  We rederive the number density of intervening line-of-sight haloes relative
to lens subhaloes in galaxy-galaxy strong lensing observations, where these
perturbers can generate detectable image fluctuations. Previous studies have
calculated the detection limit of a line-of-sight small-mass dark halo by
comparing the lensing deflection angles it would cause, to those caused by a
subhalo within the lens. However, this overly simplifies the difference in
observational consequences between a subhalo and a line-of-sight halo.
Furthermore, it does not take into account degeneracies between an extra
subhalo and the uncertain properties of the main lens. More in keeping with
analyses of real-world observations, we regard a line-of-sight halo as
detectable only if adding it to a smooth model generates a statistically
significant improvement in the reconstructed image. We find that the number
density of detectable line-of-sight perturbers has been overestimated by as
much as a factor of two in the previous literature. For typical lensing
geometries and configurations, very deep imaging is sensitive to twice as many
line-of-sight perturbers as subhaloes, but moderate depth imaging is sensitive
to only slightly more line-of-sight perturbers than subhaloes.
