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

Abstract

We consider a spatially flat cosmological model containing a noninteracting barotropic fluid and holographic dark energy with the future event horizon as the infrared cutoff. We parametrize the deviation of the horizon entropy from the Bekenstein--Hawking form by a smooth positive function of the horizon radius. We reduce the background evolution to a closed autonomous system and analytically describe the crossing of the phantom divide, $w=-1$. For entropies growing more slowly than the fourth power of the radius, the crossing is unique, occurs at an extremum of the event-horizon radius, and proceeds from the quintessence regime into the phantom regime. The local Chevallier-Polarski-Linder coefficient $w_{a,\mathrm{loc}}$ at the crossing is then positive, and its sign directly constrains the local entropy-scaling dimension at the horizon scale. The crossing kinematics yields an exact expression for the local entropy-scaling dimension, and a local prescription for the scale-dependent Barrow exponent in the Barrow--Tsallis entropy enables an analytic reconstruction of the entropy function. The early-time asymptotics remain consistent with the standard matter- and radiation-dominated eras, while the event-horizon consistency criterion selects physically admissible late-time trajectories leading to de Sitter states, to Type III or Big Rip singularities, or, for $δ=1$, to accelerated power-law expansion. The analytic results are illustrated numerically. Within the adopted decomposition of the total entropy, the generalized second law of thermodynamics further requires an additional dark-energy entropy in the phantom regime: the equilibrium entropy evolution implied by the Gibbs relation fails to satisfy the law already at the crossing, where the purely barotropic adiabatic description of perturbations also breaks down.