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ACT-DR6 consistent inflation in generalised entropic cosmology and $f(Q)$ gravity

T0 review · 4 major / 5 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read Reconstruction of f(Q) gravity and generalised entropies yields inflation matching ACT-DR6 with Planck-BAO.

desk verdict Solid reconstruction that produces explicit ACT-median backgrounds for f(Q) and generalised entropies, but observational claims rest only on single-field slow-roll formulae. read the letter →

arxiv 2607.06582 v1 pith:2T27VJPF submitted 2026-07-04 gr-qc astro-ph.COhep-th

Shin'ichi Nojiri , Sergei D. Odintsov This is my paper · ORCID
classification gr-qcastro-ph.COhep-th
keywords f(Q)gravitygeneralisedentropyentropiccosmologyinflationACT-DR6reconstructionscalar-tensor
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 shows how to build inflation models that match the latest ACT-DR6 constraints on the spectral index and tensor-to-scalar ratio, working inside f(Q) gravity and the equivalent cosmologies that come from generalised horizon entropies. Using a reconstruction scheme, the authors prescribe a Hubble history that starts near de Sitter and ends in radiation domination, then invert the field equations to obtain the corresponding f(Q) and the matching generalised entropy. Two explicit pure-gravity models are written down and tuned so that the slow-roll parameters at CMB exit give n_s ≈ 0.976 and r ≈ 0.012. When a scalar is added, the same Hubble histories can be realised for almost arbitrary f(Q) or any of the standard generalised entropies simply by choosing the scalar kinetic function and potential appropriately. The authors suggest that the resulting entropy may be a candidate for a unifying early-universe entropy.

What carries the argument

The reconstruction map that inverts the first Friedmann equation of f(Q) gravity for a prescribed a(t) or H(N), together with the integral correspondence that converts any such f(Q) into a generalised horizon entropy (and conversely).

What would settle it

A complete linear-perturbation calculation of the scalar and tensor power spectra in one of the explicit reconstructed f(Q) models (or its entropic dual) that yields n_s or r outside the ACT-DR6 plus Planck-BAO window for every choice of the free integration constants.

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Extended reading notes

Core claim

Explicit f(Q) models (and their equivalent generalised-entropic cosmologies) exist that realise inflationary Hubble histories whose slow-roll parameters at CMB exit satisfy the ACT-DR6 plus Planck-BAO medians; moreover, once a scalar is coupled, the same observational targets can be met for large classes of f(Q) and for any of the generalised entropies considered, by suitable choice of the scalar potential.

Load-bearing premise

That the ordinary single-field slow-roll formulae for the spectral index and tensor-to-scalar ratio, evaluated on a prescribed background, are enough to guarantee observational consistency of the full f(Q) or entropic theory.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript constructs inflationary cosmologies in f(Q) gravity and in equivalent generalised-entropic cosmologies that are claimed to be consistent with ACT-DR6 combined with Planck and BAO. Using a reconstruction scheme, the authors prescribe two background expansions H(t) (Eq. 29) and H(N) (Eq. 39), reconstruct the corresponding f(Q) (Eqs. 38, 45, 56, 66), and invert the first Friedmann equation to obtain equivalent generalised entropies (Eqs. 110–111). They further show that, once a scalar is minimally coupled, ACT-compatible backgrounds can be realised for large classes of f(Q) and for any of the generalised entropies considered by suitable choice of the scalar kinetic function and potential (Eqs. 71–81, 141–143). A new four-parameter-style entropy (Eq. 124) is also proposed to improve thermodynamic properties.

Significance. If the observational claims hold, the paper supplies explicit, closed-form f(Q) and entropic models that realise the ACT-DR6 medians, together with a systematic scalar-reconstruction recipe that works for essentially any generalised entropy in the authors’ catalogue. The algebraic reconstruction of f(Q) from a prescribed H (Eq. 26) and the entropy–f(Q) inversion (Eqs. 107–108) are formally clean and useful for the community. The scalar-coupled results are the strongest part: they show that ACT-compatible inflation is not restricted to a narrow class of f(Q) or of entropy once a potential is free. The suggestion that the reconstructed entropies may point toward a “unifying entropy of the early universe” is speculative but clearly flagged as such.

major comments (4)
  1. §V, Eqs. (48) and (59) state r=12ε, while the explicit evaluation for model (29) in Eq. (49) uses r=32/(1+e^{…})=16ε (because ε=2/(1+e^{…}) from Eq. 31). Model (39) is instead evaluated with r=12ε (Eq. 60). The two models are therefore fitted to the same ACT median r≈0.012 with inconsistent conventions. The standard single-field result is r=16ε for the Hubble-flow parameter ε=−Ḣ/H². The manuscript must adopt one consistent definition, re-derive n_s and r for both backgrounds, and re-fit the free parameters (α, the exponential factor, β, γ).
  2. The central claim that the reconstructed models “pass ACT-DR6” rests entirely on evaluating the single-field slow-roll formulae n_s−1=−2ε−η and r∝ε on a prescribed background (§V). No quadratic action, Mukhanov–Sasaki equation, or sound-speed calculation is given for f(Q) or for the entropy-derived effective fluid. The paper itself notes that the dynamical degrees of freedom of f(Q) remain unsettled (§II, citing [86,90–95]). Even if only the graviton propagates on flat space, cosmological scalar and tensor spectra can receive modifications. Without at least a sketch of the linear perturbation analysis (or a clear argument why the GR formulae remain exact), the numerical match to ACT medians is not yet a statement about the observables of the theory being advertised. This applies equally to the pure entropic models and to the scalar-coupled constructions.
  3. §VII.D: the entropies inverted from the ACT-compatible f(Q) models, Eqs. (110) and (111), fail the authors’ own “Generalised third law” and “Bekenstein–Hawking limit” conditions because of singularities at Q=−6H₀² (Eqs. 119–120). The regularisations (121)–(122) are ad hoc and are not shown to preserve the reconstructed background or the slow-roll fit. Either the regularised entropies must be re-checked against ACT, or the claim that these are viable generalised entropies of the early universe should be substantially weakened.
  4. Reheating is treated by an ad-hoc source J_p (Eq. 37) or by order-of-magnitude quantum particle creation (Eq. 112), neither of which is derived from the action or from the thermodynamic first law used to obtain the background equations. The discussion in §VII.C correctly notes the tension with the conservation law assumed in the entropy derivation, but then leaves the issue unresolved for the pure f(Q)/entropic models. A concrete, conservation-consistent reheating mechanism (or an explicit statement that the models describe only the inflationary phase) is needed for the “ACT-consistent cosmology” claim to be complete.
minor comments (5)
  1. Section heading “VIII. SUMMUR Y AND DISCUSSION” contains a typo (“SUMMUR Y”).
  2. The constant of integration C in Eqs. (38) and (45) is left free; a brief statement of how it is fixed (or that it drops out of the slow-roll observables) would help the reader.
  3. In Eq. (48) the spectral-index formula is written n_s−1=−2ε−η; a one-line reminder of the Hubble-flow definition of η (already given in Eq. 30) would avoid confusion with the potential slow-roll parameter η_V.
  4. The new entropy (124) is introduced and shown to satisfy the formal conditions, but no parameter set is exhibited that realises the ACT medians; either supply one or move the construction to an appendix as an existence proof.
  5. Several long displayed equations (e.g. 80, 81, 141, 143) would benefit from being split or from defining auxiliary functions, to improve readability.

Circularity Check

4 steps flagged · score 6.0 of 10

ACT consistency is obtained by construction: H(t)/H(N) is chosen so that standard slow-roll formulae already give the ACT medians, free parameters are fixed to those medians, and f(Q)/entropy (or scalar potentials) are reconstructed afterwards.

  1. fitted input called prediction [§V, eqs. (49)–(57) and (60)–(66)]
    "By the combined data (51), we choose n_s=0.976 and r=0.012, again. Then the equations (60) tell, 0.024=β(1+2γ e^{β(N_*-N_1)})/(1+e^{β(N_*-N_1)}), 0.012=12βγ e^{β(N_*-N_1)}/(1+e^{β(N_*-N_1)}), that is, β∼0.023×(1+e^{β(N_*-N_1)}), γ∼0.43×e^{-β(N_*-N_1)}. … An explicit form of the model is given by (45), f(Q)=…, β=0.023, γ=86, which gives e^{β(N_*-N_1)}∼0.5 imes10^{-2} and satisfies the constraint (42)."

    The free parameters of the prescribed H(N) (and likewise of H(t) via (52)–(55)) are solved so that the standard slow-roll expressions already return the ACT medians. The reconstructed f(Q) is then declared ACT-consistent. The observational numbers are inputs that fix the model, not outputs predicted by an a-priori theory.

  2. self definitional [§III eq. (26); §IV eqs. (29),(38),(39),(45); abstract]
    "By using (26), one can find the f(Q) gravity corresponding to the arbitrary expansion history of the universe given by a=a(t). … We now consider the inflation by using Eq. (26). … Explicit inflationary cosmologies of the above theory, which pass ACT-Data Release 6 (DR6) data combined with Planck and BAO, are obtained."

    f(Q) is defined by integrating the energy density that belongs to a pre-chosen a(t) or H(t). When that H is itself chosen so that ε,η match ACT, the statement “the f(Q) model passes ACT” is true by the definition of the reconstruction, not by an independent dynamical calculation.

2 more flagged steps
  1. fitted input called prediction [§VI eqs. (71)–(81); §VII.E eqs. (141)–(143); abstract]
    "For f(Q) gravity coupled with scalar, we obtain ACT inflation for large classes of f(Q) via the choice of the scalar potentials. … It is demonstrated how one can obtain ACT-DR6-consistent inflation for such a theory based on any generalised entropy considered in this paper via the choice of scalar potentials."

    Given any f(Q) (or any generalised entropy), ω(ϕ) and V(ϕ) are solved so that the exact solution is the already ACT-tuned H=η(t) or η_N(N). Consistency with ACT is therefore guaranteed by the choice of potential, not predicted by the gravity/entropy sector.

  2. self citation load bearing [§VII.C eqs. (106)–(108); abstract; citation [130]]
    "Using the correspondence between the generalised entropy and f(Q) gravity, we obtain generalised entropic inflation, which is consistent with the ACT-DR6. … In [130], we found the correspondence between f(T) and f(Q) gravities and general entropic cosmology."

    Transfer of the reconstructed ACT-consistent f(Q) to a “generalised entropic inflation” rests on the authors’ own prior correspondence paper. Without that self-citation the entropic claim has no independent derivation inside the present work.

full rationale

The paper is an honest reconstruction exercise, not a hidden tautology of the strongest kind. It repeatedly states that the reconstruction scheme realises an arbitrary expansion history (abstract; §III, eq. (26); §IV). Nevertheless the central claim—that explicit f(Q) models, their equivalent generalised entropies, and scalar-coupled versions are ACT-DR6 consistent—is forced by the inputs. The backgrounds (29) and (39) are engineered so that ε and η at the pivot already produce n_s≈0.976 and r≈0.012 via the single-field formulae (48)/(59); the free parameters α≈1.9×10², e^{-2H_0(t_*-t_0)/α}≈2.7×10³, β≈0.023, γ=86 are then fixed to the ACT+Planck+BAO medians (§V). Only after that is f(Q) reconstructed (38), (45), (56), (66) and the corresponding entropies written down (110)–(111). With a scalar the same H is realised for essentially arbitrary f(Q) or any generalised entropy by solving for ω(ϕ) and V(ϕ) (71)–(81), (141)–(143). Thus “ACT-consistent inflation” is not a prediction of a fixed theory; it is the reconstruction target. The load-bearing correspondence between generalised entropy and f(Q) is taken from the authors’ own prior work [130], but that is secondary to the reconstruction circularity. Score 6 reflects partial circularity of the fitted-input/reconstruction type without claiming the entire paper is empty.

Assumptions & free parameters 5 free parameters · 5 assumptions · 3 invented entities

The central claim rests on (i) free parameters fitted to the ACT medians, (ii) the standard reconstruction and thermodynamic-correspondence axioms already used in the authors’ earlier works, and (iii) a handful of ad-hoc entities (source terms J, a new entropy functional) introduced to close the models. No machine-checked proofs or external falsifiable predictions independent of the fit are supplied.

free parameters (5)
  • α (model (29)) = ≈1.9e2
    Fixed to ≈1.9×10² so that n_s=0.976 when the exponential factor is set to give r=0.012 (eqs. (52)–(55)).
  • e^{-(2H_0/α)(t_*-t_0)} = ≈2.7e3
    Set to ≈2.7×10³ to obtain r=0.012 from eq. (49).
  • β, γ (model (39)) = β≈0.023, γ=86
    Solved from the simultaneous system (61) to give n_s=0.976, r=0.012; explicit choice β=0.023, γ=86 (eq. (66)).
  • H_0, t_0 / N_1, N_0, ρ_0, C
    Overall scale and integration constants left free or fixed by the pivot-scale condition k=aH; they enter every reconstructed f(Q) and entropy.
  • σ_0, S_0 (new entropy (124)) = 0<σ_0≪1
    Introduced by hand to satisfy the four generalised-entropy axioms while producing a large e-folding; 0<σ_0≪1 required for sufficient inflation.
assumptions (5)
  • domain assumption Any prescribed flat FLRW expansion history a(t) can be realised by an f(Q) obtained from the integral reconstruction (26).
    Stated as the reconstruction scheme of [97] and used throughout §§III–V.
  • domain assumption The first Friedmann equation of f(Q) is in one-to-one correspondence with the thermodynamic first law applied to a generalised entropy on the apparent horizon (eqs. (100), (106)–(108)).
    Taken from the authors’ prior work [130] and used to map every reconstructed f(Q) to an entropy.
  • domain assumption The slow-roll expressions n_s−1=−2ε−η and r=12ε evaluated on the background are sufficient to confront ACT-DR6 + Planck-BAO data.
    Invoked without further derivation in §V (eqs. (48), (59)).
  • domain assumption Matter energy-momentum is covariantly conserved with respect to the Levi-Civita connection even in f(Q) gravity.
    Used to obtain the continuity equation (19)–(20) and the source J.
  • ad hoc to paper An ad-hoc source J_ρ, J_p (or quantum particle creation) can generate the radiation that ends inflation without spoiling the reconstructed background.
    Introduced in eqs. (27)–(28), (36)–(37) and discussed again in the reheating paragraphs of §VII.
invented entities (3)
  • Explicit f(Q) of eqs. (38) and (45) (and the regularised versions (121)–(122))
    purpose: Realise the two chosen ACT-compatible expansion histories.
    Obtained by direct integration of the reconstruction formula; no independent observational handle beyond the fitted n_s, r.
  • Source terms J_ρ=0, J_p (37) that create radiation at the end of inflation
    purpose: Provide an effective description of reheating inside pure f(Q)/entropic cosmology.
    Inserted by hand so that the continuity equation is violated only near t≈t_0; origin left unspecified.
  • New generalised entropy S_g=(1−σ_0)S + σ_0 S/(1+S²/S_0²) (eq. (124))
    purpose: Satisfy the four entropy axioms while allowing a large e-folding and avoiding the singularities of the reconstructed entropies (110)–(111).
    Postulated in §VII.D; parameters chosen for positivity and monotonicity, not fixed by external data.

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Pith. "Pith review of ACT-DR6 consistent inflation in generalised entropic cosmology and $f(Q)$ gravity." pith.science (2026). https://pith.science/paper/2T27VJPF

@misc{pith2026260706582,
  author       = {Pith},
  title        = {Pith review of: ACT-DR6 consistent inflation in generalised entropic cosmology and $f(Q)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2T27VJPF}},
  note         = {Machine review of arXiv:2607.06582}
}
abstract

Within the framework of generalised entropic cosmology and its equivalent $f(Q)$ gravity, we construct inflationary scenarios consistent with recent Atacama Cosmology Telescope (ACT) observations combined with Planck-BAO data. We work with $f(Q)$ gravity and use the reconstruction scheme from which one can obtain an arbitrary cosmological evolution consistent with desirable theoretical or observational considerations. Explicit inflationary cosmologies of the above theory, which pass ACT-Data Release 6 (DR6) data combined with Planck and BAO, are obtained. For $f(Q)$ gravity coupled with scalar, we obtain ACT inflation for large classes of $f(Q)$ via the choice of the scalar potentials. The entropic cosmology based on generalised entropy, which includes most of the known entropies like Tsallis, R\'{e}nyi, Barrow, etc as a particular case, is investigated. The corresponding FRW equations, depending on the parameters of generalised entropy, are presented. Using the correspondence between the generalised entropy and $f(Q)$ gravity, we obtain generalised entropic inflation, which is consistent with the ACT-DR6. At the next step, the entropic gravity coupled with scalar is considered. It is demonstrated how one can obtain ACT-DR6-consistent inflation for such a theory based on any generalised entropy considered in this paper via the choice of scalar potentials. These findings may point towards the correct unifying entropy of the early universe.

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