  Gravitational waves perturb the paths of photons, impacting both the
time-of-flight and the arrival direction of light from stars. Pulsar timing
arrays can detect gravitational waves by measuring the variations in the time
of flight of radio pulses, while astrometry missions such as Gaia can detect
gravitational waves from the time-varying changes in the apparent position of a
field of stars. Just as gravitational waves impart a characteristic correlation
pattern in the arrival times of pulses from pulsars at different sky locations,
the deflection of starlight is similarly correlated across the sky. Here we
compute the astrometric correlation patterns for the full range of polarization
states found in alternative theories of gravity, and decompose the sky-averaged
correlation patterns into vector spherical harmonics. We find that the tensor
and vector polarization states produce equal power in the electric- and
magnetic-type vector spherical harmonics, while the scalar modes produce only
electric-type correlations. Any difference in the measured electric and
magnetic-type correlations would represent a clear violation of Einstein
gravity. The angular correlations functions for the vector and scalar
longitudinal modes show the same enhanced response at small angular separations
that is familiar from pulsar timing.
