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REVIEW 3 major objections 5 minor 23 references

Phase space analysis of cosmological models on $f(R,G,T)$ gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that power-law f(R,G,T) gravity can replace Lambda-CDM as the driver of late-time cosmic acceleration.

desk verdict A routine phase-space extension to f(R,G,T) whose central closure relation is algebraically wrong, so the critical points and conclusions don't follow. read the letter →

arxiv 2505.22690 v1 pith:XOUOKPED submitted 2025-05-28 gr-qc hep-th

Elangbam Chingkheinganba Meetei , S. Surendra Singh This is my paper
classification gr-qchep-th MSC 83F0583D0534C05 PACS 98.80.-k04.50.Kd95.36.+x
keywords dynamicalsystemmodifiedf(RGT)gravityphasespaceanalysisdarkenergydensityparametersequationofstatequintessenceLambdaCDMcomparison
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 the modified-gravity theory $f(R,G,T)=\alpha R^{l}+\beta G^{m}+\gamma T^{n}$ can account for the Universe's late-time acceleration without a cosmological constant. The authors build a six-dimensional autonomous dynamical system from the cosmological field equations, isolate eight critical points, and identify four that are stable late-time attractors. For $l=2$ and a fixed phase-space variable $x=0.1$, the model gives a present deceleration parameter $q=-0.5$ and equation of state $\omega=-0.66$, a quintessence phase, with a deceleration-to-acceleration transition at $z_{\rm tr}\approx0.616$. They then compare the model's Hubble parameter and distance modulus with 77 Hubble, 15 BAO, and 1048 Pantheon data points and conclude that it behaves like $\Lambda$CDM. If correct, this would provide a modified-gravity alternative to the cosmological constant that still reproduces the standard expansion history.

What carries the argument

The load-bearing object is the six-dimensional autonomous system (28) in the dimensionless variables $(d_1,d_2,u,w,z_1,z_2)$, obtained after eliminating $v$, $y$, and $x$ with the constraint (22) and the algebraic formulas (25)-(27). The decisive identity is Eq. (27), which expresses the normalized derivative $x=\dot f_R/(H f_R)$ in terms of the other phase-space variables; every critical point, eigenvalue, and the resulting $q$ and $\omega$ values are computed from it. The model parameters $l,m,n$ enter through $a=l-1$, $b=n-1$ and through the explicit forms of $v$, $y$, and $x$, so the whole stability analysis is a consequence of this single elimination step.

What would settle it

Substitute the coordinates of a critical point from Table 1, together with the printed Eq. (27), into the constraint Eq. (22) and check whether the equality holds; the paper's fixed points should satisfy this constraint by construction. If the printed sign leaves a nonzero residual, the fixed-point table and the reported $q=-0.5$, $\omega=-0.66$ are not self-consistent, and the analysis must be repeated with the corrected coefficient $(2n+1)/(2n)$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the power-law model $f(R,G,T)=\alpha R^{l}+\beta G^{m}+\gamma T^{n}$ admits a closed autonomous phase-space description whose late-time fixed points are dominated by dark energy. Working in a flat FLRW universe and using logarithmic time $N=\ln a$, the paper defines nine dimensionless variables and reduces them to six ordinary differential equations through the constraint (22) and the algebraic relations (25)-(27). Solving the fixed-point equations gives eight critical points; under the stated parameter ranges, C, D, F, and H are stable, with C, D, and F describing pure dark-energy domination and H describing a mixed matter-dark-energy phase. With $l=2$ and $x=0.1$, the numerical evolution yields $q=-0.5$ and $\omega=-0.66$, and fixing $w=1.5$ makes $H(z)=H_0(1+z)^{2-w}$ and the distance modulus match $\Lambda$CDM against the Hubble, BAO, and Pantheon samples. The paper concludes that the model is compatible with current observations and is a possible alternative to $\Lambda$CDM.

Load-bearing premise

The whole fixed-point and stability analysis rests on the algebraic relation (27) that expresses the phase-space variable $x$ in terms of the others; if that relation is printed with the wrong sign in the $z_1$ coefficient, as a consistency check with (25) and (22) suggests, every critical point, eigenvalue, and derived cosmic parameter inherits the error.

Editorial extensions

If this is right

  • If the eight fixed points and their stability conditions are correct, the model contains a radiation era (point A) and four stable late-time attractors, so it can reproduce the standard sequence of cosmic epochs.
  • The reported $q=-0.5$ and $\omega=-0.66$ imply an accelerating quintessence phase rather than a cosmological constant, with the transition from deceleration to acceleration at $z_{\rm tr}\approx0.616$ inside the observationally favoured range.
  • With $w=1.5$, the model's Hubble parameter and distance modulus align with $\Lambda$CDM against the 77 Hubble, 15 BAO, and 1048 Pantheon data points used in the paper.
  • The state-finder diagnostics give exact $\Lambda$CDM values ($r=1$, $s=0$) at point D for all parameters, and at points E and F for special parameter choices, while other parameter regions correspond to quintessence or Chaplygin-gas behaviour.

Reading between the lines

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

  • I infer that the observational check is weaker than a full test: fixing $w=1.5$ and $x=0.1$ is not a likelihood fit over the free parameters, so a parameter-estimation run against the same data would be needed to confirm the claimed agreement.
  • I infer that the reduction's correctness hinges on Eq. (27); re-deriving the system with the consistency-required coefficient $(2n+1)/(2n)$ rather than the printed $(2n-1)/(2n)$ would shift the fixed points, stability windows, and the reported $q$ and $\omega$ values.
  • I infer that the same phase-space construction could be applied to other $f(R,G,T)$ families and to non-flat or anisotropic backgrounds, and whether the four stable dark-energy attractors survive would show whether the mechanism is generic.
  • I infer that the state-finder pair is the most discriminating observable: several critical points share $\Omega_{\rm de}=1$ but differ in $(q,\omega,r,s)$, so measuring those at low redshift could separate this theory from $\Lambda$CDM even where the Hubble diagram overlaps.
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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

3 major / 5 minor

Summary. The paper constructs a nine-variable autonomous dynamical system for f(R,G,T)=αR^l+βG^m+γT^n gravity in a flat FLRW background, reduces it to six equations, and reports eight critical points with stability conditions. It then derives the deceleration parameter, effective equation of state, statefinder parameters, and compares H(z) and the distance modulus with 77 Hubble, 15 BAO, and 1024/1048 Pantheon data, concluding that the model is a viable alternative to ΛCDM with present values q=-0.5 and ω_eff=-0.66.

Significance. If the analysis were correct, the paper would provide a useful phase-space classification of a general f(R,G,T) model and would show that a simple power-law form can produce a stable late-time dark-energy attractor compatible with ΛCDM-like expansion. The attempt to construct and reduce a nine-dimensional system is in itself a useful exercise, and the authors tabulate enough expressions for the algebra to be checked. However, the central reduction contains a concrete algebraic error that invalidates the critical-point analysis as printed, and the later 'predictions' are in large part fixed by hand-chosen values rather than derived from the dynamics. The claimed significance is therefore not currently established.

major comments (3)
  1. [§3, Eqs. (22), (25), (27)] Equation (27) is not the solution of the constraint (22) when combined with Eq. (25). Substituting v=w/l+u/m−z1/(2n) and y=(m−1)u/(w−1)^2(wx/(l−1)+2(w−2)^2) into Eq. (22) gives a numerator coefficient (2n+1)/(2n) z1, not the printed (2n−1)/(2n) z1. The discrepancy is numerically consequential: at critical point C with l=2, n=0.3, Table 1 gives w=1.775 and z1=0.50625; Eq. (27) yields x=−0.45, but the original constraint (22) evaluates to 2.69 rather than 1. With the corrected coefficient, x=1.2375, which satisfies the constraint but makes dw/dN=2.995 at the same point, so C is not a fixed point of the reduced system. Consequently, the critical points, eigenvalues, stability conditions, and the q and ω_eff values derived from them in Tables 1–2 and Section 4 are not established as printed.
  2. [§5, Figs. 10–11] The values q=−0.5 and ω_eff=−0.66 are not independent outputs of the phase-space analysis; they follow directly from the hand-chosen value w=1.50 through q=1−w and ω_eff=−(2w−1)/3. Similarly, the density-parameter evolution in Fig. 10 is generated by fixing x=0.1 and l=2. The abstract and Section 7 should not describe these numbers as predictions of the model without showing that they are selected by the dynamical equations rather than imposed by the initial conditions and parameter choices.
  3. [§6, Fig. 12] The comparison with 77 Hubble, 15 BAO, and Pantheon data is not a statistical validation of the model. The model curve H(z)=H0(1+z)^{2−w} is evaluated with w=1.50 and with H0, Ωm, and ΩΛ fixed to Planck values, and the agreement is assessed visually; no chi-square, likelihood, residuals, or parameter estimation is reported. Since the same data are commonly used to calibrate ΛCDM, the plots show at most that one particular curve lies near the ΛCDM curve, not that the f(R,G,T) model is supported over ΛCDM. The claims in Section 7 that the model is 'compatible with observational evidence' and 'a possible alternative' are therefore overstated.
minor comments (5)
  1. [Eq. (23)] In the dv/dN line, '3/2 z' appears without a subscript; it should read '3/2 z1'.
  2. [Abstract and §6] The abstract states 1024 Pantheon data points, while Section 6 states 1048; the number should be reconciled.
  3. [Throughout] The text contains several typographical and formatting artifacts, including 'Out[]=' remnants in the figure captions, 'l=01.50', 'de-Silter', 'bahaviour', and 'Fmendola' in reference [11].
  4. [Figures 1–9] The phase portraits do not specify the full set of parameter values, initial conditions, and projected variables used, which limits reproducibility; the captions should be completed.
  5. [Eqs. (47)–(48)] The statefinder parameters r and s are introduced without a derivation connecting them to the jerk and snap parameters or to the model variables; a brief derivation or reference would improve clarity.

Circularity Check

2 steps flagged · score 6.0 of 10

The claimed 'present values' q=-0.5 and omega=-0.66 are arithmetic consequences of the hand-set w=1.50, and the density-parameter 'prediction' uses current values as initial conditions; central validation is therefore partly input, though fixed-point analysis is derived.

  1. fitted input called prediction [Section 5 (Cosmic parameters), Eqs. (29) and (31); Fig. 11; Section 6]
    "The present value of q = −0.50 indicates that currently our Universe is in the phase of accelerated expansion. ... The present value of ω ≈ −0.66, indicates that our Universe is in the phase of Quintessence. ... Also, we take w = 1.50."

    By Eq. (31), q = 1 − w, and by Eq. (29), ω_eff = −(2w−1)/3. Inserting the ad hoc value w = 1.50 gives q = −0.5 and ω_eff = −0.667 exactly. The paper never determines w from the dynamical system; it fixes w as an input ('we take w = 1.50'). Therefore the headline 'present values' are not outputs of the phase-space analysis but a restatement of the chosen parameter. A different chosen w would produce a different 'prediction' by the same identities.

  2. self definitional [Section 4, Fig. 10 caption and text after Table 2]
    "The evolution of density parameters when x = 0.1 and l = 2 is shown in figure (10). It also show that the evolution is consistent with the current observation with present value calculated as Ωm ≈ 0.315, Ωr ≈ 0.171 and Ωde ≈ 0.514. ... with initial condition d1 = 0.315, d2 = 0.171 and w = 1.50"

    The quoted 'present values' are precisely the imposed initial conditions for d1 and d2. Since Ωm = d1, Ωr = d2 and Ωde = 1 − d1 − d2 in Eq. (30), the output at N = 0 reproduces the input by construction. Calling this agreement with observation 'calculated' is circular: the agreement was put in through the initial data, not derived from the model's attractor structure.

full rationale

The paper's phase-space construction is a genuine derivation: nine dimensionless variables are defined, the field equations are manipulated into an autonomous system, fixed points are computed, and stability is analyzed. That part is not circular by itself. However, the paper's headline cosmological results are not derived from that system. The present q and omega values come from fixing w = 1.50 and then using the definitions q = 1 − w and ω_eff = −(2w−1)/3; they are exact algebraic identities from the chosen parameter. Likewise, the density-parameter plot is initialized at d1 = 0.315 and d2 = 0.171 and then the same numbers are reported as the 'present' matter and radiation densities, so the agreement with observation is imposed through initial conditions. The observational comparison in Section 6 also uses the same pre-chosen w = 1.50 together with Planck-fixed H0, Ωm and ΩΛ, and reports visual alignment without parameter estimation or a goodness-of-fit statistic; while that is not strictly circular (the data could have disagreed), it is a presentation of hand-chosen input as model validation. No load-bearing self-citation chain was found: the cited prior works by the same authors are contextual, and the dynamical-system formalism is standard. The possible algebraic inconsistency in Eq. (27), which appears to prevent the printed fixed points from satisfying constraint (22), is a correctness issue rather than a circularity issue; it would undermine the fixed-point table but does not change the fact that the q, omega and density-parameter 'predictions' reduce to chosen inputs.

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

The central result rests on a constraint closure that is internally inconsistent, plus hand-fixed variables w=1.50 and x=0.1. It also assumes conservation laws for matter and radiation and a specific trace sign convention that are not derived in the f(R,G,T) context.

free parameters (4)
  • w (dimensionless variable R/(6H^2)) = 1.50
    Fixed by hand in Sections 5 and 6. It directly determines q=1-w=-0.5 and omega_eff=-(2w-1)/3=-0.66, and sets H(z)=H0(1+z)^0.5 for the data comparison.
  • x (dimensionless variable fR_dot/(H fR)) = 0.1
    Fixed to 0.1 in Section 5 to produce the density-parameter evolution in Fig. 10 and the quoted present values Omega_m~0.315, Omega_r~0.171, Omega_de~0.514.
  • l (power of Ricci scalar in action) = 2 in Sections 5 and 6; 1.5 in some phase portraits
    The values q, omega, density parameters, and stability conditions depend on l; the choice is not fit to data, it is a model parameter selected for display.
  • m and n (powers of Gauss-Bonnet and trace terms) = varied by hand in phase portraits, e.g., m=2, n=2
    Stability of critical points C, D, G, and H depends on m and n; the paper explores preferred examples but does not fit them.
assumptions (4)
  • domain assumption Flat FLRW metric and separate conservation of dust and radiation (Eqs. 5, 10, 11).
    The dynamical system is built on the flat FLRW metric; the paper assumes rho_m and rho_r are separately conserved without deriving this from the f(R,G,T) field equations, where fT terms can alter conservation.
  • ad hoc to paper Trace relation used in Eq. (25): v = w/l + u/m - z1/(2n).
    This relation ties T^n and fT rho_m with a specific sign. No derivation is shown, and the printed sign is inconsistent with Eq. (22) when carried through to Eq. (27).
  • standard math Constraint Eq. (22) can be used to eliminate x and reduce the system to six ODEs.
    Eliminating a dependent variable via a first integral is standard, but it requires algebraic consistency; the paper's printed relations violate it.
  • ad hoc to paper The non-variable terms in Eq. (21), including eta, correctly close the system.
    eta = fR_ddot/(H^2 fR) is listed but never expressed in the variables, and dx/dN is omitted from the reduced system without a consistency proof.

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Pith. "Pith review of Phase space analysis of cosmological models on $f(R,G,T)$ gravity." pith.science (2026). https://pith.science/paper/XOUOKPED

@misc{pith2026250522690,
  author       = {Pith},
  title        = {Pith review of: Phase space analysis of cosmological models on $f(R,G,T)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XOUOKPED}},
  note         = {Machine review of arXiv:2505.22690}
}
abstract

In the present article, we developed a dynamical system in the context of modified $f(R,G,T)$ gravity, where $R$, $G$ and $T$ are Ricci scalar, Gauss-Bonnet term and energy-momentum tensor respectively. Development of the dynamical system is done by first defining 9 dimensionless variables and formulate a ordinary differential equations by taking derivative of the variables with respect to logarithmic time $N=\log a(t)$, where $a(t)$ is the scale factor. This formulation is applied to the model $f(R,G,T)=\alpha R^{l}+\beta G^{m} + \gamma T^{n} $, and its solutions and their corresponding stabilities are analysed in details. From the plot of density parameters vs $N=-log(1+z)$, we conclude that our Universe is currently dominated by dark energy, which is compatible with current observation. Taking $l=2$, cosmic parameters such as deceleration parameter $(q)$, equation of state $(\omega)$ and state finder parameters are also discussed by fixing one dimensionless variable, showing our Universe's expansion is accelerating with the present value of $q=-0.5$. Present value of $\omega=-0.66$ suggests that our Universe is in Quintessence phase. Lastly, the validity of the model with respect to the $\Lambda$CDM model is checked by using 77 Hubble, 15 BAO and 1024 pantheon datasets, which implies that our model is aligned with the $\Lambda$CDM model's bahaviour.

Figures

Figures reproduced from arXiv: 2505.22690 by the authors.

Figure 1
Figure 1. Phase portrait with l = 2 This critical point exists for any value of the model’s parameters. The eigenvalues of this point are −1, 1, 4 −4(2m−1), −3(n−1), and 4−3n. As there exists at least one positive and negative eigenvalues, the present critical point is an unstable saddle point. Cosmic parameters such as density parameters, equation of state and deceleration parameter are given by Ωm = 0, Ωr = 1, Ωde = 0, ωde … view at source ↗
Figure 2
Figure 2. Phase portrait with l = 2 and n = 1 The Hubble parameter and the scale factor are given by H = 1 3nt 2l + C , a = K  3nt 2l + C  2l 3n , (36) where C and K are the constants of integration. From (35), it is clear that this point represents the phase of the accel￾erated expansion and further suggests that the Universe is dominated by dark energy for 2l > 3n. For l = 1.5 and n = 0.32, ωef f = −0.787 and q = −0.68, w… view at source ↗
Figure 3
Figure 3. Phase portrait with l = 3, m = 2 and n = 2 The Hubble parameter and the scale factor are given by H = C, a = KeCt , (38) where C and K are the constants of integration. From (37), it is clear that this point represents the phase of the accel￾erated expansion and further suggests that the Universe is dominated by dark energy. This CP represents a de-Sitter model of the Universe. The corresponding phase portrait when … view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Phase portrait with l = 1.1 Cosmic parameters such as density parameters, equation of state and deceleration parameter are given by Ωm = 0, Ωr = −2 + 8l − 5l 2 l 2 , Ωde = 2 − 8l + 6l 2 l 2 , ωde = 20 3 l − l 2 2 − 8l + 6l 2 , ωef f = −(3l − 4) 3l , q = −(l − 2) l . (3…
Figure 5
Figure 5. Figure 5: Phase portrait with l = 1.5 The Hubble parameter and the scale factor are given by H = 1 (l−2)t (2l−1)(l−1) + C , a = K  (l − 2)t (2l − 1)(l − 1)+C (2l−1)(l−1) (l−2) , (42) where C and K are the constants of integration. From (41), it is clear that the Universe is do…
Figure 6
Figure 6. Figure 6: Phase portrait with l = 2 and n = 1.5 Here one of the eigenvalues is positive, so this CP cannot be a stable point. This CP is unsta￾ble if n > 1, 2m > l and l > 3n+1 6n−4 satisfy￾ing −3+27l−88l 2+48l 3−18n+110ln−124l 2n+48l 3n−27n 2+87ln2−72l 2n 2+12l 3n 2 l−1 < 0 or …
Figure 7
Figure 7. Figure 7: Phase portrait with l = 2 and n = 1.5 where C and K are the constants of integration. From (43), it is clear that this point represents the phase of the accel￾erated expansion and further suggests that the Universe is dominated by dark energy for 2l > 3n + 1. For l = 4…
Figure 8
Figure 8. Figure 8: Phase portrait with l = 0.8 Out[ ]= 1.0 1.5 2.0 2.5 3.0 -0.8 -0.6 -0.4 -0.2 0.0 t w [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Phase portrait with l = 0.7 This critical point exists for any value of the model’s pa￾rameters. The eigenvalues of this point are −1, 3(2m−l) l , 2 − 3n, −3(n − 1), − 3 4l − √ (−1+l)(−81+417l−608l 2+256l 3) 4l(l−1) , and − 3 4l + √ (−1+l)(−81+417l−608l 2+256l 3) 4l(l−…
Figure 10
Figure 10. Figure 10: Evolution of density parameters with initial condition [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: Plot of cosmic parameters with initial condition [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: Comparison with observational datasets 7 Results and discussion In this study, we consider the modified f(R, G, T ) gravity to develop a dynamical system of ordinary differential equations (ODEs). For this, we define 9 dimensionless variables as given in section (3) a…

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