Double-Source-Plane Strong Lenses

• Double-source-plane strong gravitational lenses are compound systems where a single foreground deflector lenses two background sources at different redshifts, providing an H₀-independent observable.
• The methodology uses multiplane lens equations and advanced modeling techniques (e.g., lenstronomy, stochastic variational inference) to extract cosmological parameters and mass profiles.
• Practical DSPL analyses require high-resolution imaging, detailed spectroscopy, and careful control of degeneracies and systematics such as line-of-sight effects and environmental mass.

Double-source-plane strong gravitational lenses (DSPLs) are strong-lensing systems in which a single foreground deflector lenses two background sources at different redshifts. In galaxy-scale DSPLs, the nearer source can itself act as an additional deflector for the farther source, so the relevant description is intrinsically multiplane. Their principal importance is cosmographic: comparing the lensing geometry of the two source planes yields a dimensionless distance-ratio observable, commonly written as $\beta$ or $\eta$, that is independent of $H_0$ and sensitive to $\Omega_{\rm m}$, curvature, and dark-energy parameters. At the same time, the second source plane provides an additional radial lever arm for mass-profile studies and substructure searches. The same literature that identifies this geometric appeal also emphasizes that DSPLs are rare, require high-resolution imaging and spectroscopy for precision work, and remain limited by generalized mass-sheet, line-of-sight, and environmental systematics (Sahu et al., 1 Apr 2025, Collett et al., 2014, Johnson et al., 28 Jan 2025).

1. Geometric configuration and distance-ratio observables

A DSPL consists of a lens at redshift $z_l$ and two sources at redshifts $z_{s1}$ and $z_{s2}$, with $z_{s2}>z_{s1}>z_l$. In the simplest description, the same foreground galaxy produces two sets of arcs or rings at different radii. In the fully compound case, the intermediate source is not merely a lensed emitter but also contributes deflection for the more distant source. The cosmological dependence then enters through ratios of angular-diameter distances rather than through an absolute distance scale (Collett et al., 2014, Schneider, 2014).

A standard DSPL observable is

$$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$

while an alternative convention uses

$$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$

with the explicit note that some work adopts $\eta$0. Because each angular-diameter distance scales as $\eta$1, these ratios are independent of $\eta$2. For isothermal lenses, and when the mass in the first source plane is negligible, the ratio of Einstein radii can be turned directly into the cosmological scaling factor. Several papers therefore describe DSPLs as a purely geometric or image-geometry-based probe, although that idealization is qualified elsewhere by degeneracy analyses (Sahu et al., 1 Apr 2025, Johnson et al., 28 Jan 2025, Linder, 2016).

The redshift configuration strongly affects leverage. Forecast work has argued that low-redshift lenses are especially interesting because the DSPL ratio can be more sensitive to $\eta$3 and $\eta$4 than to $\eta$5, with a near-null in the $\eta$6 dependence around $\eta$7, and with sensitivity remaining useful out to about $\eta$8. More generally, systems with small separation between lens and first source and large separation between the two source planes are repeatedly identified as cosmographically favorable (Linder, 2016, Collett et al., 2012).

2. Cosmographic use and parameter inference

The first cosmological measurement with a galaxy-scale DSPL was obtained from SDSSJ0946+1006, the “Jackpot” lens. Modeling the primary lens as an elliptical power-law mass distribution and including perturbative lensing by the first source yielded $\eta$9, implying $H_0$0 in flat $H_0$1CDM. Combined with a Planck prior in flat $H_0$2CDM, the result was $H_0$3, with a quoted improvement of about $H_0$4 relative to Planck alone (Collett et al., 2014).

Subsequent analyses have expanded the DSPL cosmography sample beyond the Jackpot system. The AGEL collaboration reported AGEL150745+052256 as only the second DSPL ever used to measure cosmological parameters and emphasized that combining it with the published model of the Jackpot lens improves the precision on $H_0$5 in $H_0$6CDM and on $H_0$7 in $H_0$8CDM by $H_0$9, while adding DSPLs to Planck enhances precision on $\Omega_{\rm m}$0 by $\Omega_{\rm m}$1 (Sahu et al., 1 Apr 2025). A later AGEL analysis of AGEL035346$\Omega_{\rm m}$2170639 inferred, for that system alone, $\Omega_{\rm m}$3 and $\Omega_{\rm m}$4 in flat $\Omega_{\rm m}$5CDM, then combined AGEL0353 with AGEL1507 and J0946 to form the largest galaxy-scale DSPL sample used for cosmological measurement to date. For the three-DSPL sample, the reported flat $\Omega_{\rm m}$6CDM constraint was $\Omega_{\rm m}$7 and $\Omega_{\rm m}$8, while DSPL + CMB gave $\Omega_{\rm m}$9 and $z_l$0, a $z_l$1 improvement in the uncertainty on $z_l$2 compared with CMB-only constraints (Bowden et al., 18 Sep 2025).

These results are generally presented as evidence of complementarity rather than stand-alone dominance. Forecast and sample papers repeatedly state that DSPL contours are nearly orthogonal to CMB and BAO in the $z_l$3 plane, or otherwise complementary to standard probes, so the main cosmographic value of current DSPL work lies in joint inference and cross-checking rather than in replacing other late-time probes (Collett et al., 2012, Collett et al., 2014, Sahu et al., 1 Apr 2025).

3. Discovery routes and survey-scale searches

Because DSPLs are extremely rare, discovery strategy is central to the field. Early spectroscopic evidence for compound systems came from the MaNGA-based SILO survey, which searched about $z_l$4 million spectra from $z_l$5 galaxies, identified $z_l$6 spectroscopically detected strong lens candidates, and found $z_l$7 compound lens candidates with two background source planes. The key advance was MaNGA’s integral-field spectroscopy: multiple spatially resolved spectra permitted grouping background emission-line detections by redshift and assessing whether they formed a plausible common source-plane geometry, something not possible in previous SLACS and BELLS single-fiber programs (Talbot et al., 2018).

A widely cited observational milestone is HSC J142449$z_l$8005322, the “Eye of Horus,” discovered in Hyper Suprime-Cam Subaru Strategic Program imaging. It was reported as the first DSPL with spectroscopic redshifts for both background sources, with $z_l$9, $z_{s1}$0, and $z_{s1}$1. Two independent modeling codes reproduced the main morphology, but a smooth potential could not reproduce two lensed sources separated by about $z_{s1}$2, so additional substructure in the lens potential was required (Tanaka et al., 2016).

Recent survey pipelines increasingly use machine learning and large-scale triage. In DES Year 6 imaging, a search based on a pre-trained Vision Transformer, Interactive Machine Learning, Space Warps citizen-science inspection, and expert classification reduced approximately $z_{s1}$3 million cutout images to $z_{s1}$4 targets of interest. Citizen scientists ruled out approximately $z_{s1}$5 as false positives, leaving $z_{s1}$6 candidates, of which $z_{s1}$7 were expert-classified as definite lenses and $z_{s1}$8 as probable lenses, with $z_{s1}$9 newly identified. A second ViT was trained specifically to find double-source-plane systems and found at least one double-source system (Gonzalez et al., 26 Jan 2025).

Euclid has already produced a dedicated DSPL candidate sample. In the Quick Release 1 data, four new galaxy-scale DSPL candidates were reported from the Strong Lensing Discovery Engine after machine-learning ranking, citizen-science classification, and expert review. These candidates—the Teapot Lens, Cosmic Dartboard, Galileo’s Lens, and Cosmic Ammonite—were modeled as preliminary multiplane systems, and the paper used LensPop to predict about $z_{s2}$0 DSPLs in the full Euclid survey, scaling to $z_{s2}$1 DSPLs in the $z_{s2}$2 Q1 area (Collaboration et al., 19 Mar 2025).

4. Modeling methodology and representative systems

DSPL analysis uses the multiplane lens equation rather than a single-plane approximation whenever the first source contributes non-negligible deflection to the second. For the two-source-plane case, one convenient form is

$z_{s2}$3

$z_{s2}$4

which makes explicit that the farther source is lensed by both the primary deflector and the mass associated with the first source plane (Collett et al., 2014).

In the Jackpot analysis, the foreground deflector was modeled as an elliptical power-law mass distribution plus external shear, while the first source was modeled as an isothermal sphere whose centroid was tied to the lensed light distribution. The authors used HST ACS F814W imaging and a semi-linear inversion with pixellated sources. This modeling choice was important because the second source is not simply a background ring behind a single lens; it is a genuinely compound configuration (Collett et al., 2014).

AGEL035346$z_{s2}$5170639 illustrates the contemporary galaxy-scale workflow. The system was modeled with lenstronomy using HST WFC3 imaging and Keck spectroscopy. The adopted mass model comprised an elliptical power law for the main deflector, a singular isothermal ellipse for the first source plane, external shear, and elliptical Sérsic light profiles for the deflector and sources. Optimization used PSO, posterior sampling used emcee, and the best-fit model had $z_{s2}$6. The inferred parameters included $z_{s2}$7 arcsec, $z_{s2}$8, $z_{s2}$9, $z_{s2}>z_{s1}>z_l$0 arcsec, and $z_{s2}>z_{s1}>z_l$1; a test including a mass sheet in the deflector plane left $z_{s2}>z_{s1}>z_l$2 essentially unchanged (Bowden et al., 18 Sep 2025).

For Euclid Q1 candidate validation, the open-source code Herculens was extended to handle multiplane lensing. The analysis used stochastic variational inference with NumPyro, an AdaBelief optimizer, and a low-rank multivariate normal guide distribution, running $z_{s2}>z_{s1}>z_l$3 iterations in $z_{s2}>z_{s1}>z_l$4 different SVI realizations per target and adopting the run with the lowest average loss. The lens light was modeled simultaneously with the arcs using five elliptical Gaussian components, after which the sources were reconstructed on regular pixel grids. Because source and lens redshifts were not yet available, the stated goal was validation of a DSPL interpretation rather than full precision inference (Collaboration et al., 19 Mar 2025).

5. Degeneracies, line-of-sight structure, and environmental systematics

A major conceptual issue is that DSPLs do not evade the mass-sheet problem simply by adding a second source plane. In the generalized two-plane case, there exists an invariance transformation that leaves image positions, relative shapes, flux ratios, and time-delay ratios unchanged, while absolute time delays scale with the same factor. In one formulation, the transformed first-plane deflection is

$z_{s2}>z_{s1}>z_l$5

with a corresponding transformation in the second plane. This generalized MST preserves the relative distribution of mass and light, so a mass-follows-light assumption does not fix the degeneracy. The paper’s central claim is that changing cosmological parameters, and thus changing $z_{s2}>z_{s1}>z_l$6, is essentially equivalent to such a transformation; DSPLs therefore do not robustly determine cosmological parameters unless the degeneracy is broken by additional information (Schneider, 2014).

Line-of-sight weak lensing adds a second layer of systematic control. Using statistically representative lines of sight from RayGalGroupSims, one study found that LOS perturbations add an extra uncertainty to $z_{s2}>z_{s1}>z_l$7 at roughly the $z_{s2}>z_{s1}>z_l$8 to $z_{s2}>z_{s1}>z_l$9 level across a Euclid-like DSPL population, and about $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$0 for the Jackpot system. The same analysis showed that the LOS shear experienced by the two source planes can differ significantly in both magnitude and direction, invalidating the common simplifying assumption that the more distant source sees a simple rescaling of the nearer-source shear. The practical recommendation was that compound lens models should include an independent external-shear term for each lens/source plane (Johnson et al., 28 Jan 2025).

Environmental mass can also be decisive. XMM-Newton observations of the Eye of Horus detected two extended X-ray sources, corresponding to a main cluster centered near the lens and a northeastern cluster about $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$1 away. The main cluster contribution to the projected mass within the Einstein radius was reported as $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$2, corresponding to $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$3–$$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$4 of the total mass within the Einstein radius, while the northeastern cluster contributed only $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$5. The conclusion was that DSPL interpretation in this system is strongly environment-sensitive and that cluster-scale mass must be included in precise lens models (Tanaka et al., 2019).

Systematics are not limited to lensing degeneracies in the narrow sense. A DSPL-based test of the distance-duality relation concluded that evolution of the lens mass-profile slope must be controlled to $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$6 times tighter fractional precision than a claimed distance-duality violation. This suggests that DSPLs are as much a laboratory for joint astrophysical and cosmological calibration as they are a direct geometric probe (Sharma et al., 2022).

6. Population forecasts, cosmological tests, and related extensions

Before the first mature galaxy-scale cosmography analyses, forecasts had already argued that DSPLs could provide useful dark-energy constraints with relatively small samples. One study estimated that constraints comparable to current data sets—quoted as $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$7 uncertainty on the dark equation of state at $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$8 CL—were possible with a handful of double source plane systems, and emphasized that the $$\beta \equiv \frac{D_{ls}(z_l,z_{s1})}{D_s(z_{s1})}\,\frac{D_s(z_{s2})}{D_{ls}(z_l,z_{s2})},$$9 degeneracy is almost orthogonal to that of CMB and BAO measurements (Collett et al., 2012). A later forecast framed DSPLs as complementary to time-delay cosmography and reported that adding DSPL distance-ratio measurements could improve the dark-energy figure of merit by about $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$0 for fewer than $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$1 low-redshift systems, with the main text giving $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$2 for a specific configuration of $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$3 systems at $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$4 precision (Linder, 2016).

More recent survey forecasts are correspondingly larger. LensPop-based Euclid and LSST studies have repeatedly used a figure of about $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$5 galaxy-scale DSPLs suitable for cosmology, while also noting that practical cosmographic samples will be smaller because usable systems require high-resolution imaging, redshift confirmation, and careful mass modeling (Sharma et al., 2022, Collaboration et al., 19 Mar 2025, Johnson et al., 28 Jan 2025). The same forecast machinery has been used not only for parameter inference but also for consistency tests of the FLRW framework and the Etherington distance-duality relation. For a simulated sample of $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$6 DSPLs, one paper reported a stand-alone flat $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$7CDM constraint of $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$8 and derived explicit curvature-consistency relations based on the DSPL ratio (Sharma et al., 2022).

A related 2026 development generalized DSPL logic to “Pseudo Double-Source Plane Lenses” (PDSPLs), defined as pairs of independent single-source-plane lenses with self-similar deflectors. The motivation was explicit: true DSPLs are exceedingly rare and fundamentally limited by the mass-sheet degeneracy, especially for small samples. In that hierarchical framework, the LSST 10-year photometric PDSPL sample was forecast to achieve $$\eta \equiv \frac{D_{ds_2}D_{s_1}}{D_{ds_1}D_{s_2}},$$9, tightening to $\eta$00 with an $\eta$01 prior $\eta$02. This does not redefine DSPLs themselves, but it shows that DSPL formalism has become a template for broader strong-lensing tomography (Sharma et al., 1 Jul 2026).

The cumulative picture is therefore dual. On the one hand, DSPLs are a rare class of compound strong lenses whose distinctive ratio observable is $\eta$03-free, cosmographically complementary, and valuable for probing mass profiles and substructure. On the other hand, the current literature treats them as precision tools only when multiplane modeling, velocity-dispersion information, line-of-sight error budgets, and environmental mass characterization are all taken seriously. This suggests that the future significance of DSPLs will depend not only on discovery rates in Euclid, DES, Rubin, and related surveys, but also on the follow-up infrastructure needed to convert visually striking two-ring systems into controlled cosmological measurements (Bowden et al., 18 Sep 2025, Collaboration et al., 19 Mar 2025, Sahu et al., 1 Apr 2025).