Binary black hole mergers: formation and populations

We review the main physical processes that lead to the formation of stellar binary black holes (BBHs) and to their merger. BBHs can form from the isolated evolution of massive binary stars. The physics of core-collapse supernovae and the process of common envelope are two of the main sources of uncertainty about this formation channel. Alternatively, two black holes can form a binary by dynamical encounters in a dense star cluster. The dynamical formation channel leaves several imprints on the mass, spin and orbital properties of BBHs.

asṁ ∝ Z β , where Z is the absolute metallicity (see e.g. Vink et al. 2001 and references therein). The most recent models suggest that β is not constant, but depends at least on the luminosity of the star (Gräfener and Hamann, 2008;Vink et al., 2011;Chen et al., 2015): the closer the luminosity L * is to the Eddington value L Edd , the higher the mass loss, basically cancelling the dependence on metallicity when L * L Edd .
In single stars, stellar winds uniquely determine the final mass of the star at the onset of collapse. If we consider a star with m ZAMS = 90 M and metallicity Z = 0.02 (i.e. approximately solar), its final mass will be only m fin ∼ 30 M ; while the same star with Z < 0.0002 has m fin 0.8 m ZAMS . The final mass of a star m fin is the strongest upper limit to the mass of the BH.

Core-collapse supernovae
Several models in the literature try to predict the final outcome of a core-collapse supernova (SN). For example, Fryer (1999) and Fryer and Kalogera (2001) suggest that if the final mass of the star is m fin 40 M , the fate of the star is to collapse to a BH directly, without supernova, because the binding energy of the outer stellar layers is too big to be overcome by the explosion. Fryer et al. (2012) elaborate on these early results proposing that the mass of the compact object depends not only on m fin but also on the final mass of the carbon-oxygen core. Alternatively, O'Connor and Ott (2011) proposed the role of the compactness parameter ξ M = M/M R(≤M )/1000 km : if the compactness is small (e.g. ξ 2.5 ≤ 0.2 − 0.4), the SN explosion is successful, otherwise we expect the star to collapse directly. All of these simplified models as well as more sophisticated ones (e.g. Ertl et al. 2016) point toward a similar direction: if the star ends its life with a large final mass, its carbon-oxygen core grows larger, its compactness is generally higher, and so on. Hence, we expect that metal-poor stars, which retain a larger fraction of their mass to the very end and develop larger cores, are more likely to collapse to BHs directly, producing larger BHs (e.g. Mapelli et al. 2009Mapelli et al. , 2010Zampieri and Roberts 2009;Belczynski et al. 2010). This simplified picture seems to agree with observations, but must be taken with several grains of salt: we need a vigorous step forward in core-collapse SN simulations and theoretical models, before we can draw robust conclusions (e.g. Burrows et al. 2018).

Pair instability
Core-collapse SNe are not the only mechanism that can end the life of a massive star. When the helium core of a star grows to ≥ 60 M and the central temperature reaches ∼ 10 9 K, electron and positron pairs are produced at an efficient rate, leading to a softening of the equation of state. The star undergoes pair instability (PI, Ober et al. 1983;Bond et al. 1984;Heger et al. 2003;Woosley et al. 2007): oxygen, neon and silicon are burned explosively and the entire star is disrupted leaving no remnant, unless its helium core is ≥ 130 M . In the latter case, the gravity of the outer layers is so big that the star collapses to a massive BH directly as an effect of PI (Heger et al., 2003). Smaller helium cores (∼ 30 − 60 M ) are associated with a less dramatic manifestation of PI: the softened equation of state drives oscillations of the core (pulsational PI, Barkat et al. 1967;Woosley et al. 2007;Chen et al. 2014;Yoshida et al. 2016); during each oscillation the star sheds some mass till it finds a new equilibrium to a lower core mass, but leaves a BH smaller than expected without pulsational PI (Woosley, 2017(Woosley, , 2019Belczynski et al., 2016a;Spera and Mapelli, 2017;Marchant et al., 2019;Stevenson et al., 2019;Renzo et al., 2020).
From the combination of PI, core-collapse SNe and stellar-wind mass loss prescriptions, we expect the mass spectrum of BHs to behave roughly as shown in Figure 1. In particular, PI is expected to carve a gap in the mass spectrum of BHs between ∼ 50(−10, +20) M and ≈ 120 − 130 M . The uncertainty on this mass gap is mainly connected with uncertainties nuclear reaction rates , on the collapse of the residual hydrogen envelope and on the role of stellar rotation . Within this framework, we predict a reasonable mass range for stellar-origin BHs to be ∼ 3 − 65 M (assuming Figure 1. Predicted compact object mass (M rem ) as a function of the zero-age main-sequence (ZAMS) mass of the progenitor star (M ZAMS ) for 11 different metallicities, ranging from Z = 2 × 10 −4 to Z = 2 × 10 −2 , as shown in the legend. The yellow area highlights the pair-instability mass gap. These models are obtained with the SEVN population synthesis code , using PARSEC evolutionary tracks (Bressan et al., 2012) and the delayed model from Fryer et al. (2012). See Spera and Mapelli (2017) for details. the most conservative value for the lower edge of pair-instability mass gap). If exotic metal-poor stars exist with mass m ZAMS > 250 M , these might directly collapse to intermediate-mass BHs (IMBHs) with mass > 100 M .

BINARY BH FORMATION IN ISOLATION
The scenario highlighted in the previous section assumes that the progenitor star is single. But gravitational waves have shown the existence of BBHs with a very short orbital separation: the initial separation of a BBH must be of the order of few ten solar radii for the BBH to merge within a Hubble time by gravitational-wave emission. This challenges our understanding of binary star evolution. A close binary star undergoes several physical processes during its life, which can completely change its final fate (see e.g. Eggleton 2006). The most important processes include mass transfer and common envelope, tides and natal kicks (Hurley et al., 2002).
Mass transfer and common envelope are crucial in this regard. After main sequence, a massive star can develop a stellar radius as large as several thousand solar radii. Hence, if this star is member of a binary system and its orbital separation is of few hundred to few thousand solar radii, the binary system undergoes Roche lobe overflow and possibly common envelope (Ivanova et al., 2013). If common envelope occurs between a BH and a giant companion star, the BH and the core of the giant star orbit about each other surrounded by the giant's envelope: they feel a strong gas drag from the envelope and lose kinetic energy, inspiralling about each other. This transfers thermal energy to the envelope, which might trigger the ejection of the envelope. If the envelope is not ejected, the binary system merges prematurely giving birth to a single BH. In contrast, if the envelope is ejected, the final binary system is composed of the BH and the core of the giant. Because of the spiral-in, the final semi-major axis of the binary is just few solar radii, much smaller than the initial one. If the naked core collapses to a BH without receiving a strong natal kick, the system becomes a BBH with a short orbital period, able to merge within a Hubble time. Unfortunately, our understanding of common envelope is still poor (see Fragos et al. 2019 for a recent simulation) and this uncertainty heavily affects our knowledge of BBH demography. The left-hand panel of Figure 2 is a schematic view of the isolated binary evolution channel through common envelope.
Several alternative scenarios to common envelope have been proposed (Marchant et al., 2016;de Mink and Mandel, 2016;Mandel and de Mink, 2016). For example, in the over-contact binary evolution, Marchant et al. (2016) show that, when two massive stars in a tight binary are fast rotators, they remain fully mixed as a result of their tidally induced high spin; in this case, the binary avoids premature merger even if it is overfilling its Roche lobe and might evolve into a tight BBH.
The isolated binary evolution scenario has several characteristic signatures. In the common envelope isolated binary evolution scenario, the masses of the two BHs span from ∼ 3 M up to ∼ 45 M (see e.g. Giacobbo and Mapelli 2018) and the mass ratios are preferentially close to 1 (although all mass ratios q = m 2 /m 1 0.1 are possible, see e.g. Giacobbo and Mapelli 2018). Most processes in binary evolution tend to produce aligned spins (e.g. Rodriguez et al. 2016), while the magnitude of the spin is basically unconstrained (but see Qin et al. 2018Qin et al. , 2019Fuller and Ma 2019 for some recent attempts to quantify spins). Mass transfer episodes and gravitational-wave decay are expected to efficiently damp eccentricity, so that almost all isolated binaries have near zero eccentricity in the LIGO-Virgo band. Finally, local merger rate densities span from a few to few thousand events Gpc −3 yr −1 , depending on the details of common envelope and natal kicks (e.g. Dominik et al. 2013;Belczynski et al. 2016b;Mapelli et al. 2017;Mapelli and Giacobbo 2018;Mapelli 2018, 2020;Neijssel et al. 2019;Tang et al. 2020;Santoliquido et al. 2020). The scenarios which include alternatives to common envelope predict an even stronger prevalence of systems with q ∼ 1, a preferred mass range ∼ 25 − 60 M (Marchant et al., 2016), high and aligned spins, zero eccentricity in the LIGO-Virgo band, and long delay times ( 3 Gyr, de Mink and Mandel 2016). Local merger rate densities are expected to be ∼ 10 Gpc −3 yr −1 , with large uncertainties.

BINARY BH FORMATION IN STAR CLUSTERS
Star clusters are among the densest places in the Universe. There is a plethora of star clusters, with their distinguishing features: i) globular clusters (Gratton et al., 2019) are old (∼ 12 Gyr) and massive systems (∼ 10 4−6 M ), ii) nuclear star clusters can be even more massive (∼ 10 7 M ) and lie at the centre of many galaxies, in some cases coexisting with the supermassive BH (Neumayer et al., 2020), iii) open clusters and young star clusters (Portegies Zwart et al., 2010) are generally less massive (up to ∼ 10 5 M ) and short lived (less than a few Gyr), but are the main birthplace of massive stars in the local Universe (Lada and Lada, 2003).
The central density of star clusters is sufficiently high ( 10 3 stars pc −3 ) and their typical velocity dispersion sufficiently low (from a few to a few tens of km s −1 , possibly with the exception of nuclear star clusters) that their central two-body relaxation time (Spitzer, 1987) is shorter than their lifetime. This has one fascinating implication: the orbits of stars and binary stars in a star cluster are constantly perturbed by dynamical encounters with other cluster members. This process affects the formation and the evolution of binary BHs in multiple ways (e.g. Portegies Zwart and McMillan 2000).
Dynamical exchanges occur when a binary system interacts with a single stellar object and the latter replaces one of the members of the binary. We have known for a long time that massive objects are more likely to acquire companions by dynamical exchanges (Hills and Fullerton, 1980). Since BHs are among the most massive objects in a star cluster, they are very efficient in forming new binaries through exchanges (e.g. Ziosi et al. 2014).
During a three-body encounter, a binary star exchanges a fraction of its internal energy with the third body. If the binary is particularly tight (hard binary), such encounters tend to harden the binary star, i.e. to increase its binding energy by reducing its semi-major axis (dynamical hardening). In the case of a BBH, this hardening might speed up the merger, because it drives the semi-major axis of the BBH in the regime where orbital decay by gravitational waves becomes efficient (see e.g. Figure 10 of Mapelli 2018). On the other hand, the least massive BBHs can even be ionized, i.e. split by strong dynamical encounters with massive intruders.
Mergers of massive stars are common in dense young star clusters, because of the short dynamical friction timescale (Portegies Zwart et al., 2010). Under some assumptions, these mergers can lead to the formation of massive BHs (m BH > 60 M ), with mass in the pair-instability gap . In star clusters, such massive BHs can acquire a companion by dynamical exchanges, leading to the formation of BBHs in the mass gap. A fast sequence of stellar mergers in the dense core of a young star cluster (also known as runaway collision, Portegies Zwart et al. 2004;Giersz et al. 2015) might even lead to the formation of intermediate-mass BHs (IMBHs), i.e. BHs with mass m BH > 100 M , especially at low metallicity (Mapelli, 2016).

Frontiers
The dynamical processes we briefly summarized above (and in the right-hand panel of Figure 2) leave a clear imprint on BBHs. First, dynamically formed BBHs extend to higher masses than isolated BBHs: they might even be in the pair-instability mass gap or in the IMBH regime Rodriguez et al., 2019). Secondly, dynamical exchanges randomize the spin direction, leading to an isotropic distribution of BH spins. In contrast, isolated BBHs have a preference for aligned spins (Rodriguez et al., 2016;Gerosa et al., 2018). Third, dynamics can trigger the merger of BBHs with non-zero eccentricity even in the LIGO-Virgo band (Samsing et al., 2014;Samsing, 2018;Rodriguez et al., 2018;Zevin et al., 2019). These signatures provide an unique opportunity to differentiate among the isolated and the dynamical formation channel when the number of gravitational-wave detections will be of the order of a few hundreds (e.g. Zevin et al. 2017;Bouffanais et al. 2019).

AUTHOR CONTRIBUTIONS
MM collected the material and wrote the review.

FUNDING
MM acknowledges financial support from the European Research Council for the ERC Consolidator grant DEMOBLACK, under contract no. 770017.