Probing Primordial Cosmology Through BBN Observational Constraints Under Extended Gravitational Dynamics

Abdul Malik Sultan; Manahil Ali; Muhammad Israr Aslam; Nazek Alessa

REVIEW 4 major objections 5 minor 72 references

Probing Primordial Cosmology Through BBN Observational Constraints Under Extended Gravitational Dynamics

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper claims that f(R,G,T) gravity—a modified theory built from Ricci curvature, the Gauss-Bonnet invariant, and matter-trace coupling—remains compatible with Big Bang nucleosynthesis for restricted parameter ranges, with a leading upp

desk verdict The four BBN bounds are new in a narrow sense, but the central constraint n<0.3721 is computed from fractional powers of a negative Gauss-Bonnet invariant with no branch choice, so the main claim is undefined as written. read the letter →

arxiv 2607.17252 v1 pith:U3QFNOFA submitted 2026-07-19 gr-qc

Abdul Malik Sultan , Manahil Ali , Muhammad Israr Aslam , Nazek Alessa This is my paper · ORCID

classification gr-qc MSC 83F0583D05 PACS 04.50.Kd98.80.Ft

keywords f(RGT)gravityBigBangnucleosynthesisfreeze-outtemperatureprimordialheliumabundanceGauss-BonnetinvariantmodifiedFriedmannequationsBBNconstraints

topics Dark Energy

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 tries to establish that f(R,G,T) gravity, a recently proposed extension of general relativity, does not spoil the standard picture of primordial nucleosynthesis. Using the BBN freeze-out temperature limit |ΔTf/Tf| < 4.7×10^{-4} and the helium-4 mass fraction Yp = 0.245 ± 0.003, it constrains four representative models. The main quantitative result is n < 0.3721 for the Gauss-Bonnet exponent in the additive model f = α1R + G^n + γ1T. If correct, the theory remains viable at early times while still allowing significant departures from standard cosmology elsewhere. The argument depends on a branch choice for fractional powers of the negative Gauss-Bonnet invariant during radiation domination.

What carries the argument

The central object is the modified Friedmann equation of f(R,G,T) gravity, combined with the freeze-out condition H(Tf) = Λ(Tf) and the BBN relation |ΔTf/Tf| ≈ (ρ/ρr) H_GR / (10 c_q T_f^5). The Gauss-Bonnet invariant during radiation domination is G = 24H²(Ḣ + H²) = −24H⁴, so each model's G^n term is effectively a power of −H⁴. The paper uses this to convert observational bounds on freeze-out temperature and helium abundance into algebraic constraints on n.

What would settle it

Evaluate the Model 1 Lagrangian with n = 0.3721 in the early universe, compute G = −24H⁴, and determine whether G^n admits a real value under the branch conventions needed to make the field equations real; if no real value exists, the derived |ΔTf/Tf| curve does not represent a real cosmological model. Alternatively, redo the derivation keeping the full Eq. (9) without linearization and check whether the n = 0.3721 crossing persists.

Watch

Extended reading notes

Core claim

For each model, the paper substitutes the modified Friedmann equation into the standard BBN freeze-out relation H(Tf) = Λ(Tf), solves for the matter energy density from the field equations, fixes a coupling using the present dark-energy density ΩDE0 ≈ 0.7, and derives |ΔTf/Tf| as a function of the exponent n. It finds that the predicted deviation crosses the observational bound at n ≈ 0.3721 for Model 1, while Models 2–4 have somewhat larger but still narrow allowed intervals. The helium mass fraction Yp curves remain inside the observed band only for restricted n ranges. The authors read these results as showing that f(R,G,T) gravity is consistent with BBN and therefore a viable extension o

Load-bearing premise

The argument assumes that expressions like (−H⁴)^n with non-integer n are real and well-defined during radiation domination, even though G = −24H⁴ is negative; the paper never states which branch it uses, so without that convention the plotted constraints are undefined.

Editorial extensions

If this is right

If the central claim is correct, f(R,G,T) gravity passes a stringent early-universe test for a restricted range of the Gauss-Bonnet exponent, meaning higher-curvature corrections are allowed but tightly bounded.
The same framework can be confronted with other early-universe probes such as the CMB, baryon-to-photon ratio, and primordial gravitational waves, as the paper itself suggests.
Models with inverse powers of G appear to be more sensitive to n, so BBN places especially narrow allowed intervals on those couplings.
The helium-4 abundance analysis independently supports the freeze-out temperature constraints for restricted parameter values.
Significant departures from standard cosmology are compatible with BBN observations within the allowed parameter regions.

Reading between the lines

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

The paper leaves unspecified how fractional powers of the negative Gauss-Bonnet invariant are defined; a natural extension would be to reformulate the models with a regularized branch, for example using |G| or an explicit complex prescription, so that the predicted expansion rate is real-valued for the quoted bounds.
The derived bounds likely shift if the assumed priors H0 = 70 and ΩDE0 = 0.7 are replaced by other observational values; recomputing the crossing points under different priors would provide a quick consistency test.
The analytic constraints rely on the approximation that the dark-energy density is negligible during BBN; checking the exact Eq. (9) rather than the linearized Eq. (26) could either tighten or relax the n bounds.
The same freeze-out machinery could be extended to deuterium and lithium abundances, not just helium-4, which would give independent and possibly stricter constraints on the model parameters.

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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 paper applies the standard BBN freeze-out constraint to four f(R,G,T) Lagrangians. What is actually new is modest: explicit closed forms for |ΔTf/Tf| and corresponding bounds on the Gauss-Bonnet exponent n for each model. The underlying machinery — the freeze-out condition (19), the helium formula (21), the bound |ΔTf/Tf|<4.7e-4 — is the same used in earlier f(T), f(Q,T), and f(G) papers, and the authors cite that literature.

The problem is that the central numerical claims are not well-defined. During radiation domination G = 24H^2(Ḣ+H^2) = -24H^4 < 0. The models contain G^n and G^{-n} with non-integer n; the constraints pick n≈0.3721, n≈0.8, etc. Equations (30)-(45) treat (-H^4)^n as if it were real. With the principal branch, (-H^4)^n = (H^4)^n e^{iπn}, which is complex for generic non-integer n. No branch or regularization is specified, and no condition n∈ℤ is imposed. So ρ, and hence ΔTf/Tf and Yp, are complex at the very points the paper claims are allowed. The dashed-line intersections in Figs. 1-4 are therefore not real observables. This is the load-bearing flaw, not a nuance.

There are also internal inconsistencies. α1 is 1e-12 in Section IV A but 1e-14 in Section V; γ2=0.5 for Fig. 2 but 1e8 for Fig. 6; γ3=1e12 in Fig. 3 but 1e8 in Fig. 7. H0 is quoted as 73.02 in Eq. (29) but set to 70 in the figures. None of these choices is fatal by itself, but the paper doesn't flag the changes, so the plots cannot be reproduced.

Credit where due: the algebra is explicit, the method is standard, and the self-citations to the authors' earlier BBN work are relevant, not padding. The paper would be salvageable if the authors pick a real branch (or restrict n), redo the plots, and make the parameter choices consistent.

Who should read it? Someone collecting model-specific BBN bounds for f(R,G,T) might want the formulas after they are fixed. In its present form, I would not send it to a referee: the main result is undefined. I'd desk reject or ask for major revision first.

Referee Report

4 major / 5 minor

Summary. The paper derives modified Friedmann equations for the f(R,G,T) gravity framework and uses BBN observational constraints, |ΔT_f/T_f|<4.7×10^-4 and Y_p=0.245±0.003, to constrain the exponent n in four representative models: f=α1R+G^n+γ1T, f=α2RT+γ2G^n, f=α3RT/G^n+γ3, and f=α4R+β1T/G^n+γ4. The main numerical findings are bounds such as n<0.3721 for Model 1 (Fig. 1, Eq. (32)), with analogous constraints for Models 2–4. The paper concludes that broad regions of parameter space are consistent with BBN, supporting the viability of f(R,G,T) gravity. However, the central equations use fractional powers of the Gauss-Bonnet invariant G during radiation domination, where G<0, and no branch or reality prescription is given; parameter values also change between sections. These issues make the reported constraints not well-defined as real predictions.

Significance. If the derivation were sound, the paper would provide useful BBN constraints on a recent f(R,G,T) framework, extending a standard test to four explicit models. The manuscript gives analytic expressions for the freeze-out temperature shift and compares them with a quoted observational bound, which is a potentially falsifiable procedure. However, the central numerical constraints are computed from expressions involving (−H^4)^n with non-integer n, which are complex-valued without an additional branch specification. The paper also uses inconsistent parameter values between sections. These are load-bearing defects: the main claim of consistency with BBN is not supported by the current analysis.

major comments (4)

[§IV, Eqs. (28)–(45), Fig. 1] The models contain G^n and G^{-n} with non-integer n. In radiation domination a(t)∼t^{1/2}, so G=24H^2(\dot H+H^2)=−24H^4<0. For n=0.3721, n=0.8, etc., (−H^4)^n has no real value; the principal branch gives (−H^4)^n=(H^4)^n e^{iπn}. None of Eqs. (30)–(45) specifies a branch or imposes reality of the right-hand side of Eq. (9). Thus ρ, and hence |ΔT_f/T_f| in Eq. (32) and its analogues, is complex-valued at the very parameter values the constraints select. The intersections in Figs. 1–4, including n≈0.3721, are therefore not defined real observables. A branch or regularization, together with a justification of the reality of the effective energy density, must be supplied before any BBN bound can be drawn.
[§IV A–D vs §V, Figs. 1–8] Parameter values for the same models change between sections. Model 1 uses α1=10^-12 in the Fig. 1 analysis but α1=10^-14 in Fig. 5; Model 2 uses γ2=0.5 in §IV B but γ2=10^8 in Fig. 6; Model 3 uses γ3=10^12 in §IV C but γ3=10^8 in Fig. 7. In addition, H0 is quoted as 73.02±1.79 km s^-1 Mpc^-1 near Eq. (29), while the numerical analysis sets H0=70. Since Eqs. (32), (37), (41), and (45) depend explicitly on these constants, the reported curves and bounds are not uniquely defined and are not reproducible from the text.
[§III, Eqs. (23)–(26)] The derivation of the central bound uses two different expansion-rate relations. Eq. (23) writes H=H_GR(1+ρ/ρ_r), while Eq. (24) uses H=H_GR(√(1+ρ_DE/ρ_r)−1). These are not equivalent; they differ at leading order by a factor of 2. Eq. (26) and all subsequent constraints are based on the square-root form, yet Eq. (23) is also presented as the defining relation. The mismatch should be resolved because it changes the inferred n-bounds by an O(1) factor. This is not a purely typographical issue: the choice of relation directly affects the claimed constraints.
[§IV around Eq. (29)] The paper assumes that during BBN the effective dark-energy density ρ remains constant and equal to its present value ρ_DE0. This is an input assumption, not a consequence of the field equations; in f(R,G,T) gravity the effective density defined by Eq. (9) can vary with H, T, and the model parameters. Since this assumption enters directly into the derivation of Eqs. (32), (37), (41), and (45), it must be justified from the model equations or by an independent physical argument. Without such justification, the constraints are conditional on an unmotivated prior.

minor comments (5)

[Throughout] There are numerous typographical and grammatical errors, e.g., 'consequen ces', 'Friedmannn Lemaître', missing superscripts in helium notation, and inconsistent use of M_p versus M_pl. The paper would benefit from careful proofreading.
[Abstract vs. §VI] The abstract claims 'broad regions of the parameter space satisfy existing nucleosynthesis constraints', while the concluding section emphasizes 'restricted parameter intervals' and 'narrow intervals'. The wording should be harmonized to avoid overstating the results.
[Eqs. (16)–(19)] Eq. (16) defines Λ_tot as the sum of forward and reverse rates, but Eq. (17) is presented as the total rate. The notation should be clarified so the reader knows which quantity enters the freeze-out condition H=Λ_tot.
[§II–IV] The dimensionful couplings α_i, β1, γ_i are introduced without units or a dimensional analysis. Since the models contain G^n with varying n, the numerical values of these parameters are not meaningful unless a consistent convention is stated.
[Fig. 5] The caption and text state that the analysis is performed at α1=10^-14, but the main Model 1 constraint in §IV A and Fig. 1 uses α1=10^-12. This inconsistency is particularly confusing because both figures are used to support the same model's viability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: BBN constraints are external inputs and model parameters are calibrated to current-universe observables, not to the BBN quantities being predicted.

full rationale

The main constraint (n < 0.3721) follows from substituting a model Lagrangian into the modified Friedmann equation (9), fixing one coupling (e.g., γ1 via Eq. 31) using the present-day Ω_DE0 and H0 in Eq. (29), and then comparing the resulting |ΔTf/Tf| expression (Eq. 32) with the externally imposed bound |ΔTf/Tf| < 4.7e-4 from Eq. (27). This is a genuine model-to-data comparison rather than a fit of the same data: the BBN bound and Yp interval come from independent observational measurements quoted in Eqs. (22) and (27), and the field equations and BBN rate formulas are taken from external references [47,48] and [23,25,51,52]. The numerous self-citations to earlier Sultan et al. papers are contextual literature references and do not supply the load-bearing derivation. The model Lagrangians are admittedly chosen by hand, but an arbitrary ansatz only becomes circular if the derived constraint is equivalent to the ansatz by construction, which is not the case here. The issue that G^n is formally complex for non-integer n when G < 0 is a serious correctness concern, but it is not a circularity of the derivation chain.

Assumptions & free parameters 11 free parameters · 5 assumptions · 0 invented entities

The burden is heavy: the results require accepting the f(R,G,T) field equations, approximate BBN formulas, a constant-DE assumption, and an unstated real-branch convention for fractional powers of a negative Gauss-Bonnet term. Model constants are hand-set or fixed through current Ω_DE0; the BBN bounds are not first-principles predictions.

free parameters (11)

n (Gauss-Bonnet exponent) = constrained: n<0.3721 (Model 1); n≈0.75-0.85 (Model 2); n≈0.74-0.86 (Model 3); n≈0.24-0.36 (Model 4)
Exponent in G^n/G^{-n} terms; freely varied in Figs. 1-4, no prior derivation; the sought constraint is a bound on n.
α1 = 10^-12 (Sec IV/Fig. 1) and 10^-14 (Sec V/Fig. 5)
Coupling of R term in Model 1; chosen by hand with inconsistent values in different sections; shifts BBN predictions.
γ1 = 1e-12 (conclusion; also Eq. 31 solves for it from Ω_DE0)
Matter-coupling constant in Model 1; set via current Ω_DE0 or by hand.
γ2 = 0.5 (Sec IV/Fig. 2) and 10^8 (Sec V/Fig. 6)
G^n coefficient in Model 2; hand-set, inconsistent values.
γ3 = 1e12 (Sec IV/Fig. 3 and conclusion) and 1e8 (Sec V/Fig. 7)
Constant term in Model 3; hand-set, inconsistent values.
α4 = 0.1
R-coupling in Model 4; hand-set.
γ4 = 1e12
Constant term in Model 4; hand-set.
α2 and β1 = not given explicitly; solved from Ω_DE0 via Eqs. (36) and (44)
Couplings in Models 2 and 4; fixed by requiring the present dark-energy density parameter, affecting the BBN bounds.
H0 = 70 (used in numerics) vs 73.02±1.79 km/s/Mpc ≈ 2.1e-42 GeV (quoted)
Used interchangeably in raw km/s/Mpc and GeV units; affects Eqs. (31)-(45) through H0^2 and H0^{4n}.
Ω_DE0 = 0.7
Used to set model constants via Eq. (29); observational input.
g* (effective relativistic d.o.f.) = ≈10
Standard BBN input, not varied; affects Tf and all bounds through Eq. (20).

assumptions (5)

domain assumption The field equations (3) and Friedmann equations (9)-(10) for f(R,G,T) from [47,48] are correct as written.
All model-specific density expressions are derived from these equations; if they are wrong, all constraints fail.
domain assumption BBN freeze-out and helium-abundance formulas (Eqs. 17-21) from [23,25,51,52] are accurate approximations.
The observational limits and ΔYp mapping rely on these standard analytic approximations; no full BBN code is run.
ad hoc to paper The effective dark-energy density ρ stays constant and equal to its present value ρ_DE0 during the BBN era.
Assumed in Section IV-A: 'Assuming that, during the BBN era, the DE density ρ stays effectively constant and equal to its present value ρDEO'; this is not derived and is not true in most evolving modified-gravity models.
ad hoc to paper The quantities (-H^4)^n are treated as real and single-valued for non-integer n.
Used in Eqs. (30)-(45) and Figs. 1-8; no branch specification; G=-24H^4<0 during radiation domination, so the action is complex-valued unless a special convention is imposed.
ad hoc to paper The exponent n can be chosen independently for each term and the action remains physically meaningful with dimensionful bare parameters.
No dimensional analysis is given for G^n with G having mass dimension 4.

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Cite this review

Pith. "Pith review of Probing Primordial Cosmology Through BBN Observational Constraints Under Extended Gravitational Dynamics." pith.science (2026). https://pith.science/paper/U3QFNOFA

@misc{pith2026260717252,
  author       = {Pith},
  title        = {Pith review of: Probing Primordial Cosmology Through BBN Observational Constraints Under Extended Gravitational Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U3QFNOFA}},
  note         = {Machine review of arXiv:2607.17252}
}

abstract

In this article, We investigate the cosmological consequences of a recently developed $f(R,G,\mathcal{T})$ gravitational framework, in which the action is formulated as a general function of the Ricci scalar $R$, the Gauss-Bonnet invariant $G$, and the trace of the energy-momentum tensor $\mathcal{T}$. As one of the most reliable probes of the physical conditions in the early universe, Big Bang nucleosynthesis offers a stringent framework for testing deviations from standard cosmology. We consider four representative models that are analyzed and constrained using observational limits on $\left|\Delta T_f/T_f\right|$ and the primordial helium mass fraction $Y_p$. The bounds obtained identify the allowed parameter regions for each model and demonstrate that significant departures from standard cosmology are compatible with nucleosynthesis observations. Our analysis shows that broad regions of the parameter space satisfy existing nucleosynthesis constraints, indicating the consistency of $f(R,G,\mathcal{T})$ gravity with the observed primordial light-element abundances and the established picture of the early universe preserving the observed abundances of light nuclei.

Figures

Figures reproduced from arXiv: 2607.17252 by the authors.

**Figure 2.** FIG. 2: Variation of the ratio [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** FIG. 3: Behavior of the ratio [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4: Variation of the ratio [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: FIG. 5: Variation of the primordial [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: FIG. 6: Dependence of the primordial [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: FIG. 7: Evolution of the [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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( 35) and then simplifying Eq

into Eq. ( 35) and then simplifying Eq. ( 26), we arrive at the ﬁnal analytical expression. ⏐ ⏐ ⏐ ⏐ ∆Tf Tf ⏐ ⏐ ⏐ ⏐ = 23n−1 3n H 4 0 ( n − 1 )( 8n − 1 ) γ2 ( − T8 f ζ 4) n ΩDEO 5cq T7 f ζ ( 6H 4 0 M 2 p ΩDEO + T4 f ζ 2( 24 n(−H 4 0 )n(n − 1)(8n − 1)γ2 − 6H 2 0 M 2 p ΩDEO ) ) . ...
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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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