Weak Lensing Approximation of Wave-optics Effects from General Symmetric Lens Profiles

TL;DR

The paper tackles the challenge of modeling wave-optics lensing of gravitational waves in the weak-lensing regime by deriving a general analytic framework for symmetric lenses. It starts from the SIS model and extends to general symmetric profiles, delivering a compact two-term decomposition of the amplification factor $F(\omega,y)$ into a GO term and a WO term, valid when $y\gg 1$ and $\omega y^2/2\gg 1$, and validates it against numerical results for SIS and NFW profiles. The approach reproduces the GO and QGO asymptotics in the high-frequency limit and remains accurate across a broad range of frequencies and impact parameters, enabling efficient lens reconstruction, delensing of standard sirens, and probing low-mass dark matter halos with minimal baryonic content. By operating in the frequency domain and avoiding time-domain regularizations, the method offers substantial computational advantages and can be extended to nested or non-symmetric lens configurations, with important implications for dark matter phenomenology and future gravitational-wave cosmology.

Abstract

Gravitational lensing of electromagnetic (EM) waves has yielded many profound discoveries across fundamental physics, astronomy, astrophysics, and cosmology. Similar to EM waves, gravitational waves (GWs) can also be lensed. When their wavelength is comparable to the characteristic scale of the lens, wave-optics (WO) effects manifest as frequency-dependent modulations in the GW waveform. These WO features encode valuable information about the lensing system but are challenging to model, especially in the weak lensing regime, which has a larger optical depth than strong lensing. We present a novel and efficient framework to accurately approximate WO effects induced by general symmetric lens profiles. Our method is validated against numerical calculations and recovers the expected asymptotic behavior in both high- and low-frequency limits. Accurate and efficient modeling of WO effects in the weak lensing regime will enable improved lens reconstruction, delensing of standard sirens, and provide a unique probe to the properties of low-mass halos with minimal baryonic content, offering new insights into the nature of dark matter.

Figures (4)

• Figure 1: Comparison between the weak lensing approximation and the full analytical results for the SIS lens model. Top: Solid blue lines show the amplification factor magnitude, $|F(\omega, y)|$, as a function of the source position $y$, for $\omega = 0.8$, 1.0, and 1.2 (left, middle, right), computed from the full analytical solution. Dash-dotted red lines represent the weak lensing approximation, and brown dashed lines indicate the GO limit, which is independent of $\omega$. As expected, $|F(\omega, y)|$ approaches the GO limit at large $\omega$ and $y$. Bottom: Dash-dotted blue lines show the absolute difference between the analytical calculation and the weak lensing approximation ($|\text{Analyt} - \text{WL}|$), representing the approximation error. For comparison, dotted orange lines show the absolute difference between the analytical result and the GO limit ($|\text{Analyt} - \text{GO}|$), characterizing the WO term amplitude. The weak lensing approximation shows excellent agreement with the analytical results, with errors consistently below 5% of the WO term amplitude for $y > 10$.
• Figure 2: Comparison between the weak lensing approximation and numerical results for the NFW lens model with $\kappa_s = 0.5$. Top: Similar to Figure \ref{['fig:SIS_all']}, but the solid blue lines show the amplification factor magnitude, $|F(\omega, y)|$, from numerical calculations performed with GRAVELAMPS2022ApJ...935...68W for $\omega = 0.8$, 1.0, and 1.2 (left, middle, right). Bottom: Same as the top panel, but all analytical results are replaced with numerical calculations from GRAVELAMPS, which serve as the benchmark for $|F(\omega, y)|$. For the NFW model, the WO features ($|\text{Num} - \text{GO}|$) at a given source position $y$ are less pronounced compared to the SIS model, reflecting its lower compactness. The weak lensing approximation remains accurate, with errors ($|\text{Num} - \text{WL}|$) limited to a few percent of the WO term amplitude for $y > 10$.
• Figure 3: Comparison between the QGO and weak lensing approximations for the normalization function $f(\omega y)$ of the WO term for the NFW model with $\kappa_s = 0.5$ and $\omega = 3$. The red dash-dotted line shows the WO term amplitude from the weak lensing approximation, while the brown dashed line shows the QGO approximation. For comparison, the WO term extracted from numerical calculations using GRAVELAMPS is shown as a blue solid line. The numerical results indicate that the QGO approximation performs better at small $y$, while the weak lensing approximation is more accurate at larger $y$, consistent with theoretical expectations.
• Figure 4: Comparison between the weak lensing approximation, QGO approximation, and numerical results for the NFW lens model with $\kappa_s = 0.5$ and $\omega = 0.1$. The layout of the top and bottom panels follows Figure \ref{['fig:NFW_all']}, but here the brown dashed line shows the amplification factor magnitude, $|F(\omega, y)|$, from the QGO approximation 2004AA...423..787T. An inset in the top panel highlights the range $4 \leq y \leq 9$. The weak lensing approximation outperforms the QGO approximation at low $\omega$, particularly at small $y$, and provides an accurate description once $\omega y^2/2 \gtrsim 10$.