Cosmographic Connection Between Cosmological And Planck Scales: The Barrow-Tsallis Entropy

Abstract

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 used our method for finding the parameters of cosmological models to evaluate the feasibility of using fractional derivatives to describe the late evolution of the Universe. The resulting relationships are exact. Therefore, the uncertainty in the relationship between the model parameters depends only on the current uncertainty in the values of the cosmographic parameters.