Proper Motions of VLBI Lenses, Inertial Frames, and the Evolution of Peculiar Velocities

Precise determinations of the image positions in quad gravitational lensesusing VLBI can be used to measure the transverse velocity of the lens galaxyand the observer. The typical proper motions are $\mu$as yr$^{-1}$, so the timescale to measure the motion is ten years. By measuring the dipole of the propermotions in an ensemble of lenses we can set limits on the deviation of theinertial frame defined by the lenses from that defined by the CMB dipole andestimate the Hubble constant. The residual proper motions after subtracting thedipole probe the evolution of peculiar velocities with redshift and can be usedto estimate the density parameter $\Omega_0$. For $N$ lenses, VLBI measurementaccuracies of $\sigma_\theta$, and a baseline of $T$ years, we estimate thatthe 2$\sigma$ limit on the rms peculiar velocity of the lens galaxies is $3100(\sigma_\theta/10\mu\{as})({yrs}/T)/N^{1/2} \kms$, and that the time requiredfor the 2--$\sigma$ limit to reach the level of the local rms peculiar velocity$v_{0,rms}$ is approximately $10 N^{-1/2}(v_{0,rms}/600\kms)(\sigma_\theta/10\mu as)$ years. For a ten year baseline and$N=10$ lenses we expect the 1$\sigma$ limit on the misalignment with the CMBdipole to be $\Delta \theta=20^{\circ}$ or equivalently to obtain an upperlimit of $\Delta H_0 /H_0 < 0.34 $.Comment: 23 pages, figures included uuencoded gzipped ps-file, submitted to the ApJ. One correction made from the original versio