12 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 Appendix C: Diagonal terms of the asymptotic moments of CMB brightness fluctuation { ℓ$_{4}$ L ℓ$_{3}$ ℓ + ℓ$_{4}$ ℓ ℓ + ℓ$_{3}$ } ≈ ( − 1) L √$_{2}$$_{ℓ}$$_{Π}$$_{ℓ}$$_{3}$ C ℓ$_{3}$ ℓ$_{4}$ ℓ$_{4}$ ℓ$_{3}$ L ( ℓ$_{4}$ − ℓ$_{3}$ $_{)}$. (C1) C$_{1}$ ( L, ℓ, ℓ$_{1}$ ) = ( − i ) ℓ Π 2 ℓ Π$_{ℓ}$$_{1}$ Π$_{ℓ}$$_{1}$$_{−}$$_{ℓ}$ β$_{ℓ}$$_{1}$$_{−}$$_{ℓ,ℓ}$$_{1}$$_{−}$$_{ℓ}$ β$_{ℓ}$$_{1}$$_{,ℓ}$$_{1}$ ( − 1) L C ( ℓ$_{1}$ − ℓ )0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{1}$ 0 ℓ 0( ℓ$_{1}$ − ℓ )0 ( − 1) L √$_{2}$$_{ℓ}$$_{Π}$$_{ℓ}$$_{1}$$_{−}$$_{ℓ}$ C ( ℓ$_{1}$ − ℓ )( ℓ$_{1}$ − ℓ ) ( ℓ$_{1}$ − ℓ )( ℓ$_{1}$ − ℓ ) L 0 . (C2) [ C ( ℓ$_{1}$ − ℓ )0 ℓ$_{1}$ 0 ℓ 0 ] 2 = [ ℓ$_{1}$ ! ℓ !( ℓ$_{1}$ − ℓ )! ] 2 (2 ℓ )!(2 ℓ$_{1}$ − 2 ℓ + 1)! (2 ℓ$_{1}$ + 1)! , (C4) C ( ℓ$_{1}$ − ℓ )0 ℓ 0 ℓ$_{1}$ 0 = ( − 1) ℓ + ℓ$_{1}$ − ( ℓ$_{1}$ − ℓ $^{)}$C ( ℓ$_{1}$ − ℓ )0 ℓ$_{1}$ 0 ℓ 0 = C ( ℓ$_{1}$ − ℓ )0 ℓ$_{1}$ 0 ℓ 0 , C ℓ$_{1}$ 0 ℓ 0( ℓ$_{1}$ − ℓ )0 = ( − 1) ℓ Π$_{ℓ}$$_{1}$ Π$_{ℓ}$$_{1}$$_{−}$$_{ℓ}$ C ( ℓ$_{1}$ − ℓ )0 ℓ$_{1}$ 0 ℓ 0 . (C3) C ( ℓ$_{1}$ − ℓ )( ℓ$_{1}$ − ℓ ) ( ℓ$_{1}$ − ℓ )( ℓ$_{1}$ − ℓ ) L 0 = (2 ℓ$_{1}$ − 2 ℓ )! √ 2 ℓ$_{1}$ − 2 ℓ + 1 (2 ℓ$_{1}$ − 2 ℓ − L )!(2 ℓ$_{1}$ − 2 ℓ + L + 1)! . (C5) C$_{1}$ ( L, ℓ, ℓ$_{1}$ ) = Π 2 $_{ℓ}$Π$_{ℓ}$$_{1}$ Π$_{ℓ}$$_{3}$ √$_{2}$$_{ℓ}$ [ ℓ$_{1}$ ! ℓ ! ℓ$_{3}$ ! ] 2 (2 ℓ )!(2 ℓ$_{3}$ + 1)! (2 ℓ$_{1}$ + 1)! √ (2 ℓ$_{3}$ )! (2 ℓ$_{3}$ − L )! (2 ℓ$_{3}$ )! (2 ℓ$_{3}$ + L + 1)! δ$_{ℓ}$$_{3}$$_{,ℓ}$$_{1}$$_{−}$$_{ℓ}$ . (C6) ˜ Δ LM ℓ$_{1}$ $_{,ℓ$_{1}$}$( k, τ ) = ∑ ℓ C$_{1}$ ( L, ℓ, ℓ$_{1}$ ) j$_{ℓ}$ ( k Δ τ ) ˜ Δ LM ℓ$_{1}$ − ℓ, ℓ$_{1}$ − $_{ℓ}$( k, τ$_{s}$ ) , (C7) { ℓ$_{4}$ L ℓ$_{3}$ ℓ$_{1}$ ℓ ℓ$_{2}$ } ≈ ( − 1) ℓ$_{4}$ + L + ℓ$_{1}$ − ℓ √$_{2}$$_{ℓ}$$_{Π}$$_{ℓ}$$_{3}$ C ℓ$_{3}$ ( ℓ$_{1}$ − ℓ ) ℓ$_{4}$ ( ℓ$_{2}$ − ℓ ) L ( ℓ$_{1}$ − ℓ$_{2}$ $_{)}$. (D1) | ℓ$_{1}$ − ℓ | ≤ ℓ$_{3}$ ℓ$_{2}$ ℓ ℓ$_{4}$ | ℓ$_{1}$ − ℓ$_{2}$ | ≤ L. ℓ$_{1}$ − ℓ = ℓ$_{3}$ ℓ$_{2}$ ℓ = ℓ$_{4}$ ℓ$_{1}$ − ℓ$_{2}$ = L. 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 Now using equation (8.4.10) from [48], Using equations (8.5.34) and (8.5.42) from [48], we get and Thus Substituting this in equation (31), the evolution equation becomes where ℓ, ℓ$_{1}$ >> | ℓ$_{1}$ − ℓ | . In equation (34), with e = 0 i.e. ℓ = R From conditions of Clebsch-Gordon coefficients Appendix D: Off-diagonal terms of the asymptotic moments of CMB brightness fluctuation 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.