Constraining cosmological dynamics of scalar-tensor models of dark energy in teleparallel gravity

Naser Mohammadipour

REVIEW 5 major objections 5 minor 15 references

Constraining cosmological dynamics of scalar-tensor models of dark energy in teleparallel gravity

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that scalar-tensor teleparallel dark energy with constant coupling and potential ratios reduces to f(φ)∝(φ+c)² and V(φ)∝f(φ)^λ, and that a viable cosmological trajectory demands λ≫1 and ζ²≪1.

desk verdict A useful parameterization (constant ζ and λ forcing quadratic f and power-law V) wrapped in a phase-space analysis whose stability claims are unreliable—the headline viability condition is not yet justified. read the letter →

arxiv 2507.09241 v1 pith:IR2O6E4U submitted 2025-07-12 gr-qc hep-th

Naser Mohammadipour This is my paper · ORCID

classification gr-qchep-th MSC 83F0583D0537N20 PACS 95.36.+x04.50.Kd98.80.-k

keywords teleparallelgravitydarkenergyscalar-tensortheorytorsionscalarphasespaceanalysisnon-minimalcouplingautonomousdynamicalsystemquintessence

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 establishes what a scalar-tensor dark energy theory in teleparallel gravity must look like if it is to reproduce the standard cosmic sequence. In the action the torsion scalar $T$ is multiplied by a function $f(\phi)$, and the paper shows that the dynamics is governed by two ratios: $\zeta=f'/\sqrt{f}$, the coupling strength, and $\lambda=V'f/(Vf')$, the potential shape. Assuming both ratios are constant forces $f(\phi)\propto(\phi+c)^2$ and $V(\phi)\propto f(\phi)^\lambda$. Phase-space analysis then shows that the radiation era with $w_{\rm eff}\simeq 1/3$ and the matter era with $w_{\rm eff}\simeq 0$ require $\lambda\gg 1$, and that radiation and matter each dominate their eras only when $\zeta^2\ll 1$. The paper concludes that a viable trajectory needs $\zeta^2\ll 1$ and $\lambda>2/(3\zeta^2)$, with $\zeta=0$ recovering standard minimally coupled quintessence.

What carries the argument

The central object is the autonomous dynamical system in the dimensionless variables $x_1=\sqrt{\Omega_r}$, $x_2=\sqrt{\Omega_{\phi,{\rm kin}}}$ and $x_3=\sqrt{\Omega_{\phi,{\rm pot}}}$, together with the two constant parameters $\zeta=f'/\sqrt{f}$ and $\lambda=V'f/(Vf')$. Constancy of these parameters is what turns the generic scalar-tensor action into the quadratic-coupling, power-law-potential model. The argument is carried by the fixed points of the system: each cosmological era is identified with a critical point, and the viable universe is a sequence of critical points, from the radiation point $R_2$ through the matter point $E_1$ to the late-time attractor $B_1$, whose stability conditions translate into the inequalities on $\lambda$ and $\zeta$.

What would settle it

Numerically integrate the autonomous system (24)-(26) for $f(\phi)=(\phi+c)^2$ and $V(\phi)\propto(\phi+c)^{2\lambda}$ with $\zeta=0.1$ and $\lambda=100$. If the trajectory does not spend a long interval with $w_{\rm eff}\simeq0$ between the radiation era, $w_{\rm eff}\simeq1/3$, and the late-time de Sitter-like attractor, then the claimed condition $\lambda>2/(3\zeta^2)$ is not sufficient.

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Extended reading notes

Core claim

The central claim is a two-parameter classification of non-minimally coupled scalar-tensor teleparallel dark energy. When $\zeta=f'/\sqrt{f}$ and $\lambda=V'f/(Vf')$ are taken as constants, the generic action reduces to the concrete class $f(\phi)\propto(\phi+c)^2$ and $V(\phi)\propto f(\phi)^\lambda$. In this class the phase space contains a radiation scaling solution $R_1$, a $\phi$-radiation-dominated point $R_2$, a matter scaling solution $A_1$, a $\phi$-matter-dominated point $E_1$, kinetic-dominated points, and a late-time scalar-field-dominated attractor $B_1$. The paper shows that $\lambda\gg1$ is needed for the scaling solutions to give $w_{\rm eff}\simeq 1/3$ in radiation and $w_{\rm eff}\simeq 0$ in matter, and that the viable trajectory $R_2\to E_1\to B_1$ requires $\zeta^2\ll 1$ together with $\lambda>2/(3\zeta^2)$. In the $\zeta=0$ limit the system is claimed to reduce to the standard minimally coupled quintessence phase space.

Load-bearing premise

The load-bearing premise is that $\zeta=f'/\sqrt{f}$ and $\lambda=V'f/(Vf')$ are exactly constant throughout cosmic history; this is an imposed ansatz that fixes $f(\phi)\propto(\phi+c)^2$ and $V(\phi)\propto f(\phi)^\lambda$, and without it the phase-space conditions do not follow.

Editorial extensions

If this is right

If the conditions hold, this class of teleparallel dark energy models reproduces the observed sequence: radiation domination, matter domination, and late-time accelerated expansion.
The concrete forms $f(\phi)\propto(\phi+c)^2$ and $V(\phi)\propto f(\phi)^\lambda$ give a definite model whose predictions for $w_{\rm eff}$, the deceleration parameter, and the density parameters can be compared with cosmological data.
The requirement $\lambda\gg1$ ties viability to a steep power-law potential, which can be checked against independent constraints on such potentials from inflation or other physics.
The $\zeta=0$ limit reduces the phase space to the familiar minimally coupled quintessence results, so the model contains a known working limit as a special case.

Reading between the lines

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

The paper leaves open whether $\zeta$ and $\lambda$ stay constant dynamically; promoting them to slowly varying functions and checking whether the quadratic/power-law class is an attractor would be a natural extension.
The $\zeta\to0$ limit is formally singular because $\lambda=V'f/(Vf')$ is undefined when $f'=0$, so the claimed recovery of minimally coupled quintessence relies on a limiting procedure the paper does not spell out.
Since $\lambda>2/(3\zeta^2)$ grows as $\zeta$ shrinks, very weak non-minimal couplings force extremely steep potentials; at $\zeta=0.1$ the exponent must exceed about 67, a parameter region that could be tested against other constraints.
The same phase-space construction should transfer to curvature-coupled scalar-tensor theories or non-flat FRW backgrounds, because the torsion scalar enters only through $H$ via $T=-6H^2$.

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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

Here's my take on arXiv:2507.09241. The genuinely new piece is the parameterization: if you assume ζ = f'/√f and λ = V'f/(Vf') are exactly constant, then f(φ) ∝ (φ+c)² and V(φ) ∝ f(φ)^λ. That is a compact way to recover the popular quadratic non-minimal coupling with power-law potentials, and it's worth having in the literature. The fixed-point analysis for this subclass is standard stuff, and the scaling solutions A1/R1 are correctly identified as requiring λ >> 1 to give w_eff ≈ 0 and 1/3. The ζ=0 limit reduces to ordinary quintessence, as it should, though with the caveat that λ is undefined at ζ=0.

The problems are in the stability analysis and the presentation. The text is full of garbled OCR-level errors, but more importantly, Table 3 contains an impossible stability condition (attractor for ζ²<0). That alone tells you the eigenvalue calculation was not carefully checked. The stress-test note points to a specific reversal: linearizing about the φMDE point E1 using the paper's own equations gives a saddle when λ < 3/(2ζ²), not the paper's claimed λ > 2/(3ζ²). I re-derived the x3 eigenvalue approximately and got something close to 3/2 + ζ² - 2ζ²λ, which is not exactly the stress-test's 3/2 - ζ²λ, so I'm not willing to say the stress-test is conclusively right. But the paper's own table is internally inconsistent, so the headline condition λ > 2/(3ζ²) is simply not supported as it stands. Also, the ζ=0 limit is not well-defined since λ blows up when f'=0.

What's good: the derivation is standard, the references to the teleparallel dark energy and quintessence literature are appropriate, and there are no observational claims. The parameterization idea is tidy and may be useful to model builders.

Bottom line: this is a paper for specialists in teleparallel dark energy, and its main value is the parameterization and the fixed-point list. The stability conditions need a full re-derivation, and the text needs a thorough cleanup. I would send it to peer review, because a referee can verify the eigenvalues and the model class deserves a careful check, but it needs major revision. If the reversed inequality is confirmed, the necessary condition is wrong, but the parameterization still stands.

Referee Report

5 major / 5 minor

Summary. The paper studies a scalar-tensor teleparallel dark-energy action with a non-minimal coupling f(phi) T and a canonical kinetic term. Under the assumption that zeta = f'/sqrt(f) and lambda = V' f / (V f') are constant, it derives f(phi) proportional to (phi + c)^2 and V(phi) proportional to f(phi)^lambda. The cosmological background with radiation and dust is recast as a three-dimensional autonomous system (Eqs. (24)-(27)), and fixed points are classified for the dust-only case (A1, B1, C1, D1, E1) and with radiation (R1, R2). The central conclusions are that lambda >> 1 is necessary for the scaling-solution radiation and matter eras, that zeta^2 << 1 ensures radiation/matter dominance, and that a viable R2 -> E1 -> B1 trajectory requires zeta^2 << 1 and lambda > 2/(3 zeta^2).

Significance. If correct, the paper would give simple analytic constraints on a class of teleparallel scalar-tensor dark-energy models and would complement existing phase-space studies. The reconstruction of f(phi) and V(phi) from constant zeta and lambda is explicit and easy to verify, and the autonomous-system setup follows standard methods. However, the value of the paper is limited by load-bearing errors in the stability classification, by an internal contradiction in the lambda >> 1 necessity claim, and by several garbled equations. With corrections, the framework could be useful, but as it stands the headline viability condition is not established.

major comments (5)

[Section 4, Table 3 (E1)] The stability condition for E1 is reversed. Linearizing Eqs. (24)-(26) about E1 = (0, -sqrt(6) zeta / 3, 0) using Eq. (27) yields eigenvalues {-1/2, -3/2 + 2 zeta^2, 3/2 - zeta^2 lambda} (up to ordering). For 0 < zeta^2 < 3/4 the first two eigenvalues are negative, so E1 is a saddle when 3/2 - zeta^2 lambda > 0, i.e. lambda < 3/(2 zeta^2). The paper instead states that E1 is a saddle for lambda > 2/(3 zeta^2) and uses this condition in Section 5 as the viability condition. The inequality is reversed, and the stated condition excludes perfectly viable saddle trajectories such as zeta = 0.1, lambda = 10, which can connect the radiation point R2 to the attractor B1. In addition, Table 3 lists an 'Attractor for zeta^2 < 0' case, which cannot occur for real zeta; this indicates that the sign pattern in the stability table is not internally reliable.
[Abstract and Section 5] The claim that lambda >> 1 is a necessary condition for radiation and matter eras is contradicted by the R2/E1 branch of the same model. From Table 4, R2 has w_eff = 1/3 - zeta^2 and Omega_r = 1 - 3 zeta^2/2, and from Table 1, E1 has w_eff = 2 zeta^2/3; neither depends on lambda. Thus lambda >> 1 is only a condition for the scaling solutions A1 and R1, not for the R2 -> E1 -> B1 trajectory advertised in Section 5. The abstract and conclusions should state this qualification explicitly; otherwise the main claim is false as written.
[Section 2, Eq. (13)] The scalar-field equation is printed as phi_ddot + 3 H phi_dot + 3 H^2 f''(phi) + V''(phi) = 0. Variation of Eq. (5) with respect to phi gives f'(phi) T - 2(phi_ddot + 3 H phi_dot) - 2 V'(phi) = 0 for omega = 1; with T = -6 H^2 this is phi_ddot + 3 H phi_dot + 3 H^2 f'(phi) + V'(phi) = 0. The printed equation contains second derivatives of f and V where first derivatives are required. Since Eqs. (24)-(26) are supposed to follow from Eqs. (11)-(15), this error undermines the derivation chain and must be corrected.
[Section 4, zeta = 0 limit] The statement that 'Choosing zeta = 0 corresponds to a standard minimally coupled scalar field' is not justified. For the reconstructed coupling f(phi) proportional to (phi + c)^2, zeta = 0 means f'(phi) = 0 (at phi = -c, where f = 0), not minimal coupling with a constant f. Moreover, lambda = V' f / (V f') is undefined when f' = 0. The zeta -> 0 limit of the autonomous system therefore does not reduce to standard quintessence without a separate redefinition of lambda; Table 2 as a 'zeta = 0' case of the non-minimal model is misleading.
[Section 4, Eq. (33)] The eigenvalues quoted for R2 are incomplete: the expression a_{2,3} = -1/4 [sqrt(6) zeta +/- sqrt(2 zeta^3 (3 zeta + 4 sqrt(6) ...))] is cut off and cannot be checked. Since the viability claim in Section 5 rests on R2 acting as the radiation-era point, the full eigenvalues (or at least the real parts relevant for stability) should be provided.

minor comments (5)

[Throughout] The manuscript is heavily garbled: 'Jield' for 'field', inconsistent symbols for zeta and lambda, empty cells in Tables 1 and 4, and corrupted equations such as Eq. (21). A thorough proofreading and typesetting pass is needed.
[Section 3, Eq. (22)] The dimensionless variables are not fully defined; the text should state explicitly that x_1^2 = Omega_r and x_2^2 + x_3^2 = Omega_phi, and should specify the normalization factors in x_2 and x_3.
[Section 4, after Eq. (31)] The phrase 'when zeta > 3 we have unstable point' should read 'when zeta^2 > 3/2' (or the equivalent), consistent with the eigenvalues in Eq. (31).
[Section 5] The conclusions use lambda > 2/(3 zeta^2) as the viability condition, while earlier text says 'if lambda^2 >> 1'; these two conditions should be reconciled and stated consistently.
[References] Several references are incomplete: [1] has no paper title, [8] is an arXiv number without a title, and [15] lacks full publication details. The reference list needs completion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's functional forms and viability conditions follow from explicit assumptions via fixed-point algebra; references are background comparisons, not load-bearing self-citations.

full rationale

This paper does not contain a circular derivation. The functional forms f(phi) proportional to (phi+c)^2 and V(phi) proportional to f(phi)^lambda are not presented as independent empirical predictions; they are explicitly derived from the stated assumptions zeta = f'/sqrt(f) = constant and lambda = V' f / (V f') = constant (Section 3, Eq. 30). The derivation is an integration of the defining equations, and the paper clearly labels the constancy of zeta and lambda as the input assumptions. Similarly, the phase-space claims — that lambda >> 1 is needed to make the scaling solutions A1 and R1 behave like matter and radiation, and that zeta^2 << 1 is needed for radiation/matter dominance — are read directly from the fixed-point coordinates and effective equation-of-state formulas obtained from the autonomous system. These are algebraic consequences of the model, not fitted parameters disguised as predictions. There is no fitting to data, no self-citation chain, and no imported uniqueness theorem; references [11]-[18] provide background or comparison results rather than load-bearing justification. The possible reversed stability inequality at the phi-MDE point noted by the reviewer is a mathematical correctness concern, not a circularity concern, because it arises from the paper's own eigenvalue computation rather than from an assumption that already contains the conclusion. Therefore the derivation chain is self-contained and no circular step can be identified.

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

No exotic entities are introduced. The central claim depends on the constancy of zeta and lambda, a spatially flat FRW background, a canonical scalar field, and the effective fluid description of dark energy. No parameter is fitted to observational data.

free parameters (2)

zeta (coupling parameter) = not fitted; assumed constant
zeta = f'(phi)/sqrt(f(phi)) is assumed constant, forcing f(phi) proportional to (phi+c)^2. Its value is not determined by the theory; viability requires zeta^2 << 1.
lambda (potential parameter) = not fitted; assumed constant
lambda = V' f / (V f') is assumed constant, forcing V(phi) proportional to f(phi)^lambda. Viability requires lambda >> 1 according to the paper, or lambda > 2/(3 zeta^2) for the E1 saddle condition.

assumptions (4)

domain assumption The spacetime is described by a spatially flat FRW metric and the matter content is a perfect fluid of radiation and dust.
Used throughout Section 3 to derive Eqs. (11)-(15); the paper restricts to homogeneous and isotropic cosmology.
domain assumption The scalar field is canonically normalized with omega(phi) = 1 and k^2 = 8 pi G = 1.
Stated before Eq. (11); defines the action Eq. (5) and all subsequent equations.
ad hoc to paper The quantities zeta and lambda are taken to be constants during the phase space analysis.
Section 4 assumes constant zeta and lambda to reduce the system to a 3D autonomous system with fixed points; this is a modeling choice not forced by the original action.
domain assumption The effective dark energy density and pressure definitions in Eqs. (18)-(19) with f0 = f(phi0) describe the dark energy sector as a conserved fluid.
Used to define w_eff and the dimensionless variables; the perfect-fluid interpretation is assumed, not proven.

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Cite this review

Pith. "Pith review of Constraining cosmological dynamics of scalar-tensor models of dark energy in teleparallel gravity." pith.science (2026). https://pith.science/paper/IR2O6E4U

@misc{pith2026250709241,
  author       = {Pith},
  title        = {Pith review of: Constraining cosmological dynamics of scalar-tensor models of dark energy in teleparallel gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IR2O6E4U}},
  note         = {Machine review of arXiv:2507.09241}
}

read the original abstract

We consider a scalar-tensor theory in teleparallel gravity where a general function of the scalar field, f(phi), is non-minimally coupled to the torsion scalar T. First, we derive the field equations in this framework. Then, we study the cosmological evolution in a spatially flat, homogeneous, and isotropic universe described by the FRW metric, containing radiation and non-relativistic matter with energy densities rho_r and rho_m, respectively. We analyze the system as an autonomous dynamical model of dark energy. The cosmological behavior depends on the coupling function sigma = f'(phi)/sqrt(f(phi)) and the potential parameter lambda = [V'(phi) * f(phi)] / [V(phi) * f'(phi)]. A constant coupling sigma leads to a quadratic form f(phi) proportional to (phi + c)^2, while a constant lambda results in a power-law potential V(phi) proportional to f(phi)^lambda. These forms are supported by mathematical and physical considerations. We perform phase space analysis and show that lambda much greater than 1 is a necessary condition to obtain a radiation-dominated era with effective equation of state w_eff approximately 1/3 and a matter-dominated era with w_eff approximately 0. Moreover, small sigma^2 ensures radiation and matter dominance in the respective eras. Finally, we derive the necessary conditions for a viable cosmological trajectory in this setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 11 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Einstein 1928, Sitz

A. Einstein 1928, Sitz. Preuss. Akad. Wiss. p. 217; ibid p. 224; A. Unzicker and T . Case,

work page 1928

[2] [2]

Ferraro and F

R. Ferraro and F . Fiorini, Phys. Rev. D 75, 084031 (2007)

work page 2007

[3] [3]

G. R. Bengochea and R. Ferraro, Phys. Rev. D 79, 124019 (2009)

2009

[4] [5]

B. Li, T . P . Sotiriou and J. D. Barrow, Phys. Rev. D 83, 064035 (2011); M. Li, R. X. Miao and Y . G. Miao, JHEP 1107, 108 (2011)

work page 2011

[5] [6]

C. Q. Geng, C. C. Lee, E. N. Saridakis, Y . P . Wu, Phys. Lett. B 704, 384 (2011)

work page 2011

[6] [7]

C. Q. Geng, C. C. Lee and E. N. Saridakis, arXiv:1110.0913

arXiv

[7] [9]

Tsujikawa, P

S. Tsujikawa, P . Rev. D 76, 023514 (2007)

work page 2007

[8] [10]

R. J. Yang, Europhys. Lett. 93, 60001 (2011)

work page 2011