Constraining binary mergers in AGN disks using the non-observation of lensed gravitational waves

TL;DR

The paper investigates whether BBHs merging in AGN disks can be probed via lensing by the central SMBH. It derives the lensing probability for BBHs at radius r as $f_{lensing|AGN} \approx \frac{1}{2 r}$ after marginalizing over inclination and uses a Bayesian upper bound on $f_{AGN}$ from non-detection. It quantifies how many detections are required to rule out migration-trap–like disk regions and to bound disk sizes, while highlighting demagnification effects and model limitations of thin disks. The study provides a practical link between GW lensing and BBH formation channels in AGN environments, enabling constraints with current LVK data and powerful tests with future detectors.

Abstract

The dense and dynamic environments within active galactic nuclei (AGN) accretion disks may serve as prolific birthplaces for binary black holes (BBHs) and one possible origin for some of the BBHs detected by gravitational-wave (GW) observatories. We show that a considerable fraction of the BBH in AGN disks will be strongly lensed by the central supermassive black hole (SMBH). Thus, the non-observation of lensed GW signals can be used to constrain the fraction of BBH binaries residing in AGN disks. The non-detection of lensing with current ${\cal O}(100)$ detections will be sufficient to start placing constraints on the fraction of BBHs living within accretion disks near the SMBH. In the next-generation detectors era, with ${\cal O}(10^5)$ BBH observations and no lensed events, we will be able to rule out most migration traps as dominant birthplaces of BBH mergers; moreover, we will be able to constrain the minimum size of the accretion disk. On the other hand, should AGNs constitute a major formation channel, lensed events from AGNs will become prominent in the future.

Figures (4)

• Figure 1: Schematic diagram of the AGN-BBH system. Diagram showing a BBH (left) orbiting an SMBH (center). The grey ellipse is the orbital plane of the BBH around the SMBH, defined by the orbital angular momentum $\vu*L$. The accretion disk is modelled as a thin disk, coplanar with the orbital plane. The observer is located at $\va*N$ and represented by the green vector; its continuation below the plane is the green dashed line, and they form the optical axis. The inclination angle is $\iota$, and the projection of $\va*N$ onto the orbital plane defines the $x$-axis, from which the azimuthal angle of the BBH, $\phi_S$, is measured. The angles subtended by the source and the SMBH, and the source and the observer are denoted by $\gamma$, and $\beta$, respectively.
• Figure 2: BBH lensing probability as a function of $R_{\rm orb}$. The dashed blue and grey lines represent the lensing probability (Eq. \ref{['eq:conditional_prob']}) at fixed inclinations $\iota = \pi/2$ and $9 \pi / 20$, respectively. The former follows the $1 / \sqrt{r}$ trend, and the latter has a finite radius cut-off, at around $80\,R_{\rm Sch}$. Shown in red is the marginalised probability over inclination (Eq. \ref{['eq:marg_prob']}, or \ref{['eq:exact_f_lensing_AGN']}), which matches very closely $1 / (2 r)$ (black dash-dotted line, Eq. \ref{['eq:prob_approx']}), except for at small radius (for $r \lesssim 10$). Vertical dotted lines indicate the locations of some migration traps mentioned in the literature ( e.g.Bellovary2016:MigrationPeng2021:LastMigrationThompson2005:AGNdiskSirko2003:AGNdisk), where a majority of binary black holes are expected to migrate to and merge.
• Figure 3: Constraints on the fraction of AGN-BBH events based on the non-detection of lensed events. The left and right panels show the 90th percentile upper bound of $f_{\rm AGN}$ as a function of the number observed BBH events $N_{\rm obs}$ and the radial distance $r$ from the SMBH, respectively. The hatched area represents the $f_{\rm AGN}$ that is ruled out by the non-detection of lensed events. In the left panel, each curve corresponds to a different fixed radial distribution of BBH around the SMBH; in particular, two grey and blue curves (hatched from the top-right and the top-left, and small circles) represent BBHs from fixed radii $r =$ 10, 245 and 1000, respectively. The orange (circles) curve accounts for the whole disk case by assuming the number of BBHs grows linearly with the radius ( i.e. BBHs distributed uniformly on the disk, with $2 \leqslant r \leqslant 10^4$). The right panel shows the disk inner region that will be excluded as the dominant BBHs production site, for three different numbers of observed events, as well as for the number of mergers growing linearly at $N_{\rm obs}=100$ (orange, circles). Note that in the last case, the $x$-axis represents the maximum disk radius, $2 \leqslant r \leqslant r_{\rm max}$, with the same meaning as in Fig. \ref{['fig:f_AGN_disk_constraint']}. It is worth noting that, following from the definition Eq. \ref{['eq:fraction']}, $f_{\rm AGN}^{{\rm upper}, 90\%}$ is 0.9 for regions with no constraint. Finally, the vertical dotted lines denote the trap locations reported in Bellovary2016:Migration and Peng2021:LastMigration, at $r \in \{7,\ 24.5,\ 245,\ 331\}$ respectively.
• Figure 4: Constraints on the fraction of AGN-BBH events when they are distributed over the entire disk. The 90th percentile upper bound of $f_{\rm AGN}$ is shown as a function of the maximum AGN disk radius, assuming four disk models and with $10^5$ observations, in the upper panel. The hatched area represents the $f_{\rm AGN}$ that is ruled out by the non-detection of lensed events. The orange curve denotes the same simple disk model as in Fig. \ref{['fig:f_AGN_constraint']}. The blue curve assumes the BBHs follow the radial distribution given by Eq. (19) of grobner_binary_2020. The last two, dark and light grey, have their radial distributions constructed from the density and height profiles from Sirko2003:AGNdisk (SG) and Thompson2005:AGNdisk (TQM), respectively. In the bottom panel, we show the several radial distributions we employed, in log-log scale, normalised to $r_{\rm max} = 10^5$, following the same color codes as above.