Non-singular bounce solutions in Myrzakulov $f(R,T)$ gravity

Almira Sergazina; Andreas Lymperis; Gulgassyl Nugmanova

REVIEW 5 major objections 4 minor 1 cited by

Non-singular bounce solutions in Myrzakulov $f(R,T)$ gravity

T0 review · 5 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Myrzakulov $F(R,T)$ gravity can host nonsingular bouncing universes whose scalar perturbation spectrum is scale invariant, with amplitude $\sigma G/(36\pi)$.

desk verdict Reconstructing u and v to force a bounce is fine as an exercise, but the variational setup has a v(a,ä) inconsistency and the perturbation analysis is imported, not derived. read the letter →

arxiv 2501.18524 v1 pith:H52FQT4K submitted 2025-01-30 gr-qc astro-ph.CO

Andreas Lymperis , Gulgassyl Nugmanova , Almira Sergazina This is my paper · ORCID

classification gr-qcastro-ph.CO PACS 04.50.Kd98.80.-k95.36.+x

keywords MyrzakulovF(RT)gravitybouncingcosmologynullenergyconditionviolationmatterbounceMukhanov-Sasakiformalismscalarperturbationsprimordialpowerspectrumnon-specialconnection

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 argues that Myrzakulov $F(R,T)$ gravity — a modified gravity built on a connection that carries both curvature and torsion — can produce nonsingular bouncing cosmologies without exotic matter. By choosing the two functions $u$ and $v$ that parametrize the non-special connection, the theory's effective energy density and pressure can violate the null energy condition, which is the necessary condition for a bounce. The authors reconstruct $u$ and $v$ for a simple bounce and for a matter bounce, and both scale factors satisfy the Friedmann equations exactly. They then use the Mukhanov-Sasaki formalism to evolve scalar perturbations through the bounce and obtain a scale-invariant power spectrum $P_\zeta = \sigma G/(36\pi)$. If correct, this makes Myrzakulov gravity a candidate framework for a singularity-free early universe whose perturbation spectrum is compatible with observations.

What carries the argument

The load-bearing object is the non-special connection of Myrzakulov gravity, whose curvature and torsion scalars are written as $R=R^{(LC)}+u$ and $T=T^{(W)}+v$, with $u$ and $v$ free functions of the scale factor and its derivatives. Choosing $u$ and $v$ changes the effective energy density $\rho_{MG}$ and pressure $p_{MG}$ in the Friedmann equations, so the connection parametrization acts as a designer effective fluid that can violate the null energy condition. For perturbations, the argument is carried by the Mukhanov-Sasaki equation $v_k''+(c_s^2 k^2-z''/z)v_k=0$ with $z=a\sqrt{\rho_t+p_t}/(c_s H)$, which the paper applies to the bounce geometry to convert quantum vacuum fluctuations in the contracting phase into classical curvature perturbations after horizon exit.

What would settle it

Derive the quadratic action for scalar perturbations directly from the action (7) for the linear case $F(R,T)=R+T$; if the resulting perturbation equation contains extra terms involving $u_{\ddot a}$ or the non-special connection that are absent from Eq. (30), then the predicted power spectrum for the matter bounce will differ from $P_\zeta=\sigma G/(36\pi)$, and the scale-invariance conclusion would rest on an inapplicable equation.

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Extended reading notes

Core claim

The central claim is that in the linear case $F(R,T)=R+\lambda T$, with $\lambda=1$ for the explicit solutions, the non-special connection's parametrization functions $u(a,\dot a,\ddot a)$ and $v(a,\dot a)$ can be chosen so that the modified Friedmann equations admit nonsingular bounce solutions. For the basic bounce $a(t)=a_B(1+\sigma t^2)$, the choices $\beta=24a_B/5$ and $\delta=-24(a_B+5)/5$ satisfy the equations, with $\gamma$ and $\epsilon$ free. For the matter bounce $a(t)=a_B(1+3\sigma t^2/2)^{1/3}$, the choice $v=6\ddot a/a+\epsilon \dot a/a+\zeta/a^3$ works. In both cases the effective gravitational sector violates the null energy condition near the bounce. For the matter bounce, the paper further claims that scalar perturbations obey the standard Mukhanov-Sasaki equation, that the contracting phase begins in the Bunch-Davies vacuum, and that the resulting primordial power spectrum is scale invariant with amplitude $\sigma G/(36\pi)$.

Load-bearing premise

The load-bearing premise is that the standard Mukhanov-Sasaki perturbation equation, with its usual definition of $z$, applies unchanged to this higher-derivative theory; the paper introduces that equation in Section IV without deriving it from the action, and if the transfer is invalid the computed scale-invariant spectrum does not follow.

Editorial extensions

If this is right

If the central claim holds, Myrzakulov $F(R,T)$ gravity provides a singularity-free early-universe scenario in which the null energy condition is violated by the geometry itself, not by exotic matter.
The matter-bounce realization inherits the standard matter-bounce property of a nearly scale-invariant scalar power spectrum, here with the concrete amplitude $\sigma G/(36\pi)$.
The reconstruction method gives a model-building recipe: for a desired nonsingular scale factor, the free connection functions $u$ and $v$ can be solved from the Friedmann equations, so other bounce profiles can be treated in the same way.
Because perturbations remain finite through the bounce and match onto a scale-invariant spectrum, the scenario can in principle be compared with CMB observations in the same way as inflationary models.

Reading between the lines

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

I would expect the same reconstruction technique to reproduce bouncing solutions known from other modified gravities, since the free functions $u$ and $v$ effectively encode the full background dynamics of a designer fluid.
A consequence the authors do not spell out is that the connection parametrization is likely degenerate with the matter content: many different pairs $(u,v)$ may produce identical background histories, so distinguishing this theory from other NEC-violating models will require the perturbation spectra.
A natural testable extension is to compute the tensor power spectrum and the tensor-to-scalar ratio in the same formalism; if the torsion-carrying connection modifies tensor perturbations differently from scalars, the consistency relation would separate Myrzakulov gravity from standard matter bounce.
The scale-invariance result should be checked for $F(R,T)$ beyond the linear choice $R+\lambda T$; the higher-derivative dependence of $u$ on $\ddot a$ may introduce additional terms in the quadratic action that alter the Mukhanov-Sasaki variable.

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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

You should know: this paper claims non-singular bounces in Myrzakulov F(R,T) gravity by reconstructing the connection parametrization functions u and v, plus a scale-invariant scalar spectrum from the Mukhanov-Sasaki equation. The background reconstruction is explicit, but there is a concrete internal inconsistency that undermines the main equations as written. The stress-test note is correct: in Sec. II B the action is varied under the assumption v = v(a,ȧ), so the resulting Friedmann equations (14)–(15) contain no v_ä terms. Yet the reconstruction ansätze (23) and (28) put ä/a inside v. That changes the variation: terms proportional to v_ä and its time derivatives are missing from ρ_MG and p_MG. This isn't a typo; it propagates into the NEC expression (29) and the z variable (31). The claimed bounce solutions are not solutions of the equations as derived. The perturbation section is weaker still. Equation (30) is simply announced as the governing equation. In a higher-derivative theory with a non-special connection, the Mukhanov-Sasaki variable, sound speed, and z must be derived from the quadratic action; none of that is done. The power spectrum (38) therefore hangs on an unverified assumption. The conclusions claim stability, but no stability analysis exists anywhere in the paper. The NEC check is also asserted rather than shown: Eq. (29) is an expression, not a demonstrated inequality for the constructed solutions. What is genuinely new is the specific polynomial ansätze for u and v in this particular gravity (eqs. 22–28) and the algebraic observation that, if one ignores the variation issue, they formally satisfy the Friedmann equations. That is a niche contribution, but it is not a viable model. The paper is honest about the reconstruction logic — the bounce is an input, not a prediction — but the internal inconsistency makes even the background example unsupported. Who is this for? Possibly someone cataloguing bounce realizations in modified gravity, but they would need to redo the variation properly and derive the perturbations before citing it as an example. I would not send this to a serious referee in its current form; the load-bearing flaw is internal and the advertised results are not supported. Desk reject, with an invitation to resubmit after deriving the correct field equations and a genuine perturbation action.

Referee Report

5 major / 4 minor

Summary. The paper studies non-singular bouncing cosmologies in Myrzakulov F(R,T) gravity, a modified theory with a connection carrying both curvature and torsion. After presenting the mini-superspace field equations, the authors choose two bounce scale factors (a 'basic bounce' and a 'matter bounce'), and reconstruct the connection parametrization functions u(a,ȧ,ä) and v(a,ȧ) so that the Friedmann equations are formally satisfied. They then assert null energy condition violation and, using the Mukhanov–Sasaki formalism with z = a√(ρ_t+p_t)/(c_s H), compute the scalar power spectrum, reporting a scale-invariant result P_ζ = σG/(36π) for the matter bounce. The paper concludes that Myrzakulov F(R,T) gravity is a viable bouncing-cosmology framework. The background reconstruction is algebraically explicit, but the perturbative analysis is imported from standard cosmology without derivation from the action of the theory.

Significance. If the claims were correct, this would be a notable extension of bouncing-cosmology results to Myrzakulov gravity, a theory that has received recent phenomenological attention. The paper's greatest strength is its explicitness: the background equations and the reconstruction ansätze are written out, and the reader can check the algebra. However, the central results are not established. The paper explicitly states that the bounce scale factors are assumed and u,v are then chosen to satisfy the Friedmann equations, so the bounce is an input rather than a prediction. More seriously, the variational derivation of the background equations is internally inconsistent with the later ansätze, and the Mukhanov–Sasaki equation is imported without deriving the quadratic action of the theory. These issues affect the claimed NEC violation, the stability statement, and the computed power spectrum, so the paper's main conclusions rest on unverified assumptions.

major comments (5)

[§II.B, Eqs. (14)–(15); §III.B, Eqs. (23) and (28)] The variation leading to Eqs. (14)–(15) is performed under the stated assumption v = v(a, ȧ), so the effective density and pressure contain partial derivatives of v only with respect to a and ȧ. However, the reconstruction ansätze (23) and (28) both contain ä/a, e.g. v = δ ä/a + ε ȧ/a in Eq. (23) and v = 6 ä/a + ε ȧ/a + ζ/a³ in Eq. (28). If v genuinely depends on ä, the Euler–Lagrange variation of the term −(a³/2κ²)λ v(a, ȧ, ä) in the Lagrangian (11) generates additional contributions proportional to v_ä and its time derivatives, contributions that are absent from Eqs. (14)–(15). The claim that the scale factors (20) and (25) satisfy the Friedmann equations is therefore not supported by the field equations as derived. Since the same effective ρ_MG and p_MG enter the NEC analysis (29) and the perturbation variable z (31), this inconsistency propagates into the central claims of the paper.
[§IV, Eqs. (30)–(31) and (38)] The Mukhanov–Sasaki equation is introduced without deriving it from the action (7). Myrzakulov F(R,T) gravity has a non-special connection and, in the actual reconstruction, higher-derivative dependences (v contains ä/a), so the standard quadratic action for curvature perturbations in general relativity cannot be assumed to carry over. In particular, the definition z = a√(ρ_t+p_t)/(c_s H) and the sound speed c_s must follow from the second-order action of the theory; none of this is provided. Consequently, the power spectrum (38), the scale-invariance conclusion, and the statement that the bounce is stable are unsupported. The authors need to derive the quadratic action, the gauge-invariant variable, and the resulting perturbation equation within this theory.
[§III.B, Eqs. (20)–(24) and (25)–(28)] The construction is reverse engineered: the bounce scale factors (20) and (25) are fixed by hand, and the functions u and v are then chosen as polynomial ansätze so that the Friedmann equations hold by construction, with free parameters (γ, ε, ζ) left arbitrary and the bounce energy density ρ_B fixed by the reconstruction. While reconstruction is a recognized technique in modified gravity, the paper presents no independent physical principle that selects these forms or fixes the free parameters. As presented, the results should not be described as 'obtaining' or 'producing' the bounce from the theory; they are consistency conditions for an assumed scale factor. This does not by itself invalidate the paper, but it substantially weakens the claimed viability of Myrzakulov gravity as a bounce model.
[§IV, Eqs. (32)–(38) and §V] The stability claim is not demonstrated. The paper computes z''/z for the simple bounce and states that it is 'well-behaved', but it does not analyze the evolution of the curvature perturbation ζ through the bounce: there is no matching of the growing and decaying solutions, no check of the sign of c_s², no verification that the perturbations remain finite across the bounce, and no treatment of the transition from the contracting-phase solution (37) to the post-bounce phase. Therefore the concluding sentence that the bounce is 'stable and free from singularities' is not supported by the perturbative analysis in the manuscript.
[§III.B, Eq. (29)] The claim that the null energy condition is violated is not checked. Equation (29) is a general expression for κ²(ρ_MG+p_MG), but the authors do not substitute the reconstructed u and v for either the basic bounce (20) or the matter bounce (25) to show that the expression becomes negative near the bounce. The sentence 'As we observe, the NEC violation is satisfied' is therefore unsupported by any explicit calculation; the following sentence merely restates that a bounce requires ρ_eff + p_eff < 0. Moreover, Eq. (29) inherits the variational inconsistency of Eqs. (14)–(15) if v depends on ä, so the right-hand side is not a reliable expression for the NEC combination.

minor comments (4)

[Abstract and title] The abstract and title use 'f(R,T)' while the body consistently uses 'F(R,T)'; the notation should be standardized.
[Abstract] There is a typo 'with in' in the abstract ('nonsingular bounce with in the framework'); the same typo appears in the body ('start form the torsional' in §I). The manuscript needs a proofreading pass.
[§III.B, Eq. (20)] The line following Eq. (20) writes ȧ(t) = 2σta_B, which is correct, but the sentence 'a(t) = a_B(1+σt²)' repeats the definition without explaining the role of a_B; consider clarifying that a_B is the scale factor at t=0.
[§III.B, Eqs. (22)–(24)] The identification ρ_B ≡ ρ_m(a_B) = 144/(5κ²) σ a_B³ is stated without derivation; since this is a matching condition used in the reconstruction, a brief derivation would help the reader verify the result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bounce is an openly declared reconstruction and the perturbation result, though underived from the Myrzakulov action, is not an output that reduces to its own input.

full rationale

The paper clearly states that it will "reconstruct the deformation functions u and v such that they satisfy the Friedmann equations under the bounce scale factor" (Sec. III B), so the scale factors in Eqs. (20) and (25) are inputs chosen for the reconstruction, not predictions derived from the theory. Verifying that chosen ansatze make the Friedmann equations hold is an existence argument, not a circular derivation. The NEC discussion in Eq. (29) is similarly a consistency check on the reconstructed effective fluid, not a separately predicted result. The perturbation section (Sec. IV) states the standard Mukhanov-Sasaki equation (30) without deriving it from the action (7), and the matter-bounce spectrum (38) is the standard result for the assumed scale factor; this is an omitted-derivation or assumption gap, not a circular step, because no fitted parameter is renamed as a prediction and the spectrum is not used to define u or v. I do flag two serious non-circular concerns: (i) the ansatze (23) and (28) define v with an a-dot-dot/a term even though the variation leading to (14)-(15) assumed v = v(a, a-dot), making the background verification internally inconsistent; and (ii) the Mukhanov-Sasaki equation is imported without checking its validity in a higher-derivative, non-special-connection theory. These undermine the paper's reliability but do not constitute a reduction of any claimed prediction to its own input by construction. No load-bearing self-citation is present. Hence the circularity score is 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The background derivation depends on the adopted mini-superspace equations, the chosen perfect-fluid matter Lagrangian, and especially the unproven transfer of the Mukhanov-Sasaki formalism to this higher-derivative theory. The u and v ansätze are free-form inputs, making the bounce a reconstruction rather than a prediction.

free parameters (6)

sigma (bounce rate parameter) = not fixed; enters the scale factor and the power spectrum amplitude Pζ ∝ σG
Controls the bounce timescale and the amplitude of the computed spectrum; chosen by hand in the scale factor ansatz (20) and (25).
a_B (scale factor at bounce) = not fixed
Scale factor normalization at t=0; appears in the reconstructed coefficients β, δ and in ρB.
gamma = arbitrary constant
Coefficient in the u/v ansatz (Eqs. 22 and 27); left free by the reconstruction.
epsilon = arbitrary constant
Coefficient in the v ansatz (Eqs. 23 and 28); left free by the reconstruction.
zeta = related to ρB by ρB = (ζ + 4σ a_B^3)/(2κ^2)
Matter-bounce ansatz coefficient that sets the matter energy density at the bounce.
lambda = set to 1
Coupling constant in F(R,T) = R + λT; fixed by hand for simplicity and not varied.

assumptions (5)

domain assumption The mini-superspace action (11) and Friedmann equations (12)-(13) from Saridakis et al. [103] correctly describe Myrzakulov F(R,T) gravity with F = R + λT.
The paper adopts these equations without re-derivation; all background results rest on them.
domain assumption The matter sector is a perfect fluid with Lm = -ρm(a), and the matter and modified-gravity energy densities are separately conserved (Eq. 16).
Used in the Friedmann equations and in defining ρMG and pMG in Section II.B.
ad hoc to paper The standard Mukhanov-Sasaki equation (30) with z = a√(ρ_t + p_t)/(c_s H) governs scalar perturbations in this modified gravity.
No quadratic action or perturbation derivation is provided for Myrzakulov F(R,T); this is an unproven transfer of standard cosmological perturbation theory.
domain assumption The sound speed c_s ≈ 1 during the matter-dominated contracting phase.
Assumed for the perturbation solution (36)-(37) without computation from the theory's action.
ad hoc to paper The background bounce scale factors (20) and (25) are acceptable and the reconstructed u,v have no additional physical constraints.
The reconstruction imposes only the Friedmann equations; no Hamiltonian analysis, boundary conditions, or consistency conditions are given.

invented entities (1)

Ad hoc connection parametrization functions u(a, ˙a, ¨a) and v(a, ˙a) with polynomial forms
purpose: To force the Friedmann equations to admit the desired bounce scale factors; they encode the effects of the non-special connection.
The specific forms (22)-(23) and (27)-(28) are chosen by hand; no independent or falsifiable handle is provided outside the constructed examples.

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Pith. "Pith review of Non-singular bounce solutions in Myrzakulov $f(R,T)$ gravity." pith.science (2026). https://pith.science/paper/H52FQT4K

@misc{pith2026250118524,
  author       = {Pith},
  title        = {Pith review of: Non-singular bounce solutions in Myrzakulov $f(R,T)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H52FQT4K}},
  note         = {Machine review of arXiv:2501.18524}
}

abstract

We investigate the realization of a nonsingular bounce within the framework of Myrzakulov $ F(R,T) $ gravity. This modified gravitational theory uses a non-special connection that combines both curvature and torsion, giving rise to an effective sector that can easily satisfy the violation of the null energy condition. We suitably choose the functions that parametrize the connection in order to be able to produce simple and matter bounce scale factors at the background level. Finally, we examine the evolution of scalar perturbations through the bounce using the Mukhanov-Sasaki formalism, and we calculate the power spectrum.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

This scale factor satisﬁes the bouncing conditions, namely ˙ a(t) = 2 σtaB > 0 for t > 0 (expansion), and ˙a(t) < 0 for t < 0 (contraction)

Basic bounce A simple and commonly studied bouncing scale factor that ensures a n on-singular evolution of the universe is given by the form: a(t) = aB ( 1 + σt2) , (20) where aB is the scale factor at the bounce point and σ is a positive constant controlling the bounce duration. This scale factor satisﬁes the bouncing conditions, namely ˙ a(t) = 2 σtaB >...

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Matter bounce Let us now proceed to the realization of matter bounce. This solutio n has attracted the interest of the community from the time the works [ 121, 122] appeared, since it can give rise to an almost scale-invariant power s pectrum of primordial perturbations [ 123, 124]. 5 In this case the scale factor is a(t) = aB ( 1 + 3 2 σt2 ) 1/ 3 , (25) ...

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