Non-flat Universe with Tsallis holographic dark energy

We investigated evolution of metric and density perturbations for Tsallis model of holographic dark energy with energy density $\sim L^{2γ- 4}$, where $L$ is length of event horizon or inverse Hubble parameter and $γ$ is parameter of non-additivity close to $1$. Because holographic dark energy is not an ordinary cosmological fluid but a phenomena caused by boundaries of the universe, the ordinary analysis for perturbations is not suitable. One needs to consider perturbations of the future event horizon. For realistic values of parameters it was discovered that perturbations of dark energy don't grow infinitely but vanish or freeze. We also considered the case of realistic interaction between holographic dark energy and matter and showed that in this case perturbations also can asymptotically freeze with time.