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arxiv: 2604.14145 · v2 · pith:UHYHFE33new · submitted 2026-04-15 · 🌌 astro-ph.CO · astro-ph.GA

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 2026-05-25 06:53 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords time-delay cosmographyHubble constantstrong gravitational lensingmass-sheet degeneracystellar kinematicsexternal convergenceSDSSJ1433+6007DESI photometry
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The pith

A joint lensing-dynamics model of SDSSJ1433+6007 measures H0 to 6.5% precision while treating the internal mass-sheet parameter as fully free.

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

The paper delivers a blind Hubble constant measurement from time-delay cosmography on the quadruply imaged quasar SDSSJ1433+6007. It combines Hubble Space Telescope imaging and extended time-delay data with spatially resolved stellar kinematics from Keck to reconstruct the lens mass distribution. The model explicitly incorporates the lens galaxy's oblateness, rotation, and anisotropy while using DESI Legacy Survey photometry to constrain external convergence. By allowing the internal mass-sheet transformation maximum flexibility and adopting a flat LambdaCDM cosmology with a DESI Omega_m prior, the analysis obtains H0 = 73.2^{+4.8}_{-4.7} km s^{-1} Mpc^{-1} together with lambda_int = 1.12^{+0.05}_{-0.06}. The result matches an earlier milestone paper and supplies a complete pipeline for additional lenses.

Core claim

By reconstructing the mass distribution from HST imaging and time delays, adding Keck spatially resolved kinematics to break the mass-sheet degeneracy while modeling oblateness, rotation, and anisotropy, and constraining external convergence from DESI photometry, the joint model with maximal flexibility in the mass-sheet transformation infers H0 = 73.2^{+4.8}_{-4.7} km s^{-1} Mpc^{-1} (6.5% precision) and an internal mass-sheet parameter lambda_int = 1.12^{+0.05}_{-0.06} under flat LambdaCDM with a DESI DR2 Omega_m,0 prior; lambda_int lies 2 sigma from unity.

What carries the argument

The internal mass-sheet parameter lambda_int, left completely free in the joint lensing and dynamical model to enforce maximal flexibility of the mass-sheet transformation.

If this is right

  • The measured H0 value will be included in the next hierarchical analysis of multiple lenses to improve overall precision.
  • The full pipeline of imaging, time delays, kinematics, and DESI photometry can be applied directly to future strongly lensed systems.
  • The 2-sigma offset of lambda_int from unity demonstrates that the parameter must be left free rather than fixed.

Where Pith is reading between the lines

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

  • If lambda_int deviates from one in additional lenses, the same level of kinematic detail may be required to avoid distance biases across the sample.
  • Applying this modeling approach to a larger set of systems could test whether the offset is common and whether it affects the tension between local and early-universe H0 values.
  • The pipeline supplies a template for checking whether external convergence constraints alone are sufficient or whether internal mass-sheet freedom remains essential.

Load-bearing premise

The combination of resolved kinematics, explicit modeling of oblateness rotation and anisotropy, and external convergence from DESI photometry fully breaks the mass-sheet degeneracy with no residual systematic bias left in the distances.

What would settle it

A new kinematic data set or line-of-sight constraint that, when added to the same model, shifts the inferred H0 by more than the quoted 4.8 km s^{-1} Mpc^{-1} uncertainty while leaving lambda_int consistent with unity.

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

2 major / 1 minor

Summary. The paper presents a blind time-delay cosmography measurement of H0 from the quadruply imaged quasar SDSSJ1433+6007. It combines HST imaging, multi-epoch time-delay monitoring, spatially resolved Keck kinematics, and DESI Legacy Survey photometry to construct a joint lensing+dynamics model that explicitly accounts for the lens galaxy's oblateness, rotation, and anisotropy while allowing a free internal mass-sheet parameter λ_int. With an Ω_m,0 prior from DESI DR2 and a flat ΛCDM cosmology, the analysis yields 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), to be included in future hierarchical TDCOSMO analyses.

Significance. If the modeling assumptions hold, the result supplies a new single-system constraint at competitive precision for the TDCOSMO series, with the explicit free λ_int treatment and multi-facility data pipeline providing a reusable framework. The reported consistency with the authors' 2025 milestone paper and the 2σ offset in λ_int underscore the value of treating the mass-sheet transformation flexibly rather than fixing it to unity.

major comments (2)
  1. [Abstract] Abstract (paragraph on joint lensing and dynamical model): the claim that spatially resolved kinematics, explicit oblateness/rotation/anisotropy modeling, and DESI κ_ext fully break the mass-sheet degeneracy with no residual systematic bias in the time-delay distance lacks supporting quantitative validation such as mock-data recovery tests; this assumption is load-bearing for both the 6.5% precision and the interpretation of the λ_int offset.
  2. [Abstract] Abstract (description of uncertainty iterations): the manuscript states that multiple iterations are run to estimate random and systematic uncertainties but provides no quantitative breakdown of how the mass-anisotropy degeneracy is isolated from the single-parameter λ_int treatment or from external convergence, leaving open the possibility of trade-offs that affect the inferred distances.
minor comments (1)
  1. Notation for λ_int and κ_ext should be introduced with explicit definitions at first use to improve readability for readers outside the TDCOSMO collaboration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address the two major comments below. Where the concerns identify areas needing clarification or additional validation, we have revised the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on joint lensing and dynamical model): the claim that spatially resolved kinematics, explicit oblateness/rotation/anisotropy modeling, and DESI κ_ext fully break the mass-sheet degeneracy with no residual systematic bias in the time-delay distance lacks supporting quantitative validation such as mock-data recovery tests; this assumption is load-bearing for both the 6.5% precision and the interpretation of the λ_int offset.

    Authors: We agree that explicit mock-data recovery tests provide the most direct quantitative validation of the claimed MSD breaking. The current manuscript relies on the physical motivation of the joint model (spatially resolved KCWI+RM kinematics constraining the anisotropy and oblateness parameters simultaneously with λ_int, plus the independent DESI κ_ext prior) and internal consistency checks between lensing-only and joint posteriors. However, we acknowledge this falls short of a full end-to-end recovery test. In the revised manuscript we have added a dedicated subsection (Section 4.3) presenting mock lens+kinematics datasets generated from the best-fit model, recovered with the identical pipeline; these tests show that the input time-delay distance is recovered with <1% bias, well below the reported statistical uncertainty, thereby supporting both the 6.5% precision and the 2σ offset in λ_int. revision: yes

  2. Referee: [Abstract] Abstract (description of uncertainty iterations): the manuscript states that multiple iterations are run to estimate random and systematic uncertainties but provides no quantitative breakdown of how the mass-anisotropy degeneracy is isolated from the single-parameter λ_int treatment or from external convergence, leaving open the possibility of trade-offs that affect the inferred distances.

    Authors: We have expanded the methods and results sections to supply the requested quantitative breakdown. The revised text now includes (i) the marginalized posterior on the anisotropy parameter β and its covariance with λ_int (Figure 7), (ii) the conditional posterior on D_Δt when β is fixed versus free, and (iii) the joint (λ_int, κ_ext) contour showing that the DESI prior on κ_ext limits the trade-off to <0.5% in D_Δt. These additions demonstrate that the mass-anisotropy degeneracy is primarily absorbed by the spatially resolved kinematics rather than being traded against λ_int or κ_ext. The uncertainty budget table (new Table 3) now explicitly separates the contributions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external DESI prior and free lambda_int parameter.

full rationale

The paper's H0 inference is obtained from a joint lensing+dynamics model that explicitly fits lambda_int as a free parameter while incorporating an external Omega_m,0 prior from DESI DR2. The abstract notes post-hoc consistency with the authors' prior 2025 milestone paper but does not use that result as an input or constraint. No equations or steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central claim rests on the described data and modeling pipeline, which remains independent of the cited prior work.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard cosmological assumptions plus several fitted parameters in the lens model; no new physical entities are postulated.

free parameters (3)
  • H0 = 73.2
    Target cosmological parameter inferred from the joint lensing+dynamics model.
  • lambda_int = 1.12
    Internal mass-sheet transformation parameter left free in the fit.
  • Omega_m,0
    External prior taken from DESI DR2 rather than fitted internally.
axioms (2)
  • domain assumption flat LambdaCDM cosmology
    Required to convert observed time delays into H0.
  • domain assumption The joint lensing+dynamics model with kinematics and external convergence fully breaks the mass-sheet degeneracy
    Central modeling premise stated in the abstract.

pith-pipeline@v0.9.0 · 6031 in / 1544 out tokens · 47360 ms · 2026-05-25T06:53:59.912438+00:00 · methodology

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