Future Rip Scenarios in Fractional Holographic Dark Energy

In this paper, we investigate the occurrence of late-time cosmological singularities, namely, the rip scenarios within the framework of interacting Fractional Holographic Dark Energy (FHDE). We start our investigation with the Granda-Oliveros (GO) cutoff, i.e., $L=(\gamma H^{2}+\delta\dot{H})^{-\frac{1}{2}}$, and highlight the range of allowed $\alpha$ (L\'evy's index) values for which big, little and pseudo rip can occur. In particular, we highlight the occurrence of a big rip for fractional values of the L\'evy's index in the allowed range $1<\alpha\leq2$. Moreover, we conclude that the occurrence of a pseudo-rip requires L\'evy's index to be $\alpha>2$. Therefore, we reject the possibility of pseudo-rip within the GO cutoff. Furthermore, we demonstrate that the occurrence of the little rip in FHDE equipped with a GO cutoff is rather contrived and requires a specific functional form of the IR cutoff $L\sim(\gamma H^{2}+g(H))^{-\frac{1}{2}}$, which belongs to a larger class of Nojiri-Odintsov (NO) cutoffs. To extend our perspective beyond the GO cutoff, we investigate the interacting FHDE framework equipped with the Hubble cutoff, i.e., $L=H^{-1}$, in developing an ansatz-based approach to the little and pseudo-rip singularities as they fail to appear in the GO cutoff. Within this approach, we invoke the expression of the Hubble parameter, $H(t)$, which corresponds to the little and pseudo-rip, into the cosmological parameters such as the Equation of State (EoS) and Squared Sound Speed (SSS) as a function of cosmic time $t$. We produce numerical plots of these parameters in both linear and non-linear $Q$ regimes, which supplement our theoretical findings. In summary, our results highlight the occurrence of little and pseudo-rip singularities within a Hubble cutoff for a non-linear $Q$ term within the FHDE framework.