Minkowski Tensors in Three Dimensions - Probing the Anisotropy Generated by Redshift Space Distortion

TL;DR

This paper demonstrates how Minkowski tensor statistics can detect and quantify anisotropy caused by redshift space distortions in three-dimensional Gaussian random fields, providing a new tool for cosmological analysis.

Contribution

It introduces Minkowski tensor statistics in three dimensions, applies them to Gaussian fields, and shows their effectiveness in measuring anisotropy from redshift space distortions.

Findings

Minkowski tensors reveal anisotropy in redshift space distorted fields.

The tensors' diagonal components differ under anisotropy, indicating preferred directions.

The method can constrain the redshift space distortion parameter A5.

Abstract

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 $\Upsilon \equiv f/b$.