11 { L J L ′ ℓ ′ ℓ$_{1}$ ℓ } ≈ ( − 1) L + J + ℓ $^{′}$− ℓ$_{1}$ √$_{2}$$_{ℓ}$$_{1}$ Π$_{L}$ ′ C L $^{′}$( ℓ $^{′}$− ℓ$_{1}$ ) L ( ℓ − ℓ$_{1}$ ) J ( ℓ ′ − ℓ ) ≈ ( − 1) J + ℓ + ℓ ′ √$_{2}$$_{ℓ}$$_{1}$ Π$_{J}$ C J ( ℓ − ℓ $^{′}$) L ( ℓ − ℓ$_{1}$ ) L ′ ( ℓ$_{1}$ − ℓ ′ $_{)}$. (A9) ( A J N 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 ′ } $_{J N}$. (A10) function as given in equation (9.9.1) from [48] Using the above relation, the asymptotic limit to the BipoSH coefficients turn out to be Appendix B: Generalized evolution equation for statistical isotropy breakdown Starting with equation (29), putting P$_{ℓ}$ ( ˆ k · ˆ n ) = 4 π 2 ℓ +1 ∑ m Y ∗ $_{ℓm}$( ˆ k ) Y$_{ℓm}$ (ˆ n ) and evaluating the double integral using the equation the most general evolution equation becomes ∫ d Ω$_{ˆ}$ $_{n}$Y$_{ℓm}$ (ˆ n ) Y$_{ℓ}$$_{1}$$_{m}$$_{1}$ (ˆ n ) Y$_{ℓ}$$_{2}$$_{m}$$_{2}$ (ˆ n ) ∗ = 1 √$_{4}$$_{π}$ Π$_{ℓℓ}$$_{1}$ Π$_{ℓ}$$_{2}$ C ℓ$_{2}$ 0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{2}$m$_{2}$ ℓmℓ$_{1}$$_{m$_{1}$}$, (B1) ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) = 1 β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ∑ ℓ ( − i ) ℓ Π 2 ℓ j$_{ℓ}$ ( k Δ τ ) ∑ ℓ$_{3}$ ℓ$_{4}$L ′ M ′ β$_{ℓ}$$_{3}$$_{ℓ}$$_{4}$ ˜ Δ L $^{′}$M ′ ℓ$_{3}$ℓ$_{4}$ ( k, τ$_{s}$ ) Π$_{ℓ}$$_{1}$$_{ℓ}$$_{4}$ Π$_{ℓ}$$_{2}$$_{ℓ}$$_{3}$ C g , (B2) C g = C ℓ$_{3}$ 0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{2}$ 0 ℓ 0 ℓ$_{4}$ 0 ∑ m$_{1}$m$_{2}$ C LM ℓ$_{1}$m$_{1}$ℓ$_{2}$m$_{2}$ ∑ m$_{3}$m$_{4}$ C L $^{′}$M ′ ℓ$_{3}$m$_{3}$ℓ$_{4}$m$_{4}$ ∑ m C ℓ$_{3}$m$_{3}$ ℓmℓ$_{1}$m$_{1}$ C ℓ$_{2}$m$_{2}$ ℓmℓ$_{4}$$_{m$_{4}$}$. (B3) ∑ mm$_{1}$m$_{2}$m$_{3}$m$_{4}$ C L $^{′}$M ′ ℓ$_{3}$m$_{3}$ℓ$_{4}$m$_{4}$ C ℓ$_{3}$m$_{3}$ ℓ$_{1}$m$_{1}$ℓm C LM ℓ$_{1}$m$_{1}$ℓ$_{2}$m$_{2}$ C ℓ$_{2}$m$_{2}$ ℓmℓ$_{4}$m$_{4}$ = δ$_{L}$ ′ $_{L}$δ$_{M}$ ′ $_{M}$( − 1) ℓ$_{1}$ + ℓ + ℓ$_{4}$ + L $^{′}$Π$_{ℓ}$$_{2}$$_{ℓ}$$_{3}$ { ℓ$_{1}$ ℓ ℓ$_{3}$ ℓ$_{4}$ L ′ ℓ$_{2}$ } , (B4) ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) = ∑ ℓℓ$_{3}$ℓ$_{4}$ ( − i ) ℓ Π$_{ℓℓℓ}$$_{1}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{3}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ( − 1) ℓ$_{3}$ + ℓ$_{4}$ + L j$_{ℓ}$ ( k Δ τ ) ˜ Δ LM ℓ$_{3}$ $_{ℓ$_{4}$}$( k, τ$_{s}$ ) C ℓ$_{3}$ 0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{2}$ 0 ℓ 0 ℓ$_{4}$ 0 { ℓ$_{1}$ ℓ ℓ$_{3}$ ℓ$_{4}$ L ℓ$_{2}$ } = ∑ ℓℓ$_{3}$ℓ$_{4}$ ( − i ) ℓ Π$_{ℓℓℓ}$$_{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$_{3}$ } . (B5) C ( L, ℓ, ℓ$_{1}$, ℓ$_{2}$ ) = ( − i ) ℓ Π$_{ℓℓℓ}$$_{1}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{3}$$_{ℓ}$$_{4}$ β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ( − 1) ℓ$_{3}$ + ℓ$_{4}$ + $^{L}$C ℓ$_{3}$ 0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{2}$ 0 ℓ 0 ℓ$_{4}$ 0 { ℓ$_{1}$ L ℓ$_{2}$ ℓ$_{4}$ ℓ ℓ$_{3}$ } , (B6) ˜ Δ LM ℓ$_{1}$$_{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}$ ) . (B7) where Using the summation formula for four Clebsch-Gordon coefficients given by equation (9.1.8) from [48], equation (B2) simplifies to Substituting the multipole dependent coefficients as the generalized evolution equation for deviations of statistical isotropy in the CMB brightness fluctuation reduces to