Local patch analysis for testing statistical isotropy of the Planck convergence map

Abstract: The small but measurable effect of weak gravitational lensing on the cosmic microwave background radiation provide information about the large-scale distribution of matter in the universe. We use the all sky distribution of matter, as represented by the {\em convergence map} that is inferred from CMB lensing measurement by Planck survey, to test the fundamental assumption of Statistical Isotropy (SI) of the universe. For the analysis we use the $\alpha$ statistic that is devised from the contour Minkowski tensor, a tensorial generalization of the scalar Minkowski functional, the contour length. In essence, the $\alpha$ statistic captures the ellipticity of isofield contours at any chosen threshold value of a smooth random field and provides a measure of anisotropy. The SI of the observed convergence map is tested against the suite of realistic simulations of the convergence map provided by the Planck collaboration. We first carry out a global analysis using the full sky data after applying the galactic and point sources mask. We find that the observed data is consistent with SI. Further we carry out a local search for departure from SI in small patches of the sky using $\alpha$. This analysis reveals several sky patches which exhibit deviations from simulations with statistical significance higher than 95\% confidence level (CL). Our analysis indicates that the source of the anomalous behaviour of most of the outlier patches is inaccurate estimation of noise. We identify two outlier patches which exhibit anomalous behaviour originating from departure from SI at higher than 95\% CL. Most of the anomalous patches are found to be located roughly along the ecliptic plane or in proximity to the ecliptic poles.