arxiv: 2605.03223 · v1 · submitted 2026-05-04 · ✦ hep-ph · astro-ph.CO· cond-mat.stat-mech

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Constraining Tsallis Corrections to Photon Reheating from Electron-Positron Annihilation

Matias P. Gonzalez

Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:32 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COcond-mat.stat-mech

keywords Tsallis statisticsnonextensive thermodynamicselectron-positron annihilationphoton reheatingN_effBBN constraintsCMB dataearly universe

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The pith

Tsallis nonextensivity corrections to electron-positron annihilation are bounded by |q-1| ≤ 1.3×10^{-2} at 2σ from CMB+BAO and BBN data.

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

The paper generalizes entropy transfer during electron-positron annihilation to photons by applying Tsallis nonextensive statistics only to the lepton pairs while leaving the photon gas extensive. Generalized distribution functions derived from Curado-Tsallis constraints alter the entropy density before annihilation, which shifts the neutrino-to-photon temperature ratio and produces a correction to the effective number of relativistic species N_eff. A combined chi-squared fit to CMB, BAO, and BBN observations then yields the stated bound on the nonextensive parameter q. If this bound holds, any departure from standard Boltzmann-Gibbs extensivity must stay small throughout the MeV-temperature epoch of the early universe.

Core claim

We generalize the entropy transfer from electron-positron annihilation to photons within the Tsallis formalism by using generalized distribution functions from Curado-Tsallis constraints. The nonextensive correction is introduced only in the e^-e^+ pairs while the photon component remains extensive, which modifies the entropic degrees of freedom before annihilation and changes the temperature ratio T_ν/T_γ. The resulting shift is mapped into N_eff. A joint χ² analysis of CMB+BAO and BBN data constrains the nonextensive parameter to |q-1| ≤ 1.3×10^{-2} at 2σ, implying that departures from Boltzmann-Gibbs extensivity must remain small during the MeV era.

What carries the argument

Generalized distribution functions from Curado-Tsallis constraints applied selectively to electron-positron pairs, which deform the electromagnetic entropy density and map the change onto N_eff.

If this is right

The neutrino-to-photon temperature ratio deviates from its standard value (4/11)^{1/3} by an amount linear in (q-1).
N_eff receives a correction proportional to the deviation from extensivity.
Any Tsallis-type model of the radiation-dominated era must keep q close to 1 at temperatures near 1 MeV.
Late-time cosmological data directly limit the statistical mechanics of the electromagnetic plasma at MeV scales.

Where Pith is reading between the lines

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

Future CMB experiments with tighter N_eff errors could shrink the allowed interval on q-1 further.
The selective nonextensivity (matter but not radiation) might be tested by applying the same formalism to other MeV-era processes such as weak freeze-out.
If q is allowed to run with temperature, the bound derived here would apply only near the annihilation epoch and could be relaxed at earlier or later times.

Load-bearing premise

The nonextensive correction applies only to the electron-positron pairs and the Curado-Tsallis distributions correctly describe the entropy transfer to photons in the expanding universe.

What would settle it

A future precision measurement of N_eff that lies outside the range allowed when |q-1| is held at or below 1.3×10^{-2} would show that larger nonextensive corrections are required.

Figures

Figures reproduced from arXiv: 2605.03223 by Matias P. Gonzalez.

**Figure 2.** Figure 2: Modified neutrino to photon temperature ratio. Relative temperature ratio RT (q) = (Tν/Tγ)q/(Tν/Tγ)q=1 induced by the Tsallis deformed electromagnetic entropy transfer during electron-positron annihilation. The horizontal dashed line denotes the standard extensive value, while the vertical dotted line marks the Boltzmann-Gibbs limit q = 1. Tsallis deformation, we define the normalized temperature ratio R… view at source ↗

**Figure 3.** Figure 3: Induced shift in the effective number of relativistic species. The curve shows ∆Neff(q) obtained by propagating the modified neutrino to photon temperature ratio into Neff(q) = Nstd eff R4 T (q). The horizontal dashed line denotes the Standard Model limit ∆Neff = 0, while the vertical dotted line marks the Boltzmann-Gibbs value q = 1. The shaded bands show representative observational 1σ intervals used f… view at source ↗

**Figure 4.** Figure 4: Likelihood profile for the Tsallis parameter. Shifted profile ∆χ2 (q) obtained from the combined BBN and CMB+BAO likelihood. The vertical dotted line indicates the Boltzmann-Gibbs limit q = 1, while the vertical dash-dotted line marks the best-fit value qbest. The horizontal dashed lines show the confidence thresholds used to determine the allowed intervals for the nonextensive parameter. minations summ… view at source ↗

read the original abstract

In this work we generalize the entropy transfer from electron-positron annihilation to photons in the early Universe. The generalization is implemented within the Tsallis formalism by using generalized distribution functions derived from Curado-Tsallis constraints. Through this deformation, the entropy density of the electromagnetic sector is modified, while the photon component is kept extensive. Therefore, the nonextensive correction is introduced only in the $e^-e^+$ pairs. This affects the entropic degrees of freedom before electron-positron annihilation and consequently modifies the temperature ratio $T_\nu/T_\gamma$. The resulting correction is then mapped into the effective number of relativistic species, $N_{\rm eff}$. Finally, by performing a combined $\chi^2$ analysis using CMB$+$BAO and BBN data, we obtain the $2\sigma$ bound $|q-1|\leq 1.3\times10^{-2}$ for the nonextensive parameter. This result implies that any departure or correction from Boltzmann-Gibbs extensivity must remain small during the MeV era.

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 gives a data-driven bound on Tsallis non-extensivity in the MeV universe by tweaking only the fermion part of annihilation entropy, though the mixed-statistics assumption needs scrutiny.

read the letter

The main thing to know is that this paper derives a bound |q − 1| ≤ 1.3 × 10^{-2} at 2σ on the Tsallis parameter by deforming the entropy transfer during electron-positron annihilation in the early universe. They start from the usual reheating story where e+ e− pairs dump entropy into photons after neutrinos have decoupled. Instead of standard Fermi-Dirac statistics for the pairs, they use generalized distributions from the Curado-Tsallis constraints. Photons stay with ordinary Bose-Einstein statistics. This changes the entropy density in the electromagnetic sector before annihilation, which alters the temperature ratio T_ν / T_γ after the pairs disappear, and that feeds into a modified N_eff. They then fit this modified N_eff to a combination of CMB, BAO, and BBN data and extract the limit on q. The work is new in the sense that it applies the Tsallis deformation specifically to this annihilation step and turns it into a data-driven constraint rather than a purely theoretical exercise. The fit itself is standard and uses established datasets, so the numerical result is reproducible in principle. The potential issue is whether the selective deformation is consistent. The electrons and positrons are coupled to the photons through electromagnetic processes, so giving them different statistical mechanics while assuming they share a temperature and conserve energy overall requires some care. The abstract does not spell out how they handle the total energy density or the equilibrium condition under mixed statistics. If the paper shows that the deformation preserves the necessary relations without extra assumptions, then the bound stands; if not, the result rests on an untested split. That is the main soft spot, and it is worth checking in review. This is the kind of paper that interests people already working on non-extensive statistics applied to cosmology. A reader who follows Tsallis or other generalized thermodynamics in the early universe will find the concrete bound useful as a reference point. It is not revolutionary, but it has a clear calculation and a data constraint, so it deserves to go to a serious referee rather than being desk-rejected. I would send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The paper generalizes the entropy transfer during electron-positron annihilation in the early Universe using Tsallis non-extensive statistics via Curado-Tsallis constrained distribution functions applied only to the e⁻e⁺ pairs (photons kept extensive). This modifies the electromagnetic sector entropy density, alters the post-annihilation temperature ratio T_ν/T_γ, maps the correction into N_eff, and yields a 2σ constraint |q−1|≤1.3×10^{-2} from a combined χ² fit to CMB+BAO and BBN data, implying limited departure from Boltzmann-Gibbs extensivity in the MeV era.

Significance. If the selective Tsallis deformation is shown to be thermodynamically consistent, the work supplies a concrete cosmological bound on the non-extensivity parameter q during the epoch of photon reheating, complementing existing limits from other eras and providing a falsifiable prediction for future precision cosmology. The combined data analysis and explicit mapping to N_eff are strengths.

major comments (2)

[generalization and entropy-density derivation] The central derivation assumes that Curado-Tsallis distributions can be applied exclusively to the fermionic e⁻e⁺ component while photons obey standard Bose-Einstein statistics, with a common temperature maintained throughout annihilation. Because electromagnetic interactions couple the two species, this selective deformation requires explicit justification that energy conservation, equilibrium conditions, and the entropy transfer formula remain valid; without it, the modified T_ν/T_γ and subsequent N_eff shift rest on an unverified assumption (see the paragraph following the abstract statement of the generalization and the entropy-density section).
[mapping to T_ν/T_γ and N_eff] The mapping from the deformed entropy density to the temperature ratio and then to ΔN_eff is presented as direct, but the paper does not show an explicit check that the q-deformed distributions preserve the total energy density or the standard relation between entropy and temperature for the combined EM plasma; a concrete derivation or numerical validation of this step is needed to support the claimed bound.

minor comments (2)

[abstract] The abstract states the final bound but does not mention the specific data sets or the value of N_eff used in the fit; adding these details would improve clarity.
[early sections] Notation for the generalized distribution functions (Curado-Tsallis constraints) should be defined once at first use with an explicit equation reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable suggestions regarding the thermodynamic consistency of our approach. We address each major comment below and have made revisions to the manuscript to incorporate additional justifications and explicit derivations as requested.

read point-by-point responses

Referee: The central derivation assumes that Curado-Tsallis distributions can be applied exclusively to the fermionic e⁻e⁺ component while photons obey standard Bose-Einstein statistics, with a common temperature maintained throughout annihilation. Because electromagnetic interactions couple the two species, this selective deformation requires explicit justification that energy conservation, equilibrium conditions, and the entropy transfer formula remain valid; without it, the modified T_ν/T_γ and subsequent N_eff shift rest on an unverified assumption (see the paragraph following the abstract statement of the generalization and the entropy-density section).

Authors: We agree that the selective application of the Tsallis formalism requires careful justification due to the electromagnetic coupling between photons and electron-positron pairs. In the revised manuscript, we have added a new paragraph in the entropy-density section explaining our assumptions. The non-extensivity is applied only to the e⁻e⁺ pairs as a phenomenological model for possible deviations in the fermionic sector during the annihilation epoch, while photons are kept extensive to serve as the reference bath. We maintain a common temperature throughout, justified by the rapid thermalization rates in the MeV-era plasma. Energy conservation is preserved by determining the temperature ratio T_ν/T_γ such that the total energy density before and after annihilation matches, using the deformed distributions. The entropy transfer is computed from the generalized entropy density formula, which is consistent with the standard Boltzmann-Gibbs case when q approaches 1. These clarifications ensure the validity of the subsequent mapping. revision: yes
Referee: The mapping from the deformed entropy density to the temperature ratio and then to ΔN_eff is presented as direct, but the paper does not show an explicit check that the q-deformed distributions preserve the total energy density or the standard relation between entropy and temperature for the combined EM plasma; a concrete derivation or numerical validation of this step is needed to support the claimed bound.

Authors: We have addressed this by including an explicit verification in the revised manuscript. We provide a derivation showing that the total energy density of the electromagnetic plasma, computed as the sum of the photon contribution (standard) and the q-deformed fermion contribution, is conserved by adjusting the post-annihilation temperature. For the small values of |q-1| constrained by our analysis, the deviation from the standard energy-temperature relation is perturbative and does not affect the entropy-to-temperature mapping at the level of precision relevant for N_eff. Additionally, we have performed a numerical check confirming that the generalized entropy density satisfies the thermodynamic identity to high accuracy within our 2σ bound. This supports the robustness of our χ² analysis and the resulting constraint on q. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard parameter constraint from external data

full rationale

The paper introduces the Tsallis parameter q via generalized distribution functions applied only to e+e- pairs (photons kept extensive), derives the resulting modification to entropy density and the post-annihilation temperature ratio T_ν/T_γ, maps the correction to N_eff, and then performs a standard χ² fit to independent CMB+BAO+BBN datasets to obtain the bound |q-1|≤1.3×10^{-2}. This chain does not reduce any claimed prediction or result to the input by construction, nor does it rely on self-citations, ansatzes smuggled via prior work, or renaming of known results. The bound is a genuine data-driven constraint on the model parameter rather than a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the Tsallis parameter q is the central free parameter, with the selective application of non-extensivity to pairs only as a key modeling choice.

free parameters (1)

q
Non-extensive parameter introduced in the Tsallis formalism and constrained by the data fit; central to deforming the entropy density.

axioms (1)

domain assumption Generalized distribution functions derived from Curado-Tsallis constraints accurately implement the deformation
Invoked to modify entropy density of the electromagnetic sector while keeping photons extensive.

pith-pipeline@v0.9.0 · 5488 in / 1534 out tokens · 69248 ms · 2026-05-08T17:32:27.899607+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 30 canonical work pages · 1 internal anchor

[1]

E. W. Kolb and M. S. Turner,The Early Universe (Addison-Wesley, Redwood City, 1990)

1990
[2]

Dodelson,Modern Cosmology(Academic Press, San Diego, 2003)

S. Dodelson,Modern Cosmology(Academic Press, San Diego, 2003)

2003
[3]

Weinberg,Cosmology(Oxford University Press, Oxford, 2008)

S. Weinberg,Cosmology(Oxford University Press, Oxford, 2008)

2008
[4]

Lesgourgues, G

J. Lesgourgues, G. Mangano, G. Miele and S. Pastor,Neu- trino Cosmology(CambridgeUniversityPress,Cambridge, 2013)

2013
[5]

On Effective Degrees of Freedom in the Early Universe.Galaxies2016,4, 78, [arXiv:astro- ph.CO/1609.04979]

L. Husdal, Galaxies4, 78 (2016), doi:10.3390/galaxies4040078, arXiv:1609.04979 [astro- ph.CO]

work page doi:10.3390/galaxies4040078 2016
[6]

2005, Nuclear Physics B, 729, 221, 10.1016/j.nuclphysb.2005.09.041

G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti and P. D. Serpico, Nucl. Phys. B729, 221 (2005), doi:10.1016/j.nuclphysb.2005.09.041, arXiv:hep- ph/0506164

work page doi:10.1016/j.nuclphysb.2005.09.041 2005
[7]

P. F. de Salas and S. Pastor, JCAP07, 051 (2016), doi:10.1088/1475-7516/2016/07/051, arXiv:1606.06986 [hep-ph]

work page doi:10.1088/1475-7516/2016/07/051 2016
[8]

A precision calculation of relic neutrino decoupling,

K. Akita and M. Yamaguchi, JCAP08, 012 (2020), doi:10.1088/1475-7516/2020/08/012, arXiv:2005.07047 [hep-ph]

work page doi:10.1088/1475-7516/2020/08/012 2020
[9]

Saikawa, J

J. Froustey, C. Pitrou and M. C. Volpe, Neutrino de- coupling including flavor oscillations and primordial nu- cleosynthesis, JCAP2020, 015 (2020), doi:10.1088/1475- 7516/2020/12/015

work page doi:10.1088/1475- 2020
[10]

J. J. Bennett, G. Buldgen, P. F. de Salas, M. Drewes, S. Gariazzo, S. Pastor and Y. Y. Y. Wong, JCAP 04, 073 (2021), doi:10.1088/1475-7516/2021/04/073, arXiv:2012.02726 [hep-ph]

work page doi:10.1088/1475-7516/2021/04/073 2021
[11]

L. C. Thomas, T. Dezen, E. B. Grohs and C. T. Kishimoto, Phys. Rev. D101, 063507 (2020), doi:10.1103/PhysRevD.101.063507, arXiv:1910.14050 [astro-ph.CO]

work page doi:10.1103/physrevd.101.063507 2020
[12]

Planck 2018 Results

N. Aghanimet al.(Planck Collaboration), Astron. Astro- phys.641, A6 (2020), doi:10.1051/0004-6361/201833910, arXiv:1807.06209 [astro-ph.CO]

work page doi:10.1051/0004-6361/201833910 2020
[13]

R. H. Cyburt, B. D. Fields, K. A. Olive and T.-H. Yeh, Rev. Mod. Phys.88, 015004 (2016), doi:10.1103/RevModPhys.88.015004, arXiv:1505.01076 [astro-ph.CO]

work page doi:10.1103/revmodphys.88.015004 2016
[14]

Pitrou, A

C. Pitrou, A. Coc, J.-P. Uzan and E. Vangioni, Phys. Rept.754, 1 (2018), doi:10.1016/j.physrep.2018.04.005, arXiv:1801.08023 [astro-ph.CO]

work page doi:10.1016/j.physrep.2018.04.005 2018
[15]

The Atacama Cosmology Telescope: DR6 Constraints on Extended Cosmological Models

E. Calabreseet al.(ACT Collaboration), arXiv:2503.14454 [astro-ph.CO]

work page internal anchor Pith review arXiv
[16]

Goldstein, T.-H

S. Goldstein and J. C. Hill, arXiv:2603.13226 [astro- ph.CO]

work page arXiv
[17]

D. J. Fixsen, Astrophys. J.707, 916 (2009), doi:10.1088/0004-637X/707/2/916, arXiv:0911.1955 [astro-ph.CO]

work page doi:10.1088/0004-637x/707/2/916 2009
[18]

Possible generalization of Boltzmann-Gibbs statistics,

C. Tsallis, J. Stat. Phys.52, 479 (1988), doi:10.1007/BF01016429

work page doi:10.1007/bf01016429 1988
[19]

Tsallis,Introduction to Nonextensive Statistical Me- chanics: Approaching a Complex World(Springer, New York, 2009)

C. Tsallis,Introduction to Nonextensive Statistical Me- chanics: Approaching a Complex World(Springer, New York, 2009)

2009
[20]

E. M. F. Curado and C. Tsallis, J. Phys. A24, L69 (1991); Errata: J. Phys. A24, 3187 (1991); J. Phys. A25, 1019 (1992), doi:10.1088/0305-4470/24/2/004

work page doi:10.1088/0305-4470/24/2/004 1991
[21]

Tsallis, R

C. Tsallis, R. S. Mendes and A. R. Plastino, Physica A 261, 534 (1998), doi:10.1016/S0378-4371(98)00437-3

work page doi:10.1016/s0378-4371(98)00437-3 1998
[22]

J. A. S. Lima, R. Silva and A. R. Plastino, Phys. Rev. Lett.86, 2938 (2001), doi:10.1103/PhysRevLett.86.2938, arXiv:cond-mat/0101030

work page doi:10.1103/physrevlett.86.2938 2001
[23]

Büyükkılıç, D

F. Büyükkılıç, D. Demirhan and A. Güleç, Phys. Lett. A 197, 209 (1995), doi:10.1016/0375-9601(94)00941-H

work page doi:10.1016/0375-9601(94)00941-h 1995
[24]

Tırnaklı, F

U. Tırnaklı, F. Büyükkılıç and D. Demirhan, Phys. Lett. A245, 62 (1998), doi:10.1016/S0375-9601(98)00378-8

work page doi:10.1016/s0375-9601(98)00378-8 1998
[25]

D. F. Torres and U. Tırnaklı, Physica A261, 499 (1998), doi:10.1016/S0378-4371(98)00351-3

work page doi:10.1016/s0378-4371(98)00351-3 1998
[26]

Mitra, Eur

S. Mitra, Eur. Phys. J. C78, 66 (2018), doi:10.1140/epjc/s10052-018-5536-3, arXiv:1709.02095 [nucl-th]

work page doi:10.1140/epjc/s10052-018-5536-3 2018
[27]

J.CleymansandD.Worku,Eur.Phys.J.A48,160(2012), doi:10.1140/epja/i2012-12160-0,arXiv:1203.4343[hep-ph]

work page doi:10.1140/epja/i2012-12160-0 2012
[28]

D. F. Torres, H. Vucetich and A. Plastino, Phys. Rev. Lett.79, 1588 (1997), doi:10.1103/PhysRevLett.79.1588, arXiv:astro-ph/9705068

work page doi:10.1103/physrevlett.79.1588 1997
[29]

A. R. Plastino, A. Plastino, H. G. Miller and H. Uys, Astrophys. Space Sci.290, 275 (2004), doi:10.1023/B:ASTR.0000032529.67037.21. Matias P. Gonzalez: Constraining Tsallis Corrections to Photon Reheating from Electron-Positron Annihilation 9

work page doi:10.1023/b:astr.0000032529.67037.21 2004
[30]

Ghoshal and G

A. Ghoshal and G. Lambiase, arXiv:2104.11296 [astro- ph.CO]

work page arXiv
[31]

Jizba and G

P. Jizba and G. Lambiase, Entropy25, 1495 (2023), doi:10.3390/e25111495, arXiv:2310.19045 [gr-qc]

work page doi:10.3390/e25111495 2023
[32]

On consistency of the interacting (anti)holomorphic higher-spin sector

P. Jizba, G. Lambiase, G. G. Luciano and L. Mastrototaro, Eur. Phys. J. C84, 1076 (2024), doi:10.1140/epjc/s10052- 024-13424-y, arXiv:2403.09797 [gr-qc]

work page doi:10.1140/epjc/s10052- 2024
[33]

M. P. Gonzalez, Eur. Phys. J. C86, 15600 (2026), doi:10.1140/epjc/s10052-026-15600-8

work page doi:10.1140/epjc/s10052-026-15600-8 2026
[34]

Pisanti, A

O. Pisanti, A. Cirillo, S. Esposito, F. Iocco, G. Mangano, G. Miele and P. D. Serpico, Comput. Phys. Com- mun.178, 956 (2008), doi:10.1016/j.cpc.2008.02.015, arXiv:0705.0290 [astro-ph]

work page doi:10.1016/j.cpc.2008.02.015 2008
[35]

Pisanti, G

O. Pisanti, G. Mangano, G. Miele and P. Mazzella, JCAP 04, 020 (2021), doi:10.1088/1475-7516/2021/04/020, arXiv:2011.11537 [astro-ph.CO]

work page doi:10.1088/1475-7516/2021/04/020 2021