A closer look at the cosmological implications of the $\Lambda$HDE model

In a previous paper, we proposed a heterotic dark energy model, called $\Lambda$HDE, in which dark energy is composed of two components: cosmological constant (CC) and holographic dark energy (HDE). The aim of this work is to give a more comprehensive and systematic investigation on the cosmological implications of the $\Lambda$HDE model. Firstly, we make use of the current observations to constrain the $\Lambda$HDE model, and compare its cosmology-fit results with the results of the $\Lambda$CDM and the HDE model. Then, by combining a qualitative theoretical analysis with a quantitative numerical study, we discuss the impact of considering curvature on the cosmic evolutions of fractional HDE density $\Omega_{hde}$ and fractional CC density $\Omega_{\Lambda}$, as well as on the ultimate cosmic fate. Finally, we explore the effects of adopting different types of observational data. We find that: (1) the current observational data cannot distinguish the $\Lambda$HDE model from the $\Lambda$CDM and the HDE model; this indicates that DE may contain multiple components. (2) the asymptotic solution of $\Omega_{hde}$ and the corresponding cosmic fate in a flat universe can be extended to the case of a non-flat universe; moreover, compared with the case of a flat universe, considering curvature will make HDE closer to a phantom dark energy. (3) compared with JLA dataset, SNLS3 data more favor a phantom type HDE; in contrast, using other types of observational data have no significant impact on the cosmic evolutions of the $\Lambda$HDE model.