{"id":"e92f7a37-379a-422b-8d0a-513cf4f61ae4","arxiv_id":"2501.00084","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"An oscillating universe in Rastall gravity is presented, but the claimed cycles are not supported by the explicit scale factor and the 8.7 Gyr transit is an input parameter.","lead":"This paper uses a periodic deceleration parameter in Rastall gravity to claim a cyclic universe with phantom crossing and a cosmic transit at about 8.7 Gyr. The explicit scale factor does not support the claimed contraction phase, and the transit time is fixed by a hand-picked frequency parameter.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The explicit scale factor in Eq. (2) yields H>0 on its real domain; no contracting branch is provided, so the cyclic claim is unsupported.","rationale":"I read the paper in good faith and focused on the central claim of a cyclic universe with an oscillating Hubble parameter. The explicit solution in Eq. (2) is the foundation of the entire analysis. For the chosen parameters, m=1.55 is non-integer, so the scale factor a(t)=a0[tan(kt/2)]^(1/m) is real only on intervals where tan(kt/2)>0, i.e. 0<t<π/k (up to branch choices). On that domain, H=k/(m sin kt)>0, so the universe expands monotonically to a Big Rip. The paper neither derives nor constructs the contracting phase with H<0 that would be required for a cycle; the figures that show H oscillating are not supported by the given analytic solution. This is precisely the load-bearing weakness identified by the reader. The transit-time 'agreement' is also circular because k is chosen to force t_q=0=8.7 Gyr, but that is secondary to the absence of a real cyclic solution. The entropy limit statement, S~→-S0 rather than S0 as γ→0, is an additional error but not the main basis for rejection. Because the explicit solution does not deliver the central claim, the reader's REJECT verdict stands; no adjustment is needed.","tokens_in":13494,"tokens_out":4751,"duration_ms":46210,"concrete_test":"Compute the real branch of the scale factor a(t)=a0[tan(kt/2)]^(1/m) and H(t)=k/(m sin kt) at t=3π/(2k) for m=1.55, k=0.1, a0=1. At this time tan(kt/2)=tan(3π/4)=-1, so a(t) is not real for non-integer m, while H(t) would be negative only if sin kt<0. If no real positive a(t) yields H<0, the claimed cyclic contraction does not exist for the stated solution.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is a cyclic universe with H alternating sign (abstract, §1, Fig. 1h). The explicit solution, Eq. (2), is a(t)=a0[tan(kt/2)]^(1/m) with H=k/(m sin kt)+c. For m=1.55 (non-integer), the fractional power is real only when tan(kt/2)>0, i.e. 0<t<π/k. On this entire domain sin kt>0, so H>0 and a(t) grows monotonically from 0 to ∞ (Big Rip). The paper provides no branch, periodic extension, or alternative solution that yields H<0; the claimed oscillatory H in Fig. 1(h) is therefore asserted rather than derived. Additionally, the c in Eq. (2) is inconsistent: the given scale factor corresponds to c=0. Without a genuine contracting phase, the cyclic universe and the oscillation between positive and negative H do not exist, undermining the abstract and conclusions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":13754,"tokens_out":6110,"duration_ms":63226,"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":[{"comment":"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).","section":"Sec. 1, Eq. (2) and Fig. 1(h)"},{"comment":"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.","section":"Sec. 1, Eq. (4)"},{"comment":"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.","section":"Sec. 1, Eq. (2)"},{"comment":"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.","section":"Sec. 3, Eq. (24)"}],"minor_comments":[{"comment":"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.","section":"Sec. 1, Introduction"},{"comment":"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.","section":"Sec. 1, Eq. (2)"},{"comment":"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.","section":"Sec. 2, Eq. (14)"},{"comment":"There are several typographical errors, including \"Quintum\" for \"quintom\", \"beahvior\" for \"behavior\", and \"verses\" for \"versus\" in figure captions; these should be corrected.","section":"Sec. 3, after Eq. (24)"},{"comment":"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.","section":"Sec. 4, Eq. (27)"}],"recommendation":"reject","confidential_remarks":"The central claim of the manuscript is contradicted by its own explicit solution in Eq. (2), and the transit-time agreement is an artifact of the free parameter k. Because these issues affect the main physical conclusions, I do not see a path to revision within the scope of the current model."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThat 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.\n\nThe 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.\n\nMy 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.","headline":"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.","tokens_in":14250,"tokens_out":2747,"would_cite":false,"duration_ms":29056,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.50+h","04.20.Jb","95.35+d","98.80.Es"],"model":"deepseek-v4-flash","headline":"Rastall gravity yields a cyclic universe crossing the phantom divide","keywords":["cyclic universe","Rastall gravity","phantom divide","quintom","deceleration parameter","Big Rip","cosmic transit","modified gravity"],"falsifier":"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.","tokens_in":13316,"feed_emoji":"🌌","tokens_out":8016,"duration_ms":66849,"temperature":0.7,"pith_summary":"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.","feed_headline":"Rastall gravity model cycles the universe across the phantom divide","feed_subtitle":"The Hubble parameter alternates sign across cycles; the cosmic transit lands at 8.7 Gyr.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the Rastall field equations with non-conserved energy-momentum tensor that the whole model is built on.","marker":"[38]"},{"why":"Introduces the oscillating deceleration parameter $q=m\\cos kt-1$ and the resulting scale factor ansatz used here.","marker":"[72]"},{"why":"Establishes the Rastall gravity extension of $\\Lambda$CDM that this cyclic model is positioned within.","marker":"[61]"},{"why":"Supplies the observational transition redshift $z\\approx 0.64$ that fixes the model parameter $m=1.55$.","marker":"[73]"},{"why":"Gives the observed transition redshift range that the predicted $8.7$ Gyr transit is compared against.","marker":"[105]"},{"why":"Provides the modified Bekenstein-Hawking entropy expression used in the thermodynamic analysis.","marker":"[103]"}],"fun_headline_variants":["Rastall gravity cycles universe across phantom divide","Oscillating Rastall universe crosses phantom divide and big rip","Cyclic cosmos in Rastall gravity: phantom crossing at 8.7 Gyr","Hubble sign toggles: Rastall model crosses phantom line"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Rastall gravity cycles universe across phantom divide","Oscillating Rastall universe crosses phantom divide and big rip","Cyclic cosmos in Rastall gravity: phantom crossing at 8.7 Gyr","Hubble sign toggles: Rastall model crosses phantom line"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1335,"prompt_tokens":867,"completion_tokens":468,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":391}},"tokens_in":483,"tokens_out":468,"duration_ms":5163,"temperature":1.0,"reasoning_tokens":391,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:02:56.485860+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Rastall 1972, Generalization of the Einstein theory , Phys","cited_arxiv_id":null,"evidence_quote":"Provides the Rastall field equations with non-conserved energy-momentum tensor that the whole model is built on."},{"cited_title":"Shen and L","cited_arxiv_id":null,"evidence_quote":"Introduces the oscillating deceleration parameter $q=m\\cos kt-1$ and the resulting scale factor ansatz used here."},{"cited_title":"Akarsu et al","cited_arxiv_id":null,"evidence_quote":"Establishes the Rastall gravity extension of $\\Lambda$CDM that this cyclic model is positioned within."},{"cited_title":"Capozziello et al","cited_arxiv_id":null,"evidence_quote":"Supplies the observational transition redshift $z\\approx 0.64$ that fixes the model parameter $m=1.55$."},{"cited_title":"Y ang and Y","cited_arxiv_id":null,"evidence_quote":"Gives the observed transition redshift range that the predicted $8.7$ Gyr transit is compared against."},{"cited_title":"Bamba et al","cited_arxiv_id":null,"evidence_quote":"Provides the modified Bekenstein-Hawking entropy expression used in the thermodynamic analysis."}],"review_version":1}