  Current searches for gravitational waves from compact-binary objects are
primarily designed to detect the dominant gravitational-wave mode and assume
that the binary components have spins which are aligned with the orbital
angular momentum. These choices lead to observational biases in the observed
distribution of sources. Sources with significant spin-orbit precession or
unequal-mass-ratios, which have non-negligible contributions from sub-dominant
gravitational-wave modes, may be missed; in particular, this may significantly
suppress or bias the observed neutron star -- black hole (NSBH) population. We
simulate a fiducial population of NSBH mergers and determine the impact of
using searches that only account for the dominant-mode and aligned spin. We
compare the impact for the Advanced LIGO design, A+, LIGO Voyager, and Cosmic
Explorer observatories. We find that for a fiducial population where the spin
distribution is isotropic in orientation and uniform in magnitude, we will miss
$\sim 25\%$ of sources with mass-ratio $q > 6$ and up to $\sim 60 \%$ of highly
precessing sources $(\chi_p > 0.5)$, after accounting for the approximate
increase in background. In practice, the true observational bias can be even
larger due to strict signal-consistency tests applied in searches. The
observation of low spin, unequal-mass-ratio sources by Advanced LIGO design and
Advanced Virgo may in part be due to these selection effects. The development
of a search sensitive to high mass-ratio, precessing sources may allow the
detection of new binaries whose spin properties would provide key insights into
the formation and astrophysics of compact objects.
