# DESI-253.2534+26.8843: A New Einstein Cross Spectroscopically Confirmed with
VLT/MUSE and Modeled with GIGA-Lens

Aleksandar Cikota Gemini Observatory / NSF’s NOIRLab, Casilla 603, La Serena,
Chile  Ivonne Toro Bertolla Gemini Observatory / NSF’s NOIRLab, Casilla 603,
La Serena, Chile  Xiaosheng Huang Department of Physics & Astronomy,
University of San Francisco, 2130 Fulton Street, San Francisco, CA 94117-1080,
USA  Physics Division, Lawrence Berkeley National Laboratory, 1 Cyclotron
Road, Berkeley, CA 94720, USA  Saul Baltasar Physics Division, Lawrence
Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
Nicolas Ratier-Werbin Physics Division, Lawrence Berkeley National Laboratory,
1 Cyclotron Road, Berkeley, CA 94720, USA  William Sheu Department of Physics
& Astronomy, University of California, Los Angeles, 430 Portola Plaza, Los
Angeles, CA 90095, USA  Christopher Storfer Institute for Astronomy,
University of Hawaii, Honolulu, HI 96822-1897  Nao Suzuki Physics Division,
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA  Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa 277-8583, Japan  David J. Schlegel Physics
Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley,
CA 94720  Regis Cartier Gemini Observatory / NSF’s NOIRLab, Casilla 603, La
Serena, Chile  Simon Torres SOAR Telescope / NSF’s NOIRLab, Casilla 603, La
Serena, Chile  Stefan Cikota Centro Astronónomico Hispano en Andalucía,
Observatorio de Calar Alto, Sierra de los Filabres, 04550 Gérgal, Spain  Eric
Jullo Aix-Marseille Univ., CNRS, CNES, LAM, Marseille, France Aleksandar
Cikota, Xiaosheng Huang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Gravitational lensing provides unique insights into astrophysics and
cosmology, including the determination of galaxy mass profiles and
constraining cosmological parameters. We present spectroscopic confirmation
and lens modeling of the strong lensing system DESI-253.2534+26.8843,
discovered in the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging
Surveys data. This system consists of a massive elliptical galaxy surrounded
by four blue images forming an Einstein Cross pattern. We obtained
spectroscopic observations of this system using the Multi Unit Spectroscopic
Explorer (MUSE) on ESO’s Very Large Telescope (VLT) and confirmed its lensing
nature. The main lens, which is the elliptical galaxy, has a redshift of
$z_{L1}=0.636\pm 0.001$, while the spectra of the background source images are
typical of a starburst galaxy and have a redshift of $z_{s}=2.597\pm 0.001$.
Additionally, we identified a faint galaxy foreground of one of the lensed
images, with a redshift of $z_{L2}=0.386$. We employed the GIGA-Lens modeling
code to characterize this system and determined the Einstein radius of the
main lens to be $\theta_{E}=2.520{{}^{\prime\prime}}_{-0.031}^{+0.032}$, which
corresponds to a velocity dispersion of $\sigma$ = 379 $\pm$ 2 km s-1. Our
study contributes to a growing catalog of this rare kind of strong lensing
systems and demonstrates the effectiveness of spectroscopic integral field
unit observations and advanced modeling techniques in understanding the
properties of these systems.

Strong gravitational lensing

##  1 Introduction

Strong gravitational lensing occurs when a massive object warps space-time and
causes the path of light from a well-aligned distant source to bend, typically
resulting in multiple images. These systems are a powerful tool for
astrophysics and cosmology. They have been used to study how dark matter is
distributed in galaxies and clusters (e.g. Kochanek, 1991; Bolton et al.,
2006a; Huang et al., 2009; Jullo et al., 2010; Meneghetti et al., 2020) and
are uniquely suited to probe the low end of the dark matter mass function and
test the predictions of dark matter models beyond the local universe (e.g.,
Vegetti et al., 2010b; Sengül et al., 2022). In addition, multiply lensed
quasars and supernovae (SNe) can be used to measure time delays and the Hubble
constant, $H_{0}$ (e.g., Refsdal, 1964; Treu, 2010; Suyu et al., 2020).

When the alignment is nearly perfect and the lens mass has an elliptical
distribution, the background source would appear as quadruply lensed. These
systems are prized as they provide the strongest constraint on the lens mass
distribution. One specific example of quadruply lensed system is the Einstein
Cross, where four distinct images of the background source form a cross-like
pattern with a high degree of symmetry.

The very first Einstein Cross discovered was a lensed quasar system, QSO
2237+0305, also known as Huchra’s Lens (Huchra et al., 1985). This was
followed by the discoveries of quite a few other similar systems, some of
which were used to measure the time-delay $H_{0}$(e.g., Wong et al., 2020).
Recently, Stern et al. (2021) reported 12 quadruply imaged quasars identified
in the Gaia DR2 dataset, several of them being Einstein Crosses.

In the category of galaxy-galaxy strong lensing, there have been only a
handful confirmed cases of Einstein Crosses (e.g., Ratnatunga et al., 1995;
Bolton et al., 2006b; Bettoni et al., 2019; Napolitano et al., 2020). In
addition to constraining the mass distribution of the lensing galaxies,
careful modeling of an increasing number of such systems, covering wider
ranges of redshift and mass, also provide valuable mass profile prior for
modeling lensed quasars to obtain more accurate $H_{0}$ measurements (e.g.,
Birrer et al., 2020).

We report the confirmation of a new galaxy-galaxy lensing Einstein Cross:
DESI-253.2534+26.8843 ($\alpha$ = 16:53:00.82, $\delta$ = +26:53:03.48),
discovered by Huang et al. (2021) in the Dark Energy Spectroscopic Instrument
(DESI) Legacy Imaging Surveys Data Release 8 using deep residual neural
networks. Out of the 1312 candidates reported in Huang et al. (2021), this
system is one of the 216 most promising (Grade A) candidates. The brightness
of the lens is 23.66 g mag, 21.78 r mag, and 20.35 z mag, and the photometric
redshift reported by Zhou et al. (2021) is $z_{phot}=0.649\pm 0.035$. Compared
with other known galaxy-galaxy lensing Einstein Crosses, this system has the
highest redshift (but comparable to the lensed quasar systems used to measure
$H_{0}$) and has the largest Einstein radius.

In Section 2 we present the MUSE observations and the spectroscopic
confirmation, in Section 3 we model this system with GIGA-Lens (Gu et al.,
2022), and in Section 4 we summarize and conclude.

##  2 MUSE observations and spectroscopic confirmation

DESI-253.2534+26.8843 was observed on 2023-05-22 at 04:00h UT as part of a ESO
filler program for characterizing of galaxy-galaxy gravitational lensing
system candidates (Prog. ID 0111.A-0407, PI Cikota) with the Multi Unit
Spectroscopic Explorer (MUSE, Bacon et al. 2010), mounted at UT4 of ESO’s Very
Large Telescope (VLT) on Cerro Paranal in Chile. MUSE is an integral field
unit spectrograph with a field of view of 60 arcsec $\times$ 60 arcsec (in the
Wide Field Mode) and a spatial resolution of 0.2 arcsec. The spectral range is
from 4750 to 9350 Å with the spectral resolution ranging from R = 2000 to 4000
across the wavelength domain.

The observations were taken with 4 $\times$ 700 seconds exposures during good
seeing of $\sim 0.6^{\prime\prime}$ and thick transparency, and reduced
following standard procedures with the MUSE pipeline package version 2.2
(Weilbacher et al., 2020) that is a part of the ESO Recipe Execution Tool
(ESOREX). We also removed sky lines using the Zurich Atmosphere Purge (ZAP)
sky subtraction tool (Soto et al., 2016).

We generated images of the DESI-253.2534+26.8843 in the SDSS $g$, $r$ and $i$
bands by convolving the MUSE data cube with the SDSS $g$, $r$ and $i$
passbands. Figure 1 shows a color image of the the MUSE field and the
gravitational lens, with the source images marked as A, B, C and D.

We measured the brightness and positions of the lens and source images in the
$g$, $r$ and $i$ images with SExtractor (Bertin & Arnouts, 1996). The zero
points for the different bands have been determined based on the brightness of
SDSS catalog stars in the field. The positions and magnitudes are listed in
Table 1.

Figure 1: Left panel: Color image of DESI-253.2534+26.8843 observed with MUSE
on May 22nd, 2023. The image is a combination of the SDSS $g$ (blue), SDSS $r$
(green), and SDSS $i$ (red) images generated from the MUSE data cube. Other
galaxies with similar redshifts to the lens galaxy are indicated with G1 – G7.
The white square indicates the cutout shown in the right panel. Right panel:
Cutout with the strong gravitational lens. The images of the source forming
the Einstein Cross are indicated with A, B, C and D. Table 1:  Positions of
the gravitational lens and images

Object | $\Delta\alpha$ | $\Delta\delta$ |  SDSS $g$ |  SDSS $r$ |  SDSS $i$
---|---|---|---|---|---
|  arcseca |  arcseca | mag | mag | mag
Lens | 0 | 0 |  23.97 $\pm$ 0.09 |  22.43 $\pm$ 0.05 |  21.12 $\pm$ 0.02
A | 2.21 | 1.22 |  22.91 $\pm$ 0.08 |  22.22 $\pm$ 0.05 |  21.97 $\pm$ 0.02
B | -2.49 | -1.02 |  22.65 $\pm$ 0.08 |  21.92 $\pm$ 0.05 |  21.71 $\pm$ 0.02
C | 0.94 | -1.77 |  22.37 $\pm$ 0.08 |  21.60 $\pm$ 0.05 |  21.41 $\pm$ 0.02
D | -1.12 | 1.46 |  23.71 $\pm$ 0.08 |  23.13 $\pm$ 0.05 |  23.02 $\pm$ 0.03

Notes — a Positions are relative to the gravitational lens ($\alpha$ =
16:53:00.82, $\delta$ = +26:53:03.48).

We extracted the spectra of the lens and the four images of the source from
the MUSE data cube and determined the redshifts by manually matching prominent
emission and absorption
lines111https://classic.sdss.org/dr6/algorithms/linestable.html to the
spectra. To extract the spectra, we used a circular aperture with a 1”
diameter, which encompassed 13 spaxels, and then calculated average spectra.
The spectra are shown in Figure 2. The gravitational lens displays the
prominent Fraunhofer absorption lines: Ca II H and K at 3968 Å and 3934 Å
respectively, and the G band at 4308 Å, which undoubtedly places the lens at
the redshift of z = 0.636 $\pm$ 0.001. Furthermore, the spectral energy
distribution also shows the 4000 Å break.

All spectra of the four images of the lensed source display consistent
spectral features, which confirms that these are indeed lensed images of the
same source (Figure 2). We identified the $\lambda$ 1336Å C II, $\lambda$
1394Å Si IV, $\lambda$ 1403Å Si IV, $\lambda$1528Å Si II, $\lambda$1548Å C IV,
$\lambda$1609Å Fe II, $\lambda$1670Å Al II absorption lines, and the
$\lambda$1909Å C III emission line in the spectra. The spectral
characteristics are typical for starbursting galaxies (see e.g. Fig. 4 in
Lowenthal et al., 1997). Based on these spectral features, we are confident
that the redshift of the source is at z = 2.597 $\pm$ 0.001. We note that the
image D is fainter compared to the other images, and because of the lower
signal to noise ratio (SNR), the spectral features are not as prominent as in
the other images.

Based on the color information of the DESI Legacy Surveys data, this system
appears to be embedded in a galaxy group. We inspected other objects in the
MUSE field and found that this is indeed the case. There are 7 additional
galaxies with similar redshifts to the lens galaxy (see left panel in Figure
1): G1 at z = 0.642, G2 at z = 0.641, G3 and G6 at z = 0.636, G4 at z = 0.633,
G5 and G7 at z = 0.637. The center of the galaxy group is likely close to G1,
G2, G5 and the strong lensing galaxy. These galaxies are all passive and
display prominent Ca II H and K lines and the 4000 Å break, similar to the
spectrum of the main lens L1, except the brightest galaxy G1, which in
addition to the Ca II H and K lines and the 4000 Å break also shows a
prominent [O II] line and weak H$\beta$ and [O III] lines. A detailed analysis
of the galaxy group is out of the scope of this paper. For a discussion on
star formation quenching and differences between central and satellite
galaxies please refer to e.g. Knobel et al. (2015); Wang et al. (2018); Davies
et al. (2019) and the references therein.

Furthermore, we found that in front of the image C, there is a faint
foreground galaxy, L2 (see Figure 1). Figure 3 shows the spectrum of the
galaxy. Although the SNR is lower, there are two prominent emission lines
clearly visible, which are consistent with H$\alpha$ and the $\lambda$
$\lambda$3726,3729 Å [O II] doublet at an redshift of $z=0.386$. While the
resolution and signal to noise is not sufficient for resolving the [O II]
doublet, other identifiable features in the spectrum correspond to the
wavelengths of the H$\beta$ line, [O III] lines, and the Ca II H and K lines.

Figure 2: Top panel: MUSE spectra of the gravitational lens
DESI-253.2534+26.8843 in rest frame wavelength at z = 0.636. Middle panel:
Spectra of the four images of the lensed source in rest-frame wavelength at z
= 2.597. Bottom panels: Cutouts of the shaded wavelength ranges in the middle
panel. We applied the Savitzky-Golay smoothing filter to all spectra using a
window size of 9 resolution elements and a first order polynomial. Figure 3:
Spectrum of the Foreground galaxy L2 located in front of the lensed image C at
$z=0.386$. The left panel shows the smooth average spectrum of 15 hand
selected spaxels, and the middle and right panels display cutouts of the
shaded wavelength ranges around [O II] and H$\alpha$ lines.

##  3 Analysis

###  3.1 Lens modelling with GIGA-Lens

We use GIGA-Lens (Gu et al., 2022) to model this system. GIGA-Lens is a
Bayesian lens modeling pipeline, consisting of three steps: finding the
maximum a posteriori (MAP) for the lensing parameters via gradient descent,
determining a surrogate multidimensional Gaussian covariance matrix for these
parameters using variational inference (VI), and finally sampling with
Hamiltonian Monte Carlo. All three steps use gradient descent with automatic
differentiation and take advantage of GPU acceleration. It is robust and very
fast, typically on the order of minutes to model a system.

Our model comprises a main lens and a second lens (the faint reddish object
next to image C, L2). The masses of both lenses are modeled as singular
isothermal ellipsoid (SIE). We model the light for both the main lens and the
lensed source using the elliptical Sérsic profile. For the secondary lens
light a spherical Sérsic profile is used. This model is composed of a total of
31 parameters. All of them are defined in Table 2. We use the $g$-band image
for modeling and a 15$\times$15 pixel Gaussian with FWHM of
$0.6^{\prime\prime}$ as our PSF.

Table 2: Prior distribution used for lens modeling

Lens Mass:
$\displaystyle\begin{cases}\hfill\theta_{E}&\sim\exp\left(\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ln
2,0.25}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ln
0.3,0.25})\right)\\\\[3.0pt]
\hfill\epsilon_{1},\epsilon_{2}&\sim\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0,0.1}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0,0.1})\\\\[3.0pt]
\hfill
x&\sim\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0,0.05}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2,0.1})\\\\[3.0pt]
\hfill
y&\sim\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0,0.05}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-2.4,0.1})\\\\[3.0pt]
\gamma_{ext,1},\gamma_{ext,2}&\sim\mathcal{N}(0,0.05)\end{cases}$ Lens Light:
$\displaystyle\begin{cases}\hfill
R_{l}&\sim\exp\left(\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ln
1,0.15}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ln
0.5,0.15})\right)\\\\[3.0pt] \hfill
n_{l}&\sim\mathcal{U}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1,5}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1,5})\\\\[3.0pt]
\hfill\epsilon_{l,1},\epsilon_{l,2}&\sim\mathcal{TN}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0,0.1,-0.3,0.3}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-})\\\\[3.0pt]
\hfill
x_{l}&\sim\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0,0.05}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2.0,0.1})\\\\[3.0pt]
\hfill
y_{l}&\sim\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0,0.05}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-2.4,0.1})\\\\[3.0pt]
\hfill
I_{l}&\sim\exp\left(\mathcal{N}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\ln
25.0,0.3}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ln
25.0,0.3})\right)\end{cases}$ Source Light: $\displaystyle\begin{cases}\hfill
R_{\text{s}}&\sim\exp\left(\mathcal{N}(\ln 0.25,0.15\right)\\\\[3.0pt] \hfill
n_{s}&\sim\mathcal{U}(0.5,4)\\\\[3.0pt]
\hfill\epsilon_{s,1},\epsilon_{\text{s},2}&\sim\mathcal{TN}(0,0.15,-0.5,0.5)\\\\[3.0pt]
\hfill x_{s}&\sim\mathcal{N}(0,0.25)\\\\[3.0pt] \hfill
y_{s}&\sim\mathcal{N}(0,0.25)\\\\[3.0pt] \hfill
I_{s}&\sim\exp\left(\mathcal{N}(\ln 150.0,0.5)\right)\end{cases}$

Notes — The model consists of SIE for both lens mass profiles and external
shear. $\theta_{E}$ is the Einstein radius in arcsec. $x$ and $y$ are the mass
centers of the two lenses. $\gamma_{1,\text{ext}}$ and $\gamma_{1,\text{ext}}$
are the external shear. The parameters $\epsilon_{1}$ and $\epsilon_{2}$ are
the lens mass eccentricities, while $\epsilon_{l,1}$, $\epsilon_{l,2}$ and
$\epsilon_{s,1}$, $\epsilon_{s,2}$ are the lens and source light
eccentricities, respectively. We employ the notation
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}a}/{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}b}$
to indicate the parameters for the main lens/secondary lens. We use a
spherical Sérsic profile for the light of the secondary lens, for which
eccentricity priors are thus indicated with a dash. $R_{l}$, $R_{s}$ are
defined as the half-light radius and $n_{l}$, $n_{s}$ as the Sérsic index.
$x_{l}$, $y_{l}$ and $x_{s}$, $y_{s}$ describe the center of the light and
$I_{l}$, $I_{s}$ its intensity. Subscripts $l$, $s$ imply the parameter
belongs to the lens light profile or to the source light profile,
respectively. Here, $\mathcal{U}(a,b)$ is a uniform distribution with support
$[a,b]$, $\mathcal{N}(\mu,\sigma)$ is Gaussian with mean $\mu$ and standard
deviation $\sigma$, and $\mathcal{TN}(\mu,\sigma;a,b)$ is a truncated Gaussian
with support $[a,b]$.

We achieved very good residual and excellent sampling results. We report two
metrics that are widely used in the statistics literature to measure the
degree of convergence of our sampler: the potential scale reduction factor
(PSRF), $\hat{R}$ (Gelman & Rubin, 1992) and the effective sample size (ESS).
The $\hat{R}$ values for nearly all parameters are below 1.1, with the lens
light eccentricity the only exception (for $\epsilon_{\text{l}2}$,
$\hat{R}=1.2$). Given how faint it is, that is hard to constrain. For the same
reason, the only two parameters whose effective sample sizes (ESS) are below
2000 are $\epsilon_{2}$ and $n_{sersic}$ of the lens light. The maximum ESS is
around 24000. Our best-fit model is shown in Figure 4, with mass parameters
presented in Table 3.

Table 3: Best-fit mass parameters for the main lens L1, the secondary lens L2, and the external shear.  Parameters | Main Lens, L1 | Secondary Lens, L2
---|---|---
$\theta_{E}$ | $2.520_{-0.031}^{+0.032}$ | $0.261_{-0.027}^{+0.028}$
$\epsilon_{1}$ | $-0.365_{-0.009}^{+0.009}$ | $0.662_{-0.069}^{+0.069}$
$\epsilon_{2}$ | $-0.486_{-0.011}^{+0.011}$ | $0.304_{-0.062}^{+0.064}$
$x$ | $-0.115_{-0.011}^{+0.010}$ | $1.836_{-0.107}^{+0.096}$
$y$ | $-0.207_{-0.018}^{+0.018}$ | $-1.563_{-0.059}^{+0.073}$
$\gamma_{1,\text{ext}}$ | $-0.008_{-0.005}^{+0.006}$
$\gamma_{2,\text{ext}}$ | $-0.038_{-0.006}^{+0.006}$
Figure 4: The results for our model consisting of a main lens and a second
lens (L2, at $z=0.386$). From left to right, we show: the observed MUSE image
in $g$-band (North is up, East is left), the best-fit model with the critical
curve, the reduced residual, and the unlensed source with the caustic.

We show representative sampling results for the main lens along with
individual chains for the Einstein radius in Figure 5. The sampling for all
other parameters are equally good.

Figure 5: The corner plot shows our sampling results for the main lens and
external shear mass parameters. In the inset, four chains of HMC demonstrate
statistical consistency within the sampling of the Einstein radius, with
$\hat{R}=1.007$ (see text).

For the main lens, we determined the Einstein radius to be
$\theta_{E}={2.520^{\prime\prime}}_{-0.031}^{+0.032}$ (Table 3). This can be
converted to the velocity dispersion as follows (e.g., Narayan & Bartelmann,
1996):

$\theta_{E}=4\pi\left(\frac{\sigma_{\rm
SIE}}{c}\right)^{2}\frac{D_{L1\textnormal{-}s}}{D_{s}},$ (1)

where $D_{L1\textnormal{-}s}$ and $D_{s}$ are the angular-diameter distances
between the main lens and the source and from the observer to source,
respectively. Using the cosmological parameters from Planck Collaboration et
al. (2020), with $H_{0}$ = 67.4 $\pm$ 0.5 km s-1 Mpc-1, $\Omega_{\Lambda}$ =
0.6847 $\pm$ 0.0073, and $\Omega_{M}$ = 0.315 $\pm$ 0.007, we obtained $D_{s}$
= 1690.6 Mpc and $D_{L1-s}$ = 1026.1 Mpc, respectively. This led to a velocity
dispersion of $\sigma^{\mathrm{L1}}_{\rm SIE}=379\pm 2\,\rm{km\,s}^{-1}$ for
the main lens. This is among the most massive galactic scale strong lenses
(e.g., Collett, 2015; Bettoni et al., 2019).

The predicted magnifications are 2.40, 3.23, 4.03 and 0.81 for images A, B, C
and D, respectively, with the total being 10.47.

We now briefly discuss the secondary lens, which is the foreground galaxy L2
at $z=0.386$, located near (in projection) the image C. Our modeling approach
is informed by the following: (i) the data we use for lens modeling is from
low-resolution ground-based observations; (ii) the main lens is much more
massive than the secondary lens; and (iii) the secondary lens is near the
Einstein ring of the main lens. Instead of pursuing lens modeling with double
lens planes, we treat this interloper as an “effective” subhalo of the main
lens (Çaǧan Şengül et al., 2020). We determine its effective Einstein radius
to be $0.261{{}^{\prime\prime}}^{+0.028}_{-0.027}$ (Table 3). By treating L2
as an “effective” subhalo, its net lensing effect of the foreground galaxy L2
is reduced by a factor of $1-\beta$ (Çaǧan Şengül et al., 2020), with
$\beta=\frac{D_{L2\textnormal{-}L1}D_{s}}{D_{L1}D_{L2\textnormal{-}s}}$, where
$D_{L2\textnormal{-}L1}$ and $D_{L2\textnormal{-}s}$ are the angular diameter
distances between the two lenses and between the secondary lens and the
source, respectively (Çaǧan Şengül et al. 2020, see also Fleury et al. 2021).
In our case, $\beta=0.470$. Thus, the lensing effect has been reduced by 0.53.
Given the SIE mass model, we adjusted the value implied by Eq (1) accordingly
and determined $\sigma_{\mathrm{SIE}}^{\mathrm{L2}}$ to be
$110\,\mathrm{km\,s}^{-1}/\sqrt{0.53}=137\,\mathrm{km\,s}^{-1}$. This
indicates that it is a relatively low-mass galaxy (see, e.g., Collett 2015),
and is consistent with its low brightness.

In recent years there has been a growing interest in using gravitational
imaging (Koopmans, 2005) to detect sub-galactic scale halos with no baryons
($M_{halo}\lesssim 10^{9}M_{\odot}$). Sometimes referred to as “dark halos”,
they include both subhalos and line-of-sight (LOS) interlopers. This
potentially provides a powerful way to test dark matter models (e.g., Vegetti
et al., 2010b; Despali et al., 2018). In this approach, the detection of a
dark halo and the measurement of its mass solely rely on lens modeling.
DESI-253.2534+26.884 is a rare system with (i) a main lens that can be well
constrained (due to the Einstein Cross pattern) and (ii) a small _visible_ LOS
interloper near (in projection) the Einstein ring of the main lens. Such a
system can provide an important check for the gravitational imaging approach
for detecting and measuring the mass of LOS _dark_ halo interlopers222For the
detection of subhalos using the gravitational imaging technique, Vegetti et
al. (2010a) performed testing by applying it to SDSSJ120602.09+514229.5, which
has a visible satellite galaxy (hosted by a subhalo) at the location of the
Einstein ring formed by the main lens. But to our knowledge, such tests have
not been done for the case of interlopers. . By obtaining high resolution
imaging for this system, as well as resolved dynamical information for the
main lens and the interloper (e.g., VLT/MUSE with adaptive optics), we can
check the agreement between the mass measurements from careful lens modeling
and detailed dynamics. This can be used to establish the validity of using
gravitational imaging to detect and measure the mass of LOS sub-galactic dark
halos, which some groups claimed are better than subhalos to determine the
nature of dark matter (e.g., Sengül et al., 2022).

###  3.2 pPXF fitting

We fit the lens galaxy spectrum with the pPXF package (Cappellari 2022, see
also Cappellari & Emsellem 2004 and Cappellari 2017) with the goal to
determine the stellar velocity dispersion. Following the pPXF usage
examples333https://github.com/micappe/ppxf_examples for high redshift
galaxies, the spectrum was fitted using E-MILES stellar population synthesis
models (Vazdekis et al., 2016). The most prominent features of our spectrum
are the Ca II H and K lines, and we obtain a best-fit velocity dispersion of
$\sigma$ = 403 $\pm$ 85 km s-1. This is in good agreement with the velocity
dispersion from lens modeling.

##  4 Summary and conclusion

We present MUSE observations of the strong gravitational lens system
DESI-253.2534+26.884, which was discovered in the DESI Legacy Imaging Surveys
data using deep residual neural networks by Huang et al. (2021) and appeared
as an Einstein cross. We determined the redshift of the main lensing galaxy,
$z_{L1}=0.636\pm$ 0.001, and show that the four images of the source display
common spectroscopic features which places the source at a redshift of
$z_{s}=2.597\pm 0.001$, fully confirming this to be a strong lensing system.
We found a faint foreground galaxy located in front of the image C (see Figure
1). A careful selection of the spaxels allowed us to extract the spectrum and
determine its redshift to be $z_{L2}=0.386$. This set of redshifts especially
demonstrate the advantage IFU observations of gravitational lens systems in
contrast to long slit spectroscopy.

We modeled the gravitational lens system using GIGA-Lens (Gu et al., 2022).
The Einstein radius of the lens is
$\theta_{E}=2.520{{}^{\prime\prime}}_{-0.031}^{+0.032}$, which corresponds to
a velocity dispersion of $\sigma^{L1}_{\rm SIE}$ = 379 $\pm$ 2 km s-1. This is
consistent with the spectroscopically determined velocity dispersion of
$\sigma$ = 403 $\pm$ 85 km s-1.

To our knowledge, this is the first time a real gravitational lensing system
has been modeled with GPUs, using the GIGA-Lens pipeline. And it shows great
promise: The modeling time is $55$ sec on a single A100 GPU on the Perlmutter
supercomputer at the National Energy Research Scientific Computing Center
(NERSC). With 4 GPUs on one GPU node, we expect the time to be roughly $15$
sec (Gu et al., 2022). By comparison, for ground-based data from DES, Rojas et
al. (2021, R21) reported an average lens modeling time of 4.3 hour using
Lenstronomy (e.g., Birrer & Amara 2018). Our model is comparable in that the
main lens is modeled as SIE just as in R21. In our model, due to the presence
of the second lens, we have more parameters than R21. Yet, we have achieved
greater than two orders of magnitude speedup. This concretely demonstrates a
very promising future of modeling $\mathcal{O}(10^{5})$ of strong lensing
systems (e.g., Collett, 2015) that are expected to be discovered in the next
decade (e.g., Euclid, LSST, and the Roman Space Telescope), in a fast, robust
and scalable way.

## Acknowledgments

We thank Greg Aldering, Adam Bolton, Saul Perlmutter, and Yiping Shu for
insightful discussion. The work of A.C. 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. X.H.
acknowledges the University of San Francisco Faculty Development Fund. This
work was supported in part by the Director, Office of Science, Office of High
Energy Physics of the US Department of Energy under contract No. DE-
AC025CH11231. This research used resources of the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility operated under the same contract as above and the
Computational HEP program in The Department of Energy’s Science Office of High
Energy Physics provided resources through the “Cosmology Data Repository”
project (grant No. KA2401022). This work is based on observations collected at
the European Organisation for Astronomical Research in the Southern Hemisphere
under ESO program 0111.A-0407. The execution in the service mode of these
observations by the VLT operations staff is gratefully acknowledged.

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