Bounds on the Tsallis Parameter from a deformed Neutrino Sector in the Early Universe

TL;DR

This work tests whether Tsallis nonextensive statistics can imprint measurable deviations in the early-Universe neutrino energy density by deforming the neutrino sector with a generalized fermionic distribution $f_q(E)$. Using Curado–Tsallis constraints, the authors link the deformation parameter $q$ to a rescaling of the neutrino energy density $R_ ho^{(\xi=+1)}(q)$ and the resulting shift in the effective number of neutrinos $ ff(q)$. They perform a joint $ ff$-based likelihood analysis with BBN and CMB+$BAO$ data, obtaining percent-level constraints on $|q-1|$ (95% CL: $oxed{1.09 imes10^{-2}}$, 99% CL: $oxed{1.32 imes10^{-2}}$). The results favor $q$ very close to unity, supporting the standard Boltzmann-Gibbs framework during the relevant epoch and providing quantitative benchmarks for any nonextensive scenarios in the early Universe. Extensions to temperature-dependent deformations or other relativistic species could sharpen these bounds with future observations of $N_{ m eff}$.

Abstract

We generalize neutrino energy density content in the early universe near BBN era $T\simeq1$ MeV within Tsallis nonextensive statistics. By using Curado-Tsallis constraints we obtain generalized distribution functions $f_q(E)$. We compute the generalized thermodynamic integral for the energy density $ρ_q$. We define a reescaling $R^{(ξ= +1)}_ρ(q) = ρ_q/ρ^{\rm std}$ which is a ratio between the deformed energy density and the standard extensive case. The last was used to directly map and deform neutrino content via the effective number of neutrinos $N_{\rm eff}$. The deformation prediction was confronted against CMB$+$BAO and BBN data for $N_{\rm eff}$ by a joint/combined $χ^2$ type-fit. We obtained the constraints $|q-1|\le 1.09\times 10^{-2}$ (95\% CL) and $|q-1|\le 1.32\times 10^{-2}$ (99\% CL) from the combined analysis by numerically calculating the best value of the Tsallis parameter $q_{\rm best}$.

Figures (2)

• Figure 1: Left: Non-extensive prediction for $\Delta N_{\rm eff}(q)$ induced by a neutrino-only deformation through the rescaling $R_{\rho}^{(\xi=+1)}(q)$, using $\Delta N_{\rm eff}(q)=(R_{\rho}^{(\xi=+1)}(q)-1)\,N_{\rm eff}^{\rm std}$. The horizontal bands show the $1\sigma$ regions of $N_{\rm eff}$ from BBN and CMB$+$BAO around the standard value; values of $q$ compatible with both datasets are those for which the curve lies within the overlap. Right:$\chi^2$ profiles as a function of $q$ for BBN and CMB$+$BAO, $\chi^2_{\rm BBN}(q)$ and $\chi^2_{\rm CMB+BAO}(q)$, with vertical lines indicating the corresponding best-fit minima $q_{\rm best}^{\rm BBN}$ and $q_{\rm best}^{\rm CMB}$.
• Figure 2: Left: Profile likelihood for the combined constraint, $\Delta\chi^2_{N_{\rm eff}}(q)\equiv \chi^2_{N_{\rm eff}}(q)-\chi^2_{N_{\rm eff},{\rm min}}$, obtained under the neutrino-only rescaling scheme. Horizontal lines mark the 68%, 95%, and 99% confidence levels (one effective parameter), and the shaded bands indicate the corresponding allowed intervals around $q_{\rm best}$; the gray line indicates $q=1$. Right: Zoom around the minimum highlighting the 68% and 95% confidence levels, with the black vertical line marking $q_{\rm best}$ and the gray dotted line indicating $q=1$.