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

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 $\rho_q$. We define a reescaling $R^{(\xi = +1)}_{\rho}(q) = \rho_q/\rho^{\rm std}$ 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 $\chi^2$ 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 $q_{\rm best}$.