2 moments today to the moments at recombination using the photon distribution function. In the Boltzmann equation for photons df /dt = C [ f ], we expand the photon distribution function f ( ⃗ x, p, ˆ n, τ ) about its zero-order Bose-Einstein value T ( τ ) [46] where ⃗ p = p ˆ n . The distribution function of the photons changes with the perturbed temperature as Violation of statistical isotropy of CMB anisotropy and its measurement has been discussed in literature earlier [35–40] by defining an estimator where SI breakdown in an observed CMB anisotropy sky map is indicated by non zero value of this estimator. Studies have also been done by implementing a directional dependent inflation- ary power spectrum P ( ⃗ k ) which gives rise to off-diagonal terms in the covariance matrix [41], [42]; spontaneous breakdown of SI in the CMB by a non-linear response to long-wavelength field fluctuations that appear as a gra- dient locally to the observer [43] or locally through a modulation field [44]; incorporating an initial period of kinetic energy domination in single field inflation [45]. The perturbation to the distribution function is charac- terized by Δ ≡ δT /T termed as CMB brightness fluc- tuations henceforth. Since the perturbation Δ is small, we can expand f ( ⃗ x, p, ˆ n, τ ) keeping only terms up to first order to get B. Fluctuations of CMB photon distribution • Although a possible source of SI breakdown, resid- ual foreground contamination would need to be of order the intrinsic CMB temperature anisotropy to account for an appreciable effect [32–34]. • It would be erroneous to assume that the true CMB temperature fluctuations are completely extracted from the observed map. Observational artifacts such as non-circular beam, inhomogeneous noise correlation, residual striping patterns could be po- tential sources of SI breakdown. f ( ⃗ x, p, ˆ n, τ ) = [ exp { p T ( τ )[1 + Δ( ⃗ x, ˆ n, τ )] } − 1 ] − 1 . (1) Δ( ⃗ x, ˆ n, τ ) ≡ ( ∂f 0 ∂ ln p ) − 1 δf, (2) where f 0 is the zero-order photon distribution function. Δ( ⃗ x, ˆ n, τ ) depends on ⃗ x, ˆ n and τ and not on the mag- nitude of momentum p ; this is a valid assumption since the temperature of the plasma is very small compared to the rest energy of the electrons which undergo scatter- ing, elastic Thomson scattering has negligible effect on the magnitude of the photon momentum. In this paper we present a new formulation that relates the breakdown of SI in the CMB photon fluctuations at last scattering, and evolving them to find the effect of the modes at present epoch hence the CMB 〈 a$_{ℓm}$a$_{ℓ}$ ′ m ′ 〉 today. We also find the Bipolar spherical harmonic coef- ficients (BipoSH) [38, 39] which are linear combinations of off-diagonal elements of the covariance matrix. Bi- poSH expansion completely represents the information of the covariance matrix thus being the most general way of studying two point correlation functions of CMB anisotropy. These BipoSH coefficients are mathemati- cally complete measures of SI violation on a sphere. Perturbations to the CMB remain small at all cosmolog- ical epochs; evolution of the largest scales being in the linear regime. In solving the linear evolution equations, it is simplest to work with Fourier transforms since every Fourier mode evolves independently. In the smooth background universe, thermalized pho- tons being distributed homogeneously and isotropically, the temperature T is independent of ⃗ x and direction of propagation ˆ p respectively. To describe perturbations about this smooth universe, we allow inhomogeneities in the photon distribution and anisotropies. With statistical isotropy assumption where φ ( ⃗ k ) is the primordial density fluctuations. II. REVIEW OF STATISTICALLY ISOTROPIC CMB BRIGHTNESS FLUCTUATIONS A. Boltzmann equations, inhomogeneities and anisotropies Δ( ⃗ x, ˆ n, τ ) = ∫ d $^{3}$k (2 π ) $_{3}$e i ⃗ k · ⃗ x ˜ Δ( ⃗ k, ˆ n, τ ) φ ( ⃗ k ) , (3) ˜ Δ( ⃗ k, ˆ n, τ ) = ˜ Δ( k, ˆ k · ˆ n, τ ) = ∑ ℓ ( − i ) $^{ℓ}$(2 ℓ + 1) ˜ Δ$_{ℓ}$( k, τ ) P$_{ℓ}$ ( ˆ k · ˆ n ) = 4 π ℓm ( − i ) ℓ ˜ Δ$_{ℓ}$( k, τ ) Y$_{ℓm}$ ( ˆ k ) Y$_{ℓm}$ (ˆ n ) . (4) Before recombination, z$_{rec}$ ≈ 1100, the photons were tightly coupled to the electrons and protons; all together they can be described as a single fluid, the baryon-photon fluid. After recombination, photons free-stream from the surface of last scattering to the present epoch. Given the cosmological perturbations to the photons at recombination, one can predict the anisotropy spec- trum today. The main motivation next is to relate the ˜ Δ$_{ℓ}$( k, τ ) are the moments of the CMB brightness fluctu- ation. The monopole ˜ Δ$_{0}$ is related to the density per- turbations while the dipole ˜ Δ$_{1}$ ∝ ˆ n · ⃗ v , gives the velocity term for baryons.