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 W1
0,2, W2
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 β = f/b. We begin by reviewing
the calculation of the ensemble expectation values W1
0,2 , W2 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 W1
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 β. We estimate the Fisher information that can be extracted from the Minkowski tensors, and
find W1
0,2 is more sensitive to β than W2
0,2, and a measurement of W1
0,2 accurate to ∼1% can yield a ~4% constraint
on β. 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.
