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REVIEW 5 major objections 8 minor 1 cited by

A Joint Analysis of Strong Lensing and Type Ia Supernovae to Determine the Hubble Constant

T0 review · 5 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Combining seven time-delay lens distances, 99 Einstein-radius systems, and Pantheon+ supernovae through the cosmic distance duality relation yields a model-independent $H_0 = 70.55 \pm 7.44$ km/s/Mpc.

desk verdict A clean CDDR-based H0 combination with an honest error budget, but the GP step over D_Als/D_As ignores source-redshift and mass-profile dependence, so I read the central value as a consistency check rather than a determination. read the letter →

arxiv 2505.17262 v1 pith:RLMKT4L6 submitted 2025-05-22 astro-ph.CO

L. R. Colaço , R. F. L. Holanda , Z. C. Santana , R. Silva This is my paper
classification astro-ph.CO
keywords Hubbleconstantcosmicdistancedualityrelationstronggravitationallensingtime-delaycosmographyEinsteinradiusGaussianprocessregressionPantheon+supernovaemodel-independentcosmology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper attempts to determine the Hubble constant without assuming any cosmological model, by chaining three datasets together: seven time-delay lens distances from the TDCOSMO collaboration, 99 Einstein-radius lens systems, and the Pantheon+ supernova sample. It computes each lens's angular diameter distance as the product of a Gaussian-process reconstruction of the Einstein-radius ratio at that lens redshift and the measured time-delay distance. It then anchors the supernova unanchored luminosity distance to this angular diameter distance through the cosmic distance duality relation, obtaining $H_0 = 70.55 \pm 7.44$ km/s/Mpc at 68% confidence. The authors present this as a consistency check that sits between the Planck and SH0ES values, not as a resolution of the Hubble tension.

What carries the argument

The load-bearing identity is the product in Eq. (9): the observed time-delay angular diameter distance $D^{\mathrm{Obs}}_{A,\Delta t}(z_l,z_s)$, which is cosmology-independent, is multiplied by $D^{\mathrm{Obs}}(z_l) \equiv D_{Als}/D_{As}$, the Einstein-radius distance ratio reconstructed from 99 systems by Gaussian Process regression as a smooth function of the lens redshift $z_l$. Evaluating that reconstruction at each of the seven time-delay lens redshifts produces $D_A^l$. The same regression machinery, applied to Pantheon+ apparent magnitudes and converted with a calibrated intercept $a_B$, gives the unanchored luminosity distance $\Theta_{\mathrm{SNe}}(z_l)$; Eq. (10) joins the two through the cosmic distance duality relation. The lens mass model is fixed to a power-law profile with $\gamma = 2.1$, and the GP covariance is a squared-exponential kernel.

What would settle it

Split the 99 Einstein-radius systems by source redshift or by image multiplicity, reconstruct $D^{\mathrm{Obs}}(z_l)$ in each subgroup, and compare the curves at the seven TDCOSMO lens redshifts; a shift between subgroups larger than the reported ~24% intrinsic scatter would show that evaluating a single smoothed curve at $z_l$ misses source-dependent information and biases $H_0$. A sharper test: for a lens with multiple measured images or a known source redshift, require the GP prediction to match the actual $D^{\mathrm{Obs}}$ value, not just the smoothed mean.

Watch

Extended reading notes

Core claim

The central claim is that the product $D_A^l = D^{\mathrm{Obs}}(z_l)\,D^{\mathrm{Obs}}_{A,\Delta t}(z_l,z_s)$ gives a cosmology-independent angular diameter distance to each time-delay lens, and that the cosmic distance duality relation $D_L = (1+z)^2 D_A$ then turns unanchored supernova distances into a per-lens $H_0$ through $H_0 = \Theta_{\mathrm{SNe}}(z_l)/[(1+z_l)^2 D_A^l]$. Applied to the seven TDCOSMO systems, the seven model-free estimates combine in a MCMC posterior, with an added intrinsic scatter of about 24%, to give $H_0 = 70.55 \pm 7.435$ km/s/Mpc at 68% confidence. The paper emphasizes that this value agrees within $1\sigma$ with both the early-universe Planck value and the local SH0ES value, so the method is a viable independent consistency probe even though it does not resolve the Hubble tension.

Load-bearing premise

The load-bearing premise is that the Gaussian-process curve built from 99 Einstein-radius ratios, treated as a smooth function of lens redshift, correctly predicts $D_{Als}/D_{As}$ for each of the seven time-delay lenses, even though that ratio also depends on each system's source redshift and mass profile.

Editorial extensions

If this is right

  • If the central claim holds, every strong-lensing Einstein-radius system becomes a distance indicator: the GP-reconstructed $D^{\mathrm{Obs}}(z)$ curve, once validated, can supply angular diameter distances at arbitrary lens redshifts without fixing a cosmology.
  • The same pipeline applied to larger lens samples, such as those expected from next-generation surveys, should sharpen the $H_0$ posterior directly, since the dominant uncertainty in the current result comes from only seven time-delay lenses.
  • The result's $1\sigma$ agreement with both Planck and SH0ES means the inverse-distance-ladder route is currently a consistency check, not a tension discriminator; its real test will come from reducing the reported ~24% intrinsic scatter.
  • Because the method avoids anchoring to the CMB sound horizon or the distance ladder, it offers a third, independent rung for cross-checking $H_0$ determinations.

Reading between the lines

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

  • Beyond the paper, the Gaussian-process assumption can be tested directly by splitting the 99 Einstein-radius systems by source redshift and comparing the reconstructed curves; a systematic offset at the seven TDCOSMO lens redshifts would propagate straight into $H_0$.
  • Beyond the paper, the same product identity could be crossed with absolute distance anchors other than supernovae, such as gravitational-wave standard sirens, to see whether the CDDR consistency holds across distance indicators.
  • Beyond the paper, the seven individual $H_0$ estimates should be published separately; if they show a redshift trend, that is an early warning of CDDR violation or residual lens-model systematics that the combined value hides.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 8 minor

Summary. This paper proposes a cosmological-model-independent estimate of the Hubble constant by combining three datasets at the lens redshifts of seven TDCOSMO time-delay systems: (i) the Einstein-radius observable D^Obs = D_Als/D_As, reconstructed as a function of lens redshift by a Gaussian process trained on 99 strong-lensing systems from Cao et al. (2015); (ii) the time-delay angular diameter distance D^Obs_{A,Δt} of each of the seven TDCOSMO systems; and (iii) the unanchored luminosity distance Θ = H0 D_L from the Pantheon+ supernova sample, also reconstructed by a Gaussian process. Using the cosmic distance duality relation, the authors form seven estimates H0,i = Θ(z_l)/[(1+z_l)^2 D^Obs(z_l) D^Obs_{A,Δt}] and combine them with an MCMC likelihood that includes a fitted intrinsic scatter, obtaining H0 = 70.55 ± 7.44 km/s/Mpc (68% CL). The paper frames the result as a model-independent consistency check that lies between the Planck and SH0ES values without resolving the Hubble tension.

Significance. The core algebraic identity is sound and genuinely cosmology-independent: D^Obs is constructed to be independent of H0, and the product D^Obs · D^Obs_{A,Δt} cancels the source angular diameter distance to leave D_Al; the CDDR then converts the SNe unanchored luminosity distance into H0 per lens. This is a clever and nontrivial way to build an inverse distance ladder from strong lensing alone, and the use of the full Pantheon+ covariance matrix, including the a_B intercept uncertainty through Eq. (13), is careful. The paper is also commendably explicit about its limitations (mass-sheet degeneracy, anisotropy, no marginalization over mass-profile details) and does not overclaim about resolving the Hubble tension. The main value, if the concerns below are addressed, is a novel cross-check that is independent of any FLRW metric assumption. The current constraining power is modest (about 10.5% precision), and the unbiasedness of the central value depends on assumptions about the Gaussian-process reconstruction and the lens mass profiles that are not yet demonstrated.

major comments (5)
  1. [II.D and Eq. (9)] The Gaussian process reconstructs D^Obs ≡ D_Als/D_As as a one-dimensional function of the lens redshift z_l, but this distance ratio also depends on the source redshift z_s and on the mass-profile slope γ. Equation (9) evaluates the population-mean curve at the z_l of each of the seven TDCOSMO lenses and multiplies it by the per-system time-delay distance; the product equals D_Al only if each TDCOSMO system is representative of the 99-system sample in both z_s and γ at that z_l. The justification given at the end of Section II.B — that the source redshifts of the Einstein-radius sample are 'clustered around those in time-delay systems' — is supported only by a citation to ref. [67], with no explicit comparison of the z_s distributions at matched lens redshifts. If, for example, the TDCOSMO systems have systematically higher (or lower) source redshifts than the average of the 99 systems at the same z_l, all seven D_Al values shift in the same direction and H0 is biased by a common-mode offset that the fitted σ_int cannot remove. I request an explicit (z_l, z_s, γ) comparison between the seven TDCOSMO systems and the 99-system sample, or alternatively a two-dimensional GP in (z_l, z_s) with a marginalization over γ.
  2. [II.A, Eq. (4)] The Einstein-radius observable is computed with the power-law slope fixed at γ = 2.1 for all 99 systems, while the TDCOSMO time-delay distances used in Eq. (9) are individually derived under power-law lens models with system-specific slopes γ_pl (Table 2 of ref. [43]). Since Eq. (4) depends on γ through f(γ) and the factor (θ_ap/θ_E)^{2−γ}, a TDCOSMO system whose γ_pl differs from 2.1 yields a D^Obs estimate that is inconsistent with the mass model used for its time-delay distance; the product in Eq. (9) is then not D_Al for that system. The paper does not quote the γ_pl values or test their consistency with 2.1. The analysis should marginalize over γ (for instance with the prior from the lens-sample measurements) and report the sensitivity of H0 to this choice.
  3. [III, σ_int] The final uncertainty is dominated by a fitted intrinsic scatter σ_int ≈ 24% that is added to the covariance matrix specifically 'to obtain a χ_red ≈ 1'. This large scatter means that the seven H0 estimates from Eq. (10) are mutually inconsistent at the level of their propagated uncertainties; the quoted 68% interval H0 = 70.55 ± 7.44 km/s/Mpc is therefore determined by an ad hoc parameter rather than by the data, and a common-mode systematic would survive the enlarged error bars. The authors should report the seven individual estimates H0_i and the χ² obtained before adding σ_int, and should justify the 24% level as a physical scatter (for example, arising from the γ and environment distributions) rather than as a normalization fudge.
  4. [III, Eq. (17)] As written, the covariance C_{H0} = C_{Θ_SNe} + C_{D^Obs} + C_{D^Obs_{A,Δt}} adds matrices for quantities of different physical dimensions (Θ has units of km/s, D^Obs is dimensionless, and D^Obs_{A,Δt} has units of Mpc), and it treats the multiplicative relation of Eq. (10) as if it were additive. The correct linearized propagation of H0 ∝ Θ/(D^Obs D^Obs_{A,Δt}) requires Jacobian-weighted covariance matrices, or equivalently the computation should be done in log-space. Please correct Eq. (17), or clarify that the code propagates the covariance of the product in Eqs. (9)-(10) and present the correct expression.
  5. [II.D] The Einstein-radius training set is selected by excluding the dataset of ref. [77] after inspecting its Gaussian-process reconstruction, which the authors report as showing 'bigger uncertainties... due to an unknown trend with redshift'. Because ref. [77] is stated to contain all systems of ref. [58], choosing the training sample on the basis of the reconstructed output is a post-hoc selection that can bias the GP mean function; the authors should either adopt a pre-defined selection criterion or demonstrate that the inferred H0 and its uncertainty are stable when the [77] data are included.
minor comments (8)
  1. [II.D] Typo: 'Einstien Radius dataset' should read 'Einstein radius dataset'.
  2. [Fig. 2] The right-panel axis label 'Θ(z)[km/s/Mpc]' has unclear units and normalization; since Θ = H0 D_L has units of km/s, please verify the units and explain the 10^6 axis scaling.
  3. [III] Typo: 'C −1_H Est 0 is is the inverse' should read 'is the inverse'.
  4. [II.B / ref. [43]] The seven TDCOSMO systems' values of (z_l, z_s, D^Obs_{A,Δt}, γ_pl) are not reproduced; given their central role in Eqs. (9)-(10), a table of these inputs would improve reproducibility and would allow readers to assess the z_s and γ comparisons requested above.
  5. [Eq. (13)] The second term in Eq. (13) should be identified as (5σ_aB)^2 J, where J is the all-ones matrix, reflecting the common shift induced by the intercept uncertainty a_B.
  6. [II.A] The statement 'the observed quantity 4 is independent of the Hubble constant value' is awkward; it should cite Eq. (4) explicitly. Also, the aperture conversion to σ0 is described in one line; a brief equation or a pointer to Table 1 of ref. [58] would help.
  7. [II.D] Please state whether any of the seven TDCOSMO lenses are also members of the 99 Einstein-radius systems; if so, their Einstein-radius data enter both the GP training and the time-delay product, and this partial dependence should be discussed.
  8. [Form] The 'Keywords' and 'PACS numbers' fields are empty and should be completed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the H0 estimate is an algebraic combination of independent datasets through the CDDR, with no fitted parameter relabeled as a prediction.

full rationale

Walking the derivation chain: DObs is constructed from 99 Einstein-radius systems via Eq. 4, DObs_A,Delta_t comes from the seven TDCOSMO time-delay lenses via Eq. 8, and Eq. 9 is the algebraic identity (DAls/DAs)*(DAl*DAs/DAls)=DAl. No cosmological model enters, and no H0 value is used to build either factor. The unanchored luminosity distance Theta_SNe is obtained from Pantheon+ with the externally calibrated intercept aB, and Eq. 10 is simply the CDDR relation DL=(1+z)^2 DA rewritten for H0. The Gaussian-process hyperparameters are fitted separately to the Einstein-radius and supernova datasets, but H0 is not an input to those fits, and the additional sigma_int only inflates the error budget rather than shifting the central value. The most fragile step is evaluating the one-dimensional GP reconstruction of DObs(zl) at the seven TDCOSMO redshifts despite differences in source redshift and mass-profile slope; this is a possible systematic bias and is acknowledged in the paper's limitation statement, but it is not a circular reduction of the result to its own inputs. Self-citations such as refs. [52,56,67,79] appear as methodological references and prior applications of the same CDDR technique; none of them supplies the target H0 value or forces the final estimate by construction. The paper therefore contains no significant circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central estimate rests on two externally calibrated inputs (a_B and gamma), one environment systematic inherited from the literature (P = 20%), one post hoc variance component (sigma_int) chosen to make the fit statistically acceptable, and a GP reconstruction that assumes a z_l-only dependence. No new physical entities are introduced.

free parameters (5)
  • sigma_int (intrinsic scatter) = about 24% (relative)
    Added to the inverse covariance of the seven H0 estimates in the MCMC to force chi-squared_red about 1 (Section III). It is not a measured quantity; it inflates the quoted uncertainty and is chosen post hoc.
  • gamma (power-law slope) = 2.1 (fixed, not fitted)
    All 99 Einstein-radius systems and the PLAW time-delay distances assume a spherically symmetric power-law mass profile with a single index gamma = 2.1 (Section II A, Eq. 4). No marginalization over gamma or stellar anisotropy is performed; the central H0 depends on this choice.
  • P (environment systematic) = 20%
    Uncertainty contribution in Eq. 6 from environment effects on image separation, adopted from ref. [90]; it inflates the D^Obs errors and therefore the final H0 uncertainty.
  • a_B (SNe intercept) = 0.71273 ± 0.00176
    External intercept from Riess et al. (2016) used in Eqs. 11-12 to convert Pantheon+ apparent magnitudes into unanchored distances; the central H0 scales with a_B.
  • GP kernel hyperparameters (sigma_f, l) = optimized via marginal likelihood, not quoted
    Chosen by maximizing Eq. 16 for each reconstruction; the smoothing scale and amplitude affect the reconstructed D^Obs and Theta_SNe at the seven lens redshifts.
assumptions (5)
  • domain assumption The cosmic distance duality relation DL = (1+z)^2 DA holds exactly.
    Used in Eq. 1 and Eq. 10 to anchor the SNe unanchored luminosity distance to the lens angular diameter distance; any violation shifts the inferred H0. No free parameter for a possible violation is included.
  • ad hoc to paper The lens mass distribution is a spherically symmetric power law with the same index gamma = 2.1 for all 99 Einstein-radius systems and for the TDCOSMO time-delay lenses.
    Eq. 4 assumes this to convert Einstein radii and velocity dispersions into D^Obs; the value 2.1 is taken from refs. [78,79] rather than fitted per system. The paper does not marginalize over gamma or anisotropy (Sections II A and IV).
  • domain assumption The reconstructed D^Obs(z_l) from the 99-system sample is representative of the distance ratio for the seven TDCOSMO lenses.
    The GP in Section II D treats D^Obs as a function of z_l only, and Eq. 9 evaluates it at TDCOSMO lens redshifts, although the ratio D_Als/D_As also depends on the source redshift and on the lens mass profile.
  • domain assumption The TDCOSMO time-delay distances D^Obs_{A,Δt} based on the power-law mass model are correct, i.e., mass-sheet degeneracy and stellar anisotropy are not fully marginalized.
    The paper uses the PLAW values from Table 2 of ref. [43] and explicitly notes it does not marginalize over anisotropy models (Section IV); this is an inherited assumption from the TDCOSMO analysis.
  • domain assumption Type Ia supernovae are standardizable candles with a known intercept a_B.
    Needed to build Theta_SNe from Pantheon+ via Eq. 12; uses a_B from ref. [99] and the Pantheon+ covariance matrix.

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Cite this review

Pith. "Pith review of A Joint Analysis of Strong Lensing and Type Ia Supernovae to Determine the Hubble Constant." pith.science (2026). https://pith.science/paper/RLMKT4L6

@misc{pith2026250517262,
  author       = {Pith},
  title        = {Pith review of: A Joint Analysis of Strong Lensing and Type Ia Supernovae to Determine the Hubble Constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLMKT4L6}},
  note         = {Machine review of arXiv:2505.17262}
}
abstract

We present a cosmological model-independent determination of the Hubble constant, $H_0$, by combining time-delay measurements from seven TDCOSMO systems, Einstein radius measurements, and Type Ia Supernovae data sourced from the Pantheon+ sample. For each lens of time-delay system, we calculate the angular diameter distance $D_{A_l}$ using the product $D^{\textrm{Obs}}(z_l) \cdot D_{A,\Delta t}^{\textrm{Obs}}(z_l, z_s)$, where $D^{\textrm{Obs}}(z_l)$ is reconstructed via Gaussian Processes from 99 Einstein radius measurements, and $D_{A,\Delta t}^{\textrm{Obs}}(z_l,z_s)$ is the time-delay angular distance. We also reconstruct the unanchored luminosity distance $H_0 D_L(z_l)$ from supernova data. By using the cosmic distance duality relation validity, we anchor $D_{A_l}$ and $H_0 D_L(z_l)$ to infer $H_0 = 70.55 \pm 7.44$ km/s/Mpc (68\% CL). Our result, though not resolving the Hubble tension, offers a cosmological model-independent consistency check and highlights the potential of using strong lensing and supernovae data via the cosmic distance duality relation to constrain $H_0$.

Figures

Figures reproduced from arXiv: 2505.17262 by the authors.

Figure 1
Figure 1. FIG. 1: The seven well-studied and refined time-delay dis [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Left Panel: The GP reconstruction of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The pdf of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

Works this paper leans on

118 extracted references · 31 canonical work pages · cited by 1 Pith paper

  1. [58]

    L. R. Colaço, Universe11, 89 (2025), arXiv:2503.06189 [astro-ph.CO]

    work page Pith review arXiv 2025

  2. [77]

    Y. Chen, R. Li, Y. Shu, and X. Cao, MNRAS 488, 3745 (2019), arXiv:1809.09845 [astro-ph.CO]

    arXiv 2019

  3. [67]

    A. Rana, D. Jain, S. Mahajan, A. Mukherjee, and R. F. L. Holanda, JCAP 07, 010 (2017), arXiv:1705.04549 [astro-ph.CO]

    arXiv 2017

  4. [43]

    Birrer et al., Astron

    S. Birrer et al., Astron. Astrophys. 643, A165 (2020), arXiv:2007.02941 [astro-ph.CO]

    arXiv 2020

  5. [1]

    A. G. Riess et al., The Astronomical Journal 116, 1009–1038 (1998)

    1998

  6. [2]

    Perlmutter, G

    S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, Goo- bar, and et at.,TheAstrophysicalJournal 517,565–586 (1999)

    1999

  7. [3]

    D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata, A. G. Riess, and E. Rozo, Physics Reports 530, 87–255 (2013)

    2013

  8. [4]

    Özer and M

    M. Özer and M. Taha, Physics Letters B171, 363–365 (1986)

    1986

Show all 118 references
  1. [5]

    5 dark matter and dark energy,

    V. Sahni, “5 dark matter and dark energy,” in The Physics of the Early Universe (Springer Berlin Heidel- berg, 2004) p. 141–179

    2004

  2. [6]

    Caldera-Cabral, R

    G. Caldera-Cabral, R. Maartens, and B. M. Schaefer, JCAP 2009, 027–027 (2009)

    2009

  3. [7]

    P. A. R. Adeet al. (Planck), Astron. Astrophys. 594, A13 (2016), arXiv:1502.01589 [astro-ph.CO]

    2016 arXiv

  4. [8]

    Kamionkowski and A

    M. Kamionkowski and A. G. Riess, Annual Review of Nuclear and Particle Science73, 153–180 (2023)

    2023

  5. [9]

    Di Valentino, O

    E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Class. Quant. Grav. 38, 153001 (2021), arXiv:2103.01183 [astro-ph.CO]

    2021 arXiv

  6. [10]

    Weinberg, Reviews of Modern Physics 61, 1–23 (1989)

    S. Weinberg, Reviews of Modern Physics 61, 1–23 (1989)

    1989

  7. [11]

    Padmanabhan, Physics Reports 380, 235–320 (2003)

    T. Padmanabhan, Physics Reports 380, 235–320 (2003)

    2003

  8. [12]

    This paper employs the 2.7 Python Machine Learning GaPP4 code to conduct the Gaussian Process (GP) re- gression [102] on Type Ia supernovae and on SGL sys- tems

    As mentioned earlier, we reconstruct these two ob- servables independently at the same lens redshifts as the DObs A,∆t(zl, zs) data from SGL. This paper employs the 2.7 Python Machine Learning GaPP4 code to conduct the Gaussian Process (GP) re- gression [102] on Type Ia supern...

    2024

  9. [13]

    Zlatev, L

    I. Zlatev, L. Wang, and P. J. Steinhardt, Physical Re- view Letters 82, 896–899 (1999)

    1999

  10. [14]

    Lombriser, Classical and Quantum Gravity 40, 155005 (2023)

    L. Lombriser, Classical and Quantum Gravity 40, 155005 (2023)

    2023

  11. [15]

    Aghanimet al

    N. Aghanimet al. (Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    2020 arXiv

  12. [16]

    A. G. Riesset al., Astrophys. J. Lett.934, L7 (2022), arXiv:2112.04510 [astro-ph.CO]

    2022 arXiv

  13. [17]

    R. A. Battye, T. Charnock, and A. Moss, Physical Re- view D 91 (2015), 10.1103/physrevd.91.103508

    2015 doi

  14. [18]

    W. L. Freedman, Nature Astronomy 1 (2017), 10.1038/s41550-017-0121

    2017 doi

  15. [19]

    J. S. Alcaniz, J. P. Neto, F. S. Queiroz, D. R. da Silva, and R. Silva, Scientific Reports 12 (2022), 10.1038/s41598-022-24608-5

    2022 doi

  16. [20]

    Pogosian, M

    L. Pogosian, M. Raveri, K. Koyama, M. Martinelli, A.Silvestri, G.-B.Zhao, J.Li, S.Peirone, andA.Zucca, Nature Astronomy 6, 1484–1490 (2022)

    2022

  17. [21]

    W. J. C. da Silva and R. Silva, The European Physical Journal C81 (2021), 10.1140/epjc/s10052-021-09177-7

    2021 doi

  18. [22]

    D’Eramo, R

    F. D’Eramo, R. Z. Ferreira, A. Notari, and J. L. Bernal, JCAP 2018, 014–014 (2018)

    2018

  19. [23]

    Refsdal, MNRAS128, 307 (1964)

    S. Refsdal, MNRAS128, 307 (1964)

    1964

  20. [24]

    K. C. Wong et al., MNRAS498, 1420–1439 (2019)

    2019

  21. [25]

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

    2024 arXiv

  22. [26]

    Birrer et al., Space Sci

    S. Birrer et al., Space Sci. Rev. 220, 48 (2024), arXiv:2210.10833 [astro-ph.CO]

    2024 arXiv

  23. [27]

    Moresco et al

    M. Moresco et al. , Living Rev. Rel. 25, 6 (2022), arXiv:2201.07241 [astro-ph.CO]

    2022 arXiv

  24. [28]

    A. J. Shajib et al., ssr220, 87 (2024), arXiv:2210.10790 [astro-ph.GA]

    2024 arXiv

  25. [29]

    Lemon et al., MNRAS 520, 3305 (2023), arXiv:2206.07714 [astro-ph.GA]

    C. Lemon et al., MNRAS 520, 3305 (2023), arXiv:2206.07714 [astro-ph.GA] . 9

    2023 arXiv

  26. [30]

    Treu et al., MNRAS 481, 1041 (2018), arXiv:1808.04838 [astro-ph.CO]

    T. Treu et al., MNRAS 481, 1041 (2018), arXiv:1808.04838 [astro-ph.CO]

    2018 arXiv

  27. [31]

    S. A. Rodney, G. B. Brammer, J. D. R. Pierel, J. Richard, S. Toft, K. F. O’Connor, M. Akhshik, and K. E. Whitaker, Nature Astron. 5, 1118 (2021), arXiv:2106.08935 [astro-ph.CO]

    2021 arXiv

  28. [32]

    P. L. Kelly et al. , Science 347, 1123 (2015), arXiv:1411.6009 [astro-ph.CO]

    2015 arXiv

  29. [33]

    Courbin et al., arXiv e-prints , arXiv:1706.09424 (2017), arXiv:1706.09424 [astro-ph.IM]

    F. Courbin et al., arXiv e-prints , arXiv:1706.09424 (2017), arXiv:1706.09424 [astro-ph.IM]

    2017 arXiv

  30. [34]

    Millon et al., Astron

    M. Millon et al., Astron. Astrophys.642, A193 (2020), arXiv:2006.10066 [astro-ph.CO]

    2020 arXiv

  31. [35]

    S. H. Suyuet al., Astrophys. J. Lett.788, L35 (2014), arXiv:1306.4732 [astro-ph.CO]

    2014 arXiv

  32. [36]

    S. Suyu, P. Marshall, M. Auger, S. Hilbert, R. Bland- ford, L. Koopmans, C. Fassnacht, and T. Treu, The Astrophysical Journal 711 (2009), 10.1088/0004- 637X/711/1/201

    2009 doi

  33. [37]

    Birrer et al., MNRAS 484, 4726 (2019), arXiv:1809.01274 [astro-ph.CO]

    S. Birrer et al., MNRAS 484, 4726 (2019), arXiv:1809.01274 [astro-ph.CO]

    2019 arXiv

  34. [38]

    K. C. Wong et al., MNRAS465, 4895 (2016)

    2016

  35. [39]

    G. C.-F. Chen et al., MNRAS490, 1743 (2019)

    2019

  36. [40]

    C. E. Rusu et al., MNRAS498, 1440 (2019)

    2019

  37. [41]

    S. H. Suyu et al., MNRAS468, 2590 (2017)

    2017

  38. [42]

    A. J. Shajib et al., MNRAS 494, 6072 (2020), https://academic.oup.com/MNRAS/article- pdf/494/4/6072/33212512/staa828.pdf

    2020

  39. [44]

    Millon et al., Astron

    M. Millon et al., Astron. Astrophys.639, A101 (2020), arXiv:1912.08027 [astro-ph.CO]

    2020 arXiv

  40. [45]

    Du, J.-J

    S.-S. Du, J.-J. Wei, Z.-Q. You, Z.-C. Chen, Z.-H. Zhu, and E.-W. Liang, MNRAS 521, 4963 (2023), arXiv:2302.13887 [astro-ph.CO]

    2023 arXiv

  41. [46]

    Collett, F

    T. Collett, F. Montanari, and S. Rasanen, Phys. Rev. Lett.123, 231101 (2019), arXiv:1905.09781 [astro- ph.CO]

    2019 arXiv

  42. [47]

    Qi, J.-W

    J.-Z. Qi, J.-W. Zhao, S. Cao, M. Biesiada, and Y. Liu, MNRAS 503, 2179 (2021), arXiv:2011.00713 [astro- ph.CO]

    2021 arXiv

  43. [48]

    Liao, Phys

    K. Liao, Phys. Rev. D 99, 083514 (2019), arXiv:1904.01744 [astro-ph.CO]

    2019 arXiv

  44. [49]

    X. Gong, T. Liu, and J. Wang, Eur. Phys. J. C84, 873 (2024)

    2024

  45. [50]

    Wei and F

    J.-J. Wei and F. Melia, apj 897, 127 (2020), arXiv:2005.10422 [astro-ph.CO]

    2020 arXiv

  46. [51]

    X. Li, R. E. Keeley, A. Shafieloo, and K. Liao, As- trophys. J. 960, 103 (2024), arXiv:2308.06951 [astro- ph.CO]

    2024 arXiv

  47. [52]

    Renzi and A

    F. Renzi and A. Silvestri, Phys. Rev. D107, 023520 (2023), arXiv:2011.10559 [astro-ph.CO]

    2023 arXiv

  48. [53]

    J. E. Gonzalez, M. Ferreira, L. R. Colaço, R. F. L. Holanda, and R. C. Nunes, Phys. Lett. B857, 138982 (2024), arXiv:2405.13665 [astro-ph.CO]

    2024 arXiv

  49. [54]

    L. R. Colaço, M. Ferreira, R. F. L. Holanda, J. E. Gonzalez, and R. C. Nunes, JCAP 05, 098 (2024), arXiv:2310.18711 [astro-ph.CO]

    2024 arXiv

  50. [55]

    Perivolaropoulos, Phys

    L. Perivolaropoulos, Phys. Rev. D110, 123518 (2024), arXiv:2408.11031 [astro-ph.CO]

    2024 arXiv

  51. [56]

    K. Liao, A. Shafieloo, R. E. Keeley, and E. V. Linder, Astrophys. J. Lett. 895, L29 (2020), arXiv:2002.10605 [astro-ph.CO]

    2020 arXiv

  52. [57]

    Scolnic et al

    D. Scolnic et al. , Astrophys. J. 938, 113 (2022), arXiv:2112.03863 [astro-ph.CO]

    2022 arXiv

  53. [59]

    M. Wang, X. Fu, B. Xu, Y. Huang, Y. Yang, and Z. Lu, Eur. Phys. J. C84, 702 (2024), arXiv:2407.12250 [astro- ph.CO]

    2024 arXiv

  54. [60]

    S. Cao, M. Biesiada, R. Gavazzi, A. Piórkowska, and Z.-H. Zhu, Astrophys. J. 806, 185 (2015), arXiv:1509.07649 [astro-ph.CO]

    2015 arXiv

  55. [61]

    Kumar, A

    D. Kumar, A. Rana, D. Jain, S. Mahajan, A. Mukher- jee, and R. F. L. Holanda, JCAP 01, 053 (2022), arXiv:2107.04784 [astro-ph.CO]

    2022 arXiv

  56. [62]

    A.Favale, A.Gómez-Valent, andM.Migliaccio, (2024), arXiv:2405.12142 [astro-ph.CO]

    2024 arXiv

  57. [63]

    R. F. L. Holanda, F. S. Lima, A. Rana, and D. Jain, Eur. Phys. J. C82, 115 (2022), arXiv:2104.01614 [astro- ph.CO]

    2022 arXiv

  58. [64]

    F. S. Lima, R. F. L. Holanda, S. H. Pereira, and W. J. C. da Silva, JCAP 08, 035 (2021), arXiv:2104.06202 [astro-ph.CO]

    2021 arXiv

  59. [65]

    R. F. L. Holanda, L. R. Colaço, S. H. Pereira, and R. Silva, JCAP06, 008 (2019), arXiv:1904.01342 [astro- ph.CO]

    2019 arXiv

  60. [66]

    R. S. Gonçalves, S. Landau, J. S. Alcaniz, and R. F. L. Holanda, JCAP 06, 036 (2020), arXiv:1907.02118 [astro-ph.CO]

    2020 arXiv

  61. [68]

    T. Yang, R. F. L. Holanda, and B. Hu, Astropart. Phys. 108, 57 (2019), arXiv:1710.10929 [astro-ph.CO]

    2019 arXiv

  62. [69]

    I. M. H. Etherington, General Relativity and Gravita- tion 39 (2007)

    2007

  63. [70]

    G. F. R. Ellis, General Relativity and Gravitation39 (2007)

    2007

  64. [71]

    K. U. Ratnatunga, R. E. Griffiths, and E. J. Ostrander, aj 117, 2010 (1999), arXiv:astro-ph/9902100 [astro-ph]

    1999 arXiv

  65. [72]

    B. A. Bassett and M. Kunz, Phys. Rev. D69, 101305 (2004), arXiv:astro-ph/0312443

    2004 arXiv

  66. [73]

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

    2010 arXiv

  67. [74]

    L. V. E. Koopmans, A. Bolton, T. Treu, O. Czoske, M. W. Auger, M. Barnabè, S. Vegetti, R. Gavazzi, L. A. Moustakas, and S. Burles, apjl703, L51 (2009), arXiv:0906.1349 [astro-ph.CO]

    2009 arXiv

  68. [75]

    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]

    2013 arXiv

  69. [76]

    Barnabè, O

    M. Barnabè, O. Czoske, L. V. E. Koopmans, T. Treu, and A. S. Bolton, MNRAS415, 2215 (2011), arXiv:1102.2261 [astro-ph.CO]

    2011 arXiv

  70. [78]

    S. Cao, M. Biesiada, M. Yao, and Z.-H. Zhu, MNRAS 461, 2192 (2016), arXiv:1604.05625 [astro-ph.CO]

    2016 arXiv

  71. [79]

    L. R. Colaço, J. E. Gonzalez, and R. F. L. Holanda, Eur. Phys. J. C81, 533 (2021), arXiv:2010.04021 [astro- ph.CO]

    2021 arXiv

  72. [80]

    E. O. Ofek, H.-W. Rix, and D. Maoz, Mon. Not. Roy. Astron. Soc. 343, 639 (2003), arXiv:astro-ph/0305201

    2003 arXiv

  73. [81]

    L. R. Colaço, R. F. L. Holanda, and R. Silva, Eur. Phys. J. C 81, 822 (2021), arXiv:2004.08484 [astro-ph.CO] . 10

    2021 arXiv

  74. [82]

    Jorgensen, M

    I. Jorgensen, M. Franx, and P. Kjaergaard, MNRAS 276, 1341 (1995)

    1995

  75. [83]

    S. Cao, J. Qi, M. Biesiada, X. Zheng, T. Xu, and Z.- H. Zhu, Astrophys. J.867, 50 (2018), arXiv:1810.01287 [astro-ph.CO]

    2018 arXiv

  76. [84]

    L. R. Colaço, R. F. L. Holanda, R. C. Nunes, and J. E. Gonzalez, (2022), arXiv:2201.04073 [astro-ph.CO]

    2022 arXiv

  77. [85]

    R. F. L. Holanda, K. Bora, and S. Desai, Eur. Phys. J. C 82, 526 (2022), arXiv:2105.10988 [astro-ph.CO]

    2022 arXiv

  78. [86]

    M. H. Amante, J. Magaña, V. Motta, M. A. García- Aspeitia, and T. Verdugo, MNRAS 498, 6013 (2020), arXiv:1906.04107 [astro-ph.CO]

    2020 arXiv

  79. [87]

    Martel, P

    H. Martel, P. Premadi, and R. Matzner, Astrophys. J. 570, 17 (2002), arXiv:astro-ph/0201198

    2002 arXiv

  80. [88]

    L. R. Colaço, S. J. Landau, J. E. Gonzalez, J. Spinelly, and G. L. F. Santos, JCAP08, 062 (2022), arXiv:2204.06459 [astro-ph.CO]

    2022 arXiv

  81. [89]

    C. R. Keeton, D. Christlein, and A. I. Zabludoff, As- trophys. J. 545, 129 (2000), arXiv:astro-ph/0007288

    2000 arXiv

  82. [90]

    Christlein, apj 545, 145 (2000), arXiv:astro- ph/0006450 [astro-ph]

    D. Christlein, apj 545, 145 (2000), arXiv:astro- ph/0006450 [astro-ph]

    2000

  83. [91]

    Grillo, M

    C. Grillo, M. Lombardi, and G. Bertin, Astron. Astro- phys. 477, 397 (2008), arXiv:0711.0882 [astro-ph]

    2008 arXiv

  84. [92]

    S. Cao, Y. Pan, M. Biesiada, W. Godlowski, and Z.-H. Zhu, JCAP 2012, 016 (2012), arXiv:1105.6226 [astro- ph.CO]

    2012 arXiv

  85. [93]

    K. C. Wong et al., MNRAS 498, 1420 (2020), arXiv:1907.04869 [astro-ph.CO]

    2020 arXiv

  86. [94]

    Melia, J.-J

    F. Melia, J.-J. Wei, and X.-F. Wu, aj149, 2 (2015), arXiv:1410.0875 [astro-ph.CO]

    2015 arXiv

  87. [95]

    Pandey, M

    S. Pandey, M. Raveri, and B. Jain, Phys. Rev. D102, 023505 (2020), arXiv:1912.04325 [astro-ph.CO]

    2020 arXiv

  88. [96]

    C. S. Kochanek, MNRAS 493, 1725 (2020), arXiv:1911.05083 [astro-ph.CO]

    2020 arXiv

  89. [97]

    Treu, araa 48, 87 (2010), arXiv:1003.5567 [astro- ph.CO]

    T. Treu, araa 48, 87 (2010), arXiv:1003.5567 [astro- ph.CO]

    2010 arXiv

  90. [98]

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

    1964

  91. [99]

    A. G. Riess et al. , Astrophys. J. 826, 56 (2016), arXiv:1604.01424 [astro-ph.CO]

    2016 arXiv

  92. [100]

    P. L. Kelly et al. , Science 380, abh1322 (2023), arXiv:2305.06367 [astro-ph.CO]

    2023 arXiv

  93. [101]

    C. E. Rasmussen and C. K. I. Williams,Gaussian pro- cesses for machine learning., Adaptive computation and machine learning (MIT Press, 2006) pp. I–XVIII, 1–248

    2006

  94. [102]

    R.NarayanandM.Bartelmann,in 13th Jerusalem Win- ter School in Theoretical Physics: Formation of Struc- ture in the Universe (1996) arXiv:astro-ph/9606001

    1996 arXiv

  95. [103]

    Seikel and C

    M. Seikel and C. Clarkson, (2013), arXiv:1311.6678 [astro-ph.CO]

    2013 arXiv

  96. [104]

    Seikel, C

    M. Seikel, C. Clarkson, and M. Smith, JCAP 2012, 036 (2012), arXiv:1204.2832 [astro-ph.CO]

    2012 arXiv

  97. [105]

    J. F. Jesus, R. Valentim, A. A. Escobal, and S. H. Pereira, JCAP04, 053 (2020), arXiv:1909.00090 [astro- ph.CO]

    2020 arXiv

  98. [106]

    Balmès and P

    I. Balmès and P. S. Corasaniti, MNRAS 431, 1528 (2013), arXiv:1206.5801 [astro-ph.CO]

    2013 arXiv

  99. [107]

    X. Li, R. E. Keeley, A. Shafieloo, X. Zheng, S. Cao, M. Biesiada, and Z.-H. Zhu, Mon. Not. Roy. Astron. Soc. 507, 919 (2021), arXiv:2103.16032 [astro-ph.CO]

    2021 arXiv

  100. [108]

    Yang, Z.-K

    T. Yang, Z.-K. Guo, and R.-G. Cai, Phys. Rev. D91, 123533 (2015), arXiv:1505.04443 [astro-ph.CO]

    2015 arXiv

  101. [109]

    Foreman-Mackey, D

    D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, pasp 125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

    2013 arXiv

  102. [110]

    Theχ2 function can be writ- ten as χ2 = (H0 − H Est 0,i )C −1 H Est 0 (H0 − H Est 0,i )T , (17) where H Est 0,i are the estimates of the Hubble rate sup- ported by Eq

    to perform the plots. Theχ2 function can be writ- ten as χ2 = (H0 − H Est 0,i )C −1 H Est 0 (H0 − H Est 0,i )T , (17) where H Est 0,i are the estimates of the Hubble rate sup- ported by Eq. 10, and H0 is a free parameter. The quantity C −1 H Est 0 is is the inverse of the cova...

    2018

  103. [111]

    J. F. Jesus, R. Valentim, A. A. Escobal, S. H. Pereira, and D. Benndorf, JCAP 11, 037 (2022), arXiv:2112.09722 [astro-ph.CO]

    2022 arXiv

  104. [112]

    Lewis, (2019), arXiv:1910.13970 [astro-ph.IM]

    A. Lewis, (2019), arXiv:1910.13970 [astro-ph.IM]

    2019 arXiv

  105. [113]

    T. Liu, X. Yang, Z. Zhang, J. Wang, and M. Biesiada, Phys. Lett. B 845, 138166 (2023), arXiv:2308.15731 [astro-ph.CO]

    2023 arXiv

  106. [114]

    Renzi, N

    F. Renzi, N. B. Hogg, and W. Giarè, MNRAS 513, 4004 (2022), arXiv:2112.05701 [astro-ph.CO]

    2022 arXiv

  107. [115]

    Meylan, P

    G. Meylan, P. Jetzer, P. North, P. Schneider, C. S. Kochanek, and J. Wambsganss, eds., Saas-Fee Ad- vanced Course 33: Gravitational Lensing: Strong, Weak and Micro (2006) arXiv:astro-ph/0407232 [astro-ph]

    2006 arXiv

  108. [116]

    Khadka, S

    N. Khadka, S. Birrer, A. Leauthaud, and H. Nix, mnras 533, 795 (2024), arXiv:2404.01513 [astro-ph.CO]

    2024 arXiv

  109. [117]

    Birrer, A

    S. Birrer, A. Amara, and A. Refregier, JCAP 2016, 020 (2016), arXiv:1511.03662 [astro-ph.CO]

    2016 arXiv

  110. [118]

    T. Treu, S. H. Suyu, and P. J. Marshall, Astron. Astro- phys.Rev. 30,8(2022),arXiv:2210.15794[astro-ph.CO]

    2022 arXiv

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