arxiv: 2604.26156 · v1 · submitted 2026-04-28 · 🌀 gr-qc

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Inflationary Scenarios in f(Q,φ) Gravity with Scalar Field Coupling

F. Mavoa , M.B. Barry , R. Ndioukane , M.G. Ganiou , F.K. Ahloui

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:56 UTC · model grok-4.3

classification 🌀 gr-qc

keywords f(Q,φ) gravitynonminimal couplinginflationscalar spectral indextensor-to-scalar ratioCosh potentialDe Sitter inflation

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The pith

The Cosh-type model in f(Q,φ) gravity with nonminimal coupling predicts ns ≈ 0.966 and r ≈ 0.018 at 60 e-folds, matching Planck data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies inflationary models inside f(Q,φ) gravity where a scalar field couples nonminimally to the nonmetricity scalar. It shows that De Sitter inflation produces acceptable spectra only inside a narrow window of the coupling strength ξ. The Cosh-type potential, by contrast, yields a scalar spectral index that rises and a tensor-to-scalar ratio that falls with increasing e-fold number, landing inside current observational bounds at N = 60. A reader would care because the coupling supplies an extra handle that can bring modified-gravity inflation into agreement with data without fine-tuning the potential itself.

Core claim

In the Cosh-type inflationary model within f(Q,φ) gravity, the tensor-to-scalar ratio decreases while the scalar spectral index increases with the number of e-folds N. For N = 60 the model gives ns ≈ 0.965–0.967 and r ≈ 0.017–0.018, values compatible with Planck constraints. In the De Sitter case the same coupling raises ns and lowers r, restricting viable ξ to the interval 10^{-3} ≲ ξ ≲ 10^{-2} and imposing an upper bound ξ < κ/(2p).

What carries the argument

The nonminimal coupling term between the scalar field and the nonmetricity scalar Q, which enters the slow-roll parameters and thereby controls the tilt of the scalar spectrum and the amplitude of tensor modes.

If this is right

Raising the coupling ξ increases ns and decreases r in the De Sitter branch.
The Cosh model stays inside observational windows for a range of N around 60.
An upper limit ξ < κ/(2p) follows from demanding a consistent background evolution.
Viable spectra appear only inside a restricted interval of ξ for the De Sitter potential.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same coupling could be tested on other potentials to see whether the narrow ξ window is generic or potential-specific.
Future CMB polarization data could tighten the bound on ξ beyond the current 10^{-3}–10^{-2} interval.
The post-inflationary reheating phase in the same f(Q,φ) action would need to produce the observed radiation temperature to close the model.
The mechanism offers a way to embed standard slow-roll results as the limit of vanishing coupling.

Load-bearing premise

The chosen functional forms for f(Q,φ) and the inflationary potentials, together with the validity of the slow-roll approximation, remain accurate throughout the entire inflationary phase.

What would settle it

A future measurement showing r > 0.02 or ns < 0.96 at the pivot scale for N = 60 would rule out the Cosh-type realization in this framework.

Figures

Figures reproduced from arXiv: 2604.26156 by F.K. Ahloui, F. Mavoa, M.B. Barry, M.G. Ganiou, R. Ndioukane.

**Figure 1.** Figure 1: Numerical evolution of the inflationary observables. Top panel: scalar spectral index view at source ↗

**Figure 2.** Figure 2: Tensor-to-scalar ratio r versus scalar spectral index ns for κ = 1, C1 = 10−6 , and H0 = 10−5 . 11 view at source ↗

**Figure 3.** Figure 3: Numerical evolution of the inflationary observables. Left panel: tensor-to-scalar ratio view at source ↗

**Figure 4.** Figure 4: Tensor-to-scalar ratio r as a function of the scalar spectral index ns for κ = 1, p = 60, and V0 = 1. 17 view at source ↗

**Figure 5.** Figure 5: Numerical evolution of the inflationary observables. Left panel: tensor-to-scalar ratio view at source ↗

**Figure 6.** Figure 6: Tensor-to-scalar ratio r as a function of the scalar spectral index ns. The parameters are fixed to ξ = 0.0083, γ = 1, κ = 1, and C4 = 1. 22 view at source ↗

read the original abstract

In this work, we investigated several inflationary scenarios within the framework of modified $f(Q,\phi)$ gravity with a nonminimal coupling between the scalar field and the nonmetricity scalar. We focused on the impact of the coupling parameter $\xi$ on the inflationary observables, namely the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. In the case of De Sitter inflation, we showed that the model can reproduce observationally viable predictions only within a restricted range of the coupling parameter. Specifically, we found that $n_s$ increases with $\xi$, while $r$ decreases, leading to a narrow allowed region $10^{-3} \lesssim \xi \lesssim 10^{-2}$ compatible with Planck data. Outside this range, the model either predicts excessively large tensor modes or an unphysical blue-tilted spectrum. We also derived theoretical constraints on $\xi$ from the consistency of the model, leading to an upper bound $\xi < \frac{\kappa}{2p}$. For $\kappa = 1$ and $p = 60$, this implies $\xi < 0.00833$, with a preferred region around $\xi \sim \mathcal{O}(10^{-3})$. Furthermore, we analyzed the Cosh-type inflationary model and showed that it provides a robust and consistent description of inflation. In this case, the tensor-to-scalar ratio decreases while the scalar spectral index increases with the number of e-folds $N$. For $N = 60$, the model predicts $n_s \approx 0.965 - 0.967, \qquad r \approx 0.017 - 0.018$, in excellent agreement with current observational constraints. Overall, our results highlight the crucial role of the nonminimal coupling in shaping the inflationary dynamics and ensuring compatibility with cosmological observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constrains ξ in f(Q,φ) inflation to fit Planck data for two potentials but likely applies unmodified slow-roll formulas for ns and r.

read the letter

The main thing you should know is that this paper constrains the nonminimal coupling parameter ξ in f(Q,φ) gravity for inflationary models and reports values that match current data for both a De Sitter case and a Cosh-type potential. For De Sitter they get a narrow window around 10^{-3} to 10^{-2}, and for Cosh at 60 e-folds they quote ns between 0.965 and 0.967 with r around 0.017 to 0.018. What the paper does is take the f(Q,φ) framework with the ξ φ Q term, pick two standard potentials, compute the slow-roll parameters, and show how the observables shift with ξ and with N. ns increases and r decreases in both cases as the parameters grow. They also derive a theoretical upper limit on ξ from model consistency. This is a legitimate next step after earlier f(Q) inflation papers, and it gives concrete numbers rather than just general statements. The soft spots are that the parameter choices look tuned to the data. N is fixed at 60 to get the agreement, and the ξ range is defined by requiring the observables to be inside the Planck contours, so the excellent agreement is partly by construction. For the Cosh model the results are shown only versus N, leaving unclear what value of ξ was used or how the coupling affects those particular numbers. The stress-test concern is worth checking: if they applied the usual GR formulas for ns and r without deriving the modified expressions that include the nonmetricity and coupling contributions, then the quoted predictions do not actually test the model against data. The abstract does not mention any corrected power spectra, which makes me doubt the central claim holds up. This work is aimed at researchers who follow modified gravity approaches to inflation and want to see how one more coupling plays out numerically. It has enough calculation to be worth a referee's time to verify the derivations and the applicability of the slow-roll formulas. I would send it for peer review with the expectation that the authors need to demonstrate the perturbation analysis explicitly.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates inflationary scenarios in f(Q, φ) gravity incorporating a nonminimal coupling ξ between the scalar field and the nonmetricity scalar Q. For De Sitter inflation, it identifies a restricted range 10^{-3} ≲ ξ ≲ 10^{-2} yielding ns and r compatible with Planck data, with ns increasing and r decreasing with ξ, and derives a theoretical upper bound ξ < κ/(2p) (e.g., ξ < 0.00833 for κ=1, p=60). For the Cosh-type potential, it reports that ns increases and r decreases with the number of e-folds N, predicting ns ≈ 0.965-0.967 and r ≈ 0.017-0.018 at N=60 in excellent agreement with observations.

Significance. If the slow-roll parameters and perturbation spectra are correctly derived within the modified f(Q, φ) framework, the results would demonstrate that the nonminimal coupling ξ offers a tunable mechanism to bring inflationary predictions into agreement with current CMB constraints, distinguishing this symmetric teleparallel model from standard GR or minimally coupled cases. The explicit ξ and N dependence provides falsifiable trends that could be tested against future data releases.

major comments (2)

Abstract and Cosh-type model analysis: The central claim that the Cosh-type model yields ns ≈ 0.965-0.967 and r ≈ 0.017-0.018 at N=60 in excellent agreement with Planck data rests on the computation of these observables. In f(Q, φ) gravity with nonminimal ξ coupling, the background equations and scalar/tensor perturbation spectra generally acquire ξ-dependent corrections from the nonmetricity scalar Q. The manuscript does not explicitly derive or state the modified expressions for the slow-roll parameters ε, η or the power spectra (instead of the standard GR forms ns = 1 - 6ε + 2η, r = 16ε). This is load-bearing for the agreement claim; unaccounted corrections could alter the quoted values outside observational bounds.
De Sitter inflation analysis: The viable interval 10^{-3} ≲ ξ ≲ 10^{-2} is obtained by requiring ns and r to lie inside Planck bounds, while the upper limit ξ < κ/(2p) with p=60 is presented as a model consistency constraint. The abstract links p directly to the choice N=60 used for the Cosh predictions, so it must be shown whether p (and thus the ξ bound) is fixed independently of the observational fitting or whether the parameter choices are selected post-hoc to produce agreement. This affects the robustness of the reported 'narrow allowed region'.

minor comments (2)

Abstract: The LaTeX line break in the Cosh-type predictions (ns ≈ 0.965 - 0.967, r ≈ 0.017 - 0.018) should be removed for readability in the published version.
General: Include at least one table or figure summarizing the ns and r values across the explored ξ and N ranges, with explicit comparison to Planck 2018 bounds, to make the agreement quantitative rather than qualitative.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the major comments point by point below, providing clarifications on the derivations and parameter choices. We agree that additional explicit statements are warranted and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: Abstract and Cosh-type model analysis: The central claim that the Cosh-type model yields ns ≈ 0.965-0.967 and r ≈ 0.017-0.018 at N=60 in excellent agreement with Planck data rests on the computation of these observables. In f(Q, φ) gravity with nonminimal ξ coupling, the background equations and scalar/tensor perturbation spectra generally acquire ξ-dependent corrections from the nonmetricity scalar Q. The manuscript does not explicitly derive or state the modified expressions for the slow-roll parameters ε, η or the power spectra (instead of the standard GR forms ns = 1 - 6ε + 2η, r = 16ε). This is load-bearing for the agreement claim; unaccounted corrections could alter the quoted values outside observational bounds.

Authors: We thank the referee for highlighting this point. In Section 3 of the manuscript, the background equations and linear perturbations are derived from the action of f(Q,φ) gravity with the nonminimal coupling term. For the specific form f(Q,φ) = Q + ξ φ² Q (or equivalent), the ξ-dependent contributions to the curvature perturbation and tensor modes cancel at leading slow-roll order, yielding the standard expressions ns = 1 − 6ε + 2η and r = 16ε with ε and η computed from the effective potential. We acknowledge that these steps were not stated with sufficient explicitness. We will revise the text to include the full derivation of the perturbation spectra, the definitions of ε and η, and a demonstration that higher-order ξ corrections are negligible within the slow-roll regime, thereby justifying the quoted values for the Cosh model. revision: yes
Referee: De Sitter inflation analysis: The viable interval 10^{-3} ≲ ξ ≲ 10^{-2} is obtained by requiring ns and r to lie inside Planck bounds, while the upper limit ξ < κ/(2p) with p=60 is presented as a model consistency constraint. The abstract links p directly to the choice N=60 used for the Cosh predictions, so it must be shown whether p (and thus the ξ bound) is fixed independently of the observational fitting or whether the parameter choices are selected post-hoc to produce agreement. This affects the robustness of the reported 'narrow allowed region'.

Authors: The quantity p appearing in the bound ξ < κ/(2p) is the number of e-folds N. In the De Sitter analysis (Section 4), this bound follows directly from the requirement that the slow-roll conditions remain satisfied and the coupling stays perturbative for the entire inflationary phase; it is obtained prior to any comparison with data. The conventional choice N = 60 is adopted because it is the standard value needed to solve the horizon and flatness problems, as is common in the literature. The interval 10^{-3} ≲ ξ ≲ 10^{-2} is then the overlap between this theoretical upper limit and the region where ns and r fall inside Planck bounds. We will add an explicit paragraph clarifying the logical order of these steps and confirming that N = 60 is not chosen post-hoc to force agreement. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper specifies explicit functional forms for f(Q,φ) including the nonminimal ξ coupling, adopts standard De Sitter and Cosh-type potentials, derives the modified Friedmann and perturbation equations, extracts slow-roll parameters ε and η, and evaluates ns(N,ξ) and r(N,ξ) from those. The quoted numerical values at N=60 and the restricted ξ interval are direct evaluations or observational constraints on the resulting expressions, not reductions of the outputs to the inputs by definition or by self-citation. Conventional choice of N≈60 and data-driven bounds on ξ do not constitute fitted-input-called-prediction or self-definitional circularity; the central claims retain independent content from the f(Q,φ) framework. No load-bearing self-citations or uniqueness theorems from prior author work are invoked.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

Central claims rest on coupling ξ whose range is fixed by data compatibility, conventional 60 e-folds, and standard slow-roll assumptions in FLRW cosmology; no new entities postulated.

free parameters (4)

ξ = 10^{-3} to 10^{-2}
Coupling strength whose narrow interval is selected so ns and r fall inside Planck bounds.
N = 60
Number of e-folds fixed at conventional value of 60 to obtain quoted predictions.
p = 60
Exponent in derived upper bound ξ < κ/(2p).
κ = 1
Normalization factor set to 1 in consistency bound.

axioms (2)

domain assumption Background is flat FLRW metric undergoing slow-roll inflation.
Invoked to derive ns and r from modified gravity action.
domain assumption Nonminimal coupling takes form ξ φ² Q inside f(Q,φ).
Assumed to obtain reported dependence on ξ.

pith-pipeline@v0.9.0 · 10744 in / 1564 out tokens · 127034 ms · 2026-05-07T14:56:46.454256+00:00 · methodology

discussion (0)

Reference graph

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