Mass-Temperature relation in $\Lambda$CDM and modified gravity

Antonino Del Popolo; David F. Mota; Francesco Pace

REVIEW 4 major objections 5 minor 124 references

Mass-Temperature relation in $\Lambda$CDM and modified gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The cluster mass–temperature relation cannot separate modified gravity from standard cosmology.

desk verdict A serious semianalytic effort that probably kills the MTR as a clean MG probe, but the conclusion rests on a visual overlay across different cosmologies, so the paper needs a quantitative comparison before the claim is solid. read the letter →

arxiv 1908.07322 v1 pith:ELEJMWEF submitted 2019-08-20 astro-ph.CO

Antonino Del Popolo , Francesco Pace , David F. Mota This is my paper · ORCID

classification astro-ph.CO PACS 98.52.Wz98.65.Cw

keywords mass-temperaturerelationgalaxyclustersmodifiedgravityf(R)symmetronmodeltop-hatcollapsedynamicalfrictiontidaltorque

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper derives the mass–temperature relation of galaxy clusters—the observed scaling between X-ray temperature and total mass—using the standard $\Lambda$CDM cosmology but with a cluster-formation model that includes angular momentum from tidal torques, random angular momentum, dynamical friction, the cosmological constant, and an external-pressure correction to the virial theorem. In this improved model the relation is no longer the classical self-similar $M\propto T^{3/2}$; it bends at 3–4 keV and steepens toward lower temperatures. The paper then compares this $\Lambda$CDM prediction with mass–temperature relations that earlier modified-gravity simulations produced for $f(R)$ and symmetron theories. Its central conclusion is that the modified-gravity curves and the $\Lambda$CDM curve behave alike, especially where screening makes gravity nearly normal, so the mass–temperature relation is not a good probe for testing gravity beyond general relativity.

What carries the argument

The load-bearing object is the improved top-hat collapse equation for the radial acceleration of a cluster shell, $$\frac{dv_r}{dt} = -\frac{GM}{$r^{2}$} + \frac{$L^{2}$(r)}{$M^{2}$ $r^{3}$} + \frac{\Lambda}{3}r - \eta\frac{dr}{dt},$$ where $L(r)$ combines ordered angular momentum from tidal torques with random angular momentum and $\eta$ is the dynamical-friction coefficient. This equation is combined with a virial theorem corrected for external surface pressure, and in the continuous-formation version with a merging-halo formalism, to produce analytic expressions for $k_B T$ as a function of mass. The angular-momentum and dynamical-friction terms are mass dependent, which breaks the self-similar $M\propto T^{3/2}$ scaling and places the bend at 3–4 keV; it is this bend that makes the $\Lambda$CDM curve resemble the $f(R)$ and symmetron curves.

What would settle it

Use the same cosmology, halo mass definition, and cluster sample for all three models—$\Lambda$CDM, f(R), and symmetron—and compute a goodness-of-fit to the observed mass–temperature data; if the $\Lambda$CDM curve separates from the modified-gravity curves by more than the data scatter, the paper's central conclusion fails. A direct $\Lambda$CDM simulation that includes tidal torques and dynamical friction would independently check whether the predicted 3–4 keV bend is real.

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Extended reading notes

Core claim

The paper's central claim is that the mass–temperature relation, a standard observational scaling between the X-ray temperature of a galaxy cluster and its mass, cannot distinguish the $\Lambda$CDM model from the modified-gravity theories studied here: $f(R)$ gravity, a class of theories with a chameleon screening mechanism, and the symmetron model, a scalar-tensor theory whose fifth force decouples in high-density regions. The authors build a semianalytic cluster-formation model that includes ordered angular momentum from tidal torques, random angular momentum, dynamical friction, the cosmological constant, and an external-pressure correction to the virial theorem. In that model the relation is not the classical self-similar $M\propto T^{3/2}$; it bends at roughly 3–4 keV and steepens toward lower temperatures. When this $\Lambda$CDM curve is overlaid on the mass–temperature relations extracted from earlier modified-gravity simulations, the curves agree in the regions where screening keeps gravity close to standard, so the relation no longer looks like a clean test of gravity theories beyond general relativity.

Load-bearing premise

The conclusion rests on overlaying curves computed with different background cosmologies—the symmetron simulations use $\Omega_{m,0}=0.35$ and $h=0.65$, while the f(R) simulations and the semianalytic model use $\Omega_{m,0}\simeq0.27$ and $h=0.7$—and different mass definitions, with no quantitative rescaling or statistical test, so part of the visual agreement could come from those differences rather than from the physics.

Editorial extensions

If this is right

A measured bend or steepening in the mass–temperature relation is not by itself evidence for modified gravity, because the standard model produces the same features once realistic formation physics is included.
The classical $M\propto T^{3/2}$ self-similar relation should not be used as the general-relativity baseline; the baseline has a mass-dependent bend.
Constraints on $f(R)$ or symmetron parameters derived from the shape of the mass–temperature relation will be weak, especially for high-temperature clusters where all models coincide.
The low-temperature end, below about 1 keV, is where modified-gravity curves and data are most separated, but it is also the sparsest part of the sample, so future low-mass cluster surveys are the natural place to look for a difference.

Reading between the lines

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

An extension would redo the comparison with identical cosmological parameters and halo mass definitions for all models and add a goodness-of-fit statistic; until then the agreement is established by visual overlay rather than by a quantitative test.
The same machinery predicts a mass-dependent collapse threshold, which could be checked against observed halo mass functions or concentration–mass relations as an independent test of the bend's origin.
If the bend comes from tidal torques and dynamical friction rather than from preheating, it should correlate with cluster environment and assembly history; a search for that correlation would distinguish the explanation proposed here from earlier ones.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The thing to know: this paper extends Del Popolo's earlier semianalytic MTR work with dynamical friction, the cosmological constant, and continuous formation, and shows the ΛCDM MTR bends at ~3–4 keV in a way that looks like the f(R) and symmetron predictions from Hammami & Mota 2017. If right, the MTR stops being a useful discriminator between GR and those MG models. That's a real negative result, and I don't think it's in the previous literature.

What's genuinely good: the model is not fitted to the MG curves. The ingredients—tidal torques, ordered and random angular momentum, dynamical friction, external pressure—are physically motivated and drawn from earlier derivations. They get the known break, and the comparison is post-hoc. I also don't see circular reasoning; the authors are not using the MG simulations to set their parameters.

The soft spot is exactly where the stress-test lands. The f(R) simulation uses Ωm=0.272, h=0.7, close enough to the ΛCDM model's Ωm=0.273, h=0.7. But the symmetron simulation uses Ωm=0.35, h=0.65. Eq. (22) has T ∝ Ωm^{1/3} M^{2/3} and masses are in h^{-1} M⊙, so the symmetron curves sit roughly 9% higher in T and 8% shifted in mass relative to a common-cosmology curve. No rescaling, no error bars on the semianalytic band, no statistical test. The conclusion that the MTR is indistinguishable is therefore only as strong as the visual overlay. The footnote about mass definitions (virial vs spherical overdensity) adds another layer of ambiguity: no check that the semianalytic virial mass matches the simulation halo masses.

There are also derivation gaps: Eqs. (22), (24), (32), (35) are stated with references but not derived, so a referee cannot verify the algebra without going back through Del Popolo 2002 and Voit 2000. That's addressable, not fatal.

Bottom line: the central claim is plausible but not proven. The paper deserves a serious referee. I'd send it out, with a request for a quantitative distinguishability test and a rescaling to a common cosmology. The model-building is serious and the negative result is worth having on the record.

Referee Report

4 major / 5 minor

Summary. This paper presents a semianalytic derivation of the mass-temperature relation (MTR) for galaxy clusters in a ΛCDM background, using an improved top-hat model and a continuous-formation model. The model incorporates ordered angular momentum from tidal torques, random angular momentum, dynamical friction, a modified virial theorem with an external pressure term, and the cosmological constant. The authors find a non-self-similar MTR with a break at about 3–4 keV and a steepening toward low temperatures. They compare their ΛCDM curves with f(R) and symmetron simulation results from Hammami & Mota (2017) and conclude that the MTR is not a good probe for distinguishing modified gravity from general relativity, because the ΛCDM and modified-gravity curves are similar. Section II summarizes the two modified-gravity models and the simulations; Section III gives the semianalytic model; Section IV presents the comparison; Section V concludes.

Significance. If the central conclusion is correct, the paper would substantially qualify earlier claims that the MTR can constrain f(R) and symmetron theories, and it would highlight the importance of baryonic and dynamical effects (angular momentum, dynamical friction, external pressure) in cluster scaling relations. The model is physically motivated and includes several effects known to break self-similarity, and the paper explicitly notes the ambiguity in mass definitions. However, no machine-checked derivations or code are provided, and the key equations are asserted via references. The significance of the claim about modified gravity is currently limited by the lack of a quantitative, apples-to-apples comparison between the semianalytic model and the simulations.

major comments (4)

[Section II.C and Section IV, Figs. 1 and 2] The central comparison of the ΛCDM semianalytic model with the modified-gravity simulation curves is not performed under a common cosmology. The symmetron simulations use Ωm,0 = 0.35 and h = 0.65 (ΩΛ = 0.65, ΩCDM = 0.3, Ωb = 0.05), while the f(R) simulations and the semianalytic ΛCDM model use Ωm,0 ≈ 0.27 and h = 0.7. Since the MTR scales approximately as T ∝ Ωm,0^{1/3} M^{2/3} (see Eq. 22 and Eq. 35), and masses are quoted in h^{-1} M☉, the symmetron curves are displaced by roughly 9% in temperature and 8% in mass relative to a common-cosmology ΛCDM curve. No rescaling or quantitative comparison is performed; the conclusion that the MTR cannot distinguish modified gravity from ΛCDM rests on visual overlay. Please recompute or rescale one of the two sets of curves to a common cosmology and provide a quantitative statistic (e.g., residuals or a chi-square test) to support the claim.
[Section III, Eqs. (22), (24), (32), and (35)] The key equations of the model are presented without derivation. While references are given, the coefficients and structures of these equations directly determine the break temperature and the low-mass slope that are the basis for the comparison. In particular, the derivation of Eq. (22) from Eqs. (19) and (21) introduces the parameters ψ, ξ, reff and the angular-momentum integral without showing the algebra, so the reader cannot verify the coefficient 1.58 or the signs of the correction terms. The same applies to the continuous-formation expression in Eqs. (32)–(35), which depends on the fitting parameter m through LerchPhi functions. Please provide a derivation of these equations in an appendix or supplementary material, or indicate explicitly where each step appears in the cited references.
[Section III (following Eq. 35) and Section IV] The model contains several free parameters (ν, β, f, m, M0, λ0, μ(δ), c) that are fixed to typical values but not varied. The break at T ≈ 3 keV and the low-mass slope are the features that make the ΛCDM curve overlap with the f(R) and symmetron curves, and it is not shown whether these features are robust to the choice of parameters. For example, the parameter m controls the mass-variance evolution and enters the K(m,x) terms in Eqs. (32)–(36); a different m could shift the bend or change the slope. A sensitivity analysis is needed to establish that the claimed indistinguishability is not an artifact of a particular parameter choice.
[Section I, footnote 2, and Section IV] The authors correctly note that the mass definition is ambiguous. However, the comparison in Figs. 1 and 2 does not establish that the halo masses from the modified-gravity simulations of Ref. [74] correspond to the same mass definition as the semianalytic 'mass' used in Eqs. (22) and (35). If the simulation masses are, for example, M200c or M500c while the model uses a virial mass, the curves could be shifted horizontally by a mass-dependent amount, which could affect the apparent agreement. Please specify the mass definition used in the simulation data and demonstrate consistency with the model.

minor comments (5)

[Abstract and Section IV] The abstract states a break at 3–4 keV, while Section IV describes the break at T ≈ 3 keV and the Conclusions state 3–4 keV; please make the numbers consistent.
[Figure 1 caption] The red, blue, and green curves for the f(R) models are not identified in the caption; please add an explicit mapping to the values |fR0| = 10^{-4}, 10^{-5}, 10^{-6}. The same applies to the symmetron curves in panels (c) and (d).
[Section IV, first paragraph] The abbreviation 'HM' is used for Ref. [74] without being defined; please define it at first use.
[Equation (36)] The notation m(M) uses the same symbol m as the constant parameter m introduced in Eq. (28); this is confusing. Please use a different symbol for either the parameter or the function.
[Section IV, discussion of Fig. 1(d)] The phrase 'The previous assertion is somehow confirmed' is informal; consider rewording for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the semianalytic Lambda-CDM MTR is compared post-hoc to independent f(R)/symmetron simulations; the main caveats are cosmology mismatch and mass-definition ambiguity, which are robustness concerns, not circularity.

full rationale

The central claim is a comparison claim: the paper overlays its own semianalytic Lambda-CDM mass-temperature relation (Eqs. 24 and 35) with f(R) and symmetron N-body curves from Ref. [74] and concludes that the MTR cannot distinguish modified gravity from GR. No parameter of Eqs. (24) or (35) is fitted to the modified-gravity curves. The 8 keV normalization of Eq. (35) is inherited from earlier external MTR work (Ref. [89]), and the angular-momentum and dynamical-friction contributions are taken from independently published derivations (Refs. [35, 92, 109]). Those citations are self-citations, but they are not the target conclusion: the claimed indistinguishability is a statement about the overlaid curves, not about the internal consistency of the cited models. The f(R)/symmetron curves are externally produced simulations with stated assumptions, so importing them as benchmarks is legitimate independent support. The paper itself flags the two genuine weaknesses in Sec. II.C and the footnote in Sec. I: the symmetron simulations use Omega_m,0=0.35, h=0.65 while the Lambda-CDM model is computed with Omega_m,0=0.273, h=0.7, and the mass definition (virial versus spherical-overdensity) is acknowledged to be ambiguous. These are comparability and robustness limitations, not circular reductions: there is no equation in which the Lambda-CDM prediction is defined in terms of the modified-gravity output, no fitted parameter renamed as a prediction, and no uniqueness or ansatz argument imported from the authors' prior work to force the conclusion. The paper therefore shows no significant circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a chain of modeling assumptions inherited from prior semianalytic work: spherical collapse, tidal-torque angular momentum, a linear dynamical friction term, a modified virial theorem, and the merging-halo formation model. The comparison to modified gravity also assumes that different simulation cosmologies can be overlaid. The main parameters are not fitted to the MTR data, which is why circularity burden is low, but several parameter values are not stated in the text.

free parameters (7)

ν (external pressure parameter) = set via (ν+1)/(ν-1) ≈ 2, i.e. c≈5
Appears in modified virial theorem (Eq. 19) and in MTR formulas (Eqs. 22, 24). Fixed using Ref. [90] for a typical cluster concentration; changes normalization.
β (velocity dispersion to temperature ratio) = not specified in text
Defined in Eq. (16) and enters all MTR formulas linearly. Standard value near 1, but the adopted value is not given.
f (hot gas fraction) = not specified
Defines βtilde in Eq. (16); affects the kinetic-temperature relation normalization.
m (mass variance slope) = not specified
In the continuous formation model (Eqs. 28-36), controls the mass dependence of binding energy and thus the MTR slope.
M0 (fiducial mass) = not specified
Defines the mass scale in Eq. (28) and functions m(M), n(M); taken from Ref. [89].
λ0 and μ(δ) (dynamical friction parameters) = not specified
Enter the dynamical friction contribution in Eqs. (24), (29), (32), (35), (36); taken from Ref. [109].
c (NFW concentration) = 5
Assumed typical cluster concentration used to set ν and the surface pressure term (Eq. 20); affects normalization and slope.

assumptions (6)

standard math Virial theorem and energy conservation describe the final cluster state
Used in Eqs. (17), (19), (21) to relate kinetic, potential, angular momentum, cosmological constant, and dynamical friction energies.
domain assumption Spherical top-hat collapse with ordered and random angular momentum and dynamical friction (Eq. 11) describes cluster formation
The radial acceleration equation is the core dynamical model; it combines tidal-torque angular momentum, a Λ term, and a linear dynamical friction force.
domain assumption Kinetic energy is related to X-ray temperature via Eq. (16) with μ=0.59
Converts the mechanical energy balance into the mass-temperature relation; assumes a thermalized intracluster medium.
domain assumption The external pressure term is modeled with an isothermal velocity dispersion and an NFW profile with c≈5
Used to derive Eq. (20) and fix ν; the paper states this is typical for clusters.
domain assumption The Lacey-Cole merging-halo formalism and Voit's continuous formation model describe gradual cluster assembly
Underpins Eqs. (27)-(35) and the comparison between formation redshift and observation redshift.
domain assumption The f(R) and symmetron MTR curves from Ref. [74], obtained with different background cosmologies, can be compared directly to the ΛCDM semianalytic curve
Section II.C states the background parameters differ between the models; the comparison in Section IV overlays them without rescaling or quantitative compensation.

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Pith. "Pith review of Mass-Temperature relation in $\Lambda$CDM and modified gravity." pith.science (2026). https://pith.science/paper/ELEJMWEF

@misc{pith2026190807322,
  author       = {Pith},
  title        = {Pith review of: Mass-Temperature relation in $\Lambda$CDM and modified gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ELEJMWEF}},
  note         = {Machine review of arXiv:1908.07322}
}

abstract

We derive the mass-temperature relation using an improved top-hat model and a continuous formation model which takes into account the effects of the ordered angular momentum acquired through tidal-torque interaction between clusters, random angular momentum, dynamical friction, and modifications of the virial theorem to include an external pressure term usually neglected. We show that the mass-temperature relation differs from the classical self-similar behavior, $M \propto T^{3/2}$, and shows a break at $3--4$ keV, and a steepening with a decreasing cluster temperature. We then compare our mass-temperature relation with those obtained in the literature with $N$-body simulations for $f(R)$ and symmetron models. We find that the mass-temperature relation is not a good probe to test gravity theories beyond Einstein's general relativity, because the mass-temperature relation of the $\Lambda$CDM model is similar to that of the modified gravity theories.

Figures

Figures reproduced from arXiv: 1908.07322 by the authors.

**Figure 1.** in Ref. [116]). It is larger than the standard value at galactic masses and tends to the standard value when we move to the largest clusters. The temperature is T ∝ ∝ δc (see [89]), and then less massive clusters are hotter than more massive ones, which are characterized by a standard MTR. Besides the effect of angular momentum in changing the shape of the MTR, we must recall that another factor contributing is t… view at source ↗

**Figure 2.** FIG. 2. The MTR for [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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