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