An oscillating Rastall universe crossing the phantom divide line

Anirudh Pradhan; Archana Dixit; Nasr Ahmed

REVIEW 4 major objections 5 minor 108 references

An oscillating Rastall universe crossing the phantom divide line

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Rastall gravity yields a cyclic universe crossing the phantom divide

desk verdict The cyclic claim is contradicted by the paper's own scale factor, which gives H > 0 on its entire real domain; the transit time is parameter-fitting, and the entropy limit is wrong. read the letter →

arxiv 2501.00084 v3 pith:T24MLRF3 submitted 2024-12-30 gr-qc

Nasr Ahmed , Anirudh Pradhan , Archana Dixit This is my paper · ORCID

classification gr-qc PACS 04.50+h04.20.Jb95.35+d98.80.Es

keywords cyclicuniverseRastallgravityphantomdividequintomdecelerationparameterBigRipcosmictransitmodified

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 claims that in Rastall gravity, the oscillating deceleration parameter $q(t)=m\cos kt-1$ produces a flat, cyclic universe whose Hubble parameter alternates between positive and negative values from one cycle to the next. The scale factor grows to a future Big Rip, and the equation of state parameter crosses the phantom divide line ($\omega=-1$) from quintessence into the phantom regime, giving quintom behavior. The model places the deceleration-to-acceleration transition at approximately $8.7$ Gyr for $m=1.55$, $k=0.1$, which agrees with recent estimates of the transition redshift. If correct, the model offers a modified-gravity route to cyclic cosmology that reproduces the observed late-time kinematics and predicts a Big Rip endstate.

What carries the argument

The central object is the periodic deceleration parameter $q(t)=m\cos kt-1$, whose sign flips generate alternating deceleration and acceleration phases within each cycle. Integration of this $q$ yields the Hubble parameter and scale factor that enter the Rastall field equations, from which $\rho$, $p$, and $\omega$ are computed. The same ansatz drives the phantom divide crossing: as $\omega$ decreases below $-1$, the model enters a phantom era that ends in a Big Rip. The thermodynamic analysis uses the Rastall-modified Bekenstein-Hawking entropy expression $\tilde S = \left(\frac{1+2\gamma}{4\gamma-1}\right)S_0$ with $S_0=A/4$, tracking the horizon radius, temperature, and work density.

What would settle it

A direct calculation of $H(t)=k/(m\sin kt)$ for the stated parameters over the interval $0<t<\pi/k$ shows that $H$ is never negative; if no additional branch or periodic extension is supplied, the claimed contraction phases do not exist in the given solution. One could also compare the model's predicted $H(z)$ to cosmic chronometer or Pantheon+ data to test whether the oscillating ansatz fits as well as $\Lambda$CDM.

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Extended reading notes

Core claim

The paper's central claim is that the Rastall field equations, with a non-conserved energy-momentum tensor characterized by a coupling $\lambda$, admit a flat cyclic cosmological solution when the deceleration parameter is chosen as $q(t)=m\cos kt-1$. Integrating this ansatz gives $H(t)=k/(m\sin kt)+c$ and $a(t)=a_0[\tan(kt/2)]^{1/m}$, from which the authors derive an energy density, pressure, and equation of state that start in a radiation-like phase, pass through dust and dark energy, and cross the phantom divide into $\omega<-1$. The Hubble parameter is claimed to oscillate in sign over successive cycles, the entropy of the apparent horizon remains positive, and the deceleration-to-acceleration transition occurs at $t\approx 8.7$ Gyr for the chosen parameters $m=1.55$, $k=0.1$, consistent with the observationally inferred transition redshift. Causality is reported to hold except near the initial and future Big Rip singularities.

Load-bearing premise

The load-bearing premise is that the scale factor $a(t)=a_0[\tan(kt/2)]^{1/m}$, with $H(t)=k/(m\sin kt)$, can be interpreted as a cyclic universe that includes contracting phases with $H<0$, even though the formula as written gives $H>0$ for all $t$ in $(0,\pi/k)$ and monotonic growth to a Big Rip.

Editorial extensions

If this is right

If the model is correct, Rastall gravity admits a cyclic universe with quintom phantom-crossing behavior without introducing scalar fields.
The cosmic transit at $t\approx 8.7$ Gyr falls inside the observationally inferred range of about 7.5 to 9.8 Gyr, so the model is compatible with late-time kinematic data.
The model predicts a future Big Rip, meaning the current accelerated expansion is not the final state.
The thermodynamic analysis predicts a positive entropy with a growing branch during expansion, a nontrivial consistency check for cyclic cosmologies.

Reading between the lines

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

The paper does not exhibit the branch on which $H<0$; a natural completion would be a periodic extension that connects the Big Rip to a subsequent contracting phase, and checking whether such an extension solves the Friedmann constraints would settle the cyclic claim.
The same $q(t)$ ansatz could be transplanted to other modified theories with non-conserved matter sectors, and would be expected to produce similar phantom crossings if the mechanism is generic.
Fitting the model to Pantheon+ or cosmic chronometer data would test whether $m\approx 1.55$ and $k\approx 0.1$ are actually preferred over $\Lambda$CDM, rather than merely tuned to match one redshift.
The phantom crossing here is built in by the periodic ansatz rather than derived from field dynamics, so the model's quintom behavior is kinematic in origin; whether that counts as an explanation depends on whether the $q$ parametrization can be traced to a Lagrangian.

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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 asked for my read on this one. The rejection is right, and the stress-test note lands: the paper's central claim—a cyclic universe with H oscillating between positive and negative—is not supported by its explicit solution. The scale factor in Eq. (2), a(t)=a0[tan(kt/2)]^{1/m}, is real only for 0 < t < π/k (with m=1.55), and on that entire interval H = k/(m sin kt) > 0. There is no contracting branch, no periodic extension, and the constant c in the Hubble expression is inconsistent with the given a(t). The H(t) plot showing negative values comes from extending the solution past the domain where it is real. Without a genuine contraction phase, there is no cycle, and the phantom-crossing story becomes just a re-parametrization of a single expanding phase ending in a Big Rip.

That is not to say the paper is worthless. The thermodynamic and energy-condition analyses are standard but competently presented, and the authors are honest about the ongoing debate over whether Rastall gravity is equivalent to GR. They cite both Visser and Golovnev, which is more than many papers do.

The soft spots are severe, not minor. The entropy claim is mathematically false: the coefficient (1+2γ)/(4γ−1) evaluated at γ→0 is −1, not +1, so the stated limit does not recover S0. The "prediction" of an 8.7 Gyr transit is not a prediction—it is the result of choosing k=0.1 and m=1.55 to match the observed transition redshift and age. And the geometry is imported wholesale from Shen & Zhao (2014); Rastall only enters through redefinitions of ρ and p, with the paper itself reporting a tiny difference from GR. The novelty is thin, but the fatal issue is the internal contradiction: the model as written cannot cycle.

My bottom line: desk reject. This is not a paper a serious referee should spend time on, because the headline result fails on the authors' own equations. If they resubmit with a genuine contracting solution, a proper derivation of the entropy limit, and an actual fit instead of parameter-picking, it might become a publishable minor contribution, but not in its current form.

Referee Report

4 major / 5 minor

Summary. The paper presents a flat FLRW model in Rastall gravity with an oscillating deceleration parameter q = m cos kt − 1, and claims that this produces a cyclic universe with a Hubble parameter that alternates between positive and negative values, quintom behaviour with phantom-divide crossing, a deceleration-to-acceleration transit at about 8.7 Gyr, and a future Big Rip. The authors then analyze the apparent horizon, work density, entropy, causality, and energy conditions for this background. The central model is defined by Eq. (2), with the scale factor a(t)=a0[tan(kt/2)]^{1/m} and H=k/(m sin kt)+c, and the transition time is given by Eq. (4).

Significance. If correct, the model would be a simple Rastall-gravity example of a cyclic universe that also crosses the phantom divide and has a transition time compatible with cosmic-chronometer and supernova estimates. However, the central construction is mathematically inconsistent: the explicit scale factor does not support the claimed negative-H phases, the quoted transition time is set by an unconstrained parameter choice rather than derived, and the entropy limit used for the thermodynamics is incorrect. The paper therefore does not provide a valid demonstration of its headline claims, and the subsidiary thermodynamic and energy-condition results are anchored to an invalid background. The paper does not include machine-checked proofs or reproducible code, and the only falsifiable prediction, the 8.7 Gyr transit, is an artifact of choosing k=0.1.

major comments (4)

[Sec. 1, Eq. (2) and Fig. 1(h)] The claimed cyclic universe is not supported by the explicit solution. For m=1.55, a(t)=a0[tan(kt/2)]^{1/m} is real only when tan(kt/2)>0, i.e. on 0<t<π/k, and on that entire interval sin kt>0, so H=k/(m sin kt)>0 and a(t) increases monotonically from 0 to ∞. No branch, periodic extension, or alternative solution is given that would produce H<0, so Figure 1(h) and the abstract's statement that the Hubble parameter oscillates between positive and negative values are not consequences of Eq. (2).
[Sec. 1, Eq. (4)] The transit time t≈8.7 Gyr is not a prediction but an output of the parameter choice k=0.1. Equation (4) gives t=(1/k)cos^{−1}(1/m); for m=1.55, the choice k≈0.1002 is required to obtain 8.7 Gyr. Since k is a free parameter with no fitting or observational constraint stated, the agreement with observed transition-redshift estimates is circular rather than a test of the model.
[Sec. 1, Eq. (2)] The expression H=k/(m sin kt)+c is inconsistent with the stated scale factor. Direct differentiation of a=a0[tan(kt/2)]^{1/m} gives H=k/(m sin kt), i.e. c=0. If c≠0 is intended, the relation between H and q is not satisfied, and the subsequent plots and dynamical equations that use H are not tied to the scale factor. The manuscript should either set c=0 explicitly or derive the correct integration constant consistently.
[Sec. 3, Eq. (24)] The modified entropy formula S̃=(1+2γ)/(4γ−1)S0 does not reduce to the Bekenstein-Hawking entropy S0 in the limit γ→0, as claimed; it gives −S0, and it diverges at γ=1/4. This invalidates the stated GR-limit check and leaves the entropy analysis without the advertised physical interpretation.

minor comments (5)

[Sec. 1, Introduction] The sentence beginning "the generalized f(R, T) gravity where T is the trace..." is incomplete and should be completed or merged with the preceding sentence.
[Sec. 1, Eq. (2)] The symbol c in Eq. (2) is introduced without definition and is not used consistently; please remove it or provide a derivation that includes the integration constant properly.
[Sec. 2, Eq. (14)] Equation (14) mixes lower-case k and upper-case K in the denominator; the same symbol should be used throughout, and the relation to Eq. (8) should be checked.
[Sec. 3, after Eq. (24)] There are several typographical errors, including "Quintum" for "quintom", "beahvior" for "behavior", and "verses" for "versus" in figure captions; these should be corrected.
[Sec. 4, Eq. (27)] The causality condition is only presented graphically; an analytic statement of the domain where 0≤dp/dρ≤1 holds would strengthen the claim that causality is satisfied except near the singularities.

Circularity Check

1 steps flagged · score 6.0 of 10

The advertised 8.7 Gyr transit is not a model prediction: m=1.55 places q=0 at the observed z≈0.64 and k=0.1 is chosen so that arccos(1/m)/k equals 8.7 Gyr, which the paper then cites as agreement with observations.

fitted input called prediction [Section 1, Eq. (4); echoed in Abstract and Conclusions]
"Observations suggests that the signature change of q occurs at z = 0 . 64 for m = 1 . 55 [73, 74, 75, 76]. Since cosmic transit happens when q = 0 ( i.e. ¨a = 0), we have tq=0 = 1 k cos−1 1 m ≈ 8. 7 Gyr f or m = 1. 55, k = 0. 1 (4)"

The m and k values are not independent outputs. With q=m cos kt−1, q=0 gives z_t=[(m+1)/(m−1)]^{1/(2m)}−1; imposing the observed z_t=0.64 fixes m≈1.55. Then t_q=0=arccos(1/m)/k, and k=0.1 is picked so that t_q=0≈8.7 Gyr. The abstract's 'expected to occur at approximately 8.7 Gyr' and the conclusion's comparison with the independent 7.5–9.8 Gyr estimates are therefore not a forecast of the model but a restatement of the chosen parameters, i.e., the fitted input is renamed as a predicted epoch.

full rationale

The clearest circular step is the 8.7 Gyr transit epoch. The paper adopts the periodic deceleration ansatz q=m cos kt−1 (Eq. 2, from ref. [72]) and then integrates to H and a. For this ansatz, q=0 at t=arccos(1/m)/k; the quoted z=0.64 transition fixes m≈1.55, and k=0.1 is chosen to turn 0.87/k into 8.7 Gyr. The later 'agreement' with observational transition-age estimates is thus built in by the parameter choice rather than supplied by the model. The other headline features—'cyclic' sign flips of q and the phantom crossing of ω(t)—are consequences of the openly declared q ansatz rather than independent Rastall predictions; I do not count those as additional circularity because they are model assumptions stated as such ('ad hoc periodic parametrization ... [72]'), not derived results disguised as predictions. The claimed H<0 contraction phase is not a circularity but an internal inconsistency: the quoted scale factor a=a0[tan(kt/2)]^{1/m} gives H=k/(m sin kt)>0 on its real domain, and the '+c' Hubble form does not satisfy q=m cos kt−1 unless c=0; this is a correctness problem. The self-citations [57,58] for oscillatory Rastall solutions are present but not load-bearing, since the oscillation is imported from the q ansatz rather than from those fixed-point results. Net: one central 'prediction' reduces by construction to fitted/selected parameters, giving partial circularity score 6.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The model's assumptions are the periodic q ansatz from ref. [72], the flat FLRW metric, Rastall's non-conservative field equations, a barotropic fluid, and an imported entropy formula. All free parameters are hand-picked or chosen to match the very observations the paper later claims to predict. No new entities are introduced.

free parameters (6)

m = 1.55
Chosen so the transition redshift z_t≈0.64 in Eq. (3) matches observational estimates cited in refs [73-76]; controls the amplitude of the q oscillation.
k = 0.1
Set in Eq. (4) so that the transit time t_{q=0}≈8.7 Gyr matches the desired value; acts as the cosmic frequency.
lambda (Rastall parameter) = 1.4
Hand-picked in the figure captions; no observational or theoretical constraint given.
K = 0.01
Coupling constant in Rastall field equations, hand-picked; appears in denominators of ρ and p.
a0 = 1
Scale factor normalization in Eq. (2).
gamma = not specified
Appears in the EoS p=(γ-1)ρ and in the entropy formula (24); no value is assigned, but the entropy behavior and the γ→0 limit are discussed.

assumptions (5)

ad hoc to paper Periodic deceleration parameter q = m cos kt - 1
Introduced in Section 1 after Eq. (2), following Shen and Zhao [72]. All oscillatory behavior and phantom crossing derive from this choice.
domain assumption Flat FLRW spacetime (κ=0)
Assumed in Section 2 with the metric (7), justified by observation.
domain assumption Rastall field equations with non-conserved energy-momentum tensor
Adopted in Section 2, Eqs. (5)-(6); the paper notes the ongoing dispute about equivalence with GR.
domain assumption Perfect fluid EoS p=(γ-1)ρ with 2/3 ≤ γ ≤ 2
Stated near Eq. (13) in Section 2; used to derive ˙H.
domain assumption Modified Bekenstein-Hawking entropy S = ((1+2γ)/(4γ-1)) S0
Imported in Section 3, Eq. (24) from refs [103,104]; the claimed γ→0 limit is incorrect.

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Cite this review

Pith. "Pith review of An oscillating Rastall universe crossing the phantom divide line." pith.science (2026). https://pith.science/paper/T24MLRF3

@misc{pith2026250100084,
  author       = {Pith},
  title        = {Pith review of: An oscillating Rastall universe crossing the phantom divide line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T24MLRF3}},
  note         = {Machine review of arXiv:2501.00084}
}

abstract

A cyclic flat universe with quintom behaviour and future big rip has been presented in the framework of Rastall gravity, which is an extension of the standard $\Lambda$CDM model. The Hubble parameter oscillates periodically between positive and negative values from one cycle to the next. Cosmic transit has been simulated through an oscillating time-dependent deceleration parameter, and is expected to occur at approximately $ 8.7~~ \text{Gyr}$. The causality is satisfied all the time except near the initial singularity and the future Big Rip singularity.The apparent horizon, entropy and other thermodynamical quantities associated to the current model have been analyzed. Energy conditions have been investigated.

Figures

Figures reproduced from arXiv: 2501.00084 by the authors.

**Figure 2.** (a) Work density verses redshift. (b) Radius of the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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