  Low redshift surveys of galaxy peculiar velocities provide a wealth of
cosmological information. We revisit the idea of extracting this information by
directly measuring the redshift-space momentum power spectrum from such
surveys. We provide a comprehensive theoretical and practical framework for
estimating and fitting this from data, analogous to well understood techniques
used to measure the galaxy density power spectrum from redshift surveys. We
formally derive a new estimator, which includes the effects of shot noise and
survey geometry; we evaluate the variance of the estimator in the Gaussian
regime; we compute the optimal weights for the estimator; we demonstrate that
the measurements are Gaussian distributed, allowing for easy extraction of
cosmological parameters; and we explore the effects of peculiar velocity
measurement errors. We finish with a proof-of-concept using realistic mock
galaxy catalogues, which demonstrates that we can measure and fit both the
redshift-space galaxy density and momentum power spectra from peculiar velocity
surveys and that including the latter substantially improves our constraints on
the growth rate of structure. We also provide theoretical descriptions for
modelling the non-linear redshift-space density and momentum power spectrum
multipoles, and forecasting the constraints on cosmological parameters using
the Fisher information contained in these measurements for arbitrary weights.
These may be useful for measurements of the galaxy density power spectrum even
in the absence of peculiar velocities.
