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arxiv: 2605.29132 · v1 · pith:ZRKQTDP6new · submitted 2026-05-27 · ✦ hep-ph · astro-ph.CO· cond-mat.stat-mech

Dynamical Tsallis WIMP Freeze-Out and Residual Memory Channels in the Radiation Sector

Matias P. Gonzalez This is my paper

Pith reviewed 2026-06-29 10:31 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.COcond-mat.stat-mech
keywords Tsallis nonextensive statisticsWIMP freeze-outdark matter relic abundanceresidual memory channeleffective neutrino number N_effdynamical q parametercosmological constraints
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The pith

A Tsallis nonextensive deformation applied only to dark matter during freeze-out can carry over to the radiation sector and still match current limits on the effective neutrino number.

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

This work extends the standard calculation of WIMP dark matter relic density by allowing the Tsallis parameter q to vary with the temperature ratio x during the decoupling process. The deformation is limited to the dark matter equilibrium distribution while the radiation background stays standard, and the Boltzmann equation is solved to find the abundance at freeze-out. After that point a residual q value is fed into a memory channel that deforms the electron-positron and neutrino distributions, which changes the energy density in neutrinos and the reheating of photons. The resulting shift in N_eff is then checked against compressed CMB plus BAO data. The central finding is that for a range of dark matter masses and starting q values the predicted N_eff stays compatible with observations.

Core claim

The Tsallis parameter q is made dynamical as a function of x = m_χ/T that relaxes toward the extensive limit q = 1, but not fully before freeze-out occurs. Under the sectorial deformation the relic density is computed for various masses and initial q, yielding a residual q at freeze-out that seeds a memory channel deforming only the neutrino and electron-positron sectors. This produces a correction to N_eff whose magnitude remains within the compressed CMB+BAO constraint for the explored parameter space.

What carries the argument

Dynamical q(x) obtained from maximum entropy with Curado-Tsallis constraints, evolving toward q=1 while preserving nonextensivity until freeze-out, and the subsequent residual-memory channel into the radiation sector.

If this is right

  • Different initial q and masses produce different residual q_fo and thus different N_eff shifts.
  • The memory channel specifically alters neutrino energy density and photon reheating.
  • Compatibility with bounds holds provided photons remain extensive.
  • The scenario requires that nonextensivity is not erased before freeze-out.

Where Pith is reading between the lines

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

  • Such a mechanism could be tested by future more precise N_eff measurements from CMB experiments.
  • Extending the deformation to include photons would require new constraints from CMB spectral distortions.
  • The approach opens a path to explore nonextensive effects in other early-universe processes like baryogenesis.

Load-bearing premise

That the nonextensive effects can be isolated to dark matter during freeze-out and then selectively transferred only to the electron-positron and neutrino components of the radiation sector.

What would settle it

An observation of N_eff outside the range allowed by the model for any choice of initial q and dark matter mass would rule out the residual-memory scenario as presented.

Figures

Figures reproduced from arXiv: 2605.29132 by Matias P. Gonzalez.

Figure 1
Figure 1. Figure 1: Evolution of qχ as a function of the freeze-out vari￾able x = mχ/T. The trajectories are obtained from Eq. (7) for different initial values of the Tsallis parameter in the dark sector. Here gχ denotes the internal degrees of freedom of the dark matter particle. The upper integration limit depends on the value of qχ(x). For qχ > 1, the distribution has a power-law tail and the integration domain is unbounde… view at source ↗
Figure 2
Figure 2. Figure 2: Dynamic-q freeze-out in the dark sector for different dark matter masses mχ. Solid curves show Yχ(x), dashed curves show Yχ,eq,q(x), and markers indicate the freeze-out points xfo. The color scale denotes the initial Tsallis parameter qχ,i [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relic Density obtained from the dynamic- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relic density target curves in the (mχ, σ0) plane. For each initial value of the dark sector Tsallis parameter qχ,i, the required annihilation strength σ req 0 is obtained by imposing Ωχh 2 ≃ 0.120 after solving the dynamic-q freeze-out equation. The horizontal dashed line denotes the reference value of σ0 used in the benchmark relic den￾sity evolution. In [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Projection of the relic density target surface onto [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the effective Tsallis parameter [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Temperature evolution of the Tsallis memory channel of the electron-positron and neutrino sectors. The left [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

In this work we generalize the thermal decoupling, or freeze-out, of weakly interacting massive particle dark matter within the Tsallis nonextensive formalism. The generalization is implemented through $q$-deformed distribution functions obtained from the maximum entropy principle with Curado-Tsallis constraints. The Tsallis parameter $q$, which measures deviations from extensivity with respect to the limit $q=1$, is promoted to a dynamical quantity depending on the dimensionless variable $x=m_\chi/T$, where $m_\chi$ is the dark matter mass. This dynamical evolution is characterized by a relaxation toward extensivity, while requiring that the nonextensive deformation is not completely erased before freeze-out. We solve the Boltzmann equation assuming a sectorial deformation, where only the dark matter equilibrium abundance is generalized and the radiation background remains extensive. The relic abundance is computed for different dark matter masses and initial values of the Tsallis parameter. From this evolution, we extract the residual value $q_\chi^{\rm fo}$ at freeze-out, which is then used as the initial input for a phenomenological memory channel. This channel propagates the residual nonextensivity into the radiation sector, specifically into the electron-positron plasma and neutrinos, while photons are kept extensive in order to avoid direct tensions with CMB physics. The resulting deformation modifies the neutrino energy density and the photon reheating contribution, producing a correction to $N_{\rm eff}$. We compare the predicted values with the compressed CMB+BAO constraint on $N_{\rm eff}$ and find that the residual-memory scenario can remain phenomenologically compatible with current bounds.

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 / 0 minor

Summary. The manuscript generalizes WIMP dark matter freeze-out in the Tsallis nonextensive formalism by promoting q to a dynamical q(x) that relaxes toward extensivity (q=1). It solves the Boltzmann equation under a sectorial deformation (only DM equilibrium abundance is q-deformed; radiation remains extensive), computes relic abundances for varying m_χ and initial q, extracts the residual q_χ^fo at freeze-out, and feeds this into a phenomenological memory channel. The channel selectively deforms the e+e- and neutrino distributions while keeping photons extensive, yielding a correction to N_eff that is claimed to remain compatible with compressed CMB+BAO bounds.

Significance. If internally consistent, the work supplies a concrete mechanism linking dynamical nonextensivity in the dark sector to residual imprints on radiation observables via a memory channel, thereby offering a new phenomenological handle on Tsallis statistics in cosmology that avoids direct tension with photon-based CMB constraints. The explicit extraction of q_χ^fo and its propagation to ΔN_eff constitutes a falsifiable prediction once the modeling choices are fixed.

major comments (2)
  1. [Abstract (description of memory channel and sectorial deformation)] The central construction rests on a sectorial deformation during freeze-out (DM only) followed by selective post-freeze-out propagation into e+e- and neutrinos while photons remain strictly extensive. No explicit consistency condition—such as joint entropy maximization across sectors or conservation of the total energy-momentum tensor—is provided to justify that this selective deformation can occur without back-reaction on the photon bath or violation of the extensive radiation background maintained earlier. This assumption is load-bearing for the reliability of the extracted ΔN_eff and the compatibility statement.
  2. [Abstract (extraction of q_χ^fo and memory channel)] The residual q_χ^fo is obtained from an evolution whose initial value and relaxation law are free modeling choices; this value is then used as direct input to the memory channel. The resulting ΔN_eff is therefore shaped by construction by the same assumptions that define the evolution, raising a circularity concern that must be addressed (e.g., by showing that the compatibility window survives independent variation of the relaxation law or by deriving q(x) from a more constrained dynamical principle).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the key assumptions in our phenomenological construction. We provide point-by-point responses to the major comments and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract (description of memory channel and sectorial deformation)] The central construction rests on a sectorial deformation during freeze-out (DM only) followed by selective post-freeze-out propagation into e+e- and neutrinos while photons remain strictly extensive. No explicit consistency condition—such as joint entropy maximization across sectors or conservation of the total energy-momentum tensor—is provided to justify that this selective deformation can occur without back-reaction on the photon bath or violation of the extensive radiation background maintained earlier. This assumption is load-bearing for the reliability of the extracted ΔN_eff and the compatibility statement.

    Authors: We agree that the sectorial deformation and the selective memory channel are phenomenological assumptions without a derived consistency condition from joint entropy maximization or global energy-momentum conservation. The manuscript presents this as an exploratory model to investigate possible residual nonextensivity effects while preserving compatibility with photon-based observations. In the revised version, we will add a dedicated paragraph in the introduction and discussion sections explicitly stating these assumptions and their limitations, including a note that a more fundamental embedding would require a multi-sector Tsallis framework which is left for future work. This addresses the load-bearing nature by making the modeling choices transparent. revision: partial

  2. Referee: [Abstract (extraction of q_χ^fo and memory channel)] The residual q_χ^fo is obtained from an evolution whose initial value and relaxation law are free modeling choices; this value is then used as direct input to the memory channel. The resulting ΔN_eff is therefore shaped by construction by the same assumptions that define the evolution, raising a circularity concern that must be addressed (e.g., by showing that the compatibility window survives independent variation of the relaxation law or by deriving q(x) from a more constrained dynamical principle).

    Authors: The initial value of q and the form of the relaxation law are indeed modeling choices, as the dynamical q(x) is introduced phenomenologically. The value q_χ^fo is extracted from the solution of the Boltzmann equation under these choices and then propagated. To address the potential circularity, in the revised manuscript we will include an additional analysis varying the relaxation timescale independently over a range of values and demonstrate that the resulting ΔN_eff remains within the observational bounds for a substantial portion of the parameter space. We will also discuss the possibility of deriving the relaxation law from a more constrained principle, such as a specific interaction term, though a full derivation is beyond the current scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; phenomenological model with explicit inputs checked against external bounds

full rationale

The paper defines a dynamical q(x) with free initial value and relaxation law, solves the Boltzmann equation under an explicit sectorial-deformation assumption (DM only), extracts the resulting q_fo, and feeds it into a separate phenomenological memory channel to obtain a Delta N_eff correction that is then compared to CMB+BAO data for compatibility. This is a standard model-building and parameter-scan procedure whose output depends on the stated modeling choices; it does not reduce any claimed prediction to its inputs by construction, nor does it rely on load-bearing self-citations or uniqueness theorems. The central claim is merely that certain choices remain compatible with bounds, which is externally falsifiable and does not constitute circular reasoning.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The model rests on several modeling choices for q evolution and the memory mechanism with no independent evidence for the new constructs; free parameters include initial q and the relaxation dynamics, while axioms include the Curado-Tsallis max-entropy principle and the sectorial deformation assumption.

free parameters (2)
  • initial Tsallis parameter q
    Chosen as starting value for the dynamical evolution of q(x)
  • relaxation law for q(x)
    The specific functional dependence on x = m_χ/T and rate of approach to q=1 is a modeling choice
axioms (2)
  • domain assumption Maximum entropy principle with Curado-Tsallis constraints yields q-deformed distributions
    Used to obtain the generalized distribution functions for dark matter
  • ad hoc to paper Sectorial deformation where only dark matter is nonextensive and radiation is extensive
    Assumed to avoid tensions with CMB physics for photons
invented entities (2)
  • dynamical Tsallis parameter q(x) no independent evidence
    purpose: To model temperature-dependent deviation from extensivity that relaxes over time
    Introduced to generalize the freeze-out process
  • residual memory channel no independent evidence
    purpose: To propagate nonextensivity from dark matter to radiation sector after freeze-out
    Phenomenological construct to affect N_eff

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Reference graph

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