content_image/389235.jpg content_image/389237.jpg content_image/389245.jpg content_image/389248.jpg content_image/389234.jpg content_image/389236.jpg content_image/389238.jpg content_image/389244.jpg content_image/389246.jpg content_image/389247.jpg
correlations as 

<img src='content_image/389235.jpg'>

To check for the statistical isotropic case, we put L = 0 & M = 0 and recover equation (7) in section (II C). 

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 

<img src='content_image/389237.jpg'>

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 

<img src='content_image/389245.jpg'>

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 

<img src='content_image/389248.jpg'>

## IV. EVOLUTION IN THE FREE-STREAMING REGIME 

We find the moments of the CMB brightness fluctua- tion to be 

<img src='content_image/389234.jpg'>

starting with equation (11) and the orthonormality con- dition of BipoSH, 

<img src='content_image/389236.jpg'>

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 

<img src='content_image/389238.jpg'>

As shown in detail in Appendix B, this can be further simplified to 

<img src='content_image/389244.jpg'>

The generalized evolution equation thus can be expressed so as to structurally resemble the evolution equation for the SI case in equation (10). 

<img src='content_image/389246.jpg'>

where 

<img src='content_image/389247.jpg'>

## A. Generalized evolution equation 

