Statistical isotropy violation of the CMB brightness fluctuations

Certain anomalies at large angular scales in the cosmic microwave background (CMB) measured by WMAP have been suggested as possible evidence of breakdown of statistical isotropy (SI). SI violation of cosmological perturbations is a generic feature of ultralarge scale structure of the cosmos and breakdown of global symmetries. 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 the literature. We consider the generalized anisotropic brightness distribution fluctuations, Delta((k) over right arrow, (n) over cap, tau) (at conformal time tau) in contrast to the SI case where it is simply a function of vertical bar(k) over right arrow vertical bar and (n) over cap. The brightness fluctuations expanded in bipolar spherical harmonic (BipoSH) series can then be written as Delta(LM)(l1l2)(k, tau), where L > 0 terms encode deviations from statistical isotropy. Violation of SI encoded in the present off-diagonal elements of the harmonic space correlation < a(lm)a(l1m1)* >, equivalently, the BipoSH coefficients A(ll1)(LM), are then related to the generalized BipoSH brightness fluctuation terms at present. We study the evolution of Delta(LM)(l1l2) (k,tau) from nonzero terms Delta(LM)(l3l4) (k, tau(s)) at last scattering, in the free-streaming regime. We show that the terms with given BipoSH multipole LM evolve independently. Moreover, similar to the SI case, power at small spherical harmonic (SH) multipoles of Delta(LM)(l3l4) (k, tau(s)) at the last scattering is transferred to Delta(LM)(l1l2) (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 SI violating physics present in the baryon-photon fluid. This is illustrated for the known result of SI violating angular correlations due to the presence of a homogeneous magnetic field in the baryon-photon fluid.