Gravitational lensing of gravitational waves: A statistical perspective
  In this paper, we study the strong gravitational lensing of gravitational
waves (GWs) from a statistical perspective, with particular focus on the high
frequency GWs from stellar binary black hole coalescences. These are most
promising targets for ground-based detectors such as Advanced Laser
Interferometer Gravitational Wave Observatory (aLIGO) and the proposed Einstein
Telescope (ET) and can be safely treated under the geometrical optics limit for
GW propagation. We perform a thorough calculation of the lensing rate, by
taking account of effects caused by the ellipticity of lensing galaxies, lens
environments, and magnification bias. We find that in certain GW source rate
scenarios, we should be able to observe strongly lensed GW events once per year
($\sim1~\text{yr}^{-1}$) in the aLIGO survey at its design sensitivity; for the
proposed ET survey, the rate could be as high as $\sim80~\text{yr}^{-1}$. These
results depend on the estimate of GW source abundance, and hence can be
correspondingly modified with an improvement in our understanding of the merger
rate of stellar binary black holes. We also compute the fraction of four-image
lens systems in each survey, predicting it to be $\sim30$ per cent for the
aLIGO survey and $\sim6$ per cent for the ET survey. Finally, we evaluate the
possibility of missing some images due to the finite survey duration, by
presenting the probability distribution of lensing time delays. We predict that
this selection bias will be insignificant in future GW surveys, as most of the
lens systems ($\sim90$ per cent) will have time delays less than $\sim1$ month,
which will be far shorter than survey durations.
