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

TL;DR

This paper introduces a spherical Fast Multipole Method (sFMM) for efficient and highly accurate gravitational lensing simulations on curved skies, outperforming existing methods in speed and precision, suitable for large sky surveys.

Contribution

The paper extends the Fast Multipole Method to spherical geometry, achieving near-linear complexity and higher accuracy than traditional spherical harmonic transforms for gravitational lensing simulations.

Findings

sFMM achieves $O(N) ext{log}(N)$ complexity and $10^{-10}$ accuracy.

sFMM is faster and more accurate than Fast Spherical Harmonic Transform.

Suitable for large-scale sky surveys like Vera Rubin Observatory.

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.