content_image/438859.jpg content_image/438872.jpg content_image/438876.jpg content_image/438871.jpg content_image/438877.jpg content_image/438858.jpg content_image/438860.jpg content_image/438870.jpg
the CMB anisotropy in the presence of a homogeneous magnetic field at last scattering. 

The SH coefficients a$_{ℓm}$ can be expressed in terms of the CMB brightness fluctuations at last scattering using equation (13) and the generalized evolution equation (31) as 

<img src='content_image/438859.jpg'>

where C ( L, ℓ$_{2}$, ℓ$_{1}$, ℓ ) is defined in equation (32). 

Hence, in general 〈 a$_{ℓm}$a$_{ℓ}$ ′ m ′ 〉 correlations measured at present are related to 〈 ˜ Δ LM ℓ$_{1}$$_{ℓ$_{2}$}$( k, τ$_{s}$ ) ˜ Δ L $^{′}$M ′ ℓ$_{3}$ℓ$_{4}$ ( k, τ$_{s}$ ) 〉 correla- tion between the generalized Boltzmann fluctuations at the last scattering surface. 

In particular, SI violation encoded in the off-diagonal cor- relation 〈 a$_{ℓm}$a$_{ℓ}$ ′ m ′ 〉 (and non-zero BipoSH A LM ℓℓ ′ , L > 0) are related as 

as in equations (24, 25, 26) or more generally for different Bipolar coefficients J ′ and N ′ as in equation (22) when the power spectrum is also anisotropic. 

Using equation (11), correlations of the CMB brightness fluctuations are 

<img src='content_image/438872.jpg'>

We illustrate the generality and power of our formalism using the case for a uniform magnetic field. We show the correlations of the CMB brightness fluctuations in this particular case is sourced by the bipolar dipole ( L = 1) terms of equation (11) with ℓ$_{1}$ = ℓ$_{2}$ = 1 where 

<img src='content_image/438876.jpg'>

with M = {− 1 , 0 , +1 } . Here ( ˆ k × ˆ n )$_{M}$ is the usual cross- product written as irreducible products of the rotation group [48]. 

Using standard vector identity [48] and equation (8) 

<img src='content_image/438871.jpg'>

<img src='content_image/438877.jpg'>

the temperature correlations in presence of a uniform magnetic field as discussed in [29] (see equation A3) are given by 

<img src='content_image/438858.jpg'>

Here the bipolar dipole terms of the CMB brightness fluc- tuation ˜ Δ 1 M 11 ( k, τ$_{s}$ ) ˜ Δ 1 M ′ 11 ( k, τ$_{s}$ ) in equation (44), encap- sulates the source term due to the presence of a uniform magnetic field [29]. 

In equation (41), the bipolar dipole terms ˜ Δ 1 M 11 ( k, τ$_{s}$ ) gives rise to SH coefficients 

<img src='content_image/438860.jpg'>

The angular correlations in this case turn out to be 

<img src='content_image/438870.jpg'>

It is interesting to note that the known diagonal ( ℓ ′ = ℓ ) and off-diagonal ( ℓ ′ = ℓ ± 2) correlations in presence of a homogeneous magnetic field [29, 30], can be easily recovered in our approach. Work is in progress to relate other cases of SI violation originating in the physics at the last scattering surface using this formalism. 

The search for subtle statistical isotropy breakdown in the universe is highly motivated by numerous theoreti- cal scenarios. The fluctuations in the cosmic microwave background is the arguably the most promising observa- tional probe of the SI of the universe. The violation of SI could have its origin not only in in anisotropic primor- dial power spectrum, but also in the SI violation in the fluctuations of the baryon-photon fluid at last scattering. SI deviations generated by a general form of anisotropic primordial power spectrum for isotropic Boltzmann func- tions has been studied in the recent literature [42]. This paper includes this equally important possibility of a gen- eral SI breakdown in the CMB photon distribution func- tion. We study the generalized case of SI violation in terms of Bipolar spherical harmonic (BipoSH) bright- ness fluctuations, substantially extending the scope of origin of SI violation solely from the anisotropic primor- dial power spectrum. 

The breakdown of SI in the CMB brightness fluctu- ation results in off-diagonal terms in the SH space an- gular correlations 〈 a$_{ℓm}$a$_{ℓ}$ ′ m ′ 〉 , or, equivalently, in the co- efficients the Bipolar spherical harmonic (BipoSH) rep- resentation [38, 39]. We relate the measurable BipoSH 

## V. CONCLUSIONS 

