2023

[1]Hakim Atek

1]Institut d’Astrophysique de Paris, CNRS, Sorbonne Université, 98bis
Boulevard Arago, 75014, Paris, France 2]Centre for Astrophysics and
Supercomputing, Swinburne University of Technology, Melbourne, VIC 3122,
Australia 3]Physics Department, Ben-Gurion University of the Negev, P.O. Box
653, Be’er-Sheva 84105, Israel 4]Department of Astronomy, The University of
Texas at Austin, Austin, TX 78712, USA 5]Department of Physics and Astronomy
and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA 6]Center
for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and
Department of Physics & Astronomy, Northwestern University, IL 60201, USA
7]Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290
Versoix, Switzerland 8]Cosmic Dawn Center (DAWN), Niels Bohr Institute,
University of Copenhagen, Jagtvej 128, København N, DK-2200, Denmark 9]Kapteyn
Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV
Groningen, The Netherlands 10]Department of Astronomy, University of
Massachusetts, Amherst, MA 01003, USA 11]Department of Astronomy, Yale
University, New Haven, CT 06511, USA 12]NSF’s National Optical-Infrared
Astronomy Research Laboratory, 950 N. Cherry Avenue, Tucson, AZ 85719, USA
13]Steward Observatory, University of Arizona, Tucson, AZ 85721, USA
14]Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane,
Princeton, NJ 08544 15]Institute for Computational Science, University of
Zurich, CH-8057 Zurich, Switzerland 16]Waseda Research Institute for Science
and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1
Okubo, Shinjuku, Tokyo 169-8555, Japan 17]National Astronomical Observatory of
Japan, 2-21-1, Osawa, Mitaka, Tokyo, Japan 18] Department of Astronomy &
Astrophysics, The Pennsylvania State University, University Park, PA 16802,
USA 19]Institute for Computational & Data Sciences, The Pennsylvania State
University, University Park, PA 16802, USA 20]Institute for Gravitation and
the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
21]Department of Astronomy, University of Wisconsin, 475 N. Charter Street,
Madison, WI 53706, USA 22]Department of Physics and Astronomy, York
University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada 23]Department of
Physics and Astronomy, Tufts University, 574 Boston Ave., Medford, MA 02155,
USA 24]Department of Physics and Astronomy, Texas A&M University, College
Station, TX, 77843-4242 USA 25]George P. and Cynthia Woods Mitchell Institute
for Fundamental Physics and Astronomy, Texas A&M University, College Station,
TX, 77843-4242 USA 26]Department for Astrophysical and Planetary Science,
University of Colorado, Boulder, CO 80309, USA 27]Steward Observatory,
University of Arizona, 933 N Cherry Ave, Tucson, AZ 85721 USA 28]Departament
d’Astronomia i Astrofìsica, Universitat de València, C. Dr. Moliner 50,
E-46100 Burjassot, València, Spain 29]Department of Astronomy and
Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa
Cruz, CA 95064 USA 30]Kavli Institute for Particle Astrophysics and Cosmology
and Department of Physics, Stanford University, Stanford, CA 94305, USA

# First spectroscopic observations of the galaxies that reionized the Universe

<EMAIL_ADDRESS>Ivo Labbé  Lukas J. Furtak  Iryna Chemerynska  Seiji
Fujimoto  David J. Setton  Tim B. Miller  Pascal Oesch  Rachel Bezanson
Sedona H. Price  Pratika Dayal  Adi Zitrin  Vasily Kokorev  John R. Weaver
Gabriel Brammer  Pieter van Dokkum  Christina C. Williams  Sam E. Cutler
Robert Feldmann  Yoshinobu Fudamoto  Jenny E. Greene  Joel Leja  Michael V.
Maseda  Adam Muzzin  Richard Pan  Casey Papovich  Erica J. Nelson  Themiya
Nanayakkara  Daniel P. Stark  Mauro Stefanon  Katherine A. Suess  Bingjie Wang
Katherine E. Whitaker  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [
[  [  [  [  [  [  [  [  [  [  [

Low-mass galaxies in the early universe are believed to be the building blocks
of present-day galaxies. These fledgling systems likely played a pivotal role
in cosmic reionization, a major phase transition from neutral Hydrogen to
ionized plasma around 600-800 Myr after the Big Bangdayal18 ; robertson22 .
However, these galaxies have eluded comprehensive spectroscopic studies owing
to their extreme faintness. Here we report the first spectroscopic analysis of
8 ultra-faint galaxies during the epoch of reionization with absolute
magnitudes between $M_{\rm UV}$$\sim-17$ to $-15$ mag (down to 0.005
$L^{\star}$ finkelstein15 ; bouwens15 ). The combination of ultra-deep NIRSpec
(Near-Infrared Spectrograph) observations and the strong gravitational lensing
boost of Abell 2744 allow us to derive the first spectroscopic constraints on
the prevalence of faint galaxies and their ionizing properties during the
Universe’s first billion years. We find that faint galaxies are prodigious
producers of ionizing photons with log($\xi_{\mathrm{ion}}$/ Hz erg-1)
=$25.8\pm 0.05$, a factor of 4 larger than canonical values. This means that
the total rate of ionizing photons produced by galaxies exceeds that needed
for reionization, even for modest values of escape fraction ($f_{\rm
esc}$=5%). These findings provide robust evidence that faint galaxies were the
main drivers of cosmic reionization at $z\sim 7$.

In this paper, we combine ultra-deep JWST imaging data with ancillary Hubble
Space Telescope (HST) imaging of the gravitational lensing cluster Abell 2744
(A2744 hereafter) in order to photometrically select extremely faint galaxy
candidates in the epoch of reionization (EoR). Besides the deep HST data in 7
broadband filterslotz17 ; steinhardt20  (F435W, F606W, F814W, F105W, F125W,
F140W, F160W), we use deep multi-wavelength NIRCam (Near-Infrared Camera)
imaging as part of the JWST UNCOVER survey bezanson22 ; weaver23  of A2744
(F115W, F150W, F200W, F277W, F356W, F410M, F444W). The selection combines
several criteria to robustly constrain the photometric redshifts of the
sources. First, we applied a color-color selection, based on a flux-dropout in
the HST optical filters caused by rest-frame Lyman$-\alpha$ absorption by
residual intergalactic Hydrogen gas. This selection consolidates most of the
sources identified in the Hubble Frontier Fields (HFF) data atek18 ;
bouwens22c  at $6<z<9$. Second, we performed spectral energy distribution
(SED) fitting with the Eazy brammer22  software to estimate photometric
redshifts, assuming a flat luminosity prior and the corr_sfhz library of
stellar population templates. The allowed redshift range was set to
$0.01<z<20$. The sources have confirmed best-fit photometric solutions lying
in the $z=6-8$ range. The final selection was then performed according to the
intrinsic luminosity, combining high magnifications ($\mu\gtrsim 2$) and faint
observed luminosities in F150W, resulting in intrinsic absolute UV magnitudes
of order $M_{\rm UV}$$\gtrsim-17$.

A crucial component of the present study is the use of strong gravitational
lensing (SL) to amplify the intrinsically-faint flux of distant sources. An
accurate estimate of the magnification factor is required to retrieve the
intrinsic luminosity of sources. This step relies on a good knowledge of the
total mass distribution in the galaxy cluster. Here we use the most recent SL
model (v1.1) published for UNCOVERfurtak23b . The magnification factors for
our galaxy sample range from $\mu\sim 2$ to $\mu\sim 27$. The values are
reported in Table 1, together with 1$\sigma$ uncertainties. The second part of
the UNCOVER program consists of ultra-deep follow-up spectroscopy with the
NIRSpec instrument jacobsen22 . We used the Multi-Shutter Assembly (MSA) to
obtain multi-object spectroscopy in 7 pointings, totaling an exposure time
ranging from 2.7 to 17.4 hours. Figure 1 shows the position of these sources
in the A2744 field with the associated regions of high magnification and the
configuration of the NIRSpec slits. The extracted NIRSpec prism spectra of the
faintest sources, spanning the wavelength range $\lambda_{\rm obs}=0.6-5.3$
$\mu$m are shown in Figure 1 (see Section Methods for a detailed version). A
clear Lyman-break at rest-frame wavelength $\lambda_{\rm rest}=1216$ Å is
observed, together with multiple strong emission lines, including
H$\alpha$+[Nii], [Oiii]$\lambda\lambda 4960,5008$, H$\beta$, H$\gamma$, and
[Oii]$\lambda 3727$. Simultaneous spectral fits to the continuum and the
emission lines provide robust estimates of the spectroscopic redshifts of
these sources, which lie between $z\sim 6$ and $z\sim 7.70$ (cf. Table 1). The
spectral extraction and fitting procedure are discussed in the Methods
section.

Figure 1: Layout of the ultra-faint galaxies identified in A2744 cluster
field. The figure shows a JWST LW filters stack (F277W+F356W+F444W) with
magnification regions (at $z=7$) marked with black contours ($\mu>2$), green
($\mu>10$), and cyan ($\mu>100$), which are derived from the latest lensing
modelfurtak23b . The position of each source is marked with a red circle. Two
of the sources (12899, 16155) are predicted to be multiply imaged by the lens
model, but only the marked image of each system was targeted
spectroscopically. On the top and the bottom of the image, we show the spectra
for the four most magnified sources in our sample. For each source, the cutout
shows an RGB image around the source and the positions of each NIRSpec MSA
slitlet on top of the target. The plots show the UNCOVER NIRSpec data, with
the 2D spectrum on top of the 1D optimally extracted spectrum (black with gray
uncertainty ranges). The red lines show the best-fit model spectrum. The
observed-frame wavelengths of key emission lines are indicated as vertical
dashed lines.

Thanks to the gravitational magnification, we robustly measure extremely faint
line fluxes down to $f=5\times 10^{-21}$ erg s-1 cm-2. We also derive
intrinsic absolute magnitudes as faint as $M_{\rm UV}$$\gtrsim-15.34$ mag,
which is 2 magnitudes fainter than the deepest JWST spectroscopic surveys
bunker23b ; roberts-borsani23 ; mascia23  of the EoR so far (see Figure 2).
Assuming a characteristic magnitude of $M_{UV}^{\star}=-21.15$ mag at $z=7$
bouwens21 , these galaxies are as faint as 0.005$L^{\star}$. As such, and in
light of the steep faint-end slope of the galaxy UV luminosity function at
$z>6$ finkelstein15 ; ishigaki18 ; yue17 ; bhatawdekar19 , such galaxies
likely provide the bulk of the radiation that reionized the Universe dayal20 ;
robertson22 . We are therefore exploring a new realm of the parameter space
that enables us to characterize the nature of the most abundant population of
galaxies at $z>6$ and establish their role in cosmic reionization for the
first time.

Figure 2: Spectroscopic observations of the faintest galaxies during the epoch
of reionization. Various literature results from ground-based, HST and JWST
observations are shown with orange squares finkelstein13 ; oesch15 ; watson15
; zitrin15 ; hoag17 ; laporte17 ; laporte21 ; hashimoto18 ; tamura19 ;
topping21 ; roberts-borsani22 ; roberts-borsani23 . The blue stars represent
the spectroscopic sample of the JWST/GLASS survey presented in mascia23 . The
green circles are derived from the latest data release of the deep
spectroscopic observations of the JWST/JADES program bunker23b . The
horizontal gray line denotes the limit of the deepest JWST spectroscopic
programs.

In order to infer the stellar populations of these systems, we perform joint
spectro-photometric SED fits using the Bagpipes software package (Bayesian
Analysis of Galaxies for Physical Inference and Parameter EStimation;
Carnall2018 ; Carnall2019:VANDELS ). Accounting for the magnification, we
derive extremely-low stellar masses between log(M⋆/M⊙) =
$5.88_{-0.08}^{+0.13}$ and log(M⋆/M⊙) = $7.12_{-0.08}^{+0.07}$. (cf. Figure
7). Our results also clearly show that these galaxies harbor very young
stellar populations, with stellar ages mostly around a few million years (cf.
Table 2). This picture is also supported by their blue UV continuum slopes,
derived from our SED-fitting, in the range $\beta=[-2.07,-2.53]$. Indeed,
these values are generally indicative of a young massive stellar population
and low dust attenuation. In addition to the star-formation rate (SFR) derived
from SED-fitting, we also compute the SFR based on the H$\alpha$ recombination
line. The H$\alpha$ indicator traces massive short-lived stars on a timescale
of a few Myr, whereas the UV emission indicates an SFR averaged on a longer
timescale up to a few 100 Myr. We report high values for the SFRHα/SFRUV ratio
in the range [5-60], indicating recent bursts of star-formation in these young
systems. This is in line with their specific star-formation rates (sSFR).
Given their low stellar masses, these sources have log(sSFR /
yr-1)=[$-7.4,-6.2$], which means that they can double their stellar mass
within 2 to 20 Myr. To further characterize these sources, we measure their
sizes by fitting their morphology in the NIRCam F150W filter with a Sérsic
profilesersic63 . The derived effective radii, corrected for magnification and
taking the shear into account, vary between $r_{\rm eff}$= 30 to 300 pc.
Overall, these constraints show that these sources are small, in broad
agreement with an extrapolation to lower masses of the size-mass relation
derived at similar redshiftsholwerda15 , albeit with significant scatter. Such
small sizes and high sSFRs also supports the scenario of stochastic star
formation histories in these systems, owing to their small dynamical time and
a low gravitational potential.

Contribution to cosmic reionization

Cosmic reionization corresponds to a dramatic transition between neutral
hydrogen pervading the entire space and the complexity of the present-day
Universe, marking a major phase change of baryons in our Universe. Galaxies
have long been thought to be the best candidates for providing the required
radiation to complete the reionization process by $z=6$stark16 . Ultimately,
the ability of galaxies to reionize the Universe will depend on their
production of ionizing photon density per unit of time which is available to
ionize the intergalactic neutral gas. This quantity can be summarized by the
following relation: $\dot{n}_{\rm ion}=f_{\rm esc}$ $\xi_{\mathrm{ion}}$
$\rho_{UV}$, where $\rho_{UV}$ is the non-ionizing UV luminosity density at
1500 Å, $\xi_{\mathrm{ion}}$ is the ionizing photon production efficiency
which represents the number of ionizing photons (Lyman continuum photons, LyC)
per unit UV luminosity density, and $f_{\rm esc}$ is the fraction of this LyC
radiation that escapes the galaxy to ionize the IGMrobertson22 . It is now
well-established that faint galaxies ($M_{\rm UV}$$>-18$) are the dominant
source of UV radiation during the EoR atek15b ; finkelstein15 ; bouwens17b ;
ishigaki18 ; robertson21 , although recent JWST observations are unveiling a
number of faint active galactic nuclei (AGNs) at $3<z<7$labbe23 ; matthee23b ;
kocevski23 . The ionizing properties of these galaxies however are virtually
unknown. For example, measurements of their ionizing efficiency
$\xi_{\mathrm{ion}}$ have proven challenging even with the deepest JWST
spectroscopic surveys, which are limited to galaxies brighter than $M_{\rm
UV}$$\sim-18$fujimoto23 ; matthee23 . For the first time, we report direct
spectroscopic measurements of the ionizing efficiency $\xi_{\mathrm{ion}}$ of
the faint population of galaxies during the EoR.

In Figure 3 we present our $\xi_{\mathrm{ion}}$ measurements based on the
ratio of H$\alpha$ recombination line, which is powered by reprocessed
ionizing radiation, and the non-ionizing UV luminosity. The most striking
result is the high value of log($\xi_{\mathrm{ion}}$/ Hz erg-1) =$25.8\pm
0.05$ observed in faint ($M_{\rm UV}$$>-16.5$) galaxies, compared to the
commonly assumed canonical value of log($\xi_{\mathrm{ion}}$/ Hz erg-1
=25.2)robertson15  assumed in reionization models, or previous studies at the
EoR. For example, the measured efficiency in a population of Ly$\alpha$
emitters, which are thought to have larger ionization radiation than the
average galaxy population, is around log($\xi_{\mathrm{ion}}$/ Hz erg-1)
=25.4simmonds23 . Our measured value is consistent with the maximum values
predicted by the BPASS eldridge17 ; stanway18  stellar population models for a
dust-free galaxy with a constant star formation and a stellar age of less than
3 Myr and a 0.1$Z_{\odot}$ metallicity. Such a large ionizing efficiency in
faint galaxies implies that modest values of $f_{\rm esc}$ are sufficient for
galaxies to reionize the Universe by $z=6$. Until now, models of reionization
needed to assume large values of $f_{\rm esc}$, typically around 20%, to
accommodate the relatively low ionizing photon emissivity observed in high$-z$
galaxies. However, direct measurements of $f_{\rm esc}$ $z=0-4$ analogs,
reported typical or sample-averaged values below 10% pahl21 ; flury22a ,
albeit with a large dispersion and higher values have been observed in
individual objects. As we can see in Figure 3, we show that a volume-averaged
escape fraction as low as $f_{\rm esc}$$=5\%$ is sufficient for faint galaxies
to maintain reionization.

Figure 3: The ionizing photon production efficiency of faint galaxies during
the epoch of reionization is about a factor of 4 larger than assumed canonical
values. Consequently, only a modest $f_{\rm esc}$=5% minimum value is required
for galaxies to reionize the Universe. Our $\xi_{\mathrm{ion}}$ measurements
are marked with an orange star (light and dark shade for galaxies brighter and
fainter than $M_{\rm UV}$$=-$16.5, respectively). The gray-shaded horizontal
line represents the canonical values assumed when assessing the contribution
of galaxies to reionization. Various literature results are also shownatek22 ;
bouwens16 ; matthee17 ; nanayakkara20 ; matthee23 ; sun23 . Assuming a
fiducial UV luminosity density atek15b ; finkelstein15 ; bouwens22c , we plot
the minimum $f_{\rm esc}$ required to maintain reionization at each given
value of $\xi_{\mathrm{ion}}$(horizontal lines).

To paint a complete picture of cosmic reionization by star-forming galaxies,
we compute the spectroscopic UV luminosity function based on the present
sample. We put for the first time robust spectroscopic constraints on the
prevalence of ultra-faint galaxies. Our measurements confirm the steep faint-
end slope of the UV LF at $z\sim 7$, in agreement with the photometric UV LF
derived from HFF observationsatek18 . By integrating this UV LF down to a
faint-end limit of $M_{\rm UV}$$=-15$ mag, we determine a UV luminosity
density of log($\rho_{UV}$ / erg s-1 Mpc-3)=26.22. Now combining these two
quantities we can infer the total ionizing emissivity of galaxies for
different values of $f_{\rm esc}$, accounting for the contribution of the
ultra-faint population. The result is shown in Figure 4. Galaxies clearly
produce enough ionizing photons to maintain reionization at $z\sim 7$madau99 ;
gnedin22 , assuming on average as little as 5% of this radiation escape from
the galaxies to heat the IGM. We can go a step further by indirectly
estimating $f_{\rm esc}$ using the UV continuum slope we measured for these
galaxies. Specifically, we follow an approach pioneered by recent studies of
nearby galaxies that calibrated indirect indicators of $f_{\rm esc}$ to LyC
emission flury22b . In particular, a strong correlation is observed between
the observed $\beta_{obs}^{1550}$ slope and $f_{\rm esc}$chisholm22 . Adopting
the UV-slope $\beta$ (see Table 2) as a proxy for $f_{\rm esc}$, we infer
escape fractions within $f_{\rm esc}$=[0.045, 0.16], well in the range of
assumed values in Figure 4. Other physical properties, such as the high sSFRs,
also point to LyC-leakage in these galaxies, since this parameter is also
known to correlate with $f_{\rm esc}$flury22b . Another indirect indicator of
$f_{\rm esc}$ that has been explored in recent studies is the star formation
surface density $\Sigma_{\rm SFR}$ naidu22 ; flury22b . For these compact
sources, we measure log($\Sigma_{\rm SFR}$/ M⊙ yr-1kpc2) $\sim 0.2-2$. These
values are commonly observed in LyC leakers naidu17 ; vanzella18  and also
predictive of high $f_{\rm esc}$ according to published best-fit relations of
$f_{\rm esc}$-$\Sigma_{\rm SFR}$naidu20 .

Overall, we were able to directly measure the total ionizing photon production
of faint galaxies at $z>6$. We indirectly estimated their escape fractions,
which is the only possible way during the EoR due to the strong IGM absorption
above $z=4$inoue14 . Our results provide strong support to the scenario of
reionization by star-forming galaxies.

Figure 4: Combined spectroscopic constraints on both the prevalence of faint
galaxies and their ionizing properties demonstrate that galaxies are able to
reionize the Universe at $z\sim 7$. The total ionizing photon production rate
density derived from the present sample at $z=7$ as a function of the faint
integration limit. The blue region delimits the two cases where $f_{\rm
esc}$=5% and $f_{\rm esc}$=15%. The gray-shaded region is the threshold
required to maintain the Universe ionized at $z=7$. The gray vertical line
marks the magnitude limit of the deepest JWST spectroscopic surveys to date.
The orange vertical line shows the limit probed by this work. At this
luminosity, galaxies produce enough radiation to reionize the Universe.

In addition to the physical properties that may ease the escape of ionizing
photons from these galaxies, some physical properties, such as the
metallicity, can also indicate the ionizing properties of this population. We
measured the gas-phase metallicity using the R3 = [Oiii]/H$\beta$ line ratio
based on the most recent calibrations sanders23 . We find extremely low
metallicities, ranging from 12+log(O/H)= 6.70 to 7.46, which corresponds to 1%
to 6% of the solar metallicity. Such low metallicities are often suggestive of
strong ionizing radiation from massive starsstanway19 . At the same time,
these distant low-mass galaxies, considered as the building-blocks of present-
day galaxies, are expected to be metal-poor, owing to their supposedly
pristine gas conditions.

To summarize, these observations provide strong constraints on the most
important parameters that determine whether galaxies can drive the cosmic
reionization. Both the prevalence and the ionizing photon production are
measured for these faint sources. Measurements of $\xi_{\mathrm{ion}}$ can be
significantly affected by dust attenuationemami20 ; atek22 . This concern also
applies to our estimate of stochastic star-formation through the SFRHα/SFRUV
ratio. However, as indicated by the blue UV continuum slopes that we observe,
we expect the dust content to be small in these galaxies. This assumption is
also supported by the low Balmer decrement H$\alpha$/H$\beta$, for which we
measure an average value of $3.3\pm 0.5$. Therefore, dust attenuation should
not affect significantly these quantities. We also note that we used indirect
indicators, which come with a significant scatter, to estimate $f_{\rm esc}$,
because direct measurements of LyC at the epoch of reionization are
impossible. The stochastic nature of star formation in these low-mass galaxies
also makes the $f_{\rm esc}$ highly variable, since it mainly relies on
stellar and supernovae feedback clearing the ISM for LyC escape trebitsch17 ;
barrow20 . However, on average, hydrodynamical simulations predict higher
$f_{\rm esc}$ in lower-mass galaxiesma20 ; yeh23 . Again, based on the
ionizing photon production we estimated, modest values of $f_{\rm esc}$ around
5% are sufficient. Finally, our metallicity estimate relies on the calibration
of strong lines diagnostics, which are prone to significant uncertainties at
$z>6$. We used the most recent constraints from JWST spectroscopic studies of
$z>6$ to assess the extent of potential systematic effectsnakajima22 ;
sanders23 . On average, these estimates lie with 1 to 2$\sigma$ intervals from
each other.

The unprecedented NIR spectroscopic capabilities of JWST, coupled with the SL
magnification of A2744, has allowed us to probe the details of the ultra-faint
population of galaxies during the epoch of reionization. They are
characterized by extremely low stellar masses and metallicities, young ages,
and likely undergo bursts of star formation. Important efforts have been
devoted in the last decades to determine the nature of the sources that
reionized the Universe. While several studies have pointed to massive galaxies
naidu20  or AGN madau15 ; mitra18  providing a significant contribution, we
present here here compelling evidence supporting the scenario of faint
galaxies as the main ionizing agents driving the bulk of cosmic reionization.
These findings lay the groundwork for upcoming spectroscopic initiatives aimed
at investigating this elusive population through cluster lensing to get deeper
insights into the conditions of early galaxy formation.

## Methods

Throughout the paper, we use AB magnitudes oke83  and a standard cosmology
with H${{}_{0}}=70$ km s-1 Mpc-1, $\Omega_{\Lambda}=0.7$, and
$\Omega_{m}=0.3$.

### Observations

The UNCOVER dataset consists of both imaging and spectroscopic observations of
the lensing cluster A2744. The imaging observations and data reduction are
described in detail in the survey and catalog papers bezanson22 ; weaver23 .
Here, we briefly summarize the imaging and photometric products used in the
present paper. The NIRCamrieke23  images include short-wavelength (SW)
broadband filters (F115W, F150W, F200W), long-wavelength (LW) broadbands
(F277W, F356W, F444W), and one medium-band filter (F410M). Data were
processed, and drizzled into 0.04 arcsec pix-1 mosaics using the Grism
redshift and line analysis software for space-based spectroscopy
(Grizli111https://github.com/gbrammer/grizli; v1.6.0.dev99)brammer19 .

In terms of ancillary data, the HFF program lotz17  has obtained deep optical
and NIR observations of the core area of A2744 with the Advanced Camera for
Surveys (435W, F606W, F814W), and Wide-Field Camera Three (F105W, F125W,
F140W, F160W). A wider area around the cluster has also been covered by the
BUFFALO program steinhardt20  in almost identical broadband filters (without
F435W and F140W). All HST observations were drizzled to the same pixel scale
and the same orientation as the NIRCam mosaics.

The second part of the UNCOVER program consists of ultra-deep follow-up
spectroscopy with the NIRSpec instrument jacobsen22 . Data were obtained
between July 31st and August 2nd 2023. Observations use the Prism mode and the
Multi-Shutter Assembly (MSA) ferruit22  of NIRSpec to observe more than 650
targets. In order to optimize background subtraction, each target has been
observed with a 3-slitlet nodding strategy. Observations were split into 7
pointings, with important overlap at the center, providing total on-target
exposure times ranging from $\sim 2.7h$ to $\sim 17.4h$. The spectral
resolution is wavelength-dependent and varies between $R\sim 30$ to $R\sim
300$ over the full wavelength range $\lambda\sim 0.6-5.3\mu m$. Data were
reduced using the JWST/NIRSpec analysis software
msaexp222https://github.com/gbrammer/msaexp version 0.6.10. The processing is
based on level 2 MAST333https://archive.stsci.edu/missions-and-data/jwst
products, using the CRDS context file jwst_1100.pmap. The software performs
basic reduction steps, including flat-field, bias, 1/f noise and snowballs
correction, wavelength and photometric calibrations of individual exposure
frames heintz23 . The extraction of 1D spectra from individual exposures is
operated on inverse-weighted stack of 2D spectrum in the dispersion direction,
following an optimal extraction procedure horne86 . Then the software fits a
Gaussian profile along the cross-dispersion direction to define the 1D
extraction aperture. Finally, we compute the final deep 1D spectrum by
inverse-variance stacking the individual spectra. In order to account for slit
loss effects, we apply a wavelength- (broadband-) dependent correction factor
to re-scale the 1D spectrum to the observed NIRCam aperture photometry.

### Spectral fitting

In order to determine the spectroscopic redshift of the sources, we fit
spectral templates using msaexp, which based on the SED fitting software
EazyBrammer2008 . The code combine a set of templates to fit simultaneously
the continuum, including the Lyman break caused by the IGM absorption, and the
emission lines. For our analysis, we adopt the corr_sfhz_13 template library
which include redshift-dependent SFHs, which are known to perform better than
the default fsps_full library in recovering the true redshift weaver23 . We
search for the best-fit solution over the redshift interval $0<z<15$. The
spectroscopic redshifts are reported in Table 1. Once the best-fit redshift is
found, we refit the spectra, fixing the redshift to $z_{\mathrm{spec}}$, with
a set of spline functions to measure the continuum and Gaussians in order to
measure the emission line fluxes. Examples of the best-fit model plotted over
the observed spectrum are presented in Figure 1.

### Strong lensing

We use version v1.1 of the UNCOVER lensing modelfurtak23b , which is publicly
available in the latest UNCOVER data release DR-1444https://jwst-
uncover.github.io/DR1.html##LensingMaps. The model is based on the parametric
approach by Zitrin et al.zitrin15b  which has been re-written to be fully
analytic, i.e. not limited by a grid-resolutionpascale22 ; furtak23b . The
UNCOVER lens model of A2744 was constructed on a wealth of ground- and space-
based data, including deep HST and JWST imaging, and Multi Unit Spectroscopic
Explorerbacon10  (MUSE) spectroscopic redshifts of both cluster members and
multiple imagesmahler18 ; richard21 ; bergamini23 ; bergamini23b . It
comprises 421 cluster member galaxies identified in the $\sim
45\,\mathrm{arcmin}^{2}$ UNCOVER field-of-view and five smooth cluster-scale
dark matter (DM) halos. The model is constrained with 141 multiple images
belonging to 48 sources and achieves an image reproduction RMS of
$\Delta_{\mathrm{RMS}}=0.51^{\prime\prime}$ in the lens plane. Thanks to the
massive cluster substructures identified with UNCOVERfurtak23b , the critical
area of the cluster is $1.5\times$ larger than inferred from HFF data and the
total source plane area with $\mu>4$ of $\sim 4\,\mathrm{arcmin}^{2}$ for a
source at redshift $z_{\mathrm{s}}=6$.

### Physical properties

#### Bagpipes

We infer global physical properties from SED-fitting using the Bagpipes
software package. Before fitting, all models are convolved with the
NIRSpec/Prism instrumental resolution curve provided by the Space Telescope
Science Institute (jwst_nirspec_Prism_disp.fits), assuming that the flight
performance is 1.3 times better than stated, and is consistent with an earlier
work where a factor 1/0.7 is introduced for modeling $z>10$ galaxies curtis-
lake23 . Additionally, we fit with a wavelength-independent velocity smoothing
($0<$ log$(v_{\mathrm{smooth}})<3.3$) as a nuisance parameter. We adapt the
following model grid: bc03  stellar population models, the MILES spectral
library Sanchez-Blazquez2006 ; Falcon-Barroso2011 , CLOUDY nebular emission
models Ferland:2017 , and Charlot2000  dust model (with $0<A_{v}<5$ and
$0.3<n<2.5$ as free parameters). The stellar and gas phase metallicity are
tied to the same value, and also included as a free parameter in the range
$-2<\mathrm{log(Z/Z_{\odot})}<0.3$. The ionization parameter is also left free
in the range of $-3.5<\mathrm{log(U)}<-1.0$. We parameterize the star
formation history as a delayed-$\tau$ model (SFR$\propto^{-t/\tau}$, which can
flexibly produce rising or falling star formation histories for this range of
$\tau$ at the redshift of the sample), with the age ($-3<$log(age)$<0.48$) and
$\tau$ (0.01$<\tau$<5) as free parameters. This parameterization has been
shown to reliably recover star formation rates and the mass formed in recent
star formation, but is potentially susceptible to under-estimating stellar
masses due to outshining by the youngest stellar population Papovich2023 .
Redshift is restricted to vary in a narrow range around the best-fitting
spectroscopic redshifts ($\pm$0.1). We fit for a polynomial calibration vector
of order 2 after applying a wavelength-independent calibration to scale the
normalization of the spectrum to the photometry. The Bagpipes white noise
model is used to allow for underestimated errors up to a factor of 10. A
signal-to-noise ceiling of 20 is imposed on both our photometry and
spectroscopy to account for systematic issues with the flux calibration.
Sampling is performed via PyMultinest Buchner:2014 ; Feroz:2019 , with the
default Bagpipes convergence criteria. The most important physical properties
derived from this procedure are presented in Table 2, and example fits and
posteriors are presented in Figure 7.

#### BEAGLE

We run an additional spectral fit with the BayEsian Analysis of GaLaxy
sEdschevallard16  tool (BEAGLE) on the magnification-corrected spectra. BEAGLE
uses the latest version of the Bruzual & Charlot stellar population synthesis
modelsBruzual_Charlot_2003  and nebular emission templates computed with
CLOUDYferland13 ; gutkin16 . We then assume a Chabrierchabrier03  initial
stellar mass function (IMF), an SMC dust attenuation lawpei92 , the latest
Inoue et al. analytic IGM attenuation modelsinoue14 , and a delayed
exponential SFH as for our Bagpipes fit. All other parameters are left free to
vary with uniform or log-uniform priors: stellar mass
$\log(M/\mathrm{M}_{\odot})\in[4,10]$, current (10 Myr) SFR
$\log(\psi/\mathrm{M}_{\odot}\,\mathrm{yr}^{-1})\in[-2,4]$, maximum stellar
age $\log(t_{\mathrm{age}}/\mathrm{yr})\in[6,t_{\mathrm{universe}}]$, star-
formation e-folding time $\log(\tau/\mathrm{yr})\in[5.5,9.5]$, stellar
metallicity $\log(Z/\mathrm{Z}_{\odot})\in[-2.2,-0.3]$, effective V-band dust
attenuation optical depth $\hat{\tau}_{V}\in[0,3]$, effective galaxy-wide
ionization parameter $\log~{}U\in[-4,-1]$, gas-phase metallicity
$\log(Z_{\mathrm{gas}}/\mathrm{Z}_{\odot})\in[-2.2,-0.3]$, and dust-to-metal
mass ratio $\xi_{\mathrm{d}}\in[0.1,0.5]$.

The posterior distribution of the physical properties that we derive with
BEAGLE agree well with the Bagpipes results presented in Table 2.

### Contribution of galaxies to reionization

Using the present spectroscopically-confirmed sample of ultra-faint galaxies,
we have the opportunity to put robust constraints on the UV luminosity
function. The original NIRSpec sample has been selected in HFF studiesatek18 ;
bouwens22c , based on HST observation, in addition to three sources selected
from the UNCOVER imaging data based on their photometric redshifts. First, we
compute an initial UV LF based on the present sample binned in four magnitude
bins and a survey volume, which depends on the original selection of the
source. For the five HFF sources, we use the source plane effective
volumeatek18  as a function of the magnitude bin. We rebin the original HFF
sample to match the new magnitude bins. A scaling factor is then applied to
the LF points to match the HFF completeness-corrected counts. Finally, we
apply a correction factor based on the success rate of the spectroscopic
confirmation. For the three sources outside the HFF area, we recompute the
source plane effective survey volume using the new lensing model and assuming
a similar completeness function across the field. We perform the same exercise
of rescaling the number counts in each magnitude bin. The final UV LF is
calculated by combining all galaxies with their associated corrected effective
volume. Regarding uncertainties, we use the HFF volume uncertainties for the
HFF sources, and the updated volume uncertainties for the new sample,
respectively. We include Poisson errors and cosmic variancetrenti08 . Figure 5
shows that our measurements (orange points) are in good agreement with the
faint-end of the photometric UV LF derived from HFF observationsatek18  (gray
points).

Figure 5: The UV luminosity function as determined from our spectroscopic
sample (orange points). Also shown, the photometric determination from the HFF
data atek18 , together with the best-fit Schechter function (blue curve) and a
modified Schechter with a potential turnover (teal curve). The shaded region
of each curve represent the 1$-\sigma$ uncertainties.

In our efforts to determine whether galaxies can reionize the Universe, we
proceed to calculate another crucial parameter: the production efficiency of
ionizing radiation $\xi_{\mathrm{ion}}$. This quantity is defined as the ratio
between the LyC photon production rate in the units of s-1, and the observed
non-ionizing UV luminosity density $L_{\rm UV}$ estimated at 1500 Å in units
of erg s-1 Hz-1:

$\xi_{\rm ion}=\frac{N(H^{0})}{L_{UV}}~{}[erg^{-1}~{}Hz],$ (1)

where $N(H^{0})$ can be estimated from the H$\alpha$ Balmer lineleitherer95
assuming a case B recombination theoryosterbrock89 :

$L(H\alpha)~{}[erg~{}s^{-1}]=1.36\times(1-f_{esc})~{}10^{-12}~{}N(H^{0})~{}[s^{-1}]$
(2)

where $L(H\alpha)$ is in units of erg s-1, and $f_{\rm esc}$ is the escape
fraction of Lyman continuum radiation. In this calculation, we assume that
$f_{\rm esc}$=0, meaning that all Lyc photons are reprocessed into the Balmer
lines. The derived $\xi_{\mathrm{ion}}$ value can be considered as a lower
limit, since higher $f_{\rm esc}$ will lead to a higher $\xi_{\mathrm{ion}}$.
The H$\alpha$ emission line is not detected in source ID 18924. In this case,
we use the H$\beta$ luminosity and a case B conversion factor.

Finally, we derive constraint on the LyC escape fraction using indirect
indicators calibrated in a large sample of nearby LyC emitting galaxies. Since
the LyC emission is impossible to measure at the EoR, large efforts have been
devoted in the last two decades to determine the escape fraction in $z<4$
galaxies, and more importantly establish robust indirect methods, which can be
transferable to EoR sources. This was precisely the motivation of the recent
Low-redshift Lyman continuum survey (LzLCS)flury22a . Among the different
physical properties of LyC leakers, the observed UV continuum slope $\beta$
has been identified as a promising proxy of $f_{\rm esc}$chisholm22 . Here, we
use the UV slopes derived from the best-fit models of Bagpipes and the LzLCS
relation to infer $f_{\rm esc}$. The derived values for the present sample
range from 4.5% to 15.6%. Despite large uncertainties (around 50%), only two
sources have $f_{\rm esc}$ values that can reach below 4% at 1-sigma.

Now with all three properties in hand, we can assess the contribution of faint
galaxies to cosmic reionization. We compute the ionizing photon emissivity of
galaxies, which is the product of the total UV luminosity density
$\rho_{\mathrm{UV}}$, derived from integrating the UV LF, and the production
efficiency $\xi_{\mathrm{ion}}$. The result will depend on the faint
integration limit, which is set the faintest bin of the spectroscopic LF at
$M_{\rm UV}$=-15 mag. By multiplying this quantity by the escape fraction
$f_{\rm esc}$, we obtain the total ionizing photon rate density that is
available to ionize the IGM. Down to $M_{\rm UV}$=-15 mag, modest $f_{\rm
esc}$values around 5% are sufficient to maintain reionization.

We measure the gas-phase oxygen abundance using the strong optical lines
diagnostic. Specifically, we use the R3=log([Oiii]$\lambda$5007 /H$\beta$) and
adopt the most recent empirical calibrations at high-redshiftnakajima22 ;
nakajima23 ; sanders21 . All of our sources have robust detections of
[Oiii]$\lambda$5007 and H$\beta$. Furthermore, we also separate our sample
into two bins according to the EW(H$\beta$) at EW(H$\beta$)=100Å to account
for ionization parameter variationsnakajima22 . Also, for a given value of R3,
the calibration defines two metallicity solutions. Although these sources have
likely low metallicities, we use the
O32=log([Oiii]$\lambda\lambda$4949,5007/[Oii]$\lambda\lambda$3727,3729) ratio
to distinguish between the two branches. For most of the sources, the [Oii] is
not detected, which provides a lower-limit on O32, which is found to vary in
the range O32=[0.7-1.4]. Using the O32 metallicity indicator, the resulting
values are all compatible with the low-metallicity branch solution. The
metallicity measurements are reported in Table 2.

Figure 6: UNCOVER JWST data for galaxy 16155 at $z_{\mathrm{spec}}=6.88$. The
top panels show image cutouts in seven different filters at increasing
wavelength including ancillary HST/ACS data in F814W, and UNCOVER JWST imaging
in F115W, F150W, F200W, F277W, F356W, and F444W bands (left to right). The
central panel shows the UNCOVER NIRSpec data, with the 2D spectrum on top of
the 1D optimally extracted spectrum (black with gray uncertainty ranges). The
red lines show the best-fit msaexp template spectrum. The observed-frame
wavelengths of key emission lines are indicated as vertical dashed lines. The
bottom panels show a zoomed in version of three different parts of the
spectrum around the Ly$\alpha$ break (left), around the [Oiii]+H$\beta$
emission lines (middle) and the H$\alpha$ line (right).

### Size Measurements

Measurements of the half-light radii for all eight galaxies are performed
using the pysersic Pasha2023  package, focusing on the F150W image. For each
source, we assume a single Sersic profile and mask all nearby sources in the
photometric catalogweaver23 . The priors for the half-light radius, Sersic
index, axis ratio and position angle are all uniform and varied from $0.5-10$
pixels, $0.65-4$, $0.1-1$ and $0-2\pi$ respectively. Priors central position
and flux are represented as Gaussian distributions with the location and width
based on the photometric catalogweaver23 . A flat sky background is fit
simultaneously. The Posterior distribution is explored using the No-U Turn
(NUTS) Hoffman2014  sampler implemented in numpyro Phan2019  with 2 chains for
1,000 warm-up and sampling steps each.

Several of our objects are significantly distorted by the gravitational
lensing which means that we need to take the shear into account when deriving
the half-light radius. In order to do that, we use our SL model to derive the
tangential and radial magnifications, defined as
$\mu=\mu_{\mathrm{t}}\mu_{\mathrm{r}}$, from the deflection field at each
source’s position and redshift. Since our objects are sheared along the
tangential direction, we use the tangential component of the magnification to
correct the half-light radii.

Figure 7: Stellar population simultaneous fitting to the NIRSpec spectra and NIRCam photometry. Two representative sources (IDs 18924 and 16155) are shown. Left the best-fit Bagpipes model (red curve) is plotted over the observed NIRSpec spectrum (black curve), together with the error spectrum (gray curve). The NIRCam photometric measurements are represented with black points with their associated 1-sigma uncertainties. The right panel shows the posterior distribution function for the main physical properties. When relevant, the parameters are corrected for magnification. Table 1: Summary of the sample properties. The exposure time (4) corresponds to the total of all MSA observations for each source. The magnification factors (5) are computed at the spectroscopic redshift of the source using the most recent lensing model furtak23b . Note that while the statistical uncertainty can be relatively small in some cases, a systematic error of 10-20% should be nominally added. The photometric redshift (6) is measured with the Eazy software. The spectroscopic redshift (7) is measured from the best msaexp fit. The typical best-fit error is $\sigma_{zspec}=0.01$. Absolute magnitude $M_{\rm UV}$ (8) is measured in the rest-frame UV using the observed magnitude derived from the photometric catalogweaver23  corrected for magnification. Source | RA | Dec | Exptime | $\mu$ | $z_{phot}$ | $z_{spec}$ | $M_{\rm UV}$
---|---|---|---|---|---|---|---
| J2000 | J2000 | hours |  |  |  | AB
18924 | 3.581044 | -30.389561 | 17.4 | $26.62\pm 3.33$ | 7.88 | 7.70 | $-15.47\pm 0.08$
16155 | 3.582953 | -30.395232 | 17.4 | $11.12\pm 1.77$ | 6.70 | 6.87 | $-16.29\pm 0.08$
23920 | 3.572830 | -30.380026 | 3.7 | $3.34\pm 0.17$ | 6.35 | 6.00 | $-16.18\pm 0.10$
12899 | 3.582353 | -30.402732 | 10.2 | $13.87\pm 0.91$ | 6.64 | 6.88 | $-15.34\pm 0.11$
8613 | 3.600602 | -30.410271 | 2.7 | $9.28\pm 0.62$ | 6.53 | 6.38 | $-16.97\pm 0.04$
23619 | 3.607272 | -30.380578 | 7.5 | $1.77\pm 0.03$ | 6.68 | 6.72 | $-16.55\pm 0.16$
38335 | 3.541383 | -30.357435 | 2.7 | $2.34\pm 0.02$ | 6.41 | 6.23 | $-16.89\pm 0.13$
27335 | 3.625081 | -30.375261 | 7.5 | $1.41\pm 0.01$ | 6.88 | 6.76 | $-17.17\pm 0.08$
Table 2: Summary of the physical properties of the sample derived from SED fitting with Bagpipes. For each source (1), the median posterior and associated uncertainties from the best-fit models are given for: the stellar mass (2), the star formation rate SFR (4), the UV continuum slope $\beta$ (5), the half-mass age (6). The oxygen abundance computed from strong optical lines is also reported (7). The star formation rate derived from the H$\alpha$ emission is also reported in column (3) Source | log(M⋆/M⊙) | SFRHα | SFRUV | $\beta$ | $t_{\mathrm{50}}$ | 12+Log(O/H)
---|---|---|---|---|---|---
|  | M⊙ yr-1 | M⊙ yr-1 |  | Myr |
18924 | $5.88_{-0.08}^{+0.13}$ | $0.33\pm 0.02$ | $0.01_{-0.07}^{+0.14}$ | $-2.39_{-0.10}^{+0.12}$ | $2.23_{-0.85}^{+0.68}$ | $6.95\pm 0.15$
16155 | $6.61_{-0.06}^{+0.07}$ | $0.92\pm 0.04$ | $0.04_{-0.06}^{+0.08}$ | $-2.09_{-0.08}^{+0.07}$ | $3.96_{-0.66}^{+0.92}$ | $7.01\pm 0.19$
23920 | $6.30_{-0.03}^{+0.03}$ | $1.32\pm 0.04$ | $0.02_{-0.03}^{+0.03}$ | $-2.45_{-0.03}^{+0.03}$ | $1.12_{-0.11}^{+0.32}$ | $6.84\pm 0.06$
12899 | $6.54_{-0.19}^{+0.14}$ | $0.49\pm 0.02$ | $0.04_{-0.15}^{+0.12}$ | $-2.51_{-0.07}^{+0.09}$ | $28.66_{-11.98}^{+15.51}$ | $6.70\pm 0.15$
8613 | $7.12_{-0.08}^{+0.07}$ | $0.78\pm 0.07$ | $0.16_{-0.07}^{+0.08}$ | $-2.53_{-0.03}^{+0.04}$ | $25.73_{-6.33}^{+6.47}$ | $6.97\pm 0.18$
23619 | $6.57_{-0.06}^{+0.10}$ | $0.85\pm 0.07$ | $0.04_{-0.05}^{+0.11}$ | $-2.51_{-0.07}^{+0.13}$ | $1.08_{-0.07}^{+0.22}$ | $7.19\pm 0.20$
38335 | $6.83_{-0.20}^{+0.25}$ | $1.00\pm 0.16$ | $0.07_{-0.15}^{+0.34}$ | $-2.07_{-0.24}^{+0.29}$ | $6.45_{-2.29}^{+4.39}$ | $7.46\pm 0.32$
27335 | $6.73_{-0.08}^{+0.15}$ | $0.73\pm 0.10$ | $0.05_{-0.07}^{+0.17}$ | $-2.35_{-0.11}^{+0.22}$ | $1.56_{-0.52}^{+1.33}$ | $6.99\pm 0.18$

Data Availability The NIRCam and HST imaging data are available on the UNCOVER
webpage: https://jwst-uncover.github.io/. The NIRSpec spectroscopic data are
publicly available through the Mikulski Archive for Space
Telescopes555https://archive.stsci.edu/ (MAST), under program ID 2561.

Code Availability Astropy astropy13 ; astropy18 , Bagpipes Carnall2018 ;
Carnall2019:VANDELS , BEAGLE chevallard16 , EAzY Brammer2008 , Matplotlib
plt07 , msaexp (brammer22msa, , v0.6.10), NumPy numpy20 , NUTS nuts11 ; nuts19
, PyMultinest Buchner:2014 ; Feroz:2019 , pysersic Pasha2023 , SciPy scipy20

Acknowledgments H.A. and IC acknowledge support from CNES, focused on the JWST
mission, and the Programme National Cosmology and Galaxies (PNCG) of CNRS/INSU
with INP and IN2P3, co-funded by CEA and CNES. H.A thanks the Cosmic Dawn
Center (DAWN) for their support. DAWN is funded by the Danish National
Research Foundation under grant No. 140. A.Z. acknowledges support by Grant
No. 2020750 from the United States-Israel Binational Science Foundation (BSF)
and Grant No. 2109066 from the United States National Science Foundation
(NSF), and by the Ministry of Science & Technology, Israel. P.D. acknowledges
support from the NWO grant 016.VIDI.189.162 (“ODIN") and from the European
Commission’s and University of Groningen’s CO-FUND Rosalind Franklin program.
The work of C.C.W. 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.

Author Contributions H.A. led the analysis and article writing. L.J.F. and
A.Z. constructed the lens model and extracted lensing related quantities. S.F
produced figures. I.L. and R.B. are the PIs of the UNCOVER program. R.B. and
I.L. designed the observations and reduced the spectra. J.W. and B.W. produced
the catalogs used for target selection. P.D. provided simulations to interpret
the observational results obtained. V.K. produced line measurements. I.C.
estimated survey volumes. D.J.S. ran SED fitting analysis. T.B.M. measured the
galaxy sizes. All authors contributed to the manuscript and aided the analysis
and interpretation.

Competing Interests The authors declare that they have no competing financial
interests.

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