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 (GL) on a curved sky. The sFMM is a non-trivial extension of the Fast Multiple Method (FMM) to sphere ${\mathbb{S}}^2$, and it can accurately solve the Poisson equation with time complexity of $O(N)\log (N)$, 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 (FSHT), the sFMM is not only faster but 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.

arXiv commentAccepted by The Astrophysical Journal (ApJ)

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