content_image/320446.jpg content_image/320445.jpg content_image/320444.jpg content_image/320441.jpg content_image/320442.jpg content_image/320443.jpg content_image/320440.jpg
<img src='content_image/320446.jpg'>

in which D$_{1(}$$_{α}$$_{)}$ ( q ) and D$_{2(}$$_{α}$$_{)}$ ( q ) are integration constants, the solution up to first order becomes: 

<img src='content_image/320445.jpg'>

The perturbations outside the horizon are defined by the limit − qτ ≪ 1. At this limit, for α > 0, the term J$_{α}$ ( − qτ ) → ( − qτ 2 ) α 1 Γ( α +1) can be ignored with respect to Y$_{α}$ ( − qτ ) → − Γ( α ) π ( − qτ 2 ) − $^{α}$. On the other hand, F$_{α}$ at − qτ ≪ 1 becomes 

<img src='content_image/320444.jpg'>

with 

<img src='content_image/320441.jpg'>

<img src='content_image/320442.jpg'>

<img src='content_image/320443.jpg'>

All these lead to the following relation for the perturbations outside the horizon: 

<img src='content_image/320440.jpg'>

The parameter α will approach to 3 / 2 by applying the slow roll condition, ϵ ≪ 1. Since − qτ ≪ 1, in the slow roll inflation, the terms ( − qτ ) 4 α − 3 = ( − qτ ) 3 and ( − qτ ) 4 α − 1 = ( − qτ ) 5 don’t have any significant contribution in F$_{α}$ . Beside, if we consider the case in which the non–improved solution at τ$_{0}$ satisfies the initial condition R$_{q}$$_{(0)}$ ( τ$_{0}$ ) = 0, we obtain D$_{1(}$$_{α}$$_{)}$ ( q ) = −C$_{α}$ ( − qτ$_{0}$ ) 2 α − $^{1}$. 

