On the Prediction of Velocity Fields from Redshift Space Galaxy Samples

We present a new method for recovering the underlying velocity field from an observed distribution of galaxies in redshift space. The method is based on a kinematic Zel'dovich relation between the velocity and density fields in redshift space. This relation is expressed in a differential equation slightly modified from the usual Poisson equation, and which depends non-trivially on $\beta \equiv \Omega^{0.6}/b$. The linear equation can be readily solved by standard techniques of separation of variables by means of spherical harmonics. One can also include a term describing the ``rocket effect" discussed by Kaiser (1987). From this redshift space information alone, one can generate a prediction of the peculiar velocity field for each harmonic ($l,m$) as a function of distance. We note that for the quadrupole and higher order moments, the equation is a boundary value problem with solutions dependent on both the interior and exterior mass distribution. However, for a shell at distance $r$, the dipole, as well as the monopole, of the velocity field in the Local Group frame is fully determined by the interior mass distribution. This implies that the shear of the measured velocity field, when fit to a dipole distortion, should be aligned and consistent with the gravity field inferred from the well determined local galaxy distribution. As a preliminary application we compute the velocity dipole of distant shells as predicted from the 1.2Jy IRAS survey compared to the measured velocity dipole on shells, as inferred from a recent POTENT analysis. The coherence between the two fields is good, yielding a best estimate of $\beta= 0.6 \pm 0.2$.