Statistical Isotropy violation of the CMB brightness fluctuations

Certain anomalies at large angular scales in the cosmic microwave background measured by WMAP have been suggested as possible evidence of breakdown of statistical isotropy(SI). Most CMB photons free-stream to the present from the surface of last scattering. It is thus reasonable to expect statistical isotropy violation in the CMB photon distribution observed now to have originated from SI violation in the baryon-photon fluid at last scattering, in addition to anisotropy of the primordial power spectrum studied earlier in literature. We consider the generalized anisotropic brightness distribution fluctuations, $\Delta(\vec{k}, \hat{n}, \tau)$ (at conformal time $\tau$) in contrast to the SI case where it is simply a function of $|\vec{k}|$ and $\hat{k} \cdot \hat{n}$. The brightness fluctuations expanded in Bipolar Spherical Harmonic (BipoSH) series, can then be written as $\Delta_{\ell_1 \ell_2}^{L M}(\vec{k}, \tau)$ where $L > 0$ terms encode deviations from statistical isotropy. We study the evolution of $\Delta_{\ell_1 \ell_2}^{L M}(\vec{k}, \tau)$ from non-zero terms $\Delta_{\ell_3 \ell_4}^{L M}(\vec{k}, \tau_s)$ at last scattering. Similar to the SI case, power at small spherical harmonic (SH) multipoles of $\Delta_{\ell_3 \ell_4}^{L M}(\vec{k},\tau_s)$ at the last scattering, is transferred to $\Delta_{\ell_1 \ell_2}^{L M}(\vec{k}, \tau)$ at larger SH multipoles. The structural similarity is more apparent in the asymptotic expression for large values of the final SH multipoles. This formalism allows an elegant identification of any SI violation observed today to a possible origin in the SI violation present in the baryon-photon fluid (eg., due to the presence of significant magnetic field).