Resurgence and Hyperasymptotics in Wave Optics Astronomy

Abstract

With the discovery of gravitational waves and fast radio bursts, wave optics has become increasingly relevant in astrophysics. This paper studies the behaviour of random gravitational and plasma lenses, presenting the refractive and diffractive expansions, with higher-order terms that allow error estimates and embody the counterintuitive resurgence phenomenon. Specifically, we show that the diffractive expansion converges for a broad class of bounded lens models and provides an efficient description of interference patterns across frequency regimes. Next, building on Picard-Lefschetz techniques, we derive the full refractive expansion to arbitrary order, organising it into a transseries. Near caustics, the standard transseries is supplemented with uniform asymptotics. We study this transseries, with both Borel and hyperasymptotic resummation yielding systematic approximations to lensing integrals at all frequencies. Our results give a framework for modelling wave optics lensing near caustics and beyond the geometric optics approximation and thereby illustrate how tools from resurgence and asymptotic analysis can be applied to practical problems in astrophysics. Near caustic singularities, the post-refractive corrections diverge, while the uniform asymptotic expansion becomes accurate. We use the leading uniform approximation to derive the strong wave optics suppression of off-axis caustics, which clarifies their subdominant role.

Figures (18)

• Figure 1: The lens system displaying a ray moving from the angular position $\hat{\bm{y}}$ on the source plane $S$ to the angular position $\hat{\bm{x}}$ on the lens plane $L$ to the observer.
• Figure 2: The partial sum $I_N$ of the asymptotic series of the integral compared with the closed form expression for $I$ for $\omega = 10$ as a function of the truncation point $N$.
• Figure 3: The convergence of the diffractive expansion for the Lorentzian lens model at $x=0$ for $\omega=1,10,50$ (blue, yellow, green).
• Figure 4: The interference pattern $|I(y)=|\Psi(y)|^2$ for the one-dimensional Gaussian lens model with the parameters $\alpha =2, \mu = 0$, and $\sigma = 1$ evaluated with Picard-Lefschetz theory (black) and the diffraction expansion (red) truncated at $N=550$.
• Figure 7: The geometric approximation $I_{\text{Geometric}}$ (red) and the lens integral $|\Psi|^2$ for the one-dimensional Lorentzian lens with amplitude $\alpha=1$ as a function of the frequency $\omega = 1, 10,$ and $50$.

Theorems & Definitions (5)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5