Interacting Holographic dark energy with matter creation: A dynamical system analysis

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) $\Gamma=3\beta H$ and (2) $\Gamma=3\alpha H_{0}+3\beta H$ 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 $Q=3\gamma H \rho_{m}$. 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 $\Gamma=3\alpha H_{0}+3\beta H$. 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.