Barrow entropic dark energy: A member of generalized holographic dark energy family

The holographic cut-off, in the formalism of generalized holographic dark energy (HDE), is generalized to depend on $L_\mathrm{IR} = L_\mathrm{IR} \left( L_\mathrm{p}, \dot L_\mathrm{p}, \ddot L_\mathrm{p}, \cdots, L_\mathrm{f}, \dot L_\mathrm{f}, \cdots, a\right)$, where $L_\mathrm{p}$ and $L_\mathrm{f}$ are the particle horizon and future horizon respectively, and $a$ is the scale factor of the universe. Based on such formalism, we showed that the Barrow entropic dark energy (DE) model is equivalent to the generalized HDE where the respective holographic cut-off is determined by two ways -- (1) in terms of particle horizon and its derivative and (2) in terms of future horizon and its derivative. Interestingly, such cut-off turns out to depend up-to first order derivative of $L_\mathrm{p}$ or $L_\mathrm{f}$ respectively. Such equivalence between the Barrow entropic dark energy and the generalized HDE is extended to the scenario where the exponent of the Barrow entropy allows to vary with the cosmological expansion of the universe. In both the cases (whether the Barrow exponent is a constant or varies with the cosmological evolution), we determine effective equation of state (EoS) parameter from the generalized holographic point of view, which, by comparing with the Barrow DE EoS parameter, further ensures the equivalence between the Barrow entropic dark energy and the generalized HDE.