Proper motions of VLBI lenses, inertial frames, and the evolution of peculiar velocities

Precise determinations of the image positions in quad gravitational lenses using VLBI can be used to measure the relative transverse velocity of the lens galaxy and the observer. The typical proper motions are mu as yr(-1), so the timescale to measure the motion is 10 years. By measuring the dipole of the proper motions in an ensemble of lenses we can set limits on the deviation of the inertial frame defined by the lenses from that defined by the CMB dipole and estimate the Hubble constant. The residual proper motions after subtracting the dipole probe the evolution of peculiar velocities with redshift and can be used to estimate the density parameter Omega(0). For N lenses, VLBI measurement accuracies of sigma(0), and a baseline of T years, we estimate that the 2 sigma limit on the rms peculiar velocity of the lens galaxies is 3100(sigma(theta)/10 mu as)(yr/T)/N-1/2 km s(-1), and that the time required for the 2 sigma limit to reach the level of the local rms peculiar velocity upsilon(0,rms) is approximately 10N(-1/2)(upsilon(0,rms)/600 km s(-1))(-1)(sigma(theta)/10 mu as) yr. For a 10 yr baseline and N = 10 lenses we expect the 1 sigma limit on the misalignment with the CMB dipole to be Delta theta = 20 degrees or equivalently to obtain an upper limit of Delta H-0/H-0 < 0.34.