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 Λ 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 (1) 3 ( z ) and S (1) 4 ( z ) are studied. The second is composite null diagnostic (CND), in which the trajectories of { S (1) 3 , ϵ } and { S (1) 4 , ϵ } are investigated, where ϵ 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 ln a . 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 Λ HDE, but only have quantitative impact on HDE. (2) S (1) 4 can lead to larger differences than S (1) 3 , 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 Λ HDE; 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 Λ HDE model can be easily distinguished by using these diagnostic tools.