Limits on a Stochastic Background of Gravitational Waves from Gravitational Lensing

Abstract

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 $\sim 10^4$ 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 $\sim 10^{-8}$ arcsec per year. This corresponds to $\sim 50$ 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 $\sim 10^2$. 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 shorter, astrophysical wavelengths. We calculate the expected magnitudes of the effects of lensing by gravity waves of such wavelengths, and find that they are too small to yield interesting limits on the energy density of gravity waves.