content_image/315146.jpg content_image/315148.jpg content_image/315149.jpg content_image/315150.jpg content_image/315147.jpg content_image/315151.jpg
inflation is 

<img src='content_image/315146.jpg'>

For simplicity, we assume that ψ decays around the same time as φ , i.e. that H does not change much between the time when φ decays and the time when ψ decays. Therefore, the energy density of the Universe at the time when ψ decays is ∼ M 2 $_{P}$Γ $^{2}$. We therefore require 

Therefore we require 

and 

<img src='content_image/315148.jpg'>

<img src='content_image/315149.jpg'>

<img src='content_image/315150.jpg'>

<img src='content_image/315147.jpg'>

Substituting 〈 φ 〉 , Γ and $^{ψ}$∗ , Eqs. (4.7), (4.18) and (4.92) respectively, into Eq. (4.172) gives the constraint 

<img src='content_image/315151.jpg'>

