Future Rip Scenarios in Fractional Holographic Dark Energy

TL;DR

The paper investigates late-time rip singularities in Fractional Holographic Dark Energy (FHDE) using Granda-Oliveros (GO) and Hubble cutoffs, exploring how memory effects encoded by the Lévy index $\alpha$ affect cosmic fate. By analyzing linear and nonlinear dark-sector interactions $Q$, it derives conditions for big, little, and pseudo rip scenarios, finding that the GO cutoff permits a big rip for $1<\alpha\le 2$ while pseudo rip would require $\alpha>2$, which is excluded; little rip is not generically realized with GO unless a contrived NO-type cutoff is adopted. Switching to the Hubble cutoff and employing an ansatz for $H(t)$, the study shows that nonlinear $Q$ supports little- and pseudo-rip behaviors, whereas linear $Q$ tends to big-rip-like outcomes, with EoS and squared sound speed illustrating the associated instabilities. Overall, the work highlights the significance of non-local fractional features in FHDE for the fate of the universe and suggests further quantum cosmology and stability analyses to assess viability and observational signatures of these singularities.

Abstract

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., $L=(γH^{2}+δ\dot{H})^{-\frac{1}{2}}$, and highlight the range of allowed $α$ (Lévy'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 Lévy's index in the allowed range $1<α\leq2$. Moreover, we conclude that the occurrence of a pseudo-rip requires Lévy's index to be $α>2$. 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 $L\sim(γH^{2}+g(H))^{-\frac{1}{2}}$, 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.

Figures (6)

• Figure 1: Plots for EoS parameter $w_{\text{GO}}(t)$ for big rip within Granda-Oliveros cutoff.
• Figure 2: Plots for squared sound speed parameter $v^{2}_{\text{GO}}(t)$ for big rip within Granda-Oliveros cutoff.
• Figure 3: Plots for EoS parameter $w_{\text{HH}}(t)$ for little rip within Hubble cutoff.
• Figure 4: Plots for squared sound speed parameter $v^{2}_{\text{HH}}$ for little rip within Hubble cutoff.
• Figure 5: Plots for EoS parameter $w_{\text{HH}}(t)$ for pseudo rip within Hubble cutoff.