Diagnosing holographic type dark energy models with the Statefinder hierarchy,composite null diagnostic and w-w′pair

The main purpose of this work is to distinguish various holographic type dark energy(DE) models, including the ΛHDE, 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),.} and {S_4~(1),.} are investigated, where. is the fractional growth parameter. The last is w-w′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 Ω_(de0) and varying DE model parameter C. We find that:(1) both the Statefinder hierarchy and the CND have qualitative impact on ΛHDE, 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′} pair has qualitative impact on ΛHDE; for the case of varying Ω_(de0), the {w, w′} pair only has quantitative impact; these results are different from the cases of HDE, RDE, and NADE, in which the {w,w′} pair only has quantitative impact on these models. In conclusion, compared with HDE, RDE, and NADE, the ΛHDE model can be easily distinguished by using these diagnostic tools.