3 III. STATISTICAL ISOTROPY BREAKDOWN IN THE CMB BRIGHTNESS FLUCTUATION C. Correlations A. Generalized CMB brightness fluctuations In this paper we take into account the SI violation of the CMB anisotropy which is seeded due to the inher- ent SI breakdown in the CMB photon distribution. We consider the general form of the CMB brightness fluc- tuation, allowing for anisotropy in ˆ k , i.e., ˜ Δ( ⃗ k, ˆ n, τ ) ̸≡ ˜ Δ( k, ˆ k · ˆ n, τ ). The observed anisotropy in multipole space can be written in terms of the CMB brightness fluctuation as Using the orthonormality property of the spherical har- monics Y$_{ℓm}$ , the SH coefficients a$_{ℓm}$ become The angular correlation can be expressed as The most general CMB brightness fluctuation is not simply a function of | ⃗ k | and ˆ k · ˆ n . In this case the physical situation of anisotropic fluctuations demand the brightness fluctuations to be expanded in Bipolar Spher- ical Harmonic series (not just a Legendre series as in the statistical isotropic case). The brightness fluctuation in multipole space is Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( ⃗ k, τ ) where L > 0 term incorpo- rates deviation from statistical isotropy. Δ( ⃗ x = 0 , ˆ n, τ ) = ∑ ℓm a$_{ℓm}$Y$_{ℓm}$ (ˆ n ) . (5) a$_{ℓm}$ = 4 π ( − i ) ℓ ∫ d $^{3}$k (2 π ) 3 φ ( ⃗ k ) ˜ Δ$_{ℓ}$( k, τ ) Y ∗ $_{ℓm}$( ˆ k ) . (6) 〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 = 4 π ∫ dk k k $^{3}$P$_{0}$ ( k ) 2 π 2 | Δ$_{ℓ}$( k, τ ) | $^{2}$δ$_{ℓℓ}$ ′ δ$_{mm}$ ′ C$_{ℓ}$ = 4 π dk k P $_{0}$( k ) | Δ$_{ℓ}$( k, τ ) | 2 , (7) ˜ Δ( ⃗ k, ˆ n, τ ) = 4 π ∑ ℓ$_{1}$ℓ$_{2}$LM ( i )$^{ℓ}$ 1 + $^{ℓ}$2 2 √ ℓ$_{1}$ + ℓ$_{2}$ + 1 × ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) { Y$_{ℓ}$$_{1}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{2}$ (ˆ n ) } LM = 4 π ∑ ℓ$_{1}$ℓ$_{2}$LM β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ ) × { Y$_{ℓ}$$_{1}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{2}$ (ˆ n ) } $_{LM}$, (11) { Y$_{ℓ}$$_{1}$ ( ˆ k ) ⊗ Y$_{ℓ}$$_{2}$ (ˆ n ) } LM = ∑ m$_{1}$m$_{2}$ C LM ℓ$_{1}$m$_{1}$ℓ$_{2}$m$_{2}$ × Y$_{ℓ}$$_{1}$$_{m}$$_{1}$ ( ˆ k ) Y$_{ℓ}$$_{2}$$_{m}$$_{2}$ (ˆ n ) . ˜ Δ( ⃗ k, ˆ n, τ ) = e i ⃗ k · ˆ n ( τ − τ s ) ˜ Δ( ⃗ k, ˆ n, τ$_{s}$ ) , (8) e i ⃗ k · ˆ n Δ τ = ∑ l ( − i ) $^{l}$(2 l + 1) j$_{l}$ ( k Δ τ ) P$_{l}$ ( ˆ k · ˆ n ) , (9) where correlation of the primordial density fluctuations 〈 φ ( ⃗ k ) φ $^{∗}$( ⃗ k $^{′}$) 〉 = P$_{0}$ ( k ) δ ( ⃗ k − ⃗ k $^{′}$) and P $_{0}$( k ) = k $^{3}$P$_{0}$ ( k ) / 2 π 2 is the primordial power spectrum per logarithmic interval generated by inflationary model and the second term is the radiative transport kernel in the post-recombination universe given by cosmological parameters. The evolution of ˜ Δ( ⃗ k, ˆ n, τ ) in the free-streaming regime can be written as where β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ = ( i ) $^{ℓ}$1 + $^{ℓ}$2 2 √$_{ℓ}$$_{1}$ + ℓ$_{2}$ + 1 has been defined for convenient notational simplicity. The tensor product in BipoSH function is defined as where τ is well inside the free-streaming regime i.e. τ$_{s}$ < τ < τ$_{0}$ , τ$_{s}$ and τ$_{0}$ being the conformal time at last scattering and today respectively [47]. Using the expansion D. Evolution of CMB brightness fluctuation in the free-streaming regime B. Angular Correlations The pre-factors in equation (11) are the normalization terms associated with the CMB brightness fluctuation. The deviations from statistical isotropy is associated with non-zero values of L, L > 0. To check for the statistical isotropy limit i.e. L = 0 & M = 0, we use equation (8.5.1) from [48] and defining Δ τ = τ − τ$_{s}$ , the evolution equation for ˜ Δ$_{ℓ}$( k, τ ) can be written as j$_{l}$ ( k Δ τ ) and P$_{l}$ ( ˆ k · ˆ n ) are the ℓ th order spherical Bessel function and Legendre Polynomial, respectively. Equa- tion (10) is the well known “free streaming ” equa- tion in CMB literature [47]. C LM ℓ$_{1}$m$_{1}$ℓ$_{2}$m$_{2}$ are the Clebsch- Gordan coefficient which satisfies the triangle inequalities ([48]) putting a constraint | ℓ$_{1}$ − ℓ$_{2}$ | ≤ L ≤ ℓ$_{1}$ + ℓ$_{2}$ and m$_{1}$ + m$_{2}$ = M . to recover equation (4) in section (II B). C 00 ℓ$_{1}$m$_{1}$ℓ$_{2}$m$_{2}$ = ( − 1) ℓ$_{1}$ − m$_{1}$ δ$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ δ$_{m}$$_{1}$ − m$_{2}$ √$_{2}$$_{ℓ}$$_{1}$ + 1 , (12) ˜ Δ$_{ℓ}$( k, τ ) = ∑ lℓ ′ ( − i ) ℓ + ℓ $^{′}$− l (2 ℓ ′ + 1) j$_{l}$ ( k Δ τ ) × [ C l 0 ℓ ℓ ′ $_{0}$] 2 ˜ Δ$_{ℓ}$ ′ ( k, τ$_{s}$ ) . (10) Starting with equation (5) and the Fourier Transform relation from equation (3), the SH coefficients a$_{ℓm}$ for the