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

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 $W^{0,2}_{1}$, $W^{0,2}_{2}$ 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 $\langle W^{0,2}_{1} \rangle$, $\langle W^{0,2}_{2} \rangle $ 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 $\langle W^{0,2}_{1} \rangle $, $\langle W^{0,2}_{2} \rangle $. 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 $W^{0,2}_{1}$ is more sensitive to $\beta$ than $W^{0,2}_{2}$, and a measurement of $W^{0,2}_{1}$ 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.