Minkowski Tensors in Redshift Space -- Beyond the Plane Parallel Approximation

Abstract

The Minkowski tensors (MTs) can be used to probe anisotropic signals in a field, and are well suited for measuring the redshift space distortion (RSD) signal in large scale structure catalogs. We consider how the linear RSD signal can be extracted from a field without resorting to the plane parallel approximation. A spherically redshift space distorted field is both anisotropic and inhomogeneous. We derive expressions for the two point correlation functions that elucidate the inhomogeneity, and then explain how the breakdown of homogeneity impacts the volume and ensemble averages of the tensor Minkowski functionals. We construct the ensemble average of these quantities in curvilinear coordinates and show that the ensemble and volume averages can be approximately equated, but this depends on our choice of definition of the volume average of a tensor and the radial distance between the observer and field. We then extract the tensor Minkowski functionals from spherically redshift space distorted, Gaussian random fields and gravitationally evolved dark matter density fields at $z=0$ to test if we can successfully measure the Kaiser RSD signal. For the dark matter field we find a significant, $\sim 10\%$ anomalous signal in the MT component parallel to the line of sight that is present even on large scales $R_{\rm G} \gtrsim 15 \, {\rm Mpc}$, in addition to the Kaiser effect. This is due to the line of sight component of the MT being significantly contaminated by the Finger of God effect, which can be approximately modelled by an additional damping term in the cumulants.