Pith. sign in

REVIEW 2 major objections 5 minor 72 references

Galaxy-scale strong lenses plus model-independent BAO distances find no clear deviation from general relativity on kiloparsec scales.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-13 21:02 UTC pith:WVBZY3VO

load-bearing objection Clean DESI-DR2 + SGL test of γ_PPN that really does drop H0/rd/w; result is GR-consistent under the preferred mass model, with the usual power-law systematic as the only soft spot. the 2 major comments →

arxiv 2603.21127 v1 pith:WVBZY3VO submitted 2026-03-22 astro-ph.CO

Testing General Relativity on Galactic Scales via DESI-BAO and Strong Lensing: Circumventing Assumptions on the Hubble Constant, Sound Horizon, and Dark Energy

Hengyu Wu , Tonghua Liu , Chenggang Shao This is my paper
classification astro-ph.CO
keywords general relativitystrong gravitational lensingbaryon acoustic oscillationsPPN parametermodel-independent reconstructionDESIgalactic scales
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tests whether Einstein gravity still holds for ordinary galaxies by combining 120 strong-lensing systems with BAO angular-scale data. Angular diameter distances to lenses and sources are reconstructed from the BAO ruler without fixing the Hubble constant, the sound horizon, or any dark-energy model, so the test does not inherit those cosmological assumptions. Under a constant-density-slope lens model the post-Newtonian parameter γ_PPN is found consistent with the general-relativity value of 1 at the 1–2σ level; a redshift-evolving slope model loosens the constraint and produces mild tension for one reconstruction method. The result matters because solar-system tests of gravity are extremely precise while galactic and cosmological tests remain only ~20 percent accurate; a clean, prior-independent check on kiloparsec scales therefore closes an important gap.

Core claim

Current galaxy-scale strong-lensing and BAO data are consistent with general relativity: under the preferred constant-slope mass model the PPN parameter is γ_PPN = 1.102^{+0.148}_{-0.125} (ANN reconstruction) and 1.150^{+0.139}_{-0.118} (cubic spline), both compatible with the GR value of unity at 1–2σ. No statistically significant evidence for departures from Einstein gravity is found on scales of a few to ~10 kpc.

What carries the argument

The distance ratio Ds/Dls reconstructed solely from BAO angular scales θ_BAO(z) via Dratio = θ_l / (θ_l - θ_s). Because the sound-horizon scale cancels, every quantity entering the strong-lensing likelihood becomes independent of H0, rd and the dark-energy equation of state.

Load-bearing premise

Every early-type lens is assumed to follow a single power-law total-mass profile whose slope is either constant or linear in redshift, with fixed luminous-matter slope and a Gaussian prior on orbital anisotropy; if real galaxies systematically deviate from this family, the inferred γ_PPN shifts by amounts comparable to the quoted errors.

What would settle it

A larger sample of well-resolved early-type lenses whose independently measured mass-density slopes deviate from the constant or linear-in-redshift power-law family would shift the joint posterior of γ_PPN away from 1 beyond the present 2σ contours.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Solar-system-level tests of GR can now be complemented by a cosmology-independent galactic-scale check that does not rely on assumed dark-energy models.
  • The statistical preference for a constant rather than redshift-evolving density slope implies that present early-type lens samples do not require mass-profile evolution with redshift.
  • Future BAO and strong-lensing surveys can tighten the same prior-free pipeline without reintroducing H0 or sound-horizon priors.
  • Any modified-gravity theory that predicts |γ_PPN - 1| ≳ 0.15 on kiloparsec scales is already in tension with the present data under the preferred mass model.

Where Pith is reading between the lines

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

  • Because the sound-horizon scale cancels in the distance ratio, the same BAO reconstruction can be reused for other ratio-based cosmological tests (curvature, time-delay distances) without reintroducing absolute-distance systematics.
  • The mild tension that appears only when redshift evolution of the density slope is allowed suggests residual mass-model systematics rather than new gravitational physics; improved stellar-kinematics mapping of individual lenses would isolate the effect.
  • The method supplies a natural consistency check for any future dark-energy or modified-gravity analysis that uses strong lenses as distance indicators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper constructs a cosmological-model-independent test of GR on galactic scales by combining non-parametric reconstructions of the BAO angular scale θ_BAO(z) (ANN and cubic spline, using SDSS/BOSS/eBOSS and DESI DR2) with 120 early-type strong-lensing systems. Angular-diameter distance ratios Ds/Dls are obtained from geometry alone via Dratio = θl/(θl − θs) (Eqs. 18–20), without H0, rd or a dark-energy equation of state. These ratios enter the standard gravitational-dynamical mass equality (Eqs. 1–11) under two power-law lens models (constant slope P1 and redshift-linear P2), yielding joint MCMC constraints on γ_PPN and the mass-profile parameters. Under the BIC-preferred P1 model both reconstructions give γ_PPN consistent with unity at 1–2σ; the mild ~2.5σ tension appears only under the disfavored P2 model. The authors conclude that current galaxy-scale data remain consistent with GR on kiloparsec scales.

Significance. If the result holds, the work supplies a clean, observationally anchored check of GR that deliberately removes the usual H0/rd/w priors that can bias gravity tests. The algebraic independence of Dratio from cosmological parameters (Eqs. 18–20), the dual non-parametric reconstructions that agree, the explicit BIC model comparison, and the per-system residual plots versus zl and RE are concrete strengths. The analysis also quantifies how strongly the inferred γ_PPN depends on the adopted lens mass model, which is useful for the community even if the final constraints remain at the ~15 % level. The framework is readily extensible to larger DESI, Euclid and CSST samples.

major comments (2)
  1. §II.A, Eqs. (5)–(6) and the surrounding text: the entire inference of γ_PPN rests on a single power-law total-density family (constant or linear in zl) with fixed luminous slope δ and a Gaussian prior β = 0.18 ± 0.13. The abstract and §III correctly note that the constraints “exhibit a clear dependence on the adopted lens mass model,” yet no quantitative robustness test against alternative profiles (e.g., NFW+stellar, broken power-law, or free δ) is provided. Because a systematic shift in the mass model of the size already seen between P1 and P2 would move γ_PPN by amounts comparable to the reported 1σ errors, a short sensitivity study or an explicit statement of the residual model systematic is needed before the claim of “no evidence for deviations” can be regarded as fully load-bearing.
  2. §II.D, Eq. (18): the conversion from reconstructed θ_BAO to Dratio assumes a spatially flat universe. While the paper repeatedly emphasizes independence from H0, rd and w, flatness is an additional geometric prior that is not varied or marginalized. A brief quantification of how a small |Ωk| ~ 0.01 would propagate into Dratio (and therefore into γ_PPN) would close this residual model dependence.
minor comments (5)
  1. Abstract and first paragraph of §I: “All the quantities” begins with a capital A mid-sentence; several other capitalization and spacing inconsistencies appear (e.g., “1 σ” vs “1σ”).
  2. Fig. 1 caption and §II.C: the distinction between “raw” (ANN) and “cleaned” (cubic spline) BAO data is mentioned but never defined; a short clause explaining what cleaning was applied would improve reproducibility.
  3. Eq. (12) and the sentence that follows: the total uncertainty formula is written twice in slightly different notation; a single consistent expression would avoid confusion.
  4. Table I: BIC values are given to three decimals while the parameter uncertainties are asymmetric; reporting ΔBIC relative to the preferred model would make the model-comparison statement in §III.C easier to read.
  5. Figs. 4–5: the vertical axis label “PPN” should be “γ_PPN” for consistency with the rest of the manuscript; a few outlier systems (e.g., SDSS J1352+3216) are discussed in the text but not marked on the plots.

Circularity Check

0 steps flagged

No significant circularity: BAO-derived distance ratios are algebraically independent of H0/rd/w and of the fitted γ_PPN; the latter is a free parameter constrained by velocity-dispersion residuals under an external mass-model family.

full rationale

The paper's central construction (Eqs. 14, 18–20) obtains Dratio = θl/(θl-θs) directly from non-parametric reconstructions of observed θ_BAO(z). The sound-horizon scale rd cancels identically, and no Hubble constant or dark-energy equation of state enters. These ratios are then inserted into the predicted aperture velocity dispersion (Eq. 11) whose only free gravity parameter is γ_PPN; the latter is fitted jointly with the mass-slope parameters via the χ^{2} of Eq. 15. Nothing in this chain defines γ_PPN in terms of itself or renames a fitted quantity as a prediction. The power-law density family (Eqs. 5–6) and the eta prior are taken from the external literature (Koopmans 2006; Chen et al. 2019) and are treated as model assumptions whose impact is quantified by comparing P1 versus P2 and by BIC; they do not force the numerical value of γ_PPN. The SGL catalogue itself is an observational compilation, not a self-derived uniqueness theorem. Consequently the reported consistency with GR is an ordinary posterior constraint, not a tautology. A score of 1 reflects only the ordinary (and non-load-bearing) reuse of a previously published lens sample and mass-model ansatz.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The claim rests on standard GR weak-field expansions, published BAO and SGL catalogues, two conventional non-parametric reconstructors, and a family of power-law lens models whose free parameters are fitted. No new physical entities are introduced; the free parameters are the usual mass-profile and PPN coefficients.

free parameters (5)
  • γ_PPN = 1.102^{+0.148}_{-0.125} (ANN, P1); 1.150^{+0.139}_{-0.118} (CS, P1); 1.315^{+0.181}_{-0.155} (ANN, P2); 1.485^{+0.193}_{
    Primary parameter of interest; fitted simultaneously with mass-profile parameters in the MCMC likelihood (Eq. 15).
  • γ0 (mass-density slope) = ≈1.97–2.23 depending on reconstruction and model
    Normalisation of the total density power-law; free in both P1 and P2 models.
  • γz (redshift evolution of slope) = −0.515^{+0.119}_{-0.130} (ANN); −0.601^{+0.100}_{-0.109} (CS)
    Linear coefficient in the P2 model γ = γ0 + γz zl; free only under P2.
  • β (orbital anisotropy) = 0.18 ± 0.13 (prior)
    Marginalised with Gaussian prior 0.18 ± 0.13 taken from local ellipticals; not fitted freely but still an external parameter that affects the dynamical mass.
  • η (aperture-correction exponent) = −0.066 ± 0.035
    Used to convert observed aperture velocity dispersions to a common physical aperture; taken from Cappellari et al. with its uncertainty propagated.
axioms (6)
  • domain assumption In the weak-field limit the metric takes the PPN form with a single free parameter γ_PPN multiplying the spatial curvature term (Eq. 2).
    Standard PPN framework; assumed throughout §II.A.
  • domain assumption Gravitational mass inside the Einstein radius equals dynamical mass (Eq. 1).
    Core consistency condition used to link lensing and kinematics.
  • domain assumption Lens galaxies obey a spherical power-law density profile for total mass and luminous matter, with constant or linearly evolving slope (Eqs. 5–6).
    Koopmans-type model adopted for all 120 systems; the paper’s own results show strong sensitivity to this choice.
  • domain assumption The Universe is spatially flat, so the distance ratio reduces to Dratio = θl / (θl − θs) (Eqs. 18–19).
    Explicitly invoked to convert reconstructed BAO angles into the lensing distance ratio without H0 or rd.
  • standard math BAO angular scale θ_BAO can be reconstructed non-parametrically by ANN or cubic spline without introducing cosmological parameters.
    Standard non-parametric regression; implemented via ReFANN and piecewise cubics.
  • ad hoc to paper A 3 % fractional systematic uncertainty on velocity dispersion adequately captures line-of-sight mass contamination.
    Adopted from Jiang & Kochanek (2007); added in quadrature (Eq. 12).

pith-pipeline@v1.1.0-grok45 · 21756 in / 3761 out tokens · 45122 ms · 2026-07-13T21:02:41.920167+00:00 · methodology

0 comments
read the original abstract

We present a cosmological model-independent framework for testing general relativity (GR) on galactic scales by combining baryon acoustic oscillation (BAO) angular scale measurements with 120 galaxy-scale strong gravitational lensing systems. Using artificial neural networks (ANNs) and cubic spline reconstruction, we reconstruct the BAO angular scale from SDSS, BOSS, eBOSS, and DESI Data Release 2 (DR2), and infer the angular diameter distances to lenses and sources. Crucially, All the quantities used in the GR test are derived from observations and are independent of cosmological parameters such as the Hubble constant, the sound horizon, or the dark energy equation of state, minimizing potential biases from model-dependent distance priors. These distances are then incorporated into the strong lensing likelihood to constrain the parameterized post-Newtonian (PPN) parameter $\gamma_{\rm PPN}$ under two lens mass models: a constant-density-slope model ($P_1$) and a redshift-evolving model ($P_2$). For the $P_1$ model, the ANN reconstruction yields $\gamma_{\rm PPN} = 1.102^{+0.148}_{-0.125}$, consistent with GR at $1\sigma$ confidence level, while the cubic spline gives $\gamma_{\rm PPN} = 1.150^{+0.139}_{-0.118}$, consistent with GR at $2\sigma$ confidence level. For the $P_2$ model, the ANN reconstruction gives $\gamma_{\rm PPN} = 1.315^{+0.181}_{-0.155}$, compatible with GR at $2\sigma$, while the spline gives $\gamma_{\rm PPN} = 1.485^{+0.193}_{-0.168}$, showing mild tension at $\sim2.5\sigma$. The constraints exhibit a clear dependence on the adopted lens mass model, underscoring the critical role of lens modeling. No significant correlation is observed between $\gamma_{\rm PPN}$ and the Einstein radius. Overall, current galaxy-scale observations are consistent with GR, providing no evidence for deviations from Einstein's theory on kiloparsec scales.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

72 extracted references · 62 linked inside Pith

  1. [1]

    F. W. Dyson, A. S. Eddington, and C. Davidson, Philo- sophical Transactions of the Royal Society of London Series A220, 291 (1920)

    1920

  2. [2]

    R. V. Pound and G. A. Rebka, Phys. Rev. Lett.4, 337 (1960)

    1960

  3. [3]

    I. I. Shapiro, Phys. Rev. Lett.13, 789 (1964)

    1964

  4. [4]

    J. H. Taylor, L. A. Fowler, and P. M. McCulloch, Nature (London)277, 437 (1979)

    1979

  5. [5]

    Song, G.-B

    Y.-S. Song, G.-B. Zhao, D. Bacon, K. Koyama, R. C. Nichol, and L. Pogosian, Phys. Rev. D84, 083523 (2011), arXiv:1011.2106 [astro-ph.CO]

    Pith/arXiv arXiv 2011

  6. [6]

    Simpson, C

    F. Simpson, C. Heymans, D. Parkinson, C. Blake, M. Kil- binger, J. Benjamin, T. Erben, H. Hildebrandt, H. Hoek- stra, T. D. Kitching, Y. Mellier, L. Miller, L. Van Waer- beke, J. Coupon, L. Fu, J. Harnois-Déraps, M. J. Hud- son, K. Kuijken, B. Rowe, T. Schrabback, E. Semboloni, S. Vafaei, and M. Velander, Mon. Not. Roy. Astron. Soc. 429, 2249 (2013), arXi...

    Pith/arXiv arXiv 2013

  7. [7]

    Blake, S

    C. Blake, S. Joudaki, C. Heymans, A. Choi, T. Erben, J. Harnois-Deraps, H. Hildebrandt, B. Joachimi, R. Naka- jima, L. van Waerbeke, and M. Viola, Mon. Not. Roy. Astron. Soc.456, 2806 (2016), arXiv:1507.03086 [astro- ph.CO]

    Pith/arXiv arXiv 2016

  8. [8]

    Wilcox, D

    H. Wilcox, D. Bacon, R. C. Nichol, P. J. Rooney, A. Teruk- ina, A. K. Romer, K. Koyama, G.-B. Zhao, R. Hood, R. G. Mann, M. Hilton, M. Manolopoulou, M. Sahlén, C. A. Collins, A. R. Liddle, J. A. Mayers, N. Mehrtens, C. J. Miller, J. P. Stott, and P. T. P. Viana, Mon. Not. Roy. Astron. Soc.452, 1171 (2015), arXiv:1504.03937 [astro- ph.CO]

    Pith/arXiv arXiv 2015

  9. [9]

    Pizzuti, B

    L. Pizzuti, B. Sartoris, S. Borgani, L. Amendola, K. Umetsu, A. Biviano, M. Girardi, P. Rosati, I. Balestra, G.B.Caminha, B.Frye, A.Koekemoer, C.Grillo, M.Lom- bardi, A. Mercurio, and M. Nonino, JCAP2016, 023 (2016), arXiv:1602.03385 [astro-ph.CO]

    Pith/arXiv arXiv 2016

  10. [10]

    Weinberg, Reviews of Modern Physics61, 1 (1989)

    S. Weinberg, Reviews of Modern Physics61, 1 (1989)

    1989

  11. [11]

    S. M. Carroll, Living Reviews in Relativity4, 1 (2001), arXiv:astro-ph/0004075 [astro-ph]

    Pith/arXiv arXiv 2001

  12. [12]

    E. J. Copeland, M. Sami, and S. Tsujikawa, International Journal of Modern Physics D15, 1753 (2006), arXiv:hep- th/0603057 [hep-th]

    arXiv 2006

  13. [13]

    T. P. Sotiriou and V. Faraoni, Reviews of Modern Physics 82, 451 (2010), arXiv:0805.1726 [gr-qc]

    Pith/arXiv arXiv 2010

  14. [14]

    Qi, R.-J

    J.-Z. Qi, R.-J. Yang, M.-J. Zhang, and W.-B. Liu, Re- search in Astronomy and Astrophysics16, 22 (2016), arXiv:1403.7287 [gr-qc]

    Pith/arXiv arXiv 2016

  15. [15]

    J.-Z. Qi, S. Cao, M. Biesiada, X. Zheng, and Z.- H. Zhu, European Physical Journal C77, 502 (2017), arXiv:1708.08603 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  16. [16]

    Y. Pan, Y. He, J.-Z. Qi, J. Li, S. Cao, T. Liu, and J. Wang, Astrophys. J.911, 135 (2021), arXiv:2103.05212 [astro-ph.CO]

    Pith/arXiv arXiv 2021

  17. [17]

    G. R. Bengochea and R. Ferraro, Phys. Rev. D79, 124019 (2009), arXiv:0812.1205 [astro-ph]

    Pith/arXiv arXiv 2009

  18. [18]

    Zhang, Y.-H

    J.-F. Zhang, Y.-H. Li, and X. Zhang, Physics Letters B 739, 102 (2014), arXiv:1408.4603 [astro-ph.CO]

    Pith/arXiv arXiv 2014

  19. [19]

    Li, J.-F

    Y.-H. Li, J.-F. Zhang, and X. Zhang, Physics Letters B 744, 213 (2015), arXiv:1502.01136 [astro-ph.CO]

    Pith/arXiv arXiv 2015

  20. [20]

    Geng, R.-Y

    J.-J. Geng, R.-Y. Guo, D.-Z. He, J.-F. Zhang, and X. Zhang, Frontiers of Physics10, 109501 (2015), arXiv:1511.06957 [astro-ph.CO]

    Pith/arXiv arXiv 2015

  21. [21]

    Zhao, J.-F

    M.-M. Zhao, J.-F. Zhang, and X. Zhang, Physics Letters B779, 473 (2018), arXiv:1710.02391 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  22. [22]

    Einstein, Science84, 506 (1936)

    A. Einstein, Science84, 506 (1936)

    1936

  23. [23]

    P. Saha, D. Sluse, J. Wagner, and L. L. R. Williams, Space Sci. Rev.220, 12 (2024), arXiv:2401.04165 [astro- ph.CO]

    Pith/arXiv arXiv 2024

  24. [24]

    Chakraborty and S

    S. Chakraborty and S. SenGupta, JCAP2017, 045 (2017), arXiv:1611.06936 [gr-qc]

    Pith/arXiv arXiv 2017

  25. [25]

    T. Liu, M. Biesiada, S. Tian, and K. Liao, Phys. Rev. D 109, 084074 (2024), arXiv:2404.05907 [gr-qc]

    Pith/arXiv arXiv 2024

  26. [26]

    Treu, Annual Review of Astronomy and Astrophysics 48, 87 (2010), arXiv:1003.5567 [astro-ph.CO]

    T. Treu, Annual Review of Astronomy and Astrophysics 48, 87 (2010), arXiv:1003.5567 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  27. [27]

    A. S. Bolton, S. Burles, L. V. E. Koopmans, T. Treu, and L. A. Moustakas, Astrophys. J.638, 703 (2006), arXiv:astro-ph/0511453 [astro-ph]

    Pith/arXiv arXiv 2006

  28. [28]

    A. S. Bolton, S. Rappaport, and S. Burles, Phys. Rev. D 74, 061501 (2006), arXiv:astro-ph/0607657 [astro-ph]

    Pith/arXiv arXiv 2006

  29. [29]

    S. Cao, X. Li, M. Biesiada, T. Xu, Y. Cai, and Z.-H. Zhu, Astrophys. J.835, 92 (2017), arXiv:1701.00357 [astro- ph.CO]

    Pith/arXiv arXiv 2017

  30. [30]

    T. E. Collett, L. J. Oldham, R. J. Smith, M. W. Auger, K. B. Westfall, D. Bacon, R. C. Nichol, K. L. Masters, 10 K. Koyama, and R. van den Bosch, Science360, 1342 (2018), arXiv:1806.08300 [astro-ph.CO]

    Pith/arXiv arXiv 2018

  31. [31]

    Liu and K

    T. Liu and K. Liao, Mon. Not. Roy. Astron. Soc.528, 1354 (2024), arXiv:2309.13608 [astro-ph.CO]

    Pith/arXiv arXiv 2024

  32. [32]

    T. Yang, B. Hu, R.-G. Cai, and B. Wang, Astrophys. J. 880, 50 (2019), arXiv:1810.00164 [astro-ph.CO]

    Pith/arXiv arXiv 2019

  33. [33]

    D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton, R. C. Nichol, R. Scranton, H.-J. Seo, M. Tegmark, Z. Zheng, S. F. Anderson, J. Annis, Bah- call,et al., Astrophys. J.633, 560 (2005), arXiv:astro- ph/0501171 [astro-ph]

    arXiv 2005

  34. [34]

    S. Cole, W. J. Percival, J. A. Peacock, P. Norberg, C. M. Baugh, C. S. Frenk, I. Baldry, J. Bland-Hawthorn, T. Bridges, R. Cannon, M. Colless, C. Collins, Couch, et al., Mon. Not. Roy. Astron. Soc.362, 505 (2005), arXiv:astro-ph/0501174 [astro-ph]

    Pith/arXiv arXiv 2005

  35. [35]

    Beutler, C

    F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley- Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson, Mon. Not. Roy. Astron. Soc.416, 3017 (2011), arXiv:1106.3366 [astro-ph.CO]

    Pith/arXiv arXiv 2011

  36. [36]

    S. Alam, M. Ata, S. Bailey, F. Beutler, D. Bizyaev, J. A. Blazek, Bolton,et al., Mon. Not. Roy. Astron. Soc.470, 2617 (2017), arXiv:1607.03155 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  37. [37]

    S. Alam, M. Aubert, S. Avila, C. Balland, J. E. Bautista, M. A. Bershady, D. Bizyaev, M. R. Blanton, A. S. Bolton, J. Bovy, J. Brinkmann, J. R. Brownstein, Burtin,et al., Phys. Rev. D103, 083533 (2021), arXiv:2007.08991 [astro- ph.CO]

    Pith/arXiv arXiv 2021

  38. [38]

    Blake, S

    C. Blake, S. Brough, M. Colless, C. Contreras, W. Couch, S. Croom, D. Croton, T. M. Davis, M. J. Drinkwater, Forster,et al., Mon. Not. Roy. Astron. Soc.425, 405 (2012), arXiv:1204.3674 [astro-ph.CO]

    Pith/arXiv arXiv 2012

  39. [39]

    Liu, Z.-H

    X.-H. Liu, Z.-H. Li, J.-Z. Qi, and X. Zhang, Astrophys. J.927, 28 (2022), arXiv:2109.02291 [astro-ph.CO]

    Pith/arXiv arXiv 2022

  40. [40]

    J.-J. Wei, Y. Chen, S. Cao, and X.-F. Wu, Astrophys. J. Lett.927, L1 (2022), arXiv:2202.07860 [astro-ph.CO]

    Pith/arXiv arXiv 2022

  41. [41]

    T. Liu, S. Wang, H. Wu, S. Cao, and J. Wang, Astrophys. J. Lett.981, L24 (2025), arXiv:2411.14154 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  42. [42]

    Millon, F

    M. Millon, F. Courbin, V. Bonvin, E. Buckley-Geer, C. D. Fassnacht, J. Frieman, P. J. Marshall, S. H. Suyu, T. Treu, T. Anguita, V. Motta, A. Agnello, J. H. H. Chan, D. C.-Y. Chao, M. Chijani, Gilman,et al., Astron. Astrophys.642, A193 (2020), arXiv:2006.10066 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  43. [43]

    T. Liu, S. Cao, and J. Wang, Phys. Rev. D111, 023524 (2025), arXiv:2406.18298 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  44. [44]

    Millon, A

    M. Millon, A. Galan, F. Courbin, T. Treu, S. H. Suyu, X. Ding, S. Birrer, G. C.-F. Chen, A. J. Shajib, D. Sluse, K. C. Wong, A. Agnello, M. W. Auger, E. J. Buckley- Geer, Chan,et al., Astron. Astrophys.639, A101 (2020), arXiv:1912.08027 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  45. [45]

    Y. Chen, R. Li, Y. Shu, and X. Cao, Mon. Not. Roy. Astron. Soc.488, 3745 (2019), arXiv:1809.09845 [astro- ph.CO]

    Pith/arXiv arXiv 2019

  46. [46]

    L. V. E. Koopmans, inEAS Publications Series, EAS Publications Series, Vol. 20, edited by G. A. Mamon, F. Combes, C. Deffayet, and B. Fort (EDP, 2006) pp. 161–166, arXiv:astro-ph/0511121 [astro-ph]

    Pith/arXiv arXiv 2006

  47. [47]

    Jorgensen, M

    I. Jorgensen, M. Franx, and P. Kjaergaard, Mon. Not. Roy. Astron. Soc.276, 1341 (1995)

    1995

  48. [48]

    Cappellari, R

    M. Cappellari, R. Bacon, M. Bureau, M. C. Damen, R. L. Davies, P. T. de Zeeuw, E. Emsellem, J. Falcón-Barroso, D. Krajnović, H. Kuntschner, R. M. McDermid, R. F. Peletier, M. Sarzi, R. C. E. van den Bosch, and G. van de Ven, Mon. Not. Roy. Astron. Soc.366, 1126 (2006), arXiv:astro-ph/0505042 [astro-ph]

    Pith/arXiv arXiv 2006

  49. [49]

    Jiang and C

    G. Jiang and C. S. Kochanek, Astrophys. J.671, 1568 (2007), arXiv:0705.3647 [astro-ph]

    Pith/arXiv arXiv 2007

  50. [50]

    L. V. E. Koopmans and T. Treu, Astrophys. J. Lett.568, L5 (2002), arXiv:astro-ph/0201017 [astro-ph]

    Pith/arXiv arXiv 2002

  51. [51]

    L. V. E. Koopmans, T. Treu, C. D. Fassnacht, R. D. Blandford, and G. Surpi, Astrophys. J.599, 70 (2003), arXiv:astro-ph/0306216 [astro-ph]

    Pith/arXiv arXiv 2003

  52. [52]

    Treu and L

    T. Treu and L. V. E. Koopmans, Astrophys. J.575, 87 (2002), arXiv:astro-ph/0202342 [astro-ph]

    Pith/arXiv arXiv 2002

  53. [53]

    Treu and L

    T. Treu and L. V. E. Koopmans, Astrophys. J.611, 739 (2004), arXiv:astro-ph/0401373 [astro-ph]

    Pith/arXiv arXiv 2004

  54. [54]

    A. J. Ruff, R. Gavazzi, P. J. Marshall, T. Treu, M. W. Auger, and F. Brault, Astrophys. J.727, 96 (2011), arXiv:1008.3167 [astro-ph.CO]

    Pith/arXiv arXiv 2011

  55. [55]

    Sonnenfeld, T

    A. Sonnenfeld, T. Treu, R. Gavazzi, S. H. Suyu, P. J. Marshall, M. W. Auger, and C. Nipoti, Astrophys. J. 777, 98 (2013), arXiv:1307.4759 [astro-ph.CO]

    Pith/arXiv arXiv 2013

  56. [56]

    Sonnenfeld, T

    A. Sonnenfeld, T. Treu, P. J. Marshall, S. H. Suyu, R. Gavazzi, M. W. Auger, and C. Nipoti, Astrophys. J.800, 94 (2015), arXiv:1410.1881 [astro-ph.GA]

    Pith/arXiv arXiv 2015

  57. [57]

    M. W. Auger, T. Treu, A. S. Bolton, R. Gavazzi, L. V. E. Koopmans, P. J. Marshall, K. Bundy, and L. A. Mous- takas, Astrophys. J.705, 1099 (2009), arXiv:0911.2471 [astro-ph.CO]

    Pith/arXiv arXiv 2009

  58. [58]

    M. W. Auger, T. Treu, A. S. Bolton, R. Gavazzi, L. V. E. Koopmans, P. J. Marshall, L. A. Moustakas, and S. Burles, Astrophys. J.724, 511 (2010), arXiv:1007.2880 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  59. [59]

    Y. Shu, A. S. Bolton, J. R. Brownstein, A. D. Montero- Dorta, L. V. E. Koopmans, T. Treu, R. Gavazzi, M. W. Auger, O. Czoske, P. J. Marshall, and L. A. Moustakas, Astrophys. J.803, 71 (2015), arXiv:1407.2240 [astro- ph.GA]

    Pith/arXiv arXiv 2015

  60. [60]

    Y. Shu, J. R. Brownstein, A. S. Bolton, L. V. E. Koop- mans, T. Treu, A. D. Montero-Dorta, M. W. Auger, O. Czoske, R. Gavazzi, P. J. Marshall, and L. A. Mous- takas, Astrophys. J.851, 48 (2017), arXiv:1711.00072 [astro-ph.GA]

    Pith/arXiv arXiv 2017

  61. [61]

    J. R. Brownstein, A. S. Bolton, D. J. Schlegel, D. J. Eisenstein, C. S. Kochanek, N. Connolly, C. Maraston, P. Pandey, S. Seitz, D. A. Wake, W. M. Wood-Vasey, J. Brinkmann, D. P. Schneider, and B. A. Weaver, Astro- phys. J.744, 41 (2012), arXiv:1112.3683 [astro-ph.CO]

    Pith/arXiv arXiv 2012

  62. [62]

    Y. Shu, A. S. Bolton, C. S. Kochanek, M. Oguri, I. Pérez- Fournon, Z. Zheng, S. Mao, A. D. Montero-Dorta, J. R. Brownstein, R. Marques-Chaves, and B. Ménard, Astro- phys. J.824, 86 (2016), arXiv:1604.01842 [astro-ph.GA]

    Pith/arXiv arXiv 2016

  63. [63]

    Y. Shu, A. S. Bolton, S. Mao, C. S. Kochanek, I. Pérez- Fournon, M. Oguri, A. D. Montero-Dorta, M. A. Cor- nachione, R. Marques-Chaves, Z. Zheng, J. R. Brown- stein, and B. Ménard, Astrophys. J.833, 264 (2016), arXiv:1608.08707 [astro-ph.GA]

    Pith/arXiv arXiv 2016

  64. [64]

    R.C.Nunes, S.K.Yadav, J.F.Jesus, andA.Bernui,Mon. Not. Roy. Astron. Soc.497, 2133 (2020), arXiv:2002.09293 [astro-ph.CO]

    Pith/arXiv arXiv 2020

  65. [65]

    Abdul Karim, J

    M. Abdul Karim, J. Aguilar, S. Ahlen, S. Alam, L. Allen, C. Allende Prieto, O. Alves, A. Anand, U. Andrade, E. Ar- mengaud, A. Aviles, S. Bailey, C. Baltay, P. Bansal, Bault, et al., Phys. Rev. D112, 083515 (2025), arXiv:2503.14738 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  66. [66]

    A. G. Adame, J. Aguilar, S. Ahlen, S. Alam, D. M. Alexan- der, M. Alvarez, O. Alves, A. Anand, U. Andrade, E. Ar- mengaud, S. Avila, Aviles,et al., JCAP2025, 021 (2025), 11 arXiv:2404.03002 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  67. [67]

    Wang, X.-J

    G.-J. Wang, X.-J. Ma, S.-Y. Li, and J.-Q. Xia, Astrophys. J., Suppl. Ser.246, 13 (2020), arXiv:1910.03636 [astro- ph.CO]

    Pith/arXiv arXiv 2020

  68. [68]

    Q. Wang, S. Cao, J. Jiang, K. Zhang, X. Jiang, T. Liu, C. Mu, and D. Cheng, Astrophys. J.987, 58 (2025), arXiv:2506.12759 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  69. [69]

    J. L. Bernal, L. Verde, and A. G. Riess, JCAP2016, 019 (2016), arXiv:1607.05617 [astro-ph.CO]

    Pith/arXiv arXiv 2016

  70. [70]

    Aylor, M

    K. Aylor, M. Joy, L. Knox, M. Millea, S. Raghunathan, and W. L. Kimmy Wu, Astrophys. J.874, 4 (2019), arXiv:1811.00537 [astro-ph.CO]

    Pith/arXiv arXiv 2019

  71. [71]

    Foreman-Mackey, D

    D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good- man, Publications of the Astronomical Society of the Pacific125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

    Pith/arXiv arXiv 2013

  72. [72]

    Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(1972)

    S. Weinberg,Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity(1972)

    1972