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. 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.

💡 Research Summary

The paper proposes a novel entropy, the Barrow‑Tsallis entropy (S_BT), which merges Barrow’s fractal deformation of the horizon (parameter Δ) with Tsallis’ non‑additive statistics (parameter δ). Starting from Barrow’s entropy S_B ∝ A^{1+Δ/2} and Tsallis’ relation S_T = (S_BH)^δ, the authors construct S_BT = γ A^{(1+Δ/2)δ}. An effective Barrow exponent Δ_eff = 2(δ − 1)+Δδ is introduced, showing how the two parameters jointly modify the horizon entropy.

Using S_BT, a holographic dark‑energy density of the form ρ_de = 3β H^α is derived, where α = 4 − 2δ − δΔ. The Friedmann equations together with the continuity equations lead to expressions involving the dimensionless combinations \dot H/H^2 and \ddot H/H^3, which are identified with the cosmographic deceleration parameter q and jerk parameter j via \dot H/H^2 = −(1+q) and \ddot H/H^3 = j+3q+2. Substituting these relations yields closed‑form formulas for the model constants α and β in terms of q and j:

α(q,j) =