Double-Source-Plane Lenses (DSPLs)

• Double-Source-Plane Lenses are rare strong gravitational lensing systems where a foreground deflector lenses two distinct background sources at different redshifts.
• They provide a geometric, H₀-independent probe of cosmological parameters by leveraging the dimensionless scaling factor derived from multiplane lens equations.
• Advanced modeling, high-resolution imaging, and statistical inference break key degeneracies, enabling precision tests of dark energy and spatial curvature.

Double-Source-Plane Lenses (DSPLs) are rare but highly informative strong gravitational lensing systems in which a foreground deflector simultaneously lenses two distinct background sources at different redshifts. The configuration yields multiple images or arcs for each background source, and the interplay between the lens and the two source planes encodes precise information about cosmological distance ratios. Critically, DSPLs provide a geometric, $H_0$-independent probe of the expansion history, spatial curvature, and the structure of lens galaxies, and have become a focal point for constraining dark energy and testing the cosmological framework with minimal astrophysical systematics.

1. Geometric and Mathematical Framework

The DSPL configuration involves an observer, a principal lens at redshift $z_l$, a nearer source at $z_{s1}$, and a more distant source at $z_{s2}$ ($z_l < z_{s1} < z_{s2}$). Each source is multiply imaged through the gravitational potential of the lens. The lens mapping for source planes must be treated using the multi-plane lens equation: $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$ where $D_{i,j}$ denotes the angular-diameter distance between redshifts $z_i$ and $z_j$. The cosmologically critical observable is the scaling factor (often denoted $\beta$ or $z_l$0), defined as: $z_l$1 For a singular isothermal sphere, the ratio of the Einstein radii of the two sources approximates $z_l$2. This dimensionless ratio is independent of $z_l$3 due to the cancellation of the common scaling in the distance ratios, directly linking observed image geometry to $z_l$4 in cosmological models (Collett et al., 2012, Sharma et al., 2022, Collett et al., 2014).

2. Cosmological Applications and Parameter Sensitivity

DSPLs are uniquely sensitive to the shape of the distance–redshift relation, not to its absolute normalization. Measuring $z_l$5 with percent-level precision gives leverage over the matter density parameter $z_l$6 and dark energy equation-of-state parameter $z_l$7, and allows direct tests of spatial curvature and the Etherington distance-duality relation (Sharma et al., 2022, Sharma et al., 2022, Bowden et al., 18 Sep 2025). Single DSPLs already return $z_l$8 constraints at the 15–30% level, while moderate-sized samples ($z_l$9–100) drive uncertainties in $z_{s1}$0 to $z_{s1}$1 (Collett et al., 2012, Bowden et al., 18 Sep 2025). The degeneracy direction of DSPLs in the $z_{s1}$2 plane is nearly orthogonal to that from the CMB and BAO, underpinning their complementarity (Collett et al., 2012, Sharma et al., 2022, Collett et al., 2014).

The spatial-curvature consistency equation derived from DSPLs reads: $z_{s1}$3 with $z_{s1}$4 the comoving angular diameter distance, enabling model-independent curvature tests (Sharma et al., 2022).

3. Degeneracies and Systematic Uncertainties

Critical degeneracies in single-plane strong lensing—most notably the mass-sheet degeneracy (MST)—affect lensing observables by allowing a transformation of the projected mass that rescales images without changing observables for a single source. DSPLs partially break this degeneracy, as two source planes introduce distinct geometric responses to the same mass-sheet transformation (Collett et al., 2014, Johnson et al., 28 Jan 2025, Johnson et al., 2 Feb 2026). The addition of a second background source enables simultaneous determination of the lens mass-profile slope and cosmological scaling factor, as $z_{s1}$5 is more robust to the MST than are time-delay distances (Collett et al., 2014, Tanaka et al., 2016, Johnson et al., 28 Jan 2025, Johnson et al., 2 Feb 2026).

A principal limitation arises from line-of-sight (LOS) density fluctuations, which induce a $z_{s1}$6–$z_{s1}$7 scatter in $z_{s1}$8 measurements. For current and next-generation surveys, this LOS term is subdominant relative to measurement uncertainties but must be explicitly included in error budgets for percent-level cosmography (Johnson et al., 28 Jan 2025, Johnson et al., 2 Feb 2026). LOS shear at each source plane generally differs in both amplitude and orientation, necessitating separate external shear parameters in multiplane models (Johnson et al., 28 Jan 2025).

4. Observational Realizations and Survey Yields

DSPLs are observationally rare, with only a handful of galaxy-scale systems spectroscopically confirmed to date, but ongoing and planned wide-area surveys (Euclid, LSST, CSST) are predicted to discover $z_{s1}$9 such lenses (Collaboration et al., 19 Mar 2025, Wu et al., 21 Jan 2026, Sharma et al., 2022). Automated pipelines employing convolutional neural networks followed by expert human vetting enable statistical studies and first population-level constraints (Collaboration et al., 19 Mar 2025). High-resolution imaging (HST, JWST, ELTs) and spectroscopic follow-up are necessary for arc resolution and redshift determination. For the most thoroughly modeled examples, such as AGEL035346–170639, AGEL150745+052256, and SDSSJ0946+1006 (“the Jackpot”), robust lens mass profiles, source redshifts, and velocity dispersions have led to per-lens uncertainties in $z_{s2}$0 of $z_{s2}$1, yielding $z_{s2}$2 constraints competitive with standard candles (Collett et al., 2014, Sahu et al., 1 Apr 2025, Bowden et al., 18 Sep 2025).

Survey simulations predict the following for Euclid-era DSPL science:

Survey	Expected DSPL Sample Size	Typical $z_{s2}$3
Euclid Wide	$z_{s2}$41700	$z_{s2}$5
LSST	$z_{s2}$61000–2000	$z_{s2}$7
CSST (UDF)	Tens	$z_{s2}$8

The best systems for early-stage cosmology have large redshift lever arms and thus small $z_{s2}$9, maximizing sensitivity to $z_l < z_{s1} < z_{s2}$0 and $z_l < z_{s1} < z_{s2}$1 (Wu et al., 21 Jan 2026).

5. Lens Modelling and Statistical Inference

State-of-the-art modelling methodologies include parametric (e.g., elliptical power-law, SIE) and pixel-based lens inversion codes, enforcing full multiplane lens equations and marginalizing over lens and source parameters alongside cosmological parameters (Tanaka et al., 2016, Bowden et al., 18 Sep 2025). The likelihood combines pixel-level image fits with prior distributions over cosmological, lens, and source parameters, often sampled with MCMC (e.g., emcee). Hierarchical Bayesian analysis of multiple systems enables improved population-level constraints (Wu et al., 21 Jan 2026).

Mass modelling for DSPL cosmography requires:

• Simultaneous fits to all source arcs with full multiplane ray-tracing.
• Treatment of LOS convergence and separate external shears per source.
• Kinematic data to constrain lens mass normalization and break remaining local degeneracies.
• Model selection between pure power-law, composite (stars+halo), and models with explicit substructure where required by observed image splitting (Tanaka et al., 2016, Bowden et al., 18 Sep 2025).

6. Cosmological Constraints and Complementarity

The first high-precision DSPLs (SDSSJ0946+1006, AGEL150745+052256, AGEL035346–170639) yield direct constraints on $z_l < z_{s1} < z_{s2}$2 and reveal the effectiveness of combining these measurements with the CMB. For example, combining DSPL and Planck yields $z_l < z_{s1} < z_{s2}$3 (Jackpot), a $z_l < z_{s1} < z_{s2}$4 precision gain over Planck-only (Collett et al., 2014, Sahu et al., 1 Apr 2025, Bowden et al., 18 Sep 2025). With current samples of $z_l < z_{s1} < z_{s2}$5 galaxy-scale DSPLs, joint constraints tighten the $z_l < z_{s1} < z_{s2}$6 uncertainty by $z_l < z_{s1} < z_{s2}$7 compared to a single system and by $z_l < z_{s1} < z_{s2}$8 when combined with Planck. Forecasts for $z_l < z_{s1} < z_{s2}$9 gold-standard Euclid or LSST DSPLs show dark-energy figure-of-merit (FOM) gains by a factor 3–4 beyond CMB+SN alone, with independent detection of evolving dark energy density out to $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$0 and model-independent curvature tests to $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$1 (Sharma et al., 2022, Sharma et al., 2022).

DSPL constraints are nearly orthogonal in degeneracy direction to those from Type Ia supernovae, BAO, and CMB, providing critical complementarity within joint analyses (Collett et al., 2012, Collett et al., 2014, Sharma et al., 2022).

7. Prospects, Requirements, and Future Directions

Ongoing progress in survey automation, machine learning classifier development, and follow-up spectroscopy is expanding the known sample size from a handful to thousands of candidates (Collaboration et al., 19 Mar 2025, Wu et al., 21 Jan 2026). Realizing the full statistical power of DSPL cosmography will require:

• High-resolution imaging to resolve arcs and measure precise Einstein radii.
• Spectroscopic redshift confirmation for both sources and the lens to $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$2.
• Dynamical measurements (velocity dispersions) to anchor lens mass models.
• Systematic control of line-of-sight effects and lens mass-profile evolution to sub-percent precision.
• Parallel development of modeling codes permitting pixelized reconstruction, multiplane shear, and external convergence terms (Johnson et al., 28 Jan 2025, Sharma et al., 2022, Johnson et al., 2 Feb 2026).
• Large-area surveys (Euclid, LSST, CSST) followed by targeted high-SNR spectroscopy for clean cosmological samples.
• Hierarchical Bayesian and joint-likelihood inference schemes for scaling precision with $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$3 (Wu et al., 21 Jan 2026, Sharma et al., 2022).

Expected outcomes include $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$4 precision on constant or evolving $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$5, $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$6 uncertainty on $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$7, direct FLRW-consistency checks, and precise calibration of lens population mass profiles and dark-matter substructure (Sharma et al., 2022, Bowden et al., 18 Sep 2025). The imminent expansion to $$\begin{align} \boldsymbol{\theta}_{S1} &= \boldsymbol{\theta} - \frac{D_{l,s1}}{D_{o,s1}}\,\hat{\alpha}_1(D_l \boldsymbol{\theta}), \ \boldsymbol{\theta}_{S2} &= \boldsymbol{\theta} - \frac{D_{l,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}) - \frac{D_{s1,s2}}{D_{o,s2}}\hat{\alpha}_1(D_l \boldsymbol{\theta}_{S1}), \end{align}$$8 DSPLs will establish them as a prime cosmographic standard, offering a geometry-based probe essentially orthogonal to the CMB and standard candles, and a stringent test of the physical underpinnings of the standard cosmological model.