content_image/444595.jpg content_image/444596.jpg content_image/444598.jpg content_image/444597.jpg content_image/444599.jpg content_image/444600.jpg content_image/444601.jpg content_image/444602.jpg content_image/444603.jpg content_image/444604.jpg
## Appendix C: Diagonal terms of the asymptotic moments of CMB brightness fluctuation 

In equation (34), taking e = 0 i.e. ℓ = R and putting ℓ$_{1}$ = R + ℓ$_{4}$ & ℓ$_{2}$ = R + ℓ$_{3}$ , we can write equation (35) as 

<img src='content_image/444595.jpg'>

From the properties of the Clebsch-Gordon coefficient, we get ℓ$_{3}$ = ℓ$_{4}$ and hence ℓ$_{1}$ = ℓ$_{2}$ . Thus in equation (31) substituting ℓ$_{3}$ = ℓ$_{1}$ − ℓ , we get the factor L 

<img src='content_image/444596.jpg'>

Now using equation (8.4.10) from [48], 

<img src='content_image/444598.jpg'>

Using equations (8.5.34) and (8.5.42) from [48], we get 

and 

<img src='content_image/444597.jpg'>

<img src='content_image/444599.jpg'>

Thus 

<img src='content_image/444600.jpg'>

Substituting this in equation (31), the evolution equation becomes 

<img src='content_image/444601.jpg'>

where ℓ, ℓ$_{1}$ >> | ℓ$_{1}$ − ℓ | . 

## Appendix D: Off-diagonal terms of the asymptotic moments of CMB brightness fluctuation 

In equation (34), with e = 0 i.e. ℓ = R 

From conditions of Clebsch-Gordon coefficients 

<img src='content_image/444602.jpg'>

<img src='content_image/444603.jpg'>

For off-diagonal terms of ˜ Δ LM ℓ$_{1}$ ,ℓ$_{2}$ we consider the minimum values of ℓ$_{3}$ , ℓ$_{4}$ & L with neither equal to 0. 

<img src='content_image/444604.jpg'>

