  While the majority of massive stars have a stellar companion, most pulsars
appear to be isolated. Taken at face value, this suggests that most massive
binaries break apart due to strong natal kicks received in supernova
explosions. However, the observed binary fraction can still be subject to
strong selection effects, as monitoring of newly discovered pulsars is rarely
carried out for long enough to conclusively rule out multiplicity. Here, we use
the second Gaia Data Release (DR2) to search for companions to 1534
rotation-powered pulsars with positions known to better than 0.5 arcseconds. We
find 22 matches to known pulsars, including one not reported elsewhere, and 8
new possible companions to young pulsars. We examine the photometric and
kinematic properties of these systems and provide empirical relations for
identifying Gaia sources with potential millisecond pulsar companions. Our
results confirm that the observed multiplicity fraction is small. However, we
show that the number of binaries below the sensitivity of Gaia and radio timing
in our sample could still be significantly higher. We constrain the binary
fraction of young pulsars to be $f_{\rm young}^{\rm true}\leq 5.3(8.3)\%$ under
realistic(conservative) assumptions for the binary properties and current
sensitivity thresholds. For massive stars ($\geq 10$ M$_{\odot}$) in
particular, we find $f_{\rm OB}^{\rm true}\leq 3.7\%$ which sets a firm
independent upper limit on the galactic neutron-star merger rate, $\leq
7.2\times 10^{-4}$ yr$^{-1}$. Ongoing and future projects such as the
CHIME/pulsar program, MeerTime, HIRAX and ultimately the SKA, will
significantly improve these constraints in the future.
