Phantom-Divide Crossing in Barrow-Tsallis Holographic Dark Energy with a Scale-Dependent Barrow Exponent

Abstract

Phantom-divide crossing is studied in spatially flat cosmologies with standard Friedmann equations, separately conserved holographic dark energy and a barotropic fluid, and the future event horizon as the infrared cutoff. The horizon entropy and dark-energy density are prescribed by $S_h=S_{\mathrm{BH}}F(L)$ and ${\rho }_X\propto F(L)/L^2$, where $F(L)$ is smooth, positive and single-valued, with $\chi =\mathrm{d}\ln F/\mathrm{d}\ln L$. For $w_b>-1$, $0<{\Omega }_X<1$ and $\chi <2$ along the trajectory, at most one phantom-divide crossing is possible. If a crossing occurs, dark energy passes from the quintessence regime into the phantom regime at a maximum of the horizon radius. The crossing kinematics determines the local entropy-scaling dimension and the sign of the local Chevallier-Polarski-Linder coefficient. Integrating the locally prescribed scale-dependent Barrow exponent reconstructs the Barrow-Tsallis entropy function. A necessary and sufficient event-horizon consistency criterion selects admissible solutions, whose late-time asymptotics include de Sitter states, Type III and Big Rip singularities, and accelerated power-law expansion under the corresponding parameter conditions. A numerical comparison with a constant Barrow exponent distinguishes the occurrence of a crossing from the effect of exponent variation on late-time evolution. With an adiabatic barotropic fluid, additive total entropy and an equilibrium Gibbs relation for dark energy, the generalized second law fails at the kinematic crossing for positive barotropic-fluid entropy and finite nonzero dark-energy temperature. The divergence of $c_a^2$ shows that the purely barotropic adiabatic description of perturbations breaks down. This divergence alone does not establish physical instability.

arXiv comment14 pages, 2 figures, 1 table