# Gravitational Lensing of Gravitational Waves: Effects of Different Lens Models

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## 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 ML and source position y. In the geometrical-optics approximation (GO), lensing in these models produces at most two discrete images which can be parameterized by two image parameters, the flux ratio I and time delay ∆td between images. In the macrolensing regime for which ∆td is large compared to the time T they spend within the sensitivity band of GW detectors, it is natural to parameterize 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 ∆td can be inverted to constrain ML and y for any lens model. Previously unexplored, we use image parameters to determine the detectability of gravitational lensing of GW in the microlensing regime (∆td ≪ T ) and find that for GW events with signal-to-noise ratios ρ and total mass M , lensing should in principle be identifiable for flux ratios I ≳ 2ρ−2 and time delays ∆td ≳ M −1. We also study GW lensing using non-axisymmetric lens models, such as the singular isothermal ellipsoid (SIE) lens. The use of SIE lens model is motivated by observational constraints on the shapes of galaxies and their dark matter halos, as well as the existence of EM strong lensing configurations with more than 2 images. An SIE can produce four images when the source and lens are well aligned in projection, the cross-section for which depends on the lens ellipticity. In case of the SIE lens, the GO becomes invalid when the source position is near the non-analytic regions, such as the cusps and folds, of the source plane. Therefore, for the first time, the quasi-geometrical optics approximation (QGO) is employed in order to quantify the breakdown of the GO. Using QGO we calculate the lower bound on the lens mass, below which the GO is invalid. We also perform match-filtering analysis between GW sources (lensed by an SIE, producing four images) and (i) unlensed templates and (ii) two types of two-image lensed templates, where the images have the same or different Morse index. Analogous to axisymmetric lens models, we use image parameters to derive analytical expressions to explain the mismatch behaviors for different choices of templates. These investigations will have an impact on searches for GW lensing events in data from current and future detectors. Orbital precession is a dramatic effect induced by spin-orbit and spin-spin couplings in a compact binary source; this precession then imprints characteristic modulations in the waveform. Search efforts are underway to detect GWs with precession during the inspiral phase of the binary system. In this study, we quantify the precessional effects by introducing five new phenomenological parameters. These parameters are expected to provide a comprehensive understanding of the amplitude and frequency modulation, especially from the geometrical perspective. We also analyse the mismatch between precessing source and non-precessing templates to determine the minimum signal-to-noise ratio needed to detect precession for upcoming observing runs of LIGO and third generation detectors.