The SLACS strong lens sample, debiased. II. Lensing-only constraints on the stellar IMF and dark matter contraction in early-type galaxies

Alessandro Sonnenfeld (1) ((1) Shanghai Jiao Tong University)

REVIEW 3 major objections 4 minor 32 references

The SLACS strong lens sample, debiased. II. Lensing-only constraints on the stellar IMF and dark matter contraction in early-type galaxies

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read Using only gravitational lensing data from 59 SLACS lenses, a selection-corrected analysis constrains the stellar IMF and dark matter contraction to a degenerate ridge that rules out IMFs heavier than Salpeter and finds SLACS velocity…

desk verdict Careful lensing-only SLACS reanalysis with new selection-bias corrections; the unresolved gamma_PL sign mismatch makes the central result conditional, but it deserves a serious referee. read the letter →

arxiv 2501.02054 v1 pith:CIQETYF3 submitted 2025-01-03 astro-ph.GA

Alessandro Sonnenfeld (1) ((1) Shanghai Jiao Tong University) This is my paper · ORCID

classification astro-ph.GA

keywords stronggravitationallensingSLACSstellarinitialmassfunctiondarkmattercontractionadiabaticselectioneffectsvelocitydispersionbiasearly-typegalaxies

topics Dark Matter

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

This paper tries to constrain two things at once from gravitational lensing alone: the stellar population synthesis mismatch parameter $\alpha_{\mathrm{sps}}$ (how much heavier the true stellar mass is than a Chabrier-IMF stellar population model predicts) and the contraction efficiency $\epsilon$ of dark matter responding to baryon infall. Analysing 59 SLACS lenses with only projected lensing data, a weak lensing halo-mass prior, and a modelled selection function, the author finds that the data do not pick a unique point but pin down a degenerate ridge: either $\log\alpha_{\mathrm{sps}}=0.22$ with no contraction, or $\log\alpha_{\mathrm{sps}}=0$ with near-maximal contraction. Either way, IMFs heavier than Salpeter are ruled out. The analysis also quantifies two selection biases: ignoring selection shifts $\log\alpha_{\mathrm{sps}}$ by $0.04$ dex, and SLACS lenses' observed velocity dispersions are inflated by $5\%$ overall, split into $3\%$ from selection of intrinsically high-dispersion galaxies and $2\%$ from observational scatter. This matters because many previous conclusions about the IMF and dark matter in massive galaxies, and the use of SLACS kinematics in time-delay cosmography, rest on joint lensing-plus-dynamics analyses that have not been corrected for these selection effects.

What carries the argument

The machinery is the statistical strong lensing framework of Equation (1), which writes the probability of a lens-source pair as the product of a foreground galaxy distribution, an effective source redshift distribution, and a selection probability. The galaxy population is described by a two-component mass model: stars in a de Vaucouleurs profile with mass $\alpha_{\mathrm{sps}} M_{*}^{\mathrm{(sps)}}$, and a dark halo obtained by applying the adiabatic contraction prescription of Blumenthal et al. (1986) with efficiency $\epsilon$ to an NFW profile and then approximating the result by a generalised NFW profile matched at the half-light radius. The selection probability uses the strong lensing cross-section and the observed velocity dispersion through a lens-finding probability, which is what lets the model correct for the overdensity of high-dispersion lenses and predict the bias in the measured $\sigma_{\mathrm{ap}}$.

What would settle it

A concrete test is to measure $\gamma_{\mathrm{PL}}$ from high-quality imaging with an independent, robust method on a large sample of SLACS lenses. If the measured $\gamma_{\mathrm{PL}}$-$\Sigma_*^{(\mathrm{sps})}$ correlation remains negative once systematic biases such as PSF, source model, and azimuthal structure are controlled, the paper's model, which predicts a positive correlation, is falsified. A second check is a survey with a fully characterised selection function: the model predicts about 20 percent more lenses at $\epsilon=0.8$ than at $\epsilon=0$, so the observed lens number density would select between the two allowed scenarios.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that lensing-only data from SLACS, once the selection function is modelled, constrain the $(\alpha_{\mathrm{sps}},\epsilon)$ plane tightly enough to exclude a Salpeter-or-heavier stellar IMF, while leaving a Chabrier-like IMF with strong dark matter contraction equally plausible as a slightly sub-Salpeter IMF with no contraction. The same model yields a bias budget for the sample: SLACS lenses have intrinsically higher velocity dispersion by $3\%$ than parent-population galaxies of the same stellar mass, size, and halo mass, and their observed SDSS velocity dispersions are a further $2\%$ higher, for a total $5\%$ upward shift. This is presented as the lensing-only counterpart to earlier joint analyses, with selection effects, not stellar dynamics assumptions, carrying the difference from previous results.

Load-bearing premise

The load-bearing premise is that the assumed two-component model of each lens, spherical de Vaucouleurs stars plus a dark matter halo contracted by a single efficiency, is complete enough to describe the real galaxies; if the true mass distributions differ, the inferred $\alpha_{\mathrm{sps}}$-$\epsilon$ ridge and the 5 percent velocity-dispersion bias would shift.

Editorial extensions

If this is right

At fixed dark matter profile, the inferred $\log\alpha_{\mathrm{sps}}$ is about $0.04$ dex lower than values from joint lensing and dynamics analyses, an amount fully attributable to modelling the selection function.
The $3\%$ intrinsic and $2\%$ observational velocity-dispersion biases mean SLACS kinematics should only be interpreted with selection-aware priors, including their use in calibrating time-delay lens models for the Hubble constant.
Current measurements of the lensing-only power-law slope $\gamma_{\mathrm{PL}}$ are inconsistent with the model: they are anti-correlated with stellar density where the model predicts a positive correlation, so they cannot yet be used to break the $\alpha_{\mathrm{sps}}$-$\epsilon$ degeneracy.
Robust measurements of the radial magnification ratio, or of the number density of lenses in a well-characterised survey, could separate the two allowed scenarios; a sample of about 100 lenses would distinguish $\epsilon=0.8$ from $\epsilon=0$ if the selection function and source and foreground densities were known.

Reading between the lines

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

If the selection-debiased lensing-only result survives, then the heavy Salpeter-like stellar IMFs inferred from many joint lensing and dynamics studies of early-type galaxies are partly a selection artefact rather than a property of the galaxy population; the paper's comparison with Shajib et al. (2021) quantifies this as exactly the $0.04$ dex selection shift.
The model's positive $\gamma_{\mathrm{PL}}$-$\Sigma_*$ correlation is a falsifiable prediction; if future robust radial-magnification measurements confirm the observed negative trend, the two-component mass model or the assumption that $\alpha_{\mathrm{sps}}$ and $\epsilon$ are universal would need revision rather than the observations being dismissed.
A natural extension is to apply the same selection-aware lensing-only framework to larger forthcoming lens samples, where number density information could break the $\alpha_{\mathrm{sps}}$-$\epsilon$ degeneracy that persists here; the paper's own posterior-predicted mocks are released for exactly such tests.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Quick take on 2501.02054. The genuinely new thing is a selection-corrected, lensing-only analysis of SLACS that maps out the alpha_sps-epsilon degeneracy and shows that ignoring selection shifts log alpha_sps by 0.04 dex and produces a 5% upward bias in measured velocity dispersion. The bias number is the kind of correction people will want to apply to a decade of SLACS-based results, so this paper deserves careful reading.

What I liked: the statistical machinery is careful, the posterior predictive tests on Einstein radii pass, and the paper is unusually transparent about what it cannot do. The MCMC chains and mock samples are online — real reproducibility credit. The gamma_PL prediction is a genuine out-of-sample test: the model predicts a positive correlation between gamma_PL and stellar density, and observations from Shajib+, Etherington+, and Tan+ show an anti-correlation. The stress-test note is right that this is load-bearing, not a side remark. The paper's response is to suspect systematic errors in the measured gamma_PL (azimuthal structure, PSF, source modeling). Plausible, since the three measurement sets disagree with each other on individual lenses, but it is not independently tested. Until that suspicion is confirmed, the mass model remains under a cloud. The paper explicitly leaves this to future work, which is honest, but it makes the headline result conditional.

Two smaller soft spots. First, the prior on log alpha_sps is uniform over (0,0.3), so the solution at log alpha_sps=0 sits at the boundary; the Chabrier-IMF branch is partly a prior edge effect. Second, the 5% bias splits into 3% intrinsic selection and 2% observational scatter, but Knabel et al. (2024) find SDSS sigma underestimated by a few percent — opposite sign to the model's 2% — and the paper's 'just shifts the zero point' response is quick. The intrinsic selection bias is on firmer ground.

That said, the central inference — lensing-only data cannot individually pin down IMF and contraction, but do rule out IMFs heavier than Salpeter without halo expansion — holds up. I'd send it to a careful referee, mainly to push on the gamma_PL discrepancy and the prior sensitivity. It is a useful, honest reanalysis that the field needs.

Referee Report

3 major / 4 minor

Summary. This paper reanalyzes 59 SLACS strong lenses using only lensing observables (Einstein radii) plus a weak-lensing-based prior on the halo mass distribution of the parent sample, with the SLACS selection function modeled explicitly. The population model uses a de Vaucouleurs stellar component plus an adiabatically contracted NFW dark halo approximated by a gNFW profile, and it treats the stellar-population mismatch parameter alpha_sps and the contraction efficiency epsilon as universal across the population. The main results are a degenerate constraint in the (log alpha_sps, epsilon) plane, with log alpha_sps approximately 0.22 at epsilon = 0 and log alpha_sps approximately 0 at epsilon = 0.8; a 0.04 dex selection-induced shift in alpha_sps; and a predicted 5% upward bias of SLACS velocity dispersions relative to parent galaxies, split into 3% intrinsic selection bias and 2% observational bias. Posterior predictive checks on the Einstein radius distribution pass, but the model's predicted relation between the lensing-only power-law slope gamma_PL and stellar surface density is opposite in sign to the measurements of Shajib et al. (2021), Etherington et al. (2022), and Tan et al. (2024). The paper attributes this discrepancy to systematic errors in the measurements and leaves alternative mass models to future work.

Significance. If correct, the paper would put lensing-only determinations of the stellar IMF and dark-matter contraction on a selection-corrected footing, and would imply that previous joint lensing and dynamics analyses need a 0.04 dex selection correction before comparison with simulations or cosmography. The predicted 5% velocity-dispersion bias is also directly relevant to the use of SLACS lenses in time-delay cosmography. The paper has clear strengths: it makes the selection function an explicit part of the inference, it uses posterior predictive tests that go beyond simple chi-square checks, it releases the MCMC chains and posterior predicted mocks, and it presents the gamma_PL discrepancy as a sharp falsifiable prediction rather than hiding it. However, the central inference and the gamma_PL prediction rely on the same two-component spherical mass model, so the failure of the gamma_PL prediction is load-bearing for the (alpha_sps, epsilon) ridge and for the selection-bias estimates. The manuscript is internally consistent, but the central claim is not yet established because this discrepancy is attributed to observational systematics without an independent test.

major comments (3)

[Section 3.4 / Figure 7] The posterior predictive distribution for beta_gamma_PL assigns 0% probability to reproducing the negative beta_gamma_PL measured by Shajib et al. (2021), Etherington et al. (2022), and Tan et al. (2024). This is a load-bearing discrepancy rather than a peripheral one: gamma_PL and the Einstein radius are both projected quantities of the same total mass distribution, so a model that matches the theta_E distribution but fails the gamma_PL-Sigma_star correlation can still return biased values of the (alpha_sps, epsilon) ridge and of the selection corrections. The paper's explanation in terms of azimuthal structure, PSF, and source-modeling systematics is plausible but is not tested here. I request a concrete robustness test: either include the three gamma_PL datasets in the likelihood, with a per-dataset systematic term if needed, and show whether the posterior in Figure 2 is displaced, or show that an alternative mass model that reproduces the observed anti-correlation leaves the Einstein-radius inference unchanged. Until one of these is done, the abstract should describe the gamma_PL tension as an unresolved problem for the mass model, not as evidence against the measurements.
[Section 2.3 / Equations 8, 10, 30-31] The predicted gamma_PL is computed from the gNFW approximation to the adiabatically contracted profile, not from the exact solution of Equation 8. Figure 1 shows that the gNFW density is accurate to better than 10% at most radii, but gamma_PL depends on the second and third derivatives of the lensing potential at the Einstein radius, so a 10% density error can translate into substantially larger errors in gamma_PL and potentially in the sign of its correlation with Sigma_star. Please recompute Figure 6 for a subset of posterior draws using the exact contracted profile and report the change in beta_gamma_PL. If the sign is not robust, the posterior predictive test in Section 3.4 cannot be used as a clean falsification of the mass model, and the comparison with the three datasets should be reframed accordingly.
[Section 4 / Knabel et al. discussion] The treatment of the Knabel et al. (2024) result is too quick. Those authors find that SDSS sigma_ap values are underestimated by a few percent, which is the opposite direction of the 2% observational bias reported in Section 3.2. The paper states that a common SDSS bias would simply shift mu_sigma,0 and leave the SLACS-parent difference unchanged, but Equation 23 feeds the observed sigma_ap into the selection probability and Equation 19 calibrates S(sap) to the same SDSS measurements. Please verify this claim quantitatively in the posterior predictive mocks, for example by adding a constant -0.03 dex offset to the noisy s_obs before applying Pfind, and report whether the 5% total bias and the 3%/2% split survive. If the bias changes, the secondary claim should be qualified accordingly.

minor comments (4)

[Table 1] The row for sigma_sigma is labelled 'Scatter in gamma around the mean'; the context of Equation 19 shows this should be the scatter in log sigma_ap, not gamma.
[Section 2.5 / Equation 24] Please clarify that theta_E^(est) uses the SDSS observed sigma_ap including any systematic offset, since a constant template bias in sigma would propagate into Pfind and into the interpretation of the 2% observational bias.
[Figure 6 caption] The caption does not identify which line style or marker corresponds to each of the three datasets; adding this would improve the print legibility.
[Conclusions / Section 4] The statement that IMFs heavier than Salpeter are disfavoured should explicitly note that it holds only within the prior log alpha_sps < 0.3 and the assumed absence of halo expansion (epsilon >= 0); the abstract currently implies a stronger, prior-independent bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central (alpha_sps, epsilon) inference is driven by Einstein-radius, SPS-mass and external weak-lensing data, and the gamma_PL comparison is a genuine out-of-sample test.

full rationale

The paper's central inference on (alpha_sps, epsilon) is not defined into existence. alpha_sps is defined in Eq. 5 and constrained by Einstein radii, SPS stellar-mass measurements, and a weak-lensing halo-mass prior from Sonnenfeld et al. (2018), which is based on HSC data rather than on the target results. The selection function and source-redshift distribution from Paper I are inputs, not outputs of this analysis: they could be wrong, but their adoption is a normal use of prior work, not a circular reduction. The gamma_PL posterior predictive test in Sec. 3.4 is a genuine out-of-sample falsifier: the model predicts a positive correlation between gamma_PL and stellar surface density, while three independent datasets (Shajib et al. 2021; Etherington et al. 2022; Tan et al. 2024) show the opposite, and the paper reports that only a handful of 10000 mocks match. This demonstrates that the model is not post hoc fitted to those lensing-only slope measurements. The velocity-dispersion bias in Sec. 3.2 is a posterior prediction of the fitted model and is labeled as such; it is model-dependent, but it is not a fitted parameter renamed as an independent prediction. The paper also explicitly acknowledges that the mass model might be inaccurate and leaves that alternative to future work; that is a stated limitation and a correctness risk, not circularity. Overall, no step satisfies the evidentiary bar for circularity: no equation reduces to its own input, and no fitted quantity is presented as an independent prediction while being identical to the fit by construction.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central inference depends on a chain of modeling assumptions inherited from Paper I and standard galaxy formation practice. Free parameters are all fit within the Bayesian model with stated priors. The paper introduces no new physical entities; epsilon and the fundamental hyper-plane are parameterizations. The largest unvalidated input is the selection function factorization, followed by universal alpha_sps and epsilon and the gNFW approximation.

free parameters (5)

log alpha_sps (mean stellar population synthesis mismatch) = 0.07 +/- 0.07 marginalized; ridge from 0.22 at epsilon=0 to 0 at epsilon=0.8
Target parameter with uniform prior U(0,0.3); rescales Chabrier-based SPS stellar masses.
epsilon (dark matter contraction efficiency) = 0.49 +/- 0.30 marginalized
Target parameter with uniform prior U(0,1); interpolates between no contraction and maximal adiabatic contraction.
Fundamental hyper-plane parameters (mu_sigma,0, beta_sigma, xi_sigma, nu_sigma, sigma_sigma) = 2.352 +/- 0.014, 0.33 +/- 0.03, -0.45 +/- 0.09, 0.06 +/- 0.05, 0.035 +/- 0.009
Empirical scaling of log sigma_ap with stellar mass, size and halo mass; fit to SDSS velocity dispersions and used for selection modeling and sigma bias inference.
Halo mass distribution parameters (mu_h,0, beta_h, sigma_h) = 13.03 +/- 0.04, 1.55 +/- 0.13, 0.32 +/- 0.03
Weak lensing measurements from Sonnenfeld+2018 used as Gaussian priors and re-fit within the full model.
Lens finding selection parameters (theta_0, log a) = 0.84 +/- 0.06, 1.09 +/- 0.13
Logistic model for photometric follow-up probability as a function of estimated Einstein radius.

assumptions (7)

domain assumption NFW initial dark matter profile with Dutton and Maccio 2014 mass-concentration relation and no scatter
Sets the initial halo for adiabatic contraction in Section 2.3, Equation 9; scatter in the concentration-mass relation is neglected.
domain assumption Adiabatic contraction with circular orbits and invariant rM(r), with efficiency parameter epsilon
Blumenthal+1986 prescription used in Equation 8; not truly adiabatic or circular, acknowledged in Section 2.3.
domain assumption gNFW profile matched to the contracted profile in density and slope at r = Re approximates lensing properties
Figure 1 shows matching to better than 10 percent at most radii, but this approximation may affect lensing predictions.
ad hoc to paper alpha_sps and epsilon are universal over the population, enforced as delta functions in Equation 16
Ignores intrinsic scatter in the IMF and dark matter contraction efficiency across galaxies.
domain assumption Strong lens selection probability factorizes into geometry and source brightness factors, with source properties captured by an effective Gaussian in source redshift
Key approximation inherited from Paper I, used in Equations 1 and 21; allows dropping explicit source brightness dependence.
domain assumption Halo mass distribution is independent of epsilon and alpha_sps, and the weak lensing prior from Sonnenfeld+2018 applies
Justified in Section 2.4 by negligible differences between contracted and NFW fits to weak lensing data.
standard math Flat Lambda CDM with H0 = 70 km/s/Mpc and Omega_m = 0.3
Cosmological model used to compute angular diameter distances throughout the analysis.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The SLACS strong lens sample, debiased. II. Lensing-only constraints on the stellar IMF and dark matter contraction in early-type galaxies." pith.science (2026). https://pith.science/paper/CIQETYF3

@misc{pith2026250102054,
  author       = {Pith},
  title        = {Pith review of: The SLACS strong lens sample, debiased. II. Lensing-only constraints on the stellar IMF and dark matter contraction in early-type galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CIQETYF3}},
  note         = {Machine review of arXiv:2501.02054}
}

abstract

The Sloan Lens ACS (SLACS) is the best studied sample of strong lenses to date. Much of our knowledge of the SLACS lenses has been obtained by combining strong lensing with stellar kinematics constraints. However, interpreting stellar kinematics data is difficult: it requires reconstructing the three-dimensional structure of a galaxy and the orbits of its stars. In this work we pursued an alternative approach to the study of galaxy structure with SLACS, based purely on gravitational lensing data. The primary goal of this study is to constrain the stellar population synthesis mismatch parameter $\alpha_{sps}$, quantifying the ratio between the true stellar mass of a galaxy and that obtained with a reference stellar population synthesis model, and the efficiency of the dark matter response to the infall of baryons, $\epsilon$. We combined Einstein radius measurements from the SLACS lenses with weak lensing information from their parent sample, while accounting for selection effects. The data can be fit comparatively well by a model with $\log{\alpha_{sps}}=0.22$ and $\epsilon=0$, corresponding to an IMF slightly lighter than Salpeter and no dark matter contraction, or $\log{\alpha_{sps}}=0$ and $\epsilon=0.8$, equivalent to a Chabrier IMF and almost maximal contraction. This degeneracy could be broken with lensing-only measurements of the projected density slope, but existing data are completely inconsistent with our model. We suspect systematic errors in the measurements to be at the origin of this discrepancy. Number density constraints would also help break the degeneracy. Because of selection effects, SLACS lenses have a larger velocity dispersion than galaxies with the same projected mass distribution, and their velocity dispersion is overestimated. These two biases combined produce a $5\%$ upward shift in the observed velocity dispersion.

Figures

Figures reproduced from arXiv: 2501.02054 by the authors.

**Figure 2.** Posterior probability in the dark matter contraction and stellar population synthesis parameters. The fiducial inference is shown by the solid think contours. Filled contours show the inference obtained by ignoring selection effects altogether. Contour levels correspond to 68% and 95% enclosed probability. (ϵ, αsps) that are consistent with the data. The Jeans model prediction can match the inferred values of µσ,0,… view at source ↗

**Figure 3.** Posterior probability of the parameters describing the distribution in velocity dispersion. These are defined in Equation 19 and Equation 20. Filled purple contours: posterior probability of the model. Solid pink contours: posterior predicted fundamental hyperplane of the SLACS lenses. Dashed black contours: posterior predicted fundamental hyperplane of the SLACS lenses, based on the observed (noisy) velocity dis… view at source ↗

**Figure 4.** Posterior predicted distribution in the observed and true µσ,0 of SLACS lenses, as a function of the corresponding value of the parent population. The quantity µσ,0 describes the average logσap of galaxies with log M (sps) ∗ = 11.3, average size and average halo mass for their stellar mass. ties, our model predicts that SLACS lenses have a 3% larger velocity dispersion, compared to that of galaxies of the same stel… view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Posterior predictive test of goodness-offit. Vertical lines indicate the observed values of the four test quantities. In each panel, the percentage to the left and right of the vertical line indicate the fraction of posterior predicted samples for which the mock tes…

**Figure 6.** Figure 6: Posterior predicted distribution in the lensing-only power-law slope, as a function of stellar mass density. Simulated values of γPL have been obtained from Equation 31. The two sets of contours correspond to the posteriors obtained by fitting models with a fixed value…

**Figure 6.** Figure 6: The distributions are different because the observed samples and the observational uncertainties vary from one work to the other. In each panel, the percentages to the left and right of the vertical lines indicate the fraction of posterior predicted samples for which…

**Figure 8.** Figure 8: Posterior predicted number density of SLACS lenses, as a function of the contraction efficiency parameter. The dashed line shows the average nlens, obtained by marginalising over the uncertainty on the model parameters. their combination that allowed us to rule out la…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 19 canonical work pages

[1]

W., Treu, T., Bolton, A

Auger, M. W., Treu, T., Bolton, A. S., et al. 2009, ApJ, 705, 1099

2009
[2]

W., Treu, T., Gavazzi, R., et al

Auger, M. W., Treu, T., Gavazzi, R., et al. 2010b, ApJ, 721, L163 Barnabè, M., Czoske, O., Koopmans, L. V . E., Treu, T., & Bolton, A. S. 2011, MNRAS, 415, 2215

work page 2011
[3]

J., Galan, A., et al

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

2020
[4]

R., Faber, S

Blumenthal, G. R., Faber, S. M., Flores, R., & Primack, J. R. 1986, ApJ, 301, 27

1986
[5]

S., Burles, S., Koopmans, L

Bolton, A. S., Burles, S., Koopmans, L. V . E., Treu, T., & Moustakas, L. A. 2006, ApJ, 638, 703

work page 2006
[6]

J., et al

Cautun, M., Benítez-Llambay, A., Deason, A. J., et al. 2020, MNRAS, 494, 4291

2020
[7]

R., Schaye, J., Kay, S

Duffy, A. R., Schaye, J., Kay, S. T., et al. 2010, MNRAS, 405, 2161

2010
[8]

Dutton, A. A. & Macciò, A. V . 2014, MNRAS, 441, 3359

2014

Show all 32 references

[9]

A., van den Bosch, F

Dutton, A. A., van den Bosch, F. C., Dekel, A., & Courteau, S. 2007, ApJ, 654, 27

2007
[10]

W., Massey, R., et al

Etherington, A., Nightingale, J. W., Massey, R., et al. 2022, MNRAS, 517, 3275

2022
[11]

W., Massey, R., et al

Etherington, A., Nightingale, J. W., Massey, R., et al. 2023, MNRAS, 521, 6005

2023
[12]

2024, A&A, 692, A87

Galan, A., Vernardos, G., Minor, Q., et al. 2024, A&A, 692, A87

2024
[13]

Y ., Kravtsov, A

Gnedin, O. Y ., Kravtsov, A. V ., Klypin, A. A., & Nagai, D. 2004, ApJ, 616, 16

2004
[14]

R., Sluse, D., Van de Vyvere, L., et al

Gomer, M. R., Sluse, D., Van de Vyvere, L., et al. 2023, A&A, 679, A128

2023
[15]

Hyde, J. B. & Bernardi, M. 2009, MNRAS, 394, 1978

2009
[16]

2024, arXiv e-prints, arXiv:2409.10631

Knabel, S., Treu, T., Cappellari, M., et al. 2024, arXiv e-prints, arXiv:2409.10631

2024 arXiv
[17]

Kochanek, C. S. 2020, MNRAS, 493, 1725

2020
[18]

Kochanek, C. S. 2021, MNRAS, 501, 5021

2021
[19]

Koopmans, L. V . E., Treu, T., Bolton, A. S., Burles, S., & Moustakas, L. A. 2006, ApJ, 649, 599

2006
[20]

2015, MNRAS, 446, 493

Posacki, S., Cappellari, M., Treu, T., Pellegrini, S., & Ciotti, L. 2015, MNRAS, 446, 493

2015
[21]

S., Bower, R

Schaller, M., Frenk, C. S., Bower, R. G., et al. 2015, MNRAS, 451, 1247

2015
[22]

J., Treu, T., Birrer, S., & Sonnenfeld, A

Shajib, A. J., Treu, T., Birrer, S., & Sonnenfeld, A. 2021, MNRAS, 503, 2380

2021
[23]

J., Treu, T., et al

Sheu, W., Shajib, A. J., Treu, T., et al. 2024, arXiv e-prints, arXiv:2408.10316

2024 arXiv
[24]

2018, MNRAS, 474, 4648

Sonnenfeld, A. 2018, MNRAS, 474, 4648

2018
[25]

2024, A&A, 690, A325, (Paper I)

Sonnenfeld, A. 2024, A&A, 690, A325, (Paper I)

2024
[26]

& Cautun, M

Sonnenfeld, A. & Cautun, M. 2021, A&A, 651, A18

2021
[27]

W., et al

Sonnenfeld, A., Leauthaud, A., Auger, M. W., et al. 2018, MNRAS, 481, 164

2018
[28]

2013, ApJ, 777, 98

Sonnenfeld, A., Treu, T., Gavazzi, R., et al. 2013, ApJ, 777, 98

2013
[29]

J., et al

Sonnenfeld, A., Treu, T., Marshall, P. J., et al. 2015, ApJ, 800, 94

2015
[30]

Y ., Shajib, A

Tan, C. Y ., Shajib, A. J., Birrer, S., et al. 2024, MNRAS, 530, 1474

2024
[31]

W., Koopmans, L

Treu, T., Auger, M. W., Koopmans, L. V . E., et al. 2010, ApJ, 709, 1195 Van de Vyvere, L., Sluse, D., Gomer, M. R., & Mukherjee, S. 2022, A&A, 663, A179

2010
[32]

2024, A&A, 690, A390 Acknowledgements

Zhou, Q., Sonnenfeld, A., & Hoekstra, H. 2024, A&A, 690, A390 Acknowledgements. This work was supported by the National Key R&D Pro- gram of China (No. 2023YFA1607800, 2023YFA1607802). Article number, page 10 of 10

2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

W., Treu, T., Bolton, A

Auger, M. W., Treu, T., Bolton, A. S., et al. 2009, ApJ, 705, 1099

2009

[2] [2]

W., Treu, T., Gavazzi, R., et al

Auger, M. W., Treu, T., Gavazzi, R., et al. 2010b, ApJ, 721, L163 Barnabè, M., Czoske, O., Koopmans, L. V . E., Treu, T., & Bolton, A. S. 2011, MNRAS, 415, 2215

work page 2011

[3] [3]

J., Galan, A., et al

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

2020

[4] [4]

R., Faber, S

Blumenthal, G. R., Faber, S. M., Flores, R., & Primack, J. R. 1986, ApJ, 301, 27

1986

[5] [5]

S., Burles, S., Koopmans, L

Bolton, A. S., Burles, S., Koopmans, L. V . E., Treu, T., & Moustakas, L. A. 2006, ApJ, 638, 703

work page 2006

[6] [6]

J., et al

Cautun, M., Benítez-Llambay, A., Deason, A. J., et al. 2020, MNRAS, 494, 4291

2020

[7] [7]

R., Schaye, J., Kay, S

Duffy, A. R., Schaye, J., Kay, S. T., et al. 2010, MNRAS, 405, 2161

2010

[8] [8]

Dutton, A. A. & Macciò, A. V . 2014, MNRAS, 441, 3359

2014