Bending of Light by Gravity Waves

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Abstract
We describe the statistical properties of light rays propagating though a random sea of gravity waves and compare with the case for scalar metric perturbations from density inhomogeneities. For scalar fluctuations, the deflection angle grows as the square root of the path length D in the manner of a random walk, and the rms displacement of a ray from the unperturbed trajectory grows as D3/2. For gravity waves the situation is very different. The mean square deflection angle remains finite and is dominated by the effect of the metric fluctuations at the ends of the ray, and the mean square displacement grows only as the logarithm of the path length. In terms of power spectra, the displacement for scalar perturbations has P(k) ∝ 1/k4 while, for gravity waves, the trajectories of photons have P(k) ∝ 1/k, which is a scale-invariant or "flicker noise" process, and departures from rectilinear motion are suppressed relative to the scalar case by a factor of ~(λ/D)3/2, where λ is the characteristic scale of the metric fluctuations and D is the path length. This result casts doubt on the viability of some recent proposals for detecting or constraining the gravity wave background by astronomical measurements.
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