# Compact Binary Coalescences: Astrophysical Processes and Lessons Learned

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## Abstract

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## 1. Introduction

**Merging compact-object binaries**are binary systems composed of two compact objects that are so close to each other to merge via gravitational wave (GW) emission within the age of the Universe. The members of such binaries can be white dwarfs (WDs), neutron stars (NSs), black holes (BHs), and their combinations, e.g., neutron star-black hole binary (NSBH) systems. These systems have been investigated for decades by many authors, who predicted their existence through theoretical studies that go from the formation and evolution of the stellar progenitors to accurate numerical relativity simulations of the final merger phase [1,2,3,4,5,6,7,8,9].

**Hulse–Taylor binary**is remarkably consistent with a GW-induced shrinking. This system, which is expected to merge in $\sim 300$ Myr, provided not only an additional confirmation of the Einstein’s theory of general relativity, but it also suggested to us that there might be not just one, but a population of binary neutron stars (BNSs) that can merge in relatively short times via GW emission.

**GW150914**, was attributed to the coalescence of two stellar-mass BHs with masses ${m}_{1}={36}_{-4}^{+5}\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and ${m}_{2}={29}_{-4}^{+4}\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ [12,13]1. The event carried many scientific implications with itself and it laid the foundations of a new way to investigate the Universe by allowing us to access data never collected before.

**the masses of the BHs**: we did not expect to detect stellar BHs with masses ≳ $20\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$.

**isolated binary channel**, two progenitor stars are bound since their formation; they evolve, and then turn into (merging) compact objects at the end of their life, without experiencing any kind of external perturbation [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. This scenario is driven by single and binary stellar evolution processes, and it is sometimes referred to as the “field” scenario, because it assumes that binaries are born in low-density environments, i.e., that they evolve in isolation. In contrast, in the

**dynamical channel**, two compact objects get very close to each other after one (or more) gravitational interactions with other stars or compact objects. This evolutionary scenario is quite common in dense stellar environments (e.g., star clusters), and it is driven mainly by stellar dynamics [27,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61]. In reality, the two formation pathways might have a strong interplay. In star clusters, the orbital parameters of binaries might be perturbed by many passing-by objects. Dynamical interactions might be strong enough to eject the stellar binary from the cluster and to trigger the merger event in the field. Such an apparently isolated merger would not have occurred if the progenitor stars had evolved in isolation. Such

**hybrid scenarios**blur the line between the dynamical and the isolated binary channel, and they have already been investigated by various authors [62,63,64,65].

**uncertainties and degeneracies of the astrophysical models**. Single-star evolutionary tracks, the strength of stellar winds (especially for massive stars at low metallicity), core-collapse and pair-instability supernova (PISN), the orbital parameters of binary stars at birth, binary mass transfer, compact-object birth kicks, stellar mergers, tidal interactions, common envelope (CE), and GW recoil, are only part of the uncertain ingredients of the unknown recipe of merging binaries. In contrast, stellar dynamics is simple and elegant, but developing accurate and fast algorithms for the long-term evolution of tight binaries is challenging. Furthermore, studying the evolution of small-scale systems (2 bodies within ∼${10}^{-6}$ pc) in large star clusters (≳${10}^{5}$ objects within a few pc) is computationally intensive [66,67,68,69,70,71,72,73,74,75].

**the GW catalog is growing faster than our theoretical understanding**of merging compact-object binaries. At the time of writing, we have already achieved an historic breakthrough: we have just started talking about a population of BHs. Indeed, the latest Gravitational Wave Transient Catalog (GWTC) reports ∼90 events2, mostly BBH mergers, and the count is expected to significantly increase in the next years, at even faster rates than ever because new ground-based interferometers will be operational and the existing ones will increase their sensitivity [79,80].

**GW190814**(see Section 5.1) is an event with very asymmetric masses, a merger that most theoretical models find very difficult to explain [81]. Furthermore, the lightest member is a mystery compact object with an uncertain nature: it can be the heaviest NS or the lightest BH ever observed and its mass falls right into the lower mass gap (see Section 2.7).

**GW190521**(see Section 5.2) is the event with the heaviest BHs, with at least one of the two falling in the upper mass gap (see Section 2.9) [82,83]. Its merger product, a BH with mass ${148}_{-16}^{+28}\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, is the first confirmation of the existence of intermediate-mass BHs.

**GW200105_162426**and

**GW200115_042309**(see Section 5.4) are the first NSBHs ever observed [84].

**GW170817**is associated with a merger of two NSs and it is the only event observed not only through GWs but also throughout the whole electromagnetic spectrum, a crucial milestone for

**multi-messenger astronomy**[85]. There are also 5 events with preference for

**negatively aligned spins**with respect to the orbital angular momentum of the binary, including the mentioned GW200115_042309. Spins and their in-plane components might provide important insights on the formation channels (see Section 2.11). The BH mass distribution and the inferred BBH merger rate make the current scenario even more complex. The former seems to have statistically significant

**substructures**, that is, it shows up as clumpy, with BHs that tend to accumulate at chirp masses3 $\mathcal{M}=8\phantom{\rule{0.166667em}{0ex}},14\phantom{\rule{0.166667em}{0ex}},27\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, whereas the latter increases with redshift [80,86,87,88].

## 2. Single Stars

**population-synthesis simulations**. To conduct statistical studies on stellar populations and their compact objects, we should follow self-consistently the evolution of any possible type of single and/or binary star from its formation to its death, and possibly beyond. This is prohibitive if we consider that simulating the evolution of just one star from the main sequence until core collapse might take days (if the complex underlying algorithms converge). Thus, for fast population-synthesis studies, the evolution of single stars is approximated through either (i) fitting formulas to detailed stellar evolution calculations (e.g., [35]) or (ii) the interpolation of look-up tables containing pre-evolved stellar evolution tracks for different stars at various metallicity (e.g., [22]). Binary stellar evolution (see Section 3) and other additional processes (e.g., supernova explosions) are generally added through analytical prescriptions on top of the single-star approximations. Fast population-synthesis codes are currently the main resource available to study compact objects from single and binary stars.

#### 2.1. Overview

**nickel-56**is the heaviest element that stars can produce efficiently through nuclear fusion (${}_{26}^{52}\mathrm{Fe}{+}_{2}^{4}\mathrm{He}\to {\phantom{\rule{0.166667em}{0ex}}}_{28}^{56}\mathrm{Ni}+\gamma $).

**zero-level uncertainty**to our understanding of the mass spectrum of compact objects. On the one hand, the uncertainty stems from theoretical modeling of detailed stellar evolution processes, such as convection, dredge up, wind mass loss, and nuclear reaction rates [25,92,93,94,95,96]. On the other hand, the limits also depend on other stellar parameters, such as rotation and chemical composition. These uncertainties affect the masses of stellar cores, which, in turn, have an impact on the nature and abundance of compact remnants that stars may form.

#### 2.2. The Chandrasekhar Limit

**Chandrasekhar mass limit**. Its precise value depends on the average molecular weight per electron, which, in turn, depends on the chemical composition of the WD. For a typical CO or helium WD, the Chandrasekhar limit is ${M}_{\mathrm{c}}\simeq 1.44\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$.

#### 2.3. The Tolman-Oppenheimer-Volkoff Limit

**the Tolman–Oppenheimer–Volkoff (TOV) limit**[101,102]. In this case, support against gravity is provided by the degenerate pressure of a neutron gas. However, unlike in the case of the degenerate electron gas in WDs, neutron-neutron interactions become a crucial (but very uncertain) ingredient to include in the equation of state. Thus, the TOV limit reflects our uncertainties on the NS equation of state. In principle, depending on the adopted equation of state, the TOV limit can be anywhere from $0.5\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ to $3\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ [17,103,104,105,106,107,108,109,110,111,112,113,114]. The observations of NS masses ≳$1\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ (e.g., those in binary pulsars, such as PSR B1913 + 16) ruled out the softest equations of state, placing the TOV limit at ${M}_{\mathrm{TOV}}\simeq 1.5$–$3\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$. The detection of the GW signal from merging NSs can also be used to constrain the maximum NS mass. In fact, tidal deformations during the last phase of the inspiral affect the properties of the gravitational waveform, which can then be linked to the NS equation of state. The analysis of GW170817 data constrains ${M}_{\mathrm{TOV}}\lesssim 2.3\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ [115,116]. Stellar rotation may also play a role, with rigidly rotating NSs having about 25% larger allowed masses [117,118,119,120].

#### 2.4. The Role of Stellar Winds

**Stellar winds**, especially for massive stars, are uncertain, and even a factor of 2 uncertainty (typical for state-of-the-art models, e.g., [96]) might have important consequences on the nature and mass spectrum of compact objects.

**interaction between radiation and matter in stellar atmospheres**. The idea that the outer layers of stars could expand was introduced already at the beginning of the 20th century by Saha [121]. Saha [121] suggested that radiation could be absorbed by matter in the solar atmosphere through an inelastic impact, with a resulting forward velocity of $h\nu /mc$, where h is the Plank length, $\nu $ is the frequency of the photon and m the rest mass of matter. We now know that the strongly anisotropic and continuous component of photons from the innermost layers constantly exchanges energy and momentum with free electrons, ions, atoms and dust grains in stellar atmospheres. The momentum equation, considering only a radial direction of the radiation (1D problem), reads

**CAK theory**), that is

**stellar winds in hot massive stars are**

**line driven**.

**metals**—elements heavier than helium—to stellar winds became increasingly important. Indeed, hydrogen and helium have very few spectral lines in the UV (i.e., the radiation peak frequency of hot massive stars), thus their contribution is expected to be minimal compared to that coming from metals, which have crowded line spectra in the UV band. From CAK theory, it was already clear that

**stellar winds are quenched at low metallicity**, that is the mass fraction of metals in a stellar layer. Denoting the mass loss by winds as ${\dot{M}}_{*}$ and the metallicity as Z, ${\dot{M}}_{*}\propto {Z}^{x}$ with x ranging from $\sim 0.5$ [126] to $\sim 0.9$ [127].

**Fe-group elements**, which are extremely efficient absorbers because their complex atomic structure allows for millions of different lines. At lower metallicity and in the outer (supersonic) wind, the main contribution comes from CNO elements. Furthermore, the recombination of Fe IV to Fe III for ${T}_{\mathrm{eff}}$ going from $\sim \mathrm{27,500}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ to $\sim \mathrm{22,500}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ gives a significant boost to mass loss, despite ${\dot{M}}_{*}\propto {T}_{\mathrm{eff}}$ in this temperature range (bi-stability jump).

**Fe-group elements even for Wolf–Rayet (WR) stars**, though the dependence on Z cannot be described as a single power-law. The atmospheres of WR stars are self-enriched with metals, (e.g., carbon), so the latter can sustain the mass loss of WR stars for $Z\lesssim {10}^{-3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{Z}}_{\odot}$, where, indeed, ${\dot{M}}_{*}$ becomes insensitive to metallicity [130,131]. The mass loss prescriptions developed by Vink et al. [129] and Vink and de Koter [130] are the ones adopted by most state-of-the-art stellar evolution codes. A summary of the prescriptions is given in Section 4 of Vink [96].

**clumpy winds**. Recent works that do not adopt this assumption predict mass loss rates lower than [129], though the metallicity dependence is remarkably similar (e.g., [132]).

**approach the Eddington limit**during their evolution might experience enhanced mass loss, which may even become insensitive to metallicity and occur in the form of pulsations [99,133,134,135,136,137,138]. Our knowledge of such continuum-driven winds is hampered by the uncertainties in modeling the interaction between winds and radiation-dominated envelopes.

**homogeneity**of stellar winds. Several observations seem to suggest that winds are clumpy, though the clumps’ formation mechanism and evolution is still under debate [139,140]. The geometry, clumpiness level, and nature of clumps (i.e., optically thin or think) are also uncertain, but they might have a significant impact on stellar winds (e.g., [141]).

#### 2.5. Core-Collapse Supernovae

**enrichment of neutrons**, which eventually form a very compact degenerate structure that can halt the collapse.

**bounce-shock mechanism**(e.g., [145]). During the collapse phase, the core contraction is not self-similar: only the innermost part of the core contracts all together (

**homologous core**, $\sim 0.5\u20131\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$). When the density in the homologous core rises to $\sim 2.7\times {10}^{14}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$, the neutron degeneracy pressure would be high enough to sustain the structure against collapse, though the core overshoots its equilibrium state and when $\rho \simeq 5\times {10}^{14}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$ the repulsive nuclear force makes the core bounce back, creating a

**shock wave**. The latter might carry enough energy to eject the stellar envelope and power a prompt explosion. However, the shock dissipates most of its energy while travelling outwards, through the infalling material, until it stalls at about hundreds of kilometers from the center, well within the Fe core, failing to produce a successful SN.

**neutrino-driven mechanism**(e.g., [146]). At central densities $\sim 5\times {10}^{9}\phantom{\rule{0.166667em}{0ex}}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$, the mean free path of neutrinos is comparable to the dimension of the homologous core. At higher densities ($\sim {10}^{11}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$) neutrinos are basically trapped in the core and they start a congestion that results in the stall of the neutronization process at $\sim {10}^{12}\phantom{\rule{0.166667em}{0ex}}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$. The latter completes only seconds after the collapse, when most of the very high-energy neutrinos have had time to escape the core and to deposit part of their energy in the material behind the former shock wave. The rise in pressure in the layer between the proto-NS and the shock wave might revive the latter and power a successful explosion.

**convection-enhanced neutrino-driven mechanism**, e.g., [147,148,149]).

#### 2.6. Electron-Capture SNe

**electron-capture supernova (ECSN)**, and its lack of non-radial asymmetries has implication for the strength of SN kicks (see Section 2.10).

#### 2.7. Compact Remnants and the Lower Mass Gap

**direct collapse**). As such, fallback is a key ingredient to understand the mass spectrum of both NSs and BHs, but constraining it is very challenging.

**energy-driven models**) or modeled through the expansion of a hard surface placed at a specified mass-cut, generally at the outer border of the iron core (

**piston-driven**). In both cases, the convective-enhanced neutrino energy becomes a parameter of the models, and it is generally calibrated using the observed SN luminosities and ${}^{56}\mathrm{Ni}$-yields. Following this approach, many authors tried to identify the key parameters of the stellar structure that

**drive the explodability of stars**and the amount of fallback.

**rapid**model), and (ii) the explosion is delayed (∼seconds) and its main engine becomes the standing accretion shock instability (

**delayed**model). In both models, fallback has a huge impact on the masses of remnants from stars with $10\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}\lesssim {\mathrm{M}}_{\mathrm{ZAMS}}\lesssim 30\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$. Both models predict direct collapse for ${m}_{\mathrm{CO}}\ge 11\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, but the rapid model also for $6\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}\le {m}_{\mathrm{CO}}\le 7\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, which corresponds to $22\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}\lesssim {\mathrm{M}}_{\mathrm{ZAMS}}\lesssim 26\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ The latter happens because the rapid mechanism occurs in ∼100 ms, i.e., when the infalling ram pressure can be still high enough to prevent a successful explosion. Thus, the rapid approach is more prone to a failed explosion than the delayed model, and it is more sensitive to the compactness of the innermost star’s regions. Specifically, in Fryer et al. [26], the stellar models with $22\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}\lesssim {\mathrm{M}}_{\mathrm{ZAMS}}\lesssim 26\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ develop mixing instabilities and more compact structures during the latest evolutionary stages, causing a failed rapid SN and a

**gap**in the remnants mass spectrum between $\sim 3\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and $\sim 5\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, a dearth which seems in agreement with observations (the

**lower mass gap**[17,18,166]). While this argument might suggest a preference for the rapid model, the approach followed by Fryer et al. [26] is simplified and sensitive to the details of the stellar late-stage burning phases. Thus, the existence of the lower mass gap is still matter of debate and there are no conclusive results on the topic.

**compactness parameter**, ${\xi}_{2.5}$6 [168], provides better insights than ${m}_{\mathrm{CO}}$ on the explodability of a star. Ugliano et al. [167] simulations revealed that BHs can form via direct collapse for ${\mathrm{M}}_{\mathrm{ZAMS}}\lesssim 20\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and that successful SNe are possible for $20\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}\lesssim {\mathrm{M}}_{\mathrm{ZAMS}}\lesssim 40\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$. This happens because stellar structure does not vary monotonically with ${\mathrm{M}}_{\mathrm{ZAMS}}$, and the SN explosion is sensitive to such variations. Rather than a ${\xi}_{2.5}$-threshold, it is the

**local maxima**in the ${\xi}_{2.5}\u2013{\mathrm{M}}_{\mathrm{ZAMS}}$ plane that increase the probability of failed SNe, thus, islands of explodability appear for ${\mathrm{M}}_{\mathrm{ZAMS}}\lesssim 40{\mathrm{M}}_{\odot}$, while direct collapse is dominant for ${\mathrm{M}}_{\mathrm{ZAMS}}\gtrsim 40\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$. Specifically, stars with $22\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}\lesssim {\mathrm{M}}_{\mathrm{ZAMS}}\lesssim 26\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ tend to have higher ${\xi}_{2.5}$ than neighboring stars and this creates a dearth of remnants with mass between $\sim 2\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and $6.5\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$. While this finding qualitatively agrees with the rapid model of Fryer et al. [26], in Ugliano et al. [167] the gap is naturally produced through a wide range of explosion timescales (from 0.1 s to 1 s) that depend only on the structure of the progenitor at the onset of collapse.

**two-parameter model**to predict the explodability of stars. The first parameter is the normalized enclosed mass for a dimensionless entropy per nucleon of $s=4$, ${M}_{4}$. This is a good proxy for the proto-NS mass, which corresponds roughly to the iron-core mass at the onset of collapse. The other parameter, ${\mu}_{4}$, is the mass derivative at $R\left({M}_{4}\right)$. The advantage of the two-parameter model is that the quantities $x\equiv {M}_{4}{\mu}_{4}$ and $y\equiv {\mu}_{4}$ are strongly connected to the mass accretion rate of the stalling shock and to the neutrino luminosity, respectively, so they are expected to capture the physics of the neutrino-driven explosion better than ${\xi}_{2.5}$. Ertl et al. [173] predicted successful (failed) SNe for ${\mu}_{4}<{y}_{\mathrm{sep}}\left(x\right)$ (${\mu}_{4}>{y}_{\mathrm{sep}}\left(x\right)$), where ${y}_{\mathrm{sep}}\left(x\right)={k}_{1}x+{k}_{2}$, and ${k}_{1}\in \left[0.19;0.28\right]$ and ${k}_{2}\in \left[0.043;0.058\right]$, depending on the adopted set of progenitor stars. Ertl et al. [173] (see also [174]) confirmed the presence of islands of explodability, the prevalence of direct collapse for ${\mathrm{M}}_{\mathrm{ZAMS}}\gtrsim 30\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, and that fallback SNe are quite rare (i.e., a gap of compact objects with mass between $\sim 2\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and $5\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$).

**they should be taken with a grain of salt**, as all the features might be either confirmed or gone by the time we will have a realistic framework for 3D explosion models, which still need improvements and should be considered only as provisional (e.g., [176]).

#### 2.8. Core-Collapse SNe in Population Synthesis Calculations

#### 2.9. Pair-Instability SNe and the Upper Mass Gap

**the upper mass gap**, as opposed to the lower mass gap which corresponds to a dearth of observations of compact objects with mass between $\sim 2.5\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and $\sim 5\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ [17,18,166] (see also Section 2.7). The pulsational pair-instability supernova (PPISN) [180] and the PISN [181] are the main mechanisms behind the formation of the upper mass gap.

**create electron-positron pairs**[180,182,183,184]. This process converts energy (gamma photons) into rest mass (electrons and positrons), thus it lowers radiation pressure and it triggers stellar collapse. In stars with helium core masses between $\sim 34\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and $\sim 64\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, the collapse is reversed by oxygen- or silicon- core burning, which shows up as a pulse and makes the core expand and cool. The flash is not energetic enough to disrupt the star and the core begins a series of contractions and expansions (stellar pulsations) that

**significantly enhance mass loss**, especially from the outermost stellar layers, and continue until the entropy becomes low enough to avoid the pair instability and stabilize the core until the core-collapse SN explosion. Such pulsational instabilities are referred to as pulsational pair-instability supernova [180,181,185]. In contrast, in stars with helium core masses between $\sim 64\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ and $\sim 130\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, the first pulse is energetic enough to

**completely disrupt the entire star**(i.e., PISN [186,187,188])7. Stars with helium cores above $\sim 130\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ experience a rapid pair instability-induced collapse but the energy released by nuclear burning is not enough to reverse the collapse before photodisintegration (endothermic) becomes the dominant photon-interaction mechanism [181]. Thus, the direct collapse to a massive BH (mass $\gtrsim 130\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$) becomes unavoidable.

**local maximum**which corresponds to the

**lower edge of the upper mass gap**(${M}_{\mathrm{low}}$) [63,97,189,190,191,192,193]. Pair creation triggers direct collapse for stars with helium core masses ≳ $130\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ (i.e., ${\mathrm{M}}_{\mathrm{ZAMS}}\gtrsim 300\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ for $Z\simeq {10}^{-3}$), thus these stars form massive (≳$130\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$) BHs. This BH mass corresponds to a

**local minimum**of the ${m}_{\mathrm{BH}}\u2013{\mathrm{M}}_{\mathrm{ZAMS}}$ curve, for ${\mathrm{M}}_{\mathrm{ZAMS}}\gtrsim 300\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, and it is referred to as

**the upper edge of the upper mass gap**(${M}_{\mathrm{high}}$) [97,194,195,196].

^{12}C($\alpha $, $\gamma $)

^{16}O reaction rate.

**hydrogen envelope of the progenitor star**is ejected during the collapse, leaving a remnant with the mass of about the helium core, or if it also contributes to the BH’s growth [172,174,200,201,202]. The collapse of the hydrogen envelope gives an uncertainty on ${M}_{\mathrm{low}}$ of about $20\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ [170].

**the temperature dependence of the**(e.g., [204]). While changing the

^{12}C($\alpha $, $\gamma $)^{16}O reaction rate^{12}C($\alpha $, $\gamma $)

^{16}O rate has no significant impact on the stellar structure, it governs the relative amount of oxygen with respect to carbon in the core. Low

^{12}C($\alpha $, $\gamma $)

^{16}O rates translate into large carbon reservoirs at the end of helium-core burning and into a prolonged carbon-burning phase, which contributes to suppress pair production, stabilize the oxygen core, and delay the latter ignition. In contrast, high

^{12}C($\alpha $, $\gamma $)

^{16}O rates imply significant carbon depletion in favor of oxygen, which ignites explosively just after the helium burning phase. This has a strong impact on the upper mass gap because low (high)

^{12}C($\alpha $, $\gamma $)

^{16}O rates push ${M}_{\mathrm{low}}$ and ${M}_{\mathrm{high}}$ towards higher (lower) values [192,193,198,205,206]. Furthermore, massive stars with low

^{12}C($\alpha $, $\gamma $)

^{16}O rates might experience significant dredge up which tends to stabilize the oxygen core even further. In this scenario, if very low ($-3\sigma $)

^{12}C($\alpha $, $\gamma $)

^{16}O rates are considered together with the collapse of the hydrogen envelope, the upper mass gap might even disappear [206,207]. Conservatively, the impact of

^{12}C($\alpha $, $\gamma $)

^{16}O rates on both ${M}_{\mathrm{low}}$ and ${M}_{\mathrm{high}}$ is about $15\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$.

#### 2.9.1. Piling-Up BHs

**an excess of BHs (pile-up)**with masses in about the same range [97,177,189,210].

**kink in the ${m}_{\mathrm{BH}}\u2013{\mathrm{M}}_{\mathrm{ZAMS}}$ curve are highly uncertain**since they depend on many ingredients including metallicity, stellar winds, the details of the growth of helium cores inside massive stars, nuclear reaction rates (e.g.,

^{12}C($\alpha $, $\gamma $)

^{16}O), and the collapse of the hydrogen envelope.

**model we adopted**to make Figure 5 has an additional kink in A, which also piles up BHs at $33\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$, even without PPISNe. The pile-up at $66.3\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$ (kink D) disappears when PPISNe are considered because the latter force ${m}_{\mathrm{BH}}\lesssim 55\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$. Therefore, besides depending on metallicity, the existence and position of the kinks A, B, C, and D is strongly model-dependent.

#### 2.9.2. Populating the Gap

**astrophysical processes that can form BHs in the mass gap**, without contradicting its existence (e.g., [52,56,58,207,211,212,213,214,215,216,217,218,219,220,221,222,223]).

#### 2.10. SNe Asymmetries and Kicks

**fairly large spatial velocities**, which can be as high as thousands of kilometers per second. Such values are too large to be explained through Blaauw kicks [224] from SN explosions in binary systems. Thus, some, if not all, compact objects should receive quite high

**kicks at birth**[225]. Kicks have a huge impact on merging compact objects if they are members of isolated binaries (e.g., change of orbital parameters, unbinding the binary) or if they reside in dense stellar environments (e.g., ejections).

**asymmetries in the SN ejecta**can impart high kicks to newly-born compact objects (e.g., [229]). The kicks can vary from $\sim 10\phantom{\rule{0.166667em}{0ex}}\mathrm{km}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ to $\sim 1000\phantom{\rule{0.166667em}{0ex}}\mathrm{km}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$, depending mainly on the steepness of the density profile at the outer edge of the stellar core (i.e., compactness), and on the stochastic variations of non-radial instabilities associated with the SN engine. Shallow (steeper) density profiles are more (less) prone to SN shock stalling, thus neutrinos will be able to interact with more (less) material and produce more- (less-)asymmetric ejecta (e.g., [230]). As already discussed in Section 2.6, the progenitors of

**electron-capture SNe**have very steep density profiles, so the explosion is expected to impart

**low kicks**to NSs (tens of $\mathrm{km}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$, e.g., [231]). Similar results have been obtained for

**ultra-stripped stars**, which might form during mass transfer in close binaries [232,233,234].

**unified approach**derived from momentum-conserving arguments, inspired by Bray and Eldridge [239], Bray and Eldridge [241]. Independent of the progenitor, the nature of compact remnant, and the SN explosion engine, the birth kick (${v}_{\mathrm{k}}$) is expressed as

#### 2.11. Spins

**hierarchical mergers**, wherein GW events are produced by second or higher-generation BHs formed from the coalescence of BBHs, rather than from SN explosions of their progenitor stars. Repeated mergers of BHs can thus produce higher and higher spinning remnants, and naively one might expect to achieve maximally spinning BHs, i.e., BHs with dimensionless spin $\chi \simeq 1$. However, this is not the case, because the spin of the final remnant depends also also on the spin of the progenitor BHs and their relative orientation with respect to the binary angular momentum vector. Spins anti-aligned with the binary angular momentum will subtract from the total angular momentum budget of the final BH. Therefore, we expect hierarchical mergers to produce, after several generations of mergers, BHs with an average of $\chi \simeq 0.7$ [190,211,273,274,275,276,277]. The latter is true for nearly-equal mass mergers of higher generation BHs. On the other hand, if many first generation BHs coalesce into a single, massive merger product (as massive BH runaway formation scenarios, e.g., see [278]), the final BH spin will decrease on average. This is because, at next-to-leading order, the decrease in BH spin is proportional to the mass ratio of the binary, and thus on average the spin distribution of merger products will decrease after asymmetric mergers (e.g., [279]). Any hierarchical merger scenario, requires mechanisms to assemble higher-generation BHs into merging binaries, which will be described later in Section 4.

## 3. Binary Stars

**most BH and NS progenitors are not isolated**, but members of binaries, triples, and even quadruple stellar systems. Studying the interactions between close stars is crucial to understand the evolutionary histories of GW mergers.

#### 3.1. Stellar Tides

**the point mass approximation is not enough**to describe its motion, because finite-size effects (i.e.,

**tidal forces**) become significant. An elegant derivation of the equations of motion for a binary affected by tides can be found in [31,289]. The main idea behind these equations is that the star is deformed by its companion, generating a gravitational quadrupole moment. Due to dissipation sources in the stellar interior, the response of the quadrupole moment is not instantaneous with respect to the tidal field. This delay, called time-lag, allows the coupling between the rotational and orbital angular momenta, in addition to the dissipation of orbital energy in the stellar interior.

**the equilibrium tide**. For this reason, to quantify the tidal dissipation of evolved stars, we need to characterize the timescale of the convective motion, which is the eddie turnover timescale ${\tau}_{\mathrm{conv}}$. This time scale can be calculated in several ways, either from the bulk properties of the star e.g., [35,291] or from the mixing length parameters adopted in the stellar models [292].

**dynamical tide**, is generally modeled following Hut [31], who relies on the ideas of quadrupole deformation and time lag, which are more suitable to describe the equilibrium tide. Nonetheless, just like the equilibrium tide, the tidal dissipation constants of the dynamical tide depend on the details of the stellar structure. The dissipation rate of the dynamical tide scales linearly with a dimensionless tidal torque constant, named ${E}_{2}$, which must be calculated from the stellar density profile e.g., [294,295]. Tabulated values for ${E}_{2}$ were provided by Zahn [293], and were later fitted as a function of stellar mass by Hurley et al. [35], to use in population-synthesis codes. More recent fitting formulae can be found in Qin et al. [265], which, in turn, are based on the ones of Yoon and Cantiello [135]. An alternative formulation for the dynamical tide, which avoids entirely the tidal torque constant ${E}_{2}$, was proposed by Kushnir et al. [296].

**circularize eccentric binaries**, shrinking their semimajor axis. Second, they tend to

**spin-up stars in close binaries**, synchronizing their rotation period to the orbital period, and aligning the spin directions with the angular momentum vector of the binary. Both effects are especially important in the context of GWs. Specifically, tidal spin-up can change both magnitude and orientation of the spins of compact objects with respect to the orbital angular momentum vector, and GW observations may give us insights into these two parameters (see also Section 2.11).

**radically change the structure and evolution of a star**. Tidal spin-up in a close binary introduces rotational mixing of the stellar interior, which tends to flatten its chemical composition gradient. For very close massive binaries, rotational mixing drives large-scale Eddington–Sweet circulations [300,301], so that the entire star is fully mixed. These stars undergo

**chemical homogeneous evolution (CHE)**, which has been proposed as a formation pathway for BBHs [36,39,40,41,302,303]. Chemically-homogeneous stars skip entirely the evolved giant phase because they do not develop a core-envelope boundary. Since such stars remain compact even during the post-MS phases, they can evolve very close to each other without merging via unstable mass transfer (Section 3.2). Therefore, CHE can produce BBHs that merge within the age of the Universe. Because this scenario involves tight binaries with synchronized spins, it predicts BH mergers with large aligned spins. It also favors high BH masses (>20 ${\mathrm{M}}_{\odot}$) and nearly equal mass ratios ($q\simeq 1$).

#### 3.2. Mass Loss, Mass Transfer and Accretion

**The wind accretion rate**can be calculated using the Bondi and Hoyle [306] accretion model. Given a binary with eccentricity e, donor wind speed ${v}_{\mathrm{w}}$, and mean orbital velocity ${v}_{\mathrm{circ}}=\sqrt{G\phantom{\rule{0.166667em}{0ex}}({m}_{1}+{m}_{2})/a}$, Hurley et al. [35] approximate the mass accretion rate as:

**Roche lobe overflow**. If the stellar radius is relatively large compared to the size of the binary, the external layers of the star may be stripped out by the gravity of the companion star and the centrifugal force of the binary motion. The region in space where this occurs is approximated by the

**Roche lobe**, the equipotential surface shaped like two tear-drops that surround both stars, with the two lobes connected by a saddle point at the center (also known as the first Lagrangian point, ${L}_{1}$) [309,310,311]. In general, Roche-lobe overflow can be caused by either the primary star entering the giant phase and increasing in radius, or by the shrinking of the binary orbit due to tides.

**Roche radius**. A convenient analytic approximation for ${R}_{\mathrm{L}}$ was given by Eggleton [312]:

**conservative mass transfer**. The material that is lost during

**non-conservative mass transfer**will carry out not only mass but also angular momentum from the binary.

**the response of the donor star to mass transfer is crucial**to predict the evolution of binary stars but it is also very challenging since it requires an in-depth knowledge of the internal structure of the star and possibly non-equilibrium solutions for it.

**nuclear timescale**but well below the

**thermal timescale**for mass transfer, that is

**thermal timescale**, but remains below the

**dynamical timescale**, that is

**$\zeta $ coefficients**:

#### Approximate Solutions for Population Synthesis Simulations

**critical mass ratio**, ${q}_{\mathrm{crit}}$. Indeed, if we assume that mass transfer is conservative, i.e., $M={m}_{1}+{m}_{2}=\mathrm{const}.$, and that the total angular momentum of the binary (J) is conserved as well, then

#### 3.3. Common Envelope

**The CE is a key process in the formation of GW events**from isolated binary stars, because it can shrink binary separations by a factor of hundreds, decreasing the coalescence time of compact-object binaries [43,44,266,318,319,320,321]. In particular, Dominik et al. [37] found that the coalescence time distribution of post-CE compact object binaries approximately follows a log-uniform distribution, $n\left({t}_{\mathrm{GW}}\right)\propto {t}_{\mathrm{GW}}^{-1}$. The reason for this peculiar scaling comes from Equation (9), which imposes ${t}_{\mathrm{GW}}\propto {a}^{4}$. If we assume that the distribution of semimajor axis of post-CE binaries follows a power-law as in

**$\alpha $–$\lambda $ model**[30,32,331], which is based on energy balance considerations. The main idea of this approach is to compare the orbital energy of the binary at the onset of CE with the binding energy of the envelope. By comparing these two energies, it is possible to determine whether or not the binary will survive the CE and to estimate the final size of the binary.

#### 3.4. Supernovae: Blaauw and Velocity Kicks

**Blaauw kick**. This unbinding mechanism was originally proposed to explain runaway O- and B-type stars, because the stars inherit a fraction of the binary orbital velocity following the breakup [224].

**the asymmetries in the SN ejecta**(see Section 2.10). The asymmetries can also affect the spin of the newly-born compact remnant [151], by changing its magnitude and orientation.

## 4. Stellar Dynamics

**dynamical scenario**involves the formation and hardening of binaries through few-body encounters in stellar clusters. However, in recent years, other forms of dynamical scenarios have been proposed, which involve not only massive star clusters but also small multiple systems like triples and quadruples.

#### 4.1. Dense Stellar Environments

**two-body relaxation**. Two-body relaxation leads star clusters to develop a high density core surrounded by a low-density stellar halo. Given their relatively high central density ($\gtrsim {10}^{3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}{\mathrm{pc}}^{-3}$) [355,356,357,358,359], the cores of star clusters are the ideal environments for extreme dynamical interactions between stars and binary stars, including stellar collisions, which are unlikely to occur in the galactic field [57,356].

**globular clusters (GCs)**are typically old systems (∼Universe’s age, ∼12 Gyr), very massive (≥10${}^{4}\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$) and dense (central density ${\rho}_{\mathrm{c}}\ge {10}^{4}\phantom{\rule{0.166667em}{0ex}}{\mathrm{M}}_{\odot}$). They are evolved systems which do not contain gas, dust or young stars. Because of their mass and high central density, many studies for the dynamical formation of compact-object binaries focus on GCs [50,51,54,55,275,319,360,361,362,363,364,365,366,367,368,369].

**young dense star clusters (YDSCs)**are relatively young (<100 Myr) systems, thought to be the most common birthplace of massive stars [355,356]. The central density of YDSCs can be as high as that of GCs, although YDSCs have smaller sizes. Some YDSCs can have comparable masses to present-days GCs and, for this reason, they are thought to be close relatives of the ancient progenitors of GCs. However, because of the stellar mass loss during their evolution, YDSCs are not massive enough to evolve into present-day GCs.

**open clusters (OCs)**are irregular star clusters composed of 10—a few ${10}^{3}$ stars. They are generally younger than GCs and they may still contain gas from the molecular cloud from which they formed. Studies of compact binary mergers in young and open clusters include [27,51,57,58,59,62,63,214,370,371,372,373,374]. Finally,

**nuclear star clusters (NSCs)**reside in the nuclei of galaxies. Nuclear star clusters are rather common in galaxies, including our own [375,376]. NSCs are usually more massive and denser than GCs, and may host a super-massive black hole (SMBH) at their center. Regardless of the presence of a NSC, the environment close to SMBHs can also be a site for formation of GW progenitors. Cusps or disks of BHs may form around SMBHs, where they can interact with other compact objects [377,378,379,380,381]. In AGN, BHs can be trapped by the SMBH accretion disk, wherein they can migrate and merge [212,285,380,382,383,384,385,386,387,388,389,390,391,392].

**Stellar evaporation**is instead the inevitable consequence of two-body relaxation.

#### 4.2. Two-Body Relaxation and the Gravothermal Instability

**crossing time**, is:

**evaporation**. In OCs and YDSCs, the expansion of the cluster accelerates the disruption of the cluster due to the tidal field of the galaxy. This runaway process is called

**gravothermal catastrophe**, and in physical terms it is a consequence of the negative heat capacity typical of every self-gravitating system [394]. A system with negative heat capacity loses energy and becomes hotter. To become hotter, a self-gravitating system contracts so that its velocity dispersion (i.e., the average speed of the stars) increases.

**dynamical friction and energy equipartition**.

#### 4.3. Dynamical Friction, Energy Equipartition and Mass Segregation

**Dynamical friction is a drag force**that acts on massive bodies that travel through a medium of less massive objects. The gravity of the massive body attracts the lighter ones, which form a wake behind it. The overdensity of light bodies tends to decelerate the motion of the massive one via a gravitational drag. The massive body decelerates until it is finally at rest with respect to the lighter bodies. The timescale of dynamical friction for a body of mass M is [395,396,397]:

**mass segregation**, is characterized by a varying mass spectrum across the cluster radius, with heavier stars sinking to the cluster’s core and the lighter ones crowding the halo. A more dramatic rearrangement of the massive stars in the cluster can be caused by energy equipartition, or rather, the lack of it.

**Energy equipartition**is the tendency for stars to equalize their average kinetic energy

#### 4.4. Halting Core Collapse with Binaries

**binaries can considered as a reservoir of kinetic energy**. The kinetic energy released through three-body encounters can be used to reverse core collapse.

**hard**if its internal energy ${E}_{\mathrm{bin}}$ is greater than the average kinetic energy ${E}_{\mathrm{k}}$ of neighboring stars, while it is

**soft**in the opposite case. On average, subsequent encounters make hard binaries harder (i.e., their semimajor axis shrinks), while soft binaries tend to become softer (i.e., wider semimajor axis) until they break up [45]. It is worth noting that hardness is a property of the binary relative to its environment. Due to the higher velocity dispersion, the same binary in the core of a cluster might be soft, whereas in the halo it would be hard.

#### 4.5. Forming Merging Compact-Object Binaries

**binary hardening**in stellar clusters is a direct consequence of the core collapse and the gravothermal catastrophe, and it is argued to be one of the most critical processes for the formation of BBHs. Shrinking the semimajor axis of compact object binaries can dramatically shorten the coalescence time of binaries, because the GW coalescence timescale scales as ${t}_{\mathrm{GW}}\propto {a}^{4}$ (Equation (9)). Another important consequence of three-body encounters is that they tend to excite the orbital eccentricity of binaries. In fact, the probability distribution of binary eccentricities after a three-body encounter is close to a thermal distribution ($N\left(e\right)\propto e$), and can be even super-thermal in the case of low angular momentum encounters [45,408,409,410,411,412,413]. The orbital eccentricity has an even greater impact on the GW coalescence timescale, because, for eccentricities close to 1, the coalescence timescale shortens as ${t}_{\mathrm{GW}}\propto {(1-{e}^{2})}^{7/2}$ [414]. An example of the effects of three-body encounters on binary coalescence times is given in Figure 8. In this simulated three-body encounter, a binary and a single meet on a hyperbolic orbit with a small impact parameter and a velocity at infinity of $0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{km}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$, which is typical of small OCs. The initial binary is not close enough to merge within the age of the Universe. However, the outgoing binary has a shorter separation and a much higher eccentricity, which will cause the binary to coalesce in about 2 Myr after the encounter.

**dynamical formation of binaries occurs in star clusters**, but most of the merger events happen from binaries that were hardened in the core and then

**ejected**[27,51,54,55,369,415].

**in-cluster mergers that can occur during few-body encounters**in the core (e.g., [211,364,416]). These mergers result from the short pericenter passages that can happen during chaotic three-body encounters, which can trigger rapid gravitational wave coalescence. These kind of mergers can be extremely rapid, due to the initial short separation between the compact objects. For this reason, binaries formed through this scenario can retain detectable eccentricities in the LVK band [417,418]. Another possible scenario for producing eccentric mergers in the LIGO band are GW captures during hyperbolic single-single interactions [379,419,420]. Because the cross section for single-single captures is extremely small compared to binaries, hyperbolic captures are likely to happen only in the most dense environments, like NSCs and GCs.

**exchanged binaries**. Statistically, the lightest body has greater chances to be ejected. For this reason, binaries formed through three-body encounters tend to have higher masses and equal mass ratios. Furthermore, even if binaries formed through dynamical interactions tend to have high eccentricities [45,408], by the time they reach the LVK band, GW emission circularizes them. Therefore, most ejected binaries are not expected to have any residual eccentricity at $>10$ Hz [13]. Figure 9 illustrates the impact of circularization of GW radiation on merging binaries. In the example, a binary with an initial eccentricity of ${e}_{0}=0.99$ will be completely circularized by the time it reaches a peak GW frequency of 10 Hz. Only binaries with an extreme eccentricity (${e}_{0}>0.999$) can retain some eccentricity at 10 Hz, but their coalescence time will be extremely small (∼days).

**GW coalescence imparts a recoil kick**to the final remnant. Depending on the mass ratio and spins of the merging BHs, these GW recoil kicks can be of the order of 100 km ${\mathrm{s}}^{-1}$, which is much higher then the escape velocity of most clusters [423,424,425,426,427,428]. Therefore, it is expected that hierarchical mergers only occur in massive stellar environments such as GCs, NSCs, and close to SMBHs [275]. Runaway hierarchical mergers are also a proposed pathway to form intermediate mass BHs from stellar mass BHs [52,276,429,430,431,432]).

#### 4.6. Small-N Systems

**triples, quadruples and higher hierarchical systems**. Hierarchical triple systems are constituted by a binary orbited by an outer object, in a stable configuration. Many stellar triple systems have been observeed so far, and it is reasonable to expect that in many triples the inner binary is composed of compact objects. In fact, massive stars are especially found in triples and higher multiples [433,434,435,436]. The fraction of stars found in multiples increases for more massive stars, up to $\sim 50$% for B-type stars [287,437,438,439]. These multiple systems may be formed primordially as a natural outcome of star formation, or may also form dynamically from few-body encounters in stellar clusters.

**von Zeipel-Kozai-Lidov (ZKL)**[440,441,442,443,444,445]. During such oscillations, the eccentricity of the inner binary can reach values very close to 1, inducing very close pericenter passages. If the inner binary is composed of compact objects, GW radiation can be efficiently emitted during these short pericenter passages, leading to the rapid coalescence of the binary. Figure 10 shows the typical evolution of a GW coalescence triggered by ZKL oscillations. There are two main effects of ZKL oscillations. First, they can accelerate the merging of compact object binaries, allowing them to merge within the age of the Universe. Secondly, they excite very high eccentricity in the inner compact binary, which can affect the GW emission waveform, and therefore can be potentially inferred by GW observations at different frequencies [446,447,448].

#### 4.7. Hybrid Scenarios

**Stellar evolution and stellar dynamics are inseparable processes**that are active at the same time. Indeed, besides evolving in complete isolation, stellar binaries can be found in star clusters as well, where they can be affected by close encounters just like compact-object binaries. These stellar binaries can therefore experience processes from both the binary evolution pathway (e.g., mass transfers, CEs) and the dynamical pathway (e.g., exchanges, excitation of eccentricity). For instance, in young star clusters, some BBHs are formed via CE evolution of dynamically-assembled main sequence binaries that, at some point of their life, are ejected from the cluster, and merge in the field, appearing as if they had evolved in complete isolation. They can contribute to the merger rate more than dynamically assembled BBHs [62,63,214,474]. These binaries undergo a CE phase, like in the isolated channel, but they also undergo dynamical interactions before and after collapsing to BHs. The CE phase might even be triggered by such dynamical interactions, so that the same binaries would not have merged without the crucial contribution of stellar dynamics. Some specific scenarios require elements from both channels. For example, three-body encounters of tidally spun-up, post-common-envelope binaries have been proposed as a mechanism to produce BBH with misaligned spins [64].

## 5. Astrophysical Insights from Exceptional Gravitational-Wave Events

#### 5.1. GW190814

#### 5.2. GW190521

^{12}C($\alpha $, $\gamma $)

^{16}O rate is adopted ($-2.5\sigma $ with respect to the fiducial value of Farmer et al. [193]), the isolated-binary channel becomes a plausible formation pathway for GW190521. [219] found that GW190521-like systems can be formed from population III (Pop III) binaries, but only for stellar evolution models with a small convective overshooting parameter.

#### 5.3. GW190412

#### 5.4. GW200105_162426 and GW200115_042309

## 6. Summary

**evolution of massive stars**(e.g., clumpiness and porosity of stellar winds, energy transport in radiation-dominated envelopes, overshooting) – in the next decades, the James Webb Space Telescope and the Extremely Large Telescope will provide crucial insights on the evolution of massive stars, especially those at low metallicity; (ii) the

**SN explosion mechanism**(e.g., explodability dependence on progenitor’s structure, fallback amount, lower mass gap, pulsational pair-instability SNe, asymmetries in the ejecta) – improvements in self-consistent, 3D simulations will help us to shed light on the SN engine and its byproducts (e.g., compact remnants, yields); (iii)

**binary evolution processes**(especially common envelope and the response of donors/accretors to mass transfer) – comparisons with self-consistent, hydrodynamic simulations of binary stars will be crucial to calibrate the main (free) parameters in population-synthesis simulations; (iv)

**direct**

**N**

**-body simulations**of star clusters (e.g., they inherit the uncertainties on single and binary stellar evolution processes, and they are computationally challenging) – new software, natively developed for state-of-the-art computing accelerators (e.g., Graphics Processing Units), and coupled with up-to-date population-synthesis codes, will give us the opportunity to accurately study merging compact objects in very dense stellar environments, possibly up to the regime of dwarf galaxies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AGN | active galactic nuclei |

BBH | binary black hole |

BH | black hole |

BNS | binary neutron star |

CO | carbon-oxygen |

CE | common envelope |

CHE | chemical homogeneous evolution |

GC | globular cluster |

GW | gravitational wave |

GWTC | Gravitational Wave Transient Catalog |

ECSN | electron-capture supernova |

LIGO | Laser Interferometer Gravitational-wave Observatory |

LVC | LIGO-Virgo Collaboration |

LVK | LIGO-Virgo-KAGRA |

NS | neutron star |

NSBH | neutron star-black hole binary |

NSC | nuclear star cluster |

OC | open cluster |

PISN | pair-instability supernova |

Pop III | population III |

PPISN | pulsational pair-instability supernova |

SMBH | super-massive black hole |

SN | supernova |

TOV | Tolman–Oppenheimer–Volkoff |

WD | white dwarf |

YDSC | young dense star cluster |

ZAMS | zero age main sequence |

ZKL | von Zeipel-Kozai-Lidov |

## Notes

1 | Throughout this work, we will use the symbol ${\mathrm{M}}_{\odot}$ to refer to the Sun’s mass. |

2 | |

3 | $\mathcal{M}=\frac{{\left({m}_{1}{m}_{2}\right)}^{3/5}}{{\left({m}_{1}+{m}_{2}\right)}^{1/5}}$. |

4 | In cool supergiants (${T}_{\mathrm{eff}}<{10}^{4}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$) the mechanism responsible for winds is the absorption of photons by dust grains, i.e., dust- (or continuum-) driven winds. |

5 | It is worth noting that wind mass loss does not depend only on metallicity, but also on luminosity, effective temperature, stellar mass, and the velocity of wind at infinity. |

6 | ${\xi}_{m}=\frac{m}{R\left(m\right)}$, where m is a threshold mass in ${\mathrm{M}}_{\odot}$ and $R\left(m\right)$ is the radius enclosing m, in units of 1000 km. |

7 | It is worth noting that a PISN is driven by a thermonuclear explosion, i.e., very different from neutrino-driven core-collapse SNe. |

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