Measuring the Statistical Isotropy of the Cosmic Microwave Background Anisotropy

The statistical expectation values of the temperature fluctuations of the cosmic microwave background (CMB) are assumed to be preserved under the rotations of the sky. This assumption of the statistical isotropy (SI) of the CMB anisotropy should be observationally verified since detection of a violation of SI could have profound implications for cosmology. We propose a set of measures, κ ℓ (ℓ = 1, 2, 3, ...), for detecting a violation of SI in an observed CMB anisotropy sky map indicated by nonzero κ ℓ . We define an estimator for the κ ℓ spectrum and analytically compute its cosmic bias and cosmic variance. The results match those obtained by measuring κ ℓ using simulated sky maps. Nonzero (bias-corrected) κ ℓ larger than the SI cosmic variance will imply a violation of SI. The SI measure proposed in this Letter is an appropriate statistic to investigate a preliminary indication of SI violation in the recently released Wilkinson Microwave Anisotropy Probe data.