Cosmological Scaling Solutions

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Cosmological Scaling Solutions

• Zong-Kuan Guo
• (advisor Yuan-Zhong Zhang)
• Institute of Theoretical Physics, Chinese Academy
  of Sciences,
• P.O. Box 2735, Beijing 100080, P.R.China

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Outline

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

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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

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• We consider the action
• The evolution equation

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

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

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The phase-space analysis

• We consider a scalar field with an exponential
  potential
• We introduce dimensionless variables

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

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• Critical points

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Multiple Positive and Negative Exponential
Potentials

• The model I

Z.K.Guo et al, Phys. Lett. B568 (2003) 1-7 .
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Cross-coupling Exponential Potential

• The model II

Z.K.Guo et al, Phys. Lett. B576 (2003) 12-17.
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Multiple Tachyon Fields with Inverse Square
Potentials

• The model III

Z.K.Guo et al, JCAP 07 (2004) 013.
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The evolution equations
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Conclusions

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

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Thank you!