The formation mechanisms of merging binary black holes (BBHs) observed by the LIGO-Virgo-KAGRA collaboration remain uncertain. Detectable eccentricity provides a powerful diagnostic for distinguishing between different formation channels, but resolving their eccentricity distributions requires the detection of a large number of eccentric mergers. Future gravitational wave detectors such as the Einstein Telescope and Cosmic Explorer will detect tens of thousands of BBH mergers out to redshifts z ≥ 10, making it critical to understand the redshift-dependent evolution of eccentricity distributions. We simulate this evolution for two key channels: dynamical assembly in globular clusters (GCs), which leads to rapid, eccentric mergers; and hierarchical triples in the field, where three-body dynamics can induce eccentricity in the inner binary. When considering all BBH mergers, the GC channel dominates overall, consistent with previous studies. However, when focusing on mergers with detectable eccentricity in next-generation detectors, we find that hierarchical triples dominate the eccentric merger rate at 0≤ z ≤ 4, with GC mergers becoming competitive at higher redshifts. Across all model variations, eccentric mergers in the local Universe (z≲ 1) have significant contributions from field triples, challenging the common view that such systems primarily form in dense environments. We show that, regardless of cluster and stellar evolution uncertainties, hierarchical triples contribute at least 30 per cent of eccentric mergers across a large range of redshifts.

The spin-orbit tilt angles theta 1(2) of merging stellar-mass black holes provide key insights into their astrophysical origin. Nonparametric population modeling by The LIGO, Virgo, and KAGRA Collaborations shows that the spin-orbit tilt distribution of mergers in the latest Gravitational-Wave Transient Catalog 4.0 exhibits a global peak at near-perpendicular directions cos theta 1(2)approximate to 0 . Here, we recover this feature using hierarchical Bayesian inference with parametric models that are tailored to enhance the diagnostic power about astrophysical formation channels. We find that the spin distribution of the low-mass bulk of the binary black hole merger population ( m1 less than or similar to 44.3-4.6+8.7M circle dot ) can be well modeled by a dominant Gaussian component that peaks at cos theta 1(2)approximate to 0 , possibly mixed with a subdominant isotropic component. Models that include a component with spins preferentially aligned with the orbit are disfavored by current data (with Bayes factors divided by Delta lnB divided by approximate to 1 -13) and constrain its contribution to be likely small ( xi similar to O(1)% ), although large contributions cannot yet ruled out with certainty. If these findings are reinforced by more detections, they would challenge any major contribution from the traditional isolated-binary formation scenario yielding closely aligned spins. Instead, the dominant component with near-perpendicular spins matches expectations from the evolution of isolated massive stellar triples in the galactic field, where the Lidov-Kozai effect naturally produces a unique overabundance of mergers with cos theta 1(2)approximate to 0 .

Gravitational-wave observations have revealed an excess of binary black hole mergers with primary masses near ∼ 35 M_⊙. We show that if this feature originates from dynamical formation in dense stellar systems, and if the pair-instability supernova truncates the first-generation black hole mass spectrum, then second-generation mergers inevitably produce a second peak near ∼ 70 M_⊙. This structure reflects the suppression of first-generation black holes above a characteristic mass and the accumulation of merger remnants near twice that scale. Its location is robust, whereas its amplitude depends strongly on cluster initial conditions. Using a large suite of cluster population-synthesis models, we show that current gravitational-wave data already constrain the birth properties of globular clusters, irrespective of their overall contribution to the observed population. If clusters dominate mergers above the pair-instability scale, these constraints tighten further and imply a minimum first-generation merger rate of ℛ(m_1 ≤ 50 M_⊙) ≥ 0.099 Gpc^-3 yr^-1 (99% confidence). We further show that a drop or gap in the secondary black hole mass spectrum is not a robust signature of a cluster origin for high-mass mergers within the pair-instability mass gap. A confirmed excess near ∼ 70 M_⊙ would support a dynamical origin of the ∼ 35 M_⊙ feature and provide independent evidence for a pair-instability mass gap with a lower edge at ≲ 50M_⊙.

We analyze the mass-ratio and effective-spin (χ_ eff) distributions of binary black hole mergers in the latest gravitational-wave catalog, GWTC-5, as a function of primary mass. Using hierarchical Bayesian inference with flexible Gaussian-process population models, we identify four distinct mass regions separated by sharp transitions in both mass-ratio and spin properties. Below ∼15 M_⊙, the population strongly favors equal-mass binaries and exhibits a narrow χ_ eff distribution peaked at positive values. In the range 18-30 M_⊙, the mass-ratio distribution becomes substantially flatter, while the χ_ eff distribution broadens, shifts to a peak consistent with zero, and shows tentative–but not statistically required–evidence for positive skewness. The region associated with the feature near ≃35 M_⊙ returns to a narrow χ_ eff distribution consistent with symmetry at zero and strongly favors equal-mass binaries. Above ≃ 45 M_⊙, both the mass-ratio and χ_ eff distributions broaden significantly. The inferred support of the spin distribution converges toward the range expected for binaries containing remnants of previous black hole mergers, making the highest-mass region fully consistent with a star cluster population of hierarchical mergers. The close correspondence between transitions in mass ratio and effective spin suggests that different primary-mass ranges trace distinct formation channels, with isolated binary or triple evolution likely dominating the lower-mass population and dynamical assembly becoming increasingly important at higher masses.

The evolution of star clusters is driven by stellar mass loss, two-body relaxation, and evaporation in the Galactic tidal field. Fast modeling tools are crucial for exploring diverse initial conditions and predicting cluster populations and their contribution to gravitational wave (GW) sources over cosmic timescales. We present an improved version of the clusterBHBdynamics (cBHBd) code, designed to evolve star clusters containing stars and stellar-mass black holes (BHs). We improve the treatment of evaporation in the Galactic tidal field and include the effects of metallicity and stellar mass functions. We also introduce new prescriptions for GW captures during BBH-BBH interactions and between resonant interactions due to distant encounters that increase BBH eccentricities. The updated cBHBd is validated against Cluster Monte Carlo (CMC) models and N-body simulations spanning a range of cluster properties. Seven model parameters are fitted to the CMC results with nested sampling. With the best-fit values, the evolution of the cluster mass, half-mass radius, and BH population over 13 Gyr is reproduced to within ∼10%. The new GW capture prescriptions allow cBHBd to reproduce BBH merger rates from CMC models of massive clusters (≳10^5,M_⊙) and direct N-body models of lower-mass clusters (≲10^5,M_⊙) to within ∼20%. The improved cBHBd provides a fast and flexible tool for large-scale star cluster studies. With a runtime of about one second per cluster, it enables applications such as searches for globular cluster initial conditions, stellar stream modeling, and GW population synthesis.

Pair instability should prevent the direct formation of black holes above about 50 M ⊙, creating a 'pair-instability' mass gap. Yet gravitational-wave observations have detected black holes in this mass range. These systems can be explained with uncertainties in massive-star evolution, or hierarchical mergers in stellar clusters, which are expected to produce large spins with isotropic orientations. Here we present evidence for the pair-instability mass gap in the LIGO-Virgo-KAGRA fourth transient catalogue, with a lower edge at 44 . 3 - 3.5 + 5.9 M ⊙ . We also obtain a measurement of the 12C(α, γ)16O reaction rate, yielding an S-factor of 26 8 - 116 + 195 keV b , a parameter critical for modelling helium burning and stellar evolution. The data reveal two populations: a low-spin group with no black holes above the gap, and a high-spin, isotropic group that extends across the full mass range and occupies the gap, consistent with hierarchical mergers. These findings are consistent with pair instability playing a role in shaping the black hole mass spectrum, point to a connection between gravitational-wave astronomy and nuclear astrophysics, and highlight dense stellar clusters as key environments in the growth of black holes.

We model the formation and retention of the most massive black hole (BH) in 47 Tuc using the semi-analytical code , coupling cluster evolution with binary BH dynamics and computing merger-remnant masses, spins, and gravitational-wave recoil kicks via numerical-relativity surrogate prescriptions. We evolve 80 000 cluster realisations spanning initial masses, densities, IMFs, and metallicities, in both a baseline scenario (m_ max = 130 M_⊙) and an extended-IMF scenario with ∼ 50-110 primordial BH seeds above the pair-instability gap (M_ BH∼ 130-700 M_⊙). Selecting models reproducing 47 Tuc's present-day mass and half-mass radius, we find hierarchical mergers alone yield a most massive retained BH of M_ BH∼ 45-70 M_⊙ with spin χ_ BH∼ 0.65, limited to ∼ 1-3 mergers, as second-generation remnants acquire spin χ∼ 0.7 that amplifies recoil kicks in subsequent generations. When primordial seeds are included, the retained-mass distribution becomes bimodal – in ∼ 90% of realisations all seeds are ejected, but in ∼ 10% a massive seed (M_ BH≳ 450 M_⊙) survives – while the joint mass-spin distribution is trimodal; seeds surviving via stellar-mass BH mergers retain low spin (χ≲ 0.3), whereas seed-seed mergers produce high-mass, high-spin remnants (χ∼ 0.65-0.7), yielding 90th-percentile retained masses of ∼ 500-1100 M_⊙. Both scenarios are consistent with the 3σ dynamical upper limit of 578 M_⊙. Our results favour a dark-remnant subsystem over a single massive IMBH and provide a spin-mass diagnostic testable with LIGO-Virgo-KAGRA, the Einstein Telescope, Cosmic Explorer, and LISA.

Einstein Telescope (ET) is the European project for a gravitational-wave (GW) observatory of third-generation. In this paper we present a comprehensive discussion of its science objectives, providing state-of-the-art predictions for the capabilities of ET in both geometries currently under consideration, a single-site triangular configuration or two L-shaped detectors. We discuss the impact that ET will have on domains as broad and diverse as fundamental physics, cosmology, early Universe, astrophysics of compact objects, physics of matter in extreme conditions, and dynamics of stellar collapse. We discuss how the study of extreme astrophysical events will be enhanced by multi-messenger observations. We highlight the ET synergies with ground-based and space-borne GW observatories, including multi-band investigations of the same sources, improved parameter estimation, and complementary information on astrophysical or cosmological mechanisms obtained combining observations from different frequency bands. We present advancements in waveform modeling dedicated to third-generation observatories, along with open tools developed within the ET Collaboration for assessing the scientific potentials of different detector configurations. We finally discuss the data analysis challenges posed by third-generation observatories, which will enable access to large populations of sources and provide unprecedented precision.

We analyze the effective-spin distribution of binary black hole mergers in GWTC-5.0 as a function of primary black hole mass using hierarchical Bayesian inference. We model the population as a mixture of two spin components separated by a transition mass scale inferred directly from the data. We find strong evidence for a transition at m̃ = 15.2^+4.3_-3.6 M_⊙. Mock-catalog analyses show that such a transition is unlikely to arise from finite-sample fluctuations of a mass-independent χ_ eff population and the posterior predictive distributions of χ_ eff inferred below and above the transition are clearly distinct. Below the transition mass, the effective-spin distribution is narrow, peaks at a small positive value χ_ eff>0, but also shows significant support for negative χ_ eff. Above the transition, the distribution is broader and its peak shifts to values consistent with χ_ eff≃0, making its support at both positive and negative χ_ eff roughly similar. These findings suggest that the dominant merger population concentrated around 10 M_⊙ is statistically distinct from the rest and that it arises from a different formation channel. We show that this low-mass population is broadly consistent with formation from massive stellar multiples in the field: it may either arise from isolated binary star evolution but only if black hole natal kicks below m̃ are generally very large (≳100 km/s) or be caused by the dynamical evolution of hierarchical triples. In contrast, isolated binary evolution with standard fallback kick models cannot reproduce the support for negative χ_ eff.

We investigate the effects of prior selection on the inferred mass and spin parameters of the neutron star-black hole merger GW230529_181500. Specifically, we explore models motivated by astrophysical considerations, including massive binary and pulsar evolution. We examine mass and spin distributions of neutron stars constrained by radio pulsar observations, alongside black hole spin observations from previous gravitational-wave detections. We show that the inferred mass distribution highly depends upon the spin prior. Specifically, under the most restrictive, binary stellar evolution models, we obtain narrower distributions of masses with a black hole mass of 4.3(-0.1)(+0.1) M-circle dot and neutron star mass of 1.3(-0.03)(+0.03) M-circle dot where, somewhat surprisingly, it is the prior on component spins that has the greatest impact on the inferred mass distributions. Re-weighting using neutron star mass and spin priors from observations of radio pulsars, with black hole spins from observations of gravitational waves, yields the black hole and the neutron star masses to be 3.8(-0.6)(+0.5) and 1.4(-0.1)(+0.2) M-circle dot, respectively. The sequence of compact object formation - whether the neutron star or the black hole formed first - cannot be determined at the observed signal-to-noise ratio. However, there is no evidence that the black hole was tidally spun up.

Black holes (BHs) with masses between 3-5M circle dot, produced by a binary neutron star (BNS) merger, can further pair up with a neutron star or BH and merge again within a Hubble time. However, the astrophysical environments in which this can happen and the rate of such mergers are open questions in astrophysics. Gravitational waves may play an important role in answering these questions. In this context, we discuss the possibility that the primary of the recent LIGO-Virgo-KAGRA binary GW230529_181500 (GW230529, in short) is the product of a previous BNS merger. Invoking numerical relativity (NR)-based fitting formulas that map the binary constituents' masses and tidal deformabilities to the mass, spin, and kick velocity of the remnant BH, we investigate the potential parents of GW230529's primary. Our calculations using NR fits based on BNS simulations reveal that the remnant of a high-mass BNS merger similar to GW190425 is consistent with the primary of GW230529. This argument is further strengthened by the gravitational wave-based merger rate estimation of GW190425-like and GW230529like populations. We show that around 18% (median) of the GW190425-like remnants could become the primary component in GW230529-like mergers. The dimensionless tidal deformability parameter of the heavier neutron star in the parent binary is constrained to 67 & thorn;163 of the gravitational-wave kick imparted to the remnant, we also discuss the astrophysical environments in which these types of mergers can take place and the implications for their future observations.

Understanding binary black hole (BBH) dynamics in dense star clusters is key to interpreting the gravitational wave detections by LIGO and Virgo. Here, we perform N-body simulations of star clusters, focusing on BBH formation mechanisms, dynamical evolution and merging properties. We explore a wide parameter space of initial conditions, with cluster masses ranging from 10(4) to 10(6) M-circle dot, densities from 103 to 10(5) M(circle dot)pc(-3), and up to 100% of massive stars in binaries. We show that most BBH mergers originate from the primordial binary population rather than being dynamically assembled, and that the evolution towards merger for most of these binaries is not significantly altered by dynamical encounters. As a result, the overall number of BBH mergers from the N-body simulations is nearly identical to that obtained when the same stellar population is evolved in isolation. Contrary to theoretical expectations, nearly all dynamically formed BBH mergers occur when the binary is still bound to its host cluster, with similar or equal to 90% of all dynamical mergers occurring within the cluster core region. In about half of these mergers the binary is part of a stable black hole-triple system. In one model, stellar mergers lead to the formation of a similar or equal to 200M(circle dot) black hole, which then grows to similar or equal to 300M(circle dot) through black hole mergers. Our study highlights the importance of detailed N-body simulations in capturing the evolution of black hole populations in dense clusters and challenges conclusions based on semi-analytical and Monte Carlo methods.

Context. Dynamical interactions in star clusters are an efficient mechanism to produce the coalescing binary black holes (BBHs) that have been detected with gravitational waves (GWs). Aims. We want to understand how BBH coalescence can occur during - or after - binary-single interactions with different mass ratios. Methods. We perform gravitational scattering experiments of binary-single interactions using different mass ratios of the binary components (q(2) equivalent to m(2)/m(1) <= 1) and the incoming single (q(3) equivalent to m(3)/m(1)). We extract cross-sections and rates for (i) GW capture during resonant interactions; (ii) GW inspiral in between resonant interactions and apply the results to different globular cluster conditions. Results. We find that GW capture during resonant interactions is most efficient if q(2) similar or equal to q(3) and that the mass-ratio distribution of BBH coalescence due to inspirals is proportional to m(1)(-1)q(2.9+alpha), where alpha is the exponent of the BH mass function. The total rate of GW captures and inspirals depends mostly on m(1) and is relatively insensitive to q(2) and q(3). We show that eccentricity increase in direct (that is, non-resonant) encounters approximately doubles the rate of BBH inspirals in between resonant encounters. For a given GC mass and radius, the BBH merger rate in metal-rich GCs is approximately double that of metal-poor GCs, because of their (on average) lower BH masses (m(1)) and steeper BH mass function, yielding binaries with lower q. Conclusions. Our results enable the mass-ratio distribution of dynamically formed BBH mergers to be translated to the underlying BH mass function. The additional mechanism that leads to a doubling of the inspirals provides an explanation for the reported high fraction of in-cluster inspirals in N-body models of clusters.

We investigate black hole-star binaries formed in N-body simulations of massive, dense star clusters. We simulate 32 clusters with varying initial masses (10(4) M-circle dot to 10(6) M-circle dot), densities (1200 M(circle dot)pc(-3) to 10(5) M-circle dot pc(-3)), and metallicities (Z=0.01,0.001,0.0001). Our results reveal that star clusters produce a diverse range of BH-star binaries, with dynamical interactions leading to extreme systems characterized by large orbital separations and high black hole masses. Of the ejected BH-main sequence (BH-MS) binaries, 20 per cent form dynamically, while the rest originate from the primordial binary population initially present in the cluster. Ejected BH-MS binaries that are dynamically formed have more massive black holes, lower-mass stellar companions, and over half are in a hierarchical triple system. All unbound BH-giant star (BH-GS) binaries were ejected as BH-MS binaries and evolved into the BH-GS phase outside the cluster. Due to their lower-mass companions, most dynamically formed binaries do not evolve into BH-GS systems within a Hubble time. Consequently, only 2 of the 35 ejected BH-GS binaries are dynamically formed. We explore the formation pathways of Gaia-like systems, identifying two Gaia BH1-like binaries that formed through dynamical interactions, and two Gaia BH2-like systems with a primordial origin. We did not find any system resembling Gaia BH3, which may however be attributed to the limited sample size of our simulations.

Dynamical interactions in star clusters are an efficient mechanism to produce the coalescing binary black holes (BBHs) that have been detected with gravitational waves (GWs). We want to understand how BBH coalescence can occur during - or after - binary-single interactions with different mass ratios. We perform gravitational scattering experiments of binary-single interactions using different mass ratios of the binary components (q_2≡ m_2/m_1≤1) and the incoming single (q_3≡ m_3/m_1). We extract cross sections and rates for (i) GW capture during resonant interactions; (ii) GW inspiral in between resonant interactions and apply the results to different globular cluster conditions. We find that GW capture during resonant interactions is most efficient if q_2≃ q_3 and that the mass-ratio distribution of BBH coalescence due to inspirals is ∝ m_1^-1q^2.9+α, where α is the exponent of the BH mass function. The total rate of GW captures and inspirals depends mostly on m_1 and is relatively insensitive to q_2 and q_3. We show that eccentricity increase by non-resonant encounters approximately doubles the rate of BBH inspiral in between resonant encounters. For a given GC mass and radius, the BBH merger rate in metal-rich GCs is approximately double that of metal-poor GCs, because of their (on average) lower BH masses (m_1) and steeper BH mass function, yielding binaries with lower q. Our results enable the translating from the mass-ratio distribution of dynamically formed BBH mergers to the underlying BH mass function. The additional mechanism that leads to a doubling of the inspirals provides an explanation for the reported high fraction of in-cluster inspirals in N-body models of clusters.

Gravitational-wave observations of massive, rapidly spinning binary black holes mergers provide increasing evidence for the dynamical origin of some mergers. Previous studies have interpreted the mergers with primary mass $\gtrsim45\,M_\odot$ as being dominated by hierarchical, second-generation mergers, with rapidly spinning primaries being the products of previous black hole mergers assembled in dense stellar clusters. In this work, we reveal confident evidence of another subpopulation with rapid and isotropic spins at low mass containing the two exceptional events GW241011 and GW241110, consistent with a hierarchical merger hypothesis. Our result suggests the mass distribution of the second-generation black holes is peaked at low primary masses of $\sim16\,M_\odot$ rather than $\gtrsim45\,M_\odot$ in the pair-instability gap. Such low-mass second-generation black holes must be formed from the merger of even lighter first-generation black holes, implying that dense, metal-rich stellar environments contribute to the binary black hole population. By separating the contamination of higher-generation black holes, our result reveals the primary mass distribution of first-generation black holes formed from stellar collapse, which shows a significant dip between $\sim12\,M_\odot$ to $\sim20\,M_\odot$. This may indicate a dearth of black holes due to variation in the core compactness of the progenitor.

Stellar evolution theories predict a gap in the black hole birth mass spectrum as the result of pair instability processes in the cores of massive stars. This gap, however, is not seen in the binary black hole masses inferred from gravitational wave data. One explanation is that black holes form dynamically in dense star clusters where smaller black holes merge to form more massive black holes, populating the mass gap. We show that this model predicts a distribution of the effective and precessing spin parameters, χ_{eff} and χ_{p}, within the mass gap that is insensitive to assumptions about black hole natal spins and other astrophysical parameters. We analyze the distribution of χ_{eff} as a function of primary mass for the black hole binaries in the third gravitational wave transient catalog. We infer the presence of a high mass and isotropically spinning population of black holes that is consistent with hierarchical formation in dense star clusters and a pair-instability mass gap with a lower edge at 44_{-4}^{+6}M_{⊙}. We compute a Bayes factor B>10^{4} relative to models that do not allow for a high mass population with a distinct χ_{eff} distribution. Upcoming data will enable us to tightly constrain the hierarchical formation hypothesis and refine our understanding of binary black hole formation.

Next-generation ground-based gravitational wave observatories will observe mergers of intermediate-mass black holes (IMBHs) out to high redshift. Such IMBHs can form through runaway tidal encounters in the cores of dense stellar clusters. In this paper, we ask if the gravitational wave observation of a single merger event between two IMBHs, occurring in the aftermath of the coalescence of the clusters in which they formed, can be used to infer the properties of their host clusters, such as mass, redshift, and half-mass radius. We implement an astrophysically motivated analytic model for cluster evolution and IMBH growth, and we perform IMBH binary parameter estimation using a network of three next-generation detectors. We find that inferring the structural properties of clusters in this way is challenging due to model degeneracy. However, the posteriors on the cluster formation redshifts have relatively narrow peaks, and it may still be possible to infer the cluster formation history by measuring a whole population of IMBH binary merger events.

We use hierarchical Bayesian inference with nonparametric Gaussian process models to investigate the effective inspiral spin parameter, %eff, as a function of primary black hole mass in the third gravitationalwave transient catalog (GWTC-3). Our analysis reveals a transition in the population at a primary mass of -5M circle dot. Beyond this mass, the %eff distribution broadens, becomes consistent with being symmetric around zero, and has a median of -0.03 & thorn;0.36 -0.59 (90% credibility). These results are consistent with the presence of a pair-instability mass gap that is repopulated by black holes that are the remnant of a previous merger, formed in dense star clusters. However, asymmetric distributions skewed toward positive %eff are not excluded by current data. Below the inferred transition mass, we constrain the fraction of secondgeneration black holes to be less than or similar to 10%. These results provide model-independent support for a high-mass and high-spin population of black holes in the data, consistent with earlier work using parametric models. Imminent gravitational-wave data releases will be essential to sharpen constraints on spin symmetry and clarify the origin of the black holes.

Stellar evolution theory predicts that electron–positron pair production in the cores of massive stars triggers unstable thermonuclear explosions that prevent the direct formation of black holes above about 50M?, creating a “pair-instability gap” [1]. Yet black holes have been detected above this mass with gravitational waves; such objects might be explained with uncertainties in the physics of mas28 sive stars and stellar collapse or with hierarchical mergers of black holes in stellar clusters [2–5]. Hierarchical mergers are associated with large spins as pre30 dicted by general relativity [6–8], and isotropic spin orientations [9]. Here we present strong evidence for the pair-instability mass gap in the LIGO–Virgo– KAGRA fourth transient catalog [10], with a lower edge at 45.3+6.5 -4.8 32 M?. We also obtain a measurement of the 12C(a, ?)16O reaction rate, yielding an Sfactor of 242.5+310.4 -101.5 34 keVb, a parameter critical for modeling helium burning and stellar evolution. The new data reveal two populations: a low-spin group with no black holes above the gap, consistent with direct stellar collapse, and a high-spin, isotropic group that extends across the full mass range and occupies the gap, consistent with hierarchical mergers. These findings confirm the role of pair-instability in shaping the black hole spectrum, establish a new link between gravitational-wave astronomy and nuclear astrophysics, and highlight hierarchi41 cal mergers and star cluster dynamics as key channels in the growth of black holes [11, 12].