Abstract. We generalize the translation invariant tensor-valued Minkowski Functionals which are defined on two-dimensional flat space to the unit sphere. We apply them to level sets of random fields. The contours enclosing boundaries of level sets of random fields give a spatial distribution of random smooth closed curves. We outline a method to compute the tensor-valued Minkowski Functionals numerically for any random field on the sphere. Then we obtain analytic expressions for the ensemble expectation values of the matrix elements for isotropic Gaussian and Rayleigh fields. The results hold on flat as well as any curved space with affine connection. We elucidate the way in which the matrix elements encode information about the Gaussian nature and statistical isotropy (or departure from isotropy) of the field. Finally, we apply the method to maps of the Galactic foreground emissions from the 2015 PLANCK data and demonstrate their high level of statistical anisotropy and departure from Gaussianity.

We apply the Minkowski tensor statistics to three dimensional Gaussian random fields. Minkowski tensors contain information regarding the orientation and shape of excursion sets, that is not present in the scalar Minkowski functionals. They can be used to quantify globally preferred directions, and additionally provide information on the mean shape of subsets of a field. This makes them ideal statistics to measure the anisotropic signal generated by redshift space distortion in the low redshift matter density field. We review the definition of the Minkowski tensor statistics in three dimensions, focusing on two coordinate invariant quantities W 0,2 1 and W 0,2 2 . We calculate the ensemble average of these 3 × 3 matrices for an isotropic Gaussian random field, finding that they are proportional to products of the identity matrix and a corresponding scalar Minkowski functional. We show how to numerically reconstruct W 0,2 1 and W 0,2 2 from discretely sampled fields and apply our algorithm to isotropic Gaussian fields generated from a linear ΛCDM matter power spectrum. We then introduce anisotropy by applying a linear redshift space distortion operator to the matter density field, and find that both W 0,2 1 and W 0,2 2 exhibit a distinct signal characterised by inequality between their diagonal components. We discuss the physical origin of this signal and how it can be used to constrain the redshift space distortion parameter Υ ≡ f /b. arXiv:1805.08752v1 [astro-ph.CO]

Minkowski Tensors are tensorial generalizations of the scalar Minkowski Functionals. Due to their tensorial nature they contain additional morphological information of structures, in particular about shape and alignment, in comparison to the scalar Minkowski functionals. They have recently been used [39] to study the statistical isotropy of temperature and E mode data from the Planck satellite. The calculation in [39] relied on stereographic projection of the fields to extract the shape and alignment information. In this work, we calculate Minkowski Tensors directly on the sphere and compute the net alignment in the data, based on a recent work that extends the definition of Minkowski Tensors to random fields on curved spaces. This method circumvents numerical errors that can be introduced by the stereographic projection. We compare the resulting net alignment parameter values obtained from the frequency coadded CMB temperature data cleaned by the SMICA pipeline, 1 to those obtained from simulations that include instrumental beam effects and residual foreground and noise. We find very good agreement between the two within ≈ 1σ. We further compare the alignments obtained from the beam-convolved CMB maps at individual Planck frequencies to those in the corresponding simulations. We find no significant difference between