  The majority of massive stars are born in binaries, and most unbind upon the
first supernova. With precise proper motion surveys such as Gaia, it is
possible to trace back the motion of stars in the vicinity of young remnants to
search for ejected companions. Establishing the fraction of remnants with an
ejected companion, and the photometric and kinematic properties of these stars,
offers unique insight into supernova progenitor systems. In this paper, we
employ binary population synthesis to produce kinematic and photometric
predictions for ejected secondary stars. We demonstrate that the unbound
neutron star velocity distribution from supernovae in binaries closely traces
the input kicks. Therefore, the observed distribution of neutron star
velocities should be representative of their natal kicks. We evaluate the
probability for any given filter, magnitude limit, minimum measurable proper
motion (as a function of magnitude), temporal baseline, distance and extinction
that an unbound companion can be associated with a remnant. We compare our
predictions with results from previous companion searches, and demonstrate that
the current sample of stars ejected by the supernova of their companion can be
increased by a factor of 5-10 with Gaia data release 3. Further progress in
this area is achievable by leveraging the absolute astrometric precision of
Gaia, and by obtaining multiple epochs of deep, high resolution near-infrared
imaging with the Hubble Space Telescope, JWST and next-generation wide-field
near-infrared observatories such as Euclid or the Nancy Grace Roman Space
Telescope.
