# COOL-LAMPS IV: A Sample of Bright Strongly-Lensed Galaxies at $3<z<4$

Yunchong Zhang Department of Astronomy and Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA  Viraj Manwadkar
Department of Astronomy and Astrophysics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL 60637, USA  Department of Physics, Stanford
University, 382 Via Pueblo Mall, Stanford, CA 94305, USA  Kavli Institute for
Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University,
Stanford, CA 94305, USA  Michael D. Gladders Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL
60637, USA  Kavli Institute for Cosmological Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, USA  Gourav Khullar Department of
Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL 60637, USA  Kavli Institute for Cosmological Physics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA  Department of
Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA
15260, USA  Håkon Dahle Institute of Theoretical Astrophysics, University of
Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway  Kate A. Napier Department
of Astronomy, University of Michigan, 1085 S. University Ave, Ann Arbor, MI
48109, USA  Guillaume Mahler Centre for Extragalactic Astronomy, Durham
University, South Road, Durham DH1 3LE, UK  Institute for Computational
Cosmology, Durham University, South Road, Durham DH1 3LE, UK  Keren Sharon
Department of Astronomy, University of Michigan, 1085 S. University Ave, Ann
Arbor, MI 48109, USA  Owen S. Matthews Acuña Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL
60637, USA  Finian Ashmead  Department of Astronomy and Astrophysics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
William Cerny Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, USA  Juan Remolina Gonzàlez
Department of Astronomy, University of Michigan, 1085 S. University Ave, Ann
Arbor, MI 48109, USA  Katya Gozman Department of Astronomy, University of
Michigan, 1085 S. University Ave, Ann Arbor, MI 48109, USA  Benjamin C. Levine
Department of Astronomy and Astrophysics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL 60637, USA  Department of Physics and Astronomy,
Stony Brook University, Stony Brook, NY 11794, USA  Daniel Marohnic
Department of Astronomy and Astrophysics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL 60637, USA  Michael N. Martinez Department of
Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL 60637, USA  Kaiya Merz Department of Astronomy and Astrophysics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA  Yue
Pan Department of Astronomy and Astrophysics, University of Chicago, 5640
South Ellis Avenue, Chicago, IL 60637, USA  Jorge A. Sanchez Department of
Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL 60637, USA  Isaac Sierra  Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL
60637, USA  Emily E. Sisco Department of Astronomy and Astrophysics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA  Ezra
Sukay Department of Astronomy and Astrophysics, University of Chicago, 5640
South Ellis Avenue, Chicago, IL 60637, USA  Kiyan Tavangar Department of
Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL 60637, USA  Erik Zaborowski  Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL
60637, USA Yunchong Zhang<EMAIL_ADDRESS>
###### Abstract

We report the discovery of five bright strong gravitationally-lensed galaxies
at $3<z<4$: COOL J0101$+$2055 ($z=3.459$), COOL J0104$-$0757 ($z=3.480$), COOL
J0145$+$1018 ($z=3.310$), COOL J0516$-$2208 ($z=3.549$), and COOL J1356$+$0339
($z=3.753$). These galaxies have magnitudes of $r_{\rm AB},z_{\rm AB}<21.81$
mag and are lensed by galaxy clusters at $0.26<z<1$. This sample doubles the
number of known bright lensed galaxies with extended arcs at $3<z<4$. We
characterize the lensed galaxies using ground-based grz/giy imaging and
optical spectroscopy. We report model-based magnitudes and derive stellar
masses, dust content, and star-formation rates via stellar-population
synthesis modeling. Building lens models based on ground-based imaging, we
estimate source magnifications ranging from $\sim$29 to $\sim$180\. Combining
these analyses, we derive demagnified stellar masses ranging from $\rm
log_{10}(M_{*}/M_{\odot})\sim 9.7-11.0$ and star formation rates in the
youngest age bin ranging from $\rm log_{10}(SFR/(M_{\odot}\cdot yr^{-1}))\sim
0.4-1.6$, placing the sample galaxies on the massive end of the star-forming
main sequence in this redshift interval. In addition, three of the five
galaxies have strong Ly$\alpha$ emissions, offering unique opportunities to
study Ly$\alpha$ emitters at high redshift in future work.

Strong gravitational lensing (1643) — High-redshift galaxies (734) — Spectral
energy distribution (2129) — Galaxy evolution (594)

††facilities: Magellan Telescopes 6.5m (Clay/LDSS3C, Baade/IMACS, Baade/FIRE),
Nordic Optical Telescope 2.56m/ALFOSC††software: NumPy(Harris et al., 2020),
matplotlib (Hunter, 2007), astropy (Astropy Collaboration et al., 2022), scipy
(Virtanen et al., 2020), SEDpy (Johnson, 2019), Prospector (Johnson et al.,
2021), SourceExtractor (Bertin & Arnouts, 1996), Lenstool (Jullo et al.,
2007), GALFIT (Peng et al., 2010)

##  1 Introduction

Strongly-lensed star-forming galaxies at high redshift provide unique
opportunities to understand star formation processes in the early Universe.
Using the spectral energy distribution (SED) (see Conroy, 2013, for a review
of this method) we can infer important physical properties of the galaxies
such as the stellar mass ($M_{*}$), star-formation rate (SFR), and dust
extinction (e.g., Kennicutt, 1998; Calzetti et al., 2000; Giavalisco, 2002;
Salim et al., 2007a; Maraston et al., 2010; Walcher et al., 2011; Wilkins et
al., 2012). Strong lensing allows us to probe much smaller sub-galactic
spatial scales that would normally be inaccessible without the extra
magnification that lensing provides.

Figure 1:  RGB color images of size 30” x 30” of the lensed galaxies in our
sample. The data for CJ0101, CJ0104, CJ0516, and CJ0145 are from Magellan
LDSS3 grz imaging. The data for CJ1356 is from Hyper Supreme-Cam Subaru
Strategic Program (HSC) giy DR3 data release. The white ticks in each panel
mark the lensed images that are confirmed spectroscopically. The
spectroscopically confirmed redshift of the lensed galaxies is shown in the
upper left corner of each panel. Note that the color contrast for each of
these images has been enhanced to clearly show the lensed arcs.

Selecting strongly-lensed galaxies consists, to first order, of two methods:
deeply detailed studies of large lensing cross section lenses, or mining large
areas of sky across a wide range of foreground lens cross sections to find the
brightest examples. The former allows one to detect galaxies well below the
detection limits reached in deep field images and significantly improves the
signal-to-noise ratio (SNR) of observations compared to their un-lensed
counterparts. Pointed (i.e., pencil-beam) strong lensing surveys such as the
Hubble Frontier Fields (Lotz et al., 2017) have enabled studies (e.g., Santini
et al., 2017; Rinaldi et al., 2022) to include fainter galaxies to probe the
low stellar mass regime ($\log_{10}(M_{*}/M_{\odot})\lesssim 9.0$) (e.g.,
Alavi et al., 2016). The latter approach has yielded exquisitely bright lensed
sources where the magnification of strong lensing provides extreme spatial
resolutions of only tens of parsecs in the source plane (e.g., Johnson et al.,
2017). A particularly cogent example of the benefits of such an object is the
Sunburst Arc (Dahle et al., 2016) where lensing allows identification of a
single star-forming region that is leaking Lyman Continuum (LyC) radiation,
enabling unique opportunities to study the detailed physics of LyC escape
(e.g., Rivera-Thorsen et al., 2017; Furtak et al., 2022).

In the redshift range $3<z<4$, only five bright lensed star-forming galaxies
with extended arcs have been published to date (Smail et al., 2007; Quider et
al., 2010; Oguri et al., 2012; Bayliss et al., 2014; Rigby et al., 2018) In
this work, we present the discovery and primary analysis of five additional
UV-bright strongly-lensed galaxies between $3<z<4$, doubling the number of
known objects in the same category. This redshift range is well before the
peak of the cosmic star-formation rate density at $z\sim 2$ (Madau &
Dickinson, 2014) and the corresponding peak of the redshift distribution of
prior bright lensed galaxy samples (Bayliss et al., 2011a; Bayliss, 2012;
Stark et al., 2013; Tran et al., 2022); z=3.5 is approximately the half-time
between the plausible beginning of star formation at $z\sim 20$ and that later
peak.

The goal of this paper is to determine the physical and evolutionary
properties of these lensed galaxies by means of their photometric and UV
spectral features. The structure of this paper is as follows. In Section 2, we
introduce the discovery of objects in this sample. Section 3 describes the
photometric and spectroscopic data acquisition following the discoveries. In
Section 4, we present the results from photometric modeling, lens modeling,
and stellar population synthesis with photometric information from rest-frame
UV bands. Section 5 discusses the limitation in our analysis and compares the
initial results with other objects in the same regime.

Table 1: The names and positions of our sample galaxies. Note that we name each lensing system by taking the position of the lensing galaxy. For COOL J0104$-$0757, we choose the position of the galaxy in the North at $z\simeq 1$ as the notional coordinate of the lensing system. The relative positions of lensed images are shown in Figure 1. As a shorthand, we will coalesce the system names to the four first digits of their coordinates (e.g., COOL J0101$+$2055 becomes CJ0101). | RA (deg) | DEC (deg)
---|---|---
COOL J0101$+$2055 | 15.437643 | 20.928818
COOL J0104$-$0757 | 16.220170 | -7.952055
COOL J0145$+$1018 | 26.276362 | 10.310288
COOL J0516$-$2208 | 79.013054 | -22.146431
COOL J1356$+$0339 | 209.094447 | 3.652319

In the entire analysis, we assume the cosmological parameters derived in
Planck Collaboration et al. (2020). All magnitudes are given in the AB system.
For inferred parameters with uncertainties, we report 16th, 50th, and 84th
percentile values, unless otherwise specified.

##  2 Discovery

###  2.1 Lens candidate selection

The COOL-LAMPS project — ChicagO Optically selected strong Lenses - Located At
the Margins of Public Surveys — began as a collaboration in an undergraduate
research class, and is an effort to find strong gravitational lenses in recent
public imaging data. The lensing search is designed to find a wide variety of
lenses and lensed sources (Khullar et al., 2021; Sukay et al., 2022; Martinez
et al., 2022) but the targeted follow-up is primarily focused on lensed
sources that are photometrically at the margins of the distributions of source
color and brightness. The details and results of this search will be presented
in an upcoming publication (COOL-LAMPS Collaboration in prep.). We select
luminous red galaxies – the most massive galaxies at a given epoch – as
potential sites of lensing by making cuts in color-magnitude diagrams of
objects morphologically tagged as galaxies. For each selected line of sight,
we construct a custom $grz$ color image, tuned to emphasize faint extended
sources, with the cutout size scaled by an estimate of the galaxy richness at
the potential lens redshift, estimated from a red-sequence early-type galaxy
model. Each such image is scored from 0 (no lensing) to 3 (obvious unambiguous
lensing) by multiple visual examiners, and a second pass is made through the
resulting candidate list to cull objects for further consideration. The bulk
of the lens selection work completed by the COOL-LAMPS collaboration so far
has used the DECaLS (The Dark Energy Camera Legacy Survey, Dey et al., 2019)
datasets and all five lensing systems discussed here are taken from those
data. The names and locations of our sample galaxies are given in Table 1. The
lensed sources listed here were selected as candidates for further follow-up
due to high overall initial ranking from the lens search, coupled with blue
$r-z$ colors and redder $g-r$ colors suggestive of $z>3$ galaxies.

###  2.2 Spectroscopic confirmation

CJ0101 and CJ0104 were initially observed spectroscopically using the 2.5m
Nordic Optical Telescope and ALFOSC spectrograph on October 14, 2020. The
total exposure time for CJ0101 was 3000 seconds, and 2400 seconds for CJ0104.
In both cases we used grism #4 and a slit width of 1.3”, producing a spectral
resolution of R$\sim$300 and covering the wavelength range 3200-9600Å. Those
data measured initial source redshifts for both systems. CJ0145 and CJ1356
were first observed spectroscopically on July 20, 2021 for 1200 seconds each,
using the 1”.0 slit in echelle mode of the FIRE near-IR spectrograph (Simcoe
et al., 2013) on the Magellan I 6.5m Baade telescope; those data provided
source redshifts from multiple emission lines in both instances. CJ0516 was
observed on the same night and telescope using the IMACS optical spectrograph
(Dressler et al., 2011) for 2000 seconds in total using the f/4 camera and the
150 ln/mm grating and a 1”.0 slit ; those data also provided a source redshift
from multiple emission and absorption features.

##  3 Follow-up observations and redshifts

Figure 2: Optical/near-IR spectra of the source galaxies of CJ0101
(Magellan/IMACS) and CJ0104 (Magellan/LDSS3). (Top panels in each row): the
extracted raw spectrum (light blue) and the spectrum convolved with a median
filter (orange); (Bottom panels in each row): the noise spectrum. The fluxes
are in units of $\rm erg/(s\cdot cm^{2}\cdot\AA)$ and are normalized
arbitrarily. For each row, the extracted spectra and noise spectrum have the
same scaling for fluxes. Black vertical dashed lines indicate significant
spectral features used to infer the redshifts of the source galaxies. Grey
vertical shaded regions mark the telluric absorption bands. For CJ0101, we
mark the Ly$\alpha$ emission line with a back dashed line and for CJ0104, we
mark Ly$\alpha$ in absorption at $\rm 1215\AA$ with a grey dashed line for
reference. In the case of CJ0101, the emission line of CIII 1905 is present
and shown in an embedded panel. Note that there are several significant
absorption features that are likely due to intervening absorbers in all object
spectra. There is no spectral data in the gap in the first row due to the chip
gaps of the IMACS detector mosaic.  Figure 3: Optical/near-IR spectra of the
source galaxies of CJ0145 (Magellan/IMACS), CJ0516 (Magellan/IMACS) and CJ1356
(Magellan/LDSS3). (Top panels in each row): the extracted raw spectrum (light
blue) and the spectrum convolved with a median filter (orange); (Bottom panels
in each row): the noise spectrum. The fluxes are in units of $\rm erg/(s\cdot
cm^{2}\cdot\AA)$ and are normalized arbitrarily. For each row, the extracted
spectra and noise spectrum have the same scaling for fluxes. Black vertical
dashed lines indicate significant spectral features used to infer the
redshifts of the source galaxies. Grey vertical shaded regions mark the
telluric absorption bands. We mark the Ly$\alpha$ emission line at $\rm
1215\AA$ with back dashed lines in cases of CJ0516 and CJ1356, and Ly$\alpha$
in absorption with a grey dashed line for reference in the case of CJ0145. For
CJ1356, the emission line of CIII 1905 is present and shown in an embedded
panel. Note that there are several significant absorption features that are
likely due to intervening absorbers in all object spectra. There is no
spectral data in the gaps in the first and second rows due to the chip gaps of
the IMACS detector mosaic.

###  3.1 Imaging

We obtained broad-band optical imaging, deeper and sharper than the discovery
DECaLS data, in the grz filters using the Magellan Clay telescope with the
LDSS3-C imaging spectrograph for four of the five lensed systems. The
exception, CJ1356, is present in the Hyper Suprime-Cam Subaru Strategic
Program (HSC-SSP) DR3 data release, and we use the available giy in lieu of
additional Magellan data. The Magellan imaging data were acquired in clear
conditions, on August 13, 2020, and October 10, 2021, in seeing of 0.6” to
0.8”, with total exposure times per filter ranging from 360 to 840 seconds
selected by source brightness and sky brightness at the time of observation.
Data were processed using standard techniques in IRAF (Tody, 1986, 1993) and
PHOTPIPE (Rest et al., 2005; Garg et al., 2006; Miknaitis et al., 2007),
producing a stacked co-added image for each object and filter pair. An initial
astrometric solution was defined using astrometry.net (Lang et al., 2010);
this was further refined using custom code to match the astrometric solution
for each image precisely to the DECaLS catalogs. Photometric zeropoints were
derived by direct comparison to DECaLS, and have in most cases uncertainties
of less than 0.02 mags. The quoted uncertainty in the HSC-SSP DR3 data is
0.015 mags (Aihara et al., 2021).

Figure 1 shows color images, derived from the above data, for all five strong
lensing systems.

###  3.2 Spectroscopy

After these initial discovery observations, four of the five systems
(excluding CJ0516) were re-observed spectroscopically using either the
Magellan I 6.5m Baade telescope and IMACS, or the Magellan II 6.5m Clay
telescope and LDSS3C. The goal was to gather restframe UV spectra to
characterize the Ly$\alpha$ line and, in most instances, to place candidate
secondary images or foreground lens galaxies on the same longslit to acquire
further redshifts. Observations occurred on October 13-14 and 26 2021, and
July 22, 2022.

The follow-up spectroscopy was processed with custom scripts using standard
techniques to flatten, extract to 1D, and wavelength and flux calibrate the
data, all implemented in IRAF. All objects with apparent spectra were
extracted and calibrated, and analyzed, not always successfully, for redshift
information.

###  3.3 Redshifts

The redshifts of sources and foreground lenses as well as their uncertainties
are tabulated in Table 2. For lensed sources, we used a set of interstellar
absorption lines (S II 1259, Si II 1260, O I 1302, Si II 1304, C II 1334, Si
IV 1393, 1402, and C IV 1548, 1550) to obtain a set of redshifts for each
object. In addition, emission line CIII 1905, and absorption lines SiII 1526
and FeII 1608 are present in the spectra of some objects. See Figure 2 and
Figure 3 for details. We report the final redshift values by taking the set
average and the uncertainties by taking the standard deviation. Note that for
CJ1356, the reported redshift refers to the doubly-imaged green arcs in the
North East of the field, as marked by white bars in Figure 1 and a second
family of lensed images at $z\simeq 2.17$ is also present.

The redshifts of lenses were obtained as follows. For CJ0104 and CJ0516, we
extracted LDSS3/IMACS spectra of the brightest central galaxies (BCGs) in
these systems. The BCG redshifts and related uncertainties, which were
computed based on the spectral features, were taken as lens redshifts for
these two systems. For CJ0101 and CJ1356, we used BCG redshifts and
uncertainties reported in eBOSS (Dawson et al., 2016), which is part of the
SDSS DR16 (Ahumada et al., 2020). For CJ0145, we obtained spectroscopic
redshifts for two cluster members from our spectra and for two other members
from SDSS DR16. We computed the final redshift by taking the mean value of the
cluster member redshifts and the uncertainties by taking the standard
deviation.

Table 2: Table for source redshifts, lens redshifts and median lens magnification with 68% confidence interval. | $z_{\rm lens}$ | $z_{\rm source}$ | Magnification
---|---|---|---
CJ0101 | $0.8708\pm 0.0003$ | $3.459\pm 0.001$ | $100_{-63}^{+125}$
CJ0104 |  $1.0037\pm 0.0006$ & $0.8581\pm 0.006$ | $3.480\pm 0.002$ | $40_{-16}^{+55}$
CJ0145 | $0.463\pm 0.006$ | $3.310\pm 0.003$ | $29_{-14}^{+23}$
CJ0516 | $0.658\pm 0.001$ | $3.549\pm 0.002$ | $180_{-69}^{+173}$
CJ1356 | $0.26185\pm 0.00006$ | $3.753\pm 0.004$ | $93_{-19}^{+41}$

##  4 Analysis

Figure 4:  Selected results of parametric photometry fitting: CJ1356 (HSC $g$
band), CJ0145 ($r$ band), CJ0516 ($r$ band), CJ0101 ($r$ band) and CJ0104 ($r$
band). The left column shows the original data frames; the central column
shows the fitted model frames; the right column shows the residual frames. The
angular scale of each fitting frame is shown on the right in arc-seconds in
each row. Note that for COOL J1356+0339, the top-left region of the image is
contaminated by cosmic ray and have a series of pixels with underestimated
values. Such a feature is not prominent in other band frames for this object.
We manually identified the corrupted pixels and applied a mask for them during
the fitting process.

###  4.1 Photometric Modeling with GALFIT

To extract the photometry of the arc and relevant potential counter-images, we
used the parametric model fitting code GALFIT (Peng et al., 2010). We measured
the point-spread functions (PSFs) directly from the images by choosing bright,
unsaturated, isolated point sources near the region of interest. We used
multiple two-dimensional Sérsic (Sérsic, 1963) profiles to model each image.
The model construction proceeded iteratively with single 2D Sérsic components
being placed down on each source in the image and additional components added
until residuals were consistent with the background noise. We constructed the
initial GALFIT models in the band where the lensed arc is the most prominent
and then propagated that model with appropriate constraints to other bands.

To estimate the statistical photometric uncertainties, we re-fitted the model
multiple times by adding the final fitted GALFIT models to simulated instances
of sky flux, generated by bootstrapping the sky pixel distribution of the
image. We obtained the sky pixels by iteratively clipping at 3$\sigma$ the
pixel distribution around the image. The measured magnitude of the object in
each of these re-fits gave us the uncertainty for each object. To compute the
total magnitude uncertainty for each object, we combined the zero-point
calibration uncertainty and above statistical uncertainties in quadrature.
Reported object magnitudes were corrected for Galactic extinction. The LDSS3
imaging grz band magnitudes were corrected by values provided in the DECaLS
database (Dey et al., 2019) while the HSC imaging giy band magnitudes were
corrected by values reported in Schlafly & Finkbeiner (2011) as implemented by
the NASA/IPAC Extragalactic Database’s extinction calculator111 The NASA/IPAC
Extragalactic Database (NED) is funded by the National Aeronautics and Space
Administration and operated by the California Institute of Technology..

Table 3: The GALFIT modelled photometry magnitudes (with 68% confidence intervals) of these galaxies. Note that magnitudes are in the AB system with $grz$ imaging from the Magellan LDSS3 imager and $giy$ band imaging from Hyper Suprime-Camera (HSC). Magnitudes are for the bright arc images only (denoted by white bars in Figure 1) and magnitudes have been accounted for Galactic extinction. | g | r | z | HSC g | HSC i | HSC y
---|---|---|---|---|---|---
CJ0101 | $22.58^{+0.02}_{-0.04}$ | $21.29^{+0.03}_{-0.02}$ | $20.89^{+0.05}_{-0.04}$ | - | - | -
CJ0104 | $21.87^{+0.03}_{-0.03}$ | $20.59^{+0.06}_{-0.05}$ | $20.31^{+0.04}_{-0.05}$ | - | - | -
CJ0145 | $21.62^{+0.05}_{-0.05}$ | $20.81^{+0.04}_{-0.04}$ | $20.71^{+0.07}_{-0.09}$ | - | - | -
CJ0516 | $21.45^{+0.04}_{-0.03}$ | $20.40^{+0.02}_{-0.03}$ | $20.34^{+0.06}_{-0.05}$ | - | - | -
CJ1356 | - | - | - | $22.55^{+0.05}_{-0.06}$ | $21.69^{+0.08}_{-0.08}$ | $21.81^{+0.27}_{-0.26}$

Table 3 tabulates the GALFIT-measured photometric magnitudes and corresponding
uncertainties for each arc. The $r$-band image (HSC $g$-band image for
CJ1356), GALFIT model, and residuals of each object field are shown in Figure
4.

Figure 5: (Broad panels): Best fit fiducial SED models (black solid lines) for
all five objects via Prospector using rz/iy photometry. Best-fit photometry is
shown as black squares while observed photometry is shown as orange circles.
The uncertainty of observed photometry is represented by the length of red
bars in the orange circles. (Narrow panels): corresponding Residual ($\chi$)
values and errors. Note that everything is shown in observed wavelengths. The
flux values of the entire model were renormalized by the flux value at
$\rm\lambda=1100\AA$. The nebular line and continuum emissions were turned on
during the fitting process. However, we chose not to display the nebular line
emission in the model spectra. Table 4: Prospector Analysis: Free Parameters
in the Fiducial SED Models

Parameter | Description | Priors
---|---|---
$M_{\star}(M_{\odot})$ | Total stellar mass formed | Log10 Uniform: [109, 1013]
$\log(Z/Z_{\odot})^{*}$ | Stellar metallicity | Uniform: [-2.0, 0.2]
$\tau_{\lambda,2}$ | Diffuse dust optical depth | Tophat: [0.01, 3.00]
$\mathrm{gas}_{\log(Z/Z_{\odot})}^{*}$ | Gas-phase metallicity | Tophat: [-2, 0.2]
$\log(U)^{*}$ | Gas Ionization Parameter $U=n\gamma/n_{H}$ | Tophat: [-4.0, -1.5]
$\log(\mathrm{SFR}_{\rm ratio})$ | Ratio of the SFRs in adjacent age bins: | Student-T: mean=0.0,scale=0.3, $\nu=2$
| non-parametric SFH | Age bins: [0,50], [50,100], [100,500] and [500, age of Universe at zobs]

Note. — $*$ Considered as nuisance parameters.

###  4.2 SED Fitting with Prospector

Using the model photometry in different filters (discussed in Section 4.1), we
performed Bayesian SED fitting for the lensed arc images. We used Prospector
(Conroy et al., 2009; Conroy & Gunn, 2010; Johnson et al., 2021), a stellar
population synthesis and parameter inference framework based on Markov Chain
Monte Carlo (MCMC). Note that the arc spectra only provide a spectroscopic
redshift constraint, and were not utilized in the SED fitting process; in this
low SNR regime, the spectra mostly sample saturated absorption lines from the
ISM, which are not constrained by the FSPS libraries (Conroy et al., 2009;
Conroy & Gunn, 2010). Other than the slope of the rest-UV continuum, our
spectral dataset does not provide additional information regarding the stellar
population, which is already more reliably and robustly sampled by the
photometry.

We chose to conduct SED fitting on the observed photometry rather than
magnification-corrected photometry. We assumed that the magnification-
dependent parameters in SED fitting (such as stellar mass and star formation
rate) are not correlated with those that are not magnification-dependent (such
as metallicity and dust extinction). This assumption has been tested and
validated in previous COOL-LAMPS publications (Khullar et al., 2021).

We assumed a non-parametric star formation history (with age bins of
$[0-50],[50-100],[100-500]$ and $[500-t_{\rm source}]$ Myr in look-back time).
It is represented by a set of SFRratio values, which give the ratio of the
star formation rates in adjacent bins, and $t_{\rm source}$ is the age of the
universe in Myr at the redshift of the source (see Leja et al. 2017 and
Khullar et al. 2021 for examples of implementation.) For example, for CJ0101,
we have $t_{\rm source}=1870$ Myr at $z=3.5$ assuming a Planck Collaboration
et al. (2020) cosmology. Throughout our entire analysis, we assume a Kroupa
initial mass function (IMF) (Kroupa, 2001). In the fiducial model, we fitted
for free parameters such as total stellar mass formed in the galaxy
$\log_{10}(M_{*}/M_{\odot})$, dust attenuation $\tau_{\lambda,2}$ (diffuse
dust optical depth), stellar metallicity $\log_{10}(Z/Z_{\odot})$, gas
ionization parameter and gas-phase metallicity.

For the five primary bright arcs photometered above, the $g$-band (or HSC
$g$-band) photometry samples the Ly$\alpha$ line and/or Lyman forest. Since
the emission and absorption of Ly$\alpha$ is complex and highly dependent on
the gas geometry in the galaxies, the correlation between such processes and
star-forming activity is not established in our sample galaxies (e.g.,
Calzetti et al., 1994; Giavalisco et al., 1996; Atek et al., 2008; Sobral &
Matthee, 2019). Hence, we chose to only use $rz$ photometry for the fiducial
SED models. The dust extinction and metallicities have flat and liberal
priors, roughly covering the range allowed by the spectral model libraries.
Gas ionization parameter and gas-phase metallicity have TopHat priors of
$[-4,-1.5]$ and $[-2,0.2]$. We also included nebular continuum and line
emission in our modeling process, while the line emission was turned off in
the model spectrum.

The best fit fiducial SED models of are shown in Figure 5. The residuals are
shown in the lower panels for each fit to visually demonstrate the quality of
the best-fit SEDs. In Figure 6, we present the posterior distributions of the
best-fit parameters, both as histograms as well as pair-wise comparison via a
corner plot. Refer to Table 5 for the best-fit values of the parameters.

Since we are only fitting for $rz$ band photometry in the fiducial model, our
model explicitly captures the rest-frame UV properties of the galaxy spectrum,
which is closely related to the emission from the hot and massive stellar
populations and the dust extinction in the galaxy (e.g., see Salim et al.
2007b; Shivaei et al. 2015, and references therein). Meanwhile, due to a lack
of information input regarding the cooler and less massive stellar
populations, properties such as stellar metallicity and gas-phase metallicity
are not constrained in the model. Hence, in Figure 6, we only show the free
parameters that are constrained by the rest-frame UV observations, which are
the total stellar mass $M_{*}$, instantaneous and dust-unobscured star
formation rate (defined as the average SFR in the age bin closest to the epoch
of observation, SFRlast-bin) and dust attenuation $\tau_{\lambda,2}$. Note
that the stellar mass and SFR shown in the figure are de-magnified and the
uncertainties from lens modeling have been included in the aforementioned
results.

In addition, we experimented with a variant model with all $grz$ photometry,
with an additional free parameter that tunes the intergalactic-medium (IGM)
absorption between the arc emission and the observer (based on the IGM
absorption prescription by Madau 1995); we chose a TopHat prior of $[0.5,2]$
for IGM absorption factor. All of our variant models favor a Landau
attenuation factor that is less than expected. As shown in a selected example
of CJ0145 in Figure 7, the IGM absorption factor distribution has a 50th
percentile value less than $1.0$. However, such a favoring of less attenuation
does not impact the derived total stellar mass $M_{*}$ significantly. If we
assume the posterior distributions of $M_{*}$ to be Gaussian, the derived
$M_{*}$ from the variant models always falls in the $1\sigma$ range of the
fiducial model values.

Figure 6: The corner plot with the posterior distributions and correlations for parameters from the Prospector SED fitting analysis for all the objects in this sample. The contours represent the 1$\sigma$(solid) and 2$\sigma$(dashed) confidence intervals. For visual clarity, one of the five systems is plotted with filled contours. Given the limitations of fitting models to a few data points, the measured stellar masses and star-formation rates span a significant range of values that significantly overlap. Refer to Section 4.2 for details on the fitted parameters. Refer to Table 5 for the numerical values on these best-fit parameters. Figure 7: The comparison between SED models of CJ0145 in which the IGM factor is fixed as $1$ (blue solid contour) or a free parameter during fitting (orange line contour). The IGM absorption factor distribution peaks around $0.5$. Such a favoring of less attenuation does not impact the derived total stellar mass $M_{*}$ significantly. Table 5: The best-fit values and corresponding 68% confidence intervals from SED fitting of demagnified stellar mass ($M_{\star}$), demagnified instantaneous star formation rate (SFR), and the diffuse optical depth ($\tau_{\lambda,2}$) for galaxies in this sample. | $\log_{10}(M_{\star}/M_{\odot})$ | $\log_{10}(\rm SFR/(M_{\odot}\cdot yr^{-1}))$ | $\tau_{\lambda,2}$
---|---|---|---
CJ0101 | $10.59^{+0.51}_{-0.55}$ | $1.35^{+0.56}_{-0.66}$ | $0.76^{+0.22}_{-0.33}$
CJ0104 | $10.86^{+0.44}_{-0.49}$ | $1.59^{+0.51}_{-0.61}$ | $0.61^{+0.21}_{-0.25}$
CJ0145 | $10.49^{+0.38}_{-0.44}$ | $1.22^{+0.43}_{-0.55}$ | $0.28^{+0.25}_{-0.19}$
CJ0516 | $9.69^{+0.35}_{-0.40}$ | $0.42^{+0.41}_{-0.49}$ | $0.22^{+0.15}_{-0.14}$
CJ1356 | $9.73^{+0.60}_{-0.60}$ | $0.44^{+0.64}_{-0.71}$ | $0.44^{+0.49}_{-0.30}$

###  4.3 Lens Modeling and Magnification

Figure 8: Lens models built with ground-based imaging. Different sets of
positional constraints are denoted with different initial indexing and colors.
The critical curve at the primary source redshift is in red (the additional
inner critical curve for CJ1356 is for the second image family at lower
redshift) and the caustic curve is in yellow. The sizes of cyan ellipses show
the relative scale of halo masses. The field of view is chosen to best present
each lensing system. Note that the second lensed image family in CJ1356 at
$z=2.17$ is marked in orange.

To model the cluster-scale mass distribution that is responsible for the
observed strong gravitational lensing, we used a parametric approach using
LENSTOOL. In LENSTOOL (Jullo et al. 2007), the mass distribution was modeled
by dual pseudo-isothermal ellipsoids (dPIEs, Elíasdóttir et al. 2007) whose
parameters and uncertainties were inferred through an MCMC approach. We
closely followed the strong lens modeling process and model optimization
strategy described in Sharon et al. (2020) and use the statistical
uncertainties of our model following a Chi statistique as in Mahler et al.
(2020). We selected the astrometric constraints for our modeling as multiple
images of lensed features, identified from the color images. The best-fit
model is defined as the one that minimizes the scatter between these observed
image locations and their model-predicted counterparts in the image plane. In
addition to the cluster-scale mass potentials, we fitted for galaxy scale
potentials whose normalization and radii parameters were determined by
observed luminosity scaling relations (Limousin et al., 2005); cluster
galaxies were selected from the DECaLS DR9 catalog, as objects with extended
morphologies whose colors are consistent with the expected red sequence at
each lens redshift. The morphological parameters used for setting up galaxy
scale potentials, namely position, ellipticity, and position angle were also
taken from the DECALS DR9 catalog.

In Figure 8, we over-plot the positional constraints, individual fitted halos
in the lens model, and critical curves on top of the imaging in the filter
band in which the arc is the brightest. Each constraint family is labeled with
the same color and the same first digit. The redshifts of the sources and lens
are provided in Table 2.

The lens models of CJ0101, CJ0145, and CJ0516 are the classic 3-image arc
configuration in which the image plane critical curve goes through the arc
twice and a counter-image is potentially present. Note that although we do not
have spectroscopic confirmations of the counter images in the above three
lensing systems, the best-fit model of CJ0516 requires the placement of a
fourth constraint on the counter image candidate. The lens models of CJ1356
and CJ0104 are more complicated and less obvious, and we provide a more
detailed description of these systems below.

####  4.3.1 COOL J1356+0339

In the lensing system of CJ1356, we find two lensed image families at two
spectroscopically confirmed redshifts. The two images located in the north-
east of the BCG (marked by constraints indices $1.1,1.2,3.1,3.2$) are
spectroscopically confirmed to have a redshift at $z=3.753$. In addition, two
objects in the north-west (constraint $1.3$) and one object in the south-west
(constraint $1.4$) are likely to be candidate lensed images from the same
source as the $z=3.753$ images, given their similar colors.

We spectroscopically confirmed a second lensed image family at $z=2.17$, which
is composed of the image in the north-east (constraint $2.3$) and the image in
the south-east (constraint $2.1$). There is a third image in the south-east
(constraint $2.2$) for which we do not yet have spectroscopic confirmation.
But given its color and geometrical location, we are confident that it belongs
to the image family at $z=2.17$.

In the modeling process, we experimented with several astrometric constraint
configurations with and without considering the candidate lensed images. The
optimized lens model of this system requires the astrometric constraints
placement shown in Figure 8. The model further suggests that only one of the
two candidate sources in the north-west is likely from the green source family
($z=3.753$).

####  4.3.2 COOL J0104-0757

In the lensing system of CJ0104, our best lens model suggests the images are
primarily lensed by a galaxy-scale or a low mass cluster-scale halo (shown as
the north ellipse “G1” in cyan in Figure 8) and a cluster-scale halo (shown as
the south ellipse “G2” in cyan in Figure 8). The galaxy hosted in the north
galaxy-scale halo is spectroscopically confirmed at $z=1.003$. The BCG of the
cluster-scale halo is spectroscopically confirmed at $z=0.858$.

The arc image (constraint $1.1,1.2,2.1,2.2,3.1,3.2$) in the north-east and the
image (constraint $1.3,2.3,3.3$) in the south are spectroscopically confirmed
as belonging to the same source at $z=3.480$. Given the similar color and
location of the image (constraint $1.4,2.4,3.4$) in the south-west, it is
likely to be part of the same image family. Our initial modeling suggests the
placement of constraints $1.5$ and $1.6$. In the case of constraint $1.5$, we
do find a candidate counter-image of similar color. In the case of constraint
$1.6$, we suspect a counter-image hidden behind the foreground red cluster
member, given the hint of green color in the RGB image. The model further
suggests a candidate image located between the galaxy at $z=1.003$ and the BCG
at $z=0.858$.

The precise modeling of the mass distribution in this system requires multi-
plane lens modeling, which is not yet implemented in LENSTOOL. Nevertheless,
the purpose of lensing analysis in this study is to provide an estimation of
the magnification factor for the lensed galaxy and to assess the implication
of projecting all the mass onto a single plane. We experimented with three
instances in which all the mass potentials were uniformly placed at the lower
redshift, the upper redshift, and the mean redshift ($z=0.930$). Apart from
the lens redshifts, we used the same initial inputs and priors for free
parameters in the three models. At $z=0.858$, the model returned a median
magnification of $40_{-16}^{+55}$ for the bright arc (constraint
$1.1,1.2,2.1,2.2,3.1,3.2$). At $z=0.930$, the model returned a median
magnification of $35_{-13}^{+54}$. At $z=1.003$, the model returned a median
magnification of $39_{-14}^{+47}$ (See Section 4.3.3 for the process of
magnification estimation). The ranges of magnification factors estimated from
the three model instances are statistically consistent with each other. We
conclude that in the model of CJ0104, the single lens plane approximation does
not affect the magnification estimate, within statistical uncertainties. We
used the magnification estimated from the model at $z=0.858$ for the analysis
in this study. We show the lens model at $z=0.858$ in Figure 8.

####  4.3.3 Magnification

To estimate the mean magnification, we calculated the ratio of area in the
image plane and the source plane. For each confirmed arc image in each lensing
system, we estimated the arc area in the image plane by taking the polygons
that best describe the non-PSF convolved GALFIT modeling image of the arc. We
ray-traced its source plane counterpart using the deflection maps generated
from the lens model, which describe the projection of image plane positions
onto the source plane. The source plane image size was estimated by taking the
convex hull of pixel points deflected from the image plane. We obtained a mean
total magnification for each of the lensed galaxies by summing the
magnification of each image in the system. To estimate the uncertainties, we
repeated the above procedures with 100 realizations of deflection maps that
sample the lens model posterior probability distribution. The resulting
magnifications and related uncertainties are shown in Table 2. We applied this
as a division factor to the results of arc imaging analysis in the image
plane, converting the derived physical parameter values from SED fitting to
those in the source plane.

The magnification depends on the choice of the image plane polygon. With the
limitations of imaging data from ground-based telescopes, we could not further
refine the constraints in the lens model or the choice of polygons for
magnification estimation. The reported magnifications and uncertainties are
the best results obtained with the limited data available. For all but CJ1356
which is better constrained, typical fractional uncertainties are $-50\%$ and
$+100\%$.

##  5 Discussion and Future Work

Figure 9:  (Center Panel): The total $r$ and $z$ band magnitudes for the
brightest published lensed galaxies in the redshift interval $3<z<4$, and one
further source (CJ2129) that will be presented in a future paper. These
objects, including newly discovered galaxies discussed in this paper (denoted
by bold font), represent the extreme tail in brightness for lensed sources in
this redshift interval. Note that the photometry for CJ1356 is in HSC $i$ and
$y$ bands, however, we make an approximation of its $r$ and $z$ band
magnitudes using the $i-y$ color for demonstration purposes here. (Right
Panel) The brightness distribution in the F606W band for $3<z<4$ sources from
the MUSE spectroscopic sample of lensed sources behind 12 massive lensing
clusters (Richard et al., 2021). (Bottom Panel) The redshift distribution of
lensed sources in the CASSOWARY (Stark et al., 2013), MEGASAURA (Rigby et al.,
2018) and Bayliss et al. 2011b samples.  Figure 10: (Top) The median estimates
(shaded regions and error bars denote 68% confidence intervals) for the star-
forming galaxy stellar mass function in redshift interval $3<z<4$ from Lee et
al. (2012), Muzzin et al. (2013), Marchesini et al. (2009), Caputi et al.
(2015), and Marsan et al. (2022). The vertical dashed lines show the estimated
de-magnified stellar masses of our lensed galaxies from the maximum likelihood
SED model. (Bottom) The stellar mass $M_{\star}$ \- star formation (SFR)
relation for star-forming galaxies in the redshift interval $3<z<4$. The
shaded regions show the 1$\sigma$ contours while the star symbols show the
median value.

Figure 9 shows the $r$-band and $z$-band magnitudes of the brightest published
lensed galaxies found in the redshift interval $3<z<4$. The galaxies in bold
denote the galaxies introduced in this paper. To emphasize the lack of known
bright-lensed galaxies in this redshift interval, we show the distribution of
redshifts and brightnesses of lensed galaxies in existing samples along the
horizontal and vertical axes respectively. The distribution along the
horizontal axes shows the redshift distribution of lensed galaxies from the
CASSOWARY (Stark et al., 2013), MEGASAURA (Rigby et al., 2018) and Bayliss et
al. (2011b) samples. Furthermore, the distribution along the vertical axes
shows the distribution of the F606W magnitudes for lensed galaxies in the
redshift range $3<z<4$ from the MUSE spectroscopic sample of lensed sources
(Richard et al., 2021). Note that the F606W band and $r/z$-band comparison is
not precise but sufficient for illustration.

Another interesting feature in this lensed galaxy sample is the presence of
Ly$\alpha$, or lack thereof as seen in the spectra (Figure 2 and Figure 3).
The Ly$\alpha$ emission is important to studying the physics of LyC escape in
these galaxies. As the scattering of these photons is influenced by the
neutral hydrogen reservoir in and around the galaxy, the shape of the
Ly$\alpha$ emission line is altered and in return contains information of the
neutral hydrogen. Hence, the Ly$\alpha$ emission line profile serves as an
indirect indicator of the $f^{\rm Lyc}_{\rm esc}$, that is, the fraction of
HI-ionizing LyC photons that escape from a galaxy (e.g., Izotov et al., 2021;
Gazagnes et al., 2020). At redshifts of $z\gtrsim 4$, due to a large number of
neutral hydrogen clumps, most LyC photons are absorbed (e.g., Haardt & Madau,
1996; Cowie & Hu, 1998; Vanzella et al., 2018). However, by studying galaxies
at $z\sim 3.0$, we are able to observe some of the ionizing photons. As a
result, we can use these relatively lower redshift galaxies as analogs of
their ‘Epoch of Reionization’ predecessors to study the escape of ionizing
photons (e.g., Rivera-Thorsen et al., 2017, 2019). Understanding the cause of
the variation in presence of Ly$\alpha$, or lack thereof, will require higher
spatial resolution data, which will be enabled by space-based telescopes like
HST and JWST.

The relation between the stellar mass $M_{\star}$ and its instantaneous star
formation rate (SFR) has been proposed to be a fundamental relation of
galaxies containing information on the evolutionary state of the galaxy and
variations in the star formation histories (e.g., Matthee & Schaye, 2019).
This relationship has been observed over high orders of magnitudes in stellar
mass (Santini et al., 2009) and from redshifts $z=0$ to $z=6$ (e.g., Tasca et
al., 2015).

The bottom panel in Figure 10 shows the stellar mass - star formation rate
relation for main-sequence galaxies in the redshift interval $3<z<4$. The
figure shows a compilation of results from numerous works and shows our lensed
galaxy sample for comparison. Galaxies in this sample appear to be fairly
typical, as they lie along the main-sequence star-forming relation.
Furthermore, to contextualize these galaxies within the stellar mass function
(SMF) of star-forming galaxies in this redshift interval, the top panel in
Figure 10 shows various estimates of the SMF and the stellar masses of our
galaxy sample as vertical dashed lines. With more detailed photometry in
future work, we will have better constraints on the star formation history of
these galaxies and thus reduce the uncertainty in our stellar mass and star
formation rate estimates.

These strongly-lensed galaxies are some of the brightest galaxies found in the
redshift interval $3<z<4$ (see Figure 9). Due to their high surface brightness
(and hence high signal-to-noise ratios), even with limited ground-based
photometry, we are able to estimate some basic properties of these galaxies
and their foreground lensing systems. Note that galaxy properties like star
formation rate, stellar mass, and physical sizes scale directly with the
lensing magnification. Thus, having a robust lens model is important. Space-
based telescopes HST and JWST have high spatial resolution and thus enable the
construction of robust lens models. Recent studies of strong-lensing systems
using HST and JWST (e.g., Rivera-Thorsen et al., 2019; Wu et al., 2022; Mahler
et al., 2022; Claeyssens et al., 2022) were able to discern morphology and
star formation at the smallest scales. Furthermore, in this redshift interval,
all the rest-frame optical nebular emission lines fall in the JWST NIRSpec
band-pass. Also, the UV emission and absorption lines, which probe the massive
stellar populations and outflowing winds, are in the band-pass of ground-based
optical IFUs like MUSE (Very Large Telescope) and KCWI (W. M. Keck
Observatory). Thus, the lensed galaxies in this sample are excellent future
targets for such follow-up observations.

##  6 Summary

We report the discoveries of five bright strongly-lensed galaxies at $3<z<4$:
COOL J0101+2055 ($z_{source}=3.459$; $z_{lens}=0.871$), COOL J0104-0757
($z_{source}=3.480$; $z_{lens1}=1.004$; $z_{lens2}=0.858$), COOL J0145+1018
($z_{source}=3.310$; $z_{lens}=0.463$), COOL J0516-2208 ($z_{source}=3.549$;
$z_{lens}=0.658$), and COOL J1356+0339 ($z_{source}=3.753$; $z_{lens}=0.262$).
These galaxies have r and z band AB magnitudes brighter than $21.81$, doubling
the number of bright lensed galaxies currently known in this redshift
interval. We characterized the lensed galaxies using ground-based grz/giy
imaging and optical spectroscopy. Using Prospector, we performed stellar-
population synthesis modeling with model-based photometry magnitudes and
derived stellar masses, dust content, and star-formation rates. Using
LENSTOOL, we performed dPIE-halo-based lens mass modeling with ground-based
imaging, which implies median source magnifications in our sample $\sim
29-180$. Combining these analyses, we derived the sample galaxies’ stellar
masses ranging from $\rm log_{10}(M_{*}/M_{\odot})\sim 9.7-11.0$ and star
formation rates in the youngest age bin ranging from $\rm
log_{10}(SFR/(M_{\odot}\cdot yr^{-1}))\sim 0.4-1.6$, placing the sample
galaxies on the massive end of the star-forming main sequence. In summary, the
five galaxies presented in this sample are rare in terms of their brightness
while they are common in terms of their stellar mass, in the redshift interval
$3<z<4$. In addition, three galaxies in our sample have strong Ly$\alpha$
emission, enabling unique opportunities to study Ly$\alpha$ emitters at high
redshifts.

## Acknowledgements

We thank Jason J. Lin, Elisabeth Medina and Daniel J. Kavin Stein for their
contribution to the initial lens searches.

This research has made use of the NASA/IPAC Extragalactic Database, which is
funded by the National Aeronautics and Space Administration and operated by
the California Institute of Technology.

This paper is based on data gathered with the 6.5 m Magellan Telescopes
located at Las Campanas Observatory, Chile. Magellan observing time for this
program was granted by the time allocation committees of the University of
Chicago and the University of Michigan.

Based on observations made with the Nordic Optical Telescope, owned in
collaboration by the University of Turku and Aarhus University, and operated
jointly by Aarhus University, the University of Turku and the University of
Oslo, representing Denmark, Finland and Norway, the University of Iceland and
Stockholm University at the Observatorio del Roque de los Muchachos, La Palma,
Spain, of the Instituto de Astrofisica de Canarias. The data presented here
were obtained in part with ALFOSC, which is provided by the Instituto de
Astrofisica de Andalucia (IAA) under a joint agreement with the University of
Copenhagen and NOT.

This work is supported by The College Undergraduate program at the University
of Chicago, and the Department of Astronomy and Astrophysics at the University
of Chicago.

The Legacy Surveys consist of three individual and complementary projects: the
Dark Energy Camera Legacy Survey (DECaLS; Proposal ID 2014B-0404; PIs: David
Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID
2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey
(MzLS; Prop. ID 2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS together
include data obtained, respectively, at the Blanco telescope, Cerro Tololo
Inter-American Observatory, NSF’s NOIRLab; the Bok telescope, Steward
Observatory, University of Arizona; and the Mayall telescope, Kitt Peak
National Observatory, NOIRLab. The Legacy Surveys project is honored to be
permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a
mountain with particular significance to the Tohono O’odham Nation.

NOIRLab is operated by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation.

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the U.S. Department of Energy, the U.S. National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho
de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de
Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia,
Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energeticas, Medioambientales y Tecnologicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de
Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig
Maximilians Universitat Munchen and the associated Excellence Cluster
Universe, the University of Michigan, NSF’s NOIRLab, the University of
Nottingham, the Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, and Texas A&M University.

The Legacy Surveys imaging of the DESI footprint is supported by the Director,
Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under Contract No. DE-AC02- 05CH1123, by the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility under the
same contract; and by the U.S. National Science Foundation, Division of
Astronomical Sciences under Contract No. AST-0950945 to NOIRLab.

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# 8Dim calculations of the third barrier in 232Th and a conflict between
theory and experiment on uranium nuclei.

P. Jachimowicz  Institute of Physics, University of Zielona Góra, Szafrana 4a,
65516 Zielona Góra, Poland  M. Kowal<EMAIL_ADDRESS>National Centre for
Nuclear Research, Hoża 69, PL-00-681 Warsaw, Poland  J. Skalski  National
Centre for Nuclear Research, Hoża 69, PL-00-681 Warsaw, Poland

###### Abstract

We find the height of the third fission barrier $B_{III}$ and energy of the
third minimum $E_{III}$ in 232Th using the macroscopic - microscopic model,
very well tested in this region of nuclei. For the first time it is done on an
8-dimensional deformation hypercube. The dipole distortion is included among
the shape variables to assure that no important shapes are missed. The saddle
point is found on a lattice containing more than 50 million points by the
immersion water flow (IWF) method. The shallow third minimum,
$B_{III}-E_{III}\approx 0.36$ MeV, agrees with experimetal data of Blons et
al. This is in a sharp contrast with the status of the IIIrd minima in
232-236U: their experimental depth of $\geq 3$ MeV contradicts all realistic
theoretical predictions. We emphasize the importance of repeating the
experiment on 232Th, by a technique similar to that used in the uranium
nuclei, for settling the puzzle of the third minima in actinides.

###### pacs:

25.85.Ca,21.10.Gv,21.60.-n,27.90.+b

Search for a hyperdeformation (HD), i.e., extremely elongated shapes of atomic
nuclei (spheroid with a major to minor axis ratio of 3:1), is one of the most
difficult challenges of modern nuclear structure studies. There is still no
convincing evidence for discrete gamma transitions of HD rotational bands. The
only experimental evidence for hyperdeformation, except in light nuclei, has
been reported in the region of light actinides, in particular in uranium
isotopes 232U, 234 U and 236U, see Kra  and references therein. In these
experiments, the fission probability was studied as a function of the
excitation energy via different reactions. Strong resonances were observed and
a fine structure of some interpreted as a signature of multiple hyperdeformed
bands. Measurements of the angular distribution of the fragments arising from
the induced fission support this interpretation. Moreover, by measuring the
angular distribution of the fission fragments one can obtain some information
about the spin and $K$ \- quantum number. Due to the predicted static mass-
asymmetry of the third minima, their intrinsic parity would be broken, so the
low-energy excitations should show a pattern of alternating parity (even
nuclei) or parity doublet (odd nuclei) bands, different from that in the
second minima. Since the third minima are predicted axially symmetric, the
excitations would have an approximate (up to a Coriolis mixing, much weaker
than in the first or second minimum) $K$ quantum number.

The current experimental status of the third barrier in 232Th is different
from that in uranium isotopes. No recent data exist, the only available are
those from the prior 30 years Blons1 ; Blons2 ; Blons3 ; Blons4 . The
experiment conducted by Blons et al pointed to a shallow third minimum (no
deeper than 0.5 MeV). Only a little more than 1 MeV deep third minimum in a
neighboring nucleus 232Pa was recently reported by the Munich-Debrecen group
Csige2012 . The technique used to resolve observed resonances was the same as
in the study of uranium nuclei by the same group. Deep third minima in uranium
( $3\div 4$ MeV) disagree with all calculations Mcdonell1 ; Mcdonell2 ; Bender
; Gogny ; HFBCS ; Delar ; MollerPRC2012  except those of S. Ćwiok et al,
within the Woods-Saxon model Stefan ; Stefan1 ; Stefan2 . Self-consitent
calculations based on the Skyrme SkM* interaction do not give a hyperdeformed
minimum in 232Th Mcdonell1  at all. With the help of a temperature effect, a
slight minimum appears, but does not exceed 500 keV. Moreover, with an
increasing excitation, the depth of this minimum decreases again. Situation is
very similar in 232U and 234U nuclei Mcdonell2 . Selfconsitent calculations
with the Gogny D1S interaction do not give third minima in those nuclei Gogny
; HFBCS ; Delar  as well. Moreover, for 236U, a recent result of Möller et al
MollerPRC2012  within the macro-micro approach also does not support a deep
minimum reported in HyperU : one can read from the map (Fig. 8. in ref.
MollerPRC2012 ) that the minimum is not deeper than 300 keV, ten times smaller
than the experimental value.

In fact, hyperdeformed minima predicted in Stefan ; Stefan1 ; Stefan2  were
quite abundant: they not only appeared in nuclei where nobody saw them in
experiment, but were double, with different octupole deformations
$\beta_{3}\approx 0.3$ and 0.6. Using the same model, we have just showed in
kowskalIIImin  that both types of the IIIrd minima in uranium nuclei disappear
completely after proper nuclear shapes are considered. The more mass-
asymmetric minima were unphysical from the beginning. Their quadrupole moment
is large, $Q_{20}\simeq 170$ b, which situates them behind the barrier, closer
to the scission point. The existence of minima with a smaller mass asymmetry
(and quadrupole moments) requires a more detailed study. Finding the barrier
in many-dimensional space requires hypercube calculations. This is the case of
232Th, where the less mass-asymmetric minimum is the deeper one.

Previously, we found in kowskalIIImin  a tentative upper limit of 330 keV for
the IIIrd barrier in 232Th and 0 in even uranium isotopes 232-236U by 6Dim
hypercube calculations and by probing various trial 8Dim fission paths. The
main aim of the present study is to ascertain the prediction for 232Th,
showing that a third minimum is there. To accomplish this, an 8-dimensional
grid calculation has been done. All deformations were treated on the same
footing, without introducing any subdivision into relevant and irrelevant
subspaces. Let us emphasize that till now only 5-dimensional macroscopic-
microscopic calculations of fission saddles were available in the literature
MollerNATURE2001 ; MollerPRL2004 ; MollerPRC2012 .

There is one more important motivation for such studies. To distinguish
between resonances in the second and third minimum, the excitation energy
should be higher than the first barrier but simultaneously lower than the
second one. We show that our multidimensional calculations predict such a
possibility in 232Th, which is a natural candidate for the future experimental
study.

The energy is calculated within the microscopic-macroscopic method with a
deformed Woods-Saxon potential. Nuclear shapes are defined in terms of the
nuclear surface WS . Since we found that nonaxiality in the region of the
third barrier is less important than a proper treatment of the axially
symmetric shapes, we parameterize the surface as

$R(\theta)=c(\\{\beta\\})R_{0}(1+\sum_{\lambda=1}\beta_{\lambda 0}Y_{\lambda
0}(\theta)),$ (1)

where $c(\\{\beta\\})$ is the volume fixing factor. The macroscopic energy is
calculated using the Yukawa plus exponential model KN  with parameters
specified in WSpar . All other parameters of the model are the same as in a
number of our previous studies, e.g. in KOW10 ; kowskal ; IIbarriers . A
hyperdeformed minimum was searched by exploring energy surfaces in a region of
deformations beyond the second minimum. We generate the grid in axially
symmetric deformations:

$\displaystyle\beta_{10}$ $\displaystyle=$ $\displaystyle\ \ -0.35\ (0.05)\
0.00;\,\,\,\beta_{20}=0.55\ (0.05)\ 1.50,$ $\displaystyle\beta_{30}$
$\displaystyle=$ $\displaystyle\ \ 0.00\ (0.05)\ 0.35;\,\,\,\beta_{40}=-0.10\
(0.05)\ 0.35,$ $\displaystyle\beta_{50}$ $\displaystyle=$ $\displaystyle\ \
-0.20\ (0.05)\ 0.20;\,\,\,\beta_{60}=-0.15\ (0.05)\ 0.15,$
$\displaystyle\beta_{70}$ $\displaystyle=$ $\displaystyle\ \ -0.20\ (0.05)\
0.20;\,\,\,\beta_{80}=-0.15\ (0.05)\ 0.15.$

Numbers in the parentheses specify the step with which the calculation is done
for a given variable. Thus, we finally have the values of energy at a total of
$50803200$ grid points. On a such giant 8-dimensional grid the IWF saddle
point searching method is used.

Let us emphasize that our model, without the $\lambda=1$ term in (1), very
well reproduces first KOW10  and second barriers IIbarriers  and second minima
kowskal  in actinide nuclei. One could hope that an extrapolation to more
elongated shapes would be still valid within such a parametrization. What then
causes troubles beyond the second peak? They result from the truncation of the
expansion (1) that restricts possible shapes for large deformations. Two
shapes close to the IIIrd saddle are shown in Fig. 1: one from the 8D
calculation (nonzero $\beta_{10}$-$\beta_{80}$, in red) and the other, from
the 7D calculation $\beta_{20}-\beta_{80}$ (in blue). Even if they seem not
much different, the difference in barrier is 4 MeV. It appears that the dipole
deformation $\beta_{10}$, usually associated only with a shift of the center
of mass, effectively makes up for truncated multipoles in very deformed, mass-
asymmetric configurations. We emphasize that our program keeps automatically
the center of mass at the coordinate origin, thus $\beta_{10}$ serves only
probing somewhat different shapes.

Figure 1: Shapes $R(\theta)/R_{0})$ near to the IIIrd saddle: red line - 8D
calculation ($\beta_{10}-\beta_{80}$), blue line \- 7D calculation (without
$\beta_{10}$).  Figure 2: Potential energy surfaces
$E({\beta_{20},\beta_{40}})$ for 232Th from the 8D $\beta_{10}-\beta_{80}$
calculation, minimized over remaining degrees of freedom; upper panel -
$\beta_{10}=0$, bottom panel - $\beta_{1}=min$.

A modification of the energy landscape by including $\beta_{10}$ is shown in
Fig. 2. Since dipole is the first spherical harmonic (Eq.1), one can suspect
its pronounced effect. Indeed, that effect is significant. Not only the height
of the third saddle is clearly reduced, but the whole landscape changed. The
result indicates a large effect of new shapes, unattainable previously.
Figures show also examples of two fission paths. In the version of calculation
without the dipole (blue trajectory), after passing through the second saddle,
the nucleus falls into a deep third minimum. To be split, it needs to tunnel
through a more than 4 MeV barrier. In the case with included $\beta_{10}$ (red
trajectory), after passing the second barrier nucleus can easily split. One
can notice that $\beta_{10}$ deformation has almost no effect on the position
and height of the second saddle. Precise lowering of the third barrier by
$\beta_{10}$ may be read from Fig. 3 as 4.1 MeV. Beside the significant
reduction, one can also see a change in the barrier shape. This rather
dramatic change of the fission path in the multidimensional deformation space
has a direct effect on fission half-life.

Table 1:  Calculated and experimental features of the fission barrier in 232Th and 232U. $M$ \- ground state mass excess, $B_{I}$ \- first barrier, $E_{II}$\- second minimum $B_{II}$ \- second barrier $E_{III}$\- third minimum $B_{III}$ \- third barrier. All quantities (in MeV) are shown relative to the ground state. Theory: | $M^{th}$ | $B^{th}_{I}$ | $E^{th}_{II}$ | $B^{th}_{II}$ | $E^{th}_{III}$ | $B^{th}_{III}$
---|---|---|---|---|---|---
Experiment: | $M^{exp}$ | $B^{exp}_{I}$ | $E^{exp}_{II}$ | $B^{exp}_{II}$ | $E^{exp}_{III}$ | $B^{exp}_{III}$
232Th | 35.33 | 4.4 | 2.2 | 6.1 | 4.1 | 4.4
RIPL3  | 35.45 | 5.8 | - | 6.7 (6.2) | - | -
Blons2  | - | 4.6 | - | 5.7 | 4.0 | 4.4
232U | 34.34 | 4.5 | 3.2 | 5.7 | 0.0 | 0.0
RIPL3  | 34.61 | 5.4 | - | 5.4 (5.3) | - | -
Hyper232U  | - | 4.0 | 3.1 | 4.9 | 3.2 | 6.0

A detailed information about the structure of the fission barrier in 232Th is
collected in Table 1. The recent theoretical results and experimental data for
232U Hyper232U  are also given. It is seen that the model calculation well
reproduces essential features of the fission barrier up to superdeformed
shapes in both nuclei. The inner fission barrier $B^{th}_{I}$ has been found
according to the prescription presented in KOW10 .

Here, a comment is needed. Usually the calculated first saddles are
substantially higher than experimental values but the opposite situation takes
place in the light Thorium isotopes what is called in the literature the
”Thorium anomaly”. In our case the nonaxiality reduces the first saddle for
light Thorium only non-significantly, to a nearly proper height (see column 3
of Table 1 column 3). Thus, the mentioned anomaly does not occur in our case
(we use the modern data of Hyper232U ).

In order to determine the second peak $B^{th}_{II}$, a mass asymmetry is
included, details can be found in IIbarriers . One can see that our calculated
second minimum and second barrier reasonably agree with experiment. As already
mentioned, the experimental situation (when it comes to the depth of
hyperdeformed minima) in two isobars: 232Th and 232U is completely different.
While in 232U, the IIIrd minimum is quite deep, in 232Th it is very shallow.
Our results support the existence of a shallow third minimum in 232Th as in
old experiments of Blons and thus agree with the majority of modern
theoretical models.

A designed experiment for 232Th, by a technique described in ref Kra ; Thirolf
, would be crucial for solving the mystery of the third minima in actinides.
Particularly promising are experiments employing highly monochromatic
$\gamma$-ray beams for photofission studies. With a high quality of photon and
spectral intensity, exceeding the performance of existing facilities by
several orders of magnitude seems to be possible in the near future Thirolfexp
. If the result of Blons et al is confirmed, we will have to understand _why
between 232Th and 232U, two beta decays away, the energy landscape changes so
dramatically_. On the other hand, if in the future experiment, a depth of the
IIIrd minimum is obtained similar as that in 232U, we will have a total
contradiction between theory and experiment. _Assuming that the existing
resonances cannot be interpreted otherwise, all current meaningful theoretical
models would have to be reconstructed anew to give deep hyperdeformed minima_.

One can expect that in hyperdeformed nuclei, some high particle states at
normal deformation become occupied with increasing deformation. Since these
are the orbits that are occupied in superheavy nuclei at moderate
deformations, the whole question may have some impact on the understanding of
superheavy nuclei.

Figure 3: Energy along a sequence of smoothly elongating shapes, beginning at
the superdeformed minimum. Appropriate shapes along trajectories are shown.

In summary:

(i) We have presented for the first time an 8D hypercube calculation for
232Th. To find the third barrier on a giant grid, the IWF method has been
applied. After a proper inclusion of the dipole deformation we found the depth
of the third minimum of about 0.36 MeV. This would rather exclude
spectroscopic studies in the IIIrd well.

(ii) Including a dipole distortion lowers the third saddle by more than 4 MeV.
It seems likely that, with the shape parametrization (1), the dipole
deformation is important everywhere, where large elongation and necking is
combined with a sizable mass asymmetry. For example, it may be the case of the
Jacobi shape transition at high spins in medium-heavy nuclei.

(iii) New experimental study dedicated to hyperdeformation in 232Th seems
essential for the understanding of the third minima in actinide nuclei.

Acknowledgments This work was partially supported by Narodowe Centrum Nauki
Grant No. 2011/01/B/ST2/05131. The support of the LEA COPIGAL fund is
gratefully acknowledged.

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