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.

Figures (11)

• Figure 1: Scaling of computation time versus the total number of source points for the 1D Fresnel integral evaluated with FIONA and GLoW (SingleIntegral_C function). The lens potential used for illustration is a singular isothermal sphere.
• Figure 2: Scaling of computation time versus the total number of source points for the 2D Fresnel integral evaluated with FIONA and GLoW (Multi_Contour_C function). The lens potential is composed of 4 randomly-placed singular isothermal spheres.
• Figure 3: $F(w,y)$ for the cored isothermal sphere (CIS) with core radius $x_c = 0.05$ evaluated at $y=0.5$. The computation with FIONA takes 2.1 s for 560 uniformly spaced frequencies between $w=0.1$ and $w=100$ on 112 CPU cores. Results from GLoW are overplotted with a dashed line for validation.
• Figure 4: Contour plot $F(w,y)$ for a CIS lens with $x_c = 0.05$, inferred from 560 and 400 uniformly spaced points in $w$ and $y$, respectively. The computation takes 2.1 s on 112 CPU cores.
• Figure 5: $|F(w, y)|$ for a lens with an Elliptical Power-Law profile ($e_1=0.2$, $e_2=0.0$) and a weak background shear ($\gamma_1$=0.03, $\gamma_2$=0.01), shown over the source plane $\mathbf{y} = (y_1, y_2)$ for varying negative density slope ($\gamma = 1.2, 1.7, 2.2$) at different frequencies ($w=1, 10, 100$). The computation of results for all three lenses, evaluated on a $500 \times 500$ source grid at the three selected frequencies, takes $\sim 100$ s on a single CPU core.