Acquiring the Lefschetz thimbles: efficient evaluation of the diffraction integral for lensing in wave optics

TL;DR

This paper introduces a new, efficient numerical method for evaluating the diffraction integral in wave optics lensing by leveraging Lefschetz thimbles, making complex astrophysical calculations more feasible.

Contribution

It presents a simplified approach to compute Lefschetz thimbles with high efficiency, improving the practical application of Picard-Lefschetz theory in astrophysical wave lensing.

Findings

Demonstrates simplified examples of Lefschetz thimbles

Proposes new flow line methods for efficient computation

Enables more practical analysis of wave effects in lensing

Abstract

Evaluating the Kirchhoff-Fresnel diffraction integral is essential in studying wave effects in astrophysical lensing, but is often intractable because of the highly oscillatory integrated. A recent breakthrough was made by exploiting the Picard-Lefschetz theory: the integral can be performed along the `Lefschetz thimbles' in the complex domain where the integrand is not oscillatory but rapidly converging. The application of this method, however, has been limited by both the unfamiliar concepts involved and the low numerical efficiency of the method used to find the Lefschetz thimbles. In this paper, we give simple examples of the Lefschetz thimbles and define the `flow lines' that facilitate the understanding of the concepts. Based on this, we propose new ways to obtain the Lefschetz thimbles with high numerical efficiency, which provide an effective tool for studying wave effects in astrophysical lensing.