Ensemble Average of Three-Dimensional Minkowski Tensors of a Gaussian Random Field in Redshift Space

Abstract

We present the ensemble expectation values for the translation invariant, rank-2 Minkowski tensors in three-dimensions, for a linearly redshift space distorted Gaussian random field. The Minkowski tensors $W_1^{0,2}$, $W_2^{0,2}$ are sensitive to global anisotropic signals present within a field, and by extracting these statistics from the low redshift matter density one can place constraints on the redshift space distortion parameter $\beta =f/b$. We begin by reviewing the calculation of the ensemble expectation values $⟨W_1^{0,2}⟩$, $⟨W_2^{0,2}⟩$ for isotropic, Gaussian random fields, then consider how these results are modified by the presence of a linearly anisotropic signal. Under the assumption that all fields remain Gaussian, we calculate the anisotropic correction due to redshift space distortion in a coordinate system aligned with the line of sight, finding inequality between the diagonal elements of $⟨W_1^{0,2}⟩$, $⟨W_2^{0,2}⟩$. The ratio of diagonal elements of these matrices provides a set of statistics that are sensitive only to the redshift space distortion parameter $\beta$. We estimate the Fisher information that can be extracted from the Minkowski tensors, and find $W_1^{0,2}$ is more sensitive to $\beta$ than $W_2^{0,2}$, and a measurement of $W_1^{0,2}$ accurate to $\sim 1\%$ can yield a $\sim 4\%$ constraint on $\beta$. Finally, we discuss the difference between using the matrix elements of the Minkowski tensors directly against measuring the eigenvalues. For the purposes of cosmological parameter estimation we advocate the use of the matrix elements, to avoid spurious anisotropic signals that can be generated by the eigenvalue decomposition.

References in corpus (5)

• Topology of structure in the Sloan Digital Sky Survey: model testing
• Morphology of 21cm brightness temperature during the Epoch of Reioinization using Contour Minkowski Tensor
• Tensor Minkowski Functionals for random fields on the sphere
• Cosmological Parameter Estimation Using the Genus Amplitude - Application to Mock Galaxy Catalogs
• Topology of Large-Scale Structures of Galaxies in Two Dimensions - Systematic Effects

Cited by in corpus (8)

• Probing massive neutrinos with the Minkowski functionals of the galaxy distribution
• Application of Contour Minkowski Tensor and $\mathcal{D}$ Statistic to the Planck $E$-mode data
• Minkowski Tensors in Redshift Space -- Beyond the Plane Parallel Approximation
• The geometrical meaning of statistical isotropy of smooth random fields in two dimensions
• Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative
• Impact of Redshift Space Distortion on Persistent Homology of cosmic matter density field
• Cosmic Strings-induced CMB anisotropies in light of Weighted Morphology
• Quantifying Weighted Morphological Content of Large-Scale Structures via Simulation-Based Inference