Gravitational lensing of gravitational waves: A statistical perspective

TL;DR

This paper analyzes the statistical likelihood of strong gravitational lensing of high-frequency gravitational waves from stellar black hole mergers, predicting observable rates for current and future detectors and assessing potential observational biases.

Contribution

It provides a comprehensive statistical framework for predicting strong lensing rates of gravitational waves, including effects of galaxy ellipticity, environment, and magnification bias, with updated predictions for aLIGO and ET.

Findings

aLIGO could observe ~1 strongly lensed GW per year.

ET could observe up to ~80 strongly lensed GW per year.

Most lensing time delays are under 1 month, minimizing detection bias.

Abstract

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.