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arxiv: 2605.20393 · v1 · pith:NP5NQH6Znew · submitted 2026-05-19 · 🌌 astro-ph.CO · astro-ph.GA

The Splashback Mass Function of Galaxy Clusters from Photometric Data

Lucas Gabriel-Silva , Laerte Sodr\'e Jr This is my paper

Pith reviewed 2026-05-21 06:56 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords splashback radiusgalaxy clustersphotometric redshiftmass functionSDSSredMaPPercluster membership
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The pith

A fully photometric method measures splashback radii and masses for galaxy clusters and constructs their first observational mass function from SDSS data.

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

This paper shows how to locate the splashback radius of galaxy clusters—the physical edge separating orbiting galaxies from infalling ones—using only photometric images and colors. The authors build a probabilistic membership assignment based on position and photometric redshifts, then apply an adaptive cut that sharpens the contrast between the dense core and the outer regions. With this cleaned galaxy count profile they fit for the splashback radius in hundreds of clusters and recalibrate a scaling relation between that radius and the enclosed mass. They then apply the relation to over fifteen thousand redMaPPer clusters to produce the first splashback mass function derived entirely from photometry. The resulting abundance matches simulation expectations at the high-mass end, with shortfalls at lower masses explained by known catalog incompleteness.

Core claim

We present a fully photometric framework to measure individual cluster splashback radii and masses, and to construct an observational splashback mass function. Using Sloan Digital Sky Survey data, we develop a probabilistic cluster membership method based on radial and photometric redshift information, optimized through an adaptive probability cut that maximizes the detection significance of the cluster core relative to its outskirts. We apply this methodology to a sample of 499 galaxy clusters from the CoMaLit weak-lensing compilation and recover splashback radii from modeling cumulative galaxy number profiles. The resulting splashback radii exhibit a median ratio R_sp/R_200m ≃ 1.1. Using a

What carries the argument

The adaptive probability cut on radial and photometric redshift information, optimized to maximize core detection significance relative to the outskirts, that produces cumulative galaxy number profiles from which the splashback feature is located.

If this is right

  • Splashback radii for clusters show a median ratio R_sp/R_200m of approximately 1.1 across the sample.
  • The recalibrated M_sp–R_sp scaling relation has a shallower slope than the constant-density expectation and exhibits no significant redshift evolution from z=0.01 to z=0.8.
  • Splashback masses can be derived for more than 15,000 redMaPPer clusters in the SDSS Northern Galactic Cap.
  • The resulting splashback mass function matches simulation-based predictions at the high-mass end, with low-mass deviations consistent with known optical catalog completeness limits.

Where Pith is reading between the lines

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

  • The same photometric membership and profile technique could be applied to deeper wide-field surveys to build much larger splashback samples without spectroscopic or lensing follow-up.
  • Adopting splashback radii as the cluster boundary definition could reduce pseudo-evolution biases when using cluster abundances to constrain cosmology.
  • Testing whether the observed shallower scaling slope appears in hydrodynamical simulations would clarify whether the relation encodes new information about cluster assembly.

Load-bearing premise

The adaptive probability cut optimized to maximize core detection significance produces unbiased cumulative galaxy number profiles that accurately locate the splashback feature without significant contamination or incompleteness effects.

What would settle it

A statistically significant systematic offset between photometrically derived splashback radii and independent radii measured from spectroscopy or weak lensing on the same clusters would show that the profiles are biased.

Figures

Figures reproduced from arXiv: 2605.20393 by Laerte Sodr\'e Jr, Lucas Gabriel-Silva.

Figure 1
Figure 1. Figure 1: Distribution of redshifts (a) and masses (b) for the clusters in CoMaLit that satisfy the selection criteria described in Section 2.1. bootstrap estimation. The dispersion is normalized by the SDSS absolute error σ0. This procedure links the redshift probability to both the galaxy magnitude and the intrinsic photo-z uncertainty. The factor of 1.5 multiplying σz(m) is adopted follow￾ing L¨osch et al. (in pr… view at source ↗
Figure 2
Figure 2. Figure 2: redMaPPer clusters sky distribution in the NGC region. Color grade represents surface number density and the horizontal bar indicates a comoving scale of 100 cMpc evaluated at the median redshift of the each sample. Instead of adopting a fixed hard cut (e.g. Pf > 0.5), we implement an adaptive threshold that adjusts to clus￾ter richness. Rich clusters require a stricter cut, while poor clusters benefit fro… view at source ↗
Figure 3
Figure 3. Figure 3: Magnitude dependence of the SDSS photo-z dis￾persion in the r band. via adaptive kernel density estimation. For this method, we fix the cluster centers using their known RA, Dec, and redshift, rather than allowing the algorithm to infer them. 3.2. Splashback Feature Following the same approach as in our previous work (Gabriel-Silva & Sodr´e 2025), we model the cumula￾tive number profile of galaxy clusters … view at source ↗
Figure 6
Figure 6. Figure 6: compares the percentage error in Rphoto sp ob￾tained through different photometric membership def￾initions relative to the spectroscopic benchmark from [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of the normalized significance with probability threshold for the CoMaLit subsample. The color gradient indicates redshift. Pcut ∼ 0.4, mainly due to increasing redshift leading to lower galaxy counts as consequence of the Malmquist bias in flux-limited samples (e.g., Malmquist 1922, 1925; Teerikorpi 1997). 4.1.3. Impact on Splashback Measurements [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distributions of splashback radii, splashback masses, Rsp/R200m ratio, and enclosed splashback overdensity for the CoMaLit sample [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: Splashback radius as a function of corrected richness for the redMaPPer cluster catalog. cut, and model the cumulative number profile to esti￾mate the splashback radius. An interesting result is the clear correlation between splashback radius and cluster richness, in agreement with previous findings (Rykoff et al. 2016). However, as shown in [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Msp–Rsp relation for the CoMaLit sample, color￾coded by redshift. The black dotted line shows the best-fit relation. With the calibrated Msp–Rsp relation in hand, we ex￾tend our analysis to a much larger volume by considering the SDSS NGC region and the redMaPPer cluster cat￾alog. Applying the same selection criteria and method￾ology adopted for the CoMaLit sample, we identify 15,144 clusters in the redshi… view at source ↗
Figure 11
Figure 11. Figure 11: Median fractional uncertainty in the splashback radius as a function of cluster richness, color-coded by red￾shift. a splashback-based cluster mass function. We restrict the analysis to log(Msp/h−1 70 M⊙) > 14.0, where incom￾pleteness effects are expected to be subdominant. The resulting mass functions are shown in [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of the ratio Rsp/R200m for the redMaPPer cluster catalog. particular, accurate measurements of cluster masses are essential for discriminating between cosmological sce￾narios. However, traditional mass and radius defini￾tions based on spherical overdensities are affected by pseudo-evolution (Diemer et al. 2013), which can bias interpretations of halo growth. The splashback feature offers a ph… view at source ↗
Figure 13
Figure 13. Figure 13: Galaxy cluster splashback mass functions for the redMaPPer cluster catalog. Filled lines represent analytical predictions for the splashback mass function calibrated from simulations in Diemer (2020) and color-coded by the splashabck quantiles. through the E(z) rescaling, we find no statisti￾cally significant redshift evolution in the relation, with the best-fit redshift slope fully consistent with zero. … view at source ↗
read the original abstract

The splashback radius marks the physical boundary of galaxy clusters, separating orbiting from infalling material, and provides a halo definition free from pseudo-evolution. In this work, we present a fully photometric framework to measure individual cluster splashback radii and masses, and to construct an observational splashback mass function. Using Sloan Digital Sky Survey data, we develop a probabilistic cluster membership method based on radial and photometric redshift information, optimized through an adaptive probability cut that maximizes the detection significance of the cluster core relative to its outskirts. We apply this methodology to a sample of 499 galaxy clusters from the \textsc{CoMaLit} weak-lensing compilation and recover splashback radii from modeling cumulative galaxy number profiles. The resulting splashback radii exhibit a median ratio $R_{\mathrm{sp}}/R_{200\mathrm{m}} \simeq 1.1$, consistent with previous observational studies. Using these measurements, we recalibrate the $M_{\mathrm{sp}}$--$R_{\mathrm{sp}}$ scaling relation over a wide redshift range ($0.01 < z < 0.8$), finding a slope shallower than the constant-density expectation and no significant redshift evolution. We then apply this relation to \textsc{redMaPPer} clusters in the SDSS Northern Galactic Cap to derive splashback masses for more than $1.5\times10^4$ systems and construct the first observational splashback mass function based solely on photometric data. The resulting mass function agrees with simulation-based predictions at the high-mass end, while deviations at lower masses are consistent with known completeness limits of optical cluster catalogs. Our results demonstrate that splashback-based cluster sizes, masses, and abundances can be robustly measured in photometric surveys, enabling cosmological studies without spectroscopic or lensing data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops a photometric framework for measuring individual cluster splashback radii (R_sp) and masses using SDSS data. An adaptive probability cut on radial and photometric-redshift membership probabilities is optimized to maximize core detection significance; this is applied to 499 CoMaLit clusters to extract R_sp from modeled cumulative galaxy number profiles (median R_sp/R_200m ≃ 1.1). The M_sp–R_sp scaling relation is recalibrated over 0.01 < z < 0.8 and then used to assign splashback masses to >15 000 redMaPPer clusters, yielding the first observational splashback mass function constructed solely from photometric data. The resulting mass function matches simulation predictions at the high-mass end, with low-mass deviations attributed to catalog completeness limits.

Significance. If the adaptive membership procedure and scaling relation are shown to be unbiased, the work is significant: it demonstrates that splashback-based sizes, masses, and abundances can be obtained without spectroscopy or weak lensing, thereby enabling cosmological analyses in purely photometric surveys. The reported consistency with simulations at high mass and the explicit handling of completeness provide a concrete path toward splashback cosmology.

major comments (3)
  1. [Methods (adaptive probability cut)] Methods section (adaptive probability cut): the threshold is chosen post-hoc to maximize core detection significance relative to the outskirts. Because splashback lies in the outer regime, any differential removal or retention of galaxies at large radii can alter the slope or break location of the cumulative number profile. No test against a fixed-threshold selection or against spectroscopic membership is described, so the assumption that the resulting profiles are unbiased for R_sp extraction remains unverified and is load-bearing for the photometric-only claim.
  2. [Scaling relation and mass function construction] Scaling-relation and mass-function sections: the M_sp–R_sp relation is fitted to the same 499-cluster CoMaLit sample whose R_sp values were measured from the adaptive profiles, then applied to derive masses for the independent redMaPPer catalog. This recalibration on the validation sample introduces circularity that propagates directly into the shape and normalization of the final splashback mass function and its comparison to simulations.
  3. [Profile modeling and error propagation] Profile modeling and error propagation: the manuscript does not detail how uncertainties in the adaptive cut, photometric-redshift errors, and profile modeling are propagated into the final R_sp uncertainties or into the mass-function covariance; without this, the statistical significance of the high-mass agreement with simulations cannot be assessed.
minor comments (2)
  1. [Introduction / Scaling relation] Notation for the splashback mass M_sp and the scaling relation parameters should be defined explicitly in the text before first use, and the functional form of the relation should be written as an equation.
  2. [Results (mass function)] Figure showing the mass-function comparison would benefit from explicit indication of the completeness limit and from error bands that include the uncertainty on the recalibrated M_sp–R_sp slope.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and describe the changes planned for the revised version.

read point-by-point responses

  1. Referee: Methods section (adaptive probability cut): the threshold is chosen post-hoc to maximize core detection significance relative to the outskirts. Because splashback lies in the outer regime, any differential removal or retention of galaxies at large radii can alter the slope or break location of the cumulative number profile. No test against a fixed-threshold selection or against spectroscopic membership is described, so the assumption that the resulting profiles are unbiased for R_sp extraction remains unverified and is load-bearing for the photometric-only claim.

    Authors: We agree that explicit validation of the adaptive cut is important. While the optimization targets core significance, we recognize that effects on the outer profile must be checked. In the revised manuscript we will add direct comparisons of the adaptive selection against both a fixed probability threshold and, for the subset of clusters with available spectroscopy, against spectroscopic membership. These tests will quantify any impact on the recovered R_sp values and will be presented in an expanded Methods section. revision: yes

  2. Referee: Scaling-relation and mass-function sections: the M_sp–R_sp relation is fitted to the same 499-cluster CoMaLit sample whose R_sp values were measured from the adaptive profiles, then applied to derive masses for the independent redMaPPer catalog. This recalibration on the validation sample introduces circularity that propagates directly into the shape and normalization of the final splashback mass function and its comparison to simulations.

    Authors: The CoMaLit sample is used solely to calibrate the scaling relation from photometrically measured R_sp, while the redMaPPer catalog constitutes an independent application set with distinct selection. This is the intended calibration-then-application procedure. Nevertheless, to address the concern we will clarify the sample independence in the text, add a brief discussion of possible shared systematics, and include a note on the robustness of the high-mass agreement with simulations. revision: partial

  3. Referee: Profile modeling and error propagation: the manuscript does not detail how uncertainties in the adaptive cut, photometric-redshift errors, and profile modeling are propagated into the final R_sp uncertainties or into the mass-function covariance; without this, the statistical significance of the high-mass agreement with simulations cannot be assessed.

    Authors: We acknowledge that the propagation of these uncertainties was not described in sufficient detail. In the revised manuscript we will expand the profile-modeling and error-analysis sections to explicitly show how uncertainties arising from the adaptive cut, photometric-redshift errors, and the functional fit are propagated into individual R_sp uncertainties and into the covariance matrix of the splashback mass function. This will enable a quantitative evaluation of the agreement with simulation predictions. revision: yes

Circularity Check

1 steps flagged

Fitted M_sp-R_sp relation on 499 clusters applied to derive masses for redMaPPer mass function

specific steps
  1. fitted input called prediction [Abstract]
    "Using these measurements, we recalibrate the M_sp--R_sp scaling relation over a wide redshift range (0.01 < z < 0.8), finding a slope shallower than the constant-density expectation and no significant redshift evolution. We then apply this relation to redMaPPer clusters in the SDSS Northern Galactic Cap to derive splashback masses for more than 1.5×10^4 systems and construct the first observational splashback mass function based solely on photometric data."

    Splashback radii are recovered from the 499-cluster sample and used to fit the M_sp-R_sp relation. The fitted relation is then applied to derive the splashback masses that enter the mass function for the 15,000+ redMaPPer systems. The mass function is therefore statistically determined by the parameters of the fit to the smaller sample.

full rationale

The paper measures splashback radii photometrically for the 499 CoMaLit clusters, fits the M_sp-R_sp scaling relation to those measurements, and then applies the fitted relation to assign splashback masses to the much larger redMaPPer sample before constructing the mass function. This makes the final mass function a direct transformation of the calibration-sample fit rather than an independent measurement across the full catalog. The core radius extraction remains observationally driven, so the circularity is partial rather than total. No self-citations, self-definitions, or ansatz smuggling appear in the provided text.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central results rest on standard domain assumptions about galaxy cluster profiles and a small number of fitted elements for membership selection and scaling.

free parameters (2)
  • adaptive probability cut threshold
    Chosen to maximize detection significance of the cluster core relative to outskirts.
  • M_sp-R_sp scaling relation parameters
    Recalibrated over 0.01 < z < 0.8 from the 499-cluster sample.
axioms (1)
  • domain assumption The splashback radius is identifiable as a steepening feature in the cumulative galaxy number profile derived from photometric membership probabilities.
    Invoked when modeling the profiles to recover R_sp for each cluster.

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Works this paper leans on

77 extracted references · 77 canonical work pages · 2 internal anchors

  1. [1]

    Ade, P. A. R., Aghanim, N., Armitage-Caplan, C., et al. 2014, Astronomy & Astrophysics, 571, A16, doi: 10.1051/0004-6361/201321591

    work page doi:10.1051/0004-6361/201321591 2014

  2. [2]

    Adhikari, S., Dalal, N., & Chamberlain, R. T. 2014, Journal of Cosmology and Astroparticle Physics, 2014, 019–019, doi: 10.1088/1475-7516/2014/11/019

    work page doi:10.1088/1475-7516/2014/11/019 2014

  3. [3]

    2021, The Astrophysical Journal, 923, 37, doi: 10.3847/1538-4357/ac0bbc

    Adhikari, S., hyeon Shin, T., Jain, B., et al. 2021, The Astrophysical Journal, 923, 37, doi: 10.3847/1538-4357/ac0bbc

    work page doi:10.3847/1538-4357/ac0bbc 2021

  4. [4]

    W., Evrard, A

    Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, Annual Review of Astronomy and Astrophysics, 49, 409, doi: https: //doi.org/10.1146/annurev-astro-081710-102514

    work page doi:10.1146/annurev-astro-081710-102514 2011

  5. [5]

    A., & Cuevas, H

    Overzier, R. A., & Cuevas, H. 2021, Monthly Notices of the Royal Astronomical Society, 504, 5054, doi: 10.1093/mnras/stab1133

    work page doi:10.1093/mnras/stab1133 2021

  6. [6]

    P., Tollerud , E

    Arnaud, M., et al. 2005, Astronomy & Astrophysics, 441, 893 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068 Astropy Collaboration, Price-Whelan, A. M., Sip˝ ocz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et ...

    work page doi:10.1051/0004-6361/201322068 2005

  7. [7]

    2017, The Astrophysical Journal, 841, 18, doi: 10.3847/1538-4357/aa6ff0

    Baxter, E., Chang, C., Jain, B., et al. 2017, The Astrophysical Journal, 841, 18, doi: 10.3847/1538-4357/aa6ff0

    work page doi:10.3847/1538-4357/aa6ff0 2017

  8. [8]

    S., Wechsler, R

    Behroozi, P. S., Wechsler, R. H., Lu, Y., et al. 2014, The Astrophysical Journal, 787, 156, doi: 10.1088/0004-637x/787/2/156

    work page doi:10.1088/0004-637x/787/2/156 2014

  9. [9]

    2021, Astronomy & Astrophysics, 653, A31

    Bonoli, S., et al. 2021, Astronomy & Astrophysics, 653, A31

    work page 2021

  10. [10]

    L., & Norman, M

    Bryan, G. L., & Norman, M. L. 1998, The Astrophysical Journal, 495, 80, doi: 10.1086/305262

    work page internal anchor Pith review doi:10.1086/305262 1998

  11. [11]

    R., et al

    Carlberg, G. R., et al. 1997, The Astrophysical Journal, 478, 462

    work page 1997

  12. [12]

    2016, A&A, 595, A111, doi: 10.1051/0004-6361/201528009

    Castignani, G., & Benoist, C. 2016, A&A, 595, A111, doi: 10.1051/0004-6361/201528009

    work page doi:10.1051/0004-6361/201528009 2016

  13. [13]

    J., et al

    Cenarro, A. J., et al. 2019, Astronomy & Astrophysics, 622, A176

    work page 2019

  14. [14]

    2018, The Astrophysical Journal, 864, 83, doi: 10.3847/1538-4357/aad5e7

    Chang, C., Baxter, E., Jain, B., et al. 2018, The Astrophysical Journal, 864, 83, doi: 10.3847/1538-4357/aad5e7

    work page doi:10.3847/1538-4357/aad5e7 2018

  15. [15]

    2010, MNRAS, 406, 1379, doi: 10.1111/j.1365-2966.2010.16776.x

    Chilingarian, I. V., Melchior, A.-L., & Zolotukhin, I. Y. 2010, Monthly Notices of the Royal Astronomical Society, 405, 1409, doi: 10.1111/j.1365-2966.2010.16506.x

    work page doi:10.1111/j.1365-2966.2010.16506.x 2010

  16. [16]

    2011, MNRAS, 412, 1473, doi: 10.1111/j.1365-2966.2011.18162.x

    Chilingarian, I. V., & Zolotukhin, I. Y. 2011, Monthly Notices of the Royal Astronomical Society, 419, 1727, doi: 10.1111/j.1365-2966.2011.19837.x

    work page doi:10.1111/j.1365-2966.2011.19837.x 2011

  17. [17]

    2019, Phys

    Contigiani, O., Vardanyan, V., & Silvestri, A. 2019, Phys. Rev. D, 99, 064030, doi: 10.1103/PhysRevD.99.064030

    work page doi:10.1103/physrevd.99.064030 2019

  18. [18]

    2021, Galaxies, 9, 60, doi: 10.3390/galaxies9030060

    Contini, E. 2021, Galaxies, 9, 60, doi: 10.3390/galaxies9030060

    work page doi:10.3390/galaxies9030060 2021

  19. [19]

    Dacunha, T., Mansfield, P., & Wechsler, R. H. 2025, The Astrophysical Journal, 994, 274, doi: 10.3847/1538-4357/ae1031

    work page doi:10.3847/1538-4357/ae1031 2025

  20. [20]

    Davis, M., & Peebles, P. J. E. 1983, Astrophysical Journal, 267, 465, doi: 10.1086/160884

    work page doi:10.1086/160884 1983

  21. [21]

    2018, The Astrophysical Journal Supplement Series, 239, 35, doi: 10.3847/1538-4365/aaee8c —

    Diemer, B. 2018, The Astrophysical Journal Supplement Series, 239, 35, doi: 10.3847/1538-4365/aaee8c —. 2020, The Astrophysical Journal, 903, 87, doi: 10.3847/1538-4357/abbf52 —. 2021, The Astrophysical Journal, 909, 112, doi: 10.3847/1538-4357/abd947 —. 2022, Monthly Notices of the Royal Astronomical Society, 519, 3292, doi: 10.1093/mnras/stac3778

    work page doi:10.3847/1538-4365/aaee8c 2018

  22. [22]

    2019, The Astrophysical Journal, 871, 168, doi: 10.3847/1538-4357/aafad6

    Diemer, B., & Joyce, M. 2019, The Astrophysical Journal, 871, 168, doi: 10.3847/1538-4357/aafad6

    work page doi:10.3847/1538-4357/aafad6 2019

  23. [23]

    Diemer, B., & Kravtsov, A. V. 2014, The Astrophysical Journal, 789, 1, doi: 10.1088/0004-637x/789/1/1

    work page doi:10.1088/0004-637x/789/1/1 2014

  24. [24]

    Diemer, B., More, S., & Kravtsov, A. V. 2013, The Astrophysical Journal, 766, 25, doi: 10.1088/0004-637x/766/1/25

    work page doi:10.1088/0004-637x/766/1/25 2013

  25. [25]

    2006, Pattern Recognition Letters, 27, 861, doi: 10.1016/j.patrec.2005.10.010

    Fawcett, T. 2006, Pattern Recognition Letters, 27, 861, doi: 10.1016/j.patrec.2005.10.010

    work page doi:10.1016/j.patrec.2005.10.010 2006

  26. [26]

    W., Lang, D., & Goodman, J

    Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, Publications of the Astronomical Society of the Pacific, 125, 306, doi: 10.1086/670067

    work page doi:10.1086/670067 2013

  27. [27]

    2025, The Astrophysical Journal, 988, 149, doi: 10.3847/1538-4357/ade308

    Gabriel-Silva, L., & Sodr´ e, L. 2025, The Astrophysical Journal, 988, 149, doi: 10.3847/1538-4357/ade308

    work page doi:10.3847/1538-4357/ade308 2025

  28. [28]

    F., et al

    Giocoli, C., Palmucci, L., Lesci, G. F., et al. 2024, A&A, 687, A79, doi: 10.1051/0004-6361/202449561 18Gabriel-Silva & Sodr ´e

    work page doi:10.1051/0004-6361/202449561 2024

  29. [29]

    J., Rodriguez, F., Garc´ ıa Lambas, D., et al

    Gonzalez, E. J., Rodriguez, F., Garc´ ıa Lambas, D., et al. 2016, Monthly Notices of the Royal Astronomical Society, 465, 1348, doi: 10.1093/mnras/stw2803

    work page doi:10.1093/mnras/stw2803 2016

  30. [30]

    E., & Gott, III, J

    Gunn, J. E., & Gott, J. R. 1972, Astrophysical Journal, 176, 1, doi: 10.1086/151605

    work page doi:10.1086/151605 1972

  31. [31]

    R., Millman, K

    Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2

    work page doi:10.1038/s41586-020-2649-2 2020

  32. [32]

    Henriques, B. M. B., White, S. D. M., Thomas, P. A., et al. 2015, Monthly Notices of the Royal Astronomical Society, 451, 2663, doi: 10.1093/mnras/stv705

    work page doi:10.1093/mnras/stv705 2015

  33. [33]

    2015, Monthly Notices of the Royal Astronomical Society, 452, 998, doi: 10.1093/mnras/stv1271

    Hoshino, H., Leauthaud, A., Lackner, C., et al. 2015, Monthly Notices of the Royal Astronomical Society, 452, 998, doi: 10.1093/mnras/stv1271

    work page doi:10.1093/mnras/stv1271 2015

  34. [34]

    Hu, W., & Kravtsov, A. V. 2003, The Astrophysical Journal, 584, 702, doi: 10.1086/345846

    work page doi:10.1086/345846 2003

  35. [35]

    2001–, SciPy: Open source scientific tools for Python

    Jones, E., Oliphant, T., Peterson, P., et al. 2001–, SciPy: Open source scientific tools for Python. http://www.scipy.org/

    work page 2001

  36. [36]

    2025, arXiv e-prints, arXiv:2506.08925, doi: 10.48550/arXiv.2506.08925

    Joshi, J., Rana, D., More, S., & Klein, M. 2025, arXiv e-prints, arXiv:2506.08925, doi: 10.48550/arXiv.2506.08925

    work page doi:10.48550/arxiv.2506.08925 2025

  37. [37]

    S., Song, H., et al

    Kang, W., Hwang, H. S., Song, H., et al. 2024, ApJS, 272, 22, doi: 10.3847/1538-4365/ad390d

    work page doi:10.3847/1538-4365/ad390d 2024

  38. [38]

    V., & Borgani, S

    Kravtsov, A. V., & Borgani, S. 2012, Annual Review of Astronomy and Astrophysics, 50, 353, doi: https: //doi.org/10.1146/annurev-astro-081811-125502

    work page doi:10.1146/annurev-astro-081811-125502 2012

  39. [39]

    1993, Monthly Notices of the Royal Astronomical Society, 262, 627, doi: 10.1093/mnras/262.3.627

    Lacey, C., & Cole, S. 1993, Monthly Notices of the Royal Astronomical Society, 262, 627, doi: 10.1093/mnras/262.3.627

    work page doi:10.1093/mnras/262.3.627 1993

  40. [40]

    Lebeau, T., Ettori, S., Aghanim, N., & Sorce, J. G. 2024, A&A, 689, A19, doi: 10.1051/0004-6361/202450146

    work page doi:10.1051/0004-6361/202450146 2024

  41. [41]

    1983, ApJ, 272, 317, doi: 10.1086/161295

    Li, T.-P., & Ma, Y.-Q. 1983, ApJ, 272, 317, doi: 10.1086/161295

    work page doi:10.1086/161295 1983

  42. [42]

    2022, Astronomy and Computing, 38, 100510, doi: https://doi.org/10.1016/j.ascom.2021.100510

    Lima, E., Sodr´ e, L., Bom, C., et al. 2022, Astronomy and Computing, 38, 100510, doi: https://doi.org/10.1016/j.ascom.2021.100510

    work page doi:10.1016/j.ascom.2021.100510 2022

  43. [43]

    Lin, Y.-T., & Mohr, J. J. 2004, ApJ, 617, 879, doi: 10.1086/425412

    work page doi:10.1086/425412 2004

  44. [44]

    Malmquist, K. G. 1922, Lund Meddelanden Serie II, 22, 1 —. 1925, Lund Meddelanden Serie I, 106, 1

    work page 1922

  45. [45]

    V., & Diemer, B

    Mansfield, P., Kravtsov, A. V., & Diemer, B. 2017, ApJ, 841, 34, doi: 10.3847/1538-4357/aa7047 Mendes de Oliveira, C., et al. 2019, Monthly Notices of the Royal Astronomical Society, 489, 241

    work page doi:10.3847/1538-4357/aa7047 2017

  46. [46]

    F., Ludlow, A., & Jenkins, A

    Merritt, D., Navarro, J. F., Ludlow, A., & Jenkins, A. 2005, The Astrophysical Journal, 624, L85, doi: 10.1086/430636

    work page doi:10.1086/430636 2005

  47. [47]

    2019, MNRAS, 482, 2838, doi: 10.1093/mnras/sty2858

    Montes, M., & Trujillo, I. 2019, MNRAS, 482, 2838, doi: 10.1093/mnras/sty2858

    work page doi:10.1093/mnras/sty2858 2019

  48. [48]

    More, S., Diemer, B., & Kravtsov, A. V. 2015, The Astrophysical Journal, 810, 36, doi: 10.1088/0004-637x/810/1/36

    work page doi:10.1088/0004-637x/810/1/36 2015

  49. [49]

    2016, The Astrophysical Journal, 825, 39, doi: 10.3847/0004-637x/825/1/39

    More, S., Miyatake, H., Takada, M., et al. 2016, The Astrophysical Journal, 825, 39, doi: 10.3847/0004-637x/825/1/39

    work page doi:10.3847/0004-637x/825/1/39 2016

  50. [50]

    L., McGee, S

    Mulroy, S. L., McGee, S. L., Gillman, S., et al. 2017, Monthly Notices of the Royal Astronomical Society, 472, 3246

    work page 2017

  51. [51]

    2018, The Astrophysical Journal, 854, 120, doi: 10.3847/1538-4357/aaaab8

    Murata, R., Nishimichi, T., Takada, M., et al. 2018, The Astrophysical Journal, 854, 120, doi: 10.3847/1538-4357/aaaab8

    work page doi:10.3847/1538-4357/aaaab8 2018

  52. [52]

    2020, Publications of the Astronomical Society of Japan, 72, 64, doi: 10.1093/pasj/psaa041

    Murata, R., Sunayama, T., Oguri, M., et al. 2020, Publications of the Astronomical Society of Japan, 72, 64, doi: 10.1093/pasj/psaa041

    work page doi:10.1093/pasj/psaa041 2020

  53. [53]

    F., Frenk, C

    Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, The Astrophysical Journal, 462, 563, doi: 10.1086/177173 O’Shea, T. M., Borrow, J., O’Neil, S., & Vogelsberger, M. 2024, Dynamical friction and measurements of the splashback radius in galaxy clusters. https://arxiv.org/abs/2405.18468 O’Neil, S., Borrow, J., Vogelsberger, M., & Diemer, B. 2022, Monthly...

    work page doi:10.1086/177173 1996

  54. [54]

    2007, A&A, 464, 451, doi: 10.1051/0004-6361:20054708

    Popesso, P., Biviano, A., B¨ ohringer, H., & Romaniello, M. 2007, A&A, 464, 451, doi: 10.1051/0004-6361:20054708

    work page doi:10.1051/0004-6361:20054708 2007

  55. [55]

    2005, A&A, 433, 431, doi: 10.1051/0004-6361:20041915

    Voges, W. 2005, A&A, 433, 431, doi: 10.1051/0004-6361:20041915

    work page doi:10.1051/0004-6361:20041915 2005

  56. [56]

    H., & Schechter, P

    Press, W. H., & Schechter, P. 1974, ApJ, 187, 425, doi: 10.1086/152650

    work page doi:10.1086/152650 1974

  57. [57]

    R., & S., E

    Rykoff, E. R., & S., E. 2014, The Astrophysical Journal, 783, 80, doi: 10.1088/0004-637X/783/2/80

    work page doi:10.1088/0004-637x/783/2/80 2014

  58. [58]

    S., Rozo, E., Hollowood, D., et al

    Rykoff, E. S., Rozo, E., Hollowood, D., et al. 2016, The Astrophysical Journal Supplement Series, 224, 1, doi: 10.3847/0067-0049/224/1/1

    work page doi:10.3847/0067-0049/224/1/1 2016

  59. [59]

    2021, The Astrophysical Journal, 917, 98, doi: 10.3847/1538-4357/ac0c14

    Ryu, S., & Lee, J. 2021, The Astrophysical Journal, 917, 98, doi: 10.3847/1538-4357/ac0c14

    work page doi:10.3847/1538-4357/ac0c14 2021

  60. [60]

    2015, Monthly Notices of the Royal Astronomical Society, 450, 3665, doi: 10.1093/mnras/stu2505

    Sereno, M. 2015, Monthly Notices of the Royal Astronomical Society, 450, 3665, doi: 10.1093/mnras/stu2505

    work page doi:10.1093/mnras/stu2505 2015

  61. [61]

    2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x Labb´ e, I., van Dokkum, P., Nelson, E., et al

    Sheth, R. K., & Tormen, G. 2001, MNRAS, 323, 1, doi: 10.1046/j.1365-8711.2001.04006.x

    work page doi:10.1046/j.1365-8711.2001.04006.x 2001

  62. [62]

    2016, Monthly Notices of the Royal Astronomical Society, 459, 3711, doi: 10.1093/mnras/stw925

    Shi, X. 2016, Monthly Notices of the Royal Astronomical Society, 459, 3711, doi: 10.1093/mnras/stw925

    work page doi:10.1093/mnras/stw925 2016

  63. [63]

    J., et al

    Shin, T., Adhikari, S., Baxter, E. J., et al. 2019, Monthly Notices of the Royal Astronomical Society, 487, 2900, doi: 10.1093/mnras/stz1434 Splashback Mass Function from Photometry19

    work page doi:10.1093/mnras/stz1434 2019

  64. [64]

    2016, Monthly Notices of the Royal Astronomical Society, 466, 3103, doi: 10.1093/mnras/stw3250

    Simet, M., McClintock, T., Mandelbaum, R., et al. 2016, Monthly Notices of the Royal Astronomical Society, 466, 3103, doi: 10.1093/mnras/stw3250

    work page doi:10.1093/mnras/stw3250 2016

  65. [65]

    Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629–636, doi: 10.1038/nature03597

    work page doi:10.1038/nature03597 2005

  66. [66]

    A., Weinberg, D

    Strauss, M. A., Weinberg, D. H., Lupton, R. H., et al. 2002, The Astronomical Journal, 124, 1810–1824, doi: 10.1086/342343

    work page doi:10.1086/342343 2002

  67. [67]

    1997, Annual Review of Astronomy and Astrophysics, 35, 101, doi: 10.1146/annurev.astro.35.1.101

    Teerikorpi, P. 1997, Annual Review of Astronomy and Astrophysics, 35, 101, doi: 10.1146/annurev.astro.35.1.101

    work page doi:10.1146/annurev.astro.35.1.101 1997

  68. [68]

    Tinker, A

    Tinker, J. L., Kravtsov, A. V., Klypin, A., et al. 2008, ApJ, 688, 709, doi: 10.1086/591439

    work page internal anchor Pith review doi:10.1086/591439 2008

  69. [69]

    2017, The Astrophysical Journal, 836, 231, doi: 10.3847/1538-4357/aa5c90

    Umetsu, K., & Diemer, B. 2017, The Astrophysical Journal, 836, 231, doi: 10.3847/1538-4357/aa5c90

    work page doi:10.3847/1538-4357/aa5c90 2017

  70. [70]

    2020, The Astronomy and Astrophysics Review, 28, 7

    Umetsu, K., et al. 2020, The Astronomy and Astrophysics Review, 28, 7

    work page 2020

  71. [71]

    2009, The Astrophysical Journal, 692, 1060

    Vikhlinin, A., et al. 2009, The Astrophysical Journal, 692, 1060

    work page 2009

  72. [72]

    2025, arXiv e-prints, arXiv:2508.07232, doi: 10.48550/arXiv.2508.07232

    Walker, K., Ludlow, A., Power, C., Knebe, A., & Cui, W. 2025, arXiv e-prints, arXiv:2508.07232, doi: 10.48550/arXiv.2508.07232

    work page doi:10.48550/arxiv.2508.07232 2025

  73. [73]

    R., Tinker, J

    Wetzel, A. R., Tinker, J. L., Conroy, C., & Bosch, F. C. v. d. 2014, Monthly Notices of the Royal Astronomical Society, 439, 2687, doi: 10.1093/mnras/stu122

    work page doi:10.1093/mnras/stu122 2014

  74. [74]

    2001, A&A, 367, 27, doi: 10.1051/0004-6361:20000357

    White, M. 2001, A&A, 367, 27, doi: 10.1051/0004-6361:20000357

    work page doi:10.1051/0004-6361:20000357 2001

  75. [75]

    2024, The Astrophysical Journal, 971, 157, doi: 10.3847/1538-4357/ad57c7

    Xu, W., Shan, H., Li, R., et al. 2024, The Astrophysical Journal, 971, 157, doi: 10.3847/1538-4357/ad57c7

    work page doi:10.3847/1538-4357/ad57c7 2024

  76. [76]

    S., O’Neil, S., Shen, X., et al

    Yu, Q. S., O’Neil, S., Shen, X., et al. 2025, The Open Journal of Astrophysics, 8, doi: 10.33232/001c.146824

    work page doi:10.33232/001c.146824 2025

  77. [77]

    Zhang, Y., Adhikari, S., Edwards, L. O. V., et al. 2025, arXiv e-prints, arXiv:2509.22780, doi: 10.48550/arXiv.2509.22780 Z¨ urcher, D., & More, S. 2019, The Astrophysical Journal, 874, 184, doi: 10.3847/1538-4357/ab08e8 Lokas, E. L., & Mamon, G. A. 2001, Monthly Notices of the Royal Astronomical Society, 321, 155, doi: 10.1046/j.1365-8711.2001.04007.x ˇZ...

    work page doi:10.48550/arxiv.2509.22780 2025