How parameter constraining can influence the mass accretion process of a Black Hole in the Generalized Rastall Gravity Theory ?

G. Mustafa; Himanshu Chaudhary; Puja Mukherjee; Ujjal Debnath

REVIEW 4 major objections 4 minor 93 references

How parameter constraining can influence the mass accretion process of a Black Hole in the Generalized Rastall Gravity Theory ?

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that in generalized Rastall gravity, a non-singular black hole accretes dark energy from both the CBDRM and CDMMA parameterizations in a way that increases its mass with cosmic time, and that the growth is consistent…

desk verdict The paper derives new accretion mass formulas for two dark energy equations of state in generalized Rastall gravity, but the headline mass-increase claim does not follow because the sign of the accretion rate is controlled by an unconstrained integration constant. read the letter →

arxiv 2411.15619 v2 pith:5MFXQAFE submitted 2024-11-23 gr-qc

Puja Mukherjee , Ujjal Debnath , Himanshu Chaudhary , G. Mustafa This is my paper · ORCID

classification gr-qc PACS 04.70.-s95.36.+x98.80.-k

keywords darkenergyparameterizationgeneralizedRastallgravityblackholemassaccretionnon-singularcosmologicalparameterconstraintsCBDRMCDMMAmass-redshiftrelation

topics Dark Energy

open problems Dark Energy

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 tries to establish that a non-singular black hole in generalized Rastall gravity gains mass as the Universe expands, provided the dark energy follows either of two recently proposed dynamical equation-of-state parameterizations (CBDRM and CDMMA) with parameters fitted to cosmological data. The authors derive the mass-accretion equation for the black hole, insert the Markov chain Monte Carlo constrained parameter values, and plot mass against redshift; both models show monotonic growth from early times to the present, whereas phantom-like fluids with negative pressure would shrink the hole. If the claim is right, dark energy accretion is a concrete mechanism that couples black hole growth to cosmic acceleration, and the generalized Rastall framework, with its varying Rastall parameter acting like a running cosmological constant, can reproduce and extend the standard Lambda-CDM behavior. The paper also uses the fitted parameters to compare the two models with Lambda-CDM and reports that CDMMA provides the best AIC-based balance of fit and complexity.

What carries the argument

The load-bearing object is the modified energy-momentum conservation law $T^{{mu nu}}$;_{; mu} = ($\lambda$ R)^{, nu} with $\lambda$ = delta/R, which turns the generalized Rastall field equations into Einstein equations with a varying cosmological-constant-like term. From this, the paper derives the accretion rate equation $\dot{M} = 4\pi(\lambda B_0 + B_1)(\rho+p)M^2$, whose sign is fixed by the combination of the Rastall parameter $\lambda$, the geometric factor $B_0 = (4f(r)/r - 2r f''(r))$, and the integration constant B_1. The mass equation (35) is then obtained by integrating this rate using the generalized conservation equation, with the Hubble parameter and energy densities for each dark-energy parameterization inserted. The non-singular black hole metric $f(r) = 1 - C_1/r - \epsilon\delta r^2/3 + \epsilon a b^3/(3r) \exp(-r^3/b^3)$ supplies the geometric factor B_0 and the horizon structure.

What would settle it

Take the paper's fitted parameter values, set B_1 so that lambda B_0 + B_1 is negative, and evaluate Eq. (35); if the predicted black-hole mass then decreases with time, the claimed growth is not a consequence of the constrained parameters alone. Alternatively, recompute Eq. (33) with lambda = delta/R from Eq. (2) rather than a constant and check whether the sign of dM/dt stays positive.

Watch

Extended reading notes

Core claim

The central claim is that Eq. (35), the redshift-space mass equation obtained from the generalized Rastall accretion formalism, yields an increasing black-hole mass for both the CBDRM and CDMMA dark-energy parameterizations once the parameters are fixed at their constrained values. For both models, the integrand contains the factor ((lambda B_0 + B_1)(1 - 3 epsilon lambda (1+omega(z))))/(1 - 4 epsilon lambda) times d rho / H(z), and with the constrained dark-energy equation of state giving positive rho+p, the integral over redshift is positive, so M(z) rises from early times to the present. The comparison plots show the same monotonic increase for the standard Lambda-CDM model, and the authors interpret the growth as the signature of dark energy's accelerating influence within generalized Rastall gravity.

Load-bearing premise

The conclusion rests on the unconstrained integration constant B_1 making lambda B_0 + B_1 positive, yet the paper assigns B_1 no value or prior, so the same equations would predict mass loss for a sufficiently negative B_1.

Editorial extensions

If this is right

Under both dark-energy parameterizations, a non-singular black hole in generalized Rastall gravity gains mass as the Universe expands, so dark-energy accretion acts as a growth mechanism rather than an evaporation mechanism.
Because the fitted dark-energy models have positive rho+p, the mass growth is tied to the accelerated expansion; the same accretion equation would predict mass loss for phantom-like fluids with rho+p<0.
The constrained parameter values place both models in the region where the mass integral in Eq. (35) is positive, so the qualitative mass-increase result holds within the reported 1 sigma and 2 sigma credible intervals.
The comparison with Lambda-CDM shows the same qualitative growth but different rates, implying that dark-energy accretion does not distinguish the models by sign alone but could distinguish them by rate.
If the result is correct, the growth mechanism could leave an imprint on the late-time mass function of black holes, connecting the accretion process to cosmic acceleration in a directly observable way.

Reading between the lines

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

An implication the paper leaves implicit is that the sign of the mass change is controlled by the freely adjustable combination lambda B_0 + B_1; until B_1 is fixed by data, the same formalism can produce either growth or shrinkage for identical dark-energy models.
A testable extension would be to predict the redshift-dependent mass growth as a correction to the local black-hole mass function; future gravitational-wave or quasar observations could search for the cumulative effect.
The paper treats lambda as a constant even though Eq. (2) defines it as delta/R; a fully self-consistent extension would couple the accretion rate back to the local Ricci scalar and allow the Rastall parameter to vary through the black hole's environment.
The same mass-equation machinery could be applied to other non-singular black hole solutions or other dark-energy parameterizations, since the main empirical content is not the sign alone but the predicted growth rate as a function of redshift.

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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 useful piece here is Section IV: they take the standard spherically symmetric accretion setup, adapt it to the non-conserved stress-energy of generalized Rastall gravity, and end up with explicit mass-redshift integrals for the CBDRM and CDMMA equation-of-state parameterizations (Eqs. 36 and 38). Those specialized formulas are not in the cited literature, and the MCMC work in Section III (CC + Pantheon + DESI BAO) is competent and standard. Credit is due for laying out the cosmological side carefully and for making the black hole metric and its properties explicit.

The problem is that the central claim does not follow from their own equations. Eq. (33) gives dM/dt = 4π(λB0+B1)(ρ+p)M^2. The sign of dM/dt is controlled by (λB0+B1), not by (ρ+p) alone. B1 is introduced in Eq. (31) as an arbitrary integration constant; no value, prior, or boundary condition is ever assigned to it. B0 = 4f/r − 2rf'' depends on the metric function (26), whose constants a and b also never appear in the accretion plots. The text says "ρ+p>0 gives mass increase," which is only true if λB0+B1 > 0. For the same fitted dark-energy parameters, choosing B1 negative enough reverses the inequality, and the identical equations would predict mass decrease. Figures 3–5 therefore do not demonstrate the headline result. This is load-bearing, not cosmetic.

There is also a minor-to-moderate internal inconsistency: Table I lists CBDRM λ = 0.339 ± 0.037 with prior [0, 0.1], and ω1 = 1.756 ± 0.017 with prior [0, 0.1]; similarly CDMMA λ = −0.082 ± 0.004 with prior [0, 0.1]. Posterior means far outside the prior range suggest a mistake in prior definitions or in the sampling setup. And λ is treated as an independent free parameter even though Eq. (2) defines λ = δ/R; the paper never reconciles these.

The specialization to two EOS parameterizations is genuinely new, and the paper is honest about its lineage. But as it stands, the conclusion is not supported by the equations, so I would not accept it without major revision. For a referee: this is worth a careful read because the fix is straightforward — specify B1 (or an appropriate boundary condition) and redo the plots — and the explicit mass formulas would then be serviceable. I would send it to review with a request for major revision rather than desk-reject, since the core derivation is coherent and the gap is identifiable.

Referee Report

4 major / 4 minor

Summary. The paper studies mass accretion onto a non-singular black hole in generalized Rastall gravity, using two dynamical dark-energy equation-of-state parameterizations (CBDRM and CDMMA). The authors first constrain the model parameters with MCMC analyses of Cosmic Chronometers, Pantheon+ SNe Ia, and DESI/DES BAO data, then derive a mass-redshift relation from the modified conservation equation and plot the black-hole mass evolution for both parameterizations and for ΛCDM. The central claim, stated in the abstract and repeated in the conclusions, is that accreting either of the two dark-energy parameterizations, with constrained parameters, increases the black-hole mass during cosmic evolution.

Significance. If the central claim were established, the paper would connect observational cosmological constraints to a specific prediction for black-hole mass evolution in a modified gravity theory, which is a potentially interesting cross-area result. The manuscript has some strengths: it uses public observational datasets, employs standard Bayesian tools (polyChord and GetDist), presents explicit posterior tables and statistical model comparisons, and writes the accretion equations in a transparent algebraic form. However, the main physical conclusion is not supported by the presented calculation. The sign of the mass change is controlled by an arbitrary integration constant, and the MCMC priors appear inconsistent with the reported posteriors. Because these issues directly affect the headline prediction, the paper's claim about the 'true nature of dark energy' from accretion is not established.

major comments (4)

[Section IV, Eqs. (31)-(35)] The sign of the mass accretion rate in Eq. (33), dM/dt = 4π(λB0+B1)(ρ+p)M^2, is proportional to the product (λB0+B1)(ρ+p). The text after Eq. (33) argues that ρ+p>0 implies mass increase, but that conclusion requires λB0+B1>0. The constant B1 is introduced in Eq. (31) as an arbitrary integration constant, and no value, prior, or boundary condition is specified for it anywhere in the paper. In addition, B0 = 4f(r)/r − 2r f''(r) depends on the metric function f(r) in Eq. (26), which contains the unspecified parameters a and b. Therefore the fitted MCMC parameters in Table I do not determine the sign or magnitude of the mass evolution. For the same fitted dark-energy parameters, one can choose B1 < −λB0 and obtain a decreasing mass for identical cosmological input. Consequently, Figures 3-5 and the abstract's claim of mass increase are not consequences of the constrained parameters.
[Section III, Table I] Table I reports posterior means that lie outside the stated prior ranges: for CBDRM, λ = 0.339 ± 0.037 with prior [0, 0.1], ω1 = 1.756 ± 0.017 with prior [0, 0.1], and δ = 1.059 ± 0.540 with prior [0, 1]; for CDMMA, λ = −0.082 ± 0.004 with prior [0, 0.1]. With bounded uniform priors, a posterior distribution supported outside the prior box is impossible in a correctly implemented nested-sampling or MCMC run. This indicates either misreported priors or an error in the sampling procedure. Since the paper's stated novelty is parameter constraining, this inconsistency directly undermines the reliability of the constraints used in all subsequent accretion plots.
[Section II, Eq. (2), and Section III, Table I] The generalized Rastall theory is defined with λ = δ/R in Eq. (2), but in Section III the parameters λ and δ are treated as independent free parameters with independent priors and independent posterior means (e.g., for CBDRM, λ = 0.339 ± 0.037 and δ = 1.059 ± 0.540). No constraint enforcing λ = δ/R appears in the analysis. Because λ and δ both enter the Hubble parameter expressions in Eqs. (17) and (21) and the accretion integrand in Eqs. (36) and (38), this internal inconsistency affects the quantitative results and should be resolved before the mass-accretion conclusion can be assessed.
[Section IV, Eq. (34)] Equation (34), which states (ρ+p) = −[(1−3ϵλ(1+ω(z)))/(3H(1−4ϵλ))] ρdot, is asserted without derivation. It does not follow directly from the generalized conservation law in Eq. (7); one would need additional assumptions relating Hdot, ρ, and ω(z). Since Eq. (35) is obtained by substituting Eq. (34) into Eq. (33), this missing derivation is a load-bearing step for the mass equation and for the plotted mass evolution. The authors should either provide the derivation or state the additional relation used.

minor comments (4)

[Figures 3 and 4] Each of Figs. 3 and 4 appears to show two identical side-by-side panels. If these are duplicate panels, they should be removed; if they are meant to show different quantities or parameter choices, the panels should be labeled and the difference explained.
[Figures 3-5] The axes in Figures 3-5 are not clearly labeled with units or normalization. Equation (35) gives M in terms of M0, but the plots show unlabeled values; the authors should state whether the plotted quantity is M/M0 and specify the values used for M0, a, b, and B1.
[Section IV, paragraph after Eq. (30)] The phrase 'redial temporal component' should be 'radial temporal component'.
[Abstract and title] The title and abstract contain stylistic and typographical issues, including a missing space before the question mark in the title and several comma splices; a careful language edit is recommended.

Circularity Check

1 steps flagged · score 6.0 of 10

Eq. (33) makes the sign of dM/dt proportional to the unconstrained prefactor (λB0+B1); the paper's mass-increase conclusion is therefore chosen via B1, not derived from the MCMC-constrained dark-energy parameters.

other [Sec. IV, 'Mass accretion in generalized Rastall gravity theory', Eqs. (31), (33) and text following Eq. (33); central claim in Abstract and Conclusions.]
"M˙ = 4π (λB0 + B1)(ρ + p(ρ)) M2. ... Thus, in the phantom-filled Universe (ρ + p < 0), the mass of the black hole will gradually decrease; on the contrary, in the dark energy-filled Universe (ρ + p > 0), that mass will increase progressively. ... B0 = (4 f (r)/r − 2r f''(r)) and B1 being any integration constant whose dimension is the same as energy density."

Eq. (33) makes the sign of dM/dt equal to the sign of (λB0+B1)(ρ+p). The text infers the sign from ρ+p alone, but B1 is introduced as 'any integration constant' with no value, prior, or boundary condition, and B0 depends on the metric constants a and b from Eq. (26), also unassigned and unconstrained by the MCMC analysis. Therefore, with the same fitted CBDRM/CDMMA parameters and ρ+p>0, choosing λB0+B1<0 (e.g., a sufficiently negative B1) makes M decrease. The claimed mass increase is thus not a consequence of the constrained dark-energy parameterizations; it is an input encoded in the unconstrained prefactor. Figures 3-5 do not demonstrate the headline as a prediction.

full rationale

I find one load-bearing underdetermination that functions as circularity: the central 'mass increase' prediction is not a consequence of the MCMC-constrained EOS parameters, because Eq. (33) multiplies ρ+p by (λB0+B1), and B1 (plus a,b entering B0) is completely unconstrained. The paper's inference from ρ+p>0 to mass increase is exactly the step that is invalid without a sign choice for the prefactor; the result reduces to that arbitrary choice. Self-cited CBDRM/CDMMA parameterizations are adopted model inputs rather than derived outputs, so citing refs [19,20,52] is not by itself circular. The Visser equivalence [57] is independent support. Additional internal inconsistencies reinforce the concern but are not circularity: Table I reports posterior means outside the stated priors (CBDRM lambda=0.339±0.037 with prior [0,0.1] and omega1=1.756±0.017 with prior [0,0.1]; CDMMA lambda=-0.082±0.004 with prior [0,0.1]), and lambda and delta are fit independently despite Eq. (2) defining lambda=delta/R. Because the manuscript's headline claim is underdetermined by its fitted inputs and can be reversed by the arbitrary prefactor, the circularity score is 6.

Assumptions & free parameters 22 free parameters · 6 assumptions · 0 invented entities

The central mass equation Eq (35) is built on the generalized Rastall field equations, a specific non-singular black hole solution borrowed from ref [47], and an unconstrained integration constant B1. The cosmological parameters are fitted to data, but the accretion sign is not: it is fixed by B1 and by the choice of dark energy equation of state. This ledger shows that the plotted mass growth is not a free-standing prediction.

free parameters (22)

H0 (CBDRM) = 63.282 ± 1.623
MCMC constrained; enters Hubble parameter Eq (17) and the mass integral.
H0 (CDMMA) = 65.032 ± 1.460
MCMC constrained; enters Hubble parameter Eq (21) and the mass integral.
Omega_m0 (CBDRM) = 0.280 ± 0.005
MCMC constrained; enters the energy density and Hubble parameter.
Omega_m0 (CDMMA) = 0.279 ± 0.006
MCMC constrained; enters the energy density and Hubble parameter.
Omega_DE0 (CBDRM) = 0.720 ± 0.005
MCMC constrained; sets the dark energy density normalization.
Omega_DE0 (CDMMA) = 0.721 ± 0.006
MCMC constrained; sets the dark energy density normalization.
epsilon (CBDRM) = 13.678 ± 0.273
Rastall gravitational constant, MCMC constrained; appears in Friedmann and mass equations.
epsilon (CDMMA) = 14.614 ± 0.175
Rastall gravitational constant, MCMC constrained; appears in Friedmann and mass equations.
lambda (CBDRM) = 0.339 ± 0.037 (prior [0, 0.1])
Rastall parameter, treated as constant in the mass equation despite Eq (2); posterior mean lies outside the stated prior.
lambda (CDMMA) = -0.082 ± 0.004 (prior [0, 0.1])
Rastall parameter, treated as constant in the mass equation despite Eq (2); posterior mean lies outside the stated prior.
delta (CBDRM) = 1.059 ± 0.540 (prior [0, 1])
Non-minimal coupling parameter; enters Hubble and mass equations; posterior mean lies outside the stated prior.
delta (CDMMA) = 0.622 ± 0.026
Non-minimal coupling parameter; enters Hubble and mass equations.
omega0 (CBDRM) = -1.469 ± 0.099
EOS parameter in Eq (14); directly sets the dark energy behavior in the accretion integral.
omega1 (CBDRM) = 1.756 ± 0.017
EOS parameter in Eq (14); directly sets the dark energy behavior in the accretion integral.
omega0 (CDMMA) = -1.451 ± 0.008
EOS parameter in Eq (18); directly sets the dark energy behavior in the accretion integral.
omega1 (CDMMA) = 0.033 ± 0.003
EOS parameter in Eq (18); directly sets the dark energy behavior in the accretion integral.
omega2 (CDMMA) = 0.916 ± 0.006
EOS parameter in Eq (18); directly sets the dark energy behavior in the accretion integral.
alpha (CDMMA) = 0.068 ± 0.007
EOS parameter in Eq (18); directly sets the dark energy behavior in the accretion integral.
beta (CDMMA) = 0.851 ± 0.093
EOS parameter in Eq (18); directly sets the dark energy behavior in the accretion integral.
B1 = not stated
Integration constant in Eqs (31)-(35); its sign determines whether mass increases or decreases, and no constraint is given.
a = not stated
Parameter of the non-singular density profile Eq (23); enters B0 in the mass integral and is never specified.
b = not stated
Parameter of the non-singular density profile Eq (23); enters B0 in the mass integral and is never specified.

assumptions (6)

standard math Bianchi identity and standard FLRW kinematics
Used to obtain the modified Friedmann equations (5)-(6) in Section II without derivation.
domain assumption Generalized Rastall field equations Eq (3) and the modified conservation law Eq (1) are the correct gravitational theory
Section II, Eqs (1)-(3); the paper adopts this framework from refs [42,44,45].
domain assumption The Universe is spatially flat, isotropic, homogeneous and filled with a barotropic perfect fluid of dark matter plus dark energy
Section II, Eqs (4)-(7).
domain assumption The same parameters epsilon, delta, lambda constrained by cosmological data are valid for the local black hole accretion problem
Section III to Section IV; no scale-separation or screening argument is supplied.
ad hoc to paper The sign and magnitude of (lambda B0 + B1) allow a positive mass accretion rate for dark energy
Eqs (31)-(35); B1 is never specified, so the increase is assumed rather than derived.
domain assumption The non-singular black hole solution Eq (26) with constraints Eqs (28)-(29) is taken from ref [47] without re-derivation
Section IV, Eq (26).

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Cite this review

Pith. "Pith review of How parameter constraining can influence the mass accretion process of a Black Hole in the Generalized Rastall Gravity Theory ?." pith.science (2026). https://pith.science/paper/5MFXQAFE

@misc{pith2026241115619,
  author       = {Pith},
  title        = {Pith review of: How parameter constraining can influence the mass accretion process of a Black Hole in the Generalized Rastall Gravity Theory ?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MFXQAFE}},
  note         = {Machine review of arXiv:2411.15619}
}

abstract

Black holes, one of the greatest enigmas of our Universe, are challenging to decipher. This work is dedicated to observing the changes in the mass of a non-singular black hole with the evolution of the Universe in the generalized Rastall gravity framework, considering the effects of parameter constraining. We examine two recently developed dynamical dark-energy equation of state parameterization models: Chaudhary-Bouali-Debnath-Roy-Mustafa-type~(CBDRM)~parameterization and Chaudhary-Debnath-Mustafa-Maurya-Atamurotov-type~(CDMMA)~parameterization. Starting with the concept and fundamental equations of the generalized Rastall gravity theory, we introduce the two models along with their equations of state, energy density equations, and corresponding Hubble parameter equations. We then constrain the required parameters using Monte Carlo Markov chain (MCMC) analyses to ensure the accuracy and reliability of our study. Next, we discuss the non-singular black holes from the perspective of generalized Rastall gravity theory and some of their essential properties. Finally, we pursue the primary goal of our work: analyzing the mass accretion process. We derive the mass equation for both models in terms of the redshift function, represent the results graphically, and compare them with the standard $\Lambda$CDM model of the Universe. Our findings indicate that the accretion of both CBDRM~and~CDMMA dark energy parameterizations, considering constrained parameter values, leads to an increase in the mass of the black hole during the Universe's evolution within the generalized Rastall gravity theory, revealing the true nature of dark energy.

Figures

Figures reproduced from arXiv: 2411.15619 by the authors.

**Figure 2.** FIG. 2: The posterior distributions of the CDMMA model at 68% (1 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3: Changes in the mass of the black hole with redshift for Model-1 (CBDRM parametrization). [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: Changes in the mass of the black hole with redshift for Model-2 (CDMMA parametrization). [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

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