Cosmographic Connection Between Cosmological And Planck Scales: The Barrow-Tsallis Entropy
One of the fundamental challenges of quantum gravity is to understand how the microscopic degrees of freedom of the cosmological horizon shape the evolution of the Universe. One possible approach to this problem is based on the Barrow--Tsallis entropy. This entropy accounts for both quantum gravitational effects and the nonextensive effects inherent in any long-range interaction. By employing an inverse cosmographic reconstruction of the model parameters, we derive a relation between the Barrow parameter, which encodes the microscopic deformation of the horizon geometry, and the Tsallis parameter, which characterizes macroscopic nonextensivity. Within the IR--UV correspondence, this relation determines the scaling of the microscopic length uncertainty in terms of the current cosmographic parameters and demonstrates how long-range nonextensive effects alter the standard Karolyhazy-type scaling. We also applied our cosmographic reconstruction method to evaluate the feasibility of using fractional derivatives to describe the late evolution of the Universe. Within the assumed non-interacting power-law holographic model class, the resulting algebraic relations are exact. For this fixed model class, the observational uncertainty of the reconstructed parameter combination is determined by the current uncertainties in the cosmographic parameters; the quoted uncertainty of $\delta$ additionally includes the adopted prior on $\Delta$, but not uncertainty associated with the model choice. Propagating the observational errors of the deceleration and jerk parameters and marginalizing over a uniform prior on the Barrow parameter within the adopted interval $\Delta \in [-1,1]$, we obtain the Monte Carlo estimate $\delta = 1.11 \pm 0.57$ for the nonextensivity parameter, with the jerk parameter providing the largest observational contribution to the error budget.