# Quiescent and Active Galactic Nuclei as Factories of Merging Compact Objects in the Era of Gravitational Wave Astronomy

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## Abstract

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## 1. Introduction

## 2. Observational Evidence of Binaries in Galactic Nuclei: The Milky Way Test Case

## 3. Environmental Effects on Binary Formation in Galactic Nuclei

#### 3.1. Binaries in Galactic Nuclei: Primordial, Dynamical, or Hybrid?

#### 3.2. Nuclear Cluster Formation Processes: In Situ Versus Dry Merger

#### The Impact of Nuclear Cluster Formation Scenarios on the Population of Compact Objects in Galactic Nuclei

## 4. Early Black Hole Dynamics in Nuclear Clusters and Galactic Nuclei

`B-POP`population synthesis code [99], which combines stellar evolution models for single and binary BHs obtained with the

`MOBSE`tool [280] and semi-analytic recipes to describe the motion and pairing of BHs via dynamics. Using

`B-POP`, we considered an NC with mass ${M}_{\mathrm{NC}}=2.5\times {10}^{7}\text{}{\mathrm{M}}_{\odot}$ and assumed that a fraction ∼${10}^{-3}$ of such mass consists of BH progenitors, assuming that the underlying mass distribution follows a standard initial mass function [255]. For each BH progenitor in the NC, we retrieved the final BH mass, the lifetime, and the SN kick. We divided the time into logarithmic bins and calculated the BH progenitor position via Equation (9). As soon as the time exceeds the i-th progenitor lifetime, we turned it into a BH (assuming a metallicity $Z=0.0002$) and assigned a natal kick amplitude ${v}_{\mathrm{kick}}$, which was based on the stellar evolution recipes implemented in the

`MOBSE`population synthesis tool. Given the kick, we assumed that the newborn BH will reach a maximum distance ${r}_{\mathrm{max}}$ in a travel time ${t}_{\mathrm{tr}}={r}_{\mathrm{max}}/{v}_{\mathrm{kick}}$, and then, it comes back over a dynamical friction time. In order to simplify the visualisation of such a complex system, we present in the sketch in the left panel of Figure 4a a cartoon showing the evolution of BH progenitor stars. More quantitatively, we show in the right panel of the same figure the time evolution of the radii containing the $10\%,25\%,50\%,75\%$—referred to as Lagrangian radii—of BH mass in this simple toy model. The SN kick effect is rather small, owing to the relatively small kick amplitude compared to the NC velocity dispersion, suggesting that, in an MW-like nucleus, mass segregation is practically accomplished over a timespan of ∼${10}^{7\text{\u2013}8}$ yr.

## 5. Dynamical Formation of Black Hole and Compact Object Binaries in Galactic Nuclei

#### 5.1. Galactic Threats: What Binaries Can Survive around a Supermassive Black Hole?

_{⊙}, double-NSs can, instead, evaporate over a timescale shorter than the Hubble time, taking only ${t}_{\mathrm{ev}}=2\text{\u2013}4$ Gyr in the aforementioned example.

#### 5.2. Moving through a Swarm: Orbital Evolution of Compact Binaries in Galactic Nuclei

#### 5.3. Multiple Encounters Make Bound Pairs: How Dynamical Processes Aid Binary Formation in Galactic Nuclei

#### 5.3.1. GW Capture Binary Formation

#### 5.3.2. Three-Body Binary Formation

#### 5.3.3. Binary–Single Scatterings

- If $\Delta {E}_{\mathrm{bs}}<{E}_{\mathrm{bin}}$, the binary will harden or soften depending on the environment;
- If $\Delta {E}_{\mathrm{bs}}>{E}_{\mathrm{bin}}$, the binary will exchange one component, most likely the least-massive one, if the perturber is heavier than the binary or its components, i.e., ${m}_{\mathrm{p}}>({m}_{1}+{m}_{2})$ or at least ${m}_{\mathrm{p}}>{m}_{1,2}$.

- If ${v}_{\infty}<{v}_{c}$, the perturber cannot recede to infinity and the three bodies undergo a resonant interaction that can culminate in the exchange of one binary component if:
- –
- $\Delta {E}_{\mathrm{bs}}>{E}_{\mathrm{bin}}$;
- –
- $\Delta {E}_{\mathrm{bs}}<{E}_{\mathrm{bin}}$ and ${m}_{p}>{m}_{\mathrm{bin}}$.
- Either way, the perturber or the exchanged component recedes to infinity and possibly leaves the host system;

- If ${v}_{\infty}<{v}_{c}$, $\Delta {E}_{\mathrm{bs}}<{E}_{\mathrm{bin}}$, and ${m}_{p}<{m}_{\mathrm{bin}}$, the system undergoes a resonant interaction, which generally leads to the ejection of the lighter component;
- If ${v}_{\infty}>{v}_{c}$, the binary undergoes:
- –
- Component exchange if $\Delta {E}_{\mathrm{bs}}<{E}_{\mathrm{bin}}$;
- –
- Ionisation if $\Delta {E}_{\mathrm{bs}}>{E}_{\mathrm{bin}}$.

## 6. Secular Dynamical Effects on Binary Evolution around a Supermassive Black Hole

#### 6.1. Secular Perturbations on Black Hole Binaries in Galactic Nuclei: The Impact of a Supermassive Black Hole

**Three-Body Newtonian Limit**

#### 6.2. Other Physical Processes

**Tidal dissipation**

**Stellar evolution**

`BSE/SSE`[154] and

`MESA`[369], which are publicly available. Furthermore, once the binaries cross each other’s Roche limit, the binary stellar evolution is often followed using

`COSMIC`binary stellar evolutionary code [370].

**General relativity precession, first pN**

**Spin effects, 1.5pN**

**GW, 2.5pN**

**Resonant relaxation processes**

- Resonant relaxation

- Vector resonant relaxation

## 6.3. Initial Conditions and Unknowns

## 7. Dynamics of Black Hole Binaries in Active Galactic Nuclei

## 8. Black Hole and Neutron Star Mergers around Supermassive Black Holes: Implications for Current and Future Gravitational Wave Detections

#### 8.1. Population Properties: Masses, Mass Ratio, Spins, and Eccentricity

#### 8.1.1. Gravitational Scatterings

#### 8.1.2. Eccentric Kozai–Lidov Mechanism

#### 8.1.3. Active Galactic Nuclei

#### 8.2. Expected Merger Rate and Prediction

#### 8.3. Imprint of Galactic Nuclei on the Gravitational Wave Emission from Merging Compact Objects

#### 8.3.1. Eccentricity Variation Encoded in the Gravitational Wave Signal of Merging Compact Objects

#### 8.3.2. Supermassive Black Hole Acceleration Encoded in the Gravitational Wave Signal of Merging Compact Objects

## 9. Summary

- The dynamics plays a crucial role in determining COB formation in galactic nuclei. Three-body scatterings, involving three initially unbound objects, are likely dominant in galaxies with a large NC-to-SMBH mass ratio, but become extremely inefficient close to the SMBH. Conversely, single–single interactions that form bound pairs via GW bremsstrahlung—or GW captures—are more efficient in the SMBH’s immediate vicinity and in the nuclei with the most-massive SMBHs. However, GW captures produce short-lived binaries that merge within days or hours from their formation and have a large chance of being highly eccentric when sweeping through high-frequency detectors.
- Galaxies dominated by a quiescent SMBH can efficiently replenish their population of COBs—particularly BHs—via the accretion of massive star clusters that undergo inward migration owing to dynamical friction.
- A substantial population of primordial binaries can also play a crucial role in determining the properties of COBs in galactic nuclei, although most of them are likely destroyed by the SMBH’s tidal field.
- Once binaries start forming in galactic nuclei, their further evolution is regulated mostly by binary–single interactions, which generally promote the formation of tighter and more massive binaries, but, depending on the binary properties, can lead to their evaporation well before GW emission starts dominating the binary evolution.
- Owing to dynamical friction, or mass segregation, and dynamical interactions, COBs are expected to move through regions of the nucleus with different velocity dispersions and densities. The variation of the environment structure can dramatically affect the COB’s fate: an initially hard binary moving inward can appear soft closer to the SMBH and eventually be disrupted by interactions with other stars and COs.
- Around 20–70% of COBs formed in galactic nuclei are expected to suffer the effect of the SMBH’s gravitational field, which can cause periodic oscillations of their eccentricity called eccentric Kozai–Lidov resonances. This mechanism can significantly shorten the COB’s lifetime, possibly affecting the delay time of merging COs. The development of EKL oscillations strongly depends on the binary properties (e.g., general relativistic precession can suppress EKL), the distance to the SMBH, and the eccentricity of the COB’s orbit about the SMBH.
- In AGNs, the formation of COBs is favoured by both gaseous torques and dynamical scatterings, whose efficiency is boosted by the nearly planar configuration. The possible existence of migration traps, where inward and outward torques cancel out, makes AGNs potential factories of multiple-generation COs mergers and IMBHs.
- Mergers occurring in galactic nuclei feature some peculiar traits: a significant fraction of mergers with one component in the upper mass-gap, a non-negligible fraction of multiple-generation mergers that can affect the high -end of the BH mass distribution, and fairly misaligned spins; although, in AGNs, a noticeable fraction of high-generation mergers might have mildly aligned spins and a quite significant probability to preserve an eccentricity $e>0.1$ whilst sweeping through the frequency bands of both low- and high-frequency detectors.
- The merger rate inferred for present-day GW detectors for BBH and NS–BH binary mergers in galactic nuclei is poorly constrained owing to the many model uncertainties. For BBH mergers, models for quiescent SMBHs and AGNs predict similar estimates, which generally fall in the range ${\mathcal{R}}_{\mathrm{BBH}}={10}^{-3}-{10}^{2}\text{}{\mathrm{yr}}^{-1}{\mathrm{Gpc}}^{-3}$. For NS–BH mergers, instead, there are clear differences between the prediction for quiescent, ${\mathcal{R}}_{\mathrm{BBH}}={10}^{-5}-1\text{}{\mathrm{yr}}^{-1}{\mathrm{Gpc}}^{-3}$, and active nuclei models, ${\mathcal{R}}_{\mathrm{BBH}}=1-{10}^{3}\text{}{\mathrm{yr}}^{-1}{\mathrm{Gpc}}^{-3}$, partly owing to the relatively poor literature and the huge uncertainties.
- The presence of an SMBH in the vicinity of a merging COB can leave some imprints on the emitted GW signal that could, in principle, be detected with future detectors, among others a shift in the peak frequency for mergers occurring in the Milky Way centre, a variation in the measured redshift induced by the rapid motion of the binary around the SMBH, and the development of a GW echo produced by the scattering of the emitted GWs onto the SMBH.

**Initial conditions**: Probably the most-important unknown that mostly affects all the models is the scarce knowledge of how stars form and pair in the extreme environment of a galactic nucleus. The initial binary fraction, the initial distribution of periods and masses, and the metallicity spread in the galactic nucleus are all factors that crucially determine the COB’s properties: semimajor axis, eccentricity, and component masses.**Interplay of mechanisms**: As we have seen throughout the review, COB formation is likely regulated by many mechanisms likely operating simultaneously. However, most theoretical models focus on one specific aspect at a time. Fully self-consistent N-body simulations capable of taking into account the stellar evolution of single and binary stars, the SMBH tidal field, and potentially, the effect of an AGN disc exist, but their resolution is still too low and their computational cost too large to permit a one-to-one representation of a Milky Way-like nucleus. Simpler models relying on semi-analytic assumptions or few-body (scattering) simulations represent valid alternatives, although they sometimes neglect potentially crucial elements, such as the importance of flybys on the evolution of COBs undergoing EKL oscillations, the development of EKL resonances in binaries formed in AGNs, and the role of star formation in the actual population of COBs around an SMBH.**Observations**: From an observational perspective, the observation of young massive binaries in galactic nuclei, a larger number of GW detections, a more precise localisation of GW sources, and the future detection of inspiralling binaries in the Milky Way centre can definitely help us improve our knowledge of the processes regulating COB formation in galactic nuclei.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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