All Transverse Motion is Peculiar: Connecting the Proper Motions of Galaxies to the Matter Power Spectrum

In an isotropic and homogeneous Hubble expansion, all transverse motion is peculiar. Like the radial peculiar velocities of galaxies, transverse peculiar velocities are a means to trace the density of matter that does not rely on light tracing mass. Unlike radial peculiar velocity measurements that require precise redshift-independent distances in order to distinguish between the Hubble expansion and the observed redshift, transverse peculiar velocities can be measured using redshifts alone as a proxy for distance. Extragalactic proper motions can therefore directly measure peculiar velocities and probe the matter power spectrum. Here we develop two-point transverse velocity correlation statistics and demonstrate their dependence on the matter power spectrum. We predict the power in these correlation statistics as a function of the physical separation, angular separation, and distance of pairs of galaxies and demonstrate that the effect of large scale structure on transverse motions is best measured for pairs of objects with comoving separations less than about 50 Mpc. Transverse peculiar velocities induced by large scale structure should be observable as proper motions using long baseline radio interferometry or space-based optical astrometry.