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arxiv: 2605.18070 · v1 · pith:WRVV3SB2new · submitted 2026-05-18 · 🌀 gr-qc

Slow-roll inflation in (dual) Kaniadakis cosmology

Leila Liravi , Ahmad Sheykhi This is my paper

Pith reviewed 2026-05-20 09:58 UTC · model grok-4.3

classification 🌀 gr-qc
keywords slow-roll inflationKaniadakis entropydeformation parametermodified Friedmann equationsscalar spectral indextensor-to-scalar ratioPlanck constraintsprimordial perturbations
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The pith

Kaniadakis entropy modifications allow slow-roll inflation only for strongly suppressed deformation parameters to match Planck data.

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

This paper examines how a deformation parameter kappa generalizes the entropy in Kaniadakis cosmology and alters the Friedmann equations that govern the early universe. It derives the resulting slow-roll dynamics and computes the scalar spectral index ns, tensor spectral index, and tensor-to-scalar ratio r. The analysis shows that consistency with Planck constraints requires kappa to be very small in the standard formulation, while the dual version permits viable regions for the primordial power spectrum. These kappa corrections could leave observable imprints on early-universe perturbations if the parameter is not suppressed. The work thereby links a non-extensive thermodynamic generalization to concrete predictions testable by cosmic microwave background measurements.

Core claim

In the Kaniadakis framework the entropy is deformed by the parameter kappa, which produces modified Friedmann equations. These equations alter the slow-roll parameters and yield inflationary observables that remain compatible with Planck constraints on ns and r only when kappa is strongly suppressed in the standard case. The dual Kaniadakis formulation permits consistency with the observed scalar power spectrum within certain parameter regions, indicating that kappa-induced corrections can connect non-extensive thermodynamics to the physics of primordial perturbations.

What carries the argument

Kaniadakis entropy formula deformed by the parameter kappa, which generates modified Friedmann equations and thereby changes the slow-roll inflationary observables ns and r.

Load-bearing premise

The modified Friedmann equations from Kaniadakis entropy produce slow-roll dynamics that can be compared directly to Planck measurements of ns and r without additional unaccounted corrections.

What would settle it

A future measurement finding the tensor-to-scalar ratio r or the scalar spectral index ns outside the narrow ranges allowed by the suppressed values of kappa would rule out consistency with the model.

Figures

Figures reproduced from arXiv: 2605.18070 by Ahmad Sheykhi, Leila Liravi.

Figure 1
Figure 1. Figure 1: FIG. 1: The behaviour of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The behaviour of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The behaviour of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The behaviour of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We investigate slow-roll inflation within the framework of Kaniadakis and dual Kaniadakis cosmology, where the usual entropy formalism is generalized through a deformation parameter $\kappa$. By deriving the modified Friedmann equations and the corresponding inflationary dynamics induced by Kaniadakis entropy, we analyze the deviations from standard inflation arising from $\kappa$-corrections. We compute the scalar and tensor spectral indices, the tensor-to-scalar ratio, and examine the observational constraints on the deformation parameter. Our results show that consistency with current observational data imposes stringent bounds on the deformation parameter $\kappa$. In the standard Kaniadakis formulation, viable slow-roll inflationary scenarios compatible with the Planck constraints on the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ can be obtained, although the allowed values of $\kappa$ are strongly suppressed. For the dual Kaniadakis formulation, we find that the primordial power spectrum of scalar perturbations can remain consistent with observational data within certain parameter regions. We also verify the compatibility of the predicted scalar power spectrum with the latest Planck results and discuss the phenomenological implications of the $\kappa$-induced corrections. These findings suggest that Kaniadakis cosmology may leave potentially observable imprints on primordial perturbations, providing a possible connection between non-extensive thermodynamics and the physics of the early universe.

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.

Referee Report

2 major / 2 minor

Summary. The paper derives modified Friedmann equations from the Kaniadakis entropy (and its dual) with deformation parameter κ, studies the resulting slow-roll inflationary dynamics, computes the scalar spectral index ns, tensor-to-scalar ratio r and scalar power spectrum, and reports that small bounded values of κ yield models consistent with Planck constraints on ns and r.

Significance. If the central results hold, the work would link non-extensive thermodynamics to early-universe observables and furnish concrete bounds on κ from CMB data. The explicit derivation of background equations from the first law is a clear strength; however, the significance is limited by the absence of any verification that the perturbation sector remains unmodified.

major comments (2)
  1. [§4] §4 (slow-roll parameters and spectral indices): the expressions for ns ≈ 1−2ε−η and r ≈ 16ε are inserted directly using the κ-modified H(φ) and slow-roll parameters ε, η. No derivation or justification is given that the Mukhanov-Sasaki equation or tensor quadratic action retains its standard GR form; any κ-dependent corrections to the fluctuation Lagrangian would shift ns and r by amounts comparable to the reported bounds on κ, rendering the Planck-compatibility claim unsupported.
  2. [§5] §5 (observational constraints): the bounds on κ are obtained by requiring the computed ns and r to lie inside the Planck 1σ/2σ contours. This procedure makes the viability of the model dependent on the same data used to constrain the free parameter, without an independent consistency check or falsifiable prediction outside the fitting procedure.
minor comments (2)
  1. [§2] The distinction between the standard Kaniadakis and dual Kaniadakis formulations is introduced in the abstract and §2 but never summarized in a single comparative table or paragraph, making it difficult for readers to track which results apply to which case.
  2. [§3] Notation for the modified Hubble parameter and potential is introduced without an explicit list of all κ-dependent terms; a short appendix collecting the leading-order corrections would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review of our manuscript on slow-roll inflation in Kaniadakis cosmology. We address each of the major comments below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [§4] §4 (slow-roll parameters and spectral indices): the expressions for ns ≈ 1−2ε−η and r ≈ 16ε are inserted directly using the κ-modified H(φ) and slow-roll parameters ε, η. No derivation or justification is given that the Mukhanov-Sasaki equation or tensor quadratic action retains its standard GR form; any κ-dependent corrections to the fluctuation Lagrangian would shift ns and r by amounts comparable to the reported bounds on κ, rendering the Planck-compatibility claim unsupported.

    Authors: We appreciate the referee's observation regarding the perturbation sector. Our approach modifies the background cosmology through the Kaniadakis entropy in the first law applied to the cosmological horizon, resulting in κ-dependent Friedmann equations. The slow-roll parameters ε and η are then defined in the standard way but evaluated with the modified Hubble parameter. We have implicitly assumed that the linear perturbation equations, including the Mukhanov-Sasaki equation for scalars and the tensor modes, remain unchanged because the underlying gravitational dynamics at the perturbative level are still described by general relativity, with the entropy modification primarily affecting the thermodynamic relations at the background. However, we acknowledge that this assumption requires explicit justification. In the revised manuscript, we will add a new subsection or paragraph in §4 explaining this assumption and arguing that since the modification arises from the entropy rather than a direct change to the Einstein-Hilbert action, the quadratic action for perturbations is expected to retain its GR form to leading order. We will also note that for the small κ values allowed by data, any potential corrections would be suppressed. This addresses the concern and supports the validity of our ns and r expressions. revision: yes

  2. Referee: [§5] §5 (observational constraints): the bounds on κ are obtained by requiring the computed ns and r to lie inside the Planck 1σ/2σ contours. This procedure makes the viability of the model dependent on the same data used to constrain the free parameter, without an independent consistency check or falsifiable prediction outside the fitting procedure.

    Authors: We thank the referee for this comment on the nature of our constraints. In inflationary model building, it is standard to use CMB data to place bounds on model parameters such as κ by comparing predicted values of ns and r to observational limits. Our analysis does provide a falsifiable aspect in that the specific dependence of the observables on κ can be tested with improved precision from future experiments like CMB-S4 or LiteBIRD. Additionally, as mentioned in the abstract, we verify the compatibility of the predicted scalar power spectrum amplitude with Planck results, which serves as an independent check since the amplitude is normalized separately. In the revised version, we will expand the discussion in §5 to highlight these points and clarify that while the current bounds are derived from fitting, the model predicts a particular trajectory in the ns-r plane as a function of κ, offering testable predictions beyond the current data. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses external Planck benchmarks on independently derived background

full rationale

The paper starts from the Kaniadakis entropy formula, applies the first law to the apparent horizon to obtain modified Friedmann equations, inserts the resulting H(φ) into the standard slow-roll definitions of ε and η, and evaluates the usual ns ≈ 1−2ε−η and r ≈ 16ε expressions. These computed observables are then compared against external Planck constraints to bound the free parameter κ. Because the thermodynamic step supplies new content not present in the final data fit, the Planck comparison is an external benchmark rather than a self-referential input, and no self-citation chain or uniqueness theorem is required to close the argument, the derivation chain remains non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of the Kaniadakis entropy deformation to cosmological horizons during inflation and on the validity of the slow-roll approximation once the Friedmann equations are modified. Kappa functions as the sole adjustable parameter whose value is bounded rather than derived.

free parameters (1)
  • kappa
    Deformation parameter introduced by the Kaniadakis entropy generalization; its magnitude is constrained by requiring the predicted spectral index and tensor-to-scalar ratio to match Planck observations.
axioms (2)
  • domain assumption The Kaniadakis entropy formula applies to the apparent horizon or to the thermodynamics of the early universe and yields modified Friedmann equations.
    This assumption is invoked when the authors state that they derive the modified Friedmann equations from the generalized entropy.
  • domain assumption Slow-roll conditions remain valid after the kappa corrections are included.
    The computation of spectral indices and the tensor-to-scalar ratio presupposes that the usual slow-roll parameters can still be defined and used.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Constraints on Kaniadakis Cosmology from Starobinsky Inflation and Primordial Tensor Perturbations

    gr-qc 2026-05 unverdicted novelty 4.0

    Kaniadakis entropic cosmology modifies early-universe dynamics and is constrained by its predictions for Starobinsky inflation and the primordial tensor spectrum using current CMB and gravitational-wave observations.

Reference graph

Works this paper leans on

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    The energy density and pressure associated with the inflaton field are expressed as ρφ = 1 2 ˙φ 2 + V (φ), (25) pφ = 1 2 ˙φ 2 − V (φ)

    Standard slow-roll inflation Within the standard slow-roll framework, we con- sider a canonical scalar field φ with potential V (φ) in a flat Friedmann-Lema ˆ ıtre-Robertson-Walker (FLR W) universe. The energy density and pressure associated with the inflaton field are expressed as ρφ = 1 2 ˙φ 2 + V (φ), (25) pφ = 1 2 ˙φ 2 − V (φ). (26) The background dynamics...

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    Kaniadakis-modified slow-roll inflation We now examine inflation within the (dual) Kaniadakis-modified Friedmann framework. If we solve the first modified Friedmann equations (21) and (23) with respect to H and use the second conditions in (30), we obtain to the leading order in κ H ≃ √ 8πG 3 V ± √ 27π 2G7V 3 κ 2 64 , (50) where the plus and minus signs corresp...

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    The case n = 1 has already been studied in [3] within the Kaniadakis framework. In this work, we focus on the n = 2 case and compute both the Kaniadakis case and its dual. Therefore, all subsequent calculations are performed for n = 2. Solving Eq. (56) for φ f , we obtain φ f ≃ 1 2 √ πG ∓ 27κ 2 32V 2 0 √ π 3 G5 . (57) Based on Eq. (39), the value of N can...

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    When the Kanidakis correction term is taken into account ( κ 2 ̸= 0), as one can see from Eq. (61), we have nonzero value for parameter η. This may show that the Kaniadkis cosmology is richer compared to standard case. Therefore Eq. (61) for n = 2 can be rewritten as η(φ c) ≃ ± 9 κ 2 256V 2 0 πG 5φ 6 c . (62) Substituting φ c from Eq. (59), leads to η(φ c...

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