content_image/321406.jpg content_image/321407.jpg content_image/321405.jpg content_image/321408.jpg content_image/321409.jpg content_image/321410.jpg
which is identical to Eq. (4.80) except for the difference in the limit. For the case Γ$_{ϕ}$ ≪ H$_{TI}$ , we would be saying that 

Substituting H$_{TI}$ , Eq. (4.10), into Eq. (4.104) gives 

<img src='content_image/321406.jpg'>

<img src='content_image/321407.jpg'>

<img src='content_image/321405.jpg'>

which is identical to Eq. (4.81) except for the difference in the limit. 

As we are dealing with the thermal fluctuation of φ about φ =0 , we have 〈 δφ 〉 T = 〈 φ 〉 $_{T}$. The thermal fluctuation of φ is 

and we require 

A detailed derivation of this is given in Appendix A. 

In order to keep m$_{ψ,}$$_{eff}$ light, we require, from Section 4.4.3.4, 

<img src='content_image/321408.jpg'>

<img src='content_image/321409.jpg'>

<img src='content_image/321410.jpg'>

During the time between the end of primordial inflation and primordial in- flation reheating, T ∝ a − 3 8 and H ∝ a − 3 2 and during radiation domination, 

