$\Lambda(t)$CDM Model as a Unified Origin of Holographic and Agegraphic Dark Energy Models

Motivated by the fact that any nonzero $\Lambda$ can introduce a length scale or a time scale into Einstein's theory, $r_{\Lambda}=ct_{\Lambda}=\sqrt{3/|\Lambda|}$. Conversely, any cosmological length scale or time scale can introduce a $\Lambda (t)$, $\Lambda(t)=3/r^2_{\Lambda}(t)=3/(c^2t^2_{\Lambda}(t))$. In this letter, we investigate the time varying $\Lambda(t)$ corresponding to the length scales, including the Hubble horizon, the particle horizon and the future event horizon, and the time scales, including the age of the universe and the conformal time. It is found out that, in this scenario, the $\Lambda(t)$CDM model can be taken as the unified origin of the holographic and agegraphic dark energy models with interaction between the matter and the dark energy, where the interacting term is determined by $Q=-\dot{\rho}_{\Lambda}$. We place observational constraints on the $\Lambda(t)$CDM models originating from different cosmological length scales and time scales with the recently compiled "Union2 compilation" which consists of 557 Type Ia supernovae (SNIa) covering a redshift range $0.015\leq z \leq 1.4$. In conclusion, an accelerating expansion universe can be derived in the cases taking the Hubble horizon, the future event horizon, the age of the universe and the conformal time as the length scale or the time scale.