5 IV. EVOLUTION IN THE FREE-STREAMING REGIME A. Generalized evolution equation We find the moments of the CMB brightness fluctua- tion to be correlations as To check for the statistical isotropic case, we put L = 0 & M = 0 and recover equation (7) in section (II C). starting with equation (11) and the orthonormality con- dition of BipoSH, As shown in detail in Appendix A, the BipoSH coeffi- cients in equation (19), can be expressed using the angu- lar correlations in equation (23) for statistically isotropic primordial perturbations as ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) = 1 4 π 1 β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ∫ ∫ d Ω$_{ˆ}$ n d Ω$_{ˆ}$ k ˜ Δ( ⃗ k, ˆ n, τ ) × { Y$_{ℓ}$$_{1}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{2}$ (ˆ n ) } $_{LM}$, (27) 〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 = 4 π ∫ k $^{2}$dk 2 π 2 P$_{0}$ ( k ) ∑ ℓ$_{1}$m$_{1}$LM L ′ M ′ β$_{ℓ}$$_{1}$$_{ℓ}$ β ∗ ℓ$_{1}$ℓ ′ × ˜ Δ LM ℓ$_{1}$ℓ ( k, τ ) [ ˜ Δ L $^{′}$M ′ ℓ$_{1}$ℓ ′ ( k, τ )] ∗ × C LM ℓ$_{1}$m$_{1}$ℓm C L $^{′}$M ′ ℓ$_{1}$m$_{1}$ℓ ′ m ′ . (23) ∫∫ d Ω$_{ˆ}$ $_{n}$d Ω$_{ˆ}$ k { Y$_{ℓ}$$_{1}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{2}$ (ˆ n ) } LM { Y$_{ℓ}$$_{3}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{4}$ (ˆ n ) } ∗ L ′ M ′ = δ$_{ℓ}$$_{1}$$_{ℓ}$$_{3}$ δ$_{ℓ}$$_{2}$$_{ℓ}$$_{4}$ δ$_{LL}$ ′ δ$_{M M}$ ′ . (28) ( A JN ℓℓ ′ ) l =0 = 4 π ∫ k $^{2}$dk 2 π 2 P$_{0}$ ( k ) ∑ ℓ$_{1}$LL ′ ( − 1) ℓ$_{1}$ + L + L $^{′}$+ J × Π$_{LL}$ ′ β$_{ℓ}$$_{1}$$_{ℓ}$ β ∗ ℓ$_{1}$ℓ ′ { L J L ′ ℓ ′ ℓ$_{1}$ ℓ } × { ˜ Δ L ℓ$_{1}$ℓ ⊗ ˜ Δ L ′ ℓ$_{1}$ℓ ′ } JN . (24) ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) = 1 β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ∑ ℓℓ$_{3}$ℓ$_{4}$L ′ M ′ ( − i ) ℓ Π$_{ℓℓ}$ j$_{ℓ}$ ( k Δ τ ) β$_{ℓ}$$_{3}$$_{ℓ}$$_{4}$ × ∑ m$_{1}$m$_{2}$m$_{3}$m$_{4}$ ˜ Δ LM ℓ$_{3}$$_{ℓ$_{4}$}$( k, τ$_{s}$ ) C LM ℓ$_{1}$m$_{1}$ℓ$_{2}$m$_{2}$ C L $^{′}$M ′ ℓ$_{3}$m$_{3}$ℓ$_{4}$m$_{4}$ × ∫∫ d Ω$_{ˆ}$ n d ˆ k P$_{ℓ}$ ( ˆ k · ˆ n ) Y ∗ ℓ$_{1}$$_{m$_{1}$}$( ˆ k ) Y ∗ ℓ$_{2}$$_{m$_{2}$}$(ˆ n ) × Y$_{ℓ}$$_{3}$$_{m}$$_{3}$ ( ˆ k ) Y$_{ℓ}$$_{4}$$_{m}$$_{4}$ (ˆ n ) . (29) In the free streaming regime, using the plane wave ap- proximation as in equations (8) and (9), the most general evolution equation turns out to be The first term in braces is the Wigner-6j symbol which is related to the coefficients of transformations between dif- ferent coupling schemes of three angular momenta. These vanish unless the triangular conditions [48] are fulfilled for the triads ( LJ L $^{′}$) , ( L $^{′}$ℓ$^{′}$ℓ$_{1}$ ) , ( J ℓ $^{′}$ℓ ) and ( ℓLℓ$_{1}$ ). We consider low Bipolar deviations from SI i.e. L, L ′ and J << ℓ, ℓ $^{′}$, ℓ$_{1}$ in equation (24) and use the asymptotic relation for Wigner-6j functions given by equation (9.9.1) from [48]. We find that the asymptotic limit to these BipoSH coefficients are As shown in detail in Appendix B, this can be further simplified to The generalized evolution equation thus can be expressed so as to structurally resemble the evolution equation for the SI case in equation (10). ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) = ∑ ℓℓ$_{3}$ℓ$_{4}$ ( − i ) ℓ Π$_{ℓℓℓ}$$_{1}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{3}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ( − 1) ℓ$_{3}$ + ℓ$_{4}$ + L × j$_{ℓ}$ ( k Δ τ ) ˜ Δ LM ℓ$_{3}$ $_{ℓ$_{4}$}$( k, τ$_{s}$ ) C ℓ$_{3}$ 0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{2}$ 0 ℓ 0 ℓ$_{4}$ 0 × { ℓ$_{1}$ L ℓ$_{2}$ ℓ$_{4}$ ℓ ℓ$_{3}$ } (30) ( A JN ℓℓ ′ ) l =0 ≈ 4 π ∫ k $^{2}$dk 2 π 2 P$_{0}$ ( k ) ∑ ℓ$_{1}$LL ′ ( − 1) ℓ + ℓ $^{′}$+ ℓ$_{1}$ + L + L ′ × Π$_{LL}$ ′ √$_{2}$$_{ℓ}$$_{1}$ Π$_{J}$ β$_{ℓ}$$_{1}$$_{ℓ}$ β ∗ ℓ$_{1}$ℓ ′ C J ( ℓ − ℓ $^{′}$) L ( ℓ − ℓ$_{1}$ ) L ′ ( ℓ$_{1}$ − ℓ ′ ) × { ˜ Δ L ℓ$_{1}$ℓ ⊗ ˜ Δ L ′ ℓ$_{1}$ℓ ′ } JN . (25) ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) = ∑ ℓℓ$_{3}$ℓ$_{4}$ C ( L, ℓ, ℓ$_{1}$, ℓ$_{2}$ ) j$_{ℓ}$ ( k Δ τ ) × ˜ Δ LM ℓ$_{3}$$_{ℓ$_{4}$}$( k, τ$_{s}$ ) , (31) C ( L, ℓ, ℓ$_{1}$, ℓ$_{2}$ ) = ( − i ) ℓ Π$_{ℓℓℓ}$$_{1}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{3}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ( − 1) ℓ$_{3}$ + ℓ$_{4}$ + L × C ℓ$_{3}$ 0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{2}$ 0 ℓ 0 ℓ$_{4}$ 0 { ℓ$_{1}$ L ℓ$_{2}$ ℓ$_{4}$ ℓ ℓ$_{3}$ } . (32) ( A JN ℓℓ ) l =0 ≈ 4 π ∫ k $^{2}$dk 2 π 2 P$_{0}$ ( k ) ∑ LL ′ ( − 1) L + L $^{′}$+1 Π$_{ℓℓLL}$ ′ Π$_{J}$ × C J 0 L L ′ 0 { ˜ Δ L ℓℓ ⊗ ˜ Δ L ′ ℓℓ } JN . (26) where For diagonal brightness fluctuations i.e ℓ$_{1}$ = ℓ and ℓ$_{1}$ = ℓ $^{′}$, the BipoSH coefficients themselves turn out to be di- agonal in multipole space