Measuring the Statistical Isotropy of the Cosmic Microwave Background Anisotropy

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