Lagrangian reconstruction of Barrow holographic dark energy in interacting tachyon model

Abstract: We consider a correspondence between the tachyon dark energy model and Barrow holographic dark energy (BHDE). The latter is a modified scenario based on the application of the holographic principle with Barrow entropy instead of the usual Bekenstein-Hawking one. We reconstruct the dynamics of the tachyon scalar field $T$ in a curved Friedmann-Robertson-Walker Universe both in the presence and absence of interactions between dark energy and matter. As a result, we show that the tachyon field exhibits a non-trivial dynamics. In a flat Universe, $\dot T^2$ must always be vanishing, independently of the existence of interaction. This implies $\omega_D=-1$ for the equation-of-state parameter, which in turn can be used for modeling the cosmological constant behavior. On the other hand, for a non-flat Universe and various values of Barrow parameter, we find that $\dot T^2$ decreases monotonically for increasing $\cos(R_h/a)$ and $\cosh(R_h/a)$, where $R_h$ and $a$ are the future event horizon and the scale factor, respectively. Specifically, $\dot T^2\ge0$ for a closed Universe, while $\dot T^2<0$ for an open one, which is physically not allowed. We finally comment on the inflation mechanism and Trans-Planckian Censorship Conjecture in BHDE and discuss observational consistency of our model.