WIMP Freeze-out dynamics under Tsallis statistics

We generalize thermal WIMP (Weakly Interacting Massive Particle) freeze-out within Tsallis nonextensive statistics. Using Curado-Tsallis $q$-distributions $f_q(E;μ,T)$ we compute $q$-deformed number and energy densities, pressure, entropy density and Hubble rate, ${n_q,ρ_q,P_q,s_q,H_q}$. The Boltzmann equation is generalized accordingly to obtain the comoving abundance $Y_{χ,q}(x)$ and relic density $Ω_{χ,q}h^2$ for a dark-matter candidate $χ$ in a model-independent setup. The thermally averaged cross section is expanded as $\langleσv\rangle_q \approx a + b,\langle v_{\rm rel}^2\rangle_q$ up to $p$-wave. The freeze-out parameter $x_f(q)$ is determined from $Γ_{{\rm ann},q}(T_f)\simeq H_q(T_f)$ using a $q$-logarithmic inversion, with the expansion rate modified through ultra-relativistic rescalings $R_ρ(q)$ of the effective relativistic degrees of freedom $g_*$ and $g_{*s}$. We show that $x_f$ increases with $q$ and that QCD-threshold features propagate into $Y_{χ,q}(x)$ and $Ω_{χ,q}h^2$. We then perform two $q$-grid scans: fixing $\langleσv\rangle_q$ while varying the dark-matter mass $m_χ$, and fixing $m_χ$ while varying the $s$-wave coefficient $a$. For an $s$-wave dominated scenario we construct $χ^2$ profiles in these planes by comparing $Ω_{χ,q}h^2$ with the Planck benchmark $Ω_c h^2 = 0.120\pm 0.001$. In both cases we find a clear degeneracy in the preferred nonextensive parameter $q_{\rm best}$ along valleys in parameter space. However, fixed-mass scans (varying $\langleσv\rangle_q$) are significantly more constraining than fixed-cross-section scans, reflecting that $Ω_{χ,q}h^2$ is mainly controlled by $\langleσv\rangle_q$, so that for realistic cross sections the best-fit $q_{\rm best}$ remains close to the extensive limit $q\to 1$.

💡 Research Summary

The paper presents a systematic extension of the standard thermal freeze‑out calculation for weakly interacting massive particles (WIMPs) by embedding the whole framework in Tsallis non‑extensive statistics. The authors adopt the Curado‑Tsallis (CT) q‑distribution,
(f_q(E; \mu,T)=1/