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. Using a general method we developed for finding the parameters of cosmological models, we discovered a relationship between the parameter describing the microscopic structure of quantum foam and the parameter associated with macroscopic nonextensive effects. 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.

Figures (4)

• Figure 1: Dependence of parameter $\Delta_{eff}$ on parameters $\Delta$ and $\delta$.
• Figure 2: Barrow parameter $\Delta$ as a function of the non-extensiveness parameter $\delta$. The curve corresponds to the current cosmographic parameters $q_0 = -0.580$ and $j_0= 0.745$ derived from observational data.
• Figure 3: Relation $\delta(\Delta)$ for a fixed pair of cosmographic parameters ($q_0 = -0.580, j_0 = 0.745$). The solid curve shows the Barrow--Tsallis model and the dashed curve shows the extensive limit. The red part of the curve illustrates the distribution generated by the Monte Carlo method.
• Figure 4: The allowable region for the jerk parameter $j$ as a function of the deceleration parameter $q$ in the fractional holographic dark energy model. The shaded area represents the parameter space where the fractional exponent $\beta$ lies within the physical interval $[1, 2]$. The markers show the $\Lambda$CDM point and the DESI DR2 + DES Y5 estimateDESI2025DR2IDESI2025DR2IIDES2024SN5YRcosmoDES2024SN5YRlightcurves.