Measuring Statistical isotropy of the CMB anisotropy

Abstract

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 {\em 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, $κ_\ell$ ($\ell=1,2,3, ...$) for detecting violation of statistical isotropy in an observed CMB anisotropy sky map indicated by non zero $κ_\ell$. We define an estimator for the $κ_\ell$ spectrum and analytically compute its cosmic bias and cosmic variance. The results match those obtained by measuring $κ_\ell$ using simulated sky maps. Non-zero (bias corrected) $κ_\ell$ 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.

Figures (1)

• Figure 1: The figure shows the bias corrected 'measurement' of $\kappa_\ell$ of a SI CMB sky with a flat band power spectrum smoothed by a Gaussian beam ($l(l+1) C_l = \exp(-l^2/18^2)$. The cosmic error, $\sigma({\kappa_\ell})$, obtained using $50$ independent realizations of CMB (full) sky map match the analytic results shown by lower dotted curve with stars. The upper dotted curves separately outline the cosmic error envelope for odd multipoles (filled triangles) and for even multipoles (empty triangles) to explicitly highlight their convergence. Violation of SI will be indicated by non-zero $\kappa_\ell$ measured in an observed CMB map in excess of $\sigma({\kappa_\ell})$ given by the $C_l$ of the map. The lower dashed curve (filled squares) shows the cosmic error for ideal unit flat band power spectrum ($l(l+1) C_l =1$) with no beam smoothing. The curve falls off roughly at $1/\ell$ at large $\ell$.