1. Cosmological Scaling Solutions
Zong-Kuan Guo
(advisor: Yuan-Zhong Zhang)
Institute of Theoretical Physics, Chinese Academy of Sciences,
P.O. Box 2735, Beijing , P.R.China

2. Outline Cosmological Scaling Solutions
Multiple Positive and Negative Exponential Potentials
Cross-coupling Exponential Potential
Multiple Tachyon Felds
Conclusions

3. Cosmological Scaling Solutions
Why do we study cosmological scaling solutions?
Power-law inflation
Dark energy
What is cosmological scaling solutions?
Potential and kinetic energies remain proportional

4. We consider the action
The evolution equation

5. The corresponding equation of state parameter
Substituting Eq.(4) into Eq.(1) gives

6. From Eq.(2), we can get
Substituting Eq.(6) into Eq.(1) gives
Using Eq.(3), Eq.(5) and Eq.(6), we get

7. The phase-space analysis
We consider a scalar field with an exponential potential
We introduce dimensionless variables

8. The autonomous system
The constraint equation (1) becomes
Critical points correspond to power-law solution

9. Critical points:

10. Multiple Positive and Negative Exponential Potentials
The model I
Z.K.Guo et al, Phys. Lett. B568 (2003) 1-7 .

12. Cross-coupling Exponential Potential
The model II
Z.K.Guo et al, Phys. Lett. B576 (2003)

14. Multiple Tachyon Fields with Inverse Square Potentials
The model III
Z.K.Guo et al, JCAP 07 (2004) 013.

15. The evolution equations:

18. Conclusions The phase-space analysis
We have showed existence and stability of the cosmological scaling solution
We present the parameter space
The coincidence problem

19. Thank you!