  When traveling from their source to the observer, gravitational waves can get
deflected by massive objects along their travel path. When the lens is massive
enough and the source aligns closely with the line-of-sight to the lens, the
wave undergoes strong lensing, leading to several images with the same
frequency evolution. These images are separated in time, magnified, and can
undergo an overall phase shift. Searches for strongly-lensed gravitational
waves are already ongoing. In essence, such searches look for events
originating from the same sky location and having the same detector frame
parameters. However, the agreement between these quantities can also happen by
chance, leading to an important confusion background. To reduce the overlap
between background and foreground, one can include lensing models, enforcing
the lensing parameters to also be consistent with our expectations. In
principle, such models should decrease the confusion background. However, when
doing realistic searches, one does not know which model is the correct one. The
use of an incorrect model could lead to the non-detection of genuinely lensed
events. In this work, we investigate under realistic conditions how one can
identify lensed events in the unlensed background. We focus on the impact of
the addition of a model for the lens density profile and investigate the effect
of potential errors in the modelling. We show that it is extremely difficult to
identify lensing confidently without the addition of a lens model. We also show
that slight variations in the profile of the lens model are tolerable but a
model with incorrect assumptions about the underlying lens population causes
significant errors compared to assuming no lens model at all. We also suggest
some strategies to improve the confidence in the detection of strong lensing
for gravitational waves.
