Precise LIGO lensing rate predictions for binary black holes

We show how LIGO is expected to detect coalescing binary black holes at $z>1$ that are lensed by the intervening galaxy population. Gravitational magnification, $\mu$, strengthens gravitational-wave signals by $\sqrt{\mu }$ without altering their frequencies, which if unrecognized leads to an underestimate of the event redshift and hence an overestimate of the binary mass. High magnifications can be reached for coalescing binaries, because the region of intense gravitational-wave emission during coalescence is so small ($\sim 100\mathrm{km}$), permitting very close projections between lensing caustics and gravitational-wave events. Our simulations use the current LIGO event-based mass function and incorporate accurate waveforms convolved with the LIGO power spectral density. Importantly, we include the detection dependence on sky position and orbital orientation, which for the LIGO configuration translates into a wide spread in observed redshifts and chirp masses. Currently, we estimate a detectable rate of lensed events $0.{06}_{-0.02}^{+0.02}{\mathrm{yr}}^{-1}$ that rises to $5_{-3}^{+5}{\mathrm{yr}}^{-1}$ at LIGO design sensitivity limit, depending on the high redshift rate of black hole coalescence.

Figure 1

Predicted rates in different stages of LIGO using the mass function from Ref. [22]. The solid line shows the overall rate of events, and the dashed line shows the rate of lensed events. Even though the rate of lensed events only contributes a small fraction of the overall rate of events, lensed events may be observed frequently in the advanced detector era.

Figure 2

The differential rate as a function of the source redshift $z_s$. The solid lines show the differential rate of overall events, and the dashed lines show the differential rate of lensed events. Red, blue, and cyan lines represent different stages of LIGO. The magenta solid line shows the differential rate of overall events with averaged $\mathrm{\Theta }$, for comparison with Ref. [13].

Figure 3

The differential rate as a function of the observed chirp mass ${\mathcal{M}}_c$ for all events (solid) and for lensed events (dashed) at different stages of LIGO’s sensitivity. Lensing introduces a sharp transition in the differential rate that corresponds to the intrinsic chirp mass distribution, where the model uncertainty shown here covers the range of power-law slope and normalization constrained to date by the current LIGO detections [22].

Figure 4

The ratio of lensed events to total events as a function of the observed chirp mass ${\mathcal{M}}_c$ and SNR $\rho$. At constant $\rho$, lensing starts to dominates for certain high-mass regions. High-chirp-mass events are dominated by lensing, with a transition mass that is lower for higher SNR, reflecting the maximum distance to which unlensed events can be detected.