Application of Contour Minkowski Tensor and $mathcal{D}$ Statistic to the Planck $E$-mode data

We test the statistical isotropy (SI) of the $E$-mode polarization of the Cosmic Microwave Background (CMB) radiation observed by the Planck satellite using two statistics, namely, the $alpha$ estimator that is derived from the contour Minkowski Tensor (CMT), and the Directional statistic ($mathcal{D}$ statistic). The $alpha$ estimator obtained from the CMT provides information about the alignment of structures and can be used to infer statistical properties such as Gaussianity and SI of random fields. The $mathcal{D}$ statistic is based on detecting preferred directionality shown by vectors defined by the field. We compute $alpha$ and $mathcal{D}$ statistic for the low resolution component separated SMICA $E$-mode map of CMB polarization, and compare with the values calculated using FFP10 SMICA simulations. We find good agreement between the Planck data and SMICA simulations for both $alpha$ estimator and $mathcal{D}$ statistic.