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.

Abstract. Tensor Minkowski Functionals (TMFs) are tensorial generalizations of the usual Minkowski Functionals which are scalar quantities. We introduce them here for use in cosmological analysis, in particular to analyze the Cosmic Microwave Background (CMB) radiation. They encapsulate information about the shapes of structures and the orientation of distributions of structures. We focus on one of the TMFs, namely W 1,1 2 , which is the (1, 1) rank tensor generalization of the genus. The ratio of the eigenvalues of the average of W 1,1 2 over all structures, α, encodes the net orientation of the structures; and the average of the ratios of the eigenvalues of W 1,1 2 for each structure, β, encodes the net intrinsic anisotropy of the structures. We have developed a code that computes W 1,1 2 , and from it α and β, for a set of structures on the 2-dimensional Euclidean plane. We use it to compute α and β as functions of chosen threshold levels for simulated Gaussian and isotropic CMB temperature and E mode fields. We obtain the value of α to be one for both temperature and E mode, which means that we recover the statistical isotropy of density fluctuations that we input in the simulations. We find that the standard inflationary ΛCDM predicts that the level of intrinsic anisotropy of hotspot and coldspot structures in the CMB fields is quantified by β ∼ 0.62. Then we compute α and β for temperature and E mode data from the PLANCK mission. We find that the temperature field agrees with the standard ΛCDM prediction of no net orientation within 3 − σ. However, we find that the structures in E mode data have a net orientation that deviates from the theoretical expectation at 14−σ. The possible origin of this deviation may be due to instrumental effects or other sources and needs to be investigated further. For the net intrinsic anisotropy of structures we obtain values of β for both temperature and E mode that are consistent with the expectations from the standard ΛCDM simulations. Accurate measurements of α and β can be used to test the standard model of cosmology and to search for deviations from it.

We apply the Minkowski Tensor statistics to two dimensional slices of the three dimensional matter density field. The Minkowski Tensors are a set of functions that are sensitive to directionally dependent signals in the data, and furthermore can be used to quantify the mean shape of density fields. We begin by reviewing the definition of Minkowski Tensors and introducing a method of calculating them from a discretely sampled field. Focusing on the statistic W 1,1 2 -a 2 × 2 matrix -we calculate its value for both the entire excursion set and for individual connected regions and holes within the set. To study the morphology of structures within the excursion set, we calculate the eigenvalues λ 1 , λ 2 for the matrix W 1,1 2 of each distinct connected region and hole and measure their mean shape using the ratio β ≡ λ 2 /λ 1 . We compare both W 1,1 2 and β for a Gaussian field and a smoothed density field generated from the latest Horizon Run 4 cosmological simulation, to study the effect of gravitational collapse on these functions. The global statistic W 1,1 2 is essentially independent of gravitational collapse, as the process maintains statistical isotropy. However, β is modified significantly, with overdensities becoming relatively more circular compared to underdensities at low redshifts. When applying the statistics to a redshift-space distorted density field, the matrix W 1,1 2 is no longer proportional to the identity matrix and measurements of its diagonal elements can be used to probe the large-scale velocity field.