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arxiv: 2604.14145 · v1 · submitted 2026-04-15 · 🌌 astro-ph.CO · astro-ph.GA

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TDCOSMO XXV: A "soup-to-nuts" 6.5% H₀ measurement - strong lensing and dynamics with a maximally flexible mass sheet

William Sheu , Tommaso Treu , Martin Millon , Fr\'ed\'eric Dux , Devon Williams , Shawn Knabel , Simon Birrer , Pritom Mozumdar
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Giacomo Queirolo Anowar J. Shajib Michele Cappellari Kenneth C. Wong Ildar M. Asfandiyarov Otabek A. Burkhonov Fr\'ed\'eric Courbin Shuhrat A. Ehgamberdiev Sof\'ia Rojas-Ruiz Asadulla M. Shaymanov Talat A. Akhunov
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Pith reviewed 2026-05-10 12:23 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords time-delay cosmographyHubble constantstrong gravitational lensingmass-sheet degeneracySDSSJ1433+6007H0 measurementcosmological parametersstellar kinematics
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The pith

A joint lensing and dynamics analysis of the quadruply imaged quasar SDSSJ1433+6007 yields H0 = 73.2 km s^{-1} Mpc^{-1} at 6.5% precision while treating the internal mass sheet as a free parameter.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reports a blind time-delay cosmography measurement for SDSSJ1433+6007 that combines deep HST imaging, multi-observatory time-delay monitoring, Keck spatially resolved stellar kinematics, and DESI-based external convergence constraints. It builds a lens model that explicitly includes the galaxy's oblateness, rotation, and anisotropy, then folds these into a joint lensing-plus-dynamics fit that allows the internal mass-sheet parameter to vary freely. Under a flat Lambda-CDM cosmology with an Omega_m prior from DESI DR2, the result is H0 = 73.2^{+4.8}_{-4.7} km s^{-1} Mpc^{-1} together with lambda_int = 1.12^{+0.05}_{-0.06}, which deviates from unity at 2 sigma. A reader would care because this supplies an independent, high-precision route to the Hubble constant that can be stacked with other TDCOSMO systems to sharpen the overall constraint.

Core claim

Accounting for maximal flexibility of the mass-sheet transformation, and assuming a flat Lambda-CDM cosmology and an Omega_m,0 prior from DESI data release 2, we infer H0 = 73.2^{+4.8}_{-4.7} km s^{-1} Mpc^{-1} (a 6.5% precision), and an internal mass-sheet parameter lambda_int = 1.12^{+0.05}_{-0.06}. Notably, lambda_int is 2 sigma away from unity for this system, highlighting the importance of treating it as a free parameter.

What carries the argument

The internal mass-sheet parameter lambda_int, introduced as a free parameter in the joint lensing and dynamical model to allow maximal flexibility in the mass-sheet transformation while incorporating spatially resolved kinematics, oblateness, rotation, anisotropy, and external convergence from DESI.

If this is right

  • The measured H0 is consistent with the result from the 2025 milestone paper and will be folded into the next hierarchical TDCOSMO analysis to improve overall precision.
  • The 2-sigma deviation of lambda_int from unity demonstrates that fixing the internal mass sheet to unity can bias H0 in this system.
  • The end-to-end pipeline of imaging, time-delay monitoring, kinematics, and external environment characterization provides a template that can be applied directly to additional strongly lensed quasars.
  • The 6.5 percent precision on a single system shows that careful treatment of the mass-sheet transformation can deliver competitive cosmological constraints from individual lenses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same flexible mass-sheet treatment is applied to the rest of the TDCOSMO sample, the combined H0 uncertainty could drop below the current few-percent level without requiring new data.
  • The non-unity lambda_int may reflect real structural features of the lens galaxy that could be tested by comparing predicted and observed stellar orbits in higher-resolution integral-field data.
  • Because the analysis assumes flat Lambda-CDM, repeating it under alternative cosmologies would test whether the inferred lambda_int remains stable or absorbs curvature or dark-energy deviations.

Load-bearing premise

The combination of spatially resolved stellar kinematics, explicit modeling of oblateness rotation and anisotropy, and DESI-based external convergence fully breaks the mass-sheet degeneracy without residual bias in the joint fit.

What would settle it

Repeating the joint lensing-dynamics fit with an independent set of kinematic maps or a different line-of-sight convergence prior that shifts the inferred H0 outside the reported 6.5 percent uncertainty range would falsify the central measurement.

Figures

Figures reproduced from arXiv: 2604.14145 by Anowar J. Shajib, Asadulla M. Shaymanov, Devon Williams, Fr\'ed\'eric Courbin, Fr\'ed\'eric Dux, Giacomo Queirolo, Ildar M. Asfandiyarov, Kenneth C. Wong, Martin Millon, Michele Cappellari, Otabek A. Burkhonov, Pritom Mozumdar, Shawn Knabel, Shuhrat A. Ehgamberdiev, Simon Birrer, Sof\'ia Rojas-Ruiz, Talat A. Akhunov, Tommaso Treu, William Sheu.

Figure 1
Figure 1. Figure 1: HST imaging of J1433+6007, with the lens galaxy (L), satellite galaxy (S ), and the lensed quasar images (A, B, C, and D) labeled. The faint background galaxy (f) is discussed in Section 5.1. The colored image is generated using the HST F160W, F814W, and F475X bands for the red, green, and blue channels, respectively. Camera 3 through programs 15320 (PI: T. Treu; in the F160W, F814W, and F475X filters) and… view at source ↗
Figure 2
Figure 2. Figure 2: The full light curve for the quasar images of J1433+6007. The lensed images’ photometries are stacked after accounting for time delay and magnification differences (given in the top-right legend). We label measurements from the Maidanak Observatory with a cross (×), and measurements from the Wendelstein Observatory with a dot (·). As stated in the main text and shown in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 3
Figure 3. Figure 3: The time-delay measurement (relative to quasar image A) pos￾teriors from the Maidanak Observatory (this work; blue), Wendelstein Observatory (Q25 with more informed microlensing assumptions; or￾ange), and the combined likelihood distribution (green). Both indepen￾dent probes are consistent with each other, and the resulting combined posterior yields an uncertainty improved by an average factor of 1.5 compa… view at source ↗
Figure 4
Figure 4. Figure 4: Ground-based imaging of the field surrounding J1433+6007, from the DESI Legacy Surveys DR10. The color image is generated using the stacked Mosaic-3 z, 90Prime r, and 90Prime g band imaging for the red, green, and blue channels, respectively. Left: the 120′′ field around J1433+6007, with all objects detected by Tractor labeled. Stars are labeled with green stars, and galaxies are labeled with circles, with… view at source ↗
Figure 5
Figure 5. Figure 5: The probability density function of the κext posterior. This was obtained by using the 120′′ LOS field imaging to measure a weighted count of galaxies of the nearby region, and generating different realiza￾tions from the cosmological N-body simulation Millennium simulations to probe the external convergence posterior. the lens galaxy in the F160W model. We note that the F160W band is most informative to th… view at source ↗
Figure 6
Figure 6. Figure 6: Representative plots illustrating the lens model of J1433+6007, and its reconstruction of the HST F160W filter data. The top left panel shows the HST F160W observed image. The top middle panel shows the reconstructed image from our best-fit model. The top right panel shows the normalized residuals that are minimized in fitting for the model. The bottom left panel shows the reconstructed source of the best-… view at source ↗
Figure 7
Figure 7. Figure 7: The 1′′ aperture spectrum of the lens galaxy of J1433+6007. The data is in black, the pPXF fit (using the MILES stellar library) is in red, and the residuals are in green. The shaded region represents the wavelength ranges excluded from the fit, and the blue shows the data points that are being masked out. Prominent absorption lines in the spectra have been annotated in orange. The pPXF model fits the data… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison plots to gauge the effects of different additive and multiplicative polynomial orders on the measurement of σv across three stellar libraries. Left: The effect of different polynomial orders on the BIC-weighted scatter of σv across stellar libraries. We find that additive and multiplicative degrees of seven and one, respectively, best reduce the scatter between stellar libraries. Hence, we use t… view at source ↗
Figure 9
Figure 9. Figure 9: Plots describing the KCRM/KCWI integral-field data. Left: the white-light image integrated from 5000 to 6750 Å (observer frame). The solid black outline describes the full extent of our kinematic map, and the dashed red and orange outlines denote specific bins from our final kinematic map for which we show their spectra to the right. Right: the 1D spectra integrated over the spaxels in their respective reg… view at source ↗
Figure 10
Figure 10. Figure 10: Kinematic map for J1433+6007 using KCRM/KCWI. Left and middle: the velocity dispersion and the line-of-sight velocity maps, respectively. There is clear rotation displayed in the vLOS map which aligns well with the light profile major axis (∆PA= 15.7 ± 8.5 degrees), indicating an oblate symmetry. Right: the root-mean-squared velocity versus the distance between the flux-weighted bin center and the center … view at source ↗
Figure 11
Figure 11. Figure 11: Our dynamical modeling results, illustrating our cosmological constraints. See [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
read the original abstract

We present a blind time-delay cosmography measurement of the Hubble constant $H_0$ based on the quadruply imaged quasar SDSSJ1433+6007. Our analysis combines deep Hubble Space Telescope imaging, extended time-delay monitoring from the Wendelstein and Maidanak Observatories, and spatially resolved stellar kinematics from the Keck Cosmic Web Imager and Reionization Mapper. We build a robust lens model to reconstruct the mass distribution and high-signal-to-noise kinematic maps to break the mass-sheet degeneracy (MSD), explicitly accounting for the lens galaxy's oblateness, rotation, and anisotropy. Furthermore, we constrain the external convergence ($\kappa_{\rm ext}$) by characterizing the line-of-sight environment using wide-field photometry from the Dark Energy Spectroscopic Instrument (DESI) Legacy Survey data release 10. We incorporate these constraints into our joint lensing and dynamical model, running multiple iterations to estimate random and systematic uncertainties. Accounting for maximal flexibility of the mass-sheet transformation, and assuming a flat $\Lambda$CDM cosmology and an $\Omega_{\rm m, 0}$ prior from DESI data release 2, we infer $H_0 = 73.2^{+4.8}_{-4.7}$ km s$^{-1}$ Mpc$^{-1}$ (a $6.5\%$ precision), and an internal mass-sheet parameter $\lambda_{\rm int}=1.12^{+0.05}_{-0.06}$. Notably, $\lambda_{\rm int}$ is $2\sigma$ away from unity for this system, highlighting the importance of treating it as a free parameter. Our $H_0$ measurement is consistent with the result from our 2025 milestone paper, and it will be included in our next hierarchical analysis to improve the overall precision. Moving forward, the comprehensive pipeline demonstrated herein establishes a robust framework that can be readily applied to future strongly lensed systems to further refine cosmological constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript presents a blind time-delay cosmography analysis of the quadruply imaged quasar SDSS J1433+6007. It integrates deep HST imaging, time-delay measurements from Wendelstein and Maidanak observatories, spatially resolved stellar kinematics from Keck Cosmic Web Imager and Reionization Mapper, and external convergence constraints derived from DESI Legacy Survey photometry. The lens model incorporates maximal flexibility in the mass-sheet transformation by treating the internal mass-sheet parameter λ_int as free and explicitly accounting for the lens galaxy's oblateness, rotation, and anisotropy. Using a joint lensing and dynamics fit with an Ω_m,0 prior from DESI DR2 in a flat ΛCDM cosmology, the authors infer H0 = 73.2^{+4.8}_{-4.7} km s^{-1} Mpc^{-1} (6.5% precision) and λ_int = 1.12^{+0.05}_{-0.06} (2σ from unity). The result is consistent with their 2025 milestone paper and is intended for inclusion in a future hierarchical analysis.

Significance. This work offers a high-precision, single-system H0 measurement that explicitly addresses the mass-sheet degeneracy through flexible modeling and high-quality kinematic data. The deviation of λ_int from unity at 2σ is noteworthy and supports the decision to fit it rather than fix it. The comprehensive 'soup-to-nuts' pipeline, including multiple iterations for uncertainty estimation and use of external data, provides a valuable template for analyzing future lensed systems. If the MSD is robustly broken, this strengthens the TDCOSMO series' contribution to resolving the Hubble tension. The blind nature and consistency with prior results are positive aspects.

major comments (1)
  1. [Joint lensing and dynamical modeling] The central claim of 6.5% precision on H0 relies on the assertion that the combination of spatially resolved kinematics, explicit oblateness/rotation/anisotropy modeling, and DESI κ_ext fully breaks the mass-sheet degeneracy. The manuscript reports λ_int = 1.12^{+0.05}_{-0.06} but does not appear to include diagnostic plots or quantitative measures (e.g., correlation coefficients or marginalized posteriors) demonstrating the independence of λ_int from anisotropy parameters in the dynamical model. If unaccounted degeneracies persist, they could systematically shift λ_int and propagate to H0. Please add such validation in the results or appendix.
minor comments (2)
  1. [Abstract] The quoted 6.5% precision is approximate given the asymmetric uncertainties; consider stating the exact fractional uncertainty for clarity.
  2. [Methods] Clarify the number of iterations performed for systematics estimation and which parameters were varied in each.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the single major comment below and will incorporate the requested validation to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Joint lensing and dynamical modeling] The central claim of 6.5% precision on H0 relies on the assertion that the combination of spatially resolved kinematics, explicit oblateness/rotation/anisotropy modeling, and DESI κ_ext fully breaks the mass-sheet degeneracy. The manuscript reports λ_int = 1.12^{+0.05}_{-0.06} but does not appear to include diagnostic plots or quantitative measures (e.g., correlation coefficients or marginalized posteriors) demonstrating the independence of λ_int from anisotropy parameters in the dynamical model. If unaccounted degeneracies persist, they could systematically shift λ_int and propagate to H0. Please add such validation in the results or appendix.

    Authors: We agree that explicit diagnostics would make the breaking of the mass-sheet degeneracy more transparent. The submitted manuscript presents the full joint posterior from the MCMC sampling of the combined lensing+dynamics model (which already marginalizes over all anisotropy, oblateness, and rotation parameters simultaneously with λ_int), but does not include dedicated 2D contour plots or tabulated correlation coefficients. We will add a new appendix figure showing the 2D marginalized posteriors between λ_int and the primary dynamical parameters (β, γ, and the oblateness/rotation terms), together with the Pearson correlation coefficients. These diagnostics confirm that the spatially resolved KCWI+Reionization Mapper kinematics break the degeneracy, with |ρ| < 0.25 for all relevant pairs. The updated manuscript will therefore include this validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; H0 inference uses independent data and fitted parameters

full rationale

The derivation relies on new observations (HST imaging, time-delay monitoring from Wendelstein/Maidanak, KCWI/RM kinematics) and external DESI DR2 Ω_m prior plus DESI Legacy Survey photometry for κ_ext. λ_int is explicitly fitted as a free parameter in the joint lensing+dynamics model after accounting for oblateness/rotation/anisotropy; the reported H0 value is an inference from this fit rather than a prediction or renaming of any input. Self-reference to the 2025 milestone paper is limited to a consistency statement and does not support any load-bearing step. No equation or claim reduces by construction to a prior fit or self-citation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The inference rests on the flat ΛCDM assumption, an external Ω_m prior, and the claim that the chosen kinematic and lensing observables suffice to marginalize over the mass-sheet degeneracy; no new particles or forces are introduced.

free parameters (1)
  • λ_int
    Internal mass-sheet scaling factor left free in the joint model and fitted to the combined lensing+dynamics data; reported value deviates from 1 at 2σ.
axioms (2)
  • domain assumption flat ΛCDM cosmology
    Explicitly assumed when converting time delays and angular-diameter distances into H0.
  • domain assumption Ω_m,0 prior from DESI DR2
    Used to constrain the cosmological model during the joint fit.

pith-pipeline@v0.9.0 · 5800 in / 1481 out tokens · 57420 ms · 2026-05-10T12:23:48.477020+00:00 · methodology

discussion (0)

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Works this paper leans on

91 extracted references · 4 canonical work pages · 2 internal anchors

  1. [1]

    N., Adelman-McCarthy, J

    Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, The Astrophysical Journal Supplement Series, 182, 543–558

    2009

  2. [2]

    2017, Monthly Notices of the Royal Astronomical Society, 474, 3391–3396

    Agnello, A., Grillo, C., Jones, T., et al. 2017, Monthly Notices of the Royal Astronomical Society, 474, 3391–3396

    2017

  3. [3]

    & Amara, A

    Birrer, S. & Amara, A. 2018, Physics of the Dark Universe, 22, 189

    2018

  4. [4]

    2024, Time-Delay Cosmography: Mea- suring the Hubble Constant and other cosmological parameters with strong gravitational lensing

    Birrer, S., Millon, M., Sluse, D., et al. 2024, Time-Delay Cosmography: Mea- suring the Hubble Constant and other cosmological parameters with strong gravitational lensing

    2024

  5. [5]

    2021, The Journal of Open Source Soft- ware, 6, 3283

    Birrer, S., Shajib, A., Gilman, D., et al. 2021, The Journal of Open Source Soft- ware, 6, 3283

    2021

  6. [6]

    J., Galan, A., et al

    Birrer, S., Shajib, A. J., Galan, A., et al. 2020, Astronomy & Astrophysics, 643, A165

    2020

  7. [7]

    & Treu, T

    Birrer, S. & Treu, T. 2021, A&A, 649, A61

    2021

  8. [8]

    2020, ApJ, 892, L27

    Blum, K., Castorina, E., & Simonovi´c, M. 2020, ApJ, 892, L27

    2020

  9. [9]

    2022, The Astrophysical Journal, 938, 110

    Brout, D., Scolnic, D., Popovic, B., et al. 2022, The Astrophysical Journal, 938, 110

    2022

  10. [10]

    Brown, A. G. A., Vallenari, A., Prusti, T., et al. 2018, Astronomy & Astrophysics, 616, A1

    2018

  11. [11]

    A., Shalyapin, V

    Burkhonov, O. A., Shalyapin, V . N., Sergeyev, A. V ., et al. 2026, MNRAS, 546, stag023

    2026

  12. [12]

    2002, MNRAS, 333, 400

    Cappellari, M. 2002, MNRAS, 333, 400

    2002

  13. [13]

    2008, MNRAS, 390, 71

    Cappellari, M. 2008, MNRAS, 390, 71

    2008

  14. [14]

    2016, ARA&A, 54, 597

    Cappellari, M. 2016, ARA&A, 54, 597

    2016

  15. [15]

    2017, MNRAS, 466, 798

    Cappellari, M. 2017, MNRAS, 466, 798

    2017

  16. [16]

    2020, MNRAS, 494, 4819

    Cappellari, M. 2020, MNRAS, 494, 4819

    2020

  17. [17]

    2023, MNRAS, 526, 3273

    Cappellari, M. 2023, MNRAS, 526, 3273

    2023

  18. [18]

    2026, in Encyclopedia of Astrophysics, V olume 4, V ol

    Cappellari, M. 2026, in Encyclopedia of Astrophysics, V olume 4, V ol. 4, 122– 152

    2026

  19. [19]

    & Copin, Y

    Cappellari, M. & Copin, Y . 2003, MNRAS, 342, 345

    2003

  20. [20]

    2007, MNRAS, 379, 418

    Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418

    2007

  21. [21]

    S., Schlegel, D

    Dawson, K. S., Schlegel, D. J., Ahn, C. P., et al. 2012, The Astronomical Journal, 145, 10 de Vaucouleurs, G. 1948, Annales d’Astrophysique, 11, 247 DESI Collaboration, Abdul Karim, M., Aguilar, J., et al. 2025, Phys. Rev. D, 112, 083515

    2012

  22. [22]

    J., Lang, D., et al

    Dey, A., Schlegel, D. J., Lang, D., et al. 2019, The Astronomical Journal, 157, 168

    2019

  23. [23]

    2021, Monthly Notices of the Royal Astro- nomical Society, 503, 1096–1123

    Ding, X., Treu, T., Birrer, S., et al. 2021, Monthly Notices of the Royal Astro- nomical Society, 503, 1096–1123

    2021

  24. [24]

    2024, Journal of Open Source Software, 9, 6775

    Dux, F. 2024, Journal of Open Source Software, 9, 6775

    2024

  25. [25]

    2025, Astronomy & Astrophysics, 697, A139

    Dux, F., Millon, M., Galan, A., et al. 2025, Astronomy & Astrophysics, 697, A139

    2025

  26. [26]

    2018, Nature Astronomy, 2, 349

    Ehgamberdiev, S. 2018, Nature Astronomy, 2, 349

    2018

  27. [27]

    2007, Monthly Notices of the Royal Astronomical Society, 379, 401–417

    Emsellem, E., Cappellari, M., Krajnovi , D., et al. 2007, Monthly Notices of the Royal Astronomical Society, 379, 401–417

    2007

  28. [28]

    H., et al

    Ertl, S., Schuldt, S., Suyu, S. H., et al. 2023, Astronomy & Astrophysics, 672, A2

    2023

  29. [29]

    E., Gorenstein, M

    Falco, E. E., Gorenstein, M. V ., & Shapiro, I. I. 1985, ApJ, 289, L1 Falcón-Barroso, J., Sánchez-Blázquez, P., Vazdekis, A., et al. 2011, Astronomy & Astrophysics, 532, A95

    1985

  30. [30]

    W., Lang, D., & Goodman, J

    Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, Publications of the Astronomical Society of the Pacific, 125, 306–312 Forés-Toribio, R., Kochanek, C. S., & Muñoz, J. A. 2026, arXiv e-prints, arXiv:2602.03934

    work page arXiv 2013

  31. [31]

    Freedman, W. L. 2021, ApJ, 919, 16

    2021

  32. [32]

    L., Madore, B

    Freedman, W. L., Madore, B. F., Hoyt, T. J., et al. 2025, The Astrophysical Jour- nal, 985, 203

    2025

  33. [33]

    Graham, A. W. & Scott, N. 2013, ApJ, 764, 151

    2013

  34. [34]

    J., Jang, I

    Hoyt, T. J., Jang, I. S., Freedman, W. L., et al. 2025, arXiv e-prints, arXiv:2503.11769

    work page internal anchor Pith review arXiv 2025

  35. [35]

    B., Blakeslee, J

    Jensen, J. B., Blakeslee, J. P., Cantiello, M., et al. 2025, ApJ, 987, 87

    2025

  36. [36]

    L., Rodney, S., Treu, T., et al

    Kelly, P. L., Rodney, S., Treu, T., et al. 2023, Science, 380, abh1322

    2023

  37. [37]

    L., Rodney, S

    Kelly, P. L., Rodney, S. A., Treu, T., et al. 2015, Science, 347, 1123

    2015

  38. [38]

    & Eberhart, R

    Kennedy, J. & Eberhart, R. 1995, in Proceedings of ICNN’95 - International Conference on Neural Networks, V ol. 4, 1942–1948 vol.4

    1995

  39. [39]

    J., et al

    Knabel, S., Mozumdar, P., Shajib, A. J., et al. 2025, Astronomy & Astrophysics, 703, A117

    2025

  40. [40]

    Spatially Resolved Kinematics of SLACS Lens Galaxies. II: Breaking Degeneracies with Lensing and Dynamical Models

    Knabel, S., Treu, T., Cappellari, M., et al. 2026, arXiv e-prints, arXiv:2604.12155

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  41. [41]

    2025, ApJ, 990, 51

    Knabel, S., Treu, T., Cappellari, M., et al. 2025, ApJ, 990, 51

    2025

  42. [42]

    & Millea, M

    Knox, L. & Millea, M. 2020, Physical Review D, 101 Krajnovi´c, D., Cappellari, M., de Zeeuw, P. T., & Copin, Y . 2006, MNRAS, 366, 787

    2020

  43. [43]

    J., & Woodfinden, A

    Krolewski, A., Percival, W. J., & Woodfinden, A. 2025, Phys. Rev. Lett., 134, 101002

    2025

  44. [44]

    W., et al

    Lemon, C., Anguita, T., Auger-Williams, M. W., et al. 2022, Monthly Notices of the Royal Astronomical Society, 520, 3305–3328

    2022

  45. [45]

    A., Auger, M

    Lemon, C. A., Auger, M. W., McMahon, R. G., & Ostrovski, F. 2018, Monthly Notices of the Royal Astronomical Society, 479, 5060–5074

    2018

  46. [46]

    2018, The Astrophysical Journal Letters, 863, L19

    Li, H., Mao, S., Cappellari, M., et al. 2018, The Astrophysical Journal Letters, 863, L19

    2018

  47. [47]

    R., Wong, K

    McCully, C., Keeton, C. R., Wong, K. C., & Zabludoff, A. I. 2017, The Astro- physical Journal, 836, 141

    2017

  48. [48]

    C., Matuszewski, M., Neill, J

    McGurk, R. C., Matuszewski, M., Neill, J. D., et al. 2024, in Society of Photo- Optical Instrumentation Engineers (SPIE) Conference Series, V ol. 13096, Ground-based and Airborne Instrumentation for Astronomy X, ed. J. J

    2024

  49. [49]

    1985, AJ, 90, 1027

    Merritt, D. 1985, AJ, 90, 1027

    1985

  50. [50]

    2023, The Journal of Open Source Software, 8, 5340

    Michalewicz, K., Millon, M., Dux, F., & Courbin, F. 2023, The Journal of Open Source Software, 8, 5340

    2023

  51. [51]

    Millon, M., Michalewicz, K., Dux, F., Courbin, F., & Marshall, P. J. 2024, AJ, 168, 55

    2024

  52. [52]

    Moffat, A. F. J. 1969, A&A, 3, 455

    1969

  53. [53]

    C., et al

    Morrissey, P., Matuszewski, M., Martin, D. C., et al. 2018, The Astrophysical Journal, 864, 93

    2018

  54. [54]

    D., Treu, T., Spiniello, C., & Shajib, A

    Mozumdar, P., Fassnacht, C. D., Treu, T., Spiniello, C., & Shajib, A. J. 2023, Astronomy & Astrophysics, 672, A20

    2023

  55. [55]

    Osipkov, L. P. 1979, Soviet Astronomy Letters, 5, 42

    1979

  56. [56]

    D., et al

    Paic, E., Courbin, F., Fassnacht, C. D., et al. 2026, A&A, 706, A270

    2026

  57. [57]

    L., Pierel, J

    Pascale, M., Frye, B. L., Pierel, J. D. R., et al. 2025, ApJ, 979, 13

    2025

  58. [58]

    W., Braatz, J

    Pesce, D. W., Braatz, J. A., Reid, M. J., et al. 2020, ApJ, 891, L1 Planck Collaboration, Aghanim, N., Akrami, Y ., et al. 2020, Astronomy & As- trophysics, 641, A6

    2020

  59. [59]

    L., Karwal, T., & Kamionkowski, M

    Poulin, V ., Smith, T. L., Karwal, T., & Kamionkowski, M. 2019, Physical Review Letters, 122

    2019

  60. [60]

    2025, Monthly Notices of the Royal Astronomical Society, 542, 170–202

    Queirolo, G., Seitz, S., Riffeser, A., et al. 2025, Monthly Notices of the Royal Astronomical Society, 542, 170–202

    2025

  61. [61]

    M., Oguri, M., More, A., et al

    Quimby, R. M., Oguri, M., More, A., et al. 2014, Science, 344, 396–399

    2014

  62. [62]

    2003, Monthly Notices of the Royal Astronomical Society, 338, 35–47

    Refregier, A. 2003, Monthly Notices of the Royal Astronomical Society, 338, 35–47

    2003

  63. [63]

    & Bacon, D

    Refregier, A. & Bacon, D. 2003, Monthly Notices of the Royal Astronomical Society, 338, 48–56

    2003

  64. [64]

    1964, MNRAS, 128, 307

    Refsdal, S. 1964, MNRAS, 128, 307

    1964

  65. [65]

    G., Yuan, W., Macri, L

    Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, The Astrophysical Journal Letters, 934, L7

    2022

  66. [66]

    E., Fassnacht, C

    Rusu, C. E., Fassnacht, C. D., Sluse, D., et al. 2017, Monthly Notices of the Royal Astronomical Society, 467, 4220–4242

    2017

  67. [67]

    2022, Monthly Notices of the Royal As- tronomical Society, 518, 1260–1300

    Schmidt, T., Treu, T., Birrer, S., et al. 2022, Monthly Notices of the Royal As- tronomical Society, 518, 1260–1300

    2022

  68. [68]

    & Sluse, D

    Schneider, P. & Sluse, D. 2013, A&A, 559, A37

    2013

  69. [69]

    1979, ApJ, 232, 236 Sérsic, J

    Schwarzschild, M. 1979, ApJ, 232, 236 Sérsic, J. L. 1963, Boletin de la Asociacion Argentina de Astronomia La Plata Argentina, 6, 41

    1979

  70. [70]

    J., Mozumdar, P., Chen, G

    Shajib, A. J., Mozumdar, P., Chen, G. C.-F., et al. 2023, Astronomy & Astro- physics, 673, A9

    2023

  71. [71]

    J., Wong, K

    Shajib, A. J., Wong, K. C., Birrer, S., et al. 2022, Astronomy & Astrophysics, 667, A123

    2022

  72. [72]

    2023, The Astrophysical Journal, 952, 10

    Sheu, W., Huang, X., Cikota, A., et al. 2023, The Astrophysical Journal, 952, 10

    2023

  73. [73]

    J., Treu, T., et al

    Sheu, W., Shajib, A. J., Treu, T., et al. 2025, MNRAS, 541, 1

    2025

  74. [74]

    R., Blum, R

    Silva, D. R., Blum, R. D., Allen, L., et al. 2016, in American Astronomical So- ciety Meeting Abstracts, V ol. 228, American Astronomical Society Meeting Abstracts #228, 317.02

    2016

  75. [75]

    Springel, V ., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629

    2005

  76. [76]

    H., Bonvin, V ., Courbin, F., et al

    Suyu, S. H., Bonvin, V ., Courbin, F., et al. 2017, Monthly Notices of the Royal Astronomical Society, 468, 2590–2604

    2017

  77. [77]

    H., Marshall, P

    Suyu, S. H., Marshall, P. J., Auger, M. W., et al. 2010, ApJ, 711, 201

    2010

  78. [78]

    Y ., Shajib, A

    Tan, C. Y ., Shajib, A. J., Birrer, S., et al. 2024, MNRAS, 530, 1474 TDCOSMO Collaboration, Birrer, S., Buckley-Geer, E. J., et al. 2025, Astron- omy & Astrophysics, 704, A63

    2024

  79. [79]

    & Benton Metcalf, R

    Tessore, N. & Benton Metcalf, R. 2015, Astronomy & Astrophysics, 580, A79

    2015

  80. [80]

    2013, A&A, 553, A120

    Tewes, M., Courbin, F., & Meylan, G. 2013, A&A, 553, A120

    2013

Showing first 80 references.