arxiv: 2601.18652 · v3 · submitted 2026-01-26 · 🌌 astro-ph.CO · gr-qc· hep-th

Recognition: 2 theorem links

· Lean Theorem

Impact of Rastall gravity on hydrostatic mass of galaxy clusters

M. Lawrence Pattersons , Feri Apryandi , Freddy P. Zen

Authors on Pith no claims yet

Pith reviewed 2026-05-16 10:37 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-th

keywords Rastall gravitygalaxy clustershydrostatic massmass biasmodified gravitydark matter

0 comments

The pith

Rastall gravity adjusts galaxy cluster hydrostatic masses to match baryonic and lensing observations at slopes near unity.

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

This paper derives the hydrostatic mass of galaxy clusters in Rastall gravity under two scenarios. Without dark matter the modified framework reduces the calculated mass to align closely with observed baryonic mass, producing a best-fit slope of 1.07. With dark matter included the same modification reduces the usual bias against lensing mass, yielding a slope of 0.99. Statistical tests confirm that Rastall gravity supplies a workable phenomenological description of the mass discrepancy at cluster scales, although it does not always rank first among modified-gravity models by standard goodness-of-fit measures.

Core claim

In Rastall gravity the hydrostatic equilibrium equation can be solved so that the resulting mass equals the baryonic mass (slope 1.07) when no dark matter is assumed, or equals the lensing mass (slope 0.99) when dark matter is retained, by constraining a single Rastall parameter on cluster data.

What carries the argument

The hydrostatic equilibrium equation adapted to Rastall gravity, in which the Rastall parameter rescales the effective source term that links matter density to spacetime curvature.

If this is right

Without dark matter the Rastall hydrostatic mass and baryonic mass obey a linear relation with slope 1.07 ± 0.11.
With dark matter the Rastall hydrostatic mass and lensing mass obey a linear relation with slope 0.99 ± 0.26.
Goodness-of-fit statistics improve certain scaling relations yet do not place Rastall gravity ahead of every competing modified-gravity model.

Where Pith is reading between the lines

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

Cluster-scale modifications of this kind could reduce the apparent need for dark matter in mass estimates.
Repeating the analysis on larger, homogeneous cluster catalogs would test whether the Rastall parameter remains constant across different systems.
The same rescaling might be applied to other virialized objects to check consistency beyond galaxy clusters.

Load-bearing premise

The standard form of hydrostatic equilibrium can be transplanted directly into Rastall gravity at cluster scales using one universal constant parameter.

What would settle it

A statistically significant departure of the fitted slope from unity across a large, independent sample of clusters, or a Rastall parameter that cannot be fixed to a single value by the same data, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2601.18652 by Feri Apryandi, Freddy P. Zen, M. Lawrence Pattersons.

**Figure 1.** Figure 1: (a) Linear fit to the relation between the hydrostatic masses according to standard non-Rastall gravity and the observed baryonic masses, (b) the likelihood [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Figure 2: (a) Linear fit to the relation between the hydrostatic masses according to standard non-Rastall gravity and the observed lensing masses, (b) the likelihood function for parameter M in standard non-Rastall gravity, (c) linear fit to the relation between the hydrostatic masses according to Rastall gravity with λ = 1.14×10−3 and the observed baryonic masses, (d) the likelihood function for parameter M in Rast… view at source ↗

read the original abstract

Galaxy clusters are the largest virialized structures in the Universe and are predominantly dominated by dark matter. The hydrostatic mass and the mass obtained from gravitational lensing measurements generally differ, a discrepancy known as the hydrostatic mass bias. In this work, we derive the hydrostatic mass of galaxy clusters within the framework of Rastall gravity. We consider two scenarios: (i) the absence of dark matter and (ii) the presence of dark matter. In both cases, we constrain the Rastall parameter in the cluster-scale using observational data. In the first scenario, Rastall gravity effectively reduces the hydrostatic mass, bringing it closer to the observed baryonic mass. The best linear fit yields a slope $\mathbf{M}=1.07\pm0.11$, indicating a near one-to-one correspondence between the two masses. In the second scenario, Rastall gravity helps to alleviate the hydrostatic mass bias. The linear fit between the Rastall hydrostatic mass and the observed lensing mass results in a best-fit slope $\mathbf{M}=0.99\pm0.26$, which is very close to unity. We also calculate the goodness-of-fit for every fit. The statistical evaluations indicate that Rastall gravity provides a viable phenomenological framework that can improve certain aspects of the mass discrepancy problem at the level of scaling relations. However, it does not universally outperform other modified gravity model, when evaluated using standard goodness-of-fit criteria.

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.

Desk Editor's Note private letter to a colleague

Rastall gravity gives cluster mass slopes near unity after fitting its single parameter, but the hydrostatic equilibrium adaptation looks like a quick replacement rather than a full derivation from the modified equations.

read the letter

This paper applies Rastall gravity to the hydrostatic mass of galaxy clusters and reports linear fits that land close to one-to-one in both the no-dark-matter case (slope 1.07 ± 0.11) and the with-dark-matter case (slope 0.99 ± 0.26). They constrain the Rastall parameter on the same data and show the resulting goodness-of-fit numbers. That is the core output: two concrete scaling relations that look better than the usual hydrostatic bias problem under standard gravity. The work is new in taking the Rastall framework to this specific cluster-scale calculation and laying out the two scenarios side by side. The numbers are stated plainly with uncertainties, which makes the claim easy to check at a glance. They also note that the model does not beat every other modified-gravity option on standard fit criteria, which keeps the claim proportionate. The soft spot is the step that turns the Rastall field equations into a hydrostatic mass formula. Rastall changes the divergence of the stress-energy tensor, so the effective force balance for gas in spherical symmetry should carry extra gradient terms in the curvature scalar. The abstract and the reported results read as if they used a direct rescaling of the Newtonian or GR hydrostatic equation rather than deriving the full radial structure from the metric. If that is what happened, the mass reductions and the near-unity slopes rest on that assumption more than on the theory itself. Fitting the free parameter to the very data used to demonstrate the improvement adds another layer of circularity that is common in this kind of phenomenology but still needs to be stated clearly. This is for readers who track modified-gravity tests against cluster observations and want to see one more concrete example. It is not a broad resolution of the mass discrepancy, but the numbers are specific enough that a referee can verify the derivation and the data cuts. I would send it to peer review so the authors can supply the explicit hydrostatic equation and the data-selection details.

Referee Report

3 major / 2 minor

Summary. The paper claims that Rastall gravity allows derivation of hydrostatic masses for galaxy clusters in two scenarios (no dark matter and with dark matter). By fitting the Rastall parameter to observational data, it reports linear relations with slopes 1.07±0.11 (hydrostatic vs. baryonic mass) and 0.99±0.26 (Rastall hydrostatic vs. lensing mass), indicating reduced mass bias at the scaling-relation level, though goodness-of-fit metrics show it does not universally outperform other modified-gravity models.

Significance. If the hydrostatic-equilibrium adaptation is rigorously derived, the work supplies a concrete phenomenological test of Rastall gravity at cluster scales, with explicit slopes, uncertainties, and fit statistics that quantify improvement in the no-DM case and bias alleviation in the DM case. It adds to the literature on modified-gravity alternatives to dark matter by providing falsifiable numerical predictions tied to existing cluster catalogs.

major comments (3)

[Hydrostatic mass derivation (likely §3)] The hydrostatic equilibrium equation used for the Rastall case is introduced without an explicit derivation from the Rastall field equations for the metric ds² = −e^{2Φ}dt² + e^{2Λ}dr² + r²dΩ². The modified divergence of T_μν generally produces extra ∇R and λ-dependent terms in the effective gravitational acceleration for a perfect fluid; these must be shown to reduce to the adopted form or the reported mass reductions rest on an unverified assumption.
[Parameter fitting and results (likely §4)] The Rastall parameter is constrained directly from the same cluster sample whose masses are then used to demonstrate the near-unity slopes. This procedure makes the reported goodness-of-fit improvements (e.g., slopes 1.07±0.11 and 0.99±0.26) at least partly tautological; an independent validation sample or cross-check against a held-out catalog is required to substantiate the claim of genuine improvement.
[Methods and data (likely §2)] Data selection criteria, error propagation through the modified hydrostatic mass formula, and the precise observational catalogs (X-ray, lensing, etc.) are not described in sufficient detail. Without these, the quoted uncertainties and the conclusion that Rastall gravity “does not universally outperform other models” cannot be independently assessed.

minor comments (2)

[Abstract] Abstract uses boldface for the slope symbols (M=1.07±0.11); replace with standard inline math for journal consistency.
[Throughout] Ensure every equation is numbered and explicitly referenced in the text; several intermediate steps in the mass derivation appear unnumbered.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below and outline the revisions planned for the manuscript.

read point-by-point responses

Referee: The hydrostatic equilibrium equation used for the Rastall case is introduced without an explicit derivation from the Rastall field equations for the metric ds² = −e^{2Φ}dt² + e^{2Λ}dr² + r²dΩ². The modified divergence of T_μν generally produces extra ∇R and λ-dependent terms in the effective gravitational acceleration for a perfect fluid; these must be shown to reduce to the adopted form or the reported mass reductions rest on an unverified assumption.

Authors: We appreciate the referee highlighting this point. The hydrostatic equilibrium equation in Rastall gravity was obtained from the modified field equations, but we agree that the step-by-step derivation was not shown explicitly in the manuscript. In the revised version, we will add a complete derivation starting from the Rastall field equations for the given metric, explicitly demonstrating how the additional ∇R and λ-dependent terms reduce to the adopted form of the hydrostatic mass. This will make the modifications fully transparent. revision: yes
Referee: The Rastall parameter is constrained directly from the same cluster sample whose masses are then used to demonstrate the near-unity slopes. This procedure makes the reported goodness-of-fit improvements (e.g., slopes 1.07±0.11 and 0.99±0.26) at least partly tautological; an independent validation sample or cross-check against a held-out catalog is required to substantiate the claim of genuine improvement.

Authors: We acknowledge that fitting the Rastall parameter on the same sample used to evaluate the slopes introduces a degree of circularity, as the parameter is chosen to optimize the mass alignment. This is a standard phenomenological approach when constraining a free parameter at cluster scales with available data, but it limits the strength of the claim for genuine improvement. No independent held-out catalog is available in the current study. In the revision, we will add an explicit discussion of this limitation, frame the results as a demonstration rather than definitive validation, and recommend future tests with separate datasets. revision: partial
Referee: Data selection criteria, error propagation through the modified hydrostatic mass formula, and the precise observational catalogs (X-ray, lensing, etc.) are not described in sufficient detail. Without these, the quoted uncertainties and the conclusion that Rastall gravity “does not universally outperform other models” cannot be independently assessed.

Authors: We thank the referee for noting the insufficient detail in the methods. The original manuscript referenced the data sources but omitted full specifications. In the revised version, we will expand the relevant section to include the precise data selection criteria, the specific X-ray and lensing catalogs used, and a clear account of error propagation through the modified hydrostatic mass formula. This will allow independent verification of the uncertainties and the comparative goodness-of-fit statements. revision: yes

standing simulated objections not resolved

Requirement for an independent validation sample or held-out catalog to confirm the improvements, which is not available within the current dataset and analysis.

Circularity Check

1 steps flagged

Rastall parameter fitted to same cluster data produces near-unity mass slopes by construction

specific steps

fitted input called prediction [Abstract]
"we constrain the Rastall parameter in the cluster-scale using observational data. In the first scenario, Rastall gravity effectively reduces the hydrostatic mass, bringing it closer to the observed baryonic mass. The best linear fit yields a slope M=1.07±0.11, indicating a near one-to-one correspondence between the two masses. In the second scenario, Rastall gravity helps to alleviate the hydrostatic mass bias. The linear fit between the Rastall hydrostatic mass and the observed lensing mass results in a best-fit slope M=0.99±0.26"

The Rastall parameter is tuned to the identical observational mass data whose discrepancy is then 'alleviated' by the adjusted hydrostatic masses; the reported slopes near unity are therefore the direct statistical outcome of that fit rather than an independent derivation or prediction.

full rationale

The paper constrains the single Rastall parameter λ directly to the hydrostatic vs. baryonic/lensing mass data, then reports linear fits with slopes 1.07±0.11 and 0.99±0.26 as evidence of improvement. This reduces the claimed alleviation of mass bias to a fitted-input-called-prediction step. The hydrostatic equilibrium adaptation itself is presented without an explicit derivation from the Rastall field equations for the spherical metric, but the central circularity is the parameter fit. No self-citation load-bearing or ansatz smuggling is evident in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on one fitted parameter (Rastall parameter) and the assumption that the standard hydrostatic equilibrium form carries over to Rastall gravity without extra terms.

free parameters (1)

Rastall parameter = fitted to data
Constrained using observational cluster data to produce the reported mass correspondences in both scenarios.

axioms (2)

domain assumption Hydrostatic equilibrium equation holds in Rastall gravity
The derivation assumes the usual balance between pressure gradient and gravitational force can be rewritten under Rastall's modified field equations.
domain assumption Observational mass estimates are directly comparable
Baryonic mass and lensing mass are taken as ground truth for fitting and comparison.

pith-pipeline@v0.9.0 · 5559 in / 1525 out tokens · 54405 ms · 2026-05-16T10:37:00.971992+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Rastall field equation G^μ_ν + κλ R δ^μ_ν = κ T^μ_ν with free λ fitted to yield slopes M≈1.07 and 0.99
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Newtonian-limit m(r) with extra κλ/(4κλ−1) terms (Eq. 35)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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