A universal connection between lens density profiles and low-frequency wave optics in gravitational-wave lensing
Abstract
We investigate the low-frequency behavior of the amplification factor in gravitational lensing and explore how it encodes information about the density profile of the lensing object. We derive the low-frequency expansion of the amplification factor under the Born approximation for a broad class of projected density profiles. For spherically symmetric profiles that decay faster than any power law at large distances, we derive a systematic expansion of the amplification factor in powers of frequency, with the logarithmic dependence appearing only in the leading term, and show that each expansion coefficient is determined by a finite set of moments of the density profile. We then extend the analysis to profiles with power-law tails and demonstrate that such profiles induce additional non-analytic frequency dependences, including fractional powers and logarithmic terms, which directly reflect the asymptotic behavior of the density distribution. Furthermore, we investigate the effects of non-sphericity and show that contributions from the quadrupole moment appear only as higher-order corrections relative to the spherically symmetric component in the low-frequency regime. Finally, we investigate the validity of the Born approximation in the low-frequency expansion. We derive a criterion for the maximum order of the low-frequency expansion up to which the Born approximation remains dominant over the post-Born corrections.
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BibTeX
@misc{tanaka2026universal,
title = {{A universal connection between lens density profiles and low-frequency wave optics in gravitational-wave lensing}},
author = {So Tanaka and Teruaki Suyama},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09704},
eprint = {2609.09704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09704},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A universal connection between lens density profiles and low-frequency wave optics in gravitational-wave lensing
AU - So Tanaka
AU - Teruaki Suyama
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.09704
UR - https://arxiv.org/abs/2609.09704
AB - We investigate the low-frequency behavior of the amplification factor in gravitational lensing and explore how it encodes information about the density profile of the lensing object. We derive the low-frequency expansion of the amplification factor under the Born approximation for a broad class of projected density profiles. For spherically symmetric profiles that decay faster than any power law at large distances, we derive a systematic expansion of the amplification factor in powers of frequency, with the logarithmic dependence appearing only in the leading term, and show that each expansion coefficient is determined by a finite set of moments of the density profile. We then extend the analysis to profiles with power-law tails and demonstrate that such profiles induce additional non-analytic frequency dependences, including fractional powers and logarithmic terms, which directly reflect the asymptotic behavior of the density distribution. Furthermore, we investigate the effects of non-sphericity and show that contributions from the quadrupole moment appear only as higher-order corrections relative to the spherically symmetric component in the low-frequency regime. Finally, we investigate the validity of the Born approximation in the low-frequency expansion. We derive a criterion for the maximum order of the low-frequency expansion up to which the Born approximation remains dominant over the post-Born corrections.
ER -