On the detection and precise localisation of merging black holes events through strong gravitational lensing

To unlock the full spectrum of astrophysical and cosmological applications of gravitational-wave detections, it is essential to localise the associated black-hole mergers to high precision inside their host galaxies. One possible method to achieve this is to compare the properties of multiple detections of gravitationally-lensed binary black-hole merger events with the properties of strong gravitational lens systems located in the joint sky localisation of the gravitational-wave detections. In this work, we simulate the population of binary black-hole mergers lensed by galaxy-scale lenses and detectable by LIGO-Virgo-Kagra in the coming decade and the population of galaxy-scale strong lenses that will be detected by Euclid. We use these simulations to investigate the prospects for localising strongly lensed binary black-hole mergers inside the lensed galaxies of 'Euclid-like' galaxy-scale strong lenses. We find that for 20-50% of strongly lensed gravitational wave events the lens system is detectable with Euclid, if the event falls in its survey footprint. Of these, we expect to correctly identify the strongly-lensed host galaxy as likely (with posterior probability) host galaxy - based on Bayesian evidence ranking of candidate hosts - for 34.6-21.9% of quadruply-lensed gravitational-wave events when given an a-priori 1-5 deg^2 gravitational-wave-only sky localisation. For triply and doubly lensed gravitational-wave events, this becomes 29.8-14.9% and 16.4-6.6% respectively. If successfully identified, however, the localisation can be better than a fraction of the host-galaxy size, i.e. of order milli-arcseconds. A first detection in the coming decade, however, probably requires dedicated deep and high-resolution follow-ups and continued upgrades in the current and planned gravitational-wave detectors.