in which D$_{1(}$$_{α}$$_{)}$ ( q ) and D$_{2(}$$_{α}$$_{)}$ ( q ) are integration constants, the solution up to first order becomes: 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 with All these lead to the following relation for the perturbations outside the horizon: 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}$$_{)}$ ( q ) = −C$_{α}$ ( − qτ$_{0}$ ) 2 α − $^{1}$. 15 R$_{q}$ ( τ ) −→ R o $_{q}$( τ ) = − iK ′ 8 π Γ( α ) q − α [ 1 − Kπ 4 F$_{α}$$^{]}$ . (46) A$_{α}$ = 1 4 α − 3$^{(}$ 2 1 − 2 $^{α}$α Γ(1 + α ) 2 − i 2 1 − 2 $^{α}$(cos(( − 1 + α ) π )Γ(1 − α ) + α cos( απ )Γ( − α )) π Γ(1 + α ) + 2 1 − 2 α Γ( α )Γ(1 + α )$^{)}$ B$_{α}$ = 1 4 α − 1$^{(}$ 2 − 2 $^{α}$α (1 + α )Γ(1 + α ) 2 + i 2 − 1 − 2 $^{α}$cos((1 + α ) π )Γ( − 1 − α ) π Γ(1 + α ) + 2 − 1 − 2 $^{α}$(1 + 2 α ) α (1 + α )Γ( α )Γ(1 + α ) + i 2 − 1 − 2 $^{α}$(1 + 2 α ) cos(( − 1 + α ) π )Γ(1 − α ) πα (1 + α )Γ(1 + α ) + i 2 − 2 $^{α}$α cos( απ )Γ( − α ) π (1 + α )Γ(1 + α ) − 2 − 1 − 2 α Γ(1 + α )Γ(2 + α )$^{)}$ C$_{α}$ = 1 2 α − 1$^{(}$ − i − α Γ( − 1 + α ) + α $^{2}$Γ( − 1 + α ) + α $^{2}$Γ( α ) + Γ(1 + α ) − α Γ(1 + α ) 2 π ( − 1 + α ) α Γ(1 + α ) ) . F$_{α}$ = ( − q ) 3 − 2 $^{α}$( A$_{α}$ ( − qτ ) 4 α − 3 + B$_{α}$ ( − qτ ) 4 α − 1 + C$_{α}$ ( − qτ ) 2 α − $^{1}$) + D$_{1(}$$_{α}$$_{)}$ ( q ) (45) R$_{q}$$_{(1)}$ ( τ ) = K $^{′}$( − τ ) α ( H (1) α ( − qτ ) − Kπ 4 [ F$_{α}$ Y$_{α}$ ( − qτ ) + G$_{α}$ J$_{α}$ ( − qτ )]$^{)}$ . (44) G$_{α}$ = ( − q ) 3 − 2 α ∫ − qτ 0 d ( − qτ )( − qτ ) 2 α − $^{4}$Y$_{α}$ ( − qτ ) [ − qτ H (1) α − $_{1}$( − qτ ) + 2 αH (1) α ( − qτ ) + qτ H (1) α $_{+1}$( − qτ ) $^{]}$+ D$_{2(}$$_{α}$$_{)}$ ( q ) (43)