  Direct measurements of peculiar velocities of galaxies and clusters of
galaxies can in principle provide explicit information on the three dimensional
mass distribution, but this information is modulated by the fact that velocity
field is sampled at galaxy positions, and is thus probing galaxy momentum. We
derive expressions for the cross power spectrum between the density and
momentum field and the auto spectrum of the momentum field in redshift space,
by extending the distribution function method to these statistics. The
resulting momentum cross and auto power spectra in redshift space are expressed
as infinite sums over velocity moment correlators in real space, as is the case
for the density power spectrum in redshift space. We compute each correlator
using Eulerian perturbation theory (PT) and halo biasing model and compare the
resulting redshift-space velocity statistics to those measured from N-body
simulations for both dark matter and halos. We find that in redshift space
linear theory predictions for the density-momentum cross power spectrum as well
as for the momentum auto spectrum fail to predict the N-body results at very
large scales. On the other hand, our nonlinear PT prediction for these velocity
statistics, together with real-space power spectrum for dark matter from
simulations, improves the accuracy for both dark matter and halos. We also
present the same analysis in configuration space, computing the redshift-space
pairwise mean infall velocities and velocity correlation function and compare
to nonlinear PT.
