Estimation of peculiar velocity from the inverse Tully–Fisher relation

We present a method for deriving a smoothed estimate of the peculiar velocity field of a set of galaxies with measured circular velocities \eta\equiv {\rm log} \Delta v and apparent magnitudes m. The method is based on minimizing the scatter of a linear inverse Tully-Fisher relation \eta= \eta(M) where the absolute magnitude of each galaxy is inferred from its redshift z, corrected by a peculiar velocity field, M \propto m - 5\log(z-u). We describe the radial peculiar velocity field u({\bf z}) in terms of a set of orthogonal functions which can be derived from any convenient basis set; as an example we take them to be linear combinations of low order spherical harmonic and spherical Bessel functions. The model parameters are then found by maximizing the likelihood function for measuring a set of observed \eta. The predicted peculiar velocities are free of Malmquist bias in the absence of multi-streaming, provided no selection criteria are imposed on the measurement of circular velocities. This procedure can be considered as a generalized smoothing algorithm of the peculiar velocity field, and is particularly useful for comparison to the smoothed gravity field derived from full-sky galaxy redshift catalogs such as the IRAS surveys. We demonstrate the technique using a catalog of ``galaxies" derived from an N-body simulation. Increasing the resolution of the velocity smoothing beyond a certain level degrades the correlation of fitted velocities against the velocities calculated from linear theory methods, which have finite resolution, but the slope of the scatter diagram, and therefore the derived density parameter, remains fixed