Limits on a stochastic background of gravitational waves from gravitational lensing.

We compute the effects of a stochastic background of gravitational waves on multiply imaged systems or on weak lensing. There are two possible observable effects, a static relative deflection of images or shear, and an induced time-dependent shift or proper motion. We evaluate the rms magnitude of these effects for a COBE normalized, scale-invariant spectrum, which is an upper limit on spectra produced by inflation. Previous work has shown that large-scale structure may cause a relative deflection large enough to affect observations, but we find that the corresponding effect of gravity waves is smaller by ∼ 104 and so cannot be observed. This results from the oscillation in time as well as the redshifting of the amplitude of gravity waves. We estimate the magnitude of the proper motion induced by deflection of light due to large-scale structure, and find it to be ∼ 10−8 arcsec per year. This corresponds to ∼ 50 km/s at cosmological distances, which is quite small compared to typical peculiar velocities. The COBE normalized gravity wave spectrum produces motions smaller still by ∼ 102. We conclude that light deflection due to these cosmological perturbations cannot produce observable proper motions of lensed images. On the other hand, there are only a few known observational limits on a stochastic background of gravity waves at astrophysical wavelengths. High-resolution imaging of lens systems with VLBI can set limits on the relative motions of images. We show that a limit on gravity waves of Ωλ ∼10, valid for wavelengths λ greater than a few pc (up to ∼ 100 Mpc), is feasible with a ten-year observation of a four-image system.