content_image/409557.jpg content_image/409558.jpg content_image/409562.jpg content_image/409575.jpg content_image/409556.jpg content_image/409561.jpg content_image/409559.jpg content_image/409560.jpg content_image/409574.jpg
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 

<img src='content_image/409557.jpg'>

The angular correlation can be expressed as 

<img src='content_image/409558.jpg'>

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. 

## D. Evolution of CMB brightness fluctuation in the free-streaming regime 

The evolution of ˜ Δ( ⃗ k, ˆ n, τ ) in the free-streaming regime can be written 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 

<img src='content_image/409562.jpg'>

and defining Δ τ = τ − τ$_{s}$ , the evolution equation for ˜ Δ$_{ℓ}$( k, τ ) can be written as 

<img src='content_image/409575.jpg'>

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 . 

<img src='content_image/409556.jpg'>

<img src='content_image/409561.jpg'>

## C. Correlations 

## III. STATISTICAL ISOTROPY BREAKDOWN IN THE CMB BRIGHTNESS FLUCTUATION 

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 ). 

## A. Generalized CMB brightness fluctuations 

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. 

<img src='content_image/409559.jpg'>

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 

<img src='content_image/409560.jpg'>

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] 

to recover equation (4) in section (II B). 

Starting with equation (5) and the Fourier Transform relation from equation (3), the SH coefficients a$_{ℓm}$ for the 

<img src='content_image/409574.jpg'>

## B. Angular Correlations 

