arXiv subjects

Alessandro A. Trani

Publications and source records attributed to Alessandro A. Trani.

At least 19 recordsLinked to original sources

arXiv2026

Large Gravitational Wave Phase Shifts from Strong 3-body Interactions in Dense Stellar Clusters

The phase evolution of gravitational waves (GWs) can be modulated by the astrophysical environment surrounding the source, which provides a probe for the origin of individual binary black holes (BBHs) using GWs alone. We here study the evolving phase of the GW waveform derived from a large set of simulations of BBH mergers forming in dense stellar clusters through binary-single interactions. We uncover that a well-defined fraction of the assembled eccentric GW sources will have a notable GW phase shift induced by the remaining third object. The magnitude of the GW phase shift often exceeds conservative analytical estimates due to strong 3-body interactions, which occasionally results in GW sources with clearly shifted and perturbed GW waveforms. This opens up promising opportunities for current and future GW detectors, as observing such a phase shift can identify the formation environment of a BBH, as well as help to characterise the local properties of its surrounding environment.

astro-ph.HE↗

arXiv2026

The impact of envelope binding energies on the merger rate density of binary compact objects

The common envelope (CE) phase plays a key role in the formation of binary compact object systems. Its final outcome strongly depends on the envelope binding energy, but this quantity is often estimated using fitting formulas that are not fully consistent with the underlying stellar evolution models adopted in population-synthesis codes. Here, we investigate envelope binding energies across the most extensive stellar grid considered to date. Our stellar tracks, evolved with PARSEC v2.0, include hydrogen (H) -rich stars with metallicities ranging from $Z = 10^{-11}$ (Population III stars) to $Z = 0.03$, and initial masses between 2 and 2000 M$_\odot$, as well as pure-helium stars with masses from 0.36 to 350 M$_\odot$. We examine the sensitivity of the envelope binding energies to the selected core-envelope boundary definition and to different internal energy source contributions. For H-rich stars, we find that internal energy sources can alter the envelope binding energy by more than an order of magnitude, whereas the core boundary criteria play a secondary role. In contrast, for pure helium stars, the core-boundary criterion becomes the dominant factor. The envelope binding energies derived from different stellar tracks can show deviations of several orders of magnitude, with larger differences for more massive stars and higher metallicities.Finally, by implementing our new envelope binding energy prescriptions into the binary population synthesis code SEVN, we show that the predicted merger rate densities of compact binaries can differ by more than an order of magnitude compared to previous models. Our results highlight the importance of using envelope binding energies that are consistent with the underlying stellar evolution models and caution against extrapolating empirical fits beyond the considered parameter space.

astro-ph.SR↗

arXiv2026

A benchmark for binary star interaction with a supermassive black hole in general relativity

Most galaxies have supermassive black holes (SMBH) at their centres, surrounded by stars with binary systems also present in this environment. We use two schemes - post-Newtonian (PN) and a scalar perturbation to a background metric to numerically solve the three-body problem of a binary with a SMBH. We test three different PN formulations for the PN scheme: The Einstein-Infeld-Hoffman equation, pair-wise implementation of two-body PN-terms for three bodies and the Arnowitt-Deser-Misner Hamiltonian. We compare these approaches for one million solar mass and one billion solar mass black holes, and find a statistical match between the two approximations for stellar mass binary interacting with a million solar mass black hole. We also perform a statistical study for encounters with this black hole, and find that the higher order PN formulation matches with metric-with-perturbation scheme. However, we find a decrease in separation of the binary, and eccentricity variations between different schemes around the billion solar mass black hole. This behaviour is not present if binary has a large separation or is further away from the black hole due to decreased general-relativistic effects. We find that the pair-wise PN method results in a decrease in separation at pericentre in all test cases irrespective of the distance from the black hole or mass of the black hole, making this the least reliable method for solving this problem. Our work highlights the need for caution when interpreting the results in different formulations around SMBHs. This also shows that when understanding extreme mass ratio inspirals (EMRIs) using simulations, one should beware as the binary gets closer to the black hole.

astro-ph.IM↗

arXiv2026

Gravitational-Wave Signatures of Highly Eccentric Stellar-Mass Binary Black Holes in Galactic Nuclei

A significant fraction of compact-object mergers in galactic nuclei are expected to be eccentric in the Laser Interferometer Space Antenna (LISA) frequency sensitivity range, $10^{-4} - 10^{-1}\ \rm Hz$. Several compact binaries detected by the LIGO-Virgo-KAGRA Collaboration may retain hints of residual eccentricity at $\sim 10$ Hz, suggesting dynamical or triple origins for a significant fraction of the gravitational-wave-observable population. In triple systems, von-Zeipel-Lidov-Kozai oscillations perturb both the eccentricity and the argument of pericentre, $ω$, of the inner black hole binary. The latter could be fully \textit{circulating}, where $ω$ cycles through $2π$, or may \textit{librate}, with $ω$ ranges about a fixed value with small or large variation. We use \texttt{TSUNAMI}, a regularised N-body code with up to 3.5 post-Newtonian (PN) term corrections, to identify four different families of orbits: (i) circulating, (ii) small and (iii) large amplitude librating, and (iv) merging orbits. We develop and demonstrate a new method to construct gravitational waveforms using the quadrupole formula utilising the instantaneous {\it total} acceleration of each binary component in \texttt{TSUNAMI}. We show that the four orbital families have distinct waveform phenomenologies, enabling them to be distinguished if observed in LISA. The orbits are also distinguishable from an isolated binary or from a binary perturbed by a different tertiary orbit, even if the secular timescale is the same. Future burst timing models will be able to distinguish the different orbital configurations. For efficient binary formation, about $\sim 1000$ binaries can have highly eccentric, librating orbits in the Galactic Centre.

astro-ph.HE↗

arXiv2026

A young progenitor for the most common planetary systems in the Galaxy

The Galaxy's most common known planetary systems have several Earth-to-Neptune-size planets in compact orbits. At small orbital separations, larger planets are less common than their smaller counterparts by an order of magnitude. The young star V1298 Tau hosts one such compact planetary system, albeit with four planets that are uncommonly large (5 to 10 Earth radii). The planets form a chain of near-resonances that result in transit-timing variations of several hours. Here we present a multi-year campaign to characterize this system with transit-timing variations, a method insensitive to the intense magnetic activity of the star. Through targeted observations, we first resolved the previously unknown orbital period of the outermost planet. The full 9-year baseline from these and archival data then enabled robust determination of the masses and orbital parameters for all four planets. We find the planets have low, sub-Neptune masses and nearly circular orbits, implying a dynamically tranquil history. Their low masses and large radii indicate that the inner planets underwent a period of rapid cooling immediately after dispersal of the protoplanetary disk. Still, they are much less dense than mature planets of comparable size. We predict the planets will contract to 1.5-4.0 Earth radii and join the population of super-Earths and sub-Neptunes that nature produces in abundance.

astro-ph.EP↗

arXiv2025

Not all roads lead to merger: AGN disc properties influence the interactions of highly unequal mass black holes

As the number of gravitational-wave detections of black hole binaries grows, so does the diversity of proposed formation channels. The growing sample of systems with highly unequal masses, such as GW190814 with $m_1=23.2\,M_{\odot}$ and $m_2=2.59\,M_{\odot}$ -- corresponding to a mass ratio $q=0.112$ -- cannot be readily explained by isolated binary evolution and may originate through dynamical assembly in an active galactic nucleus (AGN). We investigate AGN discs capable of producing GW190814-like mergers using \texttt{pAGN} to model self-consistent AGN torques, coupled with \texttt{TSUNAMI}, a regularised N-body code including post-Newtonian terms up to 3.5 order. Suites of N-body simulations reveal possible outcomes of binary capture and merger, mean-motion resonance interactions, and other novel dynamical pathways. We develop analytical models linking the branching ratios of captures and mergers to local disc properties, applicable to black hole populations across all mass ratios. Capture probability is primarily governed by $\mathscr{B}$, the ratio of libration time to resonance-width crossing, and is well-described by a log-Gaussian, $P(\rm{capture}|\mathscr{B}) = A \exp[-(\ln \mathscr{B}-μ)^2/2σ^2]$, with $A = 0.41^{+0.04}_{-0.04}$, $μ= 1.09^{+0.08}_{-0.07}$, $σ= 1.05^{+0.08}_{-0.07}$. This fit, while an upper limit, is useful for simplified population synthesis. Finally, we explore the mass ratio AGN luminosity parameter space and find that GW190814 may be formed in a low luminosity AGN of $L_{\rm AGN}\approx 10^{43.5}\ \rm erg\ s^{-1}$. A more systematic parameter space exploration and future population studies will further test our predictions.

astro-ph.HE↗

arXiv2025

Gravitational-Wave Signatures of Highly Eccentric Stellar Binary Black-Holes in Galactic Nuclei

A significant fraction of gravitational-wave mergers are expected to be eccentric in the Laser-Interferometer-Space-Antenna (LISA) frequency band, $10^{-4} - 10^{-1}$ Hz. Several LIGO-Virgo-KAGRA events show potential hints of residual eccentricity at 10 Hz, pointing to dynamical or triple origins for part of the population, where von-Zeipel-Lidov-Kozai oscillations can perturb both the eccentricity and the inclination of the binary. Moreover, the argument of pericentre, $ω$ could be fully circulating, or librating, with a limited range for $ω$. We use TSUNAMI, a regularised N-body code with 3.5PN corrections to identify four different orbital families: (i) circulating, (ii) small-amplitude and (iii) large-amplitude librating, and (iv) merging. We develop a new method to construct gravitational-wave waveforms using the quadrupole formula from the instantaneous acceleration in TSUNAMI. The four orbital families have distinct waveform phenomenologies, enabling them to be distinguished if observed in LISA. In particular, the properties of the tertiary companion can be inferred and serve as an independent mass measurement and distinguish field triple dynamics from galactic dynamics.

astro-ph.HE↗

arXiv2025

Binary stars in the Milky Way nuclear stellar cluster

Intermediate-mass galaxies, including the Milky Way, typically host both a supermassive black hole (SMBH) and a nuclear stellar cluster (NSC). Binaries in an NSC evolve via close encounters with surrounding stars and secular processes related to the SMBH. We study moderately soft and hard binaries ($0.03$-$2.5\,\mathrm{au}$, $M \lesssim 2\,M_\odot$) initially at galactocentric radii 0.1 and 0.3 pc using three-body simulations including von Zeipel-Lidov-Kozai oscillations and tidal dissipation over $\sim 10$ Gyr. Binaries migrate both inward and outward as a consequence of kicks received in the three-body encounters. Inward migration leads to destruction via mergers and evaporation, while outward migration is a pathway to retaining intact binaries for $\gtrsim 10$ Gyr. All surviving binaries are hard and circular, but outcomes for binaries initially at the hard-soft boundary are stochastic. We find that: (i) about $0.3$ percent of evaporated binaries fall into the SMBH's loss cone, (ii) at least $1$ percent of mergers occur late enough to appear as blue straggler stars (BSSs) on the main sequence or as recently evolved red giants, (iii) about $1$ percent of binaries initially at 0.1 pc merge within the inner arcsec of the NSC, and (iv) less than about $80$ percent of field-star collisions with a binary star lead to a subsequent merger; a three-body pile up, which are relatively common in the first 1-2 Gyr and could serve as a way to form more massive BSSs in the NSC. We predict that a small fraction of binaries originate closer to the SMBH than their present-day orbits, and vice versa for evaporated binaries and BSSs. The mergers confined to the inner arcsec occur after $\gtrsim 300$ Myr, too long to be directly related to the formation of the S-stars or G-objects, but suggest that the inner arcsec is contaminated with BSSs from earlier star formation events.

astro-ph.GA↗

arXiv2025

Environmental effects in stellar mass gravitational wave sources I: Expected fraction of signals with significant dephasing in the dynamical and AGN channels

We present the first overview of the expected quantity of signals which will showcase significant gravitational wave phase shifts caused by astrophysical environments, considering the upcoming A+ and A\# LIGO/Virgo/KAGRA, Cosmic Explorer and Einstein Telescope detectors. We construct and analyse two general families of dephasing prescriptions with extensions to eccentric sources, as well as collect five specific prescriptions for the fundamental smoking gun physical mechanisms at play in the dynamical and AGN formation channel for stellar mass binary black holes: Roemer delays, tidal forces and hydrodynamical interactions. We compute the expected fraction of signals containing astrophysical dephasing, as a function of environmental properties and based on observed distributions of binary parameters. We find that next generation detectors can expect to find environmental effects in hundreds of detected signals.

astro-ph.HE↗

arXiv2025

Interactions among binary black holes in star clusters: Eccentric gravitational wave captures and triple formation

Numerical simulations of star clusters with black holes find that there is only a single dynamically active binary black hole (BBH), at odds with the theoretical expectation of ~5 dynamically formed - or, commonly referred to as three-body - BBHs in clusters with a few hundred BHs. We test the recent suggestion that this tension is because interactions among three-body BBHs were neglected in the theory. We use the public catalogue of Cluster Monte Carlo models to obtain a sample of strong BBH-BBH interactions, which we integrate using post-Newtonian equations of motion up to 3.5PN. We explore the nature of the BBHs involved in BBH-BBH interactions in star clusters, as well as the various outcomes: gravitational wave (GW) captures and the associated eccentricities at the frequencies of ground-based GW detectors, as well as BH triple formation and their contribution to BBH mergers via the ZLK mechanism. We find that almost all BBHs involved in BBH-BBH interactions are indeed three-body binaries and that BBH formation and disruption in BBH-BBH interactions occur at approximately the same rate, providing an explanation for the finding of a single dynamically active BBH in N-body models. An important implication is that the resulting rates of GW capture and triple formation are independent of uncertain initial binary properties. With the use of a population synthesis model for BBH-BBH interactions in globular clusters, we obtain a local rate of GW captures of $R=1Gpc^{-3}yr^{-1}$ as well as their eccentricity distribution and redshift dependence. We find that a BBH-BBH interaction is more likely to trigger a GW merger than a BH-BBH interaction. We also confirm that stable triples that are assembled in BBH-BBH interactions can merge via ZLK oscillations, although their merger rate is lower than GW captures. Our results will help with the interpretation of future GW signals from eccentric BBHs

astro-ph.GA↗

arXiv2025

Compact Binary Formation in Open Star Clusters III: Probability of Binary Black Holes Hidden in Gaia Black Hole Binary

The Gaia mission and its follow-up observations have discovered a few candidates of non-interacting single black holes (BHs) and visible stars, Gaia BH1, BH2, and BH3, collectively called ``astrometric BH binaries''. This paper investigates whether any of these candidates harbor binary BHs (BBHs), namely, whether any such candidates are previously undiscovered ``astrimetric BBH triples''. Focusing on open star clusters, which are promising formation sites of astrometric BH binaries, we estimate the formation rate of astrometric BBH triples through gravitational $N$-body simulations. We find a competitively high formation efficiency of astrometric BBH triples ($\sim 10^{-6} {\rm M}_\odot^{-1}$ or $\sim 10$\% of astrometric BH binaries) in low-metallicity environments but no astrometric BBH triples in solar-metallicity environments. Most of the astrometric BBH triples in our simulations were dynamically stable for $10$ Gyrs, indicating that $\sim10$\% of astrometric BH binary candidates may indeed harbor inner BBHs if they originate from open star clusters in low-metallicity environments. Astrometric BBH triples can be distinguished from astrometric BH binaries through radial velocity follow-up of the tertiary star. According to the statistics of our simulated samples, a small percent of astrometric BH binary candidates should exhibit detectable radial-velocity modulations generated by inner BBHs. Such candidates preferentially exhibit ``outer'' orbital periods of $\gtrsim 10^3$ days and moderately high ``outer'' orbital eccentricities ($\gtrsim 0.7$). Our current result will strongly motivate the search for astrometric BBH triples in the upcoming Gaia Data Release 4 and Gaia Final Data Release.

astro-ph.GA↗

arXiv2025

Stability of hierarchical triples comprising a central massive body and a tight binary: the effect of inner and outer eccentricities on the binary breakup condition

We explore the stability of gravitational triple systems comprising a central massive body and a tight binary of less massive pairs. In the present paper, we focus on improving the Hill-type stability criterion for the binary in those systems, with particular attention to the effects of the eccentricities of the inner and outer orbits. We perform direct Newtonian N-body simulations over much longer integration times than previous studies, which is essential to determine the stability and breakup timescale distributions of those systems in a reliable fashion. As a result, we obtain an empirical fitting formula of the binary breakup condition that incorporates effects of the inner and outer eccentricities, the mutual inclination of the inner and outer orbits, the mass ratios of the three bodies, and the breakup timescale.

astro-ph.SR↗

arXiv2025

The formation of mini-AGN disks around IMBHs and their dynamical implications

This study explores the formation and implications of mini-active galactic nuclei (mAGN) disks around intermediate-mass black holes (IMBHs) embedded in gas-rich globular/nuclear clusters (GCs). We examine the parameter space for stable mAGN disks, considering the influence of IMBH mass, disk radius, and gas density on disk stability. The dynamics of stars and black holes within the mAGN disk are modeled, with a focus on gas-induced migration and gas dynamical friction. These dynamical processes can lead to several potentially observable phenomena, including the alignment of stellar orbits into the disk plane, the enhancement of gravitational wave mergers (particularly IMRIs and EMRIs), and the occurrence of mili/centi-tidal disruption events (mTDEs/cTDEs) with unique observational signatures. We find that gas hardening can significantly accelerate the inspiral of binaries within the disk, potentially leading to a frequency shift in the emitted gravitational waves. Additionally, we explore the possibility of forming accreting IMBH systems from captured binaries within the mAGN disk, potentially resulting in the formation of ultraluminous X-ray sources (ULXs). The observational implications of such accreting systems, including X-ray emission, optical signatures, and transient phenomena, are discussed. Furthermore, we investigate the possibility of large-scale jets emanating from gas-embedded IMBHs in GCs. While several caveats and uncertainties exist, our work highlights the potential for mAGN disks to provide unique insights into IMBH demographics, accretion physics, and the dynamics of GCs.

astro-ph.HE↗

arXiv2024

Combining REBOUND and MESA: Dynamical Evolution of Planets Orbiting Interacting Binaries

Although planets have been found orbiting binary systems, whether they can survive binary interactions is debated. While the tightest-orbit binaries should host the most dynamically stable and long-lived circumbinary planetary systems, they are also the systems that are expected to experience mass transfer, common envelope evolution, or stellar mergers. In this study, we explore the effect of stable non-conservative mass transfer on the dynamical evolution of circumbinary planets. We present a new script that seamlessly integrates binary evolution data from the 1D binary stellar evolution code MESA into the N-body simulation code REBOUND. This integration framework enables a comprehensive examination of the dynamical evolution of circumbinary planets orbiting mass-transferring binaries, while simultaneously accounting for the detailed stellar structure evolution. In addition, we introduce a recalibration method to mitigate numerical errors from updates of binary properties during the system's dynamical evolution. We construct a reference binary model in which a $2.21\, M_\odot$ star loses its hydrogen-rich envelope through non-conservative mass transfer to the $1.76\, M_\odot$ companion star, creating a $0.38\,M_\odot$ subdwarf. We find the tightest stable orbital separation for circumbinary planets to be $\simeq 2.5$ times the binary separation after mass transfer. Accounting for tides by using the interior stellar structure, we find that tidal effects become apparent after the rapid mass transfer phase and start to fade away during the latter stage of the slow mass transfer phase. Our research provides a new framework for exploring circumbinary planet dynamics in interacting binary systems.

astro-ph.EP↗

arXiv2024

A novel category of environmental effect in gravitational waves from binaries perturbed by periodic forces

We study the gravitational wave (GW) emission of sources perturbed by periodic dynamical forces which do not cause secular evolution in the orbital elements. We construct a corresponding post-Newtonian waveform model and provide estimates for the detectability of the resulting GW phase perturbations, for both space-based and future ground-based detectors. We validate our results by performing a set of Bayesian parameter recovery experiments with post-Newtonian waveforms. We find that, in stark contrast to the more commonly studied secular dephasing, periodic phase perturbations do not suffer from degeneracies with any of the tested vacuum binary parameters. We discuss the applications of our findings to a range of possible astrophysical scenarios, finding that such periodic perturbations may be detectable for massive black hole binaries embedded in circum-binary discs, extreme mass-ratio inspirals in accretion discs, as well as stellar-mass compact objects perturbed by tidal fields. We argue that modelling conservative sub-orbital dynamics opens up a promising new avenue to detect environmental effects in binary sources of GWs that should be included in state-of-the-art waveform templates.

gr-qc↗

arXiv2024

The boring history of Gaia BH3 from isolated binary evolution

Gaia BH3 is the first observed dormant black hole (BH) with a mass of $\approx{30}$ M$_\odot$ and represents the first confirmation that such massive BHs are associated with metal-poor stars. Here, we explore the isolated binary formation channel for Gaia BH3 focusing on the old and metal-poor stellar population of the Milky Way halo. We use the MIST stellar models and our open-source population synthesis code SEVN to evolve $5.6 \times 10^8$ binaries exploring 20 sets of parameters. We find that systems like Gaia BH3 form preferentially from binaries initially composed of a massive star ($40-60$ M$_\odot$) and a low mass companion ($<1$ M$_\odot$) in a wide ($P>10^3$ days) and eccentric orbit ($e>0.6$). Such progenitor binary stars do not undergo any Roche-lobe overflow episode during their entire evolution, so that the final orbital properties of the BH-star system are determined at the core collapse of the primary star. Low natal kicks ($\lesssim$ 10~km/s) significantly favour the formation of Gaia BH3-like systems, but high velocity kicks up to $\approx 220$ km/s are also allowed. We estimate the formation efficiency for Gaia BH3-like systems in old ($t>10$ Gyr) and metal-poor ($Z<0.01$) populations to be $\sim 4 \times 10^{-8}$ M$_\odot^{-1}$ (for our fiducial model), representing $\sim 3\%$ of the whole simulated BH-star population. We expect up to $\approx 4000$ BH-star systems in the Galactic halo formed through isolated evolution, of which $\approx 100$ are compatible with Gaia BH3-like. Given the density profile of the Galactic halo we do not expect more than one at the observed distance of Gaia BH3. Our models show that, even if it was born inside a stellar cluster, Gaia BH3 is compatible with a primordial binary star that escaped from its parent cluster without experiencing significant dynamical interactions.

astro-ph.GA↗

arXiv2024

Merging Hierarchical Triple Black Hole Systems with Intermediate-mass Black Holes in Population III Star Clusters

Theoretical predictions suggest that very massive stars have the potential to form through multiple collisions and eventually evolve into intermediate-mass black holes (IMBHs) within Population III star clusters embedded in mini dark matter haloes. In this study, we investigate the long-term evolution of Population III star clusters, including models with a primordial binary fraction of $f_{\rm b}=0$ and 1, using the $N$-body simulation code PETAR. We comprehensively examine the phenomenon of hierarchical triple black holes in the clusters, specifically focusing on their merging inner binary black holes (BBHs), with post-Newtonian correction, by using the TSUNAMI code. Our findings suggest a high likelihood of the inner BBHs containing IMBHs with masses on the order of $\mathcal{O}(100)M_{\odot}$, and as a result, their merger rate could be up to $0.1{\rm Gpc}^{-3}{\rm yr}^{-3}$. The orbital eccentricities of some merging inner BBHs oscillate over time periodically, known as the Kozai-Lidov oscillation, due to dynamical perturbations. Detectable merging inner BBHs for mHz GW detectors LISA/TianQin/Taiji concentrate within $z<3$. More distant sources would be detectable for CE/ET/LIGO/KAGRA/DECIGO, which are sensitive from $\mathcal{O}(0.1)$Hz to $\mathcal{O}(100)$Hz. Furthermore, compared with merging isolated BBHs, merging inner BBHs affected by dynamical perturbations from tertiary BHs tend to have higher eccentricities, with a significant fraction of sources with eccentricities closing to 1 at mHz bands. GW observations would help constrain formation channels of merging BBHs, whether through isolated evolution or dynamical interaction, by examining eccentricities.

astro-ph.GA↗

arXiv2024

On Binary Formation from Three Initially Unbound Bodies

We explore three-body binary formation (3BBF), the formation of a bound system via gravitational scattering of three initially unbound bodies (3UB), using direct numerical integrations. For the first time, we consider systems with unequal masses, as well as finite-size and post-Newtonian effects. Our analytically derived encounter rates and numerical scattering results reproduce the 3BBF rate predicted by Goodman \& Hut (1993) for hard binaries in dense star clusters. We find that 3BBF occurs overwhelmingly through nonresonant encounters and that the two most massive bodies are never the most likely to bind. Instead, 3BBF favors pairing the two least massive bodies (for wide binaries) or the most plus least massive bodies (for hard binaries). 3BBF overwhelmingly favors wide binary formation with super-thermal eccentricities, perhaps helping to explain the eccentric wide binaries observed by Gaia. Hard binary formation is far rarer, but with a thermal eccentricity distribution. The semimajor axis distribution scales cumulatively as $a^3$ for hard and slightly wider binaries. Though mergers are rare between black holes when including relativistic effects, direct collisions occur frequently between main-sequence stars -- more often than hard 3BBF. Yet, these collisions do not significantly suppress hard 3BBF at the low velocity dispersions typical of open or globular clusters. Energy dissipation through gravitational radiation leads to a small probability of a bound, hierarchical triple system forming directly from 3UB.

astro-ph.SR↗