Kavli Institute for Cosmology, University of Cambridge, Madingley
Road, Cambridge CB3 0HA, UK  Cavendish Laboratory, University of Cambridge, 19
JJ Thomson Avenue, Cambridge CB3 0HE, UK  Mirko Curti European Southern
Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany  Stephane
Charlot Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de
Paris, 98 bis bd Arago, 75014 Paris, France  Jakob M. Helton Steward
Observatory, University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85721, USA
Santiago Arribas  Centro de Astrobiología (CAB), CSIC–INTA, Cra. de Ajalvir
Km. 4, 28850- Torrejón de Ardoz, Madrid, Spain  Rachana Bhatawdekar European
Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino Bajo del
Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain; European Space
Agency, ESA/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, NL  Andrew J. Bunker
Department of Physics, University of Oxford, Denys Wilkinson Building, Keble
Road, Oxford OX1 3RH, UK  Alex J. Cameron Department of Physics, University of
Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK  Eiichi Egami
Steward Observatory, University of Arizona, 933 N. Cherry Ave, Tucson, AZ
85721, USA  Daniel J. Eisenstein Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden St., Cambridge MA 02138 USA  Ryan Hausen Department of
Physics and Astronomy, The Johns Hopkins University, 3400 N. Charles St.
Baltimore, MD 21218  Nimisha Kumari AURA for European Space Agency, Space
Telescope Science Institute, 3700 San Martin Drive. Baltimore, MD, 21210
Roberto Maiolino Kavli Institute for Cosmology, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, UK  Cavendish Laboratory, University of
Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE, UK  Department of Physics
and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
Pablo G. Pérez-González Centro de Astrobiología (CAB), CSIC–INTA, Cra. de
Ajalvir Km. 4, 28850- Torrejón de Ardoz, Madrid, Spain  Marcia Rieke Steward
Observatory, University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85721, USA
Aayush Saxena Department of Physics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK  Department of Physics and Astronomy,
University College London, Gower Street, London WC1E 6BT, UK  Jan Scholtz
Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK  Cavendish Laboratory, University of Cambridge, 19 JJ
Thomson Avenue, Cambridge CB3 0HE, UK  Renske Smit Astrophysics Research
Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3
5RF, UK  Fengwu Sun Steward Observatory, University of Arizona, 933 N. Cherry
Ave, Tucson, AZ 85721, USA  Christina C. Williams NSF’s National Optical-
Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ
85719, USA  Christopher N. A. Willmer Steward Observatory, University of
Arizona, 933 N. Cherry Ave, Tucson, AZ 85721, USA  Chris Willott NRC Herzberg,
5071 West Saanich Rd, Victoria, BC V9E 2E7, Canada

###### Abstract

We describe new ultra-deep James Webb Space Telescope (JWST) NIRSpec PRISM and
grating spectra for the galaxies JADES-GS-z11-0
($z_{\mathrm{spec}}=11.122^{+0.005}_{-0.003}$) and JADES-GS-z13-0
($z_{\mathrm{spec}}=13.20^{+0.03}_{-0.04}$), the most distant
spectroscopically-confirmed galaxy discovered in the first year of JWST
observations. The extraordinary depth of these observations (75 hours and 56
hours, respectively) provides a unique opportunity to explore the redshifts,
stellar properties, UV magnitudes, and slopes for these two sources. For
JADES-GS-z11-0, we find evidence for multiple emission lines, including [O
II]$\lambda\lambda 3726,3729$Å and [Ne III$]\lambda 3869$Å, resulting in a
spectroscopic redshift we determine with 94% confidence. At this spectroscopic
redshift, the Lyman-$\alpha$ break in JADES-GS-z11-0 can be fit with a damped
Lyman-$\alpha$ absorber with
$\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=22.42^{+0.093}_{-0.120}$. We present
stringent upper limits on the emission line fluxes and line equivalent widths
for JADES-GS-z13-0. These results demonstrate how neutral hydrogen fraction
and Lyman-damping wings may impact the recovery of spectroscopic redshifts for
sources like these, providing insight into the overprediction of the
photometric redshifts seen for distant galaxies observed with JWST. In
addition, we analyze updated NIRCam photometry to calculate the morphological
properties of these resolved sources, and find a secondary source
$0.3^{\prime\prime}$ south of JADES-GS-z11-0 at a similar photometric
redshift, hinting at how galaxies grow through interactions in the early
Universe.

galaxies: high-redshift – galaxies: evolution – galaxies: abundances

††facilities: JWST(NIRCam, NIRSpec), HST(ACS)

##  1 Introduction

The first two years of science from the James Webb Space Telescope (JWST) have
completely transformed our understanding of galaxies in the very early
Universe. A number of studies have led to spectroscopic confirmations of
dozens of “ultra-high redshift” galaxies ($z>10$), where these sources are
seen less than $\sim 500$ Myr after the Big Bang (Robertson et al., 2023;
Curtis-Lake et al., 2023; Arrabal Haro et al., 2023a; Fujimoto et al., 2023;
Bunker et al., 2023a, b; Wang et al., 2023; Hsiao et al., 2023; Finkelstein et
al., 2023; D’Eugenio et al., 2023; Castellano et al., 2024; Zavala et al.,
2024). The spectra of these sources, as observed with the JWST near-infrared
spectrograph NIRSpec (Jakobsen et al., 2022), are varied: some show nebular UV
or optical emission lines, while many are featureless except for the
Lyman-$\alpha$ break.

The absence of emission lines in the spectra of ultra high-redshift galaxy
spectra is surprising given the large star-formation rates (SFRs) and lack of
dust predicted for these sources. Many explanations have been put forth to
explain these observations, such as a lower gas-phase metallicity (Schaerer et
al., 2022; Curti et al., 2023; Nakajima et al., 2023), a higher escape
fraction of ionizing photons in these sources (Curtis-Lake et al., 2023;
Tacchella et al., 2023a), or a bursty star formation history with a duty cycle
favoring extended periods of low star formation (Endsley et al., 2023; Looser
et al., 2023). At such large distances and low observed fluxes, faint emission
lines are difficult to discern from the noise in these spectra.

The very bright ($M_{\mathrm{UV}}=-21.5$) galaxy GN-z11 (Oesch et al., 2016;
Tacchella et al., 2023b) at $z_{\mathrm{spec}}=10.6$ was observed using the
NIRSpec PRISM and grating dispersers and the resulting UV spectrum shows
several strong lines (Bunker et al., 2023a). Many of the emission line
strengths and flux ratios in this source were dissimilar to those measured in
metal-poor star-forming galaxies in the local Universe, which has been
ascribed to stellar collisions, tidal disruption events, globular clusters, a
top-heavy initial mass function, contributions from Wolf-Rayet and
supermassive stars or the effects of a growing supermassive black hole
(Cameron et al., 2023; Senchyna et al., 2023; Kobayashi & Ferrara, 2023;
Maiolino et al., 2023; Bekki & Tsujimoto, 2023; D’Antona et al., 2023; Isobe
et al., 2023; Watanabe et al., 2024). This variety of physical phenomena
demonstrates the complexity of the UV spectra observed in ultra-high redshift
galaxies.

Recently, D’Eugenio et al. (2023) analyzed deep NIRSpec observations of JADES-
GS-z12-0 ($z_{\mathrm{spec}}=12.482\pm 0.012$), a source first discovered in
Robertson et al. (2023); Curtis-Lake et al. (2023). They found strong evidence
for C III]$\lambda\lambda$1907,1909 nebular emission, making this the highest-
redshift detection of an emission line to date. As there is only an upper
limit on the detection of the [O III]$\lambda$1666 emission line, these
authors calculate a super-solar C/O ratio for this source ([C/O]$>0.15$), in
tension with results from JWST at $z=6-9$ (Jones et al., 2023; Stiavelli et
al., 2023). Importantly, the authors present evidence for damped
Lyman-$\alpha$ absorption (DLA; e.g., Wolfe et al., 2005) in this source, in
addition to absorption from the neutral intergalactic medium (IGM) along the
line-of-sight. This potential DLA system provides insight into the physics of
the gas surrounding sources at high redshift.

The potential presence of a DLA is important for estimating the redshifts of
galaxies without emission or absorption lines, as additional DLA absorption
can bias the estimated wavelength of the IGM-driven Lyman-$\alpha$ break at
$\sim 1216$ Å (Curtis-Lake et al., 2023; Umeda et al., 2023; Heintz et al.,
2023; Willott et al., 2023). Given the high neutral gas fractions in the early
Universe (Naidu et al., 2020; Umeda et al., 2023), we expect to observe more
DLAs in galaxies at $z>10$. JADES-GS-z12-0 joins a list of other high-redshift
galaxies have been observed with evidence for a DLA, including three galaxies
at $z=9-11$ from Heintz et al. (2023). This DLA absorption can lead to an
overprediction of the spectroscopic redshift of $\Delta z\sim 0.10-0.15$, a
bias that can negatively impact the search for emission lines in these
sources. Indeed, the spectroscopic redshift derived from the Ly$\alpha$ break
by Curtis-Lake et al. (2023), $z_{\mathrm{spec}}=12.63$, is significantly
higher than what D’Eugenio et al. (2023) estimate from the emission line
detection. This bias has a larger effect on photometric redshifts, which are
used for finding these sources, deriving luminosity functions, and
understanding the evolution of the cosmic star-formation rate density.
Multiple authors have found that photometric redshifts derived for samples of
high-redshift galaxies are systematically shifted to larger $z$ by $\sim
0.2$–$0.3$ (Arrabal Haro et al., 2023a; Hainline et al., 2023; Fujimoto et
al., 2023; Finkelstein et al., 2023; Willott et al., 2023). As DLA absorption
is not accounted for in most popular photometric redshift codes, this would
naturally explain why the resulting spectroscopic redshifts are lower than the
predicted photometric redshifts, a conclusion supported by the work of
D’Eugenio et al. (2023).

In this paper, we explore two $z>10$ sources from Robertson et al. (2023) and
Curtis-Lake et al. (2023), JADES-GS-53.16476-27.77463 (hereafter, JADES-
GS-z11-0) and JADES-GS-53.14988-27.7765 (hereafter, JADES-GS-z13-0). JADES-
GS-z11-0 was originally discovered in deep Hubble imaging by Bouwens et al.
(2011) and then further discussed in Ellis et al. (2013) and Koekemoer et al.
(2013). JADES-GS-z13-0, which lies at a redshift where it was not visible to
Hubble, is the highest-redshift spectroscopically confirmed galaxy found in
the first year of JWST observations (Robertson et al., 2023; Curtis-Lake et
al., 2023). Deeper spectroscopy from NIRSpec taken as part of observations of
the JADES Origins Field (Eisenstein et al., 2023) allows us to explore the UV
properties of these distant galaxies, where we can re-evaluate their
redshifts, UV slopes and magnitudes, and the inferred stellar masses, star-
formation rates, and metallicities. In JADES-GS-z11-0, we find evidence for
multiple weak emission lines, which allows us to refine the spectroscopic
redshift estimate for this source. For JADES-GS-z13-0, even with a spectrum
with five times the observing time, we do not find evidence for any
significant UV emission lines.

We present the new observations of JADES-GS-z11-0 and JADES-GS-z13-0 in
Section 2 along with the data reduction and spectral extraction approaches. In
Section 3, we describe the details of the multiple fitting procedures we
applied to the observed NIRSpec spectra, and in Section 4, we introduce
updated NIRCam photometry for the sources. In Section 5, we present the
results of these fits: the weak emission lines detected in the JADES-GS-z11-0
spectrum, the possible causes for the lack of emission lines in JADES-
GS-z13-0, the potential existence of damped Lyman-$\alpha$ absorption in these
sources, and how this affects their photometric redshifts. We discuss these
results and conclude in Section 6. Throughout, we assume the Planck
Collaboration et al. (2020) cosmology, with $H_{0}=67.4$ km s-1 Mpc-1,
$\Omega_{\mathrm{M}}=0.315$ and $\Omega_{\Lambda}=0.685$. All magnitudes are
provided using the AB magnitude system (Oke, 1974; Oke & Gunn, 1983).

##  2 Observations and Data Reduction

The NIRSpec spectra that form the basis of this paper were taken as part of
two programs: PID 1210 (PI N. Lützgendorf) and PID 3215 (PIs D. Eisenstein and
R. Maiolino). The spectra for PID 1210 are part of JADES, and were presented
in Curtis-Lake et al. (2023) and Bunker et al. (2023b). The spectra for PID
3215 are part of the JADES Origin Field, as outlined in Eisenstein et al.
(2023). For both JADES-GS-z11-0 and JADES-GS-z13-0, the primary spectra
described in this study were observed with the NIRSpec Multi-Shutter Array
(MSA) using the PRISM/CLEAR disperser-filter combination. The wavelength range
covered by these spectra is 0.6 - 5.3 $\mu$m at a resolution of R $\sim$ 100
(Jakobsen et al., 2022). For the PID 1210 data, JADES-GS-z11-0 was observed
for a total observing time of 100.8 ksec and JADES-GS-z13-0 was observed for
33.6 ksec (Curtis-Lake et al., 2023). For PID 3215, JADES-GS-z11-0 and JADES-
GS-z13-0 were both observed for a total observing time of 168.1 ksec ($\sim$
47 hr) each, and for the present analysis, we sum the spectra from both
programs for a total observing time of 268.9 ksec ($\sim$ 75 hr) for JADES-
GS-z11-0, and 201.7 ksec ($\sim$ 56 hr) for JADES-GS-z13-0.

We supplement the NIRSpec PRISM/CLEAR observations of these sources with
NIRSpec medium-resolution spectra (R $\sim$ 1000) taken in PID 3215 using the
G140M/F070LP and G395M/F290LP disperser-filter combinations. For G140M/F070LP,
the integration times were 42.1 ksec for both galaxies, while for
G395M/F290LP, the integration times were 134.5 ksec for both galaxies. Short-
circuits in the NIRSpec MSA (Rawle et al., 2022) affected one of the five
visits, resulting in less integration time on each source than what was
requested, 168.1 ksec.

We follow the same data reduction as described in Bunker et al. (2023b) and
Carniani et al. (2023), performing background subtraction using nodding along
the 3-slitlet array, and we extracted fluxes using a 3-pixel window. We
correct for slit losses by modeling each galaxy as a point-source, and account
for the relative intra-shutter location at each nodding position and for each
different pointing (and different MSA configuration). We will discuss the
updated size properties for JADES-GS-z11-0 and JADES-GS-z13-0 further in
Section 4. To calculate the line-spread function (LSF) for the NIRSpec
observations, we followed the method outlined in de Graaff et al. (2023).

The PRISM data taken in PID 1210 and PID 3215 both employed 1400 s duration
(19 frame) PRISM sub-exposures taken in NRSIRS2 readout mode (Rauscher et al.,
2012). A total of 186 sub-exposures of JADES-GS-z11-0 and 138 sub-exposures
JADES-GS-z13-0 were taken between the two programs. These sub-exposures were
each reduced separately by the GTO pipeline, and the resulting one-dimensional
sub-spectra were combined to produce the final spectra. The official reduction
pipeline is known to occasionally leave sharp spikes in the extracted spectra
due to residual signals from noisy pixels and/or cosmic ray hits that are not
properly captured in the ramp fitting. Such spikes are eliminated by
performing iterative sigma clipping on the reduced sub-spectra on a wavelength
bin by wavelength bin basis prior to their being co-added. However, a closer
examination of the large number of sub-spectra available for these sources
prompted us to refine the standard approach somewhat. Two additional censoring
steps were introduced prior to the sigma-clipping, which served to eliminate
obviously spurious flux values that deviated from the median measured flux in
each bin by more than five times the median pipeline error estimate for the
bin. Similarly, sub-spectra bins whose pipeline error estimate exceeded five
times the median error estimate for the bin were eliminated. This was then
followed by five passes of iterative sigma-clipping that eliminated any flux
values that deviated by more than three times from the sample variance of the
surviving entries in the bin. A second change introduced was that instead of
co-adding the surviving sub-spectra entries through weighting with the inverse
of the square of the pipeline error estimate, a statistically more robust
straight averaging of the surviving entries in each wavelength bin was
performed. In the same vein, the final propagated pipeline error for the co-
added bin was calculated as the rms mean of the pipeline errors of the
surviving entries divided by the square root of their number. Altogether, this
process eliminated 7.4% of the wavelength bin entries JADES-GS-z11-0 and 12.3%
of the entries in JADES-GS-z13-0. The resulting final spectra we explore in
this work do not differ dramatically from their standard versions, but are
clearly devoid of anomalous noise spikes exceeding the actual statistical
noise present in the data.

Figure 1: 2D and 1D NIRSpec PRISM/CLEAR spectra for JADES-GS-z11-0 (top) and
JADES-GS-z13-0 (bottom), from the combined spectra from observations under PID
1210 as described in Curtis-Lake et al. (2023) and the new, deeper
observations under PID 3215. In each panel, we plot the 2D spectrum underneath
the sigma-clipped 1D spectrum. For the 1D spectrum, plotted in blue, we also
plot uncertainties in light grey. We plot the positions of UV and optical
emission lines with dashed lines at the fiducial redshifts estimated for each
source. In an insert, we show the $2^{\prime\prime}\times 2^{\prime\prime}$
JADES F444W+F200W+F090W RGB cutout with the MSA slitlets used for PID 3215
overplotted.

In Figure 1 we show the PRISM spectra for both JADES-GS-z11-0 and JADES-
GS-z13-0. In the bottom portion of each figure, we plot the 2D NIRSpec PRISM
spectrum, with the y-axis depicting the spatial offset along the slitlet. We
plot the 1D spectrum in the upper panels along with the $1\sigma$
uncertainties. We also provide our fiducial redshifts we derived for each
source (we describe how these are derived in the next section), and we show
the wavelengths of prominent ultraviolet and optical emission and absorption
features with vertical dashed lines. In an inset for each source, we plot a
$2^{\prime\prime}\times 2^{\prime\prime}$ RGB image centered on the source
created by combining the JADES NIRCam F444W, F200W, and F090W images, where
north is up and east is to the left. In each inset, we show the MSA slitlets
used in the PID 3215 observations.

As can be seen from Figure 1, the Lyman-$\alpha$ break is very strong in the
2D spectra for both sources, with no significant emission to the blue of the
break, a sharp transition, and then smoothly decreasing flux to redder
wavelengths. For JADES-GS-z11-0, we see evidence in our deeper spectrum for
multiple emission features, including [O II]$\lambda\lambda 3726,3729$Å, [Ne
III]$\lambda 3869$Å, and possibly C IV$\lambda\lambda 1548,1551$Å. For JADES-
GS-z13-0, we do not see any significant emission lines at the fiducial
redshift for the source, similar to the results from Curtis-Lake et al.
(2023), a topic we explore in Section 5.3.

##  3 Spectral Fits and Methodology

###  3.1 Estimating Redshifts

We fit the spectra for JADES-GS-z11-0 and JADES-GS-z13-0 with multiple codes
and statistical methods to explore the source redshifts and stellar
populations. In Curtis-Lake et al. (2023), the JADES-GS-z13-0 spectrum was at
a low enough signal-to-noise that the authors fit the spectrum and NIRCam
photometry together, so this current work represents the first fit to the
spectrum alone, allowing an independent check of the properties as compared to
fits to the NIRCam photometry.

To determine the redshift for each source, we searched each PRISM spectrum for
the presence of nebular emission features. To help accomplish this, we
developed a novel automated approach designed to ascertain the significance of
UV and optical emission features in NIRSpec prism spectra at a given redshift.
This method is described in more detail in Appendix A. Briefly, we start with
the combined, sigma-clipped spectra (itself generated from a number of
independent 1400s “sub-spectra”) for each source, and apply a moving boxcar
smoothing to each spectrum to estimate the continuum, which is then
subtracted. From this continuum-subtracted spectrum, we create a line flux
signal-to-noise ratio array by means of statistical bootstrapping among the
sub-exposures making up each spectrum, which allows us to explore the
potential significance of emission features found in the spectrum. We plot the
signal-to-noise ratio vs wavelength for JADES-GS-z11-0 and JADES-GS-z13-0 in
Figures 12 and 14 in the Appendix respectively. We use an “emission line comb”
to search whether there are redshifts where a significant match of lines is
found. The lines used in this search are provided in Table 3 in the Appendix.
We calculate a total probability by combining the individual probabilities for
each potential emission line, and we use these total probabilities to find
possible values for the systemic redshift for each galaxy.

For JADES-GS-z11-0, this method results in a redshift of
$z_{\mathrm{spec}}=11.122^{+0.005}_{-0.003}$, a value we state with $94$%
confidence (see Appendix A for more details). We show the probability vs
redshift plot for this galaxy in Figure 11 and describe this redshift and the
resulting lines in Section 3.2. For JADES-GS-z13-0, however, the best-fitting
redshift resulting from this method,
$z_{\mathrm{spec}}=12.922^{+0.009}_{-0.010}$, is far less likely, and is
primarily driven by a potential detection of N IV] emission. We estimate that
this solution has $56$% confidence, and we reject it in favor of the fit to
the Lyman-$\alpha$ break for this source.

Table 1: Best-Fit Spectral Parameters

Parameter | JADES-GS-z11-0 | JADES-GS-z13-0
---|---|---
R.A. (degrees) | 53.16476 | 53.14988
DEC (degrees) | -27.77463 | -27.77650
$z_{\mathrm{spec}}$ | $11.122^{+0.005}_{-0.003}$ | $13.2^{+0.03}_{-0.04}$
MUV (spectrum) | -19.32 $\pm$ 0.03 | -18.92 $\pm$ 0.05
$\beta$ (spectrum) | -2.18 $\pm$ 0.05 | -2.69 $\pm$ 0.10
BEAGLE
$\log(\hbox{{$\textnormal{M}_{\ast}$}}/\hbox{$\textnormal{M}_{\odot}$})$ | $8.3_{-0.1}^{+0.1}$ | $7.7_{-0.20}^{+0.40}$
$\log(\hbox{$\textnormal{SFR}$}/\hbox{$\textnormal{M}_{\odot}$}\hbox{$\textnormal{yr}^{-1}$})$ | $0.16_{-0.03}^{+0.03}$ | $0.15_{-0.09}^{+0.16}$
$\log(\hbox{$\textnormal{t}$}/\textnormal{yr})$ | $8.2_{-0.1}^{+0.1}$ | $7.6_{-0.5}^{+0.4}$
$\log(\hbox{$\textnormal{Z}$}/\hbox{$\textnormal{Z}_{\odot}$})$ | $-1.9_{-0.1}^{+0.1}$ | $-1.9_{-0.2}^{+0.3}$
$\log\textnormal{U}_{\scriptscriptstyle S}$ | $-2.7_{-0.3}^{+0.2}$ | $-2.9_{-0.7}^{+0.9}$
$\hat{\tau}_{\scriptscriptstyle V}$ | $0.04_{-0.02}^{+0.02}$ | $0.02_{-0.01}^{+0.03}$
$\textnormal{f}_{\mathrm{esc}}$ | - | $0.91_{-0.1}^{+0.07}$
$\beta$ | $-2.40_{-0.05}^{+0.05}$ | $-2.76_{-0.07}^{+0.08}$
Prospector
$\log(\hbox{{$\textnormal{M}_{\ast}$}}/\hbox{$\textnormal{M}_{\odot}$})$ | $8.43^{+0.06}_{-0.08}$ | $7.85^{+0.13}_{-0.17}$
$\log(\hbox{$\textnormal{SFR}$}/\hbox{$\textnormal{M}_{\odot}$}\hbox{$\textnormal{yr}^{-1}$})$ | $0.00^{+0.13}_{-0.10}$ | $0.11^{+0.04}_{-0.03}$
log(Zstars/Z⊙) | $-1.87^{+0.22}_{-0.09}$ | $-1.84^{+0.15}_{-0.12}$
log(Zgas/Z⊙) | $-0.91^{+0.05}_{-0.06}$ | $0.35^{+0.10}_{-0.08}$
$\log\textnormal{U}_{\scriptscriptstyle S}$ | $-2.21^{+0.26}_{-0.30}$ | $-1.43^{+0.31}_{-0.32}$
E(B-V) | $0.010^{+0.001}_{-0.001}$ | $0.004^{+0.002}_{-0.002}$
$\textnormal{f}_{\mathrm{esc}}$ | 0.00 | 0.00
Emission Line Fluxes and EWs
Flux CIV | $6.2\pm 4.4$ |  $<$ 8.3
Flux HeII | $<7.9$ |  $<$ 7.6
Flux OIII] | $<7.5$ |  $<$ 7.2
Flux NIII] | $<6.6$ |  $<$ 6.3
Flux CIII] | $<5.9$ |  $<$ 5.4
Flux [OII] | $4.6\pm 1.5$ |  $<$ 3.9
Flux [NeIII] | $3.4\pm 1.5$ | -
EW CIV | $3.9\pm 2.8$ |  $<$ 10.0
EW HeII | $<5.7$ |  $<$ 10.6
EW OIII] | $<5.6$ |  $<$ 10.2
EW NIII] | $<5.2$ |  $<$ 10.1
EW CIII] | $<6.0$ |  $<$ 11.6
EW [OII] | $20.8\pm 6.6$ |  $<$ 39.6
EW [NeIII] | $14.0\pm 6.3$ | -

Note. — Fluxes are provided in units of 10-20 erg s-1 cm-2, while the
equivalent width values are units of rest-frame Angstroms.

At these systemic redshifts, we estimate emission line fluxes and equivalent
widths from the continuum-subtracted spectra using a 5-wavelength bin window.
To estimate uncertainties, we repeat the entire process of combining the sub-
spectra, estimating and subtracting the continuum, and measuring the line
fluxes bootstrapped 2000 times, each time creating a combined spectrum drawn
at random from the available sub-spectra. Our estimate of the uncertainties on
the fluxes and equivalent widths is calculated from the sample variance
derived from this procedure. This is notably different from the method for
estimating line fluxes and equivalent widths used in Curtis-Lake et al. (2023)
and D’Eugenio et al. (2023), which estimates uncertainties using the NIRSpec
reduction pipeline uncertainties and 3-wavelength bin window. The resulting
bootstrap errors we estimate agree with those calculated using a covariance
matrix measured from the individual sub-spectra for each source.

###  3.2 JADES-GS-z11-0

With the deeper spectrum for JADES-GS-z11-0, we find evidence at low
significance for multiple lines in emission in the PRISM spectrum not seen in
Curtis-Lake et al. (2023). We detect the [O II]$\lambda\lambda 3726,3729$Å
with signal-to-noise ratio (SNR) = 3.1, and [Ne III]$\lambda 3869$Å with SNR =
2.2. In addition, we have tentative evidence for C IV$\lambda\lambda
1548,1551$Å with only SNR = 1.4. We list the derived line fluxes and
equivalent widths (EWs), and include 2$\sigma$ upper limits for non-detected
features, in Table 1.

To further explore the significance of these lines, we looked at the higher
resolution ($R\sim 1000$) NIRSpec G395M grating spectrum for this source,
focusing on the 4.4 - 4.8 $\mu$m region of the observed spectra, which we plot
in Figure 2. We see a similar pair of potential emission features at 4.52
$\mu$m (SNR = 2.2) and 4.69$\mu$m (SNR = 3.11), which correspond to the [O II]
and [Ne III] lines in the PRISM spectrum. We fit these features and find that
the fluxes measured from the grating spectra agree with those measured from
the PRISM spectrum within the uncertainties, with similarly low flux SNR = 2 -
3, although we do not see evidence for [O II]$\lambda$3726 in the grating
spectrum. We measure a line width (intrinsic) of $161.07\pm 70.4$ km/s from
this fit.

Figure 2: The JADES-GS-z11-0 PRISM spectrum (blue) plotted against the NIRSpec
G395M higher-resolution grating spectrum (red). Because of the difference in
resolution between the two diffraction modes, we scale the y axes differently
for each spectrum, as shown on the left and right sides of the plot. At low
significance, the [O II] and [Ne III] lines are visible in both spectra.

While we believe that the emission features at 4.52 $\mu$m and 4.69$\mu$m are
real, the lack of an observed [O II]$\lambda$3726 emission line seen in the
grating spectrum is curious. The observed [O II]$\lambda$3729 / [O
II]$\lambda$3726 flux ratio is unphysically high, and implies a very low
electron density ($n_{e}\sim 1-10$ cm-3). The large velocity dispersion we
measure from the fits to the [O II] lines arises due to the need to constrain
the [O II]$\lambda$3729 / [O II]$\lambda$3726 flux ratio to within the
physical range. If we allow the [O II]$\lambda$3726 flux to go to zero for the
fit, the measured line width is instead $\sim 98$ km/s. In addition, we
observe a positive velocity shift between the observed wavelengths for the
putative [O II] and [Ne III] features in the G395M grating and PRISM spectra,
which is likely a result of the wavelength calibration, and has been discussed
for the JADES spectroscopic releases (Bunker et al., 2023b, D’Eugenio et al.
in prep).

To explore the significance of these features, in addition to the G395M
grating spectrum reduction described in Section 2, we performed a similar
sigma clipping and boostrap reduction of the grating spectrum as was done on
the PRISM spectrum. The resulting spectrum is consistent with what we present
in Figure 2, and we observe both the [O II] and [Ne III] emission features.
Summing over a three-bin wide box, we detect [O II] in this spectrum with SNR
= 2.4 ($p=0.0124$), and [Ne III] in this spectrum with SNR = 2.79
($p=0.00261$), with a Fisher’s combined probability $p=0.000367$.

###  3.3 JADES-GS-z13-0

For JADES-GS-z13-0, the spectrum shown in the bottom panel of Figure 1 has a
spectral break at $\sim$1.8$\mu$m, with no evidence for flux blueward of this
feature, but does not show any significant emission or absorption features.
Our fiducial redshift, $z_{\mathrm{spec}}=13.2^{+0.03}_{-0.04}$, comes from a
fit to the spectrum as described in the next section. We calculate 2$\sigma$
upper limits on the line fluxes and equivalent widths at this redshift, and
provide these in Table 1. These equivalent width values are in agreement with
those presented in Curtis-Lake et al. (2023).

###  3.4 BEAGLE Fits

The first and primary code that we use to fit the spectra of these sources is
the Bayesian galaxy spectral modeling tool BEAGLE (BayEsian Analysis of GaLaxy
sEds Chevallard & Charlot, 2016). For the fits to the spectra, we follow a
similar methodology to that adopted in Curtis-Lake et al. (2023). We fit each
source using three different models, in which we vary assumptions about the
star formation history and escape fraction of ionizing photons. The motivation
is that a major challenge in interpreting the spectra of the four $z>10$
galaxies presented in Curtis-Lake et al. (2023) was the absence of detectable
emission lines. The new observations presented in this work are significantly
deeper than those presented in Curtis-Lake et al. (2023), and yet we only
detect emission lines in JADES-GS-z11-0. Explaining the absence of lines in
JADES-GS-z13-0 hence requires us to test different model hypotheses.

To estimate the redshift of JADES-GS-z13-0, we fit the spectrum using BEAGLE
and focus on the observed spectral break at 0.8 – 1.8 $\mu$m. In this fit, we
assume a constant star-formation history, fix the IGM neutral hydrogen
fraction ($\hat{x}_{\mathrm{HI}}$) to 0, and let the escape fraction of
ionizing photons vary. The resulting redshift,
$z_{\mathrm{spec}}=13.2^{+0.03}_{-0.04}$, agrees with what derived from the
Lyman-$\alpha$ break presented in Curtis-Lake et al. (2023), and we adopt this
value as the fiducial for this source.

In all our subsequent modeling, then, we adopt Gaussian priors on the redshift
of the sources centered on the spectroscopic redshifts
$z_{\mathrm{spec}}=11.122^{+0.01}_{-0.01}$ for JADES-GS-z11-0, and
$z_{\mathrm{spec}}=13.2^{+0.03}_{-0.03}$ for JADES-GS-z13-0, and with the
width of the Gaussian set to the quoted errors.

We perform a careful fit to each spectrum, pixel-by-pixel, masking the region
1150 – 1450 Å to prevent biases arising from a potential DLA in this source,
and we also include constraints on the measured EWs (including upper limits).
We use an updated version of the Bruzual & Charlot (2003) stellar population
synthesis models (see Vidal-García et al., 2017, for details), combined with
the (continuum + emission lines) photoionization models of Gutkin et al.
(2016). We assume a Chabrier (2003) initial mass function (IMF) with lower and
upper mass limits of 0.1 and $300M_{\odot}$, respectively. The model takes
into account the depletion of metals onto dust grains in the photoionized
regions of stellar birth clouds, where we fix the dust-to-metal mass ratio to
0.1. We adopt the Charlot & Fall (2000) model for dust attenuation, with the
fraction of the attenuation from the diffuse interstellar medium (ISM) fixed
at 0.4. The ionization parameter is free to vary, while the interstellar gas-
phase metallicity is set to be equal to the stellar metallicity.

For JADES-GS-z11-0, our fiducial model is based on a constant star formation
history, and is defined by six adjustable parameters: the total stellar mass
formed $\textnormal{M}_{\textnormal{tot}}$, age of the oldest stars t, stellar
metallicity $\textnormal{Z}_{\ast}$, gas ionization parameter
$\log\textnormal{U}_{\scriptscriptstyle S}$, V-band dust attenuation optical
depth $\hat{\tau}_{\scriptscriptstyle V}$, and redshift $z$. Below, we discuss
the stellar mass locked into stars $\textnormal{M}_{\ast}$, which is always
lower than the total stellar mass formed $\textnormal{M}_{\textnormal{tot}}$,
since it excludes the mass returned to the ISM by stellar winds and supernovae
(SNe) explosions, as well as the mass locked into stellar remnants. Also, we
refer to the metallicity Z, which corresponds to the stellar metallicity
$\textnormal{Z}_{\ast}$ and to the interstellar metallicity
$\textnormal{Z}_{\textsc{ism}}$, while the gas abundance of a metal further
depends on its dust depletion factor. The star formation rate SFR is computed
as the SFR averaged over the last 10 Myr of star formation.

For JADES-GS-z13-0, our fiducial model is the same as for JADES-GS-z11-0, but
with the addition of the parameter defining the escape fraction of ionizing
photons $\textnormal{f}_{\mathrm{esc}}$. The justification for adopting these
models is provided in Section 5.3 below, where we also discuss the alternative
models explored.

For our fiducial models, we plot in Figures 3 and 4 the BEAGLE predictions and
posterior probability distributions. We summarize in Table 1 the BEAGLE output
parameters from these fits. In Table 1, we additionally provide observational
estimates of both the UV slope $\beta$ and MUV (which we provide at the top
and where we use the word “spectrum” to differentiate from the value of
$\beta$ from BEAGLE), calculated directly from each spectrum given our
fiducial redshifts. To compute $\beta$, we fit the observed flux density of
each source over spectral windows defined by Calzetti et al. (1994) in the
region 1500 to 3300 Å, and we use the 1$\sigma$ flux uncertainties to estimate
the errors on the derived slope. This wavelength range was chosen so that any
additional Lyman-$\alpha$ damping would not affect the calculation of the UV
slope. To estimate MUV, we calculate the absolute magnitude for each source
through a simulated box hat filter covering the wavelengths 1400 to 1600 Å.

Figure 3: Posterior probability distributions obtained with BEAGLE for our
fiducial fits, along with the observed spectrum and model prediction, for
JADES-GS-z11-0. From left to right, the columns show the stellar mass
$\textnormal{M}_{\ast}$, star formation rate, age of the oldest stars t,
(stellar and interstellar) metallicity Z, V-band attenuation optical depth
$\hat{\tau}_{\scriptscriptstyle V}$, ionization parameter
$\log\textnormal{U}_{\scriptscriptstyle S}$, and UV slope $\beta$. The 1D
(marginal) posterior distribution of each parameter is plotted along the
diagonal, where the shaded gray regions represent the 1$\sigma$ credible
interval. The off-diagonal panels show the 2D (joint) posterior distributions,
with the shaded blue regions representing the 1, 2, and 3$\sigma$ credible
intervals. In the top panel of the inset, we show the observed spectrum (red
line), along with the model predictions (dark blue line). The model
predictions at $\lambda<1450$ Å are shown with a cyan line, to indicate that
this region was masked during the fitting. In the bottom panel of the inset,
we show the residuals in units of observed errors and the $\pm 1\sigma$ region
in grey. Figure 4: Same as in Figure 3, but for JADES-GS-z13-0 and also
including the escape fraction $\textnormal{f}_{\mathrm{esc}}$.

###  3.5 Prospector Fits

In order to explore the range of estimated galaxy parameters for these
sources, we also fit the observed spectra with the Bayesian population
synthesis code Prospector (Johnson et al., 2021). For these fits, we adopt the
MIST (Choi et al., 2016) isochrones and MILES/BaSeL stellar library (Lejeune
et al., 1997, 1998; Westera et al., 2002; Sánchez-Blázquez et al., 2006) as
implemented in the Flexible Stellar Population Synthesis (FSPS) package
(Conroy & Gunn, 2010). We mask the Lyman-$\alpha$ break region in the same
wavelength range as for the BEAGLE fits, 1150 - 1450 Å. We assume a Kroupa
(2001) IMF, which results in stellar masses that are larger on average by 6%
from those measured using a Chabrier (2003) IMF (Speagle et al., 2014). For
dust obscuration, we use the Charlot & Fall (2000) dust prescription, where
the dust obscuring the nebular emission and stars younger than 10 Myr is
modeled using a power-law attenuation, and the additional dust obscuring the
older stars is modeled with a modified Calzetti et al. (2000) law from Kriek &
Conroy (2013). We assume the Madau (1995) model to account for IGM absorption.
We allow the stellar- and gas-phase metallicity to be independent and free
parameters in the fit, and assume that the escape fraction of ionizing photons
$f_{\mathrm{esc}}=0$. For JADES-GS-z11-0 we restrict redshift to $z=11.122$,
and for JADES-GS-z13-0, we restrict redshift to $z=13.2$. For our star-
formation history, we assume a non-parametric model with 6 bins in lookback
time and the Prospector “continuity prior.” This parameterization of the SFH
is split into multiple bins, with the SFR in each bin being derived from the
ratios of those in adjacent bins (see Leja et al., 2019; Johnson et al., 2021,
for more details). For modeling the PRISM spectra, we employ the same line-
spread function as was used for the BEAGLE fits.

We show the corner plots, SEDs, and star-formation histories for the
Prospector fits in Figures 15 and 16 in the Appendix, and include the stellar
population parameters in Table 1.

##  4 NIRCam Observations and ForcePho Fits

The NIRCam photometry for the two sources comes from the JADES data taken as
of November 2023, which includes doubling the exposure time in the JADES
NIRCam filters: F090W, F115W, F150W, F200W, F277W, F335M, F356W, F410M, and
F444W. We add to this the medium-band NIRCam photometry in filters F182M,
F210M, F430M, F460M, and F480M from the JWST Extragalactic Medium Survey
(JEMS, Williams et al., 2023) as well as observations with the filters F182M,
F210M and F444W from the First Reionization Epoch Spectroscopic COmplete
Survey (FRESCO, Oesch et al., 2023) programs. We supplement the NIRCam data
observations with those from the Hubble Space Telescope’s Advanced Camera for
Surveys (HST/ACS), using updated mosaics from the Hubble Legacy Fields program
(Illingworth et al., 2013; Whitaker et al., 2019). For our purposes, we use
the HST/ACS F435W, F606W, F775W, F814W, and F850LP filters. In total, we have
observations in 5 HST/ACS filters and 14 JWST/NIRCam filters, for a total of
19 filters. Compared to the observations described for these sources in
Robertson et al. (2023), these data are significantly deeper in both the
primary JADES filters and in F182M, F210M, and F444W due to the additional
FRESCO observations not included in their analysis. We now reach 5$\sigma$
observational depths of $2.4$ nJy in the F200W mosaic (in an
$0.2^{\prime\prime}$ diameter aperture). We can compare this to the first-year
depth provided in Hainline et al. (2023) with the same aperture of $3.0$ nJy.

Because of the small sizes of these sources, we extracted fluxes using 0.2′′
diameter circular apertures, and applied an aperture correction assuming they
are point sources. In addition, we use the software ForcePho (Johnson et al.
in prep) to estimate the total fluxes of these two sources from the updated
mosaics. ForcePho models the pixel-level fluxes for sources as a combination
of Sérsic profiles, and measures these fluxes directly from the individual
NIRCam exposures. The usage of ForcePho on JADES galaxies is described in
Robertson et al. (2023). For the fits to JADES-GS-z11-0 and JADES-GS-z13-0, we
assumed a uniform prior on the Sérsic index between 0.9 and 1.1 to better
constrain the fits. We list the updated NIRCam circular aperture and resulting
ForcePho fluxes, measured half-light radii and Sérsic indices for both sources
in Table 2. We plot the marginalized and joint posterior distributions for the
half-light radius and the semiminor to semimajor axis ratios $b/a$ for both
objects in Figure 17 in the Appendix.

We compare these fluxes, as well as those measured using ForcePho, to the
observed spectra in Figure 5. On this plot, for comparison, we also include
synthetic photometry measured directly from the spectra for each source,
calculated by interpolating the spectra with the HST/ACS and JWST/NIRCam
filter curves. For JADES-GS-z11-0, the redshift of the source places the
Lyman-$\alpha$ break in the F150W band, while for JADES-GS-z13-0, the break is
between the F150W and F182M filters. The circular aperture and ForcePho fluxes
at $\lambda_{\mathrm{obs}}>2\mu$m are somewhat higher, but within the
uncertainties, than these synthetic photometric points, potentially due to
variations in background subtraction at long wavelengths. For JADES-GS-z13-0,
the circular aperture fluxes agree quite well, but the ForcePho fluxes at
$\sim 2\mu$m are slightly underpredicted compared to the spectrum by $\sim
15$%.

Figure 5: NIRSpec PRISM/CLEAR spectroscopy (light blue lines) plotted against
the ForcePho (red circles) and 0.2′′ diameter circular aperture (black
circles) NIRCam photometry for JADES-GS-z11-0 (left) and JADES-GS-z13-0
(right). In each panel, we compare to synthetic photometry measured from the
NIRSpec PRISM data. The aperture photometry agrees well with the spectrum and
synthetic photometry, while the ForcePho photometry is slightly larger than
what’s observed in the spectrum for JADES-GS-z11-0 and lower for JADES-
GS-z13-0.

We also measured the $M_{UV}$ and UV slope $\beta$ values from the ForcePho
and circular aperture photometry directly to compare with the fiducial values
from the BEAGLE fit. Using the aperture photometry, and only fitting
photometry that corresponds to the rest-frame $1500-3300$Å at the fiducial
redshifts, we measure $M_{UV}=-19.41\pm 0.11$ and $\beta=-2.1\pm 0.1$ for
JADES-GS-z11-0, and $M_{UV}=-18.79\pm 0.06$ and $\beta=-2.37\pm 0.07$ for
JADES-GS-z13-0. From the ForcePho photometry, we measure $M_{UV}=-19.49\pm
0.12$ and $\beta=-2.10\pm 0.14$ for JADES-GS-z11-0, and $M_{UV}=-18.79\pm
0.11$ and $\beta=-2.18\pm 0.13$ for JADES-GS-z13-0. The $M_{UV}$ values are
within 2$\sigma$ between the measurements from the spectrum and the
photometry, but the slopes disagree, with the photometric slopes being
significantly shallower, likely due to the uncertainty in fitting to discrete
photometric points.

Table 2: 0.2′′ Diameter Circular Aperture and ForcePho Photometry

| JADES-GS-z11-0 | JADES-GS-z13-0
---|---|---
Filter | Aperture | ForcePho | Aperture | ForcePho
F090W | $0.15\pm 0.54$ | $-0.47\pm 0.38$ | $0.7\pm 0.48$ | $-0.12\pm 0.13$
F115W | $1.08\pm 0.43$ | $0.27\pm 0.34$ | $0.35\pm 0.35$ | $0.05\pm 0.08$
F150W | $5.78\pm 0.46$ | $6.02\pm 0.38$ | $0.23\pm 0.33$ | $-0.21\pm 0.1$
F182M | $15.97\pm 1.0$ | $15.39\pm 0.68$ | $8.3\pm 1.03$ | $5.95\pm 0.28$
F200W | $16.12\pm 0.51$ | $16.65\pm 0.49$ | $8.07\pm 0.37$ | $7.05\pm 0.17$
F210M | $16.26\pm 1.2$ | $15.62\pm 0.76$ | $8.76\pm 1.18$ | $6.7\pm 0.33$
F277W | $17.38\pm 0.41$ | $17.94\pm 0.47$ | $7.34\pm 0.28$ | $6.75\pm 0.12$
F335M | $14.27\pm 0.73$ | $13.1\pm 0.88$ | $6.11\pm 0.47$ | $4.59\pm 0.24$
F356W | $15.17\pm 0.45$ | $15.88\pm 0.52$ | $6.46\pm 0.31$ | $6.14\pm 0.16$
F410M | $13.41\pm 0.71$ | $14.7\pm 0.84$ | $6.66\pm 0.49$ | $5.33\pm 0.26$
F430M | $16.76\pm 2.49$ | $15.95\pm 2.73$ | $4.34\pm 2.28$ | $-0.08\pm 1.27$
F444W | $16.39\pm 0.59$ | $17.34\pm 0.74$ | $6.65\pm 0.4$ | $6.22\pm 0.24$
F460M | $18.87\pm 3.42$ | $16.72\pm 4.04$ | $6.61\pm 3.02$ | $3.75\pm 2.09$
F480M | $17.76\pm 2.98$ | $18.43\pm 3.27$ | $0.9\pm 2.57$ | $-1.85\pm 1.33$
$r_{\mathrm{half}}$/′′ | $0.030^{+0.001}_{-0.001}$ | $0.017^{+0.001}_{-0.001}$
$n_{\mathrm{S\acute{e}rsic}}$ | $1.02^{+0.07}_{-0.06}$ | $0.99^{+0.04}_{-0.03}$
$b/a$ | $0.75^{+0.06}_{-0.06}$ | $0.64^{+0.08}_{-0.08}$

Note. — All Fluxes in nJy

##  5 Results

###  5.1 Galaxy Fit and Morphological Properties

As can be seen from Figures 3 and 4, both the BEAGLE and Prospector fits to
the PRISM spectra agree given the uncertainties, with limited evidence for
strong emission lines. We stress that the uncertainties we provide from both
fitting methods are derived entirely from the flux and model uncertainties,
and do not account for any potential systematic uncertainties that arise from
deriving galaxy parameters from fits to the UV alone.

Looking at the posterior distributions and the values in Table 1, we see that,
for JADES-GS-z11-0, the fit results in a stellar mass of
log(M∗/M⊙)$=8.3^{+0.1}_{-0.1}$, and for JADES-GS-z13-0,
log(M∗/M⊙)$=7.7^{+0.4}_{-0.2}$. This is slightly smaller than what was
measured in Curtis-Lake et al. (2023) for JADES-GS-z11-0,
log(M∗/M⊙)$=8.67^{+0.08}_{-0.13}$, while it agrees with the values presented
in that study for JADES-GS-z13-0, log(M∗/M⊙)$=7.95^{+0.19}_{-0.29}$. The
likely cause of this difference in stellar mass for JADES-GS-z11-0 is due to
the fits to the source at $\lambda_{\mathrm{obs}}>4\mu$m, where a potential
Balmer break was predicted in the spectra described in Curtis-Lake et al.
(2023). In our updated spectra and fits for JADES-GS-z11-0, we do not find
evidence for a Balmer break given the uncertainty at
$\lambda_{\mathrm{obs}}>4\mu$m. The SFR estimated from the fit to JADES-
GS-z11-0
($\log(\hbox{$\textnormal{SFR}$}/\hbox{$\textnormal{M}_{\odot}$}\hbox{$\textnormal{yr}^{-1}$})\sim
0.16$) is also slightly smaller than that reported in Curtis-Lake et al.
(2023)
($\log(\hbox{$\textnormal{SFR}$}/\hbox{$\textnormal{M}_{\odot}$}\hbox{$\textnormal{yr}^{-1}$})\sim
0.34$), while the SFR for JADES-GS-z13-0
($\log(\hbox{$\textnormal{SFR}$}/\hbox{$\textnormal{M}_{\odot}$}\hbox{$\textnormal{yr}^{-1}$})\sim
0.15$) is very similar to the previous results
($\log(\hbox{$\textnormal{SFR}$}/\hbox{$\textnormal{M}_{\odot}$}\hbox{$\textnormal{yr}^{-1}$})\sim
0.13$).

In Figure 6, we additionally plot the F200W radial profiles for JADES-GS-z11-0
and JADES-GS-z13-0. In the left panels we show the sources and the apertures
used in deriving the profiles, and in the right panels we show the radial
profiles as compared to the measured F200W mosaic PSF. For JADES-GS-z11-0, we
mask out the source JADES-GS+53.16474-27.77471, which we discuss further in
Section 5.6. Both sources are resolved beyond the extent of the PSF in this
filter out to $\sim 0.25^{\prime\prime}$, where each source is too faint to
measure a significant flux.

Figure 6: Radial profile fits to the F200W images for JADES-GS-z11-0 (top row)
and JADES-GS-z13-0 (bottom row). In the left column we show the F200W image
centered on each source, with circular apertures used in calculating the
radial profile. In the right panel, we plot the normalized radial profile for
each source with black points and error bars as compared to the F200W mosaic
PSF which we plot with a red line. For JADES-GS-z11-0, we mask a nearby source
and show this mask with a grey box.

The sizes estimated from the ForcePho fits are small, with half-light radii of
only $0.030^{\prime\prime}\pm 0.001^{\prime\prime}$ for JADES-GS-z11-0 and
$0.017^{\prime\prime}\pm 0.001^{\prime\prime}$ for JADES-GS-z13-0. The axis
ratio for JADES-GS-z11-0 is $b/a=0.75^{+0.06}_{-0.05}$ and for JADES-GS-z13-0,
$b/a=0.64^{+0.07}_{-0.08}$. These values for the half-light radii are larger
than the sizes presented in Robertson et al. (2023), likely due to the deeper
photometry and updated PSF. Robertson et al. (2023) were only able to provide
an upper limit on the size for JADES-GS-z13-0, as there was substantial
probability that the half-light radius for the source was
$0.001^{\prime\prime}$, at the lower bound of their fit, and with these
updated fits we find strong evidence that the source is resolved. The half-
light radii for the sources correspond to 119 pc at the fiducial redshift of
JADES-GS-z11-0, and 59 pc at the fiducial redshift of JADES-GS-z13-0, and
further demonstrate the very small sizes for these sources. These values are
below the FWHM of the NIRCam PSF, demonstrating that due to the dithering from
the generation of the mosaic, we are able to resolve the diameter of each
source. This is supported by the work of Robertson et al. (2023), where they
discuss how ForcePho fits to unresolved brown dwarfs in the GOODS-S field
indicated the ability to resolve sources of these sizes.

At these sizes, we can use the SFR values measured from BEAGLE to estimate the
star-formation rate surface densities for these sources, following the
definition given in Shibuya et al. (2019):

$\Sigma_{\mathrm{SFR}}[\mathrm{M}_{\odot}\mathrm{yr}^{-1}\mathrm{kpc}^{-2}]=\frac{\mathrm{SFR}_{\mathrm{UV}}/2}{\pi
r_{e}^{2}}$ (1)

Where, here, we use the half-light radius as $r_{e}$. For JADES-GS-z11-0, we
calculate $\Sigma_{\mathrm{SFR}}=16$ M⊙ yr-1 kpc-2, and for JADES-GS-z13-0, we
calculate $\Sigma_{\mathrm{SFR}}=64$ M⊙ yr-1 kpc-2. These values, which are
lower than what is presented in Robertson et al. (2023) due to the difference
in measured sizes, are still above what is seen for most starburst galaxies
out to $z\sim 2-4$ (Genzel et al., 2010; Reddy et al., 2023), and are more
similar to local ultra compact starbursts like the “green pea” galaxies
(Izotov et al., 2016a, b).

###  5.2 Ionized Gas Properties in JADES-GS-z11-0

For JADES-GS-z11-0, we detect three emission lines with SNR $>1$, C IV, [O
II], and [Ne III]. We can use these line fluxes, and the upper limits on other
strong lines, to investigate the ionization properties of this source. In
galaxies, both neon and oxygen are generated as a part of the carbon-burning
cycle in stars, and are spread through supernovae explosions (see Maiolino &
Mannucci, 2019, for a review). The ratio of the high-ionization line [Ne III]
and low-ionization line [O II] (commonly known as Ne3O2) traces mainly the
ionization state of the gas. For JADES-GS-z11-0, we measure [Ne III] / [O II]
$=0.7\pm 0.4$, a value higher (but consistent within 1$\sigma$) than that
measured for Maisie’s Galaxy ([Ne III] / [O II] $=0.3$) at
$z_{\mathrm{spec}}=11.42$ or CEERS2_588 ([Ne III] / [O II] $=0.6$) at
$z_{\mathrm{spec}}=11.04$ (Harikane et al., 2024). In addition, the value we
estimate is slightly lower (but again, consistent within 1$\sigma$) than the
value estimated for JADES-GS-z12-0 ($z_{\mathrm{spec}}=12.48$), [Ne III] / [O
II] $=0.9\pm 0.3$ (D’Eugenio et al., 2023). These values are similar to a
sample of low-redshift, low-metallicity galaxies assembled by Nakajima et al.
(2022).

We can explore other diagnostics of C/O abundance and gas photoionization
using C III] / [O II] + [Ne III] and C IV / C III]. Because we only have a
3$\sigma$ upper limit for the C III] line, we only report upper limits for
JADES-GS-z11-0: C III] / [O II] + [Ne III] $<1.1$ and C IV / C III] $>0.7$.
The upper limit for C III] / [O II] + [Ne III] is not as extreme as JADES-
GS-z12-0 (D’Eugenio et al., 2023), but is still consistent with the high end
of the values calculated from theoretical models derived from photoionization
due to star formation and AGN in Nakajima et al. (2022) and Gutkin et al.
(2016).

###  5.3 The Absence of Emission Lines in JADES-GS-z13-0

The JADES-GS-z13-0 spectrum shown in Figure 1 is notable in that we do not see
any obvious strong emission lines given our fiducial redshifts. This is
surprising given the detection of emission lines in other galaxies at $z>10$,
including JADES-GS-z11-0, in multiple galaxies in Arrabal Haro et al. (2023b),
MACS0647-JD Hsiao et al. (2023), GN-z11 (Bunker et al., 2023a), GLASS-z12
Castellano et al. (2024); Zavala et al. (2024), and JADES-GS-z12-0 in
D’Eugenio et al. (2023). The depth of the spectra in this paper puts tight
upper limits on the possible flux of any lines, as shown in Table 1.

To interpret the spectrum of JADES-GS-z13-0, we first adopted the same
fiducial model as the one used to model JADES-GS-z11-0 and discussed in
Section 3.4 above, i.e. a model with a constant star formation history and no
escape fraction of ionising photons. As shown in Figure 18 in Appendix B, this
simple model does not match well to the observed spectrum of JADES-GS-z13-0.
The model predicts a UV slope $\beta\sim-2.5$, while the data suggest a
steeper slope $\beta\sim-2.7$: the nebular continuum reddens the UV slope,
preventing the model from reaching values below $\sim 2.5$. Moreover, matching
the upper limits on the emission line EWs requires an unlikely combination of
parameters, i.e. a very low metallicity
$\log(\hbox{$\textnormal{Z}$}/\hbox{$\textnormal{Z}_{\odot}$})\lesssim-2$ and
a very low ionization parameter $\hbox{$\log\textnormal{U}_{\scriptscriptstyle
S}$}\lesssim-3$, as this suppresses the UV high-ionization lines.
Interestingly, this model predicts significant [O ii] emission,
$\textnormal{EW}(\hbox{[O\,{\sc ii]}})\sim 25$ Å, thus more stringent
constraints on [O ii] might observationally rule out this model.

We also tested the impact of the assumed star formation history and modeled
JADES-GS-z13-0 using a delayed exponential star formation plus a burst of 10
Myr duration. This model thus allows for the separation of the current star
formation rate (over the last 10 Myr) from the past star formation history,
i.e. decoupling the strength of emission lines (powered by stars younger than
10 Myr) from the UV continuum emission (powered by stars with ages up to few
$10^{8}$ yr). We plot the model predictions and posterior probability
distributions in Figure 19 in Appendix B. This model provides a formally good
fit to the data, reaching $\beta\sim-2.9$, but, again, thanks to an unlikely
combination of model parameters: emission lines are suppressed thanks to a
very low current star formation rate
($\log(\hbox{$\textnormal{SFR}$}/\hbox{$\textnormal{M}_{\odot}$}\hbox{$\textnormal{yr}^{-1}$})\lesssim-1.5$,
while the blue UV slope is produced by stars covering a narrow age range just
beyond 10 Myr (mass-weighted age between 10 and 20 Myr). This model thus
requires a very vigorous star formation that ceased precisely at the time
required for the stars emitting ionizing photons to have evolved and died by
the time of observation.

The most physically plausible model is therefore the one that allows for the
escape of ionizing photons from JADES-GS-z13-0. The large predicted escape
fraction ($\hbox{$\textnormal{f}_{\mathrm{esc}}$}\gtrsim 0.8$) and low
metallicity
($\log(\hbox{$\textnormal{Z}$}/\hbox{$\textnormal{Z}_{\odot}$})\lesssim-1.6$)
of the fiducial BEAGLE model, which we described in Section 3.4, enable the
suppression of emission lines and a blue UV slope. This model also provides
plausible value for the other parameters, namely a very low dust attenuation
($\hbox{$\hat{\tau}_{\scriptscriptstyle V}$}\lesssim 0.05$), and a mass-
weighted age of 8 – 50 Myr.

Based on the fiducial BEAGLE model, and given the high escape fraction we find
for the source, we can estimate the size of the ionized bubble surrounding
JADES-GS-z13-0 following the method described in Witstok et al. (2023) and
Mason & Gronke (2020), where the latter study demonstrates the analytical
solution for the evolution of an ionization front. In this method, we assume
that the IGM neutral fraction of hydrogen outside the bubble,
($\hat{x}_{\mathrm{HI}}$) is 1, and estimate the source emissivity from the
measured $M_{UV}$, UV slope $\beta$, and the slope of the ionizing continuum
$\alpha$. For $\alpha$, we assume a value of $\alpha=-2$ based on the NIRSpec
spectrum of a $z=7.3$ Lyman-$\alpha$ emitter in Saxena et al. (2023). The
resulting size of the ionized bubble is $\sim 0.1$ pMpc, in agreement with the
values measured for high-redshift galaxies in Umeda et al. (2023). We caution
that this size is highly uncertain, especially given that it assumes a low
hydrogen recombination rate that may not be applicable for galaxies at $z>8$.

###  5.4 DLA Fits

The redshift of a galaxy can be calculated from either a measurement of one or
multiple emission or absorption lines in the spectrum, or it can be estimated
from the observed wavelength of the Lyman-$\alpha$ break. This latter
technique is uncertain, given the potential for additional UV absorption at
high redshift, which serves to push the Lyman-$\alpha$ break to longer
wavelengths. In D’Eugenio et al. (2023), the authors observe the C
III]$\lambda\lambda 1907,1909$ emission line in the NIRSpec spectrum for
JADES-GS-z12-0, and they propose that strong Lyman damping wing absorption
with $\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}\sim 22$ cm-2 is responsible for
the observed shift between the Lyman-$\alpha$ rest wavelength and the
Lyman-$\alpha$ break for the source. This absorption would result in a slower
observed turnover of the Lyman-$\alpha$ break, and similar absorption was
observed in a sample of three galaxies at $z=9-11$ by (Heintz et al., 2023).
We note that for both GN-z11 (Bunker et al., 2023a) and for one of the two
$z>11$ sources with detected emission lines observed in Arrabal Haro et al.
(2023b), no additional absorption was necessary in their fit, although these
sources are almost a magnitude brighter in $M_{UV}$, and GN-z11 displays
Ly$\alpha$ emission.

To explore the potential need for a DLA in both JADES-GS-z11-0 and JADES-
GS-z13-0, we followed the fitting approach in D’Eugenio et al. (2023)111Based
on the publicly available python package lymana_absorption.,where we set the
redshift for JADES-GS-z11-0 to be at $z=11.122$, and let redshift be a free
parameter for JADES-GS-z13-0. We attenuated the fiducial BEAGLE fits presented
in Figures 3 and 4 with both a damped Lyman-$\alpha$ system while fixing the
IGM neutral hydrogen fraction ($\hat{x}_{\mathrm{HI}}$) to 1. Note that in
both sources, the spectra were masked in the wavelength range $\lambda$ = 1150
- 1450 Å when fitting with BEAGLE, and while the resulting fit for JADES-
GS-z11-0 does show Lyman-$\alpha$ in emission, this is not observed in either
the PRISM or grating spectra.

For the DLA fit, we tied the redshift of any potential DLA to be at the
redshift of the galaxy, and for the IGM, we follow Witstok et al. (2023) and
assume that the IGM gas is at a mean cosmic density with T = 1K, although
raising this temperature has a negligible impact on the results. We estimate
the likelihood of the fits over $\lambda$ = 1100 - 1520 Å (respectively
spanning 42 and 45 wavelength bins for JADES-GS-z11-0 and JADES-GS-z13-0) by
calculating the inverse-weighted squared residuals between the model
(convolved with the effective NIRSpec PRISM line spread function) and the
observed spectrum. We assume flat uniform priors on redshift between
$z=12.7-13.3$ for JADES-GS-z13-0, and for each source allow
$\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}$ to vary between
$\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=19.0-24.0$ with a flat prior. For
comparison, we also present a fit where we do not include an additional DLA
component. The uncertainties we use for calculating $\chi^{2}$ in the fits
were derived from the covariance matrix measured from the individual sub-
spectra for each source.

For JADES-GS-z11-0, the Lyman-$\alpha$ break implies a significantly higher
redshift than we estimate from the detected emission lines. As a result, our
fit requires additional absorption, and we estimate
$\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=22.43^{+0.10}_{-0.12}$, a column
density similar to what was measured for JADES-GS-z12-0 in D’Eugenio et al.
(2023). We plot the posterior on the column density (top), and a fit to the
observed spectrum (bottom) for JADES-GS-z11-0 in Figure 7.

Figure 7: (top) Posterior on $\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}$ and
(bottom) SED fit for JADES-GS-z11-0 where we fix the source at
$z_{\mathrm{spec}}=11.122$, while allowing the DLA HI column density to vary.
For this fit, we fix the IGM HI fraction $\hat{x}_{\mathrm{HI}}=1$. Here,
because of the spectroscopic redshift of this source, we find a DLA is
required with $\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=22.43^{+0.10}_{-0.12}$
to account for the shape of the Lyman-$\alpha$ break.

In Figure 8 we plot our fit to the JADES-GS-z13-0 spectrum. For this source,
we measure a redshift of $z_{\mathrm{spec}}=13.13^{+0.09}_{-0.13}$ when we
allow $\log{(N_{\mathrm{HI}})}$ to vary, and
$z_{\mathrm{spec}}=13.16^{+0.09}_{-0.08}$ when we don’t include a DLA. For
this source, we find from the SED fit that we do not need to include a DLA
beyond the effects of setting $\hat{x}_{\mathrm{HI}}=1$, as the best-fit
$\chi^{2}$ is not significantly improved with the addition of a DLA. Most
notably in the joint posterior in the bottom left panel of Figure 8, we can
see how redshift varies with $\log{N_{\mathrm{HI}}}$ such that at higher DLA
column densities, the best-fit redshift is lower. We note that the estimated
redshift for JADES-GS-z13-0 is lower than the fiducial redshift from the
BEAGLE fit, largely due to the effects of fixing the IGM neutral hydrogen
fraction to 1.

Figure 8: Corner plot (top) and SED fit (bottom) for JADES-GS-z13-0 where we
allow the redshift and DLA HI column density to vary while fitting the
observed spectrum. We begin with the fiducial BEAGLE fit, shown with the green
line in the bottom panel, and then explore the relationship between redshift,
IGM absorption, and DLA absorption. The $\chi^{2}$ does not change when we
include a DLA as a free parameter for this source, demonstrating that it is
not statistically preferred by the data.

These results highlight the uncertainty in estimating the redshifts for these
ultra-distant galaxies without observed emission lines. Fitting directly to
the observed spectrum without accounting for IGM absorption or a potential DLA
may result in artificially high redshifts, as was observed for both JADES-
GS-z11-0 and JADES-GS-z12-0 in Curtis-Lake et al. (2023).

###  5.5 Photometric Redshifts

In Hainline et al. (2023), the authors explore the relationship between
photometric redshift and spectroscopic redshift for a large sample of $z>8$
sources from across the JADES GOODS-S and GOODS-N footprints. They find that,
on average, their photometric redshifts overpredict the spectroscopic redshift
for these sources by $\langle
z_{\mathrm{spec}}-z_{\mathrm{phot}}\rangle=-0.26$, which has been observed for
other high-redshift surveys (Arrabal Haro et al., 2023a; Fujimoto et al.,
2023; Finkelstein et al., 2023; Willott et al., 2023). These authors have
attributed the offset to potential DLAs and the existence of a 2-photon
nuclear continuum that becomes increasingly important at high redshift. We can
better explore the origin of this discrepancy using the spectra for JADES-
GS-z11-0 and JADES-GS-z13-0.

We fit the synthetic photometry estimated from the NIRSpec PRISM spectra
(described in Section 4) using the template-fitting code EAZY (Brammer et al.,
2008) following the procedure described in Hainline et al. (2023). We let
redshift vary between $z_{\mathrm{min}}=0.01$ to $z_{\mathrm{max}}=22.0$ in
bins of $\Delta z=0.01$. As we are fitting to synthetic photometry, we don’t
calculate or use photometric offsets for the fits. The redshift corresponding
to the minimum $\chi^{2}$ ($z_{a}$) for JADES-GS-z11-0 is $z_{a}=11.8$ and for
JADES-GS-z13-0 is $z_{a}=14.0$, both in excess of our fiducial spectroscopic
redshifts.

Figure 9: EAZY fit to JADES-GS-z11-0 (left) and JADES-GS-z13-0 (right). In
each panel, the NIRSpec PRISM spectrum and uncertainty are plotted in blue,
and synthetic photometry estimated from the spectrum through the JADES NIRCam
bands are given with blue circles. In black we plot the EAZY SED corresponding
to the minimum $\chi^{2}$ of the fit (the redshift of each fit is given in the
legend), and with black squares are the EAZY photometry derived from that SED.
In the inset in each panel, we show the EAZY $P(z)$ in orange, and plot with a
black vertical line the photometric redshift $z_{a}$ and a blue dashed
vertical line the fiducial spectroscopic redshift. For each source, the
photometric redshift is biased high compared to our fiducial photometric
redshifts due to the shallower Lyman-$\alpha$ break in the observed spectrum,
which is not simulated based on the IGM model in the EAZY fit.

In Figure 9, we plot the EAZY SED corresponding to the minimum $\chi^{2}$ and
the PRISM spectra and synthetic photometry, focusing on the region around the
Lyman-$\alpha$ break for each source. In this Figure, we plot the synthetic
photometry with black points, and the EAZYtemplate photometry with black
squares. For both sources, the fit is excellent, with $\chi^{2}=0.77$ for
JADES-GS-z11-0 and $\chi^{2}=0.36$ for JADES-GS-z13-0, but in each case, the
observed Lyman-$\alpha$ break from the spectrum falls off to the blue more
gradually than the EAZY SED. In addition, for JADES-GS-z13-0, the gap between
the F150W and F182M filters makes determining a precise photometric redshift
more difficult. For galaxies at $z>12$, deep images taken with the NIRCam
F162M filter, similar to those obtained for the JADES Origins Field
(Eisenstein et al., 2023), would help with this issue. The larger problem,
however, is the possibility that DLA absorption, or a 2-photon nebular
continuum becoming increasingly important at higher redshifts, which is not
currently simulated in most photometric redshift codes, leading to photometric
redshift estimates that are biased high.

###  5.6 JADES-GS+53.16474-27.77471

Our updated, deeper NIRCam imaging provides stronger evidence of a secondary
source $\sim 0.3^{\prime\prime}$ south ($\sim 1.2$kpc at $z=11.39$) of JADES-
GS-z11-0, which can be seen in the thumbnail in Figure 1. This object, JADES-
GS+53.16474-27.77471, appears to be an F150W dropout potentially associated
with JADES-GS-z11-0, although it did not fall onto the NIRSpec MSA. In Figure
10, we plot the SED and thumbnails for this source, where we show both the
0.2′′ circular aperture and ForcePho photometry. We fit both sets of
photometry with EAZY following the procedure in Section 4, and the minimum
$\chi^{2}$ redshift is $z_{a}=12.41$ for the fit to the circular aperture
photometry, and $z_{a}=12.31$ for the fit to the ForcePho photometry. While
this redshift is potentially biased high for the same reasons as are described
in Section 5.4, there is some probability of the source being at the
spectroscopic redshift of JADES-GS-z11-0 as shown in the P(z) plot inset of
the figure. One of the primary reasons for the difference in photometric
redshifts is the redder F150W - F200W color for JADES-GS+53.16474-27.77471
($m_{\mathrm{F150W}}-m_{\mathrm{F200W}}=1.9$) as compared to JADES-GS-z11-0
($m_{\mathrm{F150W}}-m_{\mathrm{F200W}}=1.1$).

Figure 10: Photometry and EAZY fit (top panel), and NIRCam
$2^{\prime\prime}\times 2^{\prime\prime}$ thumbnails for JADES-
GS+53.16474-27.77471. This distinct source, which can be most easily seen
south of JADES-GS-z11-0 in the F277W thumbnail, has a best-fit photometric
redshift $z_{a}=12.41$ when fitting to the 0.2′′ circular aperture photometry
(black points), as represented by the blue fit in the top panel. In the inset,
we show the P(z) surface from EAZY, where we indicate our spectroscopic
redshift for JADES-GS-z11-0 with a green vertical line. The NIRCam data for
F182M, F210M, F430M, F460M, and F480M are taken from the JEMS data and are
shallower than the other filters, and non-detections are not surprising in
these filters given the flux levels of this source.

From the ForcePho fit to the source, we calculate a half-light radii of only
$0.03^{\prime\prime}\pm 0.01^{\prime\prime}$ for JADES-GS+53.16474-27.77471,
which is 109 pc at $z_{a}=12.41$ (116 pc at the redshift of JADES-GS-z11-0), a
similar size to JADES-GS-z11-0 given the uncertainties. We fit the ForcePho
photometry for this source with Prospector to estimate the stellar mass of
this potential satellite, and calculate a stellar mass of
$\log{(M_{*}/M_{\odot})}=8.0^{+0.4}_{-0.6}$.

In Hainline et al. (2023), the authors find a number of sources from across
the JADES survey at $z>8$ with complex morphologies. Many of these galaxies
have multiple knots of a similar brightness, or, like JADES-GS-z11-0 and
JADES-GS+53.16474-27.77471, a bright central knot with a fainter satellite.
Multiple galaxies at $z\sim 7-8$ have been targeted with JWST/NIRSpec as part
of JADES (Bunker et al., 2023b), which also show potential satellite galaxies
similar to JADES-GS+53.16474-27.77471. The central galaxies are bright and
compact, with markedly redder observed colors as compared to their smaller
satellites. The UV+optical spectra for these sources show evidence for strong
line emission. It would be of interest to target these satellites directly to
understand the complex interactions of high-redshift galaxies.

##  6 Discussion and Conclusions

These new spectra for JADES-GS-z11-0 and JADES-GS-z13-0 confirm that these are
among the farthest galaxies observed in the first two years after the launch
of JWST. We can compare the properties we derive from these deep spectra with
the properties for other spectroscopically-confirmed galaxies at high redshift
observed with JWST/NIRSpec.

With these new, deeper observations, JADES-GS-z11-0 joins other sources at
$z>10$ with emission line detections: MACS0647–JD ($z_{\mathrm{spec}}=10.17$,
Hsiao et al., 2023), GN-z11 ($z_{spec}=10.6$, Bunker et al., 2023a), Maisie’s
Galaxy and CEERS2_588 ($z_{spec}=11.42$ and $11.04$ respectively, Arrabal Haro
et al., 2023b), GLASS-z12 ($z_{spec}=12.34$, Castellano et al., 2024; Zavala
et al., 2024), and JADES-GS-z12-0 ($z_{spec}=12.48$, D’Eugenio et al., 2023).
MACS0647–JD, which was originally detected using HST (Coe et al., 2013), is
triply-lensed, and the NIRSpec spectrum for this source shows multiple
emission lines. Hsiao et al. (2023) estimate $M_{UV}=-20.3$ for this source.
Due to the gravitational lensing, this source has a brighter apparent
magnitude than GN-z11, a source that is more luminous with $M_{UV}=-21.5$
(Bunker et al., 2023a). Maisie’s Galaxy and CEERS2_588 are estimated to have
$M_{UV}=-20.1$ and $M_{UV}=-20.3$ Heintz et al. (2023), and GLASS-z12, at
$M_{UV}=-20.49$ (Castellano et al., 2024), is also quite bright. JADES-
GS-z12-0 ($M_{UV}=-18.23$, Curtis-Lake et al., 2023) and JADES-GS-z11-0
($M_{UV}=-19.22$) are currently the least luminous galaxies at $z>10$ detected
with emission lines.

Among the sources at $z>10$ with emission line detections, there are a variety
of Lyman-$\alpha$ profiles. The spectra for GN-z11 and CEERS2_588 do not show
evidence for additional DLA absorption. Fits to the MACS0647–JD spectrum,
however, do demonstrate a need for a damping wing, and can be explained with a
high neutral fraction ($\hat{x}_{\mathrm{HI}}=0.9$) and a small ionized bubble
around the source with a radius smaller than 1 physical Mpc (Hsiao et al.,
2023). In Heintz et al. (2023), they fit the spectrum of this source with an
absorber with $\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=22.4$, which, given
the redshift and bright $M_{UV}$ value for this source, is puzzling when
compared to GN-z11 and CEERS2_588. Heintz et al. (2023) also require an
absorber with $\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=22.2$ for Maisie’s
Galaxy. There is no discussion of any DLA in the fits to the GLASS-z12 NIRspec
spectrum shown in Castellano et al. (2024), and the spectrum for the source
shows a very sharp Lyman-$\alpha$ break, consistent with the fact that the
spectroscopic redshift for this very bright source is very similar to those
presented in Castellano et al. (2022) and Naidu et al. (2022). Fits to the
significantly fainter JADES-GS-z12-0 spectrum in D’Eugenio et al. (2023)
require $\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=22.1$ to explain the
observed Lyman-$\alpha$ profile. The value for the hydrogen column density we
derive for JADES-GS-z11-0 is similar, and well in excess of what is seen for
lower-redshift analogues as assembled in Heintz et al. (2023). Although there
are still limited sources thus far found with a need for such extreme column
densities, it appears that they are preferentially found in less luminous
sources.

We can compare JADES-GS-z13-0 with other galaxies in the literature that have
been observed to have spectra devoid of emission lines. In Wang et al. (2023),
the authors present JWST/NIRSpec PRISM spectra for two galaxies (UNCOVER-z12
and UNCOVER-z13) at $z>12$ selected from the JWST Treasure Cycle 1 UNCOVER
survey (Bezanson et al., 2022). Emission lines are not detected in either of
these spectra, and the redshifts are derived from fits to the Lyman-$\alpha$
break, with one source at $z_{\mathrm{spec}}=12.393^{+0.004}_{-0.001}$ and the
other at $z_{\mathrm{spec}}=13.079^{+0.013}_{-0.001}$. While the authors do
not provide estimates of $M_{UV}$ for these sources, they do indicate the
rest-frame absolute magnitude in the F200W filter at the spectroscopic
redshifts, $M_{\mathrm{F200W}}=-19.2\pm 0.5$ for UNCOVER-z12 and
$M_{\mathrm{F200W}}=-19.4\pm 1.8$ for UNCOVER-z13. In addition, the stellar
masses measured by Wang et al. (2023) for these sources with Prospector are
similar to what we measure for JADES-GS-z11-0 and JADES-GS-z13-0 using the
same code, but with higher SFRs. Most notably, however, the gravitational
lensing-corrected sizes estimated for these two sources (300 - 400 pc) are 3 -
8$\times$ larger than what we measure for JADES-GS-z11-0 and JADES-GS-z13-0.
The lower-redshift source in the Wang et al. (2023) sample, UNCOVER-z12, shows
evidence for multiple clumps, similar to what we observe with JADES-GS-z11-0
and JADES-GS+53.16474-27.77471.

The NIRSpec PRISM spectrum for JADES-GS-z10-0 shown in Curtis-Lake et al.
(2023) has marginal evidence for a Lyman-$\alpha$ emission line at 1.44$\mu$m,
which would put this source at $z=10.84$, higher than the fiducial redshift
those authors provide of $z_{\mathrm{spec}}=10.38$. They provide upper limits
on the equivalent widths of C III], He II, and [O II] at this redshift, and
these values are in agreement with the EW values we measure for JADES-
GS-z11-0, indicating that perhaps this is an effect of the shallower depth of
their observations.

For JADES-GS-z13-0, our modeling indicates that the lack of emission lines is
likely due to the high escape fraction of ionizing photons. Our results show
that low-metallicity models still result in detectable emission lines, even
down to $\log\,Z/Z_{\odot}\approx-2$, the lowest values explored in our grids.
Observations of other galaxies at similar redshifts which do show emission
lines reveal that these sources are metal-enriched, although still at
significantly sub-solar values ($<0.17Z/Z_{\odot}$, Bunker et al., 2023a;
D’Eugenio et al., 2023; Zavala et al., 2024), so we cannot fully rule out that
lower metallicity outside the range of our models plays a part for JADES-
GS-z13-0. Additional key information could possibly come from observing these
high-redshift sources at even longer wavelengths, for instance with JWST/MIRI,
to seek evidence for H$\beta$ and/or [O III]$\lambda$5007 emission. In
addition, it will take significantly larger populations of galaxies
spectroscopically confirmed at these redshifts to understand whether the lack
of lines is due to their SFH having a relative lull at observation, similar to
the mini-quenched galaxy seen at $z=7.3$ in Looser et al. (2023).

Our results, along with those in the literature with significant predicted DLA
absorption, demonstrate the uncertainties in estimating redshifts from UV
spectra without emission lines. The DLA fits shown for JADES-GS-z13-0 in
Figure 8 demonstrate the degeneracy between HI column density and redshift,
such that redshifts for this source could vary as much as 0.2-0.3. For samples
of sources with photometric redshifts, or those selected entirely by colors
spanning the Lyman-$\alpha$ break, this would have an effect of moving objects
to higher redshift bins, significantly affecting any recovered evolution of
the cosmic star-formation rate density or luminosity function. In addition,
these absorbers can strongly affect the recovery of the UV slope in high-
redshift objects. Indeed, new or updated codes should be developed to help
account for this absorption.

We also find evidence that both JADES-GS-z11-0 and JADES-GS-z13-0 are
spatially resolved, from both the ForcePho fits and from the radial profiles
plotted in Figure 6. These results stand in contrast to the speculation that
these sources are supermassive “dark stars” presented in Ilie et al. (2023),
as this model would require these sources to be unresolved. Our results
indicate that both JADES-GS-z11-0 and JADES-GS-z13-0 have radial profiles
significantly in excess of the PSF, and are unlikely to be unresolved stars.

In conclusion, we present significantly deeper spectra and updated photometry
for the ultra-high redshift galaxies JADES-GS-z11-0 and JADES-GS-z13-0. We
find:

  1. 1.

The PRISM spectrum for JADES-GS-z11-0, combined with a careful assessment of
the spectral errors, reveals multiple weakly detected emission lines that
indicate a redshift of $z_{\mathrm{spec}}=11.122^{+0.005}_{-0.003}$. This
redshift derived from emission lines is lower than what would be predicted
from fitting the Lyman-$\alpha$ break alone.

  2. 2.

The PRISM spectrum for JADES-GS-z13-0 does not show any emission features, and
we estimate a redshift of $z_{\mathrm{spec}}=13.20^{+0.03}_{-0.04}$ from a fit
to the Lyman-$\alpha$ break.

  3. 3.

We use both BEAGLE and Prospector to fit the spectra for these sources, and
find stellar masses that range from $\log{(M_{*}/M_{\odot})}=7.8-8.4$, with
low stellar and gas phase metallicities and little to no dust content. The UV
slopes and SFRs we derive indicate that both sources are actively star-
forming, consistent with previous results for the objects in Curtis-Lake et
al. (2023) and Robertson et al. (2023).

  4. 4.

Updated NIRCam photometry and fits provide further evidence of the small sizes
of these galaxies (half-light radii of 119 pc for JADES-GS-z11-0 and 59 pc for
JADES-GS-z13-0). Both are resolved above the PSF.

  5. 5.

We demonstrate that additional damped Lyman-$\alpha$ absorption
($\log{(N_{\mathrm{HI}}/\mathrm{cm}^{-2})}=22.43^{+0.10}_{-0.12}$) can explain
the shape of the Lyman-$\alpha$ break at the fiducial redshift for JADES-
GS-z11-0. However, we don’t find that a damped Lyman-$\alpha$ absorber is
necessary for fitting the spectrum for JADES-GS-z13-0, as its Lyman-$\alpha$
profiles can be fit with a high neutral fraction of hydrogen gas.

  6. 6.

The photometric redshifts we derive are systematically high for both of these
sources because of the treatment of the Lyman-$\alpha$ profile in the
photometric redshift code used. Higher neutral fractions and potential DLA
absorption can produce smoother Lyman-$\alpha$ break profiles that are not
accounted for, pushing the photometric redshifts higher.

  7. 7.

We uncover evidence for a secondary source $\sim 0.3^{\prime\prime}$ south of
JADES-GS-z11-0 with a similar photometric redshift. Both sources may be part
of an interacting pair similar to others seen in the early Universe in
Hainline et al. (2023).

These results demonstrate the highly complex nature of star formation and its
effect on the observed UV spectra within galaxies from the first few hundred
million years after the Big Bang. It is vital to obtain additional deep
spectra of sources at these redshifts to understand the evolution of DLA
absorption in these systems, especially given the need to update photometric
redshift and stellar population synthesis codes to account for these effects.
Future deep NIRSpec/MOS campaigns following up on samples of high-redshift
candidates will go a long way towards helping understand the way in which
galaxies drove cosmic reionization.

K.H., Z.J., B.J., J.H., B.E.R., M.R., F.W., and C.N.A.W are funded in part by
the JWST/NIRCam contract to the University of Arizona NAS5-02015. F.D.E.,
R.M., J.W., and J.S. acknowledge support by the Science and Technology
Facilities Council (STFC), by the ERC through Advanced Grant 695671 “QUENCH”,
and by the UKRI Frontier Research grant RISEandFALL. The Cosmic Dawn Center
(DAWN) is funded by the Danish National Research Foundation under grant
DNRF140. S.C acknowledges support by European Union’s HE ERC Starting Grant
No. 101040227 - WINGS. ECL acknowledges support of an STFC Webb Fellowship
(ST/W001438/1). ST acknowledges support by the Royal Society Research Grant
G125142. S.A. acknowledges support from Grant PID2021-127718NB-I00 funded by
the Spanish Ministry of Science and Innovation/State Agency of Research
(MICIN/AEI/ 10.13039/501100011033). AJB, AJC, and JC acknowledge funding from
the ”FirstGalaxies” Advanced Grant from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement No. 789056). DJE is supported as a Simons Investigator and by
JWST/NIRCam contract to the University of Arizona, NAS5-02015. Funding for
this research was provided by the Johns Hopkins University, Institute for Data
Intensive Engineering and Science (IDIES). RM also acknowledges funding from a
research professorship from the Royal Society. PGP-G acknowledges support from
grant PID2022-139567NB-I00 funded by Spanish Ministerio de Ciencia e
Innovación MCIN/AEI/10.13039/501100011033, FEDER, UE. The research of CCW is
supported by NOIRLab, which is managed by the Association of Universities for
Research in Astronomy (AURA) under a cooperative agreement with the National
Science Foundation. RS acknowledges support from a STFC Ernest Rutherford
Fellowship (ST/S004831/1).

##  Appendix A Statistical Redshift Determination for NIRSpec Prism
Observations

The NIRSpec spectra from the JWST PID 1210 and 3215 programs are notable for
their very long exposure times, and the fact that they are constructed from a
large number of nominally identical 19 frame (1400 s) NRSIRS2 mode sub-
exposures that are reduced separately by the GTO pipeline and then co-added to
produce the final spectrum. The JADES-GS-z11-0 PRISM spectrum presented in
this paper was created from of a total of 72 sub-spectra taken in PID 1210 and
114 taken in PID 3215, resulting in a total of 186 sub-spectra and a combined
exposure time of 72.3 h. The equivalent total number of sub-spectra of JADES-
GS-z13-0 is 138 (53.6 h), with 24 from PID 1210 and 114 from PID 3215.

Having this many independent sub-spectra available provides a unique
opportunity to directly measure the actual level of statistical noise present
in NIRSpec spectra and quantify the significant correlation occurring between
adjacent wavelength bins (Jakobsen et al. 2024, in preparation). In particular
it allowed us to determine the covariance matrix of the two prism spectra used
as the noise model in the two detailed fits of Section 5.4. Here we exploit
these multiple sub-spectra to perform an automated search for the systemic
redshifts of JADES-GS-z11-0 and JADES-GS-z13-0 by looking for statistically
significant redshift matches among any weak emission lines present in the
spectra.

Starting with a 674 wavelength-bin combined NIRSpec PRISM spectrum with
wavelength $\lambda(i)$ and flux $F_{\lambda}(i)$, the first step of the
process is to determine the reference continuum level $F_{\lambda}^{c}(i)$.
Since there are no obvious strong emission lines visible in the spectra of our
targets, we calculated the continuum level by boxcar-smoothing
$F_{\lambda}(i)$ with a variable box width that follows the uneven dispersion
of the PRISM spectra and varies between 9 and 87 wavelength bins across the
spectrum. Starting at the wavelength of the onset of the Gunn-Peterson trough,
a fixed five-pixel-wide window was then shifted across the spectrum, and the
strength of any narrow emission line present at the central wavelength
$\lambda(i)$ of the window was quantified by the total summed excess flux
above the continuum contained within the window, and the equivalent width of
this excess signal:

$F_{l}(i)=\sum_{j=i-2}^{i+2}(F_{\lambda}(j)-F_{\lambda}^{c}(j))\Delta\lambda(j)$
(A1)

and

$W_{\lambda}(i)=\sum_{j=i-2}^{i+2}\frac{(F_{\lambda}(j)-F_{\lambda}^{c}(j))}{F_{\lambda}^{c}(j)}\Delta\lambda(j)$
(A2)

where $\Delta\lambda(i)$ is the width of wavelength bin $i$ in the spectrum.
Statistical errors on these estimated quantities were then determined by
repeating the same measurements on 2000 bootstrapped versions of the combined
spectrum $F_{\lambda}(i)$ drawn from the 186 and 138 available sub-spectra,
and determining the sample scatter seen in $F_{l}(i)$ and $W_{\lambda}(i)$ at
each wavelength bin $i$ (cf. Efron & Hastie, 2021). The outcomes are the error
arrays $\sigma F_{l}(i)$ and $\sigma W_{\lambda}(i)$. It should be noted that
these empirical bootstrapped statistical errors should be considered as more
reliable than the errors on $F_{l}(i)$ and $W_{\lambda}(i)$ calculated from
the estimated error spectrum output by the pipeline processing, in that they
measure the actual statistical fluctuations in the quantities $F_{l}(i)$ and
$W_{\lambda}(i)$ due to all sources of noise, and implicitly take into account
the significant inter-bin correlation present in NIRSpec spectra.

Table 3: Emission lines used in the redshift search, with their ‘effective’ vacuum wavelength, i.e., averaging over spectrally unresolved multiplets.  Emission line(s) |  $\lambda_{l}$[Å ] | Emission line(s) |  $\lambda_{l}$[Å ] | Emission line(s) |  $\lambda_{l}$[Å ]
---|---|---|---|---|---
N IV]$\lambda\lambda$1483,1486 | 1486 | Mg II$\lambda\lambda$2796,2803 | 2799 | H$\gamma$ | 4342
C IV$\lambda\lambda$1548,1551 | 1549 | [O II]$\lambda\lambda$3726,3729 | 3728 | [O III]$\lambda$4363 | 4364
He II$\lambda\,$1640 | 1640 | [Ne III]$\lambda$3769 | 3870 | H$\beta$ | 4863
O III]$\lambda\lambda$1661,1666 | 1663 | [Ne III]$\lambda$3968 | 3969 | [O III]$\lambda$4959 | 4960
N III]$\lambda\lambda$1747–1754 | 1754 | H$\epsilon$ | 3970 | [O III]$\lambda$5007 | 5008
C III]$\lambda\lambda$1907,1909 | 1909 | H$\delta$ | 4103 | H$\alpha$ | 6565

The arrays $\lambda(i)$, $F_{l}(i)$, $\sigma F_{l}(i)$ together capture the
signal strength and statistical significance of any narrow emission line
potentially detected anywhere in the spectrum, and form the basis of the line
strengths reported in Section 3. A statistical search for the systemic
redshifts of JADES-GS-z11-0 and JADES-GS-z13-0 was performed by passing a
“comb” of the most common emission lines seen in high-redshift galaxy spectra
though the (signed) signal-to-noise array $SN(i)=F_{l}(i)/\sigma F_{l}(i)$,
and quantifying the statistical significance of the coincidences occurring
among the emission lines searched for as a function of redshift. That is, for
a given probed redshift $z$, for each line listed in Table 3 the signal-to-
noise ratio $SN_{l}(j)$ of any line possibly present at its redshifted
wavelength $\lambda_{l}(j)(1+z)$ was determined through interpolation in the
$\lambda(i)$, $SN(i)$ arrays. The statistical significance of each line being
present in the spectrum was then assigned the one-sided $p$-value
$p_{l}(j)=1-\Phi(SN_{l}(j))$ where $\Phi(x)$ is the cumulative normal
distribution. $p_{l}(j)$ gives the probability that the value of $SN_{l}(j)$
or greater is reached in the spectrum at the redshifted location of the
emission line under the null hypothesis that there are no emission lines
present in the spectrum. A line reaching $SN_{l}=2.0$ is therefore assigned a
$p$-value of $p_{l}=0.0227$, while a line achieving a negative value of
$SN_{l}=-2.0$ is assigned $p_{l}=0.9772$. Since we are only interested in
searching for narrow emission lines that are not all required to be physically
present in the spectrum, such asymmetric one-sided $p$-values are appropriate.

In practice, the search for weak emission lines is carried out at wavelengths
between the onset of the Gunn-Peterson trough and out to the $\lambda=5.3$
$\mu$m red cutoff of the PRISM spectra. If we find $k$ redshifted line
candidates from Table 3 falling in this wavelength interval, their individual
$p$-values are combined into a single statistic, $X_{T}$, using Fisher’s
method:

$X_{T}=-2\sum_{j=1}^{k}\ln p_{l}(j)\sim\chi^{2}_{2k}$ (A3)

Under the null hypothesis of no lines, $X_{T}$ will be Chi-squared distributed
with $2k$ degrees of freedom (Fisher, 1950).

Possible values for the systemic redshift will reveal themselves as
statistically significant peaks in $X_{T}(z)$ when evaluated over a continuous
range of plausible redshifts spanning $z=z_{GP}\pm 0.5$, where $z_{GP}$ is the
redshift of Lyman-$\alpha$ at the midpoint of the onset of the Gunn-Peterson
trough. Note that the search needs to be extended to both sides of $z_{GP}$
since the systemic redshift of the galaxy may lie below $z_{GP}$ if there is a
local Damped Lyman Alpha absorber present, and above $z_{GP}$ if the galaxy
resides in a local ionized bubble which shifts the onset of the intergalactic
absorption to shorter wavelengths.

Figure 11: The combined emission line detection $p$-value $X_{T}$ as a
function of redshift for JADES-GS-z11-0. The prominent peak at $z=11.122$ is
evident.
Figure 12: The potential line flux signal-to-noise ratio versus wavelength for
JADES-GS-z11-0. The overlay shows the locations of the emission lines searched
for at the peak redshift of $z=11.122$.

The outcome of this redshift sweep of the JADES-GS-z11-0 spectrum is shown in
Figure 11. It is evident that the combined $p$-value displays a prominent peak
at $z=11.122^{+0.005}_{-0.003}$, reaching an amplitude of
$X_{T}=54.06\sim\chi^{2}$ for 26 degrees of freedom. Figure 12 plots
$SN(i)=F_{l}(i)/\sigma F_{l}(i)$ as a function of $\lambda(i)$, with the
search lines overlaid for this peak redshift. It is seen that the $z=11.122$
peak in $X_{T}$ is made up of a set of matching weak emission lines consisting
of an [O II] line detected at 3.11$\sigma$, a pair of [Ne III] lines detected
at 2.21$\sigma$ and 0.95$\sigma$, H$\gamma$ and H$\epsilon$ detected at
1.81$\sigma$ and 1.15$\sigma$, C IV detected at 1.41$\sigma$ and C III]
detected at 0.92$\sigma$. While these lines are not overwhelmingly significant
when considered individually, when taken together they do provide good
evidence for JADES-GS-z11-0 having a systemic redshift of $z=11.122$. Given
the measured noise level in the spectrum, the probability of such a
coincidence of erroneous noise spikes giving rise to a peak reaching
$X_{T}=54.06$ occurring purely by chance is only $p_{L}=0.0022$.

Figure 13: The combined emission line detection $p$-value $X_{T}$ as a
function of redshift for JADES-GS-z13-0. The peak seen at $z=12.922$ is likely
a chance occurrence.
Figure 14: The line flux signal-to-noise ratio versus wavelength for JADES-
GS-z13-0. The overlay shows the locations of the emission lines searched for
at the peak redshift of $z=12.922$.

However, a well-known issue with this type of automated systematic search is
that it probes a range in redshift and therefore samples the $X_{T}$ statistic
multiple times when searching for peaks, thereby increasing the probability
that an extreme random excursion may be encountered by chance. This issue is
commonly referred to as the “Look-elsewere” effect. If $N$ denotes the
effective number of independent probes of $X_{T}$ carried out in the trial,
the relevant probability to consider is the global probability of the trial
encountering one or more peaks reaching a local $p$-value less or equal to the
observed peak value $p_{L}$:

$p_{G}=1-(1-p_{L})^{N}\simeq Np_{L}$ (A4)

We can attempt to estimate the number of independent probes $N$ as follows.
The line-matching in JADES-GS-z11-0 takes place at wavelengths between the
onset of the Gunn-Peterson trough at 1.5 $\mu$m and out to 5.3 $\mu$m. The
maximum spectral resolution of the NIRSpec PRISM occurs at the red end of this
range where it reaches $R=\lambda/\delta\lambda\simeq 300$ (Jakobsen et al.,
2022). Consequently, at $z_{GP}=11.4$ a change in redshift of $\delta
z\simeq\delta\lambda/\lambda(1+z)=0.041$ will cause the reddest candidate
lines contributing to $X_{T}$ to move off any features present in the
spectrum. This implies that the effective number of independent redshifts
sampled over the total $\Delta z=1.0$ range probed is approximately
$N\simeq\Delta z/\delta z=24.4$. As an alternative more stringent approach,
Bayer et al. (2021) describe a method for self-calibrating trials such as ours
for $N$ by comparing the amplitude of the highest peak to those of the second,
third and so forth highest peaks. Applying their recipe to the three highest
secondary peaks in Figure 11 yields the values $N=23.6$, $N=25.1$ and
$N=27.6$. The average value of $N=25.4$ is in remarkably good agreement with
the rough estimate above. In view of this agreement, we can be reasonably
confident that the local $p$-value of $p_{L}=0.0022$ for the peak at
$z=11.122$, corresponds to a still significant global $p$-value of
$p_{G}=Np_{L}=0.056$. This is the basis for our concluding that JADES-GS-z11-0
lies at this redshift with 94% confidence.

Our findings for JADES-GS-z13-0 are less fruitful. The $X_{T}(z)$ plot derived
from its PRISM data is shown in Figure 13. In this object, the strongest peak
at $z=12.922^{+0.009}_{-0.010}$ has an amplitude of $X_{T}=30.19$ for 16
degrees of freedom, corresponding to a local $p$-value of $p_{L}=0.017$. The
corresponding overlaid signal-to-noise plot in Figure 14 reveals that this
peak is dominated by a match between a 2.54$\sigma$ detection of N IV] and a
1.67$\sigma$ detection of [O II]. The physical plausibility of this match
aside, for $N=25.4$ the local $p$-value of this peak corresponds to a global
$p$-value of $p_{G}=0.43$, indicating that this match is likely a chance
coincidence. We therefore conclude that our search has failed to determine the
systemic redshift of JADES-GS-z13-0 on the basis of weak absorption lines in
its PRISM spectrum.

##  Appendix B Supplemental Figures

In this Section we provide figures described in the text that supplement the
analysis. In Figures 15 and 16 we plot the posteriors, SED plots, and SFHs for
JADES-GS-z11-0 and JADES-GS-z13-0 derived from Prospector as discussed in
Section 3.5. In Figure 17 we show the marginalized and joint posterior plots
for the ForcePho-derived half-light radii and axis ratios for JADES-GS-z11-0
and JADES-GS-z13-0, as discussed in Section 4. In Figures 18 and 19 we plot
corner plots and SED fits from BEAGLE for JADES-GS-z13-0, which we discuss in
Section 5.3.

Figure 15: Corner plot (left), SED fit (top), and star-formation history
(middle) for JADES-GS-z11-0 from Prospector. In the corner plot, the first
column from the left is galaxy redshift, the second column is stellar mass,
the third column is the stellar metallicity, the fourth column is gas-phase
metallicity, the fifth column is the ionization parameter of the gas, and the
sixth column is the V-band optical depth for the older ($>10$Myr) stellar
population.
Figure 16: Corner plot (left), SED fit (top), and star-formation history
(middle) for JADES-GS-z13-0 from Prospector, with columns and description as
in Figure 15.

Figure 17: ForcePho marginalized and joint posterior distribution for the
half-light radius and semiminor to semimajor axis ratios $b/a$ for JADES-
GS-z11-0 (left) and JADES-GS-z13-0 (right). Figure 18: As in Figure 4, BEAGLE
predictions and posterior probability distributions for JADES-GS-z13-0, but
assuming a model with constant star formation history and no escape of
ionizing photons. Figure 19: As in Figure 4, BEAGLE predictions and posterior
probability distributions for JADES-GS-z13-0, but assuming a model with a
delayed exponential star formation history plus a 10 Myr burst, and no escape
of ionizing photons.

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