# Statistical Isotropy violation of the CMB brightness fluctuations 

Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India. 

Moumita Aich ∗ and Tarun Souradeep † 

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). SI vi- olation of cosmological perturbations is a generic feature of ultra large 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 literature. 

We consider the generalized anisotropic brightness distribution fluctuations, Δ( ⃗ k, ˆ n, τ ) (at con- formal time τ ) in contrast to the SI case where it is simply a function of | ⃗ k | and ˆ k · ˆ n . The bright- ness fluctuations expanded in Bipolar Spherical Harmonic (BipoSH) series, can then be written as Δ LM ℓ 1 ℓ $_{2}$( k, τ ) 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 ∗ ℓ ′ m ′ 〉 , equivalently, the BipoSH coefficients A LM ℓℓ ′ , are then related to the generalized BipoSH brightness fluctuation terms at present. We study the evolution of Δ LM ℓ 1 ℓ $_{2}$( k, τ ) from non-zero terms Δ LM ℓ 3 ℓ $_{4}$( k, τ 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 Δ LM ℓ 3 ℓ $_{4}$( k, τ s ) at the last scattering, is transferred to Δ LM ℓ 1 ℓ $_{2}$( k, τ ) 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. 

The Cosmic Microwave Background (CMB) anisotropy is a very powerful observational probe of cosmology. In standard cosmology, CMB anisotropy signal is ex- pected to be statistically isotropic, i.e., statistical expec- tation values of the temperature fluctuations Δ T (ˆ n ) = ∑ $_{ℓm}$a$_{ℓm}$ Y$_{ℓm}$ (ˆ n ) are preserved under rotations of the sky. The condition for statistical isotropy (SI), in spherical harmonic space translates to a diagonal 〈 a$_{ℓm}$a ∗ ℓ ′ m ′ 〉 = C$_{ℓ}$δ$_{ℓℓ}$ ′ δ$_{mm}$ ′ where C$_{ℓ}$ is the widely used angular power spectrum of the CMB anisotropy. 

After the release of first year data of the Wilkinson Mi- crowave Anisotropy Probe (WMAP), statistical isotropy of the CMB anisotropy attracted considerable attention. The study of full sky maps from the WMAP 5 year data [1–3] and the very recent WMAP 7 year data [4], has led to some intriguing anomalies which seem to suggest that the assumption of statistical isotropy is broken on the largest angular scales [5–9]. Broken isotropy would have a profound implications for standard cosmological model as statistical isotropy underlies all cosmological inferences. 

It was pointed out that the suppression of power in the quadrupole and octopole are aligned in the form of the 

## I. INTRODUCTION 

“ axis of evil” [10–14]. Further “multipole-vector” direc- tions associated with these multipoles (and some other low multipoles as well) appear to be anomalously corre- lated [6], [15], [16]. There are indications of asymmetry in the power spectrum at low multipoles in opposite hemi- spheres, the “north-south asymmetry” [7], [8], [17–19]. Possibly related, are the results of tests of Gaussianity that show asymmetry in the amplitude of the measured genus amplitude (at about 2 to 3 σ significance) between the north and south galactic hemispheres [20–22]. Analy- sis of the distribution of extrema in WMAP sky maps has indicated non-Gaussianity, and to some extent, violation of SI [23]. 

An observed map of CMB anisotropy, Δ T (ˆ n ) obs con- tains the true CMB temperature Δ T (ˆ n ) fluctuations, convolved with the beam and instrumental noise & foreground contaminations. Breakdown of statistical isotropy can occur in any of these parts and can be cat- egorized as 

• Theoretically motivated effects which are intrinsic to the true CMB sky, Δ T (ˆ n ) include non-trivial cosmic topology [24], Bianchi models [25–28] and primordial magnetic fields [29], [30]. A recent arti- cle [31] claims that the solution to the cosmologi- cal vacuum energy can be explained as a result of the interaction of the infrared sector of the effec- tive theory of gravity with standard model fields. This theory predicts the violation of cosmological isotropy. 

