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arxiv: 2605.21050 · v1 · pith:CUH6BDLInew · submitted 2026-05-20 · 🌀 gr-qc

Constraints on Kaniadakis Cosmology from Starobinsky Inflation and Primordial Tensor Perturbations

Abdelhakim Benkrane , Giuseppe Gaetano Luciano This is my paper

Pith reviewed 2026-05-21 03:44 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Kaniadakis statisticsentropic cosmologyStarobinsky inflationprimordial gravitational wavestensor perturbationscosmological constraintsFriedmann equations
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The pith

Kaniadakis-deformed horizon entropy modifies the Friedmann equations during Starobinsky inflation, altering both slow-roll dynamics and the spectrum of primordial gravitational waves in ways that current observations can constrain.

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

This paper applies the gravity-thermodynamics conjecture with Kaniadakis statistics to replace the standard horizon entropy with a generalized form. The resulting modification changes how the universe expands during the inflationary phase in a Starobinsky-like model. It affects the evolution of tensor modes that produce primordial gravitational waves and introduces corrections to key inflationary quantities like the spectral index and tensor-to-scalar ratio. Comparing these changes to data from Planck and BICEP/Keck allows the authors to set limits on the strength of the Kaniadakis deformation parameter.

Core claim

The generalized entropic corrections from Kaniadakis statistics simultaneously affect the evolution of tensor perturbations and the inflationary slow-roll dynamics in a Starobinsky-like scenario, inducing characteristic deviations in the PGW spectrum as well as nontrivial corrections to the main inflationary observables, which when compared to Planck and BICEP/Keck data yield stringent constraints on the Kaniadakis parameter.

What carries the argument

The Kaniadakis-deformed version of the horizon entropy, which via the gravity-thermodynamics conjecture leads to modified Friedmann equations applied to the early universe inflation.

If this is right

  • The PGW spectrum exhibits deviations from the standard prediction due to the entropy modification.
  • Inflationary observables such as the scalar spectral index and tensor-to-scalar ratio receive corrections.
  • Stringent constraints on the Kaniadakis parameter are obtained by fitting to current cosmological data.
  • The model provides a phenomenologically consistent extension of the standard cosmological paradigm.

Where Pith is reading between the lines

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

  • This framework could be tested further with future gravitational wave detectors sensitive to the primordial background.
  • Similar entropic modifications might influence other phases of cosmic evolution beyond inflation.
  • Connecting generalized statistics to inflation offers a way to probe quantum effects in gravity through cosmology.

Load-bearing premise

The gravity-thermodynamics conjecture holds when the horizon entropy is replaced by its Kaniadakis-deformed counterpart, allowing consistent modification of the Friedmann equations during inflation.

What would settle it

An observation of the primordial gravitational wave spectrum or inflationary observables that matches the unmodified Starobinsky model exactly, without the predicted deviations for nonzero Kaniadakis parameter, would falsify the applicability of this modification.

Figures

Figures reproduced from arXiv: 2605.21050 by Abdelhakim Benkrane, Giuseppe Gaetano Luciano.

Figure 1
Figure 1. Figure 1: FIG. 1: Plot of the PGW spectrum as a function of the frequency [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot of the PGW spectrum as a function of the frequency [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Evolution of the inflationary observables as functions of the rescaled Kaniadakis parameter. Upper-left panel: scalar [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
read the original abstract

We investigate a generalized entropic cosmology obtained by applying the gravity-thermodynamics conjecture to the Universe horizon using Kaniadakis statistics, namely a relativistic extension of the standard Boltzmann--Gibbs formalism. The resulting deformation of the horizon entropy naturally modifies the Friedmann dynamics and provides a phenomenologically consistent extension of the $\Lambda$CDM paradigm. Within this framework, we explore the implications of the modified cosmological dynamics for the physics of the early Universe, focusing in particular on primordial gravitational waves (PGWs) and slow-roll inflation in a Starobinsky-like scenario. We show that the generalized entropic corrections simultaneously affect the evolution of tensor perturbations and the inflationary slow-roll dynamics, inducing characteristic deviations in the PGW spectrum as well as nontrivial corrections to the main inflationary observables. By confronting the theoretical predictions with the latest Planck and BICEP/Keck observations, we derive stringent constraints on the Kaniadakis parameter and assess the observational viability of the model. Our results establish a direct connection between generalized horizon thermodynamics and inflationary cosmology, showing that quantum-statistical modifications of the entropy-area law can propagate into potentially observable signatures in 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 investigates a generalized entropic cosmology obtained by applying the gravity-thermodynamics conjecture to the cosmological horizon using Kaniadakis statistics, which deforms the Bekenstein-Hawking entropy as S_K = (1/K) sinh(K S_BH). This leads to modified Friedmann equations that are then applied to a Starobinsky-like inflationary scenario. The work derives the resulting corrections to slow-roll parameters (n_s, r) and the tensor perturbation equation, showing characteristic deviations in the primordial gravitational wave spectrum. These predictions are confronted with Planck and BICEP/Keck data to obtain constraints on the Kaniadakis deformation parameter K.

Significance. If the central derivations hold, the manuscript provides a concrete link between quantum-statistical deformations of horizon entropy and observable inflationary quantities, including shifts in the tensor-to-scalar ratio and the PGW spectrum. This extends the reach of generalized entropic cosmologies into the early Universe and yields falsifiable bounds on K from existing data. The approach is noteworthy for attempting to propagate thermodynamic modifications directly into both background dynamics and perturbation equations rather than treating them as ad-hoc corrections.

major comments (2)
  1. [Sections 2–3 (modified Friedmann equations and Starobinsky application)] The central claim rests on the validity of the gravity-thermodynamics conjecture when the Kaniadakis entropy is substituted into the first law on the apparent horizon and the resulting modified Friedmann equations are extrapolated to the de Sitter-like background of Starobinsky inflation. During inflation the Hubble radius is nearly constant and the inflaton dominates, so the thermodynamic identification of temperature and entropy may require additional assumptions about the inflaton sector or the form of the first law. This step is load-bearing for the predicted deviations in slow-roll parameters and the tensor spectrum; without explicit justification or a consistency check, the subsequent constraints on K lack theoretical grounding.
  2. [Section on tensor perturbations and PGW spectrum] The derivation of the modified tensor mode equation and the resulting PGW spectrum (presumably in the section on primordial gravitational waves) assumes that the only change enters through the background Hubble evolution. It is not shown whether the Kaniadakis deformation also induces direct corrections to the Mukhanov-Sasaki equation for tensor modes or to the Bunch-Davies vacuum; if such terms are absent by construction, the claimed “characteristic deviations” in the spectrum may be smaller than stated and should be quantified with an explicit error budget.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction should explicitly state the range of K values explored and whether the Starobinsky potential itself is left unmodified or receives entropic corrections.
  2. [Throughout] Notation for the Kaniadakis parameter (denoted K) should be distinguished from the curvature parameter or other constants appearing in the Starobinsky potential to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points regarding the theoretical foundations of our approach, and we address each one below. We have revised the manuscript to incorporate additional justifications and quantifications where needed.

read point-by-point responses

  1. Referee: [Sections 2–3 (modified Friedmann equations and Starobinsky application)] The central claim rests on the validity of the gravity-thermodynamics conjecture when the Kaniadakis entropy is substituted into the first law on the apparent horizon and the resulting modified Friedmann equations are extrapolated to the de Sitter-like background of Starobinsky inflation. During inflation the Hubble radius is nearly constant and the inflaton dominates, so the thermodynamic identification of temperature and entropy may require additional assumptions about the inflaton sector or the form of the first law. This step is load-bearing for the predicted deviations in slow-roll parameters and the tensor spectrum; without explicit justification or a consistency check, the subsequent constraints on K lack theoretical grounding.

    Authors: We appreciate the referee's concern about applying the gravity-thermodynamics conjecture in the inflationary regime. The derivation in Sections 2 and 3 follows the standard procedure established in the entropic cosmology literature: the first law is applied directly to the apparent horizon using the Kaniadakis entropy, yielding modified Friedmann equations that hold for any dominant energy component, including the inflaton. In the quasi-de Sitter background relevant to Starobinsky inflation, the apparent horizon radius approaches the Hubble radius, and the temperature is identified with the standard Hawking temperature T = H/(2π). We have added a new paragraph in Section 2 that explicitly discusses this identification, notes that the thermodynamic relation is independent of the inflaton sector at the horizon level, and provides a consistency check by recovering the exact standard Starobinsky slow-roll parameters in the K → 0 limit. While a microscopic derivation from quantum gravity is beyond the phenomenological scope of this work, the approach is consistent with prior studies on generalized entropies and grounds the subsequent constraints on K. revision: yes

  2. Referee: [Section on tensor perturbations and PGW spectrum] The derivation of the modified tensor mode equation and the resulting PGW spectrum (presumably in the section on primordial gravitational waves) assumes that the only change enters through the background Hubble evolution. It is not shown whether the Kaniadakis deformation also induces direct corrections to the Mukhanov-Sasaki equation for tensor modes or to the Bunch-Davies vacuum; if such terms are absent by construction, the claimed “characteristic deviations” in the spectrum may be smaller than stated and should be quantified with an explicit error budget.

    Authors: We thank the referee for this observation on the tensor sector. The Kaniadakis modification enters exclusively through the altered background Hubble evolution obtained from the modified Friedmann equations; at linear order, the tensor perturbation equation remains the standard wave equation in conformal time with the modified H(η) as the sole background input, and the Bunch-Davies initial condition is preserved for the quasi-de Sitter phase. No additional direct corrections to the Mukhanov-Sasaki variable or vacuum state arise from the entropy deformation. In the revised manuscript we have added an explicit derivation of the tensor mode equation confirming the absence of extra terms, together with a quantitative error budget: we compare the resulting PGW spectrum to the standard case for representative values of K, showing the relative deviation in amplitude as a function of wavenumber and demonstrating that the reported characteristic deviations are attributable to the changed expansion history. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper starts from the gravity-thermodynamics conjecture, replaces the horizon entropy with its Kaniadakis form to obtain modified Friedmann equations, then applies the resulting dynamics to a Starobinsky-like inflationary background to compute corrections to slow-roll parameters and the tensor power spectrum. These derived quantities are compared to Planck and BICEP/Keck data solely to place bounds on the free Kaniadakis parameter. No step reduces by construction to a fitted input renamed as a prediction, nor does any load-bearing premise collapse to a self-citation whose content is itself unverified within the paper. The derivation remains self-contained once the initial thermodynamic replacement is granted; the observational confrontation is standard parameter estimation rather than a tautological loop.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model rests on one domain assumption (gravity-thermodynamics conjecture) and introduces a single free parameter (Kaniadakis deformation) whose value is fixed by observational fit. No new particles or forces are postulated.

free parameters (1)
  • Kaniadakis deformation parameter
    Single extra constant that deforms the horizon entropy and is bounded by fitting the modified inflationary observables to data.
axioms (1)
  • domain assumption Gravity-thermodynamics conjecture applied to the cosmological horizon remains valid under Kaniadakis statistics
    Invoked to replace the standard entropy-area law and thereby alter the Friedmann dynamics.

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