f(R, $G$, T) Gravity: Cosmological Implications and Dynamical System Analysis

G. Mustafa; Himanshu Chaudhary; Ratul Mandal; Tiberiu Harko; Ujjal Debnath

REVIEW 2 major objections 4 minor 186 references

f(R, $G$, T) Gravity: Cosmological Implications and Dynamical System Analysis

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that a specific f(R,G,T) gravity model—a mix of Gauss-Bonnet, Ricci-scalar, and matter-trace powers—fits combined cosmic data with moderate Bayesian evidence over ΛCDM and its dynamical system reproduces all standard…

desk verdict The model is new and the fitting is careful, but the stated Lagrangian is complex on the fitted trajectory, so the main statistical claim does not hold. read the letter →

arxiv 2506.07623 v3 pith:WKDM3YO5 submitted 2025-06-09 gr-qc

Ratul Mandal , Himanshu Chaudhary , Tiberiu Harko , Ujjal Debnath , G. Mustafa This is my paper

classification gr-qc MSC 83D0583F05 PACS 04.50.Kd98.80.-k

keywords f(RGT)gravityGauss-BonnetinvariantmodifieddarkenergydynamicalsystemanalysisBayesianmodelcomparisoncosmicchronometersbaryonacousticoscillations

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 particular higher-curvature theory of gravity—an action that mixes powers of the Gauss-Bonnet invariant, the Ricci scalar, and the trace of the matter stress-energy tensor—can serve as a background-level alternative to the cosmological constant. The chosen Lagrangian $f(R,G,T)=\alpha_1 G^m+\alpha_2 R^\beta-2\alpha_3\sqrt{-T}$ is fitted to combined cosmic-chronometer, Type Ia supernova, and DESI BAO data; the authors report $\ln(B_{0i})=3.5994$, which they read as moderate evidence for this model over $\Lambda$CDM. They also recast the Friedmann equations as an autonomous dynamical system and identify eight critical points whose stability reproduces radiation, matter, late acceleration, and phantom eras. If right, the model shows that geometry–matter coupling can mimic the standard cosmological history without a cosmological constant, at the price of a richer parameter space.

What carries the argument

The load-bearing object is the explicit Lagrangian $f(R,G,T)=\alpha_1 G^m+\alpha_2 R^\beta-2\alpha_3\sqrt{-T}$, where $G=R^2-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\xi\eta}R^{\mu\nu\xi\eta}$ is the Gauss-Bonnet topological invariant. The $\sqrt{-T}$ term is chosen so that dust matter is covariantly conserved, and the full form feeds into generalized Friedmann equations whose Hubble evolution is a second-order nonlinear ODE solved numerically for the fit. For the stability analysis, seven dimensionless variables $x_1,\dots,x_7$, built from $f$ and its derivatives, reduce the system to four first-order ODEs; the critical points $P_1,\dots,P_8$ and their Lyapunov linear stability supply the cosmological epochs.

What would settle it

Evaluate $G(z)=24H^2(\dot H+H^2)$ along the best-fit $H(z)$ from the paper; wherever $G<0$, check whether the numerical integration evaluated $(G)^m$ or $|G|^m$. If it integrated the former, the fitted ODE is complex-valued in the matter and radiation epochs; if the latter, the model's Lagrangian differs from the stated one. Either way, the reported Bayes factor and critical-point analysis depend on which branch was chosen.

Watch

Extended reading notes

Core claim

The central claim is that the Lagrangian density $f(R,G,T)=\alpha_1 G^m+\alpha_2 R^\beta-2\alpha_3\sqrt{-T}$ is both observationally viable and dynamically complete. A nested-sampling Bayesian fit to cosmic chronometers, Type Ia supernovae without the SHOES calibration, and DESI DR2 BAO data yields $H_0=66.1\pm3.8$ km/s/Mpc, $\Omega_{m0}=0.294\pm0.022$, and a Bayes factor $\ln(B_{0i})=3.5994$, which the paper interprets as moderate support over $\Lambda$CDM. The same model, when recast in terms of seven dimensionless phase-space variables, reduces to a four-dimensional autonomous system whose eight critical points include a radiation epoch, a matter saddle, a transition at $N\approx-0.29$, and a stable late-time accelerating attractor that can reach the phantom regime. The paper also reports the background observables $q_0\approx-0.57$, $\Omega_{m0}\approx0.24$, $j_0\approx1.05$, and $s_0\approx0$.

Load-bearing premise

The fit assumes $\alpha_1 G^m$ is a real function along the whole trajectory, but the fitted non-integer $m\approx1.95$ makes $G^m$ undefined when the Gauss-Bonnet invariant $G=24H^2(\dot H+H^2)$ turns negative during the decelerating matter and radiation epochs.

Editorial extensions

If this is right

The combined CC+SNe+BAO fit gives $H_0=66.1\pm3.8$ km/s/Mpc and $\Omega_{m0}=0.294\pm0.022$, consistent with Planck and DESI DR2 values, so the model does not obviously worsen the Hubble tension.
The Bayes factor $\ln(B_{0i})=3.5994$ places the model in the moderate-evidence band relative to $\Lambda$CDM; the paper reads this as marginal preference, not a ruling out.
The dynamical-system analysis yields a complete background history: radiation-dominated critical points, a matter-dominated saddle, transition at $N\approx-0.29$, and a late-time de Sitter or phantom attractor, so the model can in principle reproduce the full expansion history without a scalar field.
The statefinder pair $(j_0,s_0)\approx(1.05,0)$ lies near the $\Lambda$CDM values, making the model difficult to distinguish from $\Lambda$CDM on background data alone.
The model predicts a richer set of possible phases, including Chaplygin-gas, quintessence, and phantom eras, depending on the fitted parameters $m$ and $\beta$.

Reading between the lines

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

A reader should note that the fitted $m\approx1.95$ is not an integer and the Gauss-Bonnet invariant changes sign during decelerating epochs, so the reported solution depends on an unstated branch choice for $G^m$; checking whether $|G|^m$ was used would alter the likelihood surface and possibly the Bayes factor.
The same Lagrangian could be tested at the perturbation level: the paper analyzes only background dynamics, so whether the extra $f_G$ terms introduce ghost or gradient instabilities, as in earlier $f(G)$ models, remains open.
Adding a radiation pressure term would close the loop with early-universe epochs and could sharpen the predicted transition redshift and phantom crossing, which the present dust-only treatment leaves as future work.
If the model is correct, the $\sqrt{-T}$ matter coupling implies that baryonic matter is not described by a purely metric-compatible conservation law at the perturbative level; a fifth-force-like signature in structure formation would be a distinctive test.

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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

Plainly: this is a workmanlike f(R,G,T) paper. The specific power-law Lagrangian f=α1G^m+α2R^β−2α3√(−T) is new for this theory, and the authors do a real job: they derive the generalized Friedmann equations, fit to CC+SNe+DESI DR2 with PolyChord, report ln B=3.6 over ΛCDM, then reduce the system to a 4D dynamical system and classify eight critical points. The DESI DR2 constraints are current, and the derivation of the √(−T) term from the matter-conservation condition is a nice touch. The phase-space analysis is thorough, and the paper is honest that H0 comes out near 66, so the Hubble tension is not solved.

The soft spot is not minor. The Lagrangian is not real on the trajectory being fitted. From Eq. (21), G=24H^2(Ḣ+H^2)=−24qH^4. During the radiation and matter eras q>0 — their own Fig. 10b shows this — so G<0. The fitted value m≈1.95 is non-integer, so G^m is complex or undefined in the past. The same quantity enters f_G=mα1G^{m−1} and the dynamical-system variables x6, x7. The paper never states that |G|^m is meant or that m is restricted to integers. So the ODE being integrated, the likelihood, the Bayes factor, and the physical interpretation of the fixed points are not predictions of the stated real f(R,G,T) model. That is a load-bearing internal inconsistency.

Smaller issues: Table I gives β=2.544±0.021 while Fig. 1 shows 2.449 with asymmetric errors; the conclusions cite AIC/BIC but the paper does not report them; Eqs. (26) and (30) have apparent typos. No code is released, which makes reproduction harder.

I do not think the circularity concern is the real problem: the √(−T) term is fixed by conservation and the same conservation gives ρ(z), but that is model construction, not fitting to the same data. The reality issue is what sinks the current version.

The work deserves a serious referee: the core idea is salvageable by restricting m to integers or using |G|^m with justification, then rerunning the fit. I would not cite it in this form, but I would want to see the revised version. If you have a student in modified gravity, this is a good case study in checking the sign of G before integrating.

Referee Report

2 major / 4 minor

Summary. The manuscript studies a four-dimensional f(R,G,T) gravity with Lagrangian f(R,G,T)=α1G^m+α2R^β−2α3√(−T), where G is the Gauss-Bonnet invariant and T is the trace of the matter energy-momentum tensor. It derives the generalized Friedmann equations, specializes them to a pressureless dust universe, and obtains a second-order ODE for H(z) (Eq. 35). The model is fitted to cosmic-chronometer, Type Ia supernova, and DESI BAO data with the PolyChord nested sampler; the authors report a Bayes factor ln(B0i)=3.5994, which they interpret as moderate evidence in favor of the f(R,G,T) model over ΛCDM. The paper also rewrites the background equations as a four-dimensional autonomous dynamical system, identifies eight critical points, and uses linear stability analysis to associate them with radiation, matter, late-time acceleration, and phantom eras. The central claim is that this model is a viable background-level alternative to dark energy.

Significance. If the central claim were sound, the paper would be a useful contribution to modified-gravity cosmology: it provides explicit field equations, reports priors and data sets, uses the full cosmic-chronometer covariance matrix, and makes a concrete model comparison through Bayesian evidence. The derivation of the T-dependent term from the conservation law (Eqs. 31-34) is a clear and transparent step, and the use of external cosmological data is appropriate. However, the main quantitative results are undermined by an internal inconsistency: the fitted Lagrangian is not real on the decelerating part of the trajectory, so the ODE being integrated and the dynamical system being analyzed are not the ones stated. This is not a disagreement with current consensus but a mathematical flaw in the model as presented, and it invalidates the reported statistical and dynamical conclusions.

major comments (2)

[II.A, III.B, IV.B (Eqs. 21, 34, 35; Table I; Fig. 10b)] The fitted Lagrangian is not real on the trajectory over which Eq. (35) is integrated. From Eq. (21), G=24H^2(Ḣ+H^2)=-24qH^4 on an FLRW background. The paper's own Fig. 10b shows q>0 in the past, with the transition at N≈-0.29, so G<0 throughout the radiation- and matter-dominated epochs that a fit spanning z=0 to z=3 must include. Table I gives m=1.950±0.021 with a prior m∈[1,3] that does not restrict m to integers, so G^m and G^{m-1} are complex or undefined for those epochs. The term [H(z)^3(H(z)-(1+z)H'(z))]^m in Eq. (35) is (G/24)^m, and the same nonintegral power enters f_G in Eqs. (25)-(26) and the variables x6,x7 in Eqs. (39),(59). The manuscript nowhere states that |G| is used or that m must be an integer. Consequently the MCMC likelihood, the reported H(z), the Bayes factor ln(B0i)=3.5994, and the critical-point analysis are not predictions of the stated real f(R,G,T) model.
[III.B (Table I vs Fig. 1)] The reported posterior for β is internally inconsistent. Table I lists β=2.544±0.021, while Fig. 1 gives β≈2.449 with asymmetric errors (+0.021/−0.025). Since β multiplies R^β in the Lagrangian and appears throughout Eq. (35), the dynamical-system equations, and the eigenvalue expressions, the two central values cannot both describe the same fit. This makes the statistical results irreproducible even if the reality issue in the previous comment were resolved.

minor comments (4)

[II.A (Eqs. 26 and 30)] Equations (26) and (30) contain notation errors that obstruct verification: the terms "R fr" and "g" should presumably read "R f_R" and "f", and "¨f R" in Eq. (30) should read "\ddot f_R". The authors should correct these typos and re-check the subsequent derivation of Eq. (35).
[III.A] The text oscillates between describing the sampler as MCMC and as nested sampling; PolyChord is a nested-sampling algorithm, so the first sentence of Section III.A and the later statements about MCMC samples should be made consistent.
[VI (Conclusions)] The concluding section states that the statistical relevance of the results was estimated using AIC, BIC, and p values, but no AIC, BIC, or p values are reported anywhere in the paper; either these quantities should be reported or the sentence should be removed.
[Eq. (1) and throughout] The symbol G is used for Newton's gravitational constant in Eq. (1) and for the Gauss-Bonnet invariant throughout the rest of the paper; this notational conflict should be resolved, for example by writing G_N for Newton's constant.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the CC/SNe/BAO comparison is an external fit, and the dynamical-system analysis is a consistency check rather than a prediction forced by construction.

full rationale

The paper's derivation chain is not circular. After constructing the f(R,G,T) ansatz of Eq. (34) by integrating the dust conservation constraint in Eqs. (31)-(33), the authors solve the resulting second-order Hubble ODE in Eq. (35) and compare it, via nested sampling, against Cosmic Chronometer, SNe Ia, and DESI BAO data. These are external data sets; the parameters H0, Omega_m0, alpha_i, beta, m, M, and r_d are fitted to those data, so the reported H(z) and Bayes factor ln(B0i)=3.5994 are a genuine fit, not a quantity forced by construction. The dynamical-system analysis in Section IV reformulates the same generalized Friedmann equations and thus acts as a consistency check of the fitted model rather than an independent prediction; this weakens its evidentiary status but does not constitute a circular reduction. The self-citation of [153] for the f(R,G,T) field equations is a framework reference and is not the load-bearing support for the observational comparison. No equation was found in which a claimed prediction reduces to an input parameter by definition. A separate correctness concern, the realness of G^m for noninteger m during decelerating epochs with G<0, is an internal-consistency risk rather than a circularity and does not change this verdict.

Assumptions & free parameters 9 free parameters · 4 assumptions · 0 invented entities

The central claim rests on an inherited action, a flat dust FLRW background with imposed matter conservation, a hand-picked power-law Ansatz for the R and G sectors, and a numerical implementation that is not public. The free parameters are dominated by the five model parameters plus four standard cosmological nuisance parameters.

free parameters (9)

alpha_1 = 0.510 ± 0.021
Coupling of the Gauss-Bonnet term G^m in the action; fitted to the combined CC+SNe Ia+BAO data.
alpha_2 = 0.500 ± 0.026
Coupling of the curvature term R^β; fitted to the same combined data.
alpha_3 = 0.520 ± 0.029
Coupling of the matter term sqrt(-T); fitted to the same data.
beta = 2.544 ± 0.021 (Table I); Fig. 1 shows 2.449 +0.021/-0.025
Exponent of the Ricci scalar in the action; its posterior is internally inconsistent between Table I and Fig. 1.
m = 1.950 ± 0.021
Exponent of the Gauss-Bonnet invariant; noninteger value raises a reality problem when G is negative.
H0 = 66.1 ± 3.8 km/s/Mpc
Present Hubble constant; a standard nuisance parameter fitted to the data, not predicted by the model.
Omega_m0 = 0.294 ± 0.022
Present matter density parameter; fitted, not derived.
M = -19.46 ± 0.12
Absolute magnitude nuisance parameter for Type Ia supernovae; fitted.
r_d = 149.2 ± 7.5 Mpc
Sound horizon at baryon decoupling; treated as free in the BAO likelihood.

assumptions (4)

domain assumption The field equations of f(R,G,T) gravity quoted in Eq. (15) are correct.
The paper inherits them from Ref. [153] without rederivation; apparent typos in later equations (26) and (30) make this inheritance risky.
domain assumption The background is flat, homogeneous, isotropic FLRW with pressureless dust and imposed covariant conservation of matter.
These choices define the regime; the conservation condition ∇μTμν=0 is imposed by hand and forces the sqrt(-T) dependence via Eq. (31).
ad hoc to paper The Ansatz xi(R,G)=alpha_1 G^m + alpha_2 R^β is a real analytic function on the fitted trajectory.
This is a hand-picked power-law form; it is not real for noninteger m when G is negative, which the paper does not address.
domain assumption The numerical solution of Eq. (35) and the PolyChord likelihood are implemented without coding errors.
No code is provided; the displayed equations have typos and the sign convention for fractional powers is undocumented.

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Cite this review

Pith. "Pith review of f(R, $G$, T) Gravity: Cosmological Implications and Dynamical System Analysis." pith.science (2026). https://pith.science/paper/WKDM3YO5

@misc{pith2026250607623,
  author       = {Pith},
  title        = {Pith review of: f(R, $G$, T) Gravity: Cosmological Implications and Dynamical System Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WKDM3YO5}},
  note         = {Machine review of arXiv:2506.07623}
}

abstract

We consider the cosmological implications of a four-dimensional extension of the Gauss-Bonnet $f(G)$ gravity, where $G$ is the Gauss-Bonnet topological invariant, in which the Einstein-Hilbert action is replaced by an arbitrary function $f(R,G,T)$ of G, of the Ricci scalar $R$, and of the trace $T$ of the matter energy-momentum tensor. By construction, the extended Gauss-Bonnet type action involves a non-minimal coupling between matter and geometry. The field equations of the model are obtained by varying the action with respect to the metric. The generalized Friedmann equations, describing the cosmological evolution in the flat Friedmann-Lemaitre-Robertson-Walker geometry, are also presented in their general form. We investigate the cosmological evolution of the Universe in the generalized Einstein-Gauss-Bonnet theiry for a specific choice of the Lagrangian density, as given by $f(R,G,T) = \alpha_1 G^{m} + \alpha_2 R^{\beta} - 2\alpha_3 \sqrt{-T},$ where $\alpha_i$ $ i = 1, 2, 3$), $ m $, and $\beta$ are model parameters. First, the theoretical predictions of the model are compared with a set of observational data (Cosmic Chronometers, Type IA Supernovae, Baryon Acoustic Oscillations) via an MCMC analysis, which allows us to obtain constraints on the model parameters. A comparison with the predictions of the $\Lambda$CDM system is also performed. Next, the generalized Friedmann equations are reformulated as a dynamical system, and the properties of its critical points are studied by using the Lyapunov linear stability analysis. This investigation allows for the reconstruction of the Universe's history in this model, from the early inflationary era to the late accelerating phase. The statefinder diagnostic parameters for the model are also considered from the dynamical system perspective.

Figures

Figures reproduced from arXiv: 2506.07623 by the authors.

**Figure 2.** FIG. 2: Evolution of the Hubble parameter [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3: Saddle region for critical point [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4: Stable region for critical point [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: FIG. 5: Stable region for critical point [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6: Stable region for critical point [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: FIG. 7: Stable region for critical point [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

**Figure 8.** Figure 8: FIG. 8: Stable region for critical point [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

**Figure 9.** Figure 9: FIG. 9: Projection of phase space diagram corresponding to the critical points [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: FIG. 10: Evolution of various cosmological parameters for [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 11.** Figure 11: FIG. 11: Evolution of various cosmography parameters for [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]

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One may note that all the background equations are highly sensitive to the choice of functional f . In order to proceed in our study on the observa- tional data and analysis of the dynamical system of the f (R, G, T) gravity theory, it is important to consider a specific form of the functional f that satisfies all the fun- damental conditions of the theor...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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One may note that all the background equations are highly sensitive to the choice of functional f . In order to proceed in our study on the observa- tional data and analysis of the dynamical system of the f (R, G, T) gravity theory, it is important to consider a specific form of the functional f that satisfies all the fun- damental conditions of the theor...

2012

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The eigenvalues of the Jaco- 18 -1.0 -0.5 0.0 0.5 1.0 -4 -3 -2 -1 0 1 2 m β FIG

These values of the effective Eos parameter and the deceleration parameter indicate the tran- sitional epoch from the decelerated to the acceler- ated expansion era. The eigenvalues of the Jaco- 18 -1.0 -0.5 0.0 0.5 1.0 -4 -3 -2 -1 0 1 2 m β FIG. 6: Stable region for critical point P5 bian matrix are ( 2(2m − β) m − 1 , 2(β − 2), 2β − 3, 1 2 (4β − 9) ) Th...

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In terms of studying the stabil- ity features, the eigenvalues of the Jacobian matrix corresponding to P6 are given by    − 3(2m − β) β , − 3 2, ± p 256β8m2 − 864β7m2 + 1025β6m2 − 498β5m2 + 81β4m2 − 3β3m + 3β2m 4(β − 1)β3m    All the eigenvalues of the Jacobian matrix are nonzero and complex mathematical expressions of the parameters m and β. In Fig...

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Due to the presence of the positive eigenvalues E2 = 3 2, stability is not possible for this critical point

The eigenvalues of the Jaco- bian matrix corresponding to the critical point P7 are    − 9(2m − β) 2β , 3 2, − p 1444β8m2 − 5412β7m2 + 7253β6m2 − 4014β5m2 + 729β4m2 + 6β4m − 15β3m + 9β2m 8(β − 1)β3m , p 1444β8m2 − 5412β7m2 + 7253β6m2 − 4014β5m2 + 729β4m2 + 6β4m − 15β3m + 9β2m 8(β − 1)β3m   . Due to the presence of the positive eigenvalues E2 = 3 2, ...

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