WIMP Freeze-out dynamics under Tsallis statistics

TL;DR

This work extends the thermal WIMP freeze-out paradigm into Tsallis nonextensive statistics by imposing Curado–Tsallis $q$-distributions on the early-Universe plasma. It develops a model-independent framework in which $n_q$, $\rho_q$, $s_q$ and the expansion rate $H_q$ are consistently deformed via ultra-relativistic rescalings $R_\rho(q)$, while the velocity-averaged annihilation rate remains in the familiar partial-wave form $\langle\sigma v\rangle_q \approx a + b\langle v_{\rm rel}^2\rangle_q$ with $\langle v_{\rm rel}^2\rangle_q$ computed from $q$-distributed moment integrals. The $q$-Boltzmann equation yields the comoving abundance $Y_{\chi,q}(x)$ and relic density $\Omega_{\chi,q}h^2$, and the freeze-out location $x_f(q)$ is obtained from a $q$-logarithmic inversion. Numerical scans show a characteristic degeneracy: the relic density primarily tracks $\langle\sigma v\rangle_q$, with the best-fit $q$ remaining near the extensive limit $q\to1$ and a joint $(q,a,m_\chi)$ landscape organized along a valley of nearly degenerate solutions. The analysis demonstrates that nonextensivity imprints on the DM relic density mainly through the thermodynamic background and $q$-dependent moments, offering a controlled way to quantify potential departures from standard Boltzmann-Gibbs statistics in cosmology.

Abstract

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 $q\to 1$.

Figures (12)

• Figure 1: Nonextensive rescalings as functions of $q$. Radiation rescaling $R_\rho(q)$ is defined in Eq. (\ref{['eq:Rrho-exacto']}) and computed as $R_\rho=(1/6)\!\int_{0}^{z_{\max}} z^{3}\,e_q(-z)\,dz$, with $z_{\max}=1/(1-q)$ for $q<1$ and $z_{\max}\!\to\!\infty$ for $q\ge 1$. Convergence holds for $q<5/4$. Analogously, the equilibrium number-density rescaling $R_n(q)$, defined in Eq. (\ref{['eq:Rn-exacto']}), is $R_n=(1/2)\!\int_{0}^{z_{\max}} z^{2}\,e_q(-z)\,dz$ with the same $z_{\max}$ prescription; its convergence condition is $q<4/3$. Both mappings satisfy $R_{\rho,n}(1)=1$ (dash-dotted line), and the dotted vertical line marks the extensive limit $q=1$.
• Figure 2: Freeze-out for $m_\chi=100~\mathrm{GeV}$, comoving abundance $Y_{\chi,q}(x)$ versus $x_f$: where $g_{\chi} = 4$, $a = 1.825 \times 10^{-9}~\rm GeV^{-2}$ and $b = 1.05 \times 10^{-9} ~\rm GeV^{-2}$ and several values of $q$ are considered. The black dashed line is the value measured by the Planck satellite $\Omega_c h^2 = 0.120 \pm 0.001$.
• Figure 3: Relic abundance $\Omega_{\chi,q} h^2$ vs. mass $m_\chi$ for different $q$ for $g_{\chi}=4$. The shaded band and dashed line indicate, respectively, the $\pm3\sigma$ region and the central Planck value $\Omega_c h^2 = 0.120 \pm 0.001$ with the same annihilation parameters as in Fig. \ref{['fig:comovingabundance']}.
• Figure 4: Freeze-out parameter $x_f$ versus nonextensivity $q$ for fixed WIMP masses ($m_\chi=\{100,500,1000\}\,\mathrm{GeV}$). Curves are obtained by solving the $q$-generalized transcendental condition Eq. \ref{['eq:trascendent']}, $R_\rho(q)$ computed from the exact $q$-exponential, and $\langle\sigma v\rangle_q\approx a+b\,\langle v_{\rm rel}^2\rangle_q$. The grey band indicates a "canonical" range $15\leq x_f \leq 35$.
• Figure 5: Freeze-out parameter $x_f \equiv m_\chi/T_f$ as a function of $m_\chi$ for $q \in \{0.90,\,0.95,\,1.00,\,1.05,\,1.08\}$. Curves are obtained by numerically solving the $q$-generalized Boltzmann equation for $Y_{\chi,q}$ with exact $q$-exponential distributions, $\langle\sigma v\rangle_q\approx a+b\,\langle v_{\rm rel}^2\rangle_q$. The overall trend shows larger $x_f$ for larger $q$, consistent with the $q$-logarithmic inversion and the rescaled expansion rate $H_q \propto \sqrt{R_\rho(q)}$.