Can we constrain structure growth from galaxy proper motions?

Galaxy peculiar velocities can be used to trace the growth of structure on cosmological scales. In the radial direction, peculiar velocities cause redshift space distortions, an established cosmological probe, and can be measured individually in the presence of an independent distance indicator. In the transverse direction, peculiar velocities cause proper motions. In this case, however, the proper motions are too small to detect on a galaxy-by-galaxy basis for any realistic experiment in the foreseeable future, but could be detected statistically in cross-correlation with other tracers of the density fluctuations. We forecast the sensitivity for a detection of transverse peculiar velocities through the cross-correlation of a proper motion survey, modelled after existing extragalactic samples measured by Gaia, and an overlaping galaxy survey. In particular, we consider a low-redshift galaxy sample, and a higher-redshift quasar sample. We find that, while the expected cosmological signal is below the expected statistical uncertainties from current data using cross-correlations, the sensitivity can improve fast with future experiments, and the threshold for detection may not be too far away in the future. Quantitatively, we find that the signal-to-noise ratio for detection is in the range $S/N\sim0.3$, with most of the signal concentrated at low redshifts $z\lesssim0.3$. If detected, this signal is sensitive to the product of the expansion and growth rates at late times, and thus would constitute an independent observable, sensitive to both background expansion and large-scale density fluctuations.