content_image/299720.jpg content_image/299718.jpg content_image/299719.jpg content_image/299715.jpg content_image/299716.jpg
Therefore the IMSE is given by: 

<img src='content_image/299720.jpg'>

Since the right hand side of IMSE is just a small correction, we can solve it iteratively. At zeroth order of iteration, ignoring this term, the solution is Hankel functions H (1) α ( − qτ ) and H (2) α ( − qτ ). We choose the sub-horizon initial condition as exp( − iqτ ) at large − qτ . This initial condition is the result of applying WKB methods suggesting plane wave solution for initial conditions at early times, q a ≫ H , as in [18]. With the conditions: 

<img src='content_image/299718.jpg'>

for ΛCDM universe, we have the same initial condition as for the non–improved model: 

which at the limit a ( t ) → 0 causes R$_{q}$ to approach to − H δϕ$_{q}$ ˙ ¯ ϕ . 

<img src='content_image/299719.jpg'>

Since at large − qτ , H (1) α ( − qτ ) tends to √ 2 π ( − qτ ) exp( i ( − qτ ) − iα π 2 − i π $_{4}$), the first Hankel function would be an appropriate choice. After renormalizing the zeroth order solution becomes 

where K ′ = − λ √ π 4(2 π ) 3 2 $_{ϵ}$( ϵ 1 − $_{ϵ}$) − 1 1 − ϵ exp( iπα 2 + iπ 4 ) and α = ϵ − 3 2( ϵ − $_{1)}$. 

<img src='content_image/299715.jpg'>

To obtain the solution of IMSE up to the next order of iteration we have to substitute the zeroth order solution in the right hand side of (40) and solve the equation. Defining: 

<img src='content_image/299716.jpg'>

