Lensing magnification: gravitational waves from coalescing stellar-mass binary black holes

Abstract: Gravitational waves (GWs) may be magnified or de-magnified due to lensing. This phenomenon will bias the distance estimation based on the matched filtering technique. Via the multi-sphere ray-tracing technique, we study the GW magnification effect and selection effect with particular attention to the stellar-mass binary black holes (BBHs). We find that, for the observed luminosity distance $\lesssim 3~\mathrm{Gpc}$, which is the aLIGO/Virgo observational horizon limit, the average magnification keeps as unity, namely unbiased estimation, with the relative distance uncertainty $\sigma(\hat{d})/\hat{d}\simeq0.5\%\sim1\%$. Beyond this observational horizon, the estimation bias can not be ignored, and with the scatters $\sigma(\hat{d})/\hat{d} = 1\%\sim 15\%$. Furthermore, we forecast these numbers for Einstein Telescope. We find that the average magnification keeps closely as unity for the observed luminosity distance $\lesssim 90~\mathrm{Gpc}$. The luminosity distance estimation error due to lensing for Einstein Telescope is about $\sigma(\hat{d})/\hat{d} \simeq 10\%$ for the luminosity distance $\gtrsim 25~\mathrm{Gpc}$. Unlike the aLIGO/Virgo case, this sizable error is not due to the selection effect. It purely comes from the unavoidably accumulated lensing magnification. Moreover, we investigated the effects of the orientation angle and the BH mass distribution models. We found that the results are strongly dependent on these two components.