Detection Rate of Galaxy Cluster Lensed Stellar Binary Black Hole Mergers by the Third-generation Gravitational Wave Detectors

Gravitational waves (GWs) from stellar binary black hole (sBBH) mergers can be strongly gravitational lensed by intervening galaxies/galaxy clusters. Only a few works investigated the cluster-lensed sBBH mergers by adopting oversimplified models, while galaxy-lensed ones were intensively studied. In this paper, we estimate the detection rate of cluuster-lensed sBBH mergers with the third-generation GW detectors and its dependence on the lens models. We adopt detailed modeling of galaxy cluster lenses by using the mock clusters in the Synthetic Sky Catalog for Dark Energy Science with LSST (CosmoDC2) and/or approximations of the pseudo-Jaffe profile or an eccentric Navarro-Frenk-White dark matter halo plus a bright central galaxy with singular isothermal sphere profile. Considering the formation of sBBH mergers dominates by the channel of evolution of massive binary stars (EMBS), we find that the detection rate of cluster-lensed sBBHs is $\sim5-84$ yr$^{-1}$, depending on the adopted lens model and uncertainty in the merger rate density, and it is about $\sim{13_{-2.0}^{+28}}$yr$^{-1}$ if adopting relatively more realistic galaxy clusters with central main and member galaxies in the CosmoDC2 catalog, close to the estimated detection rate of sBBH mergers lensed by galaxies. In addition, we also consider the case that the production of sBBH mergers dominated by the dynamical interactions in dense stellar systems. We find that the detection rate of cluster-lensed sBBHs if from the dynamical channel is about $1.5$ times larger than that from the EMBS channel and the redshift distribution of former peaking at a higher redshift ($\sim3$) compared with that from latter ($\sim2$).