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Lessons from double-plane lenses for galaxy profiles, the Hubble tension, and the mass-sheet degeneracy

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Daniel P. Johnson, Martin Millon, Pierre Fleury, Giacomo Queirolo, Tian Li, Wolfgang J. R. Enzi, Daniel J. Ballard, Duncan Bowden
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Thomas E. Collett, Daniel Gilman

Cosmology and Nongalactic Astrophysics · Submitted 6 Nov 2026 · 19 references

Pith's verdict: The ζ12 construction is a real, clean result, but the physical mass-sheet readout rests on an untested sheet hypothesis, and the J1721 flagship number depends on an acknowledged 2 mas choice.

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arXiv ID2611.09856 (version 1)
Pith IDDYGAXFJ3, Pith's permanent record of this paper and its review
CitationDaniel Gilman, Daniel J. Ballard, Daniel P. Johnson, Duncan Bowden, Giacomo Queirolo, Martin Millon, Pierre Fleury, Thomas E. Collett, Tian Li, Wolfgang J. R. Enzi. "Lessons from double-plane lenses for galaxy profiles, the Hubble tension, and the mass-sheet degeneracy." arXiv:2611.09856 (2026). https://pith.science/paper/DYGAXFJ3
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The paper

The paper claims

This paper shows that a double-plane gravitational lens makes a combination of both lens galaxies' mass-sheet components measurable from image positions alone, requiring only a prior on cosmic expansion history.

The reading

The mass-sheet degeneracy has long limited strong-lensing cosmology: an invisible uniform sheet of mass added to a lens galaxy changes inferred distances and the Hubble constant without changing any image. This paper's central claim is that a double-plane lens — where a foreground galaxy lenses both a background galaxy and a yet more distant source, with the intermediate galaxy itself acting as a second lens — makes a specific combination of the two mass-sheet components measurable from image positions alone. The measurement needs only a prior on the relative expansion history of the universe; it does not need time delays, stellar velocity dispersions, or a value of $H_0$. Applied to the 'Einstein zig-zag' lens J1721, the Jackpot lens J0946, and AGEL0353, the method yields values of the sheet combination that mildly disfavor both planes being pure power-law mass profiles. If the constraint survives deeper data, it opens a new route to galaxy mass profiles, population-level mass-sheet statistics, and sharper $H_0$ measurements from time-delay lenses.

What carries the argument

The central object is the double-plane mass-sheet transformation (Eqs. 19–20) combined with a flexible mass model $\kappa_{\rm lens}(\theta) = \kappa_{\rm mod}(\theta) + \zeta\kappa_c(\theta)$, in which the fiducial profile $\kappa_{\rm mod}$ is fixed by the images and the sheet term satisfies $\kappa_c \approx 1$ on the probed scales and drops off beyond them (Eq. 27). The transformation shows that modelling both planes as elliptical power laws implicitly sets the sheet in each plane to zero and shifts the cost into the measured cosmological scaling factor $\tilde{\eta}$. Solving the two 'no-sheet' conditions yields the identity $\zeta_{12} \equiv \eta\zeta_1 + \zeta_2 - \zeta_1\zeta_2$ (Eq. 35), which converts a comparison of measured and reference scaling factors into a constraint on the mass-sheet combination. The reference value combines the background ratio $\bar{\eta} = D_2D_{13}/(D_{12}D_3)$ with a line-of-sight correction, and is supplied by a cosmological prior on $\Omega_{\mathrm{m}}$ together with simulated line-of-sight convergences.

Extended reading notes

Core claim

Double-plane lenses do not truly break the mass-sheet degeneracy, but they convert it into a measurable quantity. When both planes are modelled with power-law profiles, the freedom to add mass sheets in either plane is absorbed into the model's inferred cosmological scaling factor $\tilde{\eta}$; comparing that $\tilde{\eta}$ with a reference $\eta$ obtained from cosmology and line-of-sight estimates then fixes the combination $\zeta_{12} \equiv \eta\zeta_1 + \zeta_2 - \zeta_1\zeta_2$ of the two planes' mass-sheet components (Eq. 35). Imaging alone does not fix either component individually, but it eliminates large regions of mass-profile space: a positive $\zeta_{12}$, for example, rules out both lenses being 'steepening' relative to a power law. The paper obtains $\zeta_{12} = -0.22^{+0.12}_{-0.13}$ for J1721, $0.052^{+0.022}_{-0.031}$ for J0946, and $-0.191^{+0.126}_{-0.087}$ for AGEL0353, each mildly disfavouring two pure power-law planes. With an auxiliary assumption — a prior on the population scatter of sheets, or a velocity-dispersion measurement in one plane — the individual sheets can be recovered.

Load-bearing premise

The load-bearing premise is the paper's stated working hypothesis that any mass-sheet-like structure in a lens galaxy can be represented by $\kappa_{\rm lens}(\theta) = \kappa_{\rm mod}(\theta) + \zeta\kappa_c(\theta)$ with $\kappa_c \approx 1$ over exactly the scales the images probe and dropping off outside them; if real baryonic or dark-matter structures vary on those scales instead of acting as a constant sheet, the inferred $\zeta$ values are not the physical mass-sheet components they are claimed to be.

Editorial extensions

If this is right

  • Image positions alone, plus an expansion-history prior, constrain $\zeta_{12}$ in any double-plane lens — no time delays, velocity dispersions, or $H_0$ prior are required (Section 4.3).
  • For time-delay lenses with a second source plane, the extra constraint improves $H_0$ precision by up to roughly 50% for high-$\eta$ systems, and by 20–50% across forecasted Euclid- and LSST-like systems when a 5% velocity-dispersion measurement in plane 2 is included.
  • Forecasted Euclid (~1729) and LSST (~87) double-plane samples would constrain the mean and scatter of the population mass-sheet distribution to within ~0.002 and ~0.01 respectively, under the stated population assumptions.
  • The three measured systems — J1721, J0946, and AGEL0353 — each mildly disfavour both lens planes being pure power-law mass profiles, so double-plane lenses can discriminate between profile shapes such as 'steepening' versus 'flattening' relative to a power law.
  • Systems with $\eta$ well above 1 are the most constraining, and in wide-separation systems like J0946 the required sheet would have to extend far beyond the Einstein radius, connecting the mass-sheet measurement to the radial scales of dark-matter halos.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the paper leaves implicit: if survey samples deliver the forecasted sheet-population constraints, single-plane time-delay $H_0$ analyses could adopt the measured distribution as a calibrated prior, turning the mass-sheet degeneracy from a systematic into a fitted parameter.
  • Because the interpretation assumes sheets act as constant convergences across the image scales, wide-separation double-plane systems double as experiments on the radial extent of sheet-like structures, which could discriminate dark-matter models that predict extended cores.
  • A near-term check: adding J1721's extended arcs and the paper's own 9.1 mas millilensing astrometric estimate — which the paper notes leaves $\tilde{\eta}$ effectively unconstrained — will show whether the current $\zeta_{12}$ central value survives or dissolves.
  • Systems with more than two lens planes would provide internal consistency tests, since each pair of planes yields a $\zeta_{12}$ that must share the same $\zeta_1$ for the foreground plane.
Original abstract

Double-plane lenses, like single-plane lenses, are subject to the mass-sheet degeneracy, a fundamental limit to our knowledge of the deflector mass profiles from lensing. However, with a prior on the relative expansion history, unlike in the single-plane case, this degeneracy can be constrained with the modelling of lens images, without the need for time delays, velocity dispersion measurements or a prior on the Hubble constant $H_0$. We show that double-plane lenses are sensitive to a specific combination of the mass-sheet components of the two lens planes. Using literature values of the cosmological scaling factor from the double-plane lenses J0946 and AGEL0353, a measurement from the point-source images of J1721 (the ``Einstein zig-zag'' lens), and a reference value of $\Omega_\mathrm{m}$, we show the constraints that can be placed on the mass-sheet component of each of these systems. We show that the additional information brought by images of a second source can improve measurements of $H_0$ from individual time-delay lenses by up to $\sim50\%$, depending on certain key assumptions. Using forecasted Euclid and LSST double-plane lens samples, we show that these samples will enable strong constraints on the population of mass sheets in lens galaxies. Such population constraints will have implications for a range of strong-lensing science applications, in particular in constraining $H_0$, dark energy, and the dark matter profiles of galaxies.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-09: 1 request, 5,893 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.69 · human participants 0.01 · controlled comparison 0.21 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.79 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Double-plane lenses, like single-plane lenses, are subject to the mass-sheet degeneracy, a fundamental limit to our knowledge of the deflector mass profiles from lensing. However, with a prior on the relative expansion history, unlike in the single-plane case, this degeneracy can be constrained with the modelling of lens images, without the need for time delays, velocity dispersion measurements or a prior on the Hubble constant $H_0$. We show that double-plane lenses are sensitive to a specific combination of the mass-sheet components of the two lens planes. Using literature values of the cosmological scaling factor from the double-plane lenses J0946 and AGEL0353, a measurement from the point-source images of J1721 (the ``Einstein zig-zag'' lens), and a reference value of $\Omega_\mathrm{m}$, we show the constraints that can be placed on the mass-sheet component of each of these systems. We show that the additional information brought by images of a second source can improve measurements of $H_0$ from individual time-delay lenses by up to $\sim50\%$, depending on certain key assumptions. Using forecasted Euclid and LSST double-plane lens samples, we show that these samples will enable strong constraints on the population of mass sheets in lens galaxies. Such population constraints will have implications for a range of strong-lensing science applications, in particular in constraining $H_0$, dark energy, and the dark matter profiles of galaxies.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Double-plane lenses, like single-plane lenses, are subject to the mass-sheet degeneracy, a fundamental limit to our knowledge of the deflector mass profiles from lensing. background 1.00 0.07 0.07 asserted 1.00
s1 However, with a prior on the relative expansion history, unlike in the single-plane case, this degeneracy can be constrained with the modelling of lens images, without the need for time delays, velocity dispersion measurements or a prior on the Hubble constant $H_0$. contribution 0.52 0.27 0.51 asserted 0.77
s2 We show that double-plane lenses are sensitive to a specific combination of the mass-sheet components of the two lens planes. result 0.90 0.80 0.18 asserted 1.00
s3 Using literature values of the cosmological scaling factor from the double-plane lenses J0946 and AGEL0353, a measurement from the point-source images of J1721 (the ``Einstein zig-zag'' lens), and a reference value of $\Omega_\mathrm{m}$, we show the constraints that can be placed on the mass-sheet component of each of these systems. method 0.51 0.88 0.09 asserted 1.00
s4 We show that the additional information brought by images of a second source can improve measurements of $H_0$ from individual time-delay lenses by up to $\sim50\%$, depending on certain key assumptions. result 0.99 0.84 0.19 asserted 0.92
s5 Using forecasted Euclid and LSST double-plane lens samples, we show that these samples will enable strong constraints on the population of mass sheets in lens galaxies. result 0.63 0.88 0.18 asserted 0.95
s6 Such population constraints will have implications for a range of strong-lensing science applications, in particular in constraining $H_0$, dark energy, and the dark matter profiles of galaxies. interpretation 1.00 0.30 0.16 hedged possibility 0.57

What kind of paper the classifier reports this is: theoretical result 0.60 · evidence from analysis of existing data 0.87.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. — The geometry of double-plane lensing with line-of-sight perturbers (shown in green) and internal mass-sheet components (shown in gray). Light originating from a source in plane 3 experiences additional (de)focusing by the LOS perturbers, which we model as pure convergence for the purposes of this paper. In plane 1, and possibly in plane 2 as well, it is strongly lensed. absence of the planes between o and l) via the (back￾ground) angular diameter distance Dl to plane l. These angular positions θl are determined recursively from po￾sitions θm and deflection angles αˆm, where m < l…
Figure 2. — Illustrative possibilities for the mass profiles in planes 1 and 2 of a double-plane lens, assuming η = 1.021 (our fiducial value for J1721), ζ12 = 0 and both planes have been modelled with a power law mass profile, following Eq. (29), (31) and (35). If, in reality, plane 1 features a positive mass-sheet component, ζ1 > 0, which would imply a log-convex profile which grows increasingly flat as θ increases relative to the modelled power law, the profile in plane 2 must be log-concave, growing increasingly steep as θ increases (the scenario in red). If instead plane 1 features a…
Figure 3. The solid curved lines correspond to different measurements of ζ12. From imaging constraints alone, we eliminate large portions of the (ζ1, ζ2) parameter space - for example, a positive value of ζ12 is incompatible with both lenses being steepening compared to a power law, and if ζ12 = 0, the lenses must both be power laws, or have opposite-sign mass-sheet components. If the profiles are restricted to be similar, i.e. as having a limited dis￾persion in ζi , then we fix ourselves to lie near the black dotted line, and can thus use our constraint on ζ12 to constrain the profile of bot
Figure 4. — What can be learnt from a double-plane system with time delays. A H˜0 measurement is sensitive to H0 and the mass profile in the foreground plane, while a ˜η measurement is sensitive to the relative expansion history and the mass profiles in both planes. If you trust a certain fiducial cosmology, this has implications for the mass profiles of the two lens galaxies
Figure 5. , is a spectacular compound lens, featuring a sex￾tuply imaged quasar at z = 2.382 whose light is deflected by galaxies at redshifts z1 = 0.184 and z2 = 1.885, with light from the intermediate galaxy at z2 also lensed to multiple extended arcs by the first lens (Lemon et al. 2018; Dux et al. 2025). The variability of the quasar and position of the system in the sky have enabled time-delay measurements between all 6 images, and the exceedingly rare image configuration is highly sensitive to the mass profiles of both deflectors (Collett and Bacon 2016). In the following, we describe…
Figure 6. — Estimated line-of-sight contribution to the uncertainty in η, κη, and for a hypothetical H0 measurement, κ LOS 12 , for J1721, using the methods described in subsection 5.3. positions, we increase the astrometric uncertainty to 2 mas, to account for the additional uncertainty from millilensing by substructures. We return to this point in subsection 5.6. 5.2. Lens model We model the foreground lens galaxy with an EPL+shear at z1 = 0.1840, with an SIS for the back￾ground lens galaxy at z2 = 1.8845. The 4 perturbing galaxies identified in Schmidt et al. (2025) are modelled with SIS…
Figure 7. — Constraints on the mass-sheet components of J1721. In blue, we show the constraints from ζ12, which is a comparison between our ˜η measurement, a fiducial Ωm value from Planck Col￾laboration et al. (2020), and an error budget for the line of sight. In purple, we show the the constraints on the mass-sheet population scatter from subsection 5.4 for the lenses in Millon et al. (2020), using and the assumption that the lens planes in J1721 are drawn from this same population. We show the joint constraints in red. this with the measured ˜η, we obtain ζ12 = −0.22+0.12 −0.13. The…
Figure 8. — Hypothetical constraints for J1721 with a 3% velocity dispersion measurement in plane 1 and a 5% measurement in plane 2, with our measurement of ˜η = 1.0166+0.0011 −0.0011 and our population dispersion constraint from subsection 5.4 shown in purple, and the joint constraints in red. stated above, this estimate is higher than that for most systems, and comes from the highly sensitive 6-image configuration of J1721. With this value, however, ˜η is effectively unconstrained. We therefore retain the more typical 2mas estimate, but emphasise once again that our results should therefore…
Figure 9. — HST images of the DPLs studied in section 6: the “Jackpot” lens J0946 (left) and AGEL0353 (right). AGEL0353 was imaged using HST’s Wide Field Camera 3 with the F140W (red) and F200LP (blue) filters by Barone et al. (2025), the cutout for which is 6′′ in width. 6. APPLICATION TO PREVIOUSLY-STUDIED SYSTEMS Although the methods in this paper are novel, con￾straining ζ12 is convenient in that it comes “for free” with a measurement of ˜η, a cosmological prior and a line-of￾sight estimate, without the need for any additional lens modelling. In this section, we will consider two systems…
Figure 11. — Constraints on the mass-sheet components of J0946 and AGEL0353. In the left column, we show the constraints on ζ¯ 1, ζ¯ 2  from ζ12, i.e. a comparison between the measured ˜η values from Ballard et al. (2026); Bowden et al. (2025) and a reference Ωm value from Planck Collaboration et al. (2020), with an additional error budget for the line of sight, in blue. In purple, we show the constraints coming from our measured value of σζ¯ from subsection 5.4, which assumes that any mass-sheet components in the lenses in Millon et al. (2020) and the two DPL lens planes are drawn from the…
Figure 12. , we illustrate the reconstruction of the first source in J0946, as well as the light from the second source as it intersects this plane, en route to forming the image we observe in
Figure 13. — The sensitivity of ζ constraints to the η of a lens system. On the left, we show the slope implied by a ζ12 value, Eq. (35), in the (ζ1, ζ2) parameter space, for different values of the cosmological scaling factor η. As η increases, small changes in ζ1, the mass sheet in plane 1, correspond to larger and larger changes in ζ2, the mass sheet in plane 2. On the right, we show the 1σ uncertainty in ζ12 as a function of η, with no contribution from the line of sight. Different values of ση˜, the measurement uncertainty in the inferred cosmological scaling factor ˜η, are plotted as…
Figure 14. — Forecasted ζ12 constraining power of Euclid and LSST double-plane lenses. The vertical axis σζ12 is the 1σ uncertainty of the inferred ζ12 value, plotted against the true cosmological scaling factor η on the x-axis. The colours of the points correspond to the uncertainty in the fiducial η value, coming both from the uncertainty in cosmological parameters and that from the line of sight. Constraints on ζ12 are obtained by comparing these fiducial values to measured ˜η values and their forecasted uncertainties
Figure 15. — The 1σ constraining power of a sample of forecasted Euclid (in green) and LSST (in orange) DPLs on the mean and intrinsic scatter in the mass sheet parameter ζ, from ζ12 alone. For the solid, filled contours, we assume that the mass-sheet com￾ponents in the first and second planes of both galaxies are drawn from the same population, described by a Gaussian with mean µζ˜ and standard deviation σζ˜. For the dotted, empty contours, we assume that the mass-sheets in the first lens plane are described by this population, but that the impact of the mass-sheet in the second plane is…
Figure 16. — Forecasted H0 constraints with a single “Jackpot”-like lens system with time delays in one of the source planes. We assume a 2% contribution each from lens modelling (excluding the mass￾sheet) and time delays to this uncertainty. In blue, we imagine a 3% measurement of V1, in purple we imagine no velocity dispersion measurements, but a ζ12 constraint matching that in section 6 and a σζ¯ matching that in subsection 5.4. In red, we imagine all of these constraints, plus a 5% measurement of V2 from plane 2. prior (or the assumption that ˜ζ2 = 0) would provide ad￾ditional constraints…
Figure 17. — The benefit of the second plane in hypothetical Euclid and LSST-like double-plane time-delay lenses, under the two simple scenarios considered in subsection 7.2. On the vertical axis, we plot the ratio of the uncertainty H0 precision when constraints derived from the second source are added to that when they are not, with the cosmological scaling factor on the horizontal axis. The colours of the points correspond to the forecasted precision with which σζ12 could be constrained. The slight scatter around 1 for η ≈ 1 is simply a result of sampling noise in our forecast. ulation…
Figure 18. — The posterior for the fractional scatter in ζ for the sample of lenses in subsection C.2 from Millon et al. (2020). In red, we show the 1σ upper bound we can place on this scatter. REFERENCES S. Refsdal, Monthly Notices of the Royal Astronomical Society 128, 307 (1964), https://academic.oup.com/mnras/article/128/4/307/2601707. S. H. Suyu, P. J. Marshall, M. W. Auger, S. Hilbert, R. D. Blandford, L. V. E. Koopmans, C. D. Fassnacht, and T. Treu, The Astrophysical Journal 711, 201 (2010), 0910.2773. S. Birrer, M. Millon, D. Sluse, A. J. Shajib, F. Courbin, L. V. E. Koopmans, S. H. Su

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