Identifying multiple images in strongly lensed gravitational wave sources*

Strong gravitational lensing of gravitational waves (GW) occurs when the waves from a compact binary system travel near a massive object close to the line of sight. The geometrical-optics approximation is valid only when the GW frequency is much larger than the time delay between the multiple images; otherwise, wave optics must be employed. We begin with two axially symmetric lenses, the point mass and singular isothermal sphere, which can produce at most two images. When the geometrical-optics approximation is valid, we introduce two model-independent lens parameters: the flux ratio and the time delay between the GW signals. We assess the validity of geometrical optics as a function of these lens parameters. We then calculate the mismatch between the lensed and unlensed GWs. From this, we estimate the minimum signal-to-noise ratio required to distinguish lensed GW signals from unlensed signals. We then consider a singular isothermal ellipsoid lens that can produce up to four images and compute the mismatches between lensed GW signals with two or four images. This allows us to establish whether two-image templates will suffice for the identification of GW events by more complicated lenses.