Probing the Formation Environment of Strongly Lensed Black Hole Mergers: Implications for the AGN-disk Channel

TL;DR

This work develops a method to constrain the formation environment of strongly lensed binary black hole mergers by measuring both the transverse proper motion and the line-of-sight acceleration through GW phase shifts between multiple lensed images and within individual images. The approach leverages the relations $v_T^2 ∝ M/R$ and $a_L ∝ M/R^2$ for a BBH orbiting a central mass $M$ at radius $R$, enabling joint inference of $M$ and $R$ when both Doppler-induced phase shifts are detectable. A key finding is that BBH mergers in AGN disks occupy a unique region of parameter space where both φ_vel and φ_acc are observable with next-generation detectors, and that additional constraints from image localization and polarization can help break geometric degeneracies via the mass proxy $M' = M imes B(θ,i)$. The study argues that upcoming lensed GW catalogs, especially with 3G detectors and EM counterparts, will open new avenues to probe BBH origins and the physics of AGN disks, albeit with challenges in identifying and associating lensed image sets.

Abstract

The observation of multiple images from a strongly lensed gravitational wave (GW) source provides the observer with a stereoscopic view of the source. This allows for a measure of its relative proper motion by comparing the induced GW Doppler shifts between the different images. In addition, if the GW source is in a dynamical environment it will be subject to an acceleration, which will show up as a time dependent Doppler shift in each individual image. In this work we quantify for the first time how a joint detection of these effects can be used to constrain the underlying dynamics and environment of the lensed GW source. We consider a range of different astrophysical environments, from massive clusters to stellar triples, and find that binary black hole (BBH) mergers in Active Galactic Nuclei disks (AGN-disks) are particularly likely to have orbital parameters that can be constrained through our considered lensing setup. Applying these methods to the upcoming catalog of cosmologically strongly lensed GW sources will open up new possibilities for probing their origin and underlying formation mechanisms.

Figures (4)

• Figure 1: Lensing of a GW source by a galaxy cluster. The figure illustrates how two or more paths deflected by an in-between over-density, here a galaxy cluster, gives the observer several sight-lines towards the source. These essentially provides a stereoscopic view, with a corresponding baseline $\approx \Theta D_{s}$, that allows additional information about the GW source to be extracted. For example, if the GW source is moving relative to the lens and observer, the two illustrated sight-lines will have different Doppler shifts, as further clarified by the two fictitious observers, Obs. A and Obs. B. More than 2 lensed images are expected for a significant fraction of lensed GW sources, which allows for a unique measure of the transverse proper motion of the GW source - a property that usually is not measurable.
• Figure 2: GW phase shift from strongly lensed GW sources and the astrophysical landscape. The figure shows the induced GW phase shift from a lensed GW source with transverse source velocity, $v_{T}$ ($\phi_{\rm vel}$) and LOS acceleration, $a_T$ ($\phi_{\rm acc}$), in blue and red contours, respectively, as a function of $R/R_s$ (bottom $x$-axis), where $R$ is the distance to the center of the enclosed pertuber mass $M$ (y-axis), and $R_s = 2GM/c^2$. The solid blue and red regions are where $\phi_{\rm vel}$ and $\phi_{\rm acc}$ are both $>0.1$. The top axis shows the orbital velocity in units of $c$. All values are derived for our fiducial model with $m=10 M_{\odot}$, $f=5\ Hz$, and $\Theta = 25"$ as further outlined in Sec. \ref{['sec:Astrophysical Implications']}. On the figure is further included the following astrophysical systems (see Table \ref{['table:astro_sys']}): Globular cluster ( Glob.C.), Nuclear Star Cluster ( NS. C.), Milky Way Galaxy ( MW. Gal.), Galaxy Cluster ( Gal. C.), AGN-disk migration trap mergers ( AGN), and a 3-body interaction that is typical for a GC like system ( 3-Body (GC)). The orange shaded region denotes where the tidal force on the BBH evaluated at $f=5\ Hz$ is $> 0.01$ of the binary binding force, where the the green shaded area shows the region where the inspiral time of the BBH is $>0.01$ of the outer orbital time. As seen, for our considered cases, the AGN environment is the only BBH formation channel that has the potential to result in observable GW phase shifts related to both the BBH velocity and the acceleration. This allows in particular to solve for $M$ and $R$ individually, as $v^2_{T} \propto {M/R}$, and $a_L \propto M/R^2$, as further described in Sec. \ref{['sec:Astrophysical Implications']} and Sec. \ref{['sec:AGN disk Mediated Mergers']}.
• Figure 3: Strong lensing of a GW source in an AGN-disk. The figure shows an illustration of a $4$ image lens configuration (labels $1-4$), with the observer to the left ( Observer), lens-plane in the middle ( Lens Plane), and the source plane to the right ( Source Plane). The GW source is here a BBH merger ( BBH) located in an AGN disk, that is tilted by an angle $i$ w.r.t. a vector perpendicular to the LOS. The BBH is located at the time of merger at an angle $\theta$ as illustrated. It is assumed that the BBH orbits the central SMBH in a circular orbit at distance $R$, with a velocity vector here shown by the green arrow. The BBH is further assumed to be aligned with the AGN disk, such that its angular momentum vector, shown by the magenta arrow, is aligned with the AGN disk angular momentum vector. The projected, or transverse, vector components are shown in the lens-plane. At the bottom ( Aligned GW Data) is illustrated how the lensed GW signals will be Doppler shifted relative to each other as the GW source is moving around the SMBH. By comparing each of these, one can infer the projected vector components, from which the BBH environment can be constrained, as further described in Sec. \ref{['sec:AGN disk Mediated Mergers']}.
• Figure 4: Mass bias and geometric factor $\mathcal{B}$: The figure shows the factor $M'/M = \mathcal{B}(\theta, i)$, given by Eq. \ref{['eq:factorB']}, as a function of $\theta$ for a few given values of the orbital inclination angle $i$ (see Fig. \ref{['fig:lens_disk_setup']}). The angle $i$ can for some systems be inferred observationally, e.g. if the BBH is an AGN-disk (Sec. \ref{['sec:AGN disk Mediated Mergers']}). However, the angle $\theta$ is generally unknown, which leads to either upper or lower bounds on the true mass $M$ related to the measurable mass, $M'$. If the GW source and lens have been localized on the sky, one can ideally further infer $\theta$ through the geometric relation given by Eq. \ref{['eq:sinbeta2']}.