REVIEW 5 major objections 3 minor 22 references
Inflation in Myrzakulov $F(R,T)$ Gravity: A Comparative Study in Metric, Symmetric Teleparallel, and Weitzenb\"{o}ck Formalisms
T0 review · 5 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read A unified action that combines curvature and torsion can be tuned so that its inflationary predictions fall inside the Planck and BICEP/Keck bounds.
desk verdict A competent review of standard modified-gravity inflation, but the new F(R,T) section rests on equations that fail to reduce to the f(R) and f(T) limits the paper claims. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the Myrzakulov $F(R,T)$ Lagrangian—an action depending on both the Ricci scalar $R$ and the torsion scalar $T$—with the FLRW identities $R=6(2H^2+\dot H)$ and $T=-6H^2$. The paper's specific working model is $F(R,T)=R+\alpha T+\beta R^2+\gamma T^2$, whose derivatives $\nu=\partial F/\partial R$, $\tau=\partial F/\partial T$, and mixed derivative $\nu_{RT}$ enter the modified Friedmann equations as tunable geometric couplings. The slow-roll dictionary $\epsilon=-\dot H/H^2$, $\eta=\ddot\phi/(H\dot\phi)$, $n_s\approx1-6\epsilon+2\eta$, $r\approx16\epsilon$ converts the background evolution into the observables that are compared with Planck and BICEP/Keck data.
What would settle it
Vary the action $\int d^4x\sqrt{-g}[F(R,T)+L_\phi]$ under an explicit connection for which $R$ and $T$ are both nonzero, integrate the slow-roll system while keeping every term, and compare the resulting $n_s$ and $r$ for $F=R+\alpha T+\beta R^2+\gamma T^2$ with the Planck 2018 and BICEP/Keck contours.
Extended reading notes
Core claim
On its own terms, the paper establishes a comparative claim: in a flat FLRW universe, the same slow-roll formalism works in all three geometric formulations, and the hybrid action $F(R,T)=R+\alpha T+\beta R^2+\gamma T^2$ can reproduce the inflationary observables measured in the CMB. The paper states that with modest values of $\alpha,\beta,\gamma$ (of order $10^{-2}$), this model produces 50–60 e-folds of inflation, gives a scalar spectral index $n_s$ in the Planck-preferred range, and keeps the tensor-to-scalar ratio $r$ small enough for BICEP/Keck. It further claims that the model interpolates between pure $f(R)$ and pure $f(T)$ behavior, so the same geometry can tune predictions between the two limits, and that the three formalisms give distinguishable signatures in the $n_s$–$r$ plane.
Load-bearing premise
The whole inflationary prediction rests on the claim that the slow-roll Friedmann equations for F(R,T) gravity follow from the action and that one of the curvature-derivative terms can be discarded in the slow-roll limit; if that step is not correct, the computed n_s and r do not follow.
Editorial extensions
If this is right
- For the polynomial model with $\alpha,\beta,\gamma$ at the percent level, inflation can last 50–60 e-folds, long enough to solve the horizon and flatness problems.
- The hybrid action can place $n_s$ and $r$ inside the Planck 2018 and BICEP/Keck allowed regions, so the model is currently viable.
- The same scalar potential yields different $(n_s,r)$ predictions in $f(R)$, $f(T)$, $f(Q)$, and $F(R,T)$, so the three geometric pictures of gravity are in principle distinguishable by CMB measurements.
- In the warm variant, curvature–torsion couplings can act as a built-in dissipation channel, allowing inflation to transition to radiation without a separate reheating phase.
Reading between the lines
- A numerical scan over $\alpha,\beta,\gamma$ that retains the dropped $\nu_{RR}$ term would test whether the reported viable region survives; if it does, the model could be discriminated from Starobinsky inflation by future measurements of the running of $n_s$.
- The comparative tables suggest that a tensor-to-scalar ratio near the current upper bound would favor torsion-based or hybrid models, while a very small $r$ would favor non-metricity-based inflation.
- Because the same geometric couplings persist at low curvature, the polynomial $F(R,T)$ model could also be constrained by late-time cosmology, not only by inflation.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents a comparative study of scalar-field inflation in three modified-gravity settings—metric f(R), teleparallel f(T), and symmetric teleparallel f(Q)—and then extends the analysis to a hybrid F(R,T) gravity with action depending on both curvature R and torsion T. For each formalism the authors write down Friedmann-like equations in a flat FLRW background, impose slow-roll conditions, and compute the spectral index n_s and tensor-to-scalar ratio r for several potentials. The central claim is that the polynomial model F(R,T)=R+αT+βR^2+γT^2 interpolates between f(R) and f(T) behavior and can satisfy Planck and BICEP/Keck bounds on n_s and r for appropriately chosen α, β, γ. The paper also includes qualitative discussions of warm inflation and de Sitter/quasi-de Sitter solutions in F(R,T) gravity.
Significance. If the F(R,T) results were established, the paper would provide a useful organizing framework for inflationary observables across the geometric trinity of modified gravity, and the hybrid F(R,T) model would offer a phenomenologically flexible extension. The comparative review of f(R), f(T), and f(Q) inflation is a clear and mostly correct synthesis of known material, and the paper is explicit that several figures are schematic. However, the claimed new results for F(R,T) gravity are not supported by the derivations presented: the central field equations are asserted without derivation, they fail to reduce to the paper's own f(R) and f(T) limits, and the promised numerical scans are absent. As a consequence, the observational viability claim for F(R,T) inflation is not established.
major comments (5)
- [Sec. VI.A, Eqs. (94)-(95)] The Friedmann-like equations (94) and (95) are presented without derivation from the action (88), which is a load-bearing gap because all subsequent F(R,T) phenomenology rests on them. More seriously, these equations do not reduce to the paper's own limiting cases despite the claim in the text that they 'recover the usual f(R) or f(T) cosmologies.' Setting F(R,T)=f(R) in Eq. (94) gives a denominator ν−12H^2ν_RR = F_R−12H^2F_RR, whereas the paper's metric f(R) result Eq. (22) has denominator F_R and matches standard f(R) cosmology; the extra F_RR term in (94) is not present in (22). Setting F(R,T)=f(T) gives ν=0 and ν_RR=0, so both (94) and (95) have zero denominators and are singular instead of reducing to the f(T) equations (44)-(45). This internal inconsistency undermines the central derivation.
- [Sec. VI.B, Eq. (98)] The slow-roll reduction of Eq. (94) to Eq. (98) drops the 12H^2ν_RR term in the denominator without any stated justification. For the polynomial model F(R,T)=R+αT+βR^2+γT^2 discussed in Sec. VI.C, ν_RR=2β, and the paper itself proposes β of order 10^{-2} in Sec. VI.D; under slow-roll conditions with H^2 large, the dropped term need not be negligible. Because the subsequent slow-roll potentials and the n_s, r tables in Secs. VI.E and VI.G are derived from (98), this omission makes the connection between the F(R,T) action and the claimed observables unsubstantiated.
- [Sec. V.C and Sec. VI.C-VI.D] Section V.C promises that 'in subsequent sections, we will perform numerical scans over parameter space for selected models and compare predictions from each formalism to current observational bounds.' No such scans appear. Section VI.C contains only schematic figures (Figs. 3-6), and the observational-viability claim in Sec. VI.D—that 'numerical evolution of the field equations shows that for modest values of β and γ ... the model supports 50–60 e-folds of inflation'—is unsupported by any presented numerical data. This missing numerical evidence is directly load-bearing for the paper's central claim that F(R,T) models can match Planck and BICEP/Keck constraints.
- [Sec. VI.E and Sec. VI.G] The 'predictions' in the Summary Table of Sec. VI.E (e.g., n_s=1−3/N, r=16/N for λφ^4; n_s=1−2/N, r=8/N for quadratic; n_s=1−2/N, r=12/N^2 for Starobinsky-like) are the standard single-field slow-roll results for these potentials in general relativity. They do not follow from the F(R,T) Friedmann equations (94)-(95), and the F(R,T) dependence is delegated to unspecified constants c1, c2 in the comparative table in Sec. VI.G. Thus the tables do not demonstrate that the F(R,T) framework generates distinct or viable predictions; they merely restate textbook results with the geometric corrections left undetermined.
- [Sec. II.A and Sec. VI.A] The paper assumes that R=6(2H^2+\dot H) and T=−6H^2 can be simultaneously nonzero in a single spacetime, but in the teleparallel (Weitzenböck) geometry used for f(T), the Ricci scalar R vanishes identically, and in the metric (Levi-Civita) geometry, the torsion scalar T is not defined as in Eq. (90). No metric-affine connection that permits both scalars to be nonvanishing is specified in Sec. VI.A, so the geometric meaning of the hybrid F(R,T) action is unclear. This is not merely a presentation issue; it affects whether Eqs. (94)-(95) describe a consistent gravitational theory.
minor comments (3)
- [Eqs. (7), (22), (44), (51)] The κ^2 factors are inconsistent between the f(R) equations: Eq. (7) has κ^2ρ_φ in the numerator, while Eq. (22) omits κ^2, and the f(T) definitions in Eqs. (44) and (51) treat κ^2 inconsistently (one appears as 2ρ_φ with no κ^2 and the other as 1/(2κ^2) times the geometric terms). The notation should be made uniform.
- [Eqs. (91)-(93) and Eqs. (98)-(100)] The auxiliary variables are introduced as u, u_T, u_RR, u_RT in Eqs. (91)-(93), but the text later uses ν, τ, and ν_T without defining the mapping; Eq. (100) contains 'ν_T' while Eq. (95) uses ν_RT, which is a different derivative. This notational inconsistency makes the derivation hard to follow.
- [Figs. 1-6] All figures in Secs. VI.B-VI.D are explicitly schematic and do not carry quantitative content; the captions should state this clearly (some do, but Figs. 5 and 6 appear to imply actual model predictions). Since the text relies on these figures for the interpolation claim, labeling them as illustrations rather than results would improve accuracy.
Circularity Check
F(R,T) 'predictions' reduce to standard slow-roll formulas with F(R,T) corrections left as unspecified constants, and observational consistency is obtained by choosing alpha, beta, gamma rather than derived from the stated field equations.
-
other
[Section VI.A, Eqs. (94)-(95) and the paragraph after them]
"The Friedmann-like equations can now be expressed in a compact and generalized form: 3H^2 = 1/(nu - 12H^2 nu_RR) (rho_phi + 1/2(nu R + nu_T T - F) - 3H nu_dot)... These equations recover the usual f(R) or f(T) cosmologies in the respective limits F(R,T) -> f(R) and F(R,T) -> f(T)."
Substituting F(R,T)=f(R) into Eq. (94) gives nu=f_R and nu_RR=f_RR, so the denominator becomes f_R - 12H^2 f_RR, with an extra 12H^2 f_RR term compared with the paper's own f(R) result Eq. (22). Substituting F=f(T) gives nu=0, so Eqs. (94)-(95) are singular (denominators nu - 12H^2 nu_RR = 0 and nu = 0) instead of reducing to Eqs. (44)-(45). The claimed recovery of the limiting cosmologies is therefore not obtained by substitution; it is asserted. The interpolation premise on which the F(R,T) inflationary claims rest is thus an input assumption, not a derived consequence.
-
other
[Section VI.B, Eq. (98)]
"Additionally, we assume that the derivatives of the function F(R,T) with respect to R and T vary slowly. Under these approximations, the Friedmann equation becomes: 3H^2 ≈ 1/nu (V(phi) + 1/2(nu R + tau T - F)), (98)"
Eq. (98) drops the -12H^2 nu_RR term that is present in the exact Eq. (94). For the polynomial model F=R+alpha T+beta R^2+gamma T^2, nu_RR=2 beta, and during slow roll R≈12H^2, so 12H^2 nu_RR≈24 beta H^2≈2 beta R, the same order as the beta R terms kept in Eq. (104). The slow-roll system that later yields n_s and r is therefore not the slow-roll reduction of the paper's own field equation (94); the observable results are computed from a different, silently altered equation.
2 more flagged steps
-
fitted input called prediction
[Section VI.D, Observational Viability]
"Constraints from Planck and BICEP/Keck experiments impose bounds on ns and r, which can be satisfied by choosing appropriate values of alpha, beta, and gamma."
This is parameter fitting presented as prediction: alpha, beta, and gamma are free parameters with no independent determination in the paper, and the accompanying figures are explicitly 'schematic.' The claim that modest values O(10^-2) 'support 50-60 e-folds' and 'predict ns in the Planck-preferred range' is a tuning statement. The abstract's summary—'appropriately chosen parameters in F(R,T) models can produce viable and distinguishable inflationary signatures'—is exactly the same statement: the observables are placed inside the bounds by hand, so the claimed consistency is built in by parameter choice rather than derived from the model.
-
renaming known result
[Section VI.E, Summary Table of Predictions and following paragraph; Eq. (109)]
"These predictions are subject to corrections from the F(R,T) structure, including curvature-torsion cross terms, modified gravitational couplings, and changes in the Hubble evolution. Numerical modeling is needed for precise predictions and for verifying the theoretical consistency with current observational bounds."
The table lists n_s=1-3/N, r=16/N for lambda phi^4; n_s=1-2/N, r=8/N for (1/2)m^2 phi^2; and n_s=1-2/N, r=12/N^2 for the Starobinsky-like potential. These are exactly the standard single-field slow-roll formulas from Eqs. (84)-(85), with no F(R,T) input. The only F(R,T) dependence invoked is an unevaluated delta_RT in Eq. (109) and unspecified c1, c2 in the comparative table of Section VI.G. The presented F(R,T) predictions therefore rename textbook potential results as F(R,T) predictions, with the actual model corrections deferred to future numerical work.
full rationale
Score 6: the paper is not wholly circular. Sections II-IV recover standard f(R), f(T), and f(Q) slow-roll results, and Section V gives textbook potential formulas; self-citations [10]-[13] appear mainly as motivation, not as the machine-checked or independent load-bearing evidence for the quantitative claims. The circularity is concentrated in the F(R,T) part of the paper. Section VI.D fits alpha, beta, gamma to the Planck/BICEP/Keck bounds and then presents that fitted consistency as a prediction; Section VI.E tabulates ordinary single-field slow-roll n_s and r values while deferring the F(R,T) corrections to an unevaluated delta_RT and to c1, c2; the abstract's phrase 'appropriately chosen parameters ... can produce viable and distinguishable inflationary signatures' is the same tuning statement. In addition, as a correctness issue not separately scored as circularity, Eqs. (94)-(95) do not reduce to the paper's own f(R) limit Eq. (22) and are singular in the f(T) limit nu=0, and Eq. (98) drops the 12H^2 nu_RR term present in Eq. (94), so the slow-roll equations actually used for the tables are not the stated F(R,T) equations. The central 'prediction' is therefore, by construction, standard potential results plus free parameters chosen to match the data, giving partial circularity rather than a derivation from the F(R,T) action.
Assumptions & free parameters
free parameters (3)
- alpha (linear torsion coupling) =
unspecified
- beta (R^2 coefficient) =
unspecified (paper suggests ~O(10^-2))
- gamma (T^2 coefficient) =
unspecified
assumptions (5)
- domain assumption Spatially flat FLRW metric with ds^2 = -dt^2 + a(t)^2(dx^2+dy^2+dz^2).
- domain assumption Slow-roll conditions: φdot^2 << V, |φddot| << |3H φdot|, and ˙F terms negligible.
- domain assumption The scalar field is canonical and minimally coupled to F(R,T) gravity.
- ad hoc to paper Derivatives of F(R,T) with respect to R and T vary slowly during inflation.
- domain assumption R and T can be treated as independent variables and both are non-zero simultaneously in the same spacetime.
Cite this review
Pith. "Pith review of Inflation in Myrzakulov $F(R,T)$ Gravity: A Comparative Study in Metric, Symmetric Teleparallel, and Weitzenb\"{o}ck Formalisms." pith.science (2026). https://pith.science/paper/FXXQM47P
@misc{pith2026250711753,
author = {Pith},
title = {Pith review of: Inflation in Myrzakulov $F(R,T)$ Gravity: A Comparative Study in Metric, Symmetric Teleparallel, and Weitzenb\"ock Formalisms},
year = {2026},
howpublished = {\url{https://pith.science/paper/FXXQM47P}},
note = {Machine review of arXiv:2507.11753}
}
abstract
We present a unified treatment of cosmic inflation within the framework of Myrzakulov Gravity, exploring its realization in three different formalisms: metric (curvature-based), teleparallel (torsion-based), and symmetric teleparallel (non-metricity-based). For each case, we derive the corresponding field equations in a flat FLRW background, study inflationary solutions driven by a scalar field, and compute observable quantities such as the scalar spectral index \( n_s \) and the tensor-to-scalar ratio \( r \). In addition to these geometric sectors, we extend our analysis to the more general and dynamically richer Myrzakulov \( F(R,T) \) gravity, which incorporates both curvature \( R \) and torsion \( T \) in a unified action. We derive the inflationary dynamics in this hybrid model and investigate how it interpolates between pure \( f(R) \) and \( f(T) \) behaviors. The resulting framework allows for enhanced flexibility in matching Planck and BICEP/Keck observational constraints. We present analytic estimates and schematic predictions in the \( n_s \)--\( r \) plane, demonstrating that appropriately chosen parameters in \( F(R,T) \) models can produce viable and distinguishable inflationary signatures. This comparative and extended study highlights the potential of Myrzakulov Gravity and its generalizations to provide a consistent and geometrically motivated description of the early universe, with predictive power across different formulations of spacetime geometry.
Figures
Figures from the paper (5 more)
Reference graph
Works this paper leans on
-
[1]
A New Type of Isotropic Cosmological Models Without Singularity,
A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B 91, 99–102 (1980)
work page 1980
-
[2]
Inflationary universe: A possible solution to the horizon and flatness problems,
A. H. Guth, “Inflationary universe: A possible solution to the horizon and flatness problems,” Phys. Rev. D 23, 347–356 (1981)
work page 1981
-
[3]
A. D. Linde, “A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems,” Phys. Lett. B108, 389–393 (1982)
work page 1982
-
[4]
The Cosmological Constant Problem,
S. Weinberg, “The Cosmological Constant Problem,” Rev. Mod. Phys. 61, 1–23 (1989)
work page 1989
-
[5]
Cosmological constant: The weight of the vacuum,
T. Padmanabhan, “Cosmological constant: The weight of the vacuum,” Phys. Rept. 380, 235–320 (2003)
work page 2003
-
[6]
T. P. Sotiriou and V. Faraoni, “f(R) Theories of Gravity,” Rev. Mod. Phys. 82, 451–497 (2010)
work page 2010
-
[7]
A. De Felice and S. Tsujikawa, “f(R) Theories,” Living Rev. Rel. 13, 3 (2010)
work page 2010
-
[8]
f(T) Teleparallel Gravity and Cosmology,
Y. F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, “f(T) Teleparallel Gravity and Cosmology,” Rept. Prog. Phys. 79, 106901 (2016)
work page 2016
Show all 22 references
-
[9]
Teleparallel Palatini Theories,
J. B. Jim´ enez, L. Heisenberg, and T. S. Koivisto, “Teleparallel Palatini Theories,” JCAP 08, 039 (2018)
2018
-
[10]
Myrzakulov, Eur
R. Myrzakulov, Eur. Phys. J. C 72 (2012), 2203, doi:10.1140/epjc/s10052-012-2203-y [arXiv:1207.1039 [gr-qc]]
2012 arXiv
-
[11]
Metric-Affine Myrzakulov Gravity Theories: Models, Appli- cations and Theoretical Developments,
D. Momeni and R. Myrzakulov, “Metric-Affine Myrzakulov Gravity Theories: Models, Appli- cations and Theoretical Developments,” Int. J. Theor. Phys. 64, 95 (2025)
2025
-
[12]
Myrzakulov Gravity in Vielbein Formalism: A Study in Weitzenb¨ ock Spacetime,
D. Momeni and R. Myrzakulov, “Myrzakulov Gravity in Vielbein Formalism: A Study in Weitzenb¨ ock Spacetime,” Nucl. Phys. B1015, 116903 (2025)
2025
-
[13]
Einstein–Gauss–Bonnet–Myrzakulov gravity from R + F (T, G): Numerical insights and torsion–Gauss–Bonnet dynamics,
D. Momeni and R. Myrzakulov, “Einstein–Gauss–Bonnet–Myrzakulov gravity from R + F (T, G): Numerical insights and torsion–Gauss–Bonnet dynamics,” Nucl. Phys. B 1017, 116966 (2025). 40
2025
-
[14]
Warm inflation,
A. Berera, “Warm inflation,” Phys. Rev. Lett. 75, 3218 (1995)
1995
-
[15]
Warm Intermediate Inflation in F (T ) Gravity,
M. Jamil, D. Momeni, and R. Myrzakulov, “Warm Intermediate Inflation in F (T ) Gravity,” Int. J. Theor. Phys.54, 1098–1112 (2015), arXiv:1309.3269[gr-qc]
2015 arXiv
-
[16]
Warm non-minimally coupled Peccei–Quinn inflation and de Sitter Swampland conjecture,
J. Yuennan, P. Channuie, and D. Momeni, “Warm non-minimally coupled Peccei–Quinn inflation and de Sitter Swampland conjecture,” Nucl. Phys. B 1012, 116810 (2025), doi: 10.1016/j.nuclphysb.2025.116810, arXiv:2410.12296[gr-qc]
2025
-
[17]
Gravity’s rainbow effects on higher curva- ture modification of R2 inflation,
J. Yuennan, P. Channuie, and D. Momeni, “Gravity’s rainbow effects on higher curva- ture modification of R2 inflation,” Eur. Phys. J. C 84, 766 (2024), doi:10.1140/epjc/ s10052-024-13155-0 , arXiv:2405.04174[gr-qc]
2024 arXiv
-
[18]
Nonminimally-coupled warm Higgs inflation: Metric vs. Palatini formulations,
T. Eadkhong, P. Dam-O, P. Channuie, and D. Momeni, “Nonminimally-coupled warm Higgs inflation: Metric vs. Palatini formulations,” Nucl. Phys. B994, 116289 (2023), doi:10.1016/ j.nuclphysb.2023.116289, arXiv:2303.00572[gr-qc]
2023
-
[19]
Modified teleparallel gravity: Inflation without inflaton,
R. Ferraro and F. Fiorini, “Modified teleparallel gravity: Inflation without inflaton,” Phys. Rev. D 75, 084031 (2007)
2007
-
[20]
Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration,
S. Nojiri and S. D. Odintsov, “Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration,” Phys. Rev. D 68, 123512 (2003)
2003
-
[21]
Introduction to modified gravity and gravitational alternative for dark energy,
S. Nojiri and S. D. Odintsov, “Introduction to modified gravity and gravitational alternative for dark energy,” Int. J. Geom. Meth. Mod. Phys. 4, 115–146 (2007)
2007
-
[22]
f (T ) teleparallel gravity and cosmology,
Y. F. Cai, S. Capozziello, M. De Laurentis and E. N. Saridakis, “ f (T ) teleparallel gravity and cosmology,” Rept. Prog. Phys. 79, 106901 (2016)
2016
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