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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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