content_image/312628.jpg content_image/312629.jpg content_image/312630.jpg content_image/312631.jpg content_image/312632.jpg
T ∝ a − 1 and H ∝ a − $^{2}$. Therefore, if Eq. (4.108) is satisfied, then equivalent constraints for higher T and H are guaranteed to be satisfied as well. In the case of Γ$_{ϕ}$ ≳ H$_{TI}$ , by substituting Eqs. (4.8), (4.10), (4.11) and (4.92) into Eq. (4.108) we obtain the constraint 

<img src='content_image/312628.jpg'>

Rearranging this for m$_{0}$ gives the following. For α =1 and any value of n , or for α =2 and n =1 we have 

<img src='content_image/312629.jpg'>

For all other α and n combinations, the above inequality is reversed, giving 

<img src='content_image/312630.jpg'>

In the case of Γ$_{ϕ}$ ≪ H$_{TI}$ , by substituting Eqs. (4.10), (4.12) and (4.92) into Eq. (4.108) we obtain the constraint 

<img src='content_image/312631.jpg'>

Rearranging this for m$_{0}$ gives the following. For the values of n and α given in Table 4.1 we have 

<img src='content_image/312632.jpg'>

