Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology

Ahmad Sheykhi; Ava Shahbazi Sooraki

REVIEW 6 major objections 6 minor 96 references

Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology

T0 review · 6 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Yukawa cosmology is testable in the early universe: Big Bang nucleosynthesis and WIMP freeze-out together pin the strength of its extra force to a narrow interval around zero.

desk verdict Coherent entropic-gravity framework, but the headline BBN and WIMP alpha bounds contain arithmetic errors and should not be quoted as they stand. read the letter →

arxiv 2608.09390 v1 pith:VCLF2VTA submitted 2026-08-10 astro-ph.CO gr-qc

Ava Shahbazi Sooraki , Ahmad Sheykhi This is my paper · ORCID

classification astro-ph.COgr-qc PACS 98.80.Ft95.35.+d04.50.Kd

keywords YukawacosmologymodifiedgravityBigBangnucleosynthesislithiumproblemWIMPfreeze-outdarkmatterrelicabundanceapparenthorizonthermodynamicsentropic

topics Dark Matter

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 tries to establish that a Yukawa modification of Newtonian gravity—an extra force of strength α and range λ—leaves measurable fingerprints in the early universe, and that Big Bang nucleosynthesis and WIMP dark-matter freeze-out can pin α down. Starting from a Yukawa-modified entropy on the apparent horizon, the authors derive modified Friedmann equations whose leading effect is to rescale the expansion rate by 1/√(1+α). Comparing the predicted abundances of 4He, deuterium, and 7Li with observations, they find 4He and deuterium agree for -0.24≲α≲0.12, while 7Li demands a disjoint negative band, so the lithium problem persists in this model. Analyzing thermal WIMP freeze-out with the modified Hubble rate yields an independent, much tighter range, -0.017≲α≲0.018, from the observed dark-matter relic density. If correct, these results show early-universe observables can serve as complementary probes of modified gravity.

What carries the argument

The load-bearing mechanism is the Yukawa-modified horizon entropy and the amplification factor derived from it. The paper obtains S_h by equating the entropic force FΔx = TΔS to the force from the Yukawa potential Φ(r) = -GM/r (1 + $αe^{{-r/λ}}$), integrates dS_h/dR, and then feeds the result into the first law dE = TdS + WdV on the apparent horizon. This produces a modified Friedmann equation whose leading early-universe term is an effective Newton constant G/(1+α), absorbed into Z(T) = H/H_GR. For WIMP freeze-out, the same rescaling is equivalent to replacing the annihilation cross section σ0 by √(1+α)σ0, which moves the relic abundance as Ω_χh²(α) ≈ Ω_χh²(0)/√(1+α).

What would settle it

A primordial helium-4 measurement with uncertainty well below 0.004 would test whether the inferred Z ≈ 1.05 expansion rate is real; if Z moves away from 1.0475, the α band from Eq. (49) shifts. Independently, recomputing WIMP freeze-out while keeping the logarithmic term instead of using H ≈ H_GR/√(1+α) would show whether the tight window -0.017≲α≲0.018 survives the approximation.

Watch

Extended reading notes

Core claim

The central claim is that the Yukawa-corrected entropy of the apparent horizon, S_h = πR²/G - (2πα/G)$e^{{-R/λ}}$(R²+3λR+3λ²), turns the Friedmann equation into H² - Γ ln(H² + k/a²) ≈ 8πG_eff ρ/3, with effective Newton constant G/(1+α). In the early universe the logarithmic term is small, so the expansion rate is H ≈ H_GR/√(1+α). That rescaling changes the synthesis yields of light nuclei and the thermal relic abundance of WIMPs. Confronting the modified abundances with observed 4He, deuterium, and 7Li data gives mutually consistent bounds on α from 4He and deuterium but a disjoint negative band from 7Li, so no single α resolves the lithium problem. For a benchmark 100 GeV thermal WIMP, the relic-density constraint Ω_CDM h² = 0.120 ± 0.001 translates into -0.017≲α≲0.018, a window fully inside the BBN-allowed region.

Load-bearing premise

The load-bearing premise is that the entropy of the apparent horizon is the Yukawa-modified expression S_h = πR²/G - (2πα/G)$e^{{-R/λ}}$(R²+3λR+3λ²) obtained by equating the entropic force to the Yukawa force and inserting it into the first law dE = TdS + WdV, with the freeze-out analysis additionally neglecting the logarithmic term in the modified Friedmann equation.

Editorial extensions

If this is right

A Yukawa coupling outside -0.017≲α≲0.018 would over- or under-produce the dark-matter relic density for a standard thermal WIMP.
The model cannot fix the cosmological lithium problem: helium and deuterium favor small |α|, while 7Li wants α around -0.74, and the bands do not overlap.
Positive α slows the early expansion and leaves the universe hotter at a fixed cosmic time; negative α does the opposite.
The relic-density formula means a modified expansion rate is indistinguishable from a rescaling of the WIMP annihilation cross section, so freeze-out alone cannot separate the two effects.
Combining BBN and freeze-out gives a single consistent window, making early-universe data a genuine test of Yukawa-type modified gravity.

Reading between the lines

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

Because the freeze-out bound is an order of magnitude tighter than the BBN bounds, a future measurement of Ω_CDM h² with smaller error—say from CMB-S4 or the Simons Observatory—would sharpen α more than any single abundance measurement.
The paper's benchmark uses a 100 GeV s-wave WIMP; for p-wave annihilation (l = 1) the freeze-out temperature shifts slightly, and the same derivation would give a modestly different α window, a testable extension.
If the logarithmic term in the modified Friedmann equation is kept during freeze-out rather than dropped, the α window could shift; computing that next-order correction would check whether the claimed precision is stable.
The predicted relation between mχ and σ0 at fixed α, shown in the paper's Figures 5 and 6, can be compared with direct-detection and collider constraints to identify allowed WIMP models.

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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

Dear Colleague,

Quick take: the paper is a coherent entropic-gravity derivation applied to BBN and WIMP freeze-out, but the headline alpha constraints are not supported by the paper's own equations. The helium and deuterium conversions have sign/magnitude errors, and the WIMP bound is benchmark-conditional. Do not quote the numbers in this version.

What's actually new: the modified Friedmann equation from the Yukawa-corrected entropy, Eq. (27), with the logarithmic term, is not in the earlier Jusufi et al. paper, and the derivation via the first law is different from their entropic-force route. The WIMP section correctly shows that the early-universe effect is just a rescaling of the Hubble rate by 1/sqrt(1+alpha), so the freeze-out analysis reduces to a rescaled cross section; that's a clean way to see the degeneracy. The qualitative conclusion that Yukawa cosmology does not resolve the lithium problem is robust to the arithmetic errors, since the lithium-favored alpha interval is far from the helium/deuterium region regardless of sign.

Soft spots, in order of severity. First, the BBN abundance-to-Z fits are miscalculated. Eq. (43) with their own inputs gives Z = 0.9775 +/- 0.025, not 1.0475 +/- 0.105; the sign of Z-1 is wrong. That flips the sign of the preferred alpha for helium, turning alpha ~ +0.05 into alpha ~ -0.09. The deuterium conversion in Eq. (45) is also wrong (it gives Z close to 1, not 1.06). Second, Eq. (81) is an arithmetic slip: solving 0.119 <= 0.12/sqrt(1+alpha) <= 0.121 gives alpha in [-0.017, 0.017], not the printed [-0.15, 0.19]. The later Eq. (90) is consistent with the correct solution, so this one is internally inconsistent rather than load-bearing. Third, the WIMP bound is conditional: it normalizes the alpha = 0 relic density to exactly Omega h^2 = 0.12, so the result is a consistency test of a chosen benchmark, not an independent measurement of alpha. No sensitivity scans over lambda, eta10, or the WIMP mass/cross-section are given. The lambda dependence is negligible in the radiation era, so the constraints degenerate with a constant rescaling of G; that's worth saying explicitly but is not by itself a flaw.

Who it's for: readers working on entropic gravity or early-universe tests of modified gravity. With corrected arithmetic and some sensitivity scans it could be a solid subfield paper. As submitted, the central quantitative claims are unreliable.

Recommendation: send to referees, but expect major revision. A referee should request recomputation of the Z-to-alpha maps and a clear statement that the WIMP constraint is benchmark-conditional. I would not cite the current numbers.

Referee Report

6 major / 6 minor

Summary. The paper derives modified Friedmann equations for a Yukawa-type gravitational potential by combining Verlinde's entropic-force scenario with the first law of thermodynamics on the apparent horizon. It then uses the resulting early-universe expansion rate, parameterized by an amplification factor Z(T), to constrain the Yukawa coupling α from primordial 4He, deuterium, and 7Li abundances, and separately from thermal WIMP freeze-out against the Planck relic density. The paper claims mutually consistent 4He and deuterium bounds (-0.24 ≲ α ≲ 0.12), a disjoint 7Li interval (-0.76 ≲ α ≲ -0.72), a WIMP freeze-out bound (-0.017 ≲ α ≲ 0.018), and a modified time-temperature relation. The qualitative conclusion is that Yukawa cosmology is testable and that the lithium problem persists.

Significance. If the quantitative results held, this would be a useful early-Universe test of Yukawa-modified gravity, complementing solar-system and galactic constraints. The manuscript has genuine strengths: the entropy correction in Eq. (11) integrates consistently with the assumed Yukawa force, the derivation of the modified Friedmann equation is explicit, and the WIMP freeze-out analysis gives a transparent analytic rescaling of the relic abundance. However, the headline BBN constraints rest on several internal arithmetic errors that reverse the sign of the central value and change the widths by large factors. The dark-matter constraint is also conditional on a benchmark normalization. These issues make the printed central quantitative claims unreliable, although the qualitative conclusions—small |α| allowed by He/D and a persistent lithium discrepancy—are likely to survive a corrected calculation.

major comments (6)

[§III.B, Eq. (44)] Equation (44) is arithmetically and physically wrong. Solving Eq. (43) with η10 = 6 and Yp = 0.2449 ± 0.004 gives 0.2449 = 0.2485 + 0.0016[100(Z-1)], hence Z = 0.9775 ± 0.0253, not Z = 1.0475 ± 0.105. The sign matters: the observed helium mass fraction is below the standard-model prediction, so Z < 1 is required, whereas Eq. (44) has Z > 1. Since Z ≈ 1/sqrt(1+α) at the adopted λ and T, the corrected central value maps to α ≈ +0.047, opposite in sign to the paper's implicit α ≈ -0.089. The later helium bound in Eq. (49), -0.24 ≲ α ≲ 0.12, is therefore not supported; the corrected interval is approximately -0.006 ≲ α ≲ 0.103.
[§III.B, Eq. (46)] The deuterium conversion has the same sign problem. With η10 = 6, Eq. (45) reduces to YDp = 2.6(1 ± 0.06)(2-Z)^-1.6. Matching the central observed value YDp = 2.55 gives Z ≈ 0.988, not 1.062 ± 0.444. At the paper's own central value Z = 1.062, the formula would predict YDp ≈ 2.86, well above the observed value. Propagating the observational and fit uncertainties gives approximately Z = 0.988 ± 0.039, which maps to α ≈ [-0.051, 0.110], not the -0.5 ≲ α ≲ 1.5 quoted in Eq. (50).
[§III.B, Eq. (48)] The lithium value in Eq. (48) is also internally inconsistent with Eq. (47). For η10 = 6, Eq. (47) gives YLi = 4.82(1 ± 0.1)(1.5 - 0.5Z)^2. The central observed value YLi = 1.6 requires Z ≈ 1.85, not Z = 1.960025. The paper's quoted Z corresponds to a predicted lithium abundance of about 1.30, which is outside the 1σ observational range. The resulting α interval in Eq. (51) is roughly [-0.75, -0.67] rather than [-0.76, -0.72]; the lithium/He/D non-overlap conclusion may survive, but the quantitative statement in Eq. (51) needs recomputation.
[§IV.C, Eq. (81)] Equation (81) is an arithmetic error. Solving 0.119 ≤ 0.12/sqrt(1+α) ≤ 0.121 yields α ≈ [-0.017, 0.017], not [-0.15, 0.19]. This matters because Eq. (82) uses Eq. (81) to form the combined range -0.15 ≲ α ≲ 0.12. The later, more careful result in Eq. (90), -0.017 ≲ α ≲ 0.018, is consistent with the correct solution of Eq. (80), so the final WIMP constraint is not invalidated; nevertheless, the intermediate result and the combined bound in Eq. (82) are unsupported as written.
[§IV.C, Eqs. (79)-(90)] The dark-matter constraint is conditional rather than an independent measurement. The paper normalizes Ωχh2(0) = 0.12 for a chosen benchmark (mχ = 100 GeV, σ0 in the range of Eq. (78)), and then derives α from the ratio Ωχh2(α)/Ωχh2(0). Since mχ and σ0 are not fixed by external data, this is a consistency test for the benchmark, not a direct bound on α alone. The abstract's phrase 'an independent constraint' overstates this. The paper should either scan the allowed (mχ, σ0) parameter space or clearly state that the bound is benchmark-dependent; Figs. 5 and 6 show how much the implied σ0 shifts with α.
[§III.C, Figs. 1-3 and Eqs. (49)-(51)] No sensitivity scan over λ, η10, or the relevant BBN temperature is provided. All BBN bounds are evaluated at a single choice λ = 10^26 m and T = 1 MeV, and the logarithmic term in Eq. (40) is assumed negligible. The text acknowledges that the ranges 'may shift at different energy scales,' but this is asserted rather than demonstrated. Given that the central BBN constraints are already being recomputed, the authors should show at least a two-dimensional robustness check in (λ, T) before claiming that the 4He/D and 7Li intervals are mutually inconsistent across the BBN epoch.

minor comments (6)

[Abstract] The sentence 'This indicate that Yukawa cosmology cannot resolve the Lithium Problem' has a subject-verb agreement error; it should be 'This indicates'.
[Title and header] The title in the running text reads 'Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out a s Probes of Yukawa Cosmology'; the stray space in 'a s' should be removed.
[§III.B, Eq. (42)] The helium fit in Eq. (42) includes a theoretical uncertainty ±0.0006, but the propagation into Eq. (44) does not combine this uncertainty with the observational ±0.004. After the arithmetic is corrected, the error budget should be added in quadrature.
[Table I] The table lists a BBN constraint '-0.1 < α < 0.12' that does not match Eq. (49) of the text, and no reference is given for this row. The table should be reconciled with the recomputed bounds.
[Notation] The notation for the Planck mass is inconsistent: the text uses both Mp (Eq. (36) context) and MPl (Eqs. (62)-(73)), with c=1 natural units sometimes explicit. This should be unified.
[§V, Eq. (99)] In Eq. (99), the variable β is defined with GN and aB, but the preceding equation uses HGR; the connection HGR = βT² should be stated explicitly to make the substitution transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BBN and WIMP constraints are derived from external data through explicit model equations, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is transparent and self-contained: it assumes a Yukawa-modified Newton potential, derives the corresponding horizon entropy by integrating the entropic-force relation (Eqs. 8-11), obtains modified Friedmann equations from the first law of thermodynamics (Eqs. 19-29), and then confronts the resulting Z(T) factor with external BBN abundance data and the Planck relic density. The BBN constraints are genuine parameter estimations: observed 4He, D, and 7Li abundances are fed into standard empirical fits (Eqs. 42, 45, 47) to infer Z, and the theoretical Z(α) relation is then solved for α. No step defines the observable in terms of the parameter being constrained. The WIMP freeze-out analysis explicitly normalizes the standard-cosmology relic abundance to the observed central value, Ωχh²(0)=0.12, before deriving the α interval (Eqs. 79-81); this is a stated calibration and consistency check rather than a hidden fit or a prediction derived from its own input. Self-citations ([14,16,47]) supply the entropic-gravity framework, but the specific Yukawa entropy and Friedmann equations are derived in the text, not imported as a black-box ansatz or uniqueness theorem. Arithmetic or sign concerns in the Z-to-α conversions are correctness issues, not circularity. Therefore the central claims do not reduce to their inputs by construction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central derivation rests on Verlinde's entropic gravity, the generalized entropy form, the Yukawa force law, and the first law of thermodynamics on the apparent horizon. Numerical bounds additionally assume η10=6, λ~10^26 m, T=1 MeV, and a benchmark WIMP. No invented entities are introduced.

free parameters (4)

α (Yukawa coupling) = -0.017 to 0.018 (DM), -0.24 to 0.12 (He), -0.5 to 1.5 (D)
Free coupling of the model, constrained by external data; not predicted by the framework.
λ (Yukawa range) = 10^26 m (assumed)
Chosen for numerical evaluation; ranges may shift with λ, as acknowledged in Sec. III.C.
η10 (baryon-to-photon ratio) = 6
Fixed to 6 rather than the Planck value ~6.10; shifts Z-derived constraints.
WIMP benchmark (mχ, σ0) = mχ=100 GeV, σ0=(1.7-2.5)e-9 GeV^-2
Assumed thermal WIMP; the DM α bound changes for other benchmarks.

assumptions (7)

domain assumption The entropic force and thermodynamics-gravity conjecture is valid.
Used in Sec. II to connect the Yukawa force to horizon entropy and to derive the Friedmann equations.
domain assumption Apparent-horizon entropy takes the generalized form S_h=A/(4G)+S(A) (Eq. 2).
Assumed without derivation as the starting point for the entropy correction.
domain assumption The gravitational potential is Φ(r)=-(GM/r)(1+αe^{-r/λ}) (Eq. 7).
The defining model ingredient of Yukawa cosmology.
domain assumption The first law dE=T_h dS_h+W dV applies on the apparent horizon (Eq. 19).
The basis for deriving the modified Friedmann equations.
domain assumption The early Universe is flat, radiation-dominated, with Nν=3 and η10=6.
Used for BBN and freeze-out constraints; no sensitivity scan is provided.
domain assumption Standard BBN abundance fitting formulas from Refs. [59,62,63] are valid.
Used to convert the amplification factor Z into light-element abundances.
domain assumption Standard WIMP freeze-out formalism (Kolb-Turner) applies.
Used in Sec. IV to compute the relic abundance and derive the α bound.

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Cite this review

Pith. "Pith review of Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology." pith.science (2026). https://pith.science/paper/VCLF2VTA

@misc{pith2026260809390,
  author       = {Pith},
  title        = {Pith review of: Big-Bang Nucleosynthesis and WIMP Dark Matter Freeze-Out as Probes of Yukawa Cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VCLF2VTA}},
  note         = {Machine review of arXiv:2608.09390}
}

abstract

We investigate Big-Bang Nucleosynthesis (BBN) in the context of Yukawa cosmology. We first derive the modified Friedmann equations by starting from the first law of thermodynamics on the apparent horizon. Using observational data on $^4\mathrm{He}$, deuterium, and $^7\mathrm{Li}$ abundances, we place stringent bounds on the Yukawa coupling $\alpha$. The $^4\mathrm{He}$ and deuterium constraints are mutually consistent ($-0.24 \lesssim \alpha \lesssim 0.12$), while $^7\mathrm{Li}$ requires $\alpha \in [-0.76,\,-0.72]$. This indicate that Yukawa cosmology cannot resolve the Lithium Problem. We then extend our analysis to WIMP freeze-out, and show that the modified Hubble parameter alters the relic abundance, yielding an independent constraint $-0.017 \lesssim \alpha \lesssim 0.018$ from $\Omega_{\mathrm{CDM}}h^2 = 0.120 \pm 0.001$. We also derive the modified time-temperature relation, and show that the positive $\alpha$ raises the early Universe temperature. Our analysis demonstrates that BBN and dark matter relic abundance serve as complementary probes of modified gravity. Our studies confirm that Yukawa cosmology is a testable framework for early-Universe physics.

Figures

Figures reproduced from arXiv: 2608.09390 by the authors.

**Figure 1.** FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 4.** FIG. 4: Dark matter relic abundance Ω [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5: Annihilation cross section [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6: Annihilation cross section [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7: Temperature evolution as a function of cosmic time in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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