On methods of estimating cosmological bulk flows

We explore similarities and differences between several estimators of the cosmological bulk flow, | B|⁠, from the observed radial peculiar velocities of galaxies. A distinction is made between two theoretical definitions of | B| as a dipole moment of the velocity field weighted by a radial window function. One definition involves the three-dimensional (3D) peculiar velocity, while the other is based on its radial component alone. Different methods attempt at inferring | B| for either of these definitions which coincide only for the case of a velocity field which is constant in space. We focus on the Wiener Filtering (WF) and the Constrained Minimum Variance (CMV) methodologies. Both methodologies require a prior expressed in terms of the radial velocity correlation function. Hoffman et al. compute | B| in Top-Hat windows from a WF realization of the 3D peculiar velocity field. Feldman et al. infer | B| directly from the observed velocities for the second definition of | B|⁠. The WF methodology could easily be adapted to the second definition, in which case it will be equivalent to the CMV with the exception of the imposed constraint. For a prior with vanishing correlations or very noisy data, CMV reproduces the standard Maximum Likelihood estimation for | B| of the entire sample independent of the radial weighting function. Therefore, this estimator is likely more susceptible to observational biases that could be present in measurements of distant galaxies. Finally, two additional estimators are proposed.