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 mismatch between a lensed signal and unlensed templates determines whether lensing can be identified in a particular GW event. For axisymmetric lens models, the lensed signal is traditionally calculated in terms of model-dependent lens parameters such as the lens mass $M_L$ and source position $y$. We propose that it is useful to parameterize this signal instead in terms of model-independent image parameters: the flux ratio $I$ and time delay $\Delta t_d$ between images. 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. In the geometrical-optics approximation, constraints on $I$ and $\Delta t_d$ can be inverted to constrain $M_L$ and $y$ for any lens model including the point mass (PM) and singular isothermal sphere (SIS) that we consider. We use our model-independent image parameters to determine the detectability of gravitational lensing in GW signals and find that for GW events with signal-to-noise ratios $\rho$ and total mass $M$, lensing should in principle be identifiable for flux ratios $I \gtrsim 2\rho^{-2}$ and time delays $\Delta t_d \gtrsim M^{-1}$.