4 where statistical anisotropy is quantified by the BipoSH coefficients [36, 38–40], defined as a tensor product of the spherical harmonic coefficients a$_{ℓm}$ and a$_{ℓ}$ ′ m ′ , general case are The angular correlations turn out to be Here, C$_{ℓ}$ is the usual CMB power spectrum for the SI case. Directional dependent P ( ˆ k ) [42], introduces the second term in equation (18) The most general power spectrum (under statistical ho- mogeneity) depends on the direction ˆ k , where a corresponding tensor product in Bipolar har- monic space for the indices L and L ′ of CMB brightness fluctuations is defined as Further, it is useful to parametrize the directional depen- dence of ˆ k in P ( ⃗ k ) as [42] where the first term with l = 0 represents the statistical homogeneous and isotropic primordial power spectrum. For a directional dependent power spectrum, the angular correlations of temperature anisotropy can be written as In general for SI violations ( L > 0), the BipoSH coeffi- cients can be expressed using the angular correlations in equation (17) as shown in Appendix A as where The first term on the right-hand side of equation (22), is the contribution due to statistically isotropic primor- dial power spectrum i.e. l = 0 terms in equation (16). The second term gives the contribution to the Bipo- lar coefficients due to l > 0 in equation (16). The term in the first braces is the Wigner-9j symbol [48] which is related to the coefficients of transformations between different coupling schemes of four angular mo- menta and satisfies the triangular conditions for the tri- ads ( Jℓ$_{3}$ l ) , ( ℓLℓ$_{1}$ ) , ( ℓ $^{′}$L$^{′}$ℓ$_{2}$ ) , ( Jℓℓ $^{′}$) , ( ℓ$_{3}$LL $^{′}$) and ( l ℓ$_{1}$ℓ$_{2}$ ). The SI case L = 0 reduces the BipoSH coefficients in equation (22) to equation (20). where we have used the expression for integral of three spherical harmonics as in equation (5.9.4) from [48]. Here Π$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$$_{...ℓ}$ n = [(2 ℓ$_{1}$ +1)(2 ℓ$_{2}$ +1) ... (2 ℓ$_{n}$ +1)] 1 2 has been defined for convenient notational simplicity. The case L = 0 reduces equation (17) to that in the analysis [42] To evaluate the first term ( A JN ℓℓ ′ ) l =0 in equation (22), we consider statistically isotropic primordial perturbations 〈 φ ( ⃗ k ) φ $^{∗}$( ⃗ k $^{′}$) 〉 = P$_{0}$ ( k ) δ ( ⃗ k − ⃗ k $^{′}$) and express the angular 〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 = ( − 1) ℓ Π$_{ℓ}$ A 00 ℓℓ ′ δ$_{ℓℓ}$ ′ δ$_{mm}$ ′ + ( − 1) ℓ $^{′}$+ m ′ ∑ l , m A lm ℓℓ ′ C l − m ℓmℓ ′ m ′ , (18) A JN ℓℓ ′ = ( A JN ℓℓ ′ ) l =0 + √$_{4}$$_{π}$ ∫ k $^{2}$dk 2 π 2 P$_{0}$ ( k ) × ∑ l ℓ$_{1}$ℓ$_{2}$LL ′ Π l ℓ$_{1}$LL ′ ( − 1) l + L $^{′}$+ ℓ $^{′}$+ ℓ$_{1}$ − $^{ℓ$_{2}$}$β$_{ℓ}$$_{1}$$_{ℓ}$ β ∗ ℓ$_{2}$ℓ ′ × C ℓ$_{2}$ 0 ℓ$_{1}$ 0 l 0 ∑ m g lm ∑ ℓ$_{3}$m$_{3}$   $^{J}$ ℓ$_{3}$ l ℓ L ℓ$_{1}$ ℓ ′ L ′ $_{ℓ$_{2}$}$   × C JN ℓ$_{3}$m$_{3}$ lm { ˜ Δ L ℓ$_{1}$ℓ ⊗ ˜ Δ L ′ ℓ$_{2}$ℓ ′ } ℓ$_{3}$m$_{3}$ . (22) 〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 = 4 π ∫ k $^{2}$dk 2 π 2 P$_{0}$ ( k ) ∑ ℓ$_{1}$m$_{1}$LML ′ M ′ [ D$_{1}$ ˜ Δ LM ℓ$_{1}$ℓ ( k, τ ) × [ ˜ Δ L $^{′}$M ′ ℓ$_{1}$ℓ ′ ( k, τ )] ∗ + ℓ$_{2}$m$_{2}$ lm D$_{2}$ g lm ( k ) × ˜ Δ LM ℓ$_{1}$ℓ ( k, τ ) [ ˜ Δ L $^{′}$M ′ ℓ$_{2}$ℓ ′ ( k, τ )] ∗ Y ℓ$_{2}$m$_{2}$ ℓ$_{1}$m$_{1}$, lm ] , (17) D$_{1}$ = β$_{ℓ}$$_{1}$$_{ℓ}$ β ∗ ℓ$_{1}$ℓ ′ C LM ℓ$_{1}$m$_{1}$ℓm C L $^{′}$M ′ ℓ$_{1}$m$_{1}$ℓ ′ m ′ , D$_{2}$ = β$_{ℓ}$$_{1}$$_{ℓ}$ β ∗ ℓ$_{2}$ℓ ′ C LM ℓ$_{1}$m$_{1}$ℓm C L $^{′}$M ′ ℓ$_{2}$m$_{2}$ℓ ′ m ′ , Y ℓ$_{2}$m$_{2}$ ℓ$_{1}$m$_{1}$, lm = ∫ d Ω$_{ˆ}$ k Y$_{ℓ}$$_{1}$$_{m}$$_{1}$ ( k ) Y ∗ ℓ$_{2}$$_{m$_{2}$}$( k ) Y lm ( k ) = Π$_{ℓ}$$_{1}$ l √$_{4}$$_{π}$ Π$_{ℓ}$$_{2}$ C ℓ$_{2}$ 0 ℓ$_{1}$ 0 l 0 C ℓ$_{2}$m$_{2}$ ℓ$_{1}$m$_{1}$ $_{lm}$, { ˜ Δ L ℓ$_{1}$ℓ ⊗ ˜ Δ L ′ ℓ$_{2}$ℓ ′ } JN = ∑ MM ′ ( − 1) M $^{′}$C JN LML ′ − M ′ × ˜ Δ LM ℓ$_{1}$ℓ ( k, τ )[ ˜ Δ L $^{′}$M ′ ℓ$_{2}$ℓ ′ ( k, τ )] $^{∗}$. (21) P ( ⃗ k ) = P$_{0}$ ( k ) [ 1 + ∑ l > 0 l ∑ m = − l g lm ( k ) Y lm ( ˆ k ) ] , (16) A lm ℓℓ ′ = √$_{4}$$_{π}$$_{(}$ − i ) ℓ + ℓ ′ Π$_{ℓ}$Π$_{ℓ}$ ′ Π l ∫ k $^{2}$dk 2 π 2 P$_{0}$ ( k ) g lm × C l 0 ℓ ℓ ′ 0 { ˜ Δ 0 ℓℓ ⊗ ˜ Δ 0 ℓ ′ ℓ ′ } 00 , (20) 〈 φ ( ⃗ k ) φ $^{∗}$( ⃗ k $^{′}$) 〉 = P ( ⃗ k ) δ ( ⃗ k − ⃗ k $^{′}$) . (15) 〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 = (4 π ) 2 ∫ ∫ d $^{3}$k (2 π ) 3 d $^{3}$k ′ (2 π ) 3 〈 φ ( ⃗ k ) φ $^{∗}$( ⃗ k $^{′}$) 〉 × ∑ ℓ$_{1}$m$_{1}$LM ∑ ℓ$_{2}$m$_{2}$L ′ M ′ β$_{ℓ}$$_{1}$$_{ℓ}$β ∗ ℓ$_{2}$ℓ ′ × C LM ℓ$_{1}$m$_{1}$ℓm C L $^{′}$M ′ ℓ$_{2}$m$_{2}$ℓ ′ m ′ Y$_{ℓ}$$_{1}$$_{m}$$_{1}$ ( ˆ k ) Y ∗ ℓ$_{2}$$_{m$_{2}$}$( ˆ k $^{′}$) × ˜ Δ LM ℓ$_{1}$ℓ ( k, τ )[ ˜ Δ L $^{′}$M ′ ℓ$_{2}$ℓ ′ ( k $^{′}$, τ )] $^{∗}$. (14) A JN ℓℓ ′ = ∑ mm $_{′}$〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 ( − 1) m $^{′}$C JN ℓmℓ ′ − m ′ = { a$_{ℓ}$ ⊗ a$_{ℓ}$ ′ } JN A 00 ℓℓ = ( − 1) ℓ Π$_{ℓ}$ C$_{ℓ}$ (19) a$_{ℓm}$ = 4 π ∫ d $^{3}$k (2 π ) 3 φ ( ⃗ k ) ∑ ℓ$_{1}$m$_{1}$LM β$_{ℓ}$$_{1}$$_{ℓ}$ ˜ Δ LM ℓ$_{1}$ℓ ( k, τ ) × C LM ℓ$_{1}$m$_{1}$ℓm Y$_{ℓ}$$_{1}$$_{m}$$_{1}$ ( ˆ k ) . (13)