Constraining the environment of compact binary mergers with self-lensing signatures

TL;DR

This work develops a formalism to quantify GW self-lensing probabilities and imprints for BBH mergers across star clusters, nuclear clusters, and AGN disks. By computing optical depths $\tau$ for several environments and detailing wave-optics and strong-lensing imprints, the authors show that AGN disks yield the highest self-lensing probabilities ($\tau\sim0.02$), while star clusters predict far smaller rates ($\tau\lesssim10^{-5}$). They identify distinctive environmental signatures, notably a linear polarization $h_+$ in edge-on AGN-disk self-lensing and interference or multiple images in other cases, and argue that combining lensing imprints with eccentricity and polarization can constrain BBH formation channels. The findings suggest that next-generation detectors could observe a handful of self-lensed events, offering a powerful probe of compact-binary environments and dynamics near central BHs or within AGN disks.

Abstract

Gravitational waves (GWs) from coalescing binary black holes (BBHs) can come from different environments. GWs interact gravitationally with astrophysical objects, which makes it possible to use gravitational lensing by objects in the same gravitational system (self-lensing) to learn about their environments. We quantify the probability of self-lensing through the optical depth $τ$ for the main channels of detectable GWs at frequencies $f_{\rm GW}\sim (1-10^3)\,{\rm Hz}$. We then analyze the detectability of the lensing effect (imprint). In star clusters, the probability of self-lensing by stellar-mass black holes (BHs) is low, $τ\simeq10^{-7}$, even when taking into account nearby BHs in resonant interactions, $τ\simeq 10^{-5}$. Additionally, the lensing imprint of a stellar-mass lens (diffraction and interference) is too marginal to be detectable by the LIGO-Virgo-KAGRA detectors and most Einstein Telescope signals. For a massive BH lens in the center of a cluster, the probability can reach $τ\simeq 10^{-4}$ either via von Zeipel-Lidov-Kozai induced mergers of BBHs orbiting a central massive BH, or BBHs formed as GW captures in single-single interactions in the Bahcall-Wolf cusp of a nuclear cluster. For self-lensing by a supermassive BH for BBHs in the migration trap of an active galactic nucleus (AGN) disk, $τ\simeq 10^{-2}$. The imprint of these massive lenses are multiple images that are already detectable. Moreover, self-lensed signals from AGN disks have a distinct linear polarization. The probability depends on the extent of the detectability through the threshold impact parameter $y_{\rm max}$, which can increase for future detectors. We conclude that constraining the environment of BBHs is possible by combining self-lensing imprints with other waveform signatures such as eccentricity and polarization.

Figures (14)

• Figure 1: Gravitational lensing diagram to illustrate our notation. The source S is displaced from the observer-lens line-of-sight (observer at O, lens at L) by an impact parameter $\vec{\eta}=R_{\rm E}\, \vec{y}$ (Eq. \ref{['eq:y_def']}). The lens cross section $\Sigma_{\rm L}$ is shown as the grey circle of radius $R_{\rm E} \,y_{\rm max}$ at the lens plane. When the source is inside the empty dashed circle in the source plane, the GW signal has detectable lensing effects.
• Figure 2: Self-lensing optical depth $\tau$ of a BBH merger lensed by a third object, as a function of the mass of the lens $m_{\rm L}$ and the separation between the lens and the source $d_{\rm LS}$, following Eq. \ref{['eq:individual-prob-units']}. We consider the LVK sensitivity, where $y_{\rm max}\sim 1$. The Schwarzschild radius of the lens, $R_{\rm L}$, is shown as a dashed red line.
• Figure 3: Self-lensing probability $\tau$ as a function of $\sigma$, the star cluster velocity dispersion. The optical depth for self-lensing by a member of the star cluster (A), $\bar{\tau}_{\rm det}$ (Eq. \ref{['eq:optical_depth_general_plummer_BH']}), is shown as solid lines, assuming $f_{\rm L}=0.05$, $f_{r_0}=0.1$. Self-lensing of resonant interactions leading to a GW capture (B), with optical depth $\tau_{\rm res}$ (Eq. \ref{['eq:prob-resonant']}), is shown as dashed lines. Black (blue) lines correspond to LVK (ET), where we take respectively $y_{\rm max}\simeq1.5$ and $y_{\rm max}\simeq2.5$ (as detailed in Sec. \ref{['sec:detectability']}) for $m_{\rm S}=10\,{\rm M}_\odot+10\,{\rm M}_\odot$. The dotted vertical lines show a representative value of $\sigma$ for globular clusters (GC) and nuclear star clusters (NC).
• Figure 4: Diagram of a BBH-single BH interaction leading to a GW capture. A BBH (above) and a single BH (below) encounter each other, interact chaotically (resonant encounters), interchanging its components. We show the outcome we are interested in: the formation of a new BBH that merges quickly (left, S) close to the remaining single BH (right, L). This single BH can act as a lens. Not to scale: in reality, $d_{\rm LS}\sim a_0$, the initial semi-major axis of the binary.
• Figure 5: Diagram of a hyperbolic (single-single) encounter leading to a GW capture between two stellar-mass BHs close to a massive BH. The interacting BHs can end up merging and being lensed by the massive BH. Not to scale.