Constraints on R\'{e}nyi Entropy through Primordial Big-Bang Nucleosynthesis and Baryogenesis

Ahmad Sheykhi; Ava Shahbazi Sooraki

REVIEW 4 major objections 5 minor 1 cited by

Constraints on R\'{e}nyi Entropy through Primordial Big-Bang Nucleosynthesis and Baryogenesis

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A constant Rényi entropy parameter cannot fit all three light-element abundances at once, so the model does not resolve the lithium problem.

desk verdict The first BBN bounds on the Rényi parameter are a legitimate idea, but the numerical constraints do not survive because the paper evaluates them at T=10 MeV while BBN happens below 0.1 MeV, where its own expansion breaks down. read the letter →

arxiv 2507.14250 v2 pith:ODUOTTUH submitted 2025-07-18 physics.gen-ph

Ahmad Sheykhi , Ava Shahbazi Sooraki This is my paper

classification physics.gen-ph

keywords Rényientropybig-bangnucleosynthesislithiumproblemmodifiedFriedmannequationsbaryogenesisnonextensivethermodynamicsentropiccosmologyprimordialabundances

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 asks whether replacing the usual area-law entropy of the cosmological horizon with Rényi entropy can change early-universe expansion enough to matter for big-bang nucleosynthesis and baryogenesis. It derives the modified Friedmann equations, written compactly as $H^2-\alpha\ln H^2=8\pi G\rho/3$ with $\alpha=\lambda\pi/G$, and uses observed helium-4, deuterium, lithium-7, and baryon-asymmetry data to bound the Rényi parameter $\lambda$. The central result is that helium-4 and deuterium are compatible with a common $\lambda$ around $10^{-85}$, while lithium-7 requires $\lambda\sim-2\times10^{-84}$, so no constant $\lambda$ fits all three abundances at once. The paper therefore concludes that the constant-$\lambda$ Rényi model is ruled out as a complete solution to the lithium problem, though it leaves the parameter ranges close enough to encourage temperature-dependent generalizations. A baryogenesis constraint from the same framework gives a much weaker bound, $\lambda/10^{-9}\lesssim0.022$, with partial overlap with the BBN ranges.

What carries the argument

The load-bearing object is the Rényi-corrected entropy of the apparent horizon, taken to first order as $S_h=S-(\lambda/2)S^2$ with $S=A/(4G)$, applied through the first law of thermodynamics at the apparent horizon. This produces the modified Friedmann equation $H^2-\alpha\ln H^2=8\pi G\rho/3$ with $\alpha=\lambda\pi/G$; a linearized Taylor expansion in $\alpha$ then yields the amplification factor $Z(T)=H/H_{\rm GR}$ that shifts the freeze-out of nuclear reactions. The same entropy correction is re-expressed as fluctuations $\delta\rho_R$ and $\delta p_R$ that drive the Ricci-scalar derivative, $\dot R_{\rm Renyi}\neq0$, supplying the out-of-equilibrium step for baryogenesis.

What would settle it

Run a full BBN nuclear network with the exact equation $H^2-\alpha\ln H^2=8\pi G\rho/3$ for $\lambda=-2\times10^{-84}$ and check whether lithium-7 then falls from the standard factor-2-to-4 overprediction down to the observed level while helium-4 and deuterium predictions remain within their quoted uncertainties; if not, the claimed incompatibility is an artifact of the linearization.

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Extended reading notes

Core claim

Within the thermodynamics-gravity correspondence, Rényi entropy $S_h=\lambda^{-1}\ln(1+\lambda S_{\rm BH})$ changes the entropy-area relation of the apparent horizon, and after linear expansion in $\lambda$ this replaces the standard Friedmann equation by $H^2-\alpha\ln H^2 = 8\pi G\rho/3$. The paper isolates the first-order correction to the Hubble rate, $H\simeq H_{\rm GR}(1+\alpha\ln H_{\rm GR}^2/(2H_{\rm GR}^2))$, encoded in an amplification factor $Z(T)$, and feeds this into established abundance fits. Matching the observed helium-4 abundance gives $-3.18\times10^{-85}\lesssim\lambda\lesssim1.19\times10^{-85}$; deuterium gives $-1.054\times10^{-85}\lesssim\lambda\lesssim7.96\times10^{-85}$; lithium-7 gives $-2.16\times10^{-84}\lesssim\lambda\lesssim-1.84\times10^{-84}$. The helium and deuterium windows overlap while the lithium window does not, which is the paper's core finding: a single, constant Rényi parameter cannot simultaneously account for all three measured light-element abundances. The same modified dynamics also yields a nonzero time derivative of the Ricci scalar, which satisfies the out-of-equilibrium Sakharov condition and leads to a baryon-asymmetry prediction that bounds the rescaled parameter $\tilde\lambda=\lambda/10^{-9}$ by $\tilde\lambda\lesssim0.022$.

Load-bearing premise

The quoted BBN bounds assume the linearized expansion rate $H\simeq H_{\rm GR}(1+\alpha\ln H_{\rm GR}^2/(2H_{\rm GR}^2))$ with $\delta\ll C$ and $\alpha/C\ll1$ throughout the nucleosynthesis epoch, and that evaluating all three element abundances at a single freeze-out temperature $T_f=10$ MeV is adequate; if the actual conditions violate these inequalities, the $\lambda$ windows shift.

Editorial extensions

If this is right

Consistency with helium-4 and deuterium pins $\lambda$ near $10^{-85}$ in natural units, so the Rényi correction to the expansion rate during BBN is tiny and standard BBN is nearly untouched.
The lithium-7 window is separated from the helium and deuterium windows, meaning constant-$\lambda$ Rényi entropy does not explain the factor 2–4 lithium discrepancy.
The baryogenesis channel constrains $\tilde\lambda=\lambda/10^{-9}\lesssim0.022$, which is many orders of magnitude weaker than the BBN bound but confirms the parameter must be small.
Increasing $\lambda$ raises the early-universe temperature at fixed cosmic time through the modified $t(T)$ relation.
First-order truncation is self-consistent: for the allowed $\lambda$, $\lambda S\ll1$, so higher-order entropy corrections shift predictions by less than 1%.

Reading between the lines

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

A natural follow-up, suggested by the paper's outlook, is a temperature-dependent $\lambda(T)$ chosen to act mainly at the lithium production epoch; that would make the model falsifiable against the full BBN network.
If the bounds are taken at face value, any nonextensive correction of this logarithmic form is numerically negligible for later cosmological observables such as CMB anisotropies or structure formation, unless the parameter is environment-dependent.
The analysis uses a single freeze-out temperature $T_f=10$ MeV for all elements and abundance fits linearized in $Z$; embedding the same $\lambda$ into a full nuclear network would sharpen the reported ranges and test the claimed overlap.

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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

Colleague,

The genuinely new item here is the first attempt to constrain the Rényi parameter λ from BBN. That part is worth knowing about. The paper also lays out the modified Friedmann equations in a readable way and points out that the Rényi correction reproduces MOND in the Newtonian limit. It is honest about the main outcome: a constant-λ Rényi model cannot simultaneously match D, 4He, and 7Li, and the outlook says so plainly. The citation pattern is fine; prior Rényi cosmology papers and the BBN fitting formulas are credited.

The soft spots are not minor. All the λ bounds come from Eq. (35), a first-order expansion that assumes δ≪C and α/C≪1. At T=10 MeV, where the bounds are evaluated, δ/C is already of order unity for the quoted λ~10^-85. The paper itself states in section IV.D that D and He form at T~0.1 MeV and Li at T~0.03–0.06 MeV. At those temperatures the correction term scales as T^-4, so the expansion is violently invalid. Worse, for the positive λ values allowed by Eqs. (48)–(49), the exact equation (28) has no real solution at BBN temperatures: the minimum of x−α ln x lies far above H_GR^2. So the positive half of the allowed λ region corresponds to no expanding solution at all. That makes the claimed disjointness of the Li range from the He/D ranges a numerical artifact of the temperature choice, not a physical property of the model.

There are also internal mismatches. Eq. (35) states H>H_GR for α>0, but during BBN ln H^2 is negative, so positive α gives H

The underlying idea may be salvageable. A proper reanalysis at the correct BBN temperatures, or with a justified effective freeze-out temperature, might well confirm the paper's negative conclusion. But the current numerical constraints are not trustworthy, and the paper contradicts its own temperature discussion. I would not send this version to referees. I would invite a major revision that redoes the BBN calculation at realistic temperatures and checks whether the linearized expansion is valid at all.

Best.

Referee Report

4 major / 5 minor

Summary. The manuscript derives modified Friedmann equations from Rényi entropy through the thermodynamics-gravity correspondence, then constrains the Rényi parameter λ using BBN abundances of 4He, D, and 7Li, together with baryogenesis. It reports overlapping λ windows for 4He and D, a disjoint window for 7Li, and concludes that the constant-λ Rényi model cannot resolve the lithium problem. It also derives a baryogenesis bound (λ̃ ≲ 0.022) and a modified time-temperature relation in which larger λ raises early-universe temperatures.

Significance. The paper is clearly structured and provides explicit formulas, including a transparent statement in Sec. IV.D that the constant-λ model is ruled out as a complete solution. It is a useful phenomenological starting point for testing generalized entropic cosmologies with BBN data. However, the quantitative constraints are built on a linearized expansion evaluated at a single freeze-out temperature, and the numerical derivations contain internal inconsistencies; the reported bounds and the overlap/non-overlap conclusion are not currently supported.

major comments (4)

[IV.A–IV.C, Eqs. (28), (35), (37)] The BBN constraints (48)–(50) are obtained by inverting Eq. (37) at T_f = 10 MeV, as stated in the captions of Figs. 1–3, but Sec. IV.D states that D and 4He form at T ∼ 0.1 MeV and 7Li at T ∼ 0.03–0.06 MeV. Because Z(T) − 1 ∝ T^{-4} ln(GT^4), the correction is 10^8–10^10 times larger at the actual formation temperatures. The Taylor expansion leading to Eq. (35) requires δ ≪ C and α/C ≪ 1; at T = 0.1 MeV and λ ∼ 10^{-85} one has α/C ∼ 10^6, so the expansion is invalid. Moreover, for positive λ within the ranges (48)–(49), the exact equation (28), H^2 − α ln H^2 = H_GR^2, has no real solution at T = 0.1 MeV: the function has a minimum α(1 − ln α) ≈ 5 × 10^{-45} GeV^2, far above H_GR^2 ≈ 2 × 10^{-53} GeV^2. The assertion in Sec. IV.C that the non-overlap 'remains' at all relevant temperatures is therefore unsubstantiated; the quoted positive-λ ranges do not correspond to real expanding solutions at BBN temperatures.
[IV.B, Eqs. (41)–(42) and (44)–(45)] The reported Z ranges do not follow from the displayed inputs. With η10 = 6, Eq. (41) gives Z − 1 = (0.2449 − 0.2485)/0.16 = −0.0225, i.e. Z ≈ 0.978, not 1.0475. Similarly, Eq. (44) with η10 = 6 gives 2.55/2.6 = [1/(2−Z)]^{1.6}, hence Z ≈ 0.988, not 1.062. The λ windows in Eqs. (48)–(49) and the claimed 4He/D overlap rest on this arithmetic and must be recomputed.
[IV.B, Eqs. (46)–(47)] The lithium constraint is structurally circular. Eq. (46) is a numerical fit normalized so that standard BBN at η10 = 6 gives YLi = 4.82, whereas the observational input is YLi = 1.6 ± 0.3; the standard-model discrepancy factor is ∼3. Equating these two quantities forces Z ≈ 2 by construction, so the resulting lithium window is necessarily disjoint from the near-unity Z windows of 4He and D. This is a restatement of the lithium problem rather than an independent test of Rényi cosmology, and it cannot support the paper's central no-overlap conclusion.
[V, Eqs. (67)–(69)] The baryogenesis section claims 'partial overlap' between the baryogenesis bound and the BBN bounds, but this is not the case. Eq. (67) with T_D = 3.3 × 10^16 GeV, M_* = 2.4 × 10^18 GeV and g_* ≈ 106 gives η ≈ 4.5 λ in GeV units; imposing η ≲ 9.9 × 10^{-11} yields λ ≲ 2.2 × 10^{-11}. The BBN bounds in Eqs. (48)–(50) are of order λ ∼ 10^{-85}. The ranges differ by more than seventy orders of magnitude, so there is no overlap, partial or otherwise. This inconsistency, together with the temperature issue above, means the baryogenesis and BBN constraints cannot both be valid as stated.

minor comments (5)

[IV.C, Eq. (50)] Eq. (50) contains a typographical error: '−2.16×−84' should read '−2.16 × 10^{-84}'.
[IV.B] The text refers to 'Tritium 7Li'; tritium is 3H, and the intended element is lithium-7.
[Figs. 1–3] The figures would benefit from labeling the horizontal axis in units of the quoted bounds (e.g., 10^{-85}) and from indicating how the 1σ theoretical errors in Eqs. (39), (43), and (46) were propagated into Eqs. (42), (45), and (47).
[Eq. (37)] The logarithm in Eq. (37) contains a dimensionful quantity; the expression is only meaningful after choosing units (e.g., GeV), which should be stated explicitly.
[V] The effective number of relativistic degrees of freedom is denoted g ∼ 10 in Eq. (37) but g_* ≈ 106 in Sec. V; the distinction should be stated explicitly to avoid confusion.

Circularity Check

1 steps flagged · score 4.0 of 10

Lithium non-overlap claim restates the known lithium discrepancy in Rényi coordinates; the He/D and baryogenesis constraints are otherwise externally benchmarked.

renaming known result [Sec. IV.B, Eqs. (46)-(47); Sec. IV.C, Eq. (50); Fig. 3 caption]
"The numerical best fit for the abundance of 7Li is given by YLi = 4.82(1 ± 0.1)[(η10 - 3(Z - 1))/6]^2 ... By requiring consistency between • The observational Lithium abundance YLi = 1.6 ± 0.3 • The numerical best-fit prediction ... Z = 1.960025 ± 0.076675."

At η10=6 and Z=1, Eq. (46) returns YLi=4.82, which is exactly the standard-BBN lithium value whose ratio to observed YLi=1.6 defines the known lithium discrepancy (the paper quotes Li|GR/Li|obs ∈ [2.4,4.3]). Solving Eq. (46) with the observed value forces Z to the square root of this discrepancy (≈1.85-1.96), and Eq. (50) is just that Z interval mapped through monotone relation (37) at Tf=10 MeV. The conclusion that the Rényi λ range for 7Li does not overlap the D/4He ranges is thus the lithium problem restated in λ coordinates, not an independent prediction of Rényi entropy.

full rationale

The modified Friedmann equations (25)-(27) are derived internally from thermodynamics-gravity and the Rényi entropy-area relation, not from BBN. The abundance formulas (39), (43), (46) are external empirical fits to standard BBN codes, and the data (42), (45), (47) are external observations. The baryogenesis bound (69) follows from an external gravitational-baryogenesis framework and observational η. There is no load-bearing self-citation chain or imported uniqueness theorem. The one partially circular element is the lithium non-overlap conclusion: because a constant λ enters BBN only through one-to-one factor Z(T) at a fixed freeze-out temperature, the non-overlap of the Li-derived λ range with the He/D ranges is structurally equivalent to the known fact that no single constant expansion-rate rescaling fits all three light elements. The paper's single-temperature treatment (Tf=10 MeV) is a validity issue rather than circularity, and the paper itself acknowledges the ranges will vary with temperature. Overall the central He/D constraints retain independent content, so circularity is partial.

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

The model has one free parameter λ plus several auxiliary fitted quantities (Z, Tf, η10). No new entities are invented beyond the Rényi entropy model, which is already in the literature. The leading circularity is in the lithium comparison, where the fit formula encodes the lithium discrepancy.

free parameters (4)

λ (Rényi parameter) = λ ~ 10^-85 in natural units from BBN; λ ≲ 0.022 (in kB units) from baryogenesis
The central model parameter, constrained by matching Eqs. (42), (45), (47) through Z(T) in Eq. (37). Observations do not predict it; the bounds are fit to abundance data.
Amplification factor Z = Z = 1.0475 ± 0.105 (4He); 1.062 ± 0.444 (D); 1.960025 ± 0.076675 (Li)
Z is solved from each abundance matching equation. For lithium, Z is effectively a chosen parameter that encodes the lithium discrepancy, since it is forced to ~1.96 by the empirical lithium formula.
η10 (baryon density parameter) = η10 = 6
Set to the standard value in all abundance fits; not varied, but the fits in Eqs. (39), (43), (46) are linearized around this value.
Tf (freeze-out temperature) = Tf = 10 MeV
Chosen for the figures and all λ bounds. Physical neutron freezeout is ~0.8 MeV, so this choice is not justified. Different temperatures would change the λ ranges, as the paper itself notes in Section IV.C.

assumptions (4)

domain assumption The entropy of the apparent horizon is given by Rényi entropy S_h = (1/λ)ln(1+λS_BH), and its cosmological version is used to first order.
Assumed in Sections II and III. This is the central physics input of the paper.
domain assumption The thermodynamics-gravity conjecture: the first law dE = T_h dS_h + W dV holds on the apparent horizon.
Used in Section III to derive the modified Friedmann equations. This is the standard framework from Jacobson and Cai-Kim, but it is an established principle.
domain assumption Baryogenesis is generated via the gravitational baryogenesis operator (1/M*) J^μ ∂_μ R with M* = (8πG)^{-1/2}.
Adopted from Davoudiasl et al. in Section V; the analysis does not derive the operator, it relies on the prior literature for the mechanism.
standard math The specific time-temperature relation is derived from entropy conservation s(T)a^3 = const in the radiation era.
Used in Section VI; standard cosmology input.

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Pith. "Pith review of Constraints on R\'{e}nyi Entropy through Primordial Big-Bang Nucleosynthesis and Baryogenesis." pith.science (2026). https://pith.science/paper/ODUOTTUH

@misc{pith2026250714250,
  author       = {Pith},
  title        = {Pith review of: Constraints on R\'enyi Entropy through Primordial Big-Bang Nucleosynthesis and Baryogenesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ODUOTTUH}},
  note         = {Machine review of arXiv:2507.14250}
}

abstract

The R\'{e}nyi entropy, a one-parameter generalization of Boltzmann-Gibbs entropy, offers a promising framework for probing quantum gravitational effects in cosmology. By modifying the entropy-area relation of the apparent horizon, R\'{e}nyi entropy can alter the expansion dynamics of the early universe, with potential implications for Big-Bang Nucleosynthesis. In this work, we derive the modified Friedmann equations within the R\'{e}nyi entropy paradigm and investigate their impact on the primordial abundances of light elements Deuterium ($D$), Helium-4 ($_{}^{4}\textit{He}$), Lithium-7 ($_{}^{7}\textit{Li}$) and baryogenesis. Using observational constraints from Planck, primordial abundance data and observational data on baryogenesis, we put stringent bounds on the R\'{e}nyi parameter. Furthermore, we explore whether R\'{e}nyi entropy corrections could mitigate the long-standing Lithium discrepancy. This study provides the first systematic constraints on R\'{e}nyi cosmology from BBN and highlights the role of nonextensive thermodynamics in early-universe physics. Our analysis shows that the obtained ranges for the R\'{e}nyi parameter $\lambda$ exhibit a overlap for the Helium-4 and Deuterium, but this overlapping region-despite a small discrepancy-is inconsistent with the range obtained from Lithium-7. This small mismatch between the ranges raises the possibility of alleviating the \textit{Lithium Problem} in the modified cosmology scenarios. Furthermore, we present the relationship between the cosmic time and temperature within the framework of R\'{e}nyi cosmology. We observe that an increase in the R\'{e}nyi parameter increase the temperature of the early universe.

Figures

Figures reproduced from arXiv: 2507.14250 by the authors.

**Figure 2.** displays the results for primordial deuterium (D) abundance using the relation in Eq. (45). The analysis reveals the following constraint on the R´enyi parameter −1.054 × 10−85 . λ . 7.96 × 10−85 . (49) -4 -2 0 2 4 0.0 0.5 1.0 1.5 2.0 λ ZD 1.062+0.444 1.062-0.444 FIG. 2: ZD vs R´enyi parameter λ. The observational interval is reported in (45). We have set η10 = 6 and the freeze-out temperature Tf = 10MeV . The per… view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: Baryon asymmetry parameter [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5: The behavior of the temperature vs time in the radi [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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