Minkowski Tensors in Three Dimensions - Probing the Anisotropy Generated
  by Redshift Space Distortion
  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 $3 \times 3$ 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 $\Lambda$CDM 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 $\Upsilon \equiv f/b$.
