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Einstein-Gauss-Bonnet-Myrzakulov Gravity from $R + F(T, G)$: Numerical Insights and Torsion-Gauss-Bonnet Dynamics in Weitzenb\"ock Spacetime

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Pith's one-line read This paper claims that the R+F(T,G) action forms a viable modified gravity in Weitzenböck spacetime, with torsion–Gauss-Bonnet corrections driving cosmic acceleration, altering black-hole thermodynamics, and raising the neutron-star mass…

desk verdict Review-style preprint with no derivation, inconsistent central equations, and qualitative plots; not suitable for peer review. read the letter →

arxiv 2505.18285 v1 pith:262VXZFC submitted 2025-05-23 gr-qc astro-ph.COhep-th

Davood Momeni , Ratbay Myrzakulov This is my paper
classification gr-qcastro-ph.COhep-th PACS 04.50.Kd
keywords modifiedgravityteleparalleltorsionscalarGauss-BonnettermdarkenergystatefinderdiagnosticsneutronstarsWeitzenböckspacetime
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

The paper aims to establish that an action of the form $S=\int d^4x\sqrt{-g}\,[R+F(T,G)]$, with $R$ the Ricci curvature scalar, $T$ the torsion scalar of teleparallel gravity, and $G$ the Gauss-Bonnet term (a quadratic curvature invariant), defines a consistent modified theory of gravity in Weitzenböck (teleparallel) spacetime. It claims that the field equations contain explicit effective energy densities and pressures from both torsion and the Gauss-Bonnet term, and that these geometric contributions can drive early-time inflation and late-time cosmic acceleration without a cosmological constant. The same framework is then applied to statefinder diagnostics, black-hole horizons, Solar System limits, and neutron-star mass-radius relations. A sympathetic reader would care because, if the framework holds, one geometric mechanism would simultaneously address dark energy and predict heavier neutron stars than general relativity allows.

What carries the argument

The load-bearing object is the arbitrary function $F(T,G)$ in the action. The paper treats $T$ and $G$ as independent geometric scalars on a Weitzenböck (teleparallel) spacetime, varies the action with respect to the vielbein (the local frame field) and the connection, and packages the result as effective torsion and Gauss-Bonnet fluids in the modified Friedmann equations. This machinery generates the paper's concrete predictions: dynamical-system variables $x=\dot{H}/H^2$, $y=f_T/H^2$, $z=f_G/H^4$ for stability analysis, a statefinder pair $(r,s)$ built from third and second derivatives of the scale factor, and the mass-radius relation obtained by solving the hydrostatic-equilibrium equations with the same effective densities.

What would settle it

The decisive check is a direct calculation: take the action $S=\int d^4x\sqrt{-g}\,[R+F(T,G)]$ and perform the full variation with respect to the vielbein and the connection in Weitzenböck spacetime. If the resulting equations contain cross-terms involving $\partial^2F/\partial T\partial G$, or differ from Eqs. (19), (25), and (30)-(35), then the paper's equations are not the equations of its own action. On the observational side, the mass-radius curve predicts a maximum neutron-star mass above the general-relativistic bound, so a precise measurement of a neutron star that exceeds the predicted curve, or a tidal-deformability measurement that contradicts it, would settle the astrophysical claim.

Watch

Extended reading notes

Core claim

On its own terms, the central claim is that the Einstein–Gauss–Bonnet framework described by $S=\int d^4x\sqrt{-g}\,[R+F(T,G)]$ yields the modified Einstein equations $G_{\mu\nu}=\kappa\left(T^{\rm matter}_{\mu\nu}+T^{\rm torsion}_{\mu\nu}+T^{(G)}_{\mu\nu}\right)$ (Eq. (19)), and that under the Friedmann-Robertson-Walker cosmological ansatz these reduce to the Friedmann system Eqs. (30)-(35) with $\rho_{\rm torsion}=\tfrac12(TF_T-F)$, $p_{\rm torsion}=\tfrac12(TF_T-F+2\dot{T}F_T)$, $\rho_{\rm GB}=\tfrac12(GF_G-F)$, and $p_{\rm GB}=\tfrac12(GF_G-F+2\dot{G}F_G)$. The paper further claims that numerical integration of these equations produces a Hubble history starting from $H_0=70$ km/s/Mpc, statefinder tracks $(r,s)$ that leave the $\Lambda$CDM point $(1,0)$, black-hole horizon functions $A(r)$ and $B(r)$ with torsion-dependent temperatures, a Newtonian limit that survives Solar System tests, and a neutron-star mass-radius curve whose maximum mass exceeds the general-relativistic limit, consistent with observed high-mass pulsars.

Load-bearing premise

The whole framework rests on the assumption that a single spacetime can carry a well-defined torsion scalar $T$ and an independent Gauss-Bonnet term $G$, and that varying the action with respect to the frame and connection really produces the field equations written in the paper; if either geometric quantity is ill-defined or the variation yields different terms, the predicted cosmology and neutron-star results do not follow.

Editorial extensions

If this is right

  • If the field equations (19) and (30)-(35) are correct, torsion and Gauss-Bonnet form an effective dark-energy fluid, so cosmic acceleration can arise without introducing a cosmological constant by hand.
  • The statefinder parameters $(r,s)$ evolve away from $\Lambda$CDM's fixed point $(1,0)$ in a redshift-dependent way, giving high-precision cosmic surveys a geometric signature that distinguishes this theory from standard cosmology.
  • With $F(T,G)=\alpha T^n+\beta G^m$ and parameters such as $\alpha=0.1$, $\beta=0.05$, $n=m=2$, the modified potential returns to the Newtonian $1/r$ form at Solar System scales, so the theory can satisfy local gravity constraints.
  • The torsion-dependent horizon functions $A(r)$ and $B(r)$ change Hawking temperatures and horizon structure, producing testable modifications in black-hole observations.
  • The mass-radius relation predicted for neutron stars allows larger maximum masses than general relativity, offering an explanation for observed pulsars that appear to exceed the standard limit.

Reading between the lines

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

  • Editorial inference: a full variation of the action would likely produce explicit cross-coupling terms from $\partial^2F/\partial T\partial G$; computing them and testing whether they are small would settle whether the paper's effective-fluid split is the right leading-order description.
  • Editorial inference: the statefinder tracks in Figs. 1 and 2 could be converted directly into a fitting statistic against model-independent reconstructions of $H(z)$ from supernova and baryon acoustic oscillation data, turning the qualitative deviations into a quantitative viability test.
  • Editorial inference: because the theory is claimed to raise the maximum neutron-star mass, the same framework should predict the tidal deformability of neutron stars; a future gravitational-wave event could discriminate $F(T,G)$ gravity from general relativity even without resolving the mass-radius curve.
  • Editorial inference: the paper's special-case limits suggest a direct consistency check, namely verifying that sending $F(T,G)$ to $f(T)$, $f(G)$, or $f(R)$ reproduces those known equations term by term.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 5 minor

Summary. The manuscript proposes a modified gravity theory, 'Einstein-Gauss-Bonnet-Myrzakulov gravity', with action S = ∫ d^4x √−g [R+F(T,G)] in Weitzenböck spacetime. It claims to derive the field equations for this theory and then use them to study Friedmann cosmology, statefinder diagnostics, black-hole and compact-object solutions, phase-space dynamics, solar-system constraints, and neutron-star mass-radius relations. The stated goal is to demonstrate the robustness of the R+F(T,G) framework. In practice the paper is largely a review of existing F(R,T), f(T), and f(G) literature, and the derivations that would support the new claims are only sketched or entirely absent. The manuscript asserts in Section XV that the field equations have been derived and investigated, but the actual variation is never carried out and the central equations are mutually inconsistent.

Significance. The paper identifies a genuinely open problem: hybrid theories containing both torsion and Gauss-Bonnet terms require a careful variational derivation, and a correct set of field equations would be useful for cosmology and compact-object physics. The literature survey may also help readers locate related work. However, the manuscript does not solve the problem it poses. No machine-checked derivation, reproducible code, or parameter-free prediction is provided, and several of the displayed equations are internally inconsistent. The qualitative plots of statefinder parameters, potentials, and mass-radius curves are not tied to the stated field equations in any checkable way. If the central derivation were supplied and corrected, the framework might be of interest, but as it stands the paper's central claim of robustness is not established.

major comments (6)
  1. [§V.A, Eqs. (11)–(19)] The central field equations are asserted, not derived. The variational principle is stated only in the generic Euler–Lagrange forms (13)–(15); no variation of the vielbein or connection is actually computed, and the terms δF/δe and δF/δΓ are never evaluated. The symbol G in the action (11) is never defined in the Weitzenböck geometry used in the paper: Eq. (9) defines a different object, L_GB^(T) = R^2 −4R_μνR^μν + R_μναβR^μναβ + T, which is not the G of Eq. (11). The closing paragraph of §V.B asserts that the field equations have been derived, but no derivation appears in the text. Since Eqs. (19), (25), and (29)–(35) are the foundation of every subsequent plot and solution, this is a load-bearing gap.
  2. [§VI, Eq. (25) versus §V.A, Eq. (19)] The paper presents incompatible versions of the field equations from the same action. Eq. (19) includes a Gauss–Bonnet stress tensor T^(G)_μν, while Eq. (25), also labelled as the field equations of the R+F(T,G) action, contains no F_G term at all and instead has a term 1/2[F(T,G)−T]g_μν. Eq. (29) is yet another form with separate torsion and Gauss–Bonnet tensors. The paper never reconciles these expressions; a single action should yield a single set of field equations. This inconsistency directly undermines the claim that the field equations have been derived.
  3. [§VII.A, Eqs. (32)–(35)] The effective torsion and Gauss-Bonnet energy densities and pressures are imported from f(T) and f(G) cosmology without being derived from the two-variable action (11). Eq. (33) contains the term 2 RT F_T, which has dimension L^{-3} if T ~ H^2 and F has the same dimension as R; the adjacent terms have dimension L^{-2}, so the equation is dimensionally inconsistent as written. A correct variation of F(T,G) would also generate mixed F_TG terms and RG contributions, none of which appear. Consequently the modified Friedmann equations (30)–(31) are not justified.
  4. [§VII.B, Eq. (37) and following text] The dust solution is internally inconsistent. The paper sets the acceleration equation RH + H^2 + k/a^2 = 0 and then states that the Hubble parameter follows H ∼ a^{−3/2}. For k=0, H = C a^{−3/2} implies RH = −(3/2)H^2, hence RH + H^2 = −(1/2)H^2, which is not zero. The quoted scaling is the solution of RH = −(3/2)H^2, not of RH = −H^2. The claimed dust solution is therefore not a solution of the displayed equation.
  5. [§XIII, Figs. 5–6] The solar-system consistency claim is not a test of the theory. The text states that the parameters α, β, n, m 'can be tuned to ensure agreement' with the inverse-square law, and Fig. 6 shows a potential for one arbitrary parameter choice (α=0.1, β=0.05, n=m=2). No derivation of the modified Newtonian potential from the field equations is given, no comparison with actual data such as perihelion precession or Shapiro delay is performed, and no bounds are reported. A parameter that is freely tuned to force agreement cannot support the claimed compatibility with local gravity constraints.
  6. [§XIV.C, Fig. 7] The neutron-star mass-radius prediction is not supported by any calculation in the manuscript. The text asserts qualitatively that F(T,G) gravity changes the mass-radius relation and that the green curve accommodates PSR J0740+6620, but no Tolman–Oppenheimer–Volkoff equations for this theory, no equation of state, and no numerical scheme for the stellar-structure problem are given. The claim that the theory can explain high-mass neutron stars is therefore unsubstantiated.
minor comments (5)
  1. [§V.A.4, Eq. (17)] Equation (17) uses the same symbol T_μν for both the matter energy-momentum tensor and the torsional contribution; distinct notations such as T^(m)_μν and T^(tor)_μν should be used throughout.
  2. [Abstract] The abstract reads 'Einstein-Gauss-Bonnet-Myrzakulov $R = F(T, G)$ gravity', but the action used in the paper is R + F(T,G); the equals sign appears to be a typo and should be corrected.
  3. [§XII, phase-space variables] The variable z = f_G/H^4 is introduced as a dimensionless dynamical variable, but with the dimensions used elsewhere in the paper (F has dimension L^{-2}, G has dimension L^{-4}), f_G has dimension L^2 and H^4 has dimension L^{-4}, so z is not dimensionless.
  4. [Figures 1–3, 6, 7] The numerical plots are not reproducible: the specific functions F(T,G), parameter values, initial conditions, and metric functions that define 'Model A', 'Model B', the modified potential, and the mass-radius curves are not specified.
  5. [§IV] The statement that the Gauss-Bonnet term is a total divergence in four dimensions and 'does not affect the equations of motion in 4D' needs qualification, since a nonlinear f(G) or a coupled F(T,G) is nontrivial even in four dimensions.

Circularity Check

1 steps flagged · score 6.0 of 10

Solar System consistency is manufactured by tuning the free parameters α, β, n, m, so the 'prediction' of Newtonian recovery reduces to the chosen parameters; the central F(T,G) field equations are also asserted rather than derived, though that gap is a derivation failure more than a circularity.

  1. fitted input called prediction [Section XIII, Solar System Tests and Local Gravity Constraints, paragraphs around Figure 6]
    "F (T, G) = αT n + βGm, includes parameters α, β, n, m that can be tuned to ensure agreement with the known inverse-square law behavior of gravity. ... The parameters are chosen as α = 0.1, β = 0.05, and n = m = 2. The plot clearly shows that the modifications rapidly decay and converge to the Newtonian form at Solar System scales, satisfying local gravity constraints."

    The 'local gravity constraints' are not an independent test of the theory. The free constants α, β, n, m are explicitly tuned to make the F(T,G) correction fall off as desired, and Figure 6 then plots that tuned curve. No independent observable such as PPN parameters, perihelion precession, or Shapiro delay is computed. Therefore the stated conclusion that 'the model satisfies local gravity constraints' is equivalent, by construction, to the prior parameter choice; the Newtonian limit is fitted into the model rather than predicted from it.

full rationale

The strongest genuinely circular step is the Solar System consistency claim: the paper states that α, β, n, m 'can be tuned to ensure agreement' with inverse-square behavior, then presents a figure of the tuned potential as evidence that the model satisfies local gravity constraints. That is a fitted parameter being renamed as a successful prediction. The rest of the paper's derivation chain has a serious gap, but it is not circularity of the input-equals-output type: the variation of action (11) is never actually performed, G is never defined as an independent scalar in Weitzenböck spacetime (Eq. (9) proposes L_GB^(T) = R^2 − 4Rμν R^μν + Rμναβ R^μναβ + T, which is not the G used later), Eq. (25) contains no F_G term while Eqs. (34)-(35) add G F_G contributions, and no δF/δe or δF/δΓ computation is shown. These are derivation/consistency failures and are better scored as correctness risk than as circularity. The self-citations [46,47] are used only as motivation, not as load-bearing justifications, so they do not raise the score. The statefinder, Hubble, and neutron-star plots are numerical illustrations of chosen F(T,G) forms rather than independent data fits. Overall, one explicit prediction reduces by construction, so the paper is partially circular, but the central claim also has substantial unverified independent content that is incomplete rather than definitionally equivalent to its inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The ledger is nearly empty of independently grounded content. The central equations are asserted or imported from prior work; no new entities are introduced. The free parameters are the tunable coefficients of F(T,G) and a hand-chosen initial condition for the example figures.

free parameters (2)
  • α, β, n, m (coefficients in F(T,G)=αT^n+βG^m) = α=0.1, β=0.05, n=m=2 (Sec. XIII, illustrative)
    Introduced to ensure solar-system agreement; the paper explicitly says these parameters can be tuned to ensure agreement with the Newtonian limit.
  • Initial s(0) for statefinder plots = 0.1 (Fig. 2 caption)
    The figure states 'Both models start from s=0.1', a hand-chosen initial condition with no derivation; the Model A/B curves are otherwise unspecified.
assumptions (4)
  • ad hoc to paper The action S = ∫ d^4x √-g [R + F(T,G)] (Eq. 11) is a valid starting point for a theory of gravity.
    The paper postulates this action without derivation. It is the model definition.
  • domain assumption In Weitzenbock spacetime, a Gauss-Bonnet term G exists, is non-trivial, and is independent of the torsion scalar T.
    Section IV.A defines a generalized GB term by simply adding T to the curvature GB combination (Eq. 9) and never specifies how G is computed in a teleparallel geometry; this is load-bearing for the entire framework.
  • ad hoc to paper The effective energy densities and pressures of torsion and Gauss-Bonnet are given by Eqs. (32)-(35).
    These expressions are asserted without derivation from the action, and they are dimensionally inconsistent with a two-variable F(T,G) because they omit G-derivative and H-dependent terms.
  • domain assumption The form F(T,G)=αT^n+βG^m [48] is the representative model used for constraints.
    The paper imports this functional form from Kofinas and Saridakis and uses it for phase-space and solar-system constraints without deriving it.

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Cite this review

Pith. "Pith review of Einstein-Gauss-Bonnet-Myrzakulov Gravity from $R + F(T, G)$: Numerical Insights and Torsion-Gauss-Bonnet Dynamics in Weitzenb\"ock Spacetime." pith.science (2026). https://pith.science/paper/262VXZFC

@misc{pith2026250518285,
  author       = {Pith},
  title        = {Pith review of: Einstein-Gauss-Bonnet-Myrzakulov Gravity from $R + F(T, G)$: Numerical Insights and Torsion-Gauss-Bonnet Dynamics in Weitzenb\"ock Spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/262VXZFC}},
  note         = {Machine review of arXiv:2505.18285}
}
abstract

The study of modified gravity models has garnered significant attention because of their potential to provide alternative explanations for cosmological phenomena, such as the accelerated expansion of the universe and the nature of dark energy. One such model, the Einstein-Gauss-Bonnet-Myrzakulov $R + F(T, G)$ gravity (EGBMG), which incorporates the curvature $R$, torsion $T$, and the Gauss-Bonnet term $G$, offers a promising framework to explore the dynamics of the universe and its evolution. This paper delves into the theoretical and observational implications of the EGBMG model, focusing on its ability to address long-standing challenges in cosmology, including the evolution of dark energy and the transition from early-time inflationary behavior to late-time acceleration. We review recent advancements in the model, including its compatibility with observational data and its ability to provide new insights into cosmic acceleration. Through a combination of theoretical models, dynamical systems analysis, and cosmological diagnostics, we demonstrate the robustness of the EGBMG framework in explaining the large-scale structure of the universe and its accelerated expansion. This paper serves as a step toward further exploring the potential of this model to understand the fundamental forces driving Weitzenb$\"{o}$ck spacetime.

Figures

Figures reproduced from arXiv: 2505.18285 by the authors.

Figure 1
Figure 1. FIG. 1: Evolution of the statefinder parameter [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Evolution of the statefinder parameter [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Evolution of the Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: A qualitative sketch of phase-space trajectories in the ( [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison of the Newtonian potential (dashed) with the m [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Mass-radius relation for neutron stars in different gravity [PITH_FULL_IMAGE:figures/full_fig_p034_7.png]

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