Detectability of strongly lensed gravitational waves using model-independent image parameters

Abstract

Strong gravitational lensing of gravitational waves (GWs) occurs when the GWs from a compact binary system travel near a massive object. The lensed waveform is given by the product of the lensing amplification factor $F$ and the unlensed waveform. For many axisymmetric lens models such as the point mass and singular isothermal sphere that we consider, $F$ can be calculated in terms of two lens parameters, the lens mass ${M}_{L}$ and source position $y$. In the geometrical-optics approximation, lensing in these models produces at most two discrete images which can be parametrized by two image parameters, the flux ratio $I$ and time delay $\mathrm{\ensuremath{\Delta}}{t}_{d}$ between images. In the macrolensing regime for which $\mathrm{\ensuremath{\Delta}}{t}_{d}$ is large compared to the time $T$ they spend within the sensitivity band of GW detectors, it is natural to parametrize lensing searches in terms of these image parameters. The functional dependence of the lensed signal on these image parameters is far simpler, facilitating data analysis for events with modest signal-to-noise ratios, and constraints on $I$ and $\mathrm{\ensuremath{\Delta}}{t}_{d}$ can be inverted to constrain ${M}_{L}$ and $y$ for any lens model. We propose that this use of image parameters can be extended to the microlensing regime ($\mathrm{\ensuremath{\Delta}}{t}_{d}<T$) in which the two interfering images are observed as a single GW event. We use image parameters to determine the detectability of gravitational lensing in GW the microlensing regime and find that for GW events with signal-to-noise ratios $\ensuremath{\rho}$ and total mass $M$, lensing should in principle be identifiable for flux ratios $I\ensuremath{\gtrsim}2{\ensuremath{\rho}}^{\ensuremath{-}2}$ and time delays $\mathrm{\ensuremath{\Delta}}{t}_{d}\ensuremath{\gtrsim}{M}^{\ensuremath{-}1}$.