Fast Fourier Transform evaluation of the Fresnel integral for gravitational-wave lensing

Abstract: Gravitational waves (GWs) exhibit wave-optics effects when their wavelength is comparable to the scale of the gravitational lens. This may occur in lensing from galactic subhalos in GWs emitted by binary black-hole mergers, and is gaining interest as a novel probe of dark matter. Predictions for observables in these cases ultimately rely on evaluating a Fresnel integral that quantifies the effect of lensing on the amplitude of a GW at a given frequency. However, numerical evaluation of this Fresnel integral is tricky, and several algorithms and publicly available codes that implement it have been developed. Here, we show that the dependence of this integral on the lens position can be written as a two-dimensional Fourier transform. Modern FFT techniques then enable rapid evaluation at all-sky positions simultaneously for general lenses without symmetry. Vectorization of FFT routines allows for derivatives with respect to model parameters to be obtained with only incremental additional computational cost. If the lens is axisymmetric, further speedups can be achieved with recently developed techniques for non-uniform fast Hankel transforms. To demonstrate, we make available Fresnel Integral Optimization with Non-uniform trAnsforms (FIONA), an efficient and accurate code that is significantly faster than current methods for dense source grids, reaching 2-3 orders of magnitude speedups for $\sim 10^6$ GW-emitting points. As part of FIONA, we developed code that provides vectorized non-uniform fast Hankel transforms that may have other uses (e.g., calculation of cosmological two-point correlation functions) beyond those considered here.