arxiv: 2605.00054 · v1 · submitted 2026-04-29 · ⚛️ physics.gen-ph

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The R\'{e}nyi entropy and entropic cosmology

S. I. Kruglov

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Pith reviewed 2026-05-09 20:54 UTC · model grok-4.3

classification ⚛️ physics.gen-ph

keywords Rényi entropyentropic cosmologydark energyFriedmann equationsdeceleration parameterHubble parameterteleparallel gravity

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The pith

Rényi entropy for the apparent horizon produces a dark energy model that matches Planck values for present matter density and deceleration.

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

The paper develops an entropic model of cosmology in which the apparent horizon entropy takes the Rényi form rather than the usual area law. Through the established link between horizon thermodynamics and gravitational dynamics, this replacement produces generalized Friedmann equations for a flat universe containing ordinary matter. Solving these equations yields a dark energy component whose density and pressure lead to present-day values of the matter density parameter and deceleration parameter that agree with Planck satellite measurements. The same model reproduces observed Hubble rates to within five percent over a range of redshifts when the entropy parameter is fixed at a specific value. The construction is shown to be identical to a particular teleparallel gravity theory.

Core claim

Using the Rényi entropy S_R = (1/α) ln(1 + α S_BH) for the apparent horizon in a flat FLRW universe, the thermodynamics-gravity correspondence yields generalized Friedmann equations. These lead to expressions for dark energy density ρ_D and pressure p_D, from which the deceleration parameter q is computed. For appropriate values of the model parameter α, the current values Ω_m0 ≈ 0.315 and q0 ≈ -0.535 are obtained, consistent with Planck data, and the Hubble parameter agrees with observations within 5% for 0.07 ≤ z ≤ 1.75 at α ≈ 0.305 G H0². The cosmology is equivalent to teleparallel gravity with a specific F(T).

What carries the argument

The thermodynamics-gravity correspondence applied to the Rényi entropy of the apparent horizon, which modifies the first law to derive altered Friedmann equations for the scale factor evolution.

If this is right

The late-time acceleration arises from the modified entropy without new dynamical fields.
The cosmology is dual to a specific teleparallel gravity model, allowing geometric reinterpretation.
The Hubble expansion history is consistent with data over a wide redshift range at a fixed entropy parameter.
Current cosmological parameters can be reproduced to match observational constraints from Planck.

Where Pith is reading between the lines

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

If the Rényi modification reflects underlying quantum corrections to black hole entropy, this model may offer a bridge between quantum gravity and cosmology.
Similar entropy replacements could be applied to other gravitational contexts such as early-universe inflation.
The equivalence to teleparallel gravity suggests that entropic modifications might correspond to torsion-based theories more generally.

Load-bearing premise

The thermodynamics-gravity correspondence remains valid when the horizon entropy is replaced by the Rényi form, allowing the direct derivation of generalized Friedmann equations from the modified entropy.

What would settle it

A measurement of the Hubble parameter at redshift z=1.0 that deviates by more than five percent from the model's prediction at α ≈ 0.305 G H0² would falsify the claimed agreement with observational data.

Figures

Figures reproduced from arXiv: 2605.00054 by S. I. Kruglov.

**Figure 1.** Figure 1: The reduced dark energy density ρDG/b vs. the parameter H/√ b. We plotted the dimensionless variable ρDG/b versus H/√ b in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: The normalized density parameters Ωm and ΩD vs. x = H2/b. era. The solution to Eq. (14) for Ωm0 = 0.315 is given by x = 1 137 −200W−1 − 137 200e 137/200 − 137 ≈ 1.04282, (15) where W(z) is the Lambert function which obeys the equation W exp(W) = z. The W−1(z) is the lower branch of W(z) for W(z) ≤ −1. Then we obtain the entropy parameter α = bG π = GH2 0 1.04282π ≈ 0.305 GH2 0 , (16) where H0 is the… view at source ↗

**Figure 3.** Figure 3: The EoS for dark energy wD vs. the dimensionless variable x = H2/b. of Eq. (21) one has limx→∞ wD = −1 and limx→0 wD = (w − 1)/2. Thus, for the large Hubble parameter H (the small Rh), the dark energy EoS is wD = −1 which corresponds for the inflation era. To analyse the observational data it is convenient to introduce the redshift z = 1/a(t) − 1. Then using the continuity equation (5) and EoS p = wρ, we o… view at source ↗

**Figure 4.** Figure 4: The reduced Hubble rate H¯ vs. redshift z. increases the reduced Hubble parameter H¯ also increases. According to the left panel of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: The deceleration parameter q vs. x = H2/b at w = 0, 1/3, 2/3. we have two phases, the universe acceleration and deceleration. Taking into account Eq. (28), we obtain the asymptotic lim H→∞ q = 3w + 1 2 . (30) It follows from Eq. (30) that when w > −1/3 (q > 0) at large H, the universe decelerates and the universe accelerates at w < −1/3. The calculated value 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: The EoS parameter for the matter [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: The Hubble parameter H(z) in units of km/Mpc/sec vs. z. of Hubble parameters are in approximate agreement with the observational Hubble data for 0.07 ≤ z ≤ 1.75 [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60] at b = απ/G ≈ 0.959 H2 0 (α ≈ 0.305 GH2 0 ) which gives the correct value for the normalized density parameter of the matter at the current era. Some observational Hubble data for 0.07 ≤ z ≤ 1.75 are rep… view at source ↗

read the original abstract

Entropic cosmology with the R\'{e}nyi entropy of the apparent horizon $S_R=(1/\alpha)\ln(1+\alpha S_{BH})$, where $S_{BH}$ is the Bekenstein--Hawking entropy, is studied. By virtue of the thermodynamics-gravity correspondence a model of dark energy is investigated. The generalised Friedmann equations for the Friedmann--Lema\^{i}tre--Robertson--Walker spatially flat universe with the barotropic matter fluid are obtained. We compute the dark energy density $\rho_D$, pressure $p_D$ and the deceleration parameter $q$ of the universe. At some model parameters the normalized density parameter of the matter $\Omega_{m0}\approx 0.315$ and the deceleration parameter $q_0\approx -0.535$ for the current epoch, which are in the agreement with the Planck data, are found. Making use of the thermodynamics-gravity correspondence, we describe the late time of the universe acceleration. The entropic cosmology considered is equivalent to cosmology based on the teleparallel gravity with the definite function $F(T)$. The Hubble parameters are in approximate agreement (within $5$ percents) with the observational Hubble data for redshifts $0.07\leq z \leq 1.75$ at the entropy parameter $\alpha\approx 0.305~GH_0^2$.

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

Routine extension of entropic cosmology via Rényi entropy with parameter fitting and teleparallel equivalence.

read the letter

The main thing to know is that this paper takes the Rényi entropy of the apparent horizon, substitutes it into the usual thermodynamic identity for gravity, and obtains generalized Friedmann equations for a flat universe with matter. From there it derives the dark energy density and pressure, the deceleration parameter, and shows that for a particular value of the entropy parameter the model reproduces the observed matter density fraction and deceleration at the present time. It also claims an equivalence to teleparallel gravity with a specific F(T) and finds that the Hubble expansion history stays close to data over a modest redshift range. What the paper does well is to carry out the calculations explicitly and to establish the mapping to the teleparallel framework. The equivalence is a clear, reproducible relation that could be of interest to people working in modified gravity theories. The soft spots are in the fitting procedure and the foundational assumption. The values for Ω_m0 and q0 are obtained by selecting the free parameter α to match the data, which means the agreement is by construction rather than a genuine prediction. The stress-test concern is valid here: replacing the entropy in the first law assumes the temperature and flux terms remain unchanged, and without seeing the full steps it is hard to confirm that the effective dark energy equation of state follows rigorously from the modified entropy. If that step has a gap, the rest of the results rest on shaky ground. This paper is aimed at researchers who study entropic approaches to dark energy or look for connections between thermodynamic models and teleparallel gravity. It is an honest extension of existing ideas rather than a breakthrough. The derivations seem standard for the field, so the work deserves a serious referee to examine the details of the entropy substitution and the equivalence derivation. I would recommend sending it for peer review.

Referee Report

4 major / 2 minor

Summary. The manuscript studies entropic cosmology by replacing the Bekenstein-Hawking entropy of the apparent horizon with the Rényi form S_R = (1/α) ln(1 + α S_BH). Via the thermodynamics-gravity correspondence it derives generalized Friedmann equations for a flat FLRW universe containing barotropic matter, computes the resulting dark-energy density ρ_D, pressure p_D and deceleration parameter q, and reports that specific choices of the entropy parameter α and other model parameters yield Ω_m0 ≈ 0.315 and q0 ≈ −0.535, in agreement with Planck values. The model is asserted to be equivalent to teleparallel gravity with a definite F(T) function, and the Hubble parameter is stated to agree with observational data within 5 % for 0.07 ≤ z ≤ 1.75 when α ≈ 0.305 G H0².

Significance. If the substitution of the Rényi entropy into the horizon first law can be shown to produce the modified equations without additional inconsistencies, and if the numerical agreements can be demonstrated to be independent of the fitting procedure, the work would establish a concrete thermodynamic route to a specific late-time acceleration model and a direct dictionary between entropic cosmology and a particular teleparallel F(T) theory. Such a link would be of interest for exploring non-extensive entropies as origins of effective dark energy.

major comments (4)

[Derivation of generalized Friedmann equations] The derivation of the generalized Friedmann equations (the step that replaces S_BH by S_R inside the first-law relation on the apparent horizon) assumes that the Unruh temperature, apparent-horizon radius and energy flux remain exactly the same as in the Bekenstein-Hawking case and that the non-additive logarithmic form introduces no extra terms. No explicit verification is supplied that the continuity equation is preserved or that the effective ρ_D and p_D follow rigorously from this substitution; this assumption is load-bearing for every subsequent result.
[Numerical results for Ω_m0 and q0] The reported values Ω_m0 ≈ 0.315 and q0 ≈ −0.535 are obtained by choosing the entropy parameter α together with other model parameters so as to reproduce the observed numbers (as stated in the abstract). Because the agreement is achieved by construction, it does not constitute an independent test of the model and therefore does not strengthen the claim that the Rényi-entropic cosmology matches Planck data.
[Equivalence to teleparallel gravity] The claimed equivalence to teleparallel gravity with a definite F(T) is asserted but the explicit form of F(T) and the mapping steps that establish the exact correspondence are not provided; without them the equivalence remains a statement rather than a demonstrated result.
[Hubble-parameter comparison] The approximate 5 % agreement of H(z) with observational Hubble data for 0.07 ≤ z ≤ 1.75 is obtained at the same tuned value α ≈ 0.305 G H0² used for the Ω_m0 and q0 fits. An a-priori choice of α or a demonstration that the agreement persists for a range of α would be required to make this comparison a genuine test.

minor comments (2)

[Abstract and parameter definitions] The notation 'G H0²' in the abstract and parameter choice should be clarified (is G Newton’s constant?) and used consistently throughout the text and equations.
[Notation] Ensure that the symbols for the normalized matter density Ω_m0 and the present-day deceleration parameter q0 are defined once and employed uniformly in all equations and numerical statements.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and indicate planned revisions to enhance the rigor and clarity of the presentation.

read point-by-point responses

Referee: The derivation of the generalized Friedmann equations (the step that replaces S_BH by S_R inside the first-law relation on the apparent horizon) assumes that the Unruh temperature, apparent-horizon radius and energy flux remain exactly the same as in the Bekenstein-Hawking case and that the non-additive logarithmic form introduces no extra terms. No explicit verification is supplied that the continuity equation is preserved or that the effective ρ_D and p_D follow rigorously from this substitution; this assumption is load-bearing for every subsequent result.

Authors: In the standard entropic cosmology framework, the first law is applied with the modified entropy S_R while retaining the usual Unruh temperature and horizon radius, consistent with the thermodynamics-gravity correspondence. This yields the generalized equations, and the continuity equation follows directly from the definition of the energy flux. We agree an explicit verification is useful. In the revision we will insert a dedicated derivation subsection confirming that the continuity equation is preserved and detailing the steps from the first law with S_R to ρ_D and p_D. revision: yes
Referee: The reported values Ω_m0 ≈ 0.315 and q0 ≈ −0.535 are obtained by choosing the entropy parameter α together with other model parameters so as to reproduce the observed numbers (as stated in the abstract). Because the agreement is achieved by construction, it does not constitute an independent test of the model and therefore does not strengthen the claim that the Rényi-entropic cosmology matches Planck data.

Authors: We acknowledge that the quoted values result from parameter selection to match observations, which is standard for models containing free parameters such as α. The demonstration shows that viable choices exist within the Rényi-entropic framework. We will revise the text to describe these explicitly as consistency checks rather than independent predictions and will add a brief exploration of the α interval that permits agreement with Planck data. revision: partial
Referee: The claimed equivalence to teleparallel gravity with a definite F(T) is asserted but the explicit form of F(T) and the mapping steps that establish the exact correspondence are not provided; without them the equivalence remains a statement rather than a demonstrated result.

Authors: We thank the referee for highlighting this omission. The equivalence follows from equating the effective dark-energy density and pressure obtained from the Rényi entropy to the corresponding quantities in teleparallel gravity. In the revised manuscript we will supply the explicit F(T) function together with the step-by-step mapping that establishes the exact correspondence. revision: yes
Referee: The approximate 5 % agreement of H(z) with observational Hubble data for 0.07 ≤ z ≤ 1.75 is obtained at the same tuned value α ≈ 0.305 G H0² used for the Ω_m0 and q0 fits. An a-priori choice of α or a demonstration that the agreement persists for a range of α would be required to make this comparison a genuine test.

Authors: The value α ≈ 0.305 G H0² is fixed by the present-day parameters Ω_m0 and q0; the H(z) comparison then tests whether the resulting expansion history remains consistent with data at higher redshifts. We will clarify this procedure in the text and add a short robustness check showing H(z) for a narrow interval of α values around the reference choice. revision: partial

Circularity Check

0 steps flagged

No significant circularity; standard parameter fitting to external data

full rationale

The paper applies the thermodynamics-gravity correspondence to the Rényi entropy form S_R = (1/α) ln(1 + α S_BH) to obtain generalized Friedmann equations, then derives explicit expressions for ρ_D, p_D and q. Numerical values such as Ω_m0 ≈ 0.315, q0 ≈ -0.535 and H(z) agreement within 5 % are obtained by selecting model parameters (including α ≈ 0.305 G H_0²) that reproduce Planck and Hubble data. This is ordinary model calibration against external benchmarks, not a reduction of any claimed result to its own inputs by construction. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model rests on the thermodynamics-gravity correspondence applied to Rényi entropy and on the choice of a single free parameter α that is adjusted to observational numbers.

free parameters (1)

alpha
Entropy deformation parameter introduced in the Rényi formula and adjusted to produce Ω_m0 ≈ 0.315, q0 ≈ -0.535, and Hubble agreement within 5 percent.

axioms (1)

domain assumption thermodynamics-gravity correspondence holds for Rényi entropy of the apparent horizon
Invoked to obtain the generalized Friedmann equations from the modified entropy expression.

pith-pipeline@v0.9.0 · 5548 in / 1474 out tokens · 87836 ms · 2026-05-09T20:54:51.240692+00:00 · methodology

discussion (0)

Reference graph

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