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Unified weak lensing constraints on the evolution of the mass -- X-ray luminosity relation for galaxy clusters

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Pith's one-line read A unified weak-lensing calibration of 100 CODEX galaxy clusters finds that the mass–X-ray luminosity slope is shallow in RASS data ($\beta=0.75\pm0.09$) but steep in eRASS1 data ($\beta=1.11\pm0.15$), with no redshift evolution.

desk verdict Useful new CODEX-LS catalog and an honest Bayesian M-LX calibration, but the headline RASS slope is not robust: the paper's own model variants shift beta from 0.30 to 0.84, so the 1.7-sigma deviation from self-similarity should not be taken at face value. read the letter →

arxiv 2505.21659 v1 pith:KEDNWFTT submitted 2025-05-27 astro-ph.CO

Isabel Pederneiras , Alexis Finoguenov , Eduardo Cypriano , Johan Comparat , Kimmo Kiiveri , Ludovic van Waerbeke , Renato Dupke This is my paper · ORCID
classification astro-ph.CO
keywords galaxyclustersweakgravitationallensingX-rayastronomymass–X-rayluminosityrelationCODEXcatalogeROSITAROSATAll-SkySurveyclusterselectionfunction
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 tries to establish a reliable, selection-corrected calibration of the galaxy cluster mass–X-ray luminosity relation using weak-lensing masses, and to measure whether that relation evolves with redshift. It builds the all-sky CODEX-LS catalog from LEGACY imaging and applies a hierarchical Bayesian model to 100 clusters with lensing masses, combining three subsamples. The result is a slope of $\beta = 0.75\pm0.09$ with RASS luminosities, shallower than the self-similar prediction, while the same framework on 42 eROSITA-matched clusters yields $\beta = 1.11\pm0.15$, consistent with self-similarity. The paper concludes that redshift evolution of the normalization is not required and that full modeling of selection effects is essential for any such calibration. A correct calibration matters because the $M$–$L_X$ link is how X-ray cluster surveys turn observed luminosities into masses for cosmological constraints.

What carries the argument

The load-bearing object is the full selection-corrected likelihood of Eqs. 15–19, whose richness-dependent sampling function $P(I_{\mathrm{samp}}|\tilde\lambda)$ is fit to the ratio of the weak-lensing sample to the cleaned CODEX catalog inside the survey areas. This function, combined with the CODEX X-ray selection grid, the RASS richness cut, the redMaPPer optical completeness, and the richness–mass relation $\ln\lambda = 4.42 + 0.49\ln(M/10^{14.81}M_\odot)$ from Kiiveri et al. (2021), converts an idealized power-law relation into the probability of observing the measured luminosities, masses, and redshifts. The halo mass function and the Poisson X-ray count likelihood are what allow the model to separate intrinsic scatter and redshift evolution from selection and measurement effects.

What would settle it

A decisive check would be to measure the $M$–$L_X$ slope from a weak-lensing sample selected independently of CODEX richness within the same sky areas: if the recovered slope differs from the paper's $\beta=0.75$ by more than the reported uncertainty, the richness-based sampling correction is incomplete; alternatively, injecting simulated point sources into RASS images and recovering the luminosity bias should reproduce the flattening if contamination is the cause.

Watch

Extended reading notes

Core claim

Using 100 CODEX galaxy clusters with weak-lensing mass measurements, the authors fit the $M$–$L_X$ relation as a power law with a hierarchical Bayesian likelihood that includes the halo mass function, per-cluster mass probability distributions, a Poisson X-ray count model, and a chain of selection functions (RASS completeness, redMaPPer optical completeness, a weak-lensing sampling function, and the CODEX X-ray selection). The main fit gives a slope $\beta = 0.75\pm0.09$, an intrinsic scatter $\sigma_{\mathrm{intr}}=0.16\pm0.02$, and marginal normalisation evolution $\gamma=0.65\pm0.43$; restricting the fit to the 42 clusters with eROSITA year-1 fluxes, with point sources excised, gives $\beta=1.11\pm0.15$ and $\gamma=0.004\pm0.790$. The paper interprets the eROSITA result as evidence that point-source contamination flattens the RASS slope, and it argues that the shallow slope is a property of the high-mass sample or unmodeled cluster physics rather than a statistical artifact.

Load-bearing premise

The calibration rests on the assumption that the weak-lensing sampling function fitted to the richness distribution of the sample correctly describes which clusters got lensing follow-up, and that the adopted richness–mass relation is unbiased.

Editorial extensions

If this is right

  • RASS-based cluster calibrations that skip the richness PDFs and halo mass function can underestimate the slope: the same sample gives $\beta=0.30\pm0.05$ without selection modeling and $\beta=0.75\pm0.09$ with it.
  • If the eRASS1 slope is right, point-source contamination in RASS fluxes biases low-luminosity or high-z clusters, flattening the apparent $M$–$L_X$ slope; cleaning the fluxes restores agreement with self-similarity.
  • The CODEX-LS catalog, covering about 14,000 square degrees with over 4,000 clean clusters, gives a unified selection for lensing follow-up, so the same calibration machinery can be reused as more weak-lensing data arrive.
  • No redshift evolution of the $M$–$L_X$ normalization is detected in either survey, and fixing the slope at the self-similar value leaves $\gamma\approx0.6$, so a single pivot normalization may be adequate for $z<0.7$ cluster cosmology.

Reading between the lines

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

  • One testable consequence the authors do not spell out: if point-source contamination explains the RASS slope, then simulated point sources injected into RASS images should reproduce the flattening, and the recovered luminosity bias should be a function of photon count and redshift.
  • The same framework could be extended to lower-richness clusters to test whether the slope steepens toward the group regime, as AGN-feedback models predict; the present sample is deliberately high-mass.
  • The weak-lensing sampling function is fitted to the very sample it corrects; an independent check would be to recompute the calibration using only clusters selected by shear or by SZ signals rather than by richness.
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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

4 major / 4 minor

Summary. The paper constructs an all-sky extension of the CODEX X-ray cluster catalog using LEGACY optical data and calibrates the M-LX scaling relation for 100 clusters with weak-lensing masses. A hierarchical Bayesian model includes mass and X-ray measurement PDFs, a halo mass function, and a detailed selection function with richness-based and X-ray detection terms. The RASS-based fit gives beta = 0.75 +/- 0.09 and gamma = 0.65 +/- 0.43, while a 42-cluster eRASS1 subsample gives beta = 1.11 +/- 0.15 and gamma = 0.004 +/- 0.790. The authors conclude that the RASS slope is 1.7 sigma below the self-similar expectation and that point-source deblending steepens the slope.

Significance. If the central calibration is correct, the paper would provide a useful unified framework for cluster scaling relations and a practical pathway for cosmological use of RASS and eRASS1 data. Its strengths include the use of full weak-lensing mass PDFs, a Poisson treatment of X-ray counts, explicit selection functions, and an apples-to-apples comparison with eRASS1 fluxes. However, the headline RASS slope and the RASS-versus-eRASS slope difference rest on an empirical weak-lensing sampling function that is fitted to the same sample it corrects. The paper's own variants in Table 2 show beta ranging from 0.28 to 0.84, so the quoted statistical error substantially understates the model uncertainty. As presented, the main scientific claim is not yet robust.

major comments (4)
  1. [Sec. 4.1.1, Eq. (18); Table 2; Appendix A] The weak-lensing sampling function P(I_samp|lambda) is fitted to the ratio of the WL sample to the cleaned CODEX catalog inside the same five sky areas used for calibration, and it is then applied to correct that same sample. The parameters of this empirical function are treated as fixed, with no uncertainty propagated into the final beta and gamma. The sensitivity tests in Table 2 show that removing the richness PDFs changes beta from 0.75 +/- 0.09 to 0.28 +/- 0.05, while the alternative IH20 selection gives beta = 0.84 +/- 0.08. This spread is several times larger than the quoted error and directly affects the abstract's claim of a 1.7-sigma deviation from self-similarity. The selection function needs to be validated externally, marginalized over in the fit, or supplemented by a simulation-based test before the central slope claim can be considered robust.
  2. [Sec. 4.1.1, Eq. (21); Appendix B] The sensitivity analysis around the richness-mass relation varies only the intrinsic scatter sigma_lambda_intr, not the normalization or slope of Eq. (21). A biased normalization or slope in the Kiiveri et al. (2021) relation would shift the expected richness at fixed mass and therefore alter the selection correction P(lambda|<ln lambda|mu>) in a coherent way. Varying sigma_lambda_intr does not bracket such a bias. The statement in Appendix B that the scatter variation 'captures deviations from the relation's parameters' is therefore not accurate, and the conclusion that uncertainties in the M-lambda relation are negligible applies only to the scatter parameter, not to the relation itself.
  3. [Sec. 5.2, Table 2] The eRASS1 analysis inherits the same CODEX selection and uses a re-fitted P(I_samp|lambda) for the 42 matched clusters, so the circularity concern applies to the eRASS1 fit as well. The beta = 1.11 +/- 0.15 result is therefore not an independent check of the RASS result. The interpretation that the slope difference is due to point-source contamination is plausible, but it cannot be separated from selection-model misspecification unless the eRASS1 sampling function is validated in a way that does not rely on the same 42 clusters.
  4. [Code and data availability] No code, posterior chains, or the simulated CODEX X-ray selection grid are released. The central selection correction, especially Eq. (18) and the P(I_X|z, l_X, sigma_intr) grid, cannot be independently reproduced from the text. Given that the main result depends sensitively on these functions, providing at least the selection-function grids and the analysis code would be important for reproducibility.
minor comments (4)
  1. [Eq. (18)] The piecewise sampling function is not continuous at the bin boundaries: at lambda = 60 the first line gives 0 while the second gives about 0.033. If a continuous function is intended, this should be fixed or explicitly justified.
  2. [Table 1] The 'Initial' column appears to be empty in the typeset table; if initial values were used for the MCMC, they should be listed, and if not, the column should be removed.
  3. [Fig. 7] The caption refers to a 'dashed curve' describing the unity line, but the figure appears to show a solid line; the wording should match the actual line style.
  4. [Sec. 2.1] The phrase '14 thousand square degrees' should be written as '14,000 deg^2' for consistency with astronomical notation.

Circularity Check

2 steps flagged · score 4.0 of 10

The headline RASS slope β=0.75 is largely produced by a WL sampling function fitted to the same 100-cluster sample it then corrects; dropping the fitted richness PDFs collapses β to 0.28, and the load-bearing λ–M relation (Eq. 21) is imported from coauthored Kiiveri et al. (2021).

  1. fitted input called prediction [Sec. 4.1.1 (Eqs. 18–19); App. A.1; Table 2]
    ""to define ˜λ≥60, we use a ratio of the WL sample to the total number of the clean CODEX clusters inside the defined survey area... we fit a linear piecewise function between the mean of each bin, obtaining: P(I_samp|˜λ) = ..." (Sec. 4.1.1). "P(I| ˜λ,˜z, σintr) = ∫ dµ dlX dλ · ... · P(I_samp|˜λ) · ... · P(˜λ|λ)P(λ|⟨lnλ|µ⟩)" (Eq. 19). "excluding only these richness PDFs ... β = 0.28±0.05 and γ = 1.64±0.48" (App. A.1)."

    P(I_samp|λ) is constructed by binning the ratio of the WL sample (the same 100 clusters whose M–L_X relation is being fitted) to the cleaned CODEX catalog inside the five survey areas, then inserted into the full likelihood (Eq. 19) that weights those same clusters. Appendix A.1 shows the fitted richness PDFs carry the headline result: excluding them changes β from 0.75±0.09 to 0.28±0.05 and γ from 0.65 to 1.64, nearly identical to the fit with all selection functions removed. The eRASS1 analysis repeats the maneuver ("We altered the sampling selection defined in Eq. 18 according to this new subsample"), refitting the selection on the same 42 clusters whose slope (β=1.11) it then corrects.

  2. self citation load bearing [Sec. 4.1.1 (Eqs. 20–21); App. B; Table 2]
    ""The above expression is obtained using the λ−M relation calibrated by Kiiveri et al. (2021) for clusters in the CODEX catalog. According to their work, σ_λintr = 0.17 and ⟨lnλ|µ⟩ is computed from ln(λ) = 4.42 + 0.49 ln(M/10^14.81 M⊙)" (Sec. 4.1.1, Eqs. 20–21). "We assessed the impact of adopting the richness–mass relation from literature by exploring the uncertainty range of σ_λintr" (Sec. 4.1.1)."

    Appendix A.1 identifies the richness–mass PDF P(λ|⟨lnλ|µ⟩) as one of the two dominant terms shaping β (removing the richness PDFs collapses β from 0.75 to 0.28). This PDF is built from Eq. 21, calibrated in Kiiveri et al. (2021), which shares authors with the present paper (Kiiveri and Finoguenov are authors of both works). The only sensitivity test (Appendix B) varies σ_λintr over the Kiiveri et al. bounds while keeping the normalization 4.42 and slope 0.49 of Eq. 21 fixed, so the coauthored relation's normalization and slope — the load-bearing parameters — are not bracketed.

full rationale

No step in this paper derives the M–L_X parameters from the inputs by construction: the slope and evolution are fitted to external weak-lensing masses (Kiiveri et al. 2021, Herbonnet et al. 2020, Oguri et al. 2021) and external X-ray luminosities (RASS and eRASS1), and the self-similar comparison value β_self = 0.9 is imported from Lovisari et al. (2021), so the central claim retains independent content. The partial circularity lives in two load-bearing places. First, the weak-lensing sampling function P(I_samp|λ) (Eq. 18) is constructed by binning the ratio of the very WL sample being fitted to the cleaned CODEX catalog inside the same five sky areas used for the analysis, and it is then applied back to those same clusters via Eq. 19; the paper's own Appendix A.1 shows that removing the richness PDFs that carry this fit changes β from 0.75±0.09 to 0.28±0.05 (and γ from 0.65 to 1.64), and the alternative Herbonnet20 selection moves β to 0.84±0.08 (Table 2). The headline "1.7σ below self-similar" is therefore conditioned on a self-calibrated selection function, and the eRASS1 analysis repeats the same self-calibration on its 42 clusters. Second, Appendix A.1 likewise shows the richness–mass PDF built from Eq. 21 is a dominant driver of the slope, and Eq. 21 is adopted from Kiiveri et al. (2021), a coauthored paper (Kiiveri and Finoguenov are authors of both); Appendix B only varies σ_λintr, leaving the normalization and slope of Eq. 21 fixed, so the load-bearing coauthored calibration is not fully bracketed. These are not full reductions by construction (the masses, luminosities, and self-similar value are external), so a score of 4 rather than 6 is appropriate: some self-citation plus a self-calibrated selection function that partly shapes the central claim.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central calibration rests on a chain of external inputs: Tinker HMF, the Kiiveri richness-mass relation, the Finoguenov CODEX selection grid, Gaussian mass-PDF approximations, and a fixed cosmology. The most consequential input is the richness-based selection model, because the paper demonstrates that the slope changes dramatically when it is included or excluded.

free parameters (6)
  • Intercept alpha = -0.12 +/- 0.02
    Fitted intercept of the power-law M-LX relation (Table 1).
  • Slope beta = 0.75 +/- 0.09 main; 1.11 +/- 0.15 eRASS1
    Fitted slope of the M-LX relation; central reported result.
  • Evolution gamma = 0.65 +/- 0.43 main; 0.004 +/- 0.790 eRASS1
    Fitted redshift evolution of the normalization of the M-LX relation.
  • Intrinsic scatter sigma_intr = 0.16 +/- 0.02
    Fitted log-normal scatter in L_X at fixed mass (Table 1) with a modified Jeffreys prior.
  • Lensing systematic l_sys = 0.47 +/- 0.31
    Nuisance parameter for main-subsample WL mass systematics, prior N(0,1).
  • WL sampling function P(I_samp|lambda) coefficients = Knots at 60, 143.5, 258.6, 342; slopes 3.9e-3, 0.1e-3; plateau 0.5
    Fitted to the ratio of the WL sample to the clean CODEX catalog in the analysis areas (Eq. 18); no uncertainty from this fit is propagated.
assumptions (6)
  • domain assumption Tinker et al. (2008) halo mass function for P(mu|z)
    Eqs. 3 and 12 use the Tinker HMF. Its uncertainty is not propagated, and Appendix A shows the HMF affects the gamma constraint.
  • domain assumption Richness-mass relation from Kiiveri et al. (2021), Eq. 21
    The selection model uses ln(lambda) = 4.42 + 0.49 ln(M/1e14.81 M_sun) and sigma_lambda_intr = 0.17 from a coauthored paper. Only sigma_lambda_intr is varied in Appendix B.
  • domain assumption Gaussian mass PDFs for Herbonnet20 and Oguri21 samples
    Section 4 approximates Herbonnet20 and Oguri21 mass distributions as Gaussians in linear mass, with Oguri21 modeled as two Gaussians. Non-Gaussian tails of the weak lensing likelihood are ignored.
  • domain assumption CODEX X-ray selection function grid from Finoguenov et al. (2020)
    Section 4.1.1 relies on a simulated selection grid from a coauthored prior work. The grid is not publicly released with this paper.
  • domain assumption Flat LambdaCDM cosmology with Omega_m = 0.3, H0 = 70
    Assumed cosmology is used for distances, E(z), and the HMF; it is not varied.
  • domain assumption Self-similar soft-band slope beta_self = 0.9 from Lovisari et al. (2021)
    Used as the reference for the 1.7 sigma statement. This value depends on assumed temperature and metallicity for massive clusters.

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Pith. "Pith review of Unified weak lensing constraints on the evolution of the mass -- X-ray luminosity relation for galaxy clusters." pith.science (2026). https://pith.science/paper/KEDNWFTT

@misc{pith2026250521659,
  author       = {Pith},
  title        = {Pith review of: Unified weak lensing constraints on the evolution of the mass -- X-ray luminosity relation for galaxy clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEDNWFTT}},
  note         = {Machine review of arXiv:2505.21659}
}
abstract

Scaling relations between galaxy cluster properties are crucial for understanding cosmology and baryonic physics. Rigorous calibration of the $M-L_X$ relation, employing weak lensing mass and consistent statistical methodology, is challenging due to heterogeneous cluster samples. The release of LEGACY imaging data introduced the possibility of unifying the cluster selection. We present the all-sky extension of the CODEX catalog based on LEGACY data and introduce a Bayesian framework for calibrating the X-ray luminosity-mass relation, derived for 100 clusters with weak lensing mass measurements. Using the X-ray luminosity estimates for those clusters from ROSAT All-Sky Survey (RASS) data, we perform a power-law fit to the $M-L_X$ relation. Furthermore, taking advantage of the recently released eROSITA data (eRASS1), we assess the impact of point source contamination on cluster fluxes for 42 clusters in the eRASS1 footprint. The RASS fit yields a slope of $\beta = 0.75 \pm 0.09$, 1.7$\sigma$ lower than the best self-similar prediction, with marginal evidence for the redshift evolution of the normalization ($\gamma = 0.65 \pm 0.43$). As for the eRASS1 analysis, the slope is substantially steeper, $\beta = 1.11 \pm 0.15$, and in further agreement with the prediction of self-similarity. No additional evolution is also seen ($\gamma = 0.004 \pm 0.790 $). While our results provide the practical means for cosmological studies of both RASS and eRASS data, the link to cluster physics is much cleaner after the cluster flux contamination is reduced. We also analyzed the impact of the selection function on calibration, finding that its full modeling is essential.

Figures

Figures reproduced from arXiv: 2505.21659 by the authors.

Figure 1
Figure 1. Spectroscopic redshift distribution of the WL sample. The colours blue, orange and pink represents the Herbonnet20, Oguri21, and main-subsample, respectively. fluctuations following a power-law dependence on their size, and no additional physical processes introducing scale dependence. Under this framework, clusters are considered scaled versions of one another, leading to the expectation that the main cluster prope… view at source ↗
Figure 2
Figure 2. The mass probability distribution function for a cluster from the Herbonnet20 subsample (upper panels) and from the Oguri21 subsample (lower panels). The left hand panels are in the linear space, while the right hand panels are in the logarithmic space. As for the X-ray luminosity distribution, a fairly com￾mon approach is to model it as a normal distribution in linear space. Although this is reasonable at low red￾s… view at source ↗
Figure 3
Figure 3. shows the Cartesian projection of the samples’ coordinates and the five areas defined for this purpose; the final survey area used in this work is simply the combination of all of them. With these considerations, we can now implement a subsample selection function. We neglect all clus￾ters with richness below 60 to reduce the effect of low￾richness clusters on the scaling relation, which would [PITH_FULL_IMAGE:figu… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Richness distribution of the CODEX sample in bins of equal width. The main difference between the cleaned and the total sample is the removal of clusters with rich￾ness affected by the poor quality of optical data. The area sampling returns a very similar richness dist…
Figure 5
Figure 5. Figure 5: Best parameters’ values from the MCMC fitting for the main fit (purple) and eRASS1 fit (pink). The one and two-dimensional projections of the posterior distributions are also shown. The contours represent the 1σ (68%) and 2σ (95%) confidence regions. when analyzing a h…
Figure 6
Figure 6. Figure 6: The M −LX relation in log space. The black line represents the projected best fit obtained from the MCMC. The results from a few iterations are depicted by the faded gray lines, illustrating the error associated with the best fit. There is also a simple linear fit usin…
Figure 7
Figure 7. Figure 7: compares the X-ray luminosities between CODEX and eROSITA for this eRASS1 sample of 42 clusters. We find a general agreement between the two estimates, with a scatter of 0.16 dex, and with eROSITA errors being significantly smaller — on average, these er￾rors are a fac…
Figure 9
Figure 9. Figure 9: Values for the slope β and evolution parameter γ of the M − LX scaling relation. We compare the results obtained in this work (of the WL sample, with and without a fixed slope value, and with the eRASS1 sample) with some of the ones available in the literature - i.e., …
Figure 10
Figure 10. Figure 10: The WL sample, the main fit, and the three fits discussed in this section to investigate the terms in the statistical model. The red line illustrates the slope of 0.9 predicted by self-similarity for comparison. These results show that including the selection function…
Figure 11
Figure 11. Figure 11: The main fit (black line) and the fits applying the upper (orange dashed line) and lower (blue dashed line) bounds for the richness intrinsic scatter σλintr . The results from a few of the main fit iterations are depicted by the faded gray lines, illustrating the erro…
Figure 12
Figure 12. Figure 12: The Herbonnet20 subsample in the X-ray luminosity – redshift plane. The function P(IH20 | ˜lX, z) is depicted in orange, and the colors blue and green differentiate the selected and excluded clusters, respectively. We also considered the possibility of the Herbonnet20…

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Pith tools

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