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

TL;DR

This paper studies how gravitational lensing affects the measurement of gravitational wave distances from stellar-mass binary black holes, revealing that lensing introduces biases and uncertainties that depend on the observational horizon and source properties.

Contribution

It provides a detailed analysis of lensing magnification effects on GW distance estimation for current and future detectors, including the impact of source orientation and mass distribution.

Findings

Average magnification is near unity within the observational horizon for aLIGO/Virgo.

Distance estimation bias becomes significant beyond the horizon, with uncertainties up to 15%.

For Einstein Telescope, lensing causes about 10% error at large distances, independent of selection effects.

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.