The Spherical Fast Multipole Method (sFMM) for Gravitational Lensing Simulation

Abstract

In this paper, we present a spherical Fast Multipole Method (sFMM) for ray-tracing simulation of gravitational lensing on a curved sky. The sFMM is a nontrivial extension of the Fast Multiple Method to sphere , and it can accurately solve the Poisson equation with time complexity of , where N is the number of particles. It is found that the time complexity of the sFMM is near O(N) and the computational accuracy can reach 10−10 in our test. In addition, compared with the Fast Spherical Harmonic Transform, the sFMM is not only faster but also more accurate, as it has the ability to reserve high-frequency components of the density field. These merits make the sFMM an optimum method to simulate the gravitational lensing on a curved sky, which is the case for upcoming large-area sky surveys, such as the Vera Rubin Observatory and the China Space Station Telescope.