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

TL;DR

This paper applies Minkowski Tensor statistics to two-dimensional slices of the three-dimensional matter density field to quantify shape and anisotropy, revealing how gravitational collapse and redshift distortions affect these properties.

Contribution

It introduces an algorithm for constructing bounding perimeters and analyzes Minkowski Tensors to probe anisotropy and shape in cosmological density fields, including effects of gravitational collapse and redshift distortions.

Findings

Eigenvalues $\Lambda_{1,2}$ are unaffected by gravitational collapse.

Mean shape of peaks becomes more circular due to gravitational effects.

Eigenvalues $\Lambda_{1,2}$ can detect large-scale velocity fields in redshift space.

Abstract

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 ($\lambda_{1},\lambda_{2}$). The ratio of eigenvalues $\Lambda_{2}/\Lambda_{1}$ informs us of the presence of global anisotropies in the data, and $\langle \lambda_{2}/\lambda_{1} \rangle$ is a measure of the mean shape of peaks and troughs in the density field. We study these quantities for a Gaussian field, then consider how they are modified by the effect of gravitational collapse using the latest Horizon Run 4 cosmological simulation. We find $\Lambda_{1,2}$ are essentially independent of gravitational collapse, as the process maintains statistical isotropy. However, the mean shape of peaks is modified significantly - overdensities become relatively more circular compared to underdensities of the same area. When applying the statistic to a redshift space distorted density field, we find a significant signal in the eigenvalues $\Lambda_{1,2}$, suggesting that they can be used to probe the large-scale velocity field.