Minkowski Tensors in Two Dimensions - Probing the Morphology and Isotropy of the Matter and Galaxy Density Fields

Understand

We apply the Minkowski Tensor statistics to two dimensional slices of the three dimensional 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 peaks.
• We begin by introducing our algorithm for constructing bounding perimeters around subsets of a two dimensional field, and reviewing the definition of Minkowski Tensors.
• Focusing on the translational invariant statistic $W^{1,1}_{2}$ - a $2 \times 2$ matrix - we calculate its eigenvalues for both the entire excursion set ($\Lambda_{1},\Lambda_{2}$) and for individual connected regions and holes within the set ($\lambda_{1},\lambda_{2}$).