1. 报告人：李君 指导老师：杨荣佳 学校：河北大学
Discriminating dark energy models by using the statefinnder hierarchy and the growth rate of matter perturbations
报告人：李君
指导老师：杨荣佳
学校：河北大学

2. Outline
1. Introduction
2 .The Statefinder hierarchy and the growth rate of matter perturbations
3. Dark energy models and discriminations
4. Conclusions and discussions

3. INTRODUCTION
1、1998，SNIa，the universe is experiencing accelerated expansion.
2、WMAP，Planck，…，Confirm this result.
3、Dark energy is proposed to explain this acceleration: equally distributes in the Universe, its pressure is negative.

4. The simplest and most theoretically appealing candidate of dark energy is the vacuum energy (LCDM with EoS w = −1).
This scenario is consistent with most of the current astronomical observations
But suffers from the cosmological constant problem and age problem.

5. Over the past years, numerous dark energy models have been proposed, such as quintessence, phantom, k-essence, tachyon, (generalized) Chaplygin gas ((G)CG), etc.
As more and more dark energy models have been proposed,it is important to discriminate various dark energy models.
Some diagnostics have been proposed to solve this question, such as the statefinder, Om diagnostic .

6. But Om and statefinder {r,s} can not distinguish PKK from LCDM
Here we will use statefinder hierarchy and the growth rate of matter perturbations to distinguish GCG,MCG,SCG PKK from ΛCDM model.

7. THE STATEFINDER HIERARCHY AND THE GROWTH RATE OF MATTER PERTURBATIONS
The Statender hierarchy
The scale factor can be Taylor expanded around the present time t0 as follows:

8. For ΛCDM in a spatially flat LCDM, we can get:
where and
in the concordance cosmology.

9. For the other models, The deceleration parameter q,
A3, and A4 can be rewritten as

10. The Statefinder hierarchy, Sn, is defined as:
For ΛCDM:
This equation define a null diagnostic for ΛCDM, since for evolving dark energy models some of these equalities may be violated.

11. When n ≥ 3, more than one way can be adopted to define a null diagnostic, for example,one series of Statefinders can be definedas:
The second series of Statefinders are defined as:
where α is an arbitrary constant. For ΛCDM model, we have

12. The growth rate of matter perturbations
The fractional growth parameter ϵ used with statefinder can be defined as follows:
where describes the growth rate of linearized density perturbations and can be parameterized as:
where

13. DARK ENERGY MODELS AND DISCRIMINATIONS
In order to distinguish dark energy models by using Statefinder hierarchy and the growth rate of matter perturbations, we need the dimensionless matter density parameter. .
For GCG, MCG, and PKK
For SCG

14. Dark energy models
1.GCG
GCG as dark energy is characterized with a EoS, , with 0 < α < 1.
The values of parameters we take in the following content are constrained from large-scales tructure observation:
As = 0.764, = −1.436,Ωm0 =

15. 2.MCG
MCG is considered as dark energy with EoS, P = Bρ − A/ρ α, where A, B, and α are positive constants with 0 < α < 1. For B = 0, MCG reduces to the GCG model.
The best-fit values of parameters we take
are: As = 0.769, B = 0.008, α = 0.002, and Ωm0 =

16. 3.SCG SCG is a model unified dark matter and dark energy,
The best-fit values of parameters we take are: k =0.173, k0 =

17. 4.PKK PKK is a class of k-essence with Lagrangian:
, where V0 is a constant.
The best-fit values of parameters we take are: Ωm0 =0.36, k0 =

18. The fixed point at {1, 1} denotes ΛCDM
The fixed point at {1, 1} denotes ΛCDM. The solid, dash, dot, the dash-dot line represents the evolution of MCG, PKK, SCG, and GCG, respectively. The arrows show time evolution and the present epoch in different models is shown as a dot.
We use the Statefinder {S3 (1) , S4 (1)} to discriminate GCG,MCG, SCG, PKK, and ΛCDM, and find that GCG, PKK,and ΛCDM can be distinguished from each other,but PKK and SCG or MCG and ΛCDM can not be distinguished well from each other at the present epoch.

19. Using the composite diagnostic {ϵ(z) , S3 (1)} to distinguish these dark energy models, we find that GCG, MCG, SCG can be distinguished from ΛCDM or PKK, but ΛCDM and PKK can not be distinguished from each other well at the present epoch.

20. Using Statefinder {S4, S4 (1)}, MCG, PKK, SCG, and GCG can be distinguished from ΛCDM, but PKK and SCG can not be distinguished from each other at the present epoch.

21. Using Statefinder {S3(1) , S5}, MCG, PKK, SCG, GCG and ΛCDM can be distinguished from each other, but PKK and SCG can not be distinguished well at present epoch.

22. Using the composite diagnostic {ϵ(z), S4}, we find that at the present epoch MCG, SCG, and GCG can be distinguished from PKK or ΛCDM, but PKK and ΛCDM can not be distinguished well from each other.

23. Using the composite diagnostic {ϵ(z), S5(1) }, MCG,SCG, GCG, PKK, and ΛCDM can be distinguished well from each other at the present epoch.

24. We also use other pairs, such as {S3(1) , S5 (1)} , {S4, S5 (1)} , {S5, S4 (1)},
{S4 (1) , ϵ(z)}, etc., to distinguish these models;we find these models can not be distinguished well, compared with the results obtained by using pairs presented above.

25. CONCLUSIONS AND DISCUSSIONS
We find that GCG, MCG, SCG, PKK, and ΛCDM can only be distinguished well from each other at the present epoch by using the composite diagnostic
Using other combinations, some of these five dark energy models can not be distinguished.

26. 谢谢