Diagnosing holographic type dark energy models with the Statefinder hierarchy, composite null diagnostic and $w-w^{\prime }$ pair

Abstract

The main purpose of this work is to distinguish various holographic type dark energy (DE) models, including the $\hat{I}›$HDE, HDE, NADE and RDE model, by using various diagnostic tools. The first diagnostic tool is the Statefinder hierarchy, in which the evolution of Statefinder hierarchy parmeter $S_3^{(1)}(z)$ and $S_4^{(1)}(z)$ are studied. The second is composite null diagnostic (CND), in which the trajectories of $\{S_3^{(1)},\hat{I}\mu \}$ and $\{S_4^{(1)},\hat{I}\mu \}$ are investigated, where $\hat{I}\mu$ is the fractional growth parameter. The last is $w-w^{\prime }$ analysis, where $w$ is the equation of state for DE and the prime denotes derivative with respect to $lna$. In the analysis we consider two cases: varying current fractional DE density $\hat{I}{\text{c◯}}_{de0}$ and varying DE model parameter $C$. We find that: (1) Both the Statefinder hierarchy and the CND have qualitative impact on $\hat{I}›$HDE, but only have quantitative impact on HDE. (2) $S_4^{(1)}$ can lead to larger differences than $S_3^{(1)}$, while the CND pair has a stronger ability to distinguish different models than the Statefinder hierarchy. (3) For the case of varying $C$, the $\{w,w^{\prime }\}$ pair has qualitative impact on $\hat{I}›$HDE; for the case of varying $\hat{I}{\text{c◯}}_{de0}$, the $\{w,w^{\prime }\}$ pair only has quantitative impact; these results are different from the cases of HDE, RDE and NADE, in which the $\{w,w^{\prime }\}$ pair only has quantitative impact on these models. In conclusion, compared with HDE, RDE and NADE, the $\hat{I}›$HDE model can be easily distinguished by using these diagnostic tools.

12 pages, 8 figures, accepted by Sci. China Phys. Mech. Astron