arXiv:2408.08942v1 Announce Type: new

Abstract: An interacting Holographic dark energy (HDE) with different infra-red (IR) cutoffs (Hubble horizon and future event horizon) is investigated in the background dynamics of flat Friedmann Lemaitre Robertson Walker (FLRW) universe where gravitational particle creation effects via different form of particle creation rates (1) $Gamma=3beta H$ and (2) $Gamma=3alpha H_{0}+3beta H$ are considered. The created particles are considered to be pressureless Dark Matter (DM) which interacts with the HDE through a phenomenological choice of interaction term $Q=3gamma H rho_{m}$. We obtain an analytic solution of the cosmological dynamics with Hubble horizon as IR cutoff when the creation rate is taken as $Gamma=3 beta H$. We find that the interacting HDE from the Hubble horizon as the IR cutoff can give the late-time acceleration and non-interacting cannot give. On the other hand, employing the Hubble horizon and the future event as IR cutoffs for the model of HDE does not provide the analytic solution when the creation rate is taken as $Gamma=3alpha H_{0}+3beta H$. We then analyze the model separately using the dynamical systems theory. From the analysis, the model (with Hubble horizon as IR cutoff) provides two sets of critical points. One can give a late-time accelerated universe evolving in quintessence, the cosmological constant, or the phantom era. But, it does not show any matter-dominated era. On the other hand, by applying the future event as an IR cutoff, the model provides the complete evolution of the universe. It also exhibits the late-time scaling attractor gives the possible solution of the coincidence problem. Global dynamics of the model are investigated by defining the appropriate Lyapunov function. Finally, the adiabatic sound speeds of all the models have been calculated and plotted numerically to find the stability of the models.

## Interacting Holographic Dark Energy: Conclusions and Future Roadmap

The following article investigates the background dynamics of a flat Friedmann Lemaitre Robertson Walker (FLRW) universe with interacting Holographic Dark Energy (HDE) and different infra-red (IR) cutoffs. The interaction is between pressureless Dark Matter (DM) and HDE, and two different particle creation rates are considered.

The first conclusion of the study is that when the Hubble horizon is used as the IR cutoff and the particle creation rate is $Gamma=3beta H$, an analytic solution for the cosmological dynamics is obtained. In this case, the interacting HDE from the Hubble horizon can explain the late-time acceleration, unlike the non-interacting HDE.

The second conclusion is that when both the Hubble horizon and the future event horizon are used as IR cutoffs and the particle creation rate is $Gamma=3alpha H_{0}+3beta H$, an analytic solution for the cosmological dynamics is not obtained. Therefore, the authors resort to the dynamical systems theory to analyze the model separately.

By using the dynamical systems theory, the authors find that when the Hubble horizon is used as the IR cutoff, the model provides two sets of critical points. These critical points can result in a late-time accelerated universe evolving in quintessence, cosmological constant, or phantom era. However, the model does not show any matter-dominated era.

On the other hand, when the future event horizon is used as the IR cutoff, the model provides the complete evolution of the universe. It also exhibits a late-time scaling attractor, which offers a possible solution to the coincidence problem.

To further analyze the global dynamics of the model, the authors define an appropriate Lyapunov function. This allows for an investigation of the stability of the models.

Finally, the adiabatic sound speeds of all the models are calculated and plotted numerically to find their stability.

### Roadmap for Readers:

1. Introduction to interacting Holographic Dark Energy (HDE) with different IR cutoffs.

2. Explanation of the model background dynamics in a flat FLRW universe.

3. Consideration of two particle creation rates and their impact on the analytic solution.

4. Conclusions regarding the Hubble horizon as an IR cutoff and its ability to explain late-time acceleration.

5. Introduction to the use of the future event horizon as an IR cutoff and the lack of an analytic solution.

6. Analysis of the model using the dynamical systems theory.

7. Discussion of critical points and their implications for the evolution of the universe.

8. Presentation of a late-time scaling attractor as a possible solution to the coincidence problem.

9. Introduction of the Lyapunov function for investigating the global dynamics of the model.

10. Calculation and visualization of the adiabatic sound speeds to assess model stability.

### Potential Challenges:

1. The lack of an analytic solution for the model with the future event horizon as the IR cutoff may limit our ability to fully understand its dynamics.

2. The analysis using the dynamical systems theory may involve complex mathematical concepts, making it challenging for readers without a strong background in mathematical physics.

### Potential Opportunities:

1. The model with the Hubble horizon as the IR cutoff provides a solution for late-time acceleration, which aligns with observations of the universe’s expansion.

2. The model with the future event horizon as the IR cutoff offers a complete picture of the universe’s evolution and provides a potential solution to the coincidence problem.

3. The Lyapunov function and stability analysis provide insights into the long-term behavior of the model, allowing for a better understanding of its viability.

Overall, this study on interacting Holographic Dark Energy with different IR cutoffs provides valuable insights into the background dynamics of the universe. The findings highlight the importance of particle creation rates and IR cutoff choices, and offer potential explanations for late-time acceleration and the coincidence problem. While challenges exist in terms of analytic solutions and mathematical complexity, the opportunities for understanding the universe’s evolution and stability make this an intriguing field of research.