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arxiv: 2606.30835 · v1 · pith:L3ERYOD6new · submitted 2026-06-29 · 🌌 astro-ph.CO · astro-ph.GA

The Limits of Photometric Dynamics: Benchmarking Cluster Relaxation Diagnostics

Alisson P. Costa , Andre L. B. Ribeiro , Zhonglue L. Wen , Flavio R. Morais-Neto This is my paper

Pith reviewed 2026-07-01 01:35 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords galaxy clustersphotometric redshiftsdynamical staterelaxation diagnosticsAnderson-Darling testGaussian mixture modelsSDSS
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The pith

Photometric redshift errors cause velocity diagnostics to classify nearly all galaxy clusters as relaxed.

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

The paper examines how photometric redshift uncertainties distort two standard tests for whether galaxy clusters are dynamically relaxed. It applies Monte Carlo resampling to velocity distributions from 1672 SDSS clusters under Gaussian and Student-t error models, then compares outcomes against the independent Gamma morphological classification. With Gaussian errors the tests recover relaxed clusters in about 95 percent of cases but unrelaxed clusters in only 5 percent, producing a strong bias toward relaxed labels. Student-t errors raise unrelaxed recovery to 30-45 percent yet still favor relaxed classifications. This matters because upcoming photometric surveys will rely on these methods to study cluster populations at scale.

Core claim

When Gaussian photometric redshift errors are propagated through cluster velocity distributions, the Anderson-Darling and Mclust tests recover relaxed systems in roughly 95 percent of realizations but unrelaxed systems in only about 5 percent, relative to the Gamma morphological proxy; Student-t errors improve unrelaxed recovery to 30-45 percent while reducing relaxed recovery to 60-70 percent, with paired Wilcoxon tests confirming the differences are significant.

What carries the argument

The Gamma morphological proxy acts as the independent ground-truth label for true dynamical state, against which photometric realizations of the Anderson-Darling test and Gaussian mixture modeling are benchmarked via Monte Carlo error propagation.

If this is right

  • Dynamical studies based primarily on photometric data will significantly underestimate the fraction of disturbed clusters.
  • Large photometric surveys require spectroscopic calibration, outlier mitigation, and realistic mock validation to avoid biased relaxation statistics.
  • The choice of error model directly controls how completely unrelaxed clusters are recovered.

Where Pith is reading between the lines

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

  • Hybrid photometric-spectroscopic samples may be needed to measure accurate disturbed-cluster fractions for structure-formation tests.
  • The observed bias could systematically affect cosmological constraints derived from cluster abundance or merger rates.
  • Additional observables such as X-ray or weak-lensing data could be combined with photometry to reduce misclassification.

Load-bearing premise

The Gamma morphological proxy supplies an accurate and independent record of each cluster's true dynamical state.

What would settle it

An independent dynamical indicator such as X-ray morphology or weak-lensing mass maps that yields substantially different recovery fractions for the same photometric tests.

Figures

Figures reproduced from arXiv: 2606.30835 by Alisson P. Costa, Andre L. B. Ribeiro, Flavio R. Morais-Neto, Zhonglue L. Wen.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Probability of relaxed/unrelaxed state as a function of the significance ratio for the cluster sample. The red and blue curves indicate which state is most likely for each group as the ζ values change. The dashed green curve represents the reliability of the decision made for the systems based on the significance ratio. The intersection of the probability curves defines the uncertainty region where there i… view at source ↗
Figure 3
Figure 3. Figure 3: Sensitivity analysis of dynamic tests against different photometric error models. Un￾der Gaussian errors (red), both tests (AD and Mclust) show high stability (frelaxed > 0.9). Un￾der Student-t errors (blue), which simulate catastrophic outliers, a significant degradation in the ability to recover the dynamic status is observed, evidenced by the drop in medians and widening of distributions. https://doi.or… view at source ↗
Figure 4
Figure 4. Figure 4: Sensitivity analysis for unrelaxed systems (similar to [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Average recovery fraction via AD and Mclust tests for relaxed (left) and unrelaxed (right) clusters applying Gaussian and Student-t error models to photometric redshifts generated from spectroscopic measurements. The points (filled circles - AD and filled triangle - Mclust) represent the percentage efficiency of the tests in correctly classifying each dynamic group for 1000 Monte Carlo realizations of the … view at source ↗
Figure 6
Figure 6. Figure 6: Recovery fraction (Rec. Frac.) of normality tests—AD (top panels) and Mclust (bottom panels)—as a function of the number of member galaxies within r500. Results are shown separately for relaxed (Γ > 0, left) and unrelaxed (Γ < 0, right) clusters. In each panel, solid curves represent beta regression fits for Gaussian (red) and Student-t (blue) underlying distributions. The recovery fraction indicates the p… view at source ↗
read the original abstract

Galaxy clusters are key probes of cosmology and structure formation, yet their dynamical classification traditionally relies on spectroscopic redshifts, which do not scale efficiently with survey size. As large photometric surveys such as LSST become available, photometric redshifts offer an attractive alternative, but their impact on velocity-based diagnostics remains poorly constrained. We quantify the sensitivity of two Gaussianity diagnostics - the Anderson-Darling (AD) test and Gaussian mixture modeling (Mclust) - to different photometric redshift error prescriptions. Propagating Gaussian and Student-t uncertainties through SDSS photometric velocity distributions, we assess how the error model affects recovery of cluster dynamical states established by the independent $\Gamma$ morphological proxy. Using 1672 SDSS clusters with pre-existing $\Gamma$, we perform Monte Carlo resampling under Gaussian and Student-t errors, the latter mimicking heavy-tailed uncertainties and catastrophic outliers, plus a spectroscopic control experiment with mock photometric redshifts from spectroscopic data. Under Gaussian errors, relaxed clusters are recovered in ~95% of realizations, while unrelaxed ones in only ~5%, revealing a strong bias toward relaxed classifications. Student-t errors drop relaxed recovery to ~60-70% and raise unrelaxed to ~30-45%, though still incomplete. Paired Wilcoxon tests confirm these differences are significant. This has direct implications for large photometric surveys: dynamical studies based primarily on photometric data may significantly underestimate disturbed cluster fractions without robust spectroscopic calibration, outlier mitigation, and validation with realistic mock catalogs.

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

2 major / 2 minor

Summary. The paper quantifies how photometric redshift error models (Gaussian vs. Student-t) affect recovery of galaxy cluster dynamical states via Anderson-Darling and Mclust Gaussianity tests. Using Monte Carlo resampling on 1672 SDSS clusters whose states are labeled by a pre-existing Γ morphological proxy, it reports ~95% recovery of relaxed clusters and ~5% recovery of unrelaxed ones under Gaussian errors, with reduced bias (~60-70% relaxed, ~30-45% unrelaxed) under Student-t errors; Wilcoxon tests establish significance, and a spectroscopic control is performed. The central conclusion is that photometric-only dynamical studies will underestimate disturbed fractions without spectroscopic calibration.

Significance. If the Γ proxy is a reliable ground truth, the quantified bias and the demonstration that error-model choice materially changes classification rates are directly relevant to LSST-era photometric surveys. The manuscript supplies concrete Monte Carlo realizations, paired statistical tests, and a control experiment, which are strengths for reproducibility and falsifiability.

major comments (2)
  1. [Abstract, §3] Abstract and §3 (validation procedure): the recovery fractions (~95% vs ~5%) are obtained by treating the pre-existing Γ morphological labels as the reference dynamical state. No section supplies an independent check (e.g., comparison of Γ labels against spectroscopic velocity-dispersion bimodality or X-ray morphological parameters) that quantifies Γ’s own misclassification rate or possible correlation with photometric properties. This is load-bearing because the reported percentages measure fidelity to the Γ labels rather than accuracy in recovering the true dynamical state.
  2. [Methods] Methods (Monte Carlo section): the exact number of realizations, the precise prescription for drawing photometric redshifts from the reported error distributions, and whether the resampling accounts for cluster membership uncertainty or magnitude-dependent photo-z bias are not stated with sufficient detail to reproduce the Wilcoxon p-values or the 95%/5% figures.
minor comments (2)
  1. [§2] Notation for the two error models (Gaussian vs. Student-t) is introduced without an explicit equation defining the scale parameter or degrees of freedom used in the Student-t case.
  2. [§2] The sample of 1672 clusters is described only by reference to prior Γ catalogs; explicit selection cuts (redshift range, richness threshold, photo-z quality flags) should be tabulated for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the scope and limitations of our analysis. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (validation procedure): the recovery fractions (~95% vs ~5%) are obtained by treating the pre-existing Γ morphological labels as the reference dynamical state. No section supplies an independent check (e.g., comparison of Γ labels against spectroscopic velocity-dispersion bimodality or X-ray morphological parameters) that quantifies Γ’s own misclassification rate or possible correlation with photometric properties. This is load-bearing because the reported percentages measure fidelity to the Γ labels rather than accuracy in recovering the true dynamical state.

    Authors: We agree that the reported recovery fractions measure agreement with the Γ morphological proxy rather than absolute accuracy against an independently verified dynamical state. Γ is an established morphological indicator in the literature that is constructed from imaging parameters independent of the photometric redshifts under test. Our central result—that photometric redshift error models systematically bias the AD and Mclust tests toward relaxed classifications—remains valid as a demonstration of differential bias relative to this fixed reference. Nevertheless, we acknowledge the referee’s point and will add explicit discussion in §3 and the conclusions section noting that Γ itself may have non-zero misclassification rates and that our percentages should be interpreted as fidelity to Γ, not to the true dynamical state. We will also reference existing comparisons of Γ to other proxies where available. revision: yes

  2. Referee: [Methods] Methods (Monte Carlo section): the exact number of realizations, the precise prescription for drawing photometric redshifts from the reported error distributions, and whether the resampling accounts for cluster membership uncertainty or magnitude-dependent photo-z bias are not stated with sufficient detail to reproduce the Wilcoxon p-values or the 95%/5% figures.

    Authors: We thank the referee for highlighting the need for greater methodological transparency. The revised manuscript will specify: (i) 1000 Monte Carlo realizations per cluster, (ii) the exact sampling procedure (Gaussian draws using the reported photo-z uncertainty; Student-t draws with ν=3 to produce heavy tails and outliers), and (iii) that cluster membership is held fixed to the original SDSS catalog while only redshifts are resampled. Magnitude-dependent photo-z bias is not modeled because it is not provided in the SDSS photo-z catalog used; this limitation will be stated explicitly. These additions will enable exact reproduction of the Wilcoxon tests and recovery fractions. revision: yes

Circularity Check

0 steps flagged

No circularity detected; results benchmark photometric diagnostics against external independent Γ proxy

full rationale

The paper's core procedure propagates photometric redshift errors via Monte Carlo resampling and measures how often AD and Mclust diagnostics reproduce the pre-existing Γ morphological classifications. The abstract and description explicitly frame the output as 'recovery of cluster dynamical states established by the independent Γ morphological proxy' and include a spectroscopic control experiment. No equation or step defines a quantity in terms of itself, renames a fitted parameter as a prediction, or relies on a self-citation chain for the central claim. The comparison is against an external benchmark (pre-existing Γ labels from prior work), satisfying the rule that a self-contained test against external benchmarks receives score 0-2. No load-bearing step reduces to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the Gamma proxy is an independent and reliable label of dynamical state and that the two error models (Gaussian and Student-t) adequately represent real photometric redshift uncertainties; no free parameters or invented entities are described in the abstract.

axioms (2)
  • domain assumption Gamma morphological proxy accurately reflects true dynamical relaxation state independent of velocity distribution shape
    The paper uses Gamma as the reference classification against which photometric diagnostics are tested.
  • domain assumption Gaussian and Student-t error models capture the dominant photometric redshift uncertainty behavior
    Error prescriptions are propagated through the velocity distributions to test sensitivity.

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Works this paper leans on

72 extracted references · 59 canonical work pages · 5 internal anchors

  1. [1]

    , eprint =

    Postman, M.; Franx, M.; Cross, N.J.G.; Holden, B.; Ford, H.C.; Illingworth, G.D.; Goto, T.; Demarco, R.; Rosati, P .; Blakeslee, J.P .; et al. The Morphology-Density Relation in z ∼ 1 Clusters. Astrophys. J. 2005, 623, 721–741. https://doi.org/10.1086/428881

    work page doi:10.1086/428881 2005

  2. [2]

    Investigating the Relation between Galaxy Properties and the Gaussianity of the Velocity Distribution of Groups and Clusters

    de Carvalho, R.R.; Ribeiro, A.L.B.; Stalder, D.H.; Rosa, R.R.; Costa, A.P .; Moura, T.C. Investigating the Relation between Galaxy Properties and the Gaussianity of the Velocity Distribution of Groups and Clusters . Astrophys. J. 2017, 154, 96. https://doi.org/10.3847/1538-3881/aa7f2b

    work page doi:10.3847/1538-3881/aa7f2b 2017

  3. [3]

    Boosting the evolutionary picture of Cl 0024+17 and MS 0451-03: A case study at intermediate-redshift

    Costa, A.P .; Ribeiro, A.L.B.; de Carvalho, R.R.; Benavides, J.A. Boosting the evolutionary picture of Cl 0024+17 and MS 0451-03: A case study at intermediate-redshift. Mon. Not. R. Astron. Soc. 2024, 535, 1348–1363. https://doi.org/10.1093/mnras/stae2410

    work page doi:10.1093/mnras/stae2410 2024

  4. [4]

    Substructures in galaxy clusters: A comparative X-ray and photometric analysis of the REXCESS sample

    Foëx, G. Substructures in galaxy clusters: A comparative X-ray and photometric analysis of the REXCESS sample. arXiv 2019, arXiv:1901.09198

    Pith/arXiv arXiv 2019

  5. [5]

    DS+: A method for the identification of cluster substructures

    Benavides, J.A.; Biviano, A.; Abadi, M.G. DS+: A method for the identification of cluster substructures. Astron. Astrophys. 2023, 669, A147. https://doi.org/10.1051/0004-6361/202245422

    work page doi:10.1051/0004-6361/202245422 2023

  6. [6]

    Astronomical applications of distribution shape estimators

    Bird, C.M.; Beers, T.C. Astronomical applications of distribution shape estimators. AJ 1993, 105, 1596–1606. https://doi.org/10.1 086/116540

    1993

  7. [7]

    SPIDER—IX

    Ribeiro, A.L.B.; de Carvalho, R.R.; Trevisan, M.; Capelato, H.V .; La Barbera, F.; Lopes, P .A.A.; Schilling, A.C. SPIDER—IX. Classifying galaxy groups according to their velocity distribution . Mon. Not. R. Astron. Soc. 2013, 434, 784–795. https: //doi.org/10.1093/mnras/stt1071

    work page doi:10.1093/mnras/stt1071 2013

  8. [8]

    , keywords =

    Hou, A.; Parker, L.C.; Wilman, D.J.; McGee, S.L.; Harris, W.E.; Connelly, J.L.; Balogh, M.L.; Mulchaey, J.S.; Bower, R.G. Substructure in the most massive GEEC groups: Field-like populations in dynamically active groups. Mon. Not. R. Astron. Soc. 2012, 421, 3594–3611. https://doi.org/10.1111/j.1365-2966.2012.20586.x

    work page doi:10.1111/j.1365-2966.2012.20586.x 2012

  9. [9]

    Unusual A2142 supercluster with a collapsing core: Distribution of light and mass

    Einasto, M.; Gramann, M.; Saar, E.; Liivamägi, L.J.; Tempel, E.; Nevalainen, J.; Heinämäki, P .; Park, C.; Einasto, J. Unusual A2142 supercluster with a collapsing core: Distribution of light and mass. Astron. Astrophys. 2015, 580, A69. https://doi.org/10.1051/ 0004-6361/201526399

    2015

  10. [10]

    Automatic Redshift Determination by Use of Principal Component Analysis

    Glazebrook, K.; Offer, A.R.; Deeley, K. Automatic Redshift Determination by Use of Principal Component Analysis. I. Fundamen- tals. Astrophys. J. 1998, 492, 98–109. https://doi.org/10.1086/305039

    work page doi:10.1086/305039 1998

  11. [11]

    Quantifying Photometric Redshift Errors in the Absence of Spectroscopic Redshifts

    Quadri, R.F.; Williams, R.J. Quantifying Photometric Redshift Errors in the Absence of Spectroscopic Redshifts. Astrophys. J. 2010, 725, 794–802. https://doi.org/10.1088/0004-637X/725/1/794

    work page doi:10.1088/0004-637x/725/1/794 2010

  12. [12]

    HAYATE: Photometric redshift estimation by hybridizing machine learning with template fitting

    Tanigawa, S.; Glazebrook, K.; Jacobs, C.; Labbe, I.; Qin, A.K. HAYATE: Photometric redshift estimation by hybridizing machine learning with template fitting. Mon. Not. R. Astron. Soc. 2024, 530, 2012–2038. https://doi.org/10.1093/mnras/stae411

    work page doi:10.1093/mnras/stae411 2024

  13. [13]

    Template-fitting meets deep learning: Redshift estimation using physics-guided neural networks

    Ferrao, J.C.; Dias, D.; Naik, P .; D’cruz, G.; Naik, A.; Khandeparkar, S.; Dessai, M.G.F. Template-fitting meets deep learning: Redshift estimation using physics-guided neural networks. J. Astrophys. Astron. 2026, 47, 28. https://doi.org/10.1007/s12036-026-10144-5

    work page doi:10.1007/s12036-026-10144-5 2026

  14. [14]

    Predicting the Computational Cost of Deep Learning Models

    Justus, D.; Brennan, J.; Bonner, S.; McGough, A.S. Predicting the Computational Cost of Deep Learning Models. arXiv 2018, arXiv:1811.11880. https://doi.org/10.48550/arXiv.1811.11880

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1811.11880 2018

  15. [15]

    LSST Science Book, Version 2.0

    Collaboration, L.S.; Abell, P .A.; Allison, J.; Anderson, S.F.; Andrew, J.R.; Angel, J.R.P .; Armus, L.; Arnett, D.; Asztalos, S.J.; Axelrod, T.S.; et al. LSST Science Book, Version 2.0. arXiv 2009, arXiv:0912.0201v1. https://doi.org/10.3390/universe1010000 Universe 2026, 1, 0 23 of 25

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.3390/universe1010000 2009

  16. [16]

    The Astrophysical Journal , year =

    Ivezi´ c, Ž.; Kahn, S.M.; Tyson, J.A.; Abel, B.; Acosta, E.; Allsman, R.; Alonso, D.; AlSayyad, Y.; Anderson, S.F.; Andrew, J.; et al. LSST: From Science Drivers to Reference Design and Anticipated Data Products. Astrophys. J. 2019, 873, 111. https://doi.org/10.3847/1538-4357/ab042c

    work page doi:10.3847/1538-4357/ab042c 2019

  17. [17]

    Lynden-Bell, Statistical Mechanics of Violent Relaxation in Stellar Systems, Monthly Notices of the Royal Astronomical Society 136 (1) (1967) 101–121

    Lynden-Bell, D. Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 1967, 136, 101. https://doi.org/10.1093/mnras/136.1.101

    work page doi:10.1093/mnras/136.1.101 1967

  18. [18]

    A dynamical analysis of twelve clusters of galaxies.Astron

    Beers, T.C.; Gebhardt, K.; Forman, W.; Huchra, J.P .; Jones, C. A dynamical analysis of twelve clusters of galaxies.Astron. J. 1991, 102, 1581–1609

    1991

  19. [19]

    A universal velocity distribution of relaxed collisionless structures

    Hansen, S.H.; Moore, B.; Zemp, M.; Stadel, J. A universal velocity distribution of relaxed collisionless structures. J. Cosmol. Astropart. Phys. 2006, 2006, 014. https://doi.org/10.1088/1475-7516/2006/01/014

    work page doi:10.1088/1475-7516/2006/01/014 2006

  20. [20]

    Statistical tools for classifying galaxy group dynamics

    Hou, A.; Parker, L.C.; Harris, W.E.; Wilman, D.J. Statistical tools for classifying galaxy group dynamics. Astrophys. J. 2009, 702, 1199

    2009

  21. [21]

    Multimodality in galaxy clusters from SDSS DR8: Substructure and velocity distribution

    Einasto, M.; Vennik, J.; Nurmi, P .; Tempel, E.; Ahvensalmi, A.; Tago, E.; Liivamägi, L.J.; Saar, E.; Heinämäki, P .; Einasto, J.; et al. Multimodality in galaxy clusters from SDSS DR8: Substructure and velocity distribution. Astron. Astrophys. 2012, 540, A123. https://doi.org/10.1051/0004-6361/201118697

    work page doi:10.1051/0004-6361/201118697 2012

  22. [22]

    Substructure and dynamical state of 2092 rich clusters of galaxies derived from photometric data

    Wen, Z.L.; Han, J.L. Substructure and dynamical state of 2092 rich clusters of galaxies derived from photometric data. Mon. Not. R. Astron. Soc. 2013, 436, 275–293. https://doi.org/10.1093/mnras/stt1581

    work page doi:10.1093/mnras/stt1581 2092

  23. [23]

    2011, ApJS, 193, 29, doi: 10.1088/0067-0049/193/2/29

    Aihara, H.; Allende Prieto, C.; An, D.; Anderson, S.F.; Aubourg, É.; Balbinot, E.; Beers, T.C.; Berlind, A.A.; Bickerton, S.J.; Bizyaev, D.; et al. The Eighth Data Release of the Sloan Digital Sky Survey: First Data from SDSS-III. Astrophys. J. Suppl. Ser. 2011, 193, 29. https://doi.org/10.1088/0067-0049/193/2/29

    work page doi:10.1088/0067-0049/193/2/29 2011

  24. [24]

    A Catalog of 132,684 Clusters of Galaxies Identified from Sloan Digital Sky Survey III

    Wen, Z.L.; Han, J.L.; Liu, F.S. A Catalog of 132,684 Clusters of Galaxies Identified from Sloan Digital Sky Survey III. Astrophys. J. Suppl. Ser. 2012, 199, 34. https://doi.org/10.1088/0067-0049/199/2/34

    work page doi:10.1088/0067-0049/199/2/34 2012

  25. [25]

    Galaxy Clusters Identified from the SDSS DR6 and Their Properties

    Wen, Z.L.; Han, J.L.; Liu, F.S. Galaxy Clusters Identified from the SDSS DR6 and Their Properties. Astrophys. J. Suppl. Ser. 2009, 183, 197–213. https://doi.org/10.1088/0067-0049/183/2/197

    work page doi:10.1088/0067-0049/183/2/197 2009

  26. [26]

    The Galaxy Luminosity Function and Luminosity Density at Redshift z = 0.1

    Blanton, M.R.; Hogg, D.W.; Bahcall, N.A.; Brinkmann, J.; Britton, M.; Connolly, A.J.; Csabai, I.; Fukugita, M.; Loveday, J.; Meiksin, A.; et al. The Galaxy Luminosity Function and Luminosity Density at Redshift z = 0.1. Astrophys. J. 2003, 592, 819–838. https://doi.org/10.1086/375776

    work page doi:10.1086/375776 2003

  27. [27]

    The Ninth Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the SDSS-III Baryon Oscillation Spectroscopic Survey

    Ahn, C.P .; Alexandroff, R.; Allende Prieto, C.; Anderson, S.F.; Anderton, T.; Andrews, B.H.; Aubourg, É.; Bailey, S.; Balbinot, E.; Barnes, R.; et al. The Ninth Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the SDSS-III Baryon Oscillation Spectroscopic Survey. Astrophys. J. Suppl. Ser. 2012, 203, 21. https://doi.org/10.1088/...

    work page doi:10.1088/0067-0049/203/2/21 2012

  28. [28]

    Mass-Temperature Relation of Galaxy Clusters: Implications from the Observed Luminosity-Temperature Relation and X-Ray Temperature Function

    Shimizu, M.; Kitayama, T.; Sasaki, S.; Suto, Y. Mass-Temperature Relation of Galaxy Clusters: Implications from the Observed Luminosity-Temperature Relation and X-Ray Temperature Function. Astrophys. J. 2003, 590, 197–206. https://doi.org/10.1086/ 367955

    2003

  29. [29]

    The structure of star clusters

    King, I. The structure of star clusters. I. an empirical density law. Astrophys. J. 1962, 67, 471. https://doi.org/10.1086/108756

    work page doi:10.1086/108756 1962

  30. [30]

    J., et al

    Ilbert, O.; Arnouts, S.; McCracken, H.J.; Bolzonella, M.; Bertin, E.; Le Fèvre, O.; Mellier, Y.; Zamorani, G.; Pellò, R.; Iovino, A.; et al. Accurate photometric redshifts for the CFHT legacy survey calibrated using the VIMOS VLT deep survey. Astron. Astrophys. 2006, 457, 841–856. https://doi.org/10.1051/0004-6361:20065138

    work page internal anchor Pith review doi:10.1051/0004-6361:20065138 2006

  31. [31]

    Velocity dispersion and dynamical mass for 270 galaxy clusters in the Planck PSZ1 catalogue

    Ferragamo, A.; Barrena, R.; Rubino-Mart’in, J.; Aguado-Barahona, A.; Streblyanska, A.; Tramonte, D.; G’enova-Santos, R.; Hempel, A.; Lietzen, H. Velocity dispersion and dynamical mass for 270 galaxy clusters in the Planck PSZ1 catalogue. Astron. Astrophys. 2021, 655, A115. https://doi.org/10.1051/0004-6361/202140382

    work page doi:10.1051/0004-6361/202140382 2021

  32. [32]

    Velocity dispersions in galaxy clusters

    Girardi, M.; Biviano, A.; Giuricin, G.; Mardirossian, F.; Mezzetti, M. Velocity dispersions in galaxy clusters. Astrophys. J. 1993, 404, 38–50. https://doi.org/10.1086/172256

    work page doi:10.1086/172256 1993

  33. [33]

    EDF statistics for goodness of fit and some comparisons

    Stephens, M.A. EDF statistics for goodness of fit and some comparisons. J. Am. Stat. Assoc. 1974, 69, 730–737

    1974

  34. [34]

    Connecting optical and X-ray tracers of galaxy cluster relaxation

    Roberts, I.D.; Parker, L.C.; Hlavacek-Larrondo, J. Connecting optical and X-ray tracers of galaxy cluster relaxation. Mon. Not. R. Astron. Soc. 2018, 475, 4704–4716. https://doi.org/10.1093/mnras/sty131

    work page doi:10.1093/mnras/sty131 2018

  35. [35]

    Investigation of the orientation of galaxies in clusters: The importance, methods and results of research

    Pajowska, P .; Godłowski, W.; Zhu, Z.H.; Popiela, J.; Panko, E.; Flin, P . Investigation of the orientation of galaxies in clusters: The importance, methods and results of research. J. Cosmol. Astropart. Phys. 2019, 2019, 005. https://doi.org/10.1088/1475-7516/20 19/02/005

    work page doi:10.1088/1475-7516/20 2019

  36. [36]

    Integrated empirical measures and generalizations of classical goodness-of-fit statistics

    Hwang, H.K.; Kuriki, S. Integrated empirical measures and generalizations of classical goodness-of-fit statistics. Electron. J. Stat. 2025, 19, 2276–2319. https://doi.org/10.1214/25-EJS2386

    work page doi:10.1214/25-ejs2386 2025

  37. [37]

    CODEX: Role of velocity substructure in the scaling relations of galaxy clusters

    Damsted, S.; Finoguenov, A.; Clerc, N.; Davalgait˙e, I.; Kirkpatrick, C.C.; Mamon, G.A.; Ider Chitham, J.; Kiiveri, K.; Comparat, J.; Collins, C. CODEX: Role of velocity substructure in the scaling relations of galaxy clusters. Astron. Astrophys. 2023, 676, A127. https://doi.org/10.1051/0004-6361/202245308

    work page doi:10.1051/0004-6361/202245308 2023

  38. [38]

    CHEX-MATE: Exploring the kinematical properties of Planck galaxy clusters

    Pizzuti, L.; Barrena, R.; Sereno, M.; Streblyanska, A.; Ferragamo, A.; Maurogordato, S.; Cappi, A.; Ettori, S.; Pratt, G.W.; Castignani, G.; et al. CHEX-MATE: Exploring the kinematical properties of Planck galaxy clusters. Astron. Astrophys. 2025, 699, A88. https://doi.org/10.1051/0004-6361/202555417. https://doi.org/10.3390/universe1010000 Universe 2026,...

    work page doi:10.1051/0004-6361/202555417 2025

  39. [39]

    Model-based methods of classification: Using the mclust software in chemometrics

    Fraley, C.; Raftery, A.E.; et al. Model-based methods of classification: Using the mclust software in chemometrics. J. Stat. Softw. 2007, 18, 1–13

    2007

  40. [40]

    https://doi.org/10.1201/9781003277965

    Scrucca, L.; Fraley, C.; Murphy, T.; E., R.A.Model-Based Clustering, Classification, and Density Estimation Using Mclust in R; CRC: New York, NY, USA, 2023. https://doi.org/10.1201/9781003277965

    work page doi:10.1201/9781003277965 2023

  41. [41]

    The Role of Groups in Galaxy Evolution: Compelling evidence of pre-processing out to the turnaround radius of clusters

    Lopes, P .; Ribeiro, A.L.B.; Brambila, D. The Role of Groups in Galaxy Evolution: Compelling evidence of pre-processing out to the turnaround radius of clusters. Mon. Not. R. Astron. Soc. Lett. 2023, 527, L19–L25. https://doi.org/10.1093/mnrasl/slad134

    work page doi:10.1093/mnrasl/slad134 2023

  42. [42]

    Examining transitional galaxies to understand the role of clusters and their dynamical status in galaxy quenching

    Brambila, D.; Lopes, P .; Ribeiro, A.; Cortesi, A. Examining transitional galaxies to understand the role of clusters and their dynamical status in galaxy quenching. Mon. Not. R. Astron. Soc. 2023, 523, 785–801. https://doi.org/10.1093/mnras/stad1233

    work page doi:10.1093/mnras/stad1233 2023

  43. [43]

    An approach to robust Bayesian regression in astronomy

    Martin, W.; Mortlock, D.J. An approach to robust Bayesian regression in astronomy. Ras Tech. Instruments 2024, 4, rzaf035. https://doi.org/10.1093/rasti/rzaf035

    work page doi:10.1093/rasti/rzaf035 2024

  44. [44]

    Dust Spectral Energy Distributions in the Era of Herschel and Planck: A Hierarchical Bayesian-fitting Technique

    Kelly, B.C.; Shetty, R.; Stutz, A.M.; Kauffmann, J.; Goodman, A.A.; Launhardt, R. Dust Spectral Energy Distributions in the Era of Herschel and Planck: A Hierarchical Bayesian-fitting Technique. Astrophys. J. 2012, 752, 55. https://doi.org/10.1088/0004-637X/ 752/1/55

    work page doi:10.1088/0004-637x/ 2012

  45. [45]

    Nortest: Tests for Normality; R package version 1.0-4; CRAN: Kewdale, Australia, 2015

    Gross, J.; Ligges, U. Nortest: Tests for Normality; R package version 1.0-4; CRAN: Kewdale, Australia, 2015. https://doi.org/10.326 14/CRAN.package.nortest

    2015

  46. [46]

    Visualization in Astrophysics: Developing New Methods, Discovering Our Universe, and Educating the Earth

    Lan, F.; Young, M.; Anderson, L.; Ynnerman, A.; Bock, A.; Borkin, M.; Forbes, A.; Kollmeier, J.; Wang, B. Visualization in Astrophysics: Developing New Methods, Discovering Our Universe, and Educating the Earth. Comput. Graph. Forum 2021, 40, 635–663. https://doi.org/10.1111/cgf.14332

    work page doi:10.1111/cgf.14332 2021

  47. [47]

    The Integrated Violin-Box-Scatter (VBS) Plot to Visualize the Distribution of a Continuous Variable

    Gerbing, D. The Integrated Violin-Box-Scatter (VBS) Plot to Visualize the Distribution of a Continuous Variable. Stats 2024, 7, 955–966. https://doi.org/10.3390/stats7030058

    work page doi:10.3390/stats7030058 2024

  48. [48]

    Recurrence plots for the analysis of complex systems

    Marwan, N.; Romano, M.; Thiel, M.; Kurths, J. Recurrence plots for the analysis of complex systems. Phys. Rep. 2025, 438, 237–329. https://doi.org/10.1016/j.physrep.2006.11.001

    work page doi:10.1016/j.physrep.2006.11.001 2025

  49. [49]

    Violin plots as visual tools in the meta-analysis of Single-Case Experimental Designs

    Tanious, R.; Manolov, R. Violin plots as visual tools in the meta-analysis of Single-Case Experimental Designs. Methodology 2022,

    2022

  50. [50]

    https://doi.org/10.5964/meth.9209

    work page doi:10.5964/meth.9209

  51. [51]

    Is there a large sample size problem? Ophthalmic Physiol

    Armstrong, R. Is there a large sample size problem? Ophthalmic Physiol. Opt. 2019, 39, 129 –130. https://doi.org/10.1111/opo.12 618

    work page doi:10.1111/opo.12 2019

  52. [52]

    The noise error component in stochastic frontier analysis

    Papadopoulos, A. The noise error component in stochastic frontier analysis. Empir. Econ. 2022, 64, 2795–2829. https: //doi.org/10.1007/s00181-022-02339-w

    work page doi:10.1007/s00181-022-02339-w 2022

  53. [53]

    Nonparametric Statistical Methods; Wiley Series in Probability and Statistics; Wiley: Hoboken, NJ, USA, 2013

    Hollander, M.; Wolfe, D.; Chicken, E. Nonparametric Statistical Methods; Wiley Series in Probability and Statistics; Wiley: Hoboken, NJ, USA, 2013

    2013

  54. [54]

    Another Argument in Favour of Wilcoxon's Signed Rank Test

    Rosenblatt, J.; Benjamini, Y. Another Argument in Favour of Wilcoxon’s Signed Rank Test. arXiv 2013, arXiv:1311.5354. https://doi.org/10.48550/arXiv.1311.5354

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1311.5354 2013

  55. [55]

    Velocity segregation effects in galaxy clusters at 0.4 ≲ z ≲ 1.5

    Barsanti, S.; Girardi, M.; Biviano, A.; Borgani, S.; Annunziatella, M.; Nonino, M. Velocity segregation effects in galaxy clusters at 0.4 ≲ z ≲ 1.5. Astron. Astrophys. 2016, 595, A73. https://doi.org/10.1051/0004-6361/201629012

    work page doi:10.1051/0004-6361/201629012 2016

  56. [56]

    Optical substructure and BCG offsets of Sunyaev-Zel’dovich and X-ray-selected galaxy clusters

    Lopes, P .A.A.; Trevisan, M.; Laganá, T.F.; Durret, F.; Ribeiro, A.L.B.; Rembold, S.B. Optical substructure and BCG offsets of Sunyaev-Zel’dovich and X-ray-selected galaxy clusters. Mon. Not. R. Astron. Soc. 2018, 478, 5473–5490. https://doi.org/10.1093/ mnras/sty1374

    2018

  57. [57]

    Fossil group origins

    Zarattini, S.; Biviano, A.; Aguerri, J.A.L.; Girardi, M.; D’Onghia, E. Fossil group origins. XI. The dependence of galaxy orbits on the magnitude gap. Astron. Astrophys. 2021, 655, A103. https://doi.org/10.1051/0004-6361/202038722

    work page doi:10.1051/0004-6361/202038722 2021

  58. [58]

    The dynamical state of eROSITA clusters and its impact on the brightest cluster galaxy luminosity

    Zenteno, A.; Kluge, M.; Kharkrang, R.; Hernandez-Lang, D.; Damke, G.; Saro, A.; Monteiro-Oliveira, R.; Carrasco, E.R.; Salvato, M.; Comparat, J.; et al. The dynamical state of eROSITA clusters and its impact on the brightest cluster galaxy luminosity. Astron. Astrophys. 2025, 698, A171. https://doi.org/10.1051/0004-6361/202452440

    work page doi:10.1051/0004-6361/202452440 2025

  59. [59]

    Cosmological Studies from Tomographic Weak Lensing Peak Abundances and Impacts of Photo-z Errors

    Yuan, S.; Pan, C.; Liu, X.; Wang, Q.; Fan, Z. Cosmological Studies from Tomographic Weak Lensing Peak Abundances and Impacts of Photo-z Errors. Astrophys. J. 2019, 884, 164. https://doi.org/10.3847/1538-4357/ab40a5

    work page doi:10.3847/1538-4357/ab40a5 2019

  60. [60]

    Beta Regression for Modelling Rates and Proportions

    Ferrari, S.; Cribari-Neto, F. Beta Regression for Modelling Rates and Proportions. J. Appl. Stat. 2004, 31, 799–815. https: //doi.org/10.1080/0266476042000214501

    work page doi:10.1080/0266476042000214501 2004

  61. [61]

    Regression analysis of variates observed on (0, 1): Percentages, proportions and fractions

    Kieschnick, R.; McCullough, B.D. Regression analysis of variates observed on (0, 1): Percentages, proportions and fractions. Stat. Model. 2003, 3, 193–213. https://doi.org/10.1191/1471082X03st053oa

    work page doi:10.1191/1471082x03st053oa 2003

  62. [62]

    Extended Beta Regression in R: Shaken, Stirred, Mixed, and Partitioned

    Grün, B.; Kosmidis, I.; Zeileis, A. Extended Beta Regression in R: Shaken, Stirred, Mixed, and Partitioned. J. Stat. Softw. 2012, 48, 1–25. https://doi.org/10.18637/jss.v048.i11

    work page doi:10.18637/jss.v048.i11 2012

  63. [63]

    Unveiling the Dynamics in Galaxy Clusters: The Hidden Role of Low-Luminosity Galaxies in Coma

    Costa, A.P .; Ribeiro, A.L.B.; de Morais Neto, F.R.; dos Santos Junior, J. Unveiling the Dynamics in Galaxy Clusters: The Hidden Role of Low-Luminosity Galaxies in Coma. Universe 2025, 11, 82. https://doi.org/10.3390/universe11030082

    work page doi:10.3390/universe11030082 2025

  64. [64]

    J., Jing, Y

    Capozzi, D.; Collins, C.A.; Stott, J.P .; Hilton, M. The evolution of K* and the halo occupation distribution since z= 1.5: Observations versus simulations. Mon. Not. R. Astron. Soc. 2012, 419, 2821–2835. https://doi.org/10.1111/j.1365-2966.2011.19895.x. https://doi.org/10.3390/universe1010000 Universe 2026, 1, 0 25 of 25

    work page doi:10.1111/j.1365-2966.2011.19895.x 2012

  65. [65]

    2018, MNRAS, 473, 4077, doi: 10.1093/mnras/stx2656

    Pillepich, A.; Springel, V .; Nelson, D.; Genel, S.; Naiman, J.; Pakmor, R.; Hernquist, L.; Torrey, P .; Vogelsberger, M.; Weinberger, R.; et al. Simulating galaxy formation with the IllustrisTNG model. Mon. Not. R. Astron. Soc. 2018, 473, 4077–4106. https://doi.org/10.1093/mnras/stx2656

    work page internal anchor Pith review doi:10.1093/mnras/stx2656 2018

  66. [66]

    Photometric Redshift Error Estimators

    Oyaizu, H.; Lima, M.; Cunha, C.E.; Lin, H.; Frieman, J. Photometric Redshift Error Estimators. Astrophys. J. 2008, 689, 709–720. https://doi.org/10.1086/592591

    work page doi:10.1086/592591 2008

  67. [67]

    Improving Photometric Redshift Estimation for CSST Mock Catalog Using SED Templates Calibrated with Perturbation Algorithm

    Li, Y.; Fu, L.; Chen, Z.; Luo, Z.; Du, W.; Gong, Y.; Meng, X.; Lu, J.; Tang, Z.; Chen, P .; et al. Improving Photometric Redshift Estimation for CSST Mock Catalog Using SED Templates Calibrated with Perturbation Algorithm. Res. Astron. Astrophys. 2025, 25, 055021. https://doi.org/10.1088/1674-4527/adcc7e

    work page doi:10.1088/1674-4527/adcc7e 2025

  68. [68]

    Identifying Catastrophic Outlier Photometric Redshift Estimates in the COSMOS Field with Machine Learning Methods

    Dennis, M.T.; Hu, E.M.; Cowie, L.L. Identifying Catastrophic Outlier Photometric Redshift Estimates in the COSMOS Field with Machine Learning Methods. Astrophys. J. 2025, 983, 173. https://doi.org/10.3847/1538-4357/adbe62

    work page doi:10.3847/1538-4357/adbe62 2025

  69. [69]

    Improving Photometric Redshift Estimates with Training Sample Augmentation

    Moskowitz, I.; Gawiser, E.; Crenshaw, J.F.; Andrews, B.H.; Malz, A.I.; Schmidt, S.; LSST Dark Energy Science Collaboration. Improving Photometric Redshift Estimates with Training Sample Augmentation. Astrophys. J. 2024, 967, L6. https://doi.org/10.3 847/2041-8213/ad4039

    2024

  70. [70]

    Effects of Photometric Redshift Uncertainties on Weak-Lensing Tomography

    Ma, Z.; Hu, W.; Huterer, D. Effects of Photometric Redshift Uncertainties on Weak-Lensing Tomography. Astrophys. J. 2006, 636, 21–29. https://doi.org/10.1086/497068

    work page doi:10.1086/497068 2006

  71. [71]

    The LSST Dark Energy Science Collaboration (DESC) Science Requirements Document

    The LSST Dark Energy Science Collaboration.; Mandelbaum, R.; Eifler, T.; Hložek, R.; Collett, T.; Gawiser, E.; Scolnic, D.; Alonso, D.; Awan, H.; Biswas, R.; et al. The LSST Dark Energy Science Collaboration (DESC) Science Requirements Document. arXiv 2018, arXiv:1809.01669. https://doi.org/10.48550/arXiv.1809.01669

    work page doi:10.48550/arxiv.1809.01669 2018

  72. [72]

    The LSST DESC DC2 Simulated Sky Survey

    LSST Dark Energy Science Collaboration (LSST DESC).; Abolfathi, B.; Alonso, D.; Armstrong, R.; Aubourg, É.; Awan, H.; Babuji, Y.N.; Bauer, F.E.; Bean, R.; Beckett, G.; et al. The LSST DESC DC2 Simulated Sky Survey. Astrophys. J. Suppl. Ser. 2021, 253, 31. https://doi.org/10.3847/1538-4365/abd62c. Disclaimer/Publisher’s Note: The statements, opinions and d...

    work page doi:10.3847/1538-4365/abd62c 2021